【无标题】模型预测控制MPC详解（附带案例实现）
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<h2><a name="t0"></a><a id="MPC_0"></a>模型预测控制MPC详解（附带案例实现）</h2>
文章目录
写在前面本文是记录学习B站博主Dr.can的学习笔记，如有侵权请联系笔者删除此文。
1. 最优控制问题
最优控制问题就是研究在约束条件下达到最优的系统表现，通常系统的表现是综合分析的结果。比如考虑一个单输入单输出的系统（SISO），状态变量
x
x
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span>，输出为<span class="katex--inline"><span class="katex"><span class="katex-mathml">
y
y
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span></span></span></span></span>，要求其输出能跟踪预设的参考值<span class="katex--inline"><span class="katex"><span class="katex-mathml">
r
r
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span></span></span></span></span>，误差可以表示为<span class="katex--inline"><span class="katex"><span class="katex-mathml">
e
=
y
−
r
e=y-r
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.7778em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span></span></span></span></span>，那么最优控制的目标是</p>
min
∫
0
t
e
2
d
t
\min \int_0^t e^2 dt
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 2.4554em; vertical-align: -0.9119em;"></span><span class="mop">min</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5435em;"><span class="" style="top: -1.7881em; margin-left: -0.4445em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span class="" style="top: -3.8129em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.9119em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></span></p>
如果同时希望输入量
u
u
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span>也能越小越好（一般的目的是减少能耗），那最优控制的目标可以是</p>
min
∫
0
t
q
×
e
2
d
t
+
r
×
u
2
d
t
\min \int_0^t q\times e^2 dt + r\times u^2 dt
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 2.4554em; vertical-align: -0.9119em;"></span><span class="mop">min</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5435em;"><span class="" style="top: -1.7881em; margin-left: -0.4445em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span class="" style="top: -3.8129em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.9119em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.9474em; vertical-align: -0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.6667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.8641em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></span></p>
其中
q
,
r
q,r
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span></span></span></span></span>分别是权重参数，用于调节两个目标的重要性。</p>
考虑一个多输入多输出的系统（MIMO），系统的模型为：
d
X
d
t
=
A
X
+
B
U
Y
=
C
X
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-1-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mi>d</mi><mi>X</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mi>A</mi><mi>X</mi><mo>+</mo><mi>B</mi><mi>U</mi></mtd></mtr><mtr><mtd><mi>Y</mi></mtd><mtd><mi></mi><mo>=</mo><mi>C</mi><mi>X</mi></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 100%; display: inline-block; min-width: 7.647em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.281em, 1007.5em, 5.925em, -999.997em); top: -4.351em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-2"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.281em, 1007.5em, 5.925em, -999.997em); top: -4.351em; left: 50%; margin-left: -3.744em;"><span class="mtable" id="MathJax-Span-3" style="padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 7.343em; height: 0px;"><span style="position: absolute; clip: rect(2.078em, 1001.72em, 5.52em, -999.997em); top: -4.149em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.724em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.483em, 1001.72em, 4.862em, -999.997em); top: -4.554em; left: 0em;"><span class="mtd" id="MathJax-Span-4"><span class="mrow" id="MathJax-Span-5"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.483em, 1001.72em, 4.862em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.858em;"><span class="mfrac" id="MathJax-Span-6"><span style="display: inline-block; position: relative; width: 1.521em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.141em, 1001.37em, 4.154em, -999.997em); top: -4.655em; left: 50%; margin-left: -0.706em;"><span class="mrow" id="MathJax-Span-7"><span style="display: inline-block; position: relative; width: 1.369em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1001.37em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-8" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-9" style="font-family: MathJax_Math-italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.141em, 1000.86em, 4.154em, -999.997em); top: -3.288em; left: 50%; margin-left: -0.453em;"><span class="mrow" id="MathJax-Span-10"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.86em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-11" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-12" style="font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1001.52em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.521em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.76em, 4.154em, -999.997em); top: -2.782em; left: 0em;"><span class="mtd" id="MathJax-Span-22"><span class="mrow" id="MathJax-Span-23"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.402em;"><span class="mi" id="MathJax-Span-24" style="font-family: MathJax_Math-italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.205em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.154em;"></span></span><span style="position: absolute; clip: rect(2.584em, 1005.62em, 5.419em, -999.997em); top: -3.997em; left: 1.724em;"><span style="display: inline-block; position: relative; width: 5.622em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.141em, 1005.62em, 4.255em, -999.997em); top: -4.554em; left: 0em;"><span class="mtd" id="MathJax-Span-13"><span class="mrow" id="MathJax-Span-14"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1005.62em, 4.255em, -999.997em); top: -3.997em; left: 50%; margin-left: -2.782em;"><span class="mi" id="MathJax-Span-15"></span><span class="mo" id="MathJax-Span-16" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="mi" id="MathJax-Span-17" style="font-family: MathJax_Math-italic; padding-left: 0.256em;">A</span><span class="mi" id="MathJax-Span-18" style="font-family: MathJax_Math-italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-19" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="mi" id="MathJax-Span-20" style="font-family: MathJax_Math-italic; padding-left: 0.205em;">B</span><span class="mi" id="MathJax-Span-21" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.141em, 1002.89em, 4.154em, -999.997em); top: -2.782em; left: 0em;"><span class="mtd" id="MathJax-Span-25"><span class="mrow" id="MathJax-Span-26"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1002.89em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.466em;"><span class="mi" id="MathJax-Span-27"></span><span class="mo" id="MathJax-Span-28" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="mi" id="MathJax-Span-29" style="font-family: MathJax_Math-italic; padding-left: 0.256em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span class="mi" id="MathJax-Span-30" style="font-family: MathJax_Math-italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.356em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.356em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.495em; border-left: 0px solid; width: 0px; height: 3.565em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mi>d</mi><mi>X</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mi>A</mi><mi>X</mi><mo>+</mo><mi>B</mi><mi>U</mi></mtd></mtr><mtr><mtd><mi>Y</mi></mtd><mtd><mi></mi><mo>=</mo><mi>C</mi><mi>X</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">\begin{align*} \frac{dX}{dt} & = AX + BU \\ Y & = CX \end{align*}</script>
