线性系统理论
2022-06-17 11:15:17 25 举报AI智能生成
研究生线性系统理论
作者其他创作
大纲/内容
状态空间
状态方程的来源
由传递函数得状态方程
可控标准型
可观标准型
由方框图得状态方程
状态方程的变化
对角型
约当型
可控标准型
可观标准型
传递函数矩阵G(s)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="G(s)=C(sI-A)^{-1}B+D"><span></span><span></span></span>
运动分析
矩阵指数函数
性质
自身性
求导
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(e^{At})'=A*e^{At}=e^{At}*A"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="A=\left(\frac{de^{At}}{dt} \right)\rvert_{t=0}=(Ae^{At})\rvert_{t=0}=A" contenteditable="false"><span></span><span></span></span>
小技巧
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=\left(\frac{d\phi(t)}{dt} \right)\rvert_{t=0}=\left(\frac{d\psi(t)*\psi^{-1}(t_0)}{dt} \right)\rvert_{t=0}=\frac{d\psi(t)}{dt} \rvert_{t=0}*\psi^{-1}(t_0)"><span></span><span></span></span>
求逆
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(e^{At})^{-1}=e^{-At}"><span></span><span></span></span>
计算
定义法
特征值法
单根
重根
多项式法
拉氏变换法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^{At}=L^{-1}[(sI-A)^{-1}]"><span></span><span></span></span>
状态响应
<span class="equation-text" contenteditable="false" data-index="0" data-equation="t_0≠0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x(t)=\int_{t_0}^{t} e^{A(t-\tau)}Bu(\tau)\, \mathrm{d}\tau+ e^{A(t-t_0)}x_0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="t_0=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x(t)=\int_{0}^{t} e^{A(t-\tau)}Bu(\tau)\, \mathrm{d}\tau+ e^{At}x_0"><span></span><span></span></span>
脉冲响应矩阵H(t)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="H(t)=Ce^{At}B+D\delta(t)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="H(t)=L^{-1}[G(s)]"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="G(s)=L[H(t)]"><span></span><span></span></span>
时变系统
基本解阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="\psi(t) "><span></span><span></span></span>
自治方程<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dot x=Ax"><span></span><span></span></span>
n个线性无关的初始状态<span class="equation-text" contenteditable="false" data-index="0" data-equation="x'(t_0),x''(t_0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\psi(t_0)=[x'(t_0),x''(t_0)]"><span></span><span></span></span>
状态转移矩阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(t,t_0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(t,t_0)=\psi(t)*\psi^{-1}(t_0)"><span></span><span></span></span>
时不变
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(t)=e^{At}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(t,t_0)=\phi(t-t_0)=e^{A(t-t_0)}"><span></span><span></span></span>
状态转移矩阵的判断
自身性
时变
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(t_0,t_0)=\psi(t_0)*\psi^{-1}(t_0)=I"><span></span><span></span></span>
时不变
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(t_0-t_0)=e^{t_0-t_0}=I"><span></span><span></span></span>
状态响应
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x(t)=\int_{t_0}^{t} \phi(t,\tau)Bu(\tau)\, \mathrm{d}\tau+ \phi(t,t_0)x_0"><span></span><span></span></span>
能控能观
判据
能控
判据
秩判据
<span class="equation-text" contenteditable="false" data-index="0" data-equation="rankQ_c=n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q_c=[B,AB,A^2B, ...,A^{n-1}B]"><span></span><span></span></span>
对角判据
B中不包含全零行
约当判据
约当块对应的B的末行，线性无关
能控性指数<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mu"><span></span><span></span></span>
使<span class="equation-text" contenteditable="false" data-index="0" data-equation="rank(B,AB,A^2B, ...,A^{k-1}B)=n"><span></span><span></span></span> 成立的k的最小整数
单输入能控
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mu=n"><span></span><span></span></span>
多输入能控
多输入系统