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 3.8574em; vertical-align: -1.6787em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.1787em;"><span class="" style="top: -4.1787em;"><span class="pstrut" style="height: 3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3714em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span class="" style="top: -2.3527em;"><span class="pstrut" style="height: 3.3714em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">Y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.6787em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.1787em;"><span class="" style="top: -4.1787em;"><span class="pstrut" style="height: 3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0502em;">B</span><span class="mord mathnormal" style="margin-right: 0.109em;">U</span></span></span><span class="" style="top: -2.3527em;"><span class="pstrut" style="height: 3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">CX</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.6787em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p>
那么可以将上述的最优化目标改写为：
J
=
∫
0
t
E
T
Q
E
+
U
T
R
U
d
t
J = \int^t_0 E^T Q E + U^T R U dt
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0962em;">J</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4554em; vertical-align: -0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5435em;"><span class="" style="top: -1.7881em; margin-left: -0.4445em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span class="" style="top: -3.8129em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.9119em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0576em;">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0576em;">QE</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.8913em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.109em;">U</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0077em;">R</span><span class="mord mathnormal" style="margin-right: 0.109em;">U</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></span></p>
举一个实际的例子，系统的模型如下所示：
d
d
t
[
x
1
x
2
]
=
A
[
x
1
x
2
]
+
B
[
u
1
u
2
]
[
y
1
y
2
]
=
[
x
1
x
2
]
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-2-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd><mtd><mi></mi><mo>=</mo><mi>A</mi><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>+</mo><mi>B</mi><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>u</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>y</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>y</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-31" style="width: 100%; display: inline-block; min-width: 13.063em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(3.192em, 1012.56em, 8.608em, -999.997em); top: -6.123em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-32"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1012.56em, 8.608em, -999.997em); top: -6.123em; left: 50%; margin-left: -6.376em;"><span class="mtable" id="MathJax-Span-33" style="padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 12.658em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1003.34em, 8.608em, -999.997em); top: -6.123em; left: 0em;"><span style="display: inline-block; position: relative; width: 3.597em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.382em, 1003.34em, 5.115em, -999.997em); top: -5.363em; left: 0em;"><span class="mtd" id="MathJax-Span-34"><span class="mrow" id="MathJax-Span-35"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1003.34em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.82em;"><span class="mfrac" id="MathJax-Span-36"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.141em, 1000.51em, 4.154em, -999.997em); top: -4.655em; left: 50%; margin-left: -0.251em;"><span class="mi" id="MathJax-Span-37" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.141em, 1000.86em, 4.154em, -999.997em); top: -3.288em; left: 50%; margin-left: -0.453em;"><span class="mrow" id="MathJax-Span-38"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.86em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-39" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-40" style="font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1001.01em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.015em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span class="mrow" id="MathJax-Span-41"><span class="mo" id="MathJax-Span-42" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-43" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-44"><span class="mrow" id="MathJax-Span-45"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-46"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-48" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-49"><span class="mrow" id="MathJax-Span-50"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-51"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-52" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-53" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-54" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.382em, 1002.03em, 5.115em, -999.997em); top: -2.63em; left: 0em;"><span class="mtd" id="MathJax-Span-90"><span class="mrow" id="MathJax-Span-91"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1002.03em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.162em;"><span class="mrow" id="MathJax-Span-92"><span class="mo" id="MathJax-Span-93" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-94" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1000.91em, 5.115em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.91em, 4.356em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-95"><span class="mrow" id="MathJax-Span-96"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.91em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-97"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-98" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.509em;"><span class="mn" id="MathJax-Span-99" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.91em, 4.356em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-100"><span class="mrow" id="MathJax-Span-101"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.91em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-102"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-103" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.509em;"><span class="mn" id="MathJax-Span-104" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-105" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.128em;"></span></span><span style="position: absolute; clip: rect(3.192em, 1008.76em, 8.558em, -999.997em); top: -6.123em; left: 3.597em;"><span style="display: inline-block; position: relative; width: 9.013em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.382em, 1008.76em, 5.115em, -999.997em); top: -5.363em; left: 0em;"><span class="mtd" id="MathJax-Span-55"><span class="mrow" id="MathJax-Span-56"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1008.76em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -4.503em;"><span class="mi" id="MathJax-Span-57"></span><span class="mo" id="MathJax-Span-58" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="mi" id="MathJax-Span-59" style="font-family: MathJax_Math-italic; padding-left: 0.256em;">A</span><span class="mrow" id="MathJax-Span-60" style="padding-left: 0.154em;"><span class="mo" id="MathJax-Span-61" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-62" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-63"><span class="mrow" id="MathJax-Span-64"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-65"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-66" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-67" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-68"><span class="mrow" id="MathJax-Span-69"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-70"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-71" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-72" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-73" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span class="mo" id="MathJax-Span-74" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="mi" id="MathJax-Span-75" style="font-family: MathJax_Math-italic; padding-left: 0.205em;">B</span><span class="mrow" id="MathJax-Span-76" style="padding-left: 0.154em;"><span class="mo" id="MathJax-Span-77" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-78" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-79"><span class="mrow" id="MathJax-Span-80"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-81"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-82" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-83" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-84"><span class="mrow" id="MathJax-Span-85"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-86"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-87" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-88" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-89" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.382em, 1003.39em, 5.115em, -999.997em); top: -2.63em; left: 0em;"><span class="mtd" id="MathJax-Span-106"><span class="mrow" id="MathJax-Span-107"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1003.39em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.82em;"><span class="mi" id="MathJax-Span-108"></span><span class="mo" id="MathJax-Span-109" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="mrow" id="MathJax-Span-110" style="padding-left: 0.256em;"><span class="mo" id="MathJax-Span-111" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-112" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-113"><span class="mrow" id="MathJax-Span-114"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-115"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-116" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-117" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-118"><span class="mrow" id="MathJax-Span-119"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-120"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-121" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-122" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-123" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.128em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 6.128em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 6.128em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.424em; border-left: 0px solid; width: 0px; height: 5.321em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd><mtd><mi></mi><mo>=</mo><mi>A</mi><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>+</mo><mi>B</mi><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>u</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>y</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>y</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">\begin{align*} \frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} & = A \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + B \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \\ \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} & = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \end{align*}</script>
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 5.4001em; vertical-align: -2.45em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.95em;"><span class="" style="top: -4.95em;"><span class="pstrut" style="height: 3.45em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3714em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.45em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.45em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.95em;"><span class="" style="top: -4.95em;"><span class="pstrut" style="height: 3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0502em;">B</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 2.45em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p>
设置的参考值
R
R
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0077em;">R</span></span></span></span></span>为:</p>
R
=
[
r
1
r
2
]
=
[
0
0
]
R = <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-3-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>r</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>r</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-124" style="width: 100%; display: inline-block; min-width: 2.281em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.382em, 1001.98em, 5.115em, -999.997em); top: -3.997em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-125"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1001.98em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.111em;"><span class="mrow" id="MathJax-Span-126"><span class="mo" id="MathJax-Span-127" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-128" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1000.86em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-129"><span class="mrow" id="MathJax-Span-130"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-131"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-132" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-133" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-134"><span class="mrow" id="MathJax-Span-135"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-136"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-137" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-138" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-139" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.03em; border-left: 0px solid; width: 0px; height: 2.533em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>r</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>r</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\begin{bmatrix} r_1 \\ r_2 \end{bmatrix}</script> = <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-4-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-140" style="width: 100%; display: inline-block; min-width: 1.926em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.382em, 1001.62em, 5.115em, -999.997em); top: -3.997em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-141"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1001.62em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.909em;"><span class="mrow" id="MathJax-Span-142"><span class="mo" id="MathJax-Span-143" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-144" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 0.509em; height: 0px;"><span style="position: absolute; clip: rect(2.534em, 1000.46em, 4.913em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.509em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-145"><span class="mrow" id="MathJax-Span-146"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-147" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-148"><span class="mrow" id="MathJax-Span-149"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-150" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-151" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.03em; border-left: 0px solid; width: 0px; height: 2.533em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-4">\begin{bmatrix} 0 \\ 0 \end{bmatrix}</script>
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0077em;">R</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4em; vertical-align: -0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4em; vertical-align: -0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></span></p>
那么可以推导出误差
E
E
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0576em;">E</span></span></span></span></span>为：</p>
E
=
[
e
1
e
2
]
=
[
y
1
−
r
1
y
2
−
r
2
]
=
[
x
1
x
2
]