<span class="equation-text" contenteditable="false" data-index="0" data-equation="rank(B,AB,A^2B, ...,A^{n-r}B)=n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="r=rankB"><span></span><span></span></span>
能观
判据
秩判据
<span class="equation-text" contenteditable="false" data-index="0" data-equation="rankQ_o=n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q_o=\left[ \begin{array} {c} C \\ CA\\ CA^2\\ \vdots&\\ CA^{n-1} \end{array} \right]"><span></span><span></span></span>
对角判据
C中不包含全零列
约当判据
约当块对应的C的末列，线性无关
对偶系统
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{ A,B,C }"><span></span><span></span></span>是原构系统，<span class="equation-text" data-index="1" data-equation="{A_d,B_d,C_d}" contenteditable="false"><span></span><span></span></span>是对偶系统
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_d=-A^T,B_d=C^T,C_d=B^T"><span></span><span></span></span>
原构线性→对偶线性，原构时变→对偶时变
原构能控<span class="equation-text" data-index="0" data-equation="\Leftrightarrow" contenteditable="false"><span></span><span></span></span>对偶能观，原构能观<span class="equation-text" contenteditable="false" data-index="1" data-equation="\Leftrightarrow"><span></span><span></span></span>对偶能控
标准化
能控标准型
1、特征多项式<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha(s)"><span></span><span></span></span><br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha(s)=det(sI-A)=s^3+\alpha_2s^2+\alpha_1s+\alpha_0"><span></span><span></span></span>
2、变换矩阵P<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P=[A^2b,Ab,B]* \left[ {\begin {array} {ccc} 1 & 0 & 0\\ \alpha_2 & 1 & 0 \\ \alpha_1 & \alpha_2 & 1 \\ \end{array} } \right]"><span></span><span></span></span>
3、系数矩阵<span class="equation-text" data-index="0" data-equation="\bar{A}、\bar{B}、\bar{C}"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\bar{A}=\left[ {\begin {array} {ccc} 0 & 1 & 0\\ 0 & 0 & 1 \\ -\alpha_0 & -\alpha_1 & -\alpha_2 \\ \end{array} } \right]" contenteditable="false"><span></span><span></span></span>、<span class="equation-text" data-index="1" data-equation="\bar{B}=\left[ \begin{array} {c} 0 \\ 0\\1 \end{array} \right]" contenteditable="false"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="2" data-equation="\bar{C}=CP"><span></span><span></span></span>
4、状态<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{x}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{x}=P^{-1}x"><span></span><span></span></span>
验证<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{A}=P^{-1}AP"><span></span><span></span></span>
能观标准型
对偶系数矩阵
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A'=A^T,B'=B^T,C'=C^T"><span></span><span></span></span>
化为可控标准型<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overline{A'},\overline{B',}\overline{C'}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{A}=\overline{A'}^T,\bar{B}=\overline{B'}^T,\bar{C}=\overline{C'}^T"><span></span><span></span></span>
分解
能控
1、能控性判别矩阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q_k，rankQ_k=r"><span></span><span></span></span>
2、从选取Qk中选取r个线性无关的R列向量，再任取n-r与R线性无关的列向量组成P
子主题
3、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{A}=P^{-1}AP、\bar{B}=P^{-1}B、\bar{C}=CP"><span></span><span></span></span>
4、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{x}=P^{-1}x"><span></span><span></span></span>
能观
1、能控性判别矩阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q_o，rankQ_o=q"><span></span><span></span></span>
2、从选取Qo中选取q个线性无关的R行向量，再任取n-r与R线性无关的行向量组成Q
3、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{A}=QAQ^{-1}、\bar{B}=FB、\bar{C}=CQ^{-1}"><span></span><span></span></span>
4、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{x}=Qx"><span></span><span></span></span>
稳定性
分类
李氏稳定
一致稳定：与时间无关
渐进稳定：对终点有要求