E = <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-5-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>e</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>e</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-152" style="width: 100%; display: inline-block; min-width: 2.331em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.382em, 1002.03em, 5.115em, -999.997em); top: -3.997em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-153"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1002.03em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.111em;"><span class="mrow" id="MathJax-Span-154"><span class="mo" id="MathJax-Span-155" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-156" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1000.91em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.91em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-157"><span class="mrow" id="MathJax-Span-158"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.91em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-159"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-160" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-161" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.91em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-162"><span class="mrow" id="MathJax-Span-163"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.91em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-164"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-165" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-166" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-167" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.03em; border-left: 0px solid; width: 0px; height: 2.533em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>e</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>e</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">\begin{bmatrix} e_1 \\ e_2 \end{bmatrix}</script> = <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-6-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>y</mi><mn>1</mn></msub><mo>&#x2212;</mo><msub><mi>r</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>y</mi><mn>2</mn></msub><mo>&#x2212;</mo><msub><mi>r</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-168" style="width: 100%; display: inline-block; min-width: 4.457em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.382em, 1004.1em, 5.115em, -999.997em); top: -3.997em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-169"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1004.1em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -2.174em;"><span class="mrow" id="MathJax-Span-170"><span class="mo" id="MathJax-Span-171" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-172" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 2.989em; height: 0px;"><span style="position: absolute; clip: rect(2.635em, 1002.99em, 5.115em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.989em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.242em, 1002.99em, 4.356em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-173"><span class="mrow" id="MathJax-Span-174"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.242em, 1002.99em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.466em;"><span class="msubsup" id="MathJax-Span-175"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-176" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.509em;"><span class="mn" id="MathJax-Span-177" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-178" style="font-family: MathJax_Main; padding-left: 0.205em;">−</span><span class="msubsup" id="MathJax-Span-179" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-180" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-181" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.242em, 1002.99em, 4.356em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-182"><span class="mrow" id="MathJax-Span-183"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.242em, 1002.99em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.466em;"><span class="msubsup" id="MathJax-Span-184"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-185" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.509em;"><span class="mn" id="MathJax-Span-186" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-187" style="font-family: MathJax_Main; padding-left: 0.205em;">−</span><span class="msubsup" id="MathJax-Span-188" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-189" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-190" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-191" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.03em; border-left: 0px solid; width: 0px; height: 2.584em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>y</mi><mn>1</mn></msub><mo>−</mo><msub><mi>r</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>r</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">\begin{bmatrix} y_1 - r_1 \\ y_2 - r_2 \end{bmatrix}</script> = <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-7-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-192" style="width: 100%; display: inline-block; min-width: 2.432em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.382em, 1002.13em, 5.115em, -999.997em); top: -3.997em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-193"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1002.13em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.162em;"><span class="mrow" id="MathJax-Span-194"><span class="mo" id="MathJax-Span-195" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-196" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-197"><span class="mrow" id="MathJax-Span-198"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-199"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-200" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-201" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-202"><span class="mrow" id="MathJax-Span-203"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-204"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-205" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-206" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-207" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.03em; border-left: 0px solid; width: 0px; height: 2.533em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-7">\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}</script>
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0576em;">E</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4em; vertical-align: -0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4em; vertical-align: -0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4em; vertical-align: -0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></span></p>
那么上述最优控制的目标函数可以写成：
E
T
Q
E
=
[
x
1
x
2
]
T
[
q
1
0
0
q
2
]
[
x
1
x
2
]
=
q
1
x
1
2
+
q
2
x
2
2
U
T
R
U
=
[
u
1
u
2
]
T
[
r
1
0
0
r
2
]
[
u
1
u
2
]
=
r
1
u
1
2
+
r
2
u
2
2
J
=
∫
0
t
q
1
x
1
2
+
q
2
x
2
2
+
r
1
u
1
2
+
r
2
u
2
2
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display: inline-block; min-width: 20.758em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(4.761em, 1020.35em, 13.316em, -999.997em); top: -9.261em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-209"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(4.761em, 1020.35em, 13.316em, -999.997em); top: -9.261em; left: 50%; margin-left: -10.173em;"><span class="mtable" id="MathJax-Span-210" style="padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 20.201em; height: 0px;"><span style="position: absolute; clip: rect(3.85em, 1002.94em, 10.735em, -999.997em); top: -7.591em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.939em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.939em, 1002.94em, 4.356em, -999.997em); top: -6.73em; left: 0em;"><span class="mtd" id="MathJax-Span-211"><span class="mrow" id="MathJax-Span-212"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.939em, 1002.94em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.466em;"><span class="msubsup" id="MathJax-Span-213"><span style="display: inline-block; position: relative; width: 1.42em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-214" style="font-family: MathJax_Math-italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.812em;"><span class="mi" id="MathJax-Span-215" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mi" id="MathJax-Span-216" style="font-family: MathJax_Math-italic;">Q</span><span class="mi" id="MathJax-Span-217" style="font-family: MathJax_Math-italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.939em, 1002.94em, 4.154em, -999.997em); top: -3.794em; left: 0em;"><span class="mtd" id="MathJax-Span-288"><span class="mrow" id="MathJax-Span-289"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.939em, 1002.94em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.466em;"><span class="msubsup" id="MathJax-Span-290"><span style="display: inline-block; position: relative; width: 1.42em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-291" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.863em;"><span class="mi" id="MathJax-Span-292" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mi" id="MathJax-Span-293" style="font-family: MathJax_Math-italic;">R</span><span class="mi" id="MathJax-Span-294" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.66em, 4.154em, -999.997em); top: -1.01em; left: 0em;"><span class="mtd" id="MathJax-Span-365"><span class="mrow" id="MathJax-Span-366"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.66em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.301em;"><span class="mi" id="MathJax-Span-367" style="font-family: MathJax_Math-italic;">J<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 7.596em;"></span></span><span style="position: absolute; clip: rect(4.761em, 1017.26em, 13.316em, -999.997em); top: -9.261em; left: 2.939em;"><span style="display: inline-block; position: relative; width: 17.265em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.179em, 1017.26em, 5.115em, -999.997em); top: -6.73em; left: 0em;"><span class="mtd" id="MathJax-Span-218"><span class="mrow" id="MathJax-Span-219"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.179em, 1017.26em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -8.603em;"><span class="mi" id="MathJax-Span-220"></span><span class="mo" id="MathJax-Span-221" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="msubsup" id="MathJax-Span-222" style="padding-left: 0.256em;"><span style="display: inline-block; position: relative; width: 2.939em; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1002.13em, 5.115em, -999.997em); top: -3.997em; left: 0em;"><span class="mrow" id="MathJax-Span-223"><span class="mo" id="MathJax-Span-224" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-225" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-226"><span class="mrow" id="MathJax-Span-227"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-228"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-229" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-230" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-231"><span class="mrow" id="MathJax-Span-232"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-233"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-234" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-235" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-236" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -5.161em; left: 2.382em;"><span class="mi" id="MathJax-Span-237" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mrow" id="MathJax-Span-238" style="padding-left: 0.154em;"><span class="mo" id="MathJax-Span-239" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-240" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 2.736em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1000.86em, 4.913em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.86em, 4.356em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-241"><span class="mrow" id="MathJax-Span-242"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.86em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-243"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.46em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-244" style="font-family: MathJax_Math-italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-245" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-249"><span class="mrow" id="MathJax-Span-250"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-251" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(2.534em, 1000.86em, 5.115em, -999.997em); top: -3.997em; left: 1.876em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-246"><span class="mrow" id="MathJax-Span-247"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-248" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.86em, 4.356em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-252"><span class="mrow" id="MathJax-Span-253"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.86em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-254"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.46em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-255" style="font-family: MathJax_Math-italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-256" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-257" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span class="mrow" id="MathJax-Span-258" style="padding-left: 0.154em;"><span class="mo" id="MathJax-Span-259" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-260" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-261"><span class="mrow" id="MathJax-Span-262"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-263"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-264" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-265" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-266"><span class="mrow" id="MathJax-Span-267"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-268"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-269" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-270" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-271" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span class="mo" id="MathJax-Span-272" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="msubsup" id="MathJax-Span-273" style="padding-left: 0.256em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.46em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-274" style="font-family: MathJax_Math-italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-275" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-276"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-277" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-278" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-279" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-280" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="msubsup" id="MathJax-Span-281" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.46em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-282" style="font-family: MathJax_Math-italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-283" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-284"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-285" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-286" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-287" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.179em, 1017.26em, 5.115em, -999.997em); top: -3.794em; left: 0em;"><span class="mtd" id="MathJax-Span-295"><span class="mrow" id="MathJax-Span-296"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.179em, 1017.26em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -8.603em;"><span class="mi" id="MathJax-Span-297"></span><span class="mo" id="MathJax-Span-298" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="msubsup" id="MathJax-Span-299" style="padding-left: 0.256em;"><span style="display: inline-block; position: relative; width: 2.939em; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1002.13em, 5.115em, -999.997em); top: -3.997em; left: 0em;"><span class="mrow" id="MathJax-Span-300"><span class="mo" id="MathJax-Span-301" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-302" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-303"><span class="mrow" id="MathJax-Span-304"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-305"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-306" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-307" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-308"><span class="mrow" id="MathJax-Span-309"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-310"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-311" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-312" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-313" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -5.161em; left: 2.382em;"><span class="mi" id="MathJax-Span-314" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mrow" id="MathJax-Span-315" style="padding-left: 0.154em;"><span class="mo" id="MathJax-Span-316" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-317" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 2.787em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1000.86em, 4.913em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-318"><span class="mrow" id="MathJax-Span-319"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-320"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-321" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-322" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-326"><span class="mrow" id="MathJax-Span-327"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-328" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(2.534em, 1000.86em, 5.065em, -999.997em); top: -3.997em; left: 1.876em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-323"><span class="mrow" id="MathJax-Span-324"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-325" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-329"><span class="mrow" id="MathJax-Span-330"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.86em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.453em;"><span class="msubsup" id="MathJax-Span-331"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-332" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-333" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-334" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span class="mrow" id="MathJax-Span-335" style="padding-left: 0.154em;"><span class="mo" id="MathJax-Span-336" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-337" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.01em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -4.655em; left: 0em;"><span class="mtd" id="MathJax-Span-338"><span class="mrow" id="MathJax-Span-339"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-340"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-341" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-342" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.237em; left: 0em;"><span class="mtd" id="MathJax-Span-343"><span class="mrow" id="MathJax-Span-344"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1001.01em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.504em;"><span class="msubsup" id="MathJax-Span-345"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-346" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.559em;"><span class="mn" id="MathJax-Span-347" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-348" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span class="mo" id="MathJax-Span-349" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="msubsup" id="MathJax-Span-350" style="padding-left: 0.256em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-351" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-352" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-353"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-354" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-355" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-356" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-357" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="msubsup" id="MathJax-Span-358" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-359" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-360" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-361"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-362" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-363" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-364" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.331em, 1013.97em, 5.065em, -999.997em); top: -1.01em; left: 0em;"><span class="mtd" id="MathJax-Span-368"><span class="mrow" id="MathJax-Span-369"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.331em, 1013.97em, 5.065em, -999.997em); top: -3.997em; left: 50%; margin-left: -6.983em;"><span class="mi" id="MathJax-Span-370"></span><span class="mo" id="MathJax-Span-371" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="msubsup" id="MathJax-Span-372" style="padding-left: 0.256em;"><span style="display: inline-block; position: relative; width: 1.42em; height: 0px;"><span style="position: absolute; clip: rect(2.483em, 1000.96em, 5.014em, -999.997em); top: -3.997em; left: 0em;"><span class="mo" id="MathJax-Span-373" style="font-family: MathJax_Size2; vertical-align: 0.003em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.408em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.36em, 4.154em, -999.997em); top: -5.06em; left: 1.116em;"><span class="mi" id="MathJax-Span-374" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.085em; left: 0.559em;"><span class="mn" id="MathJax-Span-375" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-376" style="padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.46em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-377" style="font-family: MathJax_Math-italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-378" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-379"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-380" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-381" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-382" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-383" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="msubsup" id="MathJax-Span-384" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.46em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-385" style="font-family: MathJax_Math-italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-386" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-387"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-388" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-389" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-390" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-391" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="msubsup" id="MathJax-Span-392" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-393" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-394" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-395"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-396" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-397" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-398" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-399" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="msubsup" id="MathJax-Span-400" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 0.863em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-401" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.458em;"><span class="mn" id="MathJax-Span-402" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-403"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-404" style="font-family: MathJax_Math-italic;">u</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -4.351em; left: 0.559em;"><span class="mn" id="MathJax-Span-405" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.394em, 1000.41em, 4.154em, -999.997em); top: -3.693em; left: 0.559em;"><span class="mn" id="MathJax-Span-406" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 9.266em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 9.266em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 9.266em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -4.025em; border-left: 0px solid; width: 0px; height: 8.522em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msup><mi>E</mi><mi>T</mi></msup><mi>Q</mi><mi>E</mi></mtd><mtd><mi></mi><mo>=</mo><msup><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mi>T</mi></msup><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>q</mi><mn>1</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>q</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><msub><mi>q</mi><mn>1</mn></msub><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>q</mi><mn>2</mn></msub><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup></mtd></mtr><mtr><mtd><msup><mi>U</mi><mi>T</mi></msup><mi>R</mi><mi>U</mi></mtd><mtd><mi></mi><mo>=</mo><msup><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>u</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mi>T</mi></msup><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>r</mi><mn>1</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>r</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>u</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>r</mi><mn>2</mn></msub><msubsup><mi>u</mi><mn>2</mn><mn>2</mn></msubsup></mtd></mtr><mtr><mtd><mi>J</mi></mtd><mtd><mi></mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><msub><mi>q</mi><mn>1</mn></msub><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>q</mi><mn>2</mn></msub><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>r</mi><mn>2</mn></msub><msubsup><mi>u</mi><mn>2</mn><mn>2</mn></msubsup></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-8">\begin{align*} E^TQE & = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}^T \begin{bmatrix} q_1 & 0 \\ 0 & q_2 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = q_1 x_1^2 + q_2 x_2^2 \\ U^TRU & = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}^T \begin{bmatrix} r_1 & 0 \\ 0 & r_2 \end{bmatrix}\begin{bmatrix} u_1 \\ u_2 \end{bmatrix} = r_1 u_1^2 +r_2 u_2^2 \\ J & = \int^t_0 q_1 x_1^2 + q_2 x_2^2 + r_1 u_1^2 + r_2 u_2^2 \end{align*}</script>
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margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0077em;">R</span><span class="mord mathnormal" style="margin-right: 0.109em;">U</span></span></span><span class="" style="top: -0.8342em;"><span class="pstrut" style="height: 3.6812em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0962em;">J</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 4.059em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 4.559em;"><span class="" style="top: -6.559em;"><span class="pstrut" style="height: 3.6812em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 1.6812em;"><span class="" style="top: -3.9029em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.6277em;"><span class="pstrut" style="height: 3.6812em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 1.6812em;"><span class="" style="top: -3.9029em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="arraycolsep" style="width: 0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">0</span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span class="" style="top: -3.61em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.95em;"><span class=""></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top: 0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -0.8342em;"><span class="pstrut" style="height: 3.6812em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5435em;"><span class="" style="top: -1.7881em; margin-left: -0.4445em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span class="" style="top: -3.8129em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.9119em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0359em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0278em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8641em;"><span class="" style="top: -2.453em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 4.059em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p>
2. 什么是MPC
MPC（Model Predictive Control，模型预测控制）的基本思想是通过建立一个系统的动态模型，并在每一个控制时刻使用这个模型来预测系统未来的行为。基于这些预测，它可以生成一个优化控制序列，然后通过执行第一个控制动作来调整系统状态，接着在下一个时刻重新计算和执行。这个过程反复进行，以使系统能够在未来的一段时间内优化一个特定的性能指标。
通常来说，MPC包括以下四个基本步骤：
-
系统模型化：建立描述系统动态行为的数学模型，通常是差分方程或微分方程。
-
预测：在当前时刻基于系统状态和控制输入，使用模型预测未来一段时间内的系统响应。
-
优化：基于预测的系统响应，通过求解一个优化问题来计算最优的控制输入序列，以最大化或最小化一个性能指标（如系统响应时间、能耗等）。
-
执行：根据优化得到的控制输入序列中的第一个值，执行这个控制动作，并将实际的系统状态反馈到下一个控制周期中
在具体实施的过程中MPC主要分为下列的三个步骤：
在k时刻：
- step1：估计/测量/读取当前的系统状态；
- step2：基于
u k , u k + 1 , ⋯ , u k + N \bold{u}_k, \bold{u}_{k+1}, \cdots, \bold{u}_{k+N} </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6528em; vertical-align: -0.2083em;"></span><span class="mord"><span class="mord mathbf">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathbf">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathbf">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight" style="margin-right: 0.109em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span></span></span></span></span>来进行最优化<br> <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"> J = ∑ k N − 1 E k T Q E k + U k T R U k + E N T F E N ⏟ Terminal Cost J = \sum_k^{N-1} E_k^TQE_k + U_k^T R U_k + \underbrace{E_N^TFE_N}_{\text{Terminal Cost}} </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0962em;">J</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 3.1304em; vertical-align: -1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.8283em;"><span class="" style="top: -1.8479em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span></span></span><span class="" style="top: -3.05em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span><span class="" style="top: -4.3em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.109em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.3021em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="" style="top: -2.453em; margin-left: -0.0576em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span><span class="mord mathnormal">Q</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0576em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.1383em; vertical-align: -0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.109em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="" style="top: -2.453em; margin-left: -0.109em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span></span></span><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.247em;"><span class=""></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0077em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.109em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.109em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 2.4724em; vertical-align: -1.5811em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="" style="top: -1.4189em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">Terminal Cost</span></span></span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="svg-align" style="top: -2.105em;"><span class="pstrut" style="height: 3em;"></span><span class="stretchy" style="height: 0.548em; min-width: 1.6em;"><span class="brace-left" style="height: 0.548em;"> <svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"> <path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z">
ENTFEN
其中Terminal Cost代表了模型滑动窗口的末端的控制误差。
- step3：只取
-
u k \bold{u}_k </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.5944em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathbf">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>，进行滚动优化控制。</li></ul>3. 二次规划Quadratic Programming
为了能求解MPC问题，我们需要将其转换成二次规划（Quadratic Programming）的形式，对于二次规划已经有很多成熟的求解器了，我们只需要使用这些求解器就能顺利求解。
二次规划一般具有如下的形式：
min x x T H x + f T x subject to . . . <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-9-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd /><mtd><mi></mi><munder><mo movablelimits="true" form="prefix">min</mo><mtext mathcolor="red">\bold</mtext></munder><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow><mtext mathcolor="red">\bold</mtext><msup><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow><mi>T</mi></msup><mi>H</mi><mtext mathcolor="red">\bold</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow><mo>+</mo><msup><mi>f</mi><mi>T</mi></msup><mtext mathcolor="red">\bold</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow></mtd></mtr><mtr><mtd /><mtd><mtext>subject to</mtext><mspace width="1em" /><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-407" style="width: 100%; display: inline-block; min-width: 15.746em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.179em, 1015.39em, 5.622em, -999.997em); top: -4.149em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-408"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.179em, 1015.39em, 5.622em, -999.997em); top: -4.149em; left: 50%; margin-left: -7.743em;"><span class="mtable" id="MathJax-Span-409" style="padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 15.291em; height: 0px;"><span style="position: absolute; clip: rect(2.888em, 1000em, 5.267em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 0em; height: 0px;"><span style="position: absolute; clip: rect(3.85em, 1000em, 4.154em, -999.997em); top: -4.958em; right: 0em;"><span class="mtd" id="MathJax-Span-410"><span class="mrow" id="MathJax-Span-411"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.85em, 1000em, 4.154em, -999.997em); top: -2.883em; right: 0em;"><span class="mtd" id="MathJax-Span-440"><span class="mrow" id="MathJax-Span-441"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(2.179em, 1015.24em, 5.622em, -999.997em); top: -4.149em; left: 0em;"><span style="display: inline-block; position: relative; width: 15.291em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.939em, 1015.24em, 5.115em, -999.997em); top: -4.958em; left: 0em;"><span class="mtd" id="MathJax-Span-412"><span class="mrow" id="MathJax-Span-413"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.939em, 1015.24em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -7.641em;"><span class="mi" id="MathJax-Span-414"></span><span class="munderover" id="MathJax-Span-415" style="padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 1.673em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1001.67em, 4.154em, -999.997em); top: -3.997em; left: 0.003em;"><span class="mo" id="MathJax-Span-416" style="font-family: MathJax_Main;">min</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1001.67em, 4.407em, -999.997em); top: -3.288em; left: 0em;"><span class="mtext" id="MathJax-Span-417" style="font-size: 70.7%; font-family: MathJax_Main; color: red;">\bold</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="texatom" id="MathJax-Span-418" style="padding-left: 0.154em;"><span class="mrow" id="MathJax-Span-419"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-420" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span class="mtext" id="MathJax-Span-421" style="font-family: MathJax_Main; color: red;">\bold</span><span class="msubsup" id="MathJax-Span-422"><span style="display: inline-block; position: relative; width: 1.167em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="texatom" id="MathJax-Span-423"><span class="mrow" id="MathJax-Span-424"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-425" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.559em;"><span class="mi" id="MathJax-Span-426" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mi" id="MathJax-Span-427" style="font-family: MathJax_Math-italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span class="mtext" id="MathJax-Span-428" style="font-family: MathJax_Main; color: red;">\bold</span><span class="texatom" id="MathJax-Span-429"><span class="mrow" id="MathJax-Span-430"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-431" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-432" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="msubsup" id="MathJax-Span-433" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 1.217em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.56em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-434" style="font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.61em;"><span class="mi" id="MathJax-Span-435" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mtext" id="MathJax-Span-436" style="font-family: MathJax_Main; color: red;">\bold</span><span class="texatom" id="MathJax-Span-437"><span class="mrow" id="MathJax-Span-438"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-439" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.141em, 1006.28em, 4.356em, -999.997em); top: -2.883em; left: 0em;"><span class="mtd" id="MathJax-Span-442"><span class="mrow" id="MathJax-Span-443"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1006.28em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -3.187em;"><span class="mtext" id="MathJax-Span-444" style="font-family: MathJax_Main;">subject to</span><span class="mspace" id="MathJax-Span-445" style="height: 0em; vertical-align: 0em; width: 1.015em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-446" style="font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-447" style="font-family: MathJax_Main; padding-left: 0.154em;">.</span><span class="mo" id="MathJax-Span-448" style="font-family: MathJax_Main; padding-left: 0.154em;">.</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.154em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.154em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.154em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.392em; border-left: 0px solid; width: 0px; height: 3.359em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd></mtd><mtd><mi></mi><munder><mo movablelimits="true" form="prefix">min</mo><mtext mathcolor="red">\bold</mtext></munder><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow><mtext mathcolor="red">\bold</mtext><msup><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow><mi>T</mi></msup><mi>H</mi><mtext mathcolor="red">\bold</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow><mo>+</mo><msup><mi>f</mi><mi>T</mi></msup><mtext mathcolor="red">\bold</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mtext>subject to</mtext><mspace width="1em"></mspace><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-9">\begin{align*} & \min_\bold{x} \bold{x}^TH\bold{x} + f^T\bold{x} \\ & \text{subject to} \quad... \end{align*}</script> </span><span class="katex-html"><span class="base"><span class="strut" style="height: 3.3913em; vertical-align: -1.4457em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.9457em;"><span class="" style="top: -3.9457em;"><span class="pstrut" style="height: 2.8913em;"></span><span class="mord"></span></span><span class="" style="top: -2.1057em;"><span class="pstrut" style="height: 2.8913em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.4457em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.9457em;"><span class="" style="top: -4.0543em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.6679em;"><span class="" style="top: -2.4em; margin-left: 0em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">x</span></span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class=""><span class="mop">min</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.7em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0813em;">H</span><span class="mord mathbf">x</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8913em;"><span class="" style="top: -3.113em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span></span></span></span></span><span class="mord mathbf">x</span></span></span><span class="" style="top: -2.2143em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord">subject to</span></span><span class="mspace" style="margin-right: 1em;"></span><span class="mord">...</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.4457em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></span></p>其中
H H </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0813em;">H</span></span></span></span></span>是正定的对称矩阵，<span class="katex--inline"><span class="katex"><span class="katex-mathml"> f f </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.8889em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span></span></span></span></span>，<span class="katex--inline"><span class="katex"><span class="katex-mathml"> x \bold{x} </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.4444em;"></span><span class="mord mathbf">x</span></span></span></span></span>是向量。</p>4. MPC为什么可以转换成QP问题（推导过程）
考虑一个离散的线性系统
x ( k + 1 ) = A n × n x ( k ) + B n × p u ( k ) x n × 1 = [ x 1 x 2 ⋮ x n ] , u p × 1 = [ u 1 u 2
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