一致渐进稳定：与时间无关
大范围渐进稳定：对起点无要求
大范围渐进稳定的判据
判据1
1、V(x)正定
2、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dot V(x)"><span></span><span></span></span>负定
除原点外，<span class="equation-text" data-index="0" data-equation="\dot V(x)<0" contenteditable="false"><span></span><span></span></span>，例如：<br><span class="equation-text" contenteditable="false" data-index="1" data-equation="\dot V(x)=-2(x_1^2+x_2^2)^2"><span></span><span></span></span><br>
3、当<span class="equation-text" data-index="0" data-equation="\lVert x \rVert\to\infty" contenteditable="false"><span></span><span></span></span>时，<span class="equation-text" contenteditable="false" data-index="1" data-equation="V(x) \to\infty"><span></span><span></span></span>
判据2
1、V(x)正定
2、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dot V(x)"><span></span><span></span></span>负半定
除原点外，<span class="equation-text" data-index="0" data-equation="\dot V(x)<=0" contenteditable="false"><span></span><span></span></span>，例如：<br><span class="equation-text" data-index="1" data-equation="\dot V(x)=-x_2^2" contenteditable="false"><span></span><span></span></span>，<br><span class="equation-text" data-index="2" data-equation="\dot V(x)=-2x_1^2x_2^2" contenteditable="false"><span></span><span></span></span>，<br><span class="equation-text" contenteditable="false" data-index="3" data-equation="\dot V(x)=-2x_1^2(x_2+1)^2"><span></span><span></span></span><br>
3、<span class="equation-text" contenteditable="false" data-index="0" data-equation="V(x(t;x_0,0)) \not\equiv 0"><span></span><span></span></span>
4、当<span class="equation-text" data-index="0" data-equation="\lVert x \rVert\to\infty" contenteditable="false"><span></span><span></span></span>时，<span class="equation-text" contenteditable="false" data-index="1" data-equation="V(x) \to\infty"><span></span><span></span></span>
时不变
x=0不稳定
充分条件
V(x)一阶偏导数存在且V(0)=0
1、V(x)正定
2、<span class="equation-text" data-index="0" data-equation="\dot V(x)" contenteditable="false"><span></span><span></span></span>正定
x=0渐进稳定
充要条件：<br>特征值判据
A的所有特征值均具有负实部
反馈
极点配置
可配置的充分必要条件
系统完全能控
算法
1、判断能控
2、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha(s)=det(sI-A)=s^3+\alpha_2s^2+\alpha_1s+\alpha_0"><span></span><span></span></span>
3、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha^*(s)=(s-\lambda^*_1)(s-\lambda^*_2)(s-\lambda^*_3)=s^3+\alpha^*_2s^2+\alpha^*_1s+\alpha^*_0"><span></span><span></span></span>
4、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar k=[\alpha^*_0-\alpha_0,\alpha^*_1-\alpha_1,\alpha^*_2-\alpha_2]"><span></span><span></span></span>
5、<span class="equation-text" contenteditable="false" data-index="0" data-equation="P=[A^2b,Ab,B]* \left[ {\begin {array} {ccc} 1 & 0 & 0\\ \alpha_2 & 1 & 0 \\ \alpha_1 & \alpha_2 & 1 \\ \end{array} } \right]"><span></span><span></span></span>
6、<span class="equation-text" contenteditable="false" data-index="0" data-equation="k=\bar kP^{-1}"><span></span><span></span></span>
验证
<span class="equation-text" contenteditable="false" data-index="0" data-equation="det[s-(A-bk)]=\alpha^*(s)"><span></span><span></span></span>
观测器
全维
方案1
1、判断{A,C}能观测
2、设<span class="equation-text" data-index="0" data-equation="H=\left[ \begin{array} {c} h_1\\ h_2 \end{array} \right]" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="\alpha(s)=det(sI-(A-Hc))"><span></span><span></span></span>
3、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha^*(s)=(s-\lambda^*_1)(s-\lambda^*_2)"><span></span><span></span></span>
4、<span class="equation-text" data-index="0" data-equation="\alpha(s)=\alpha^*(s)" contenteditable="false"><span></span><span></span></span>解得<span class="equation-text" contenteditable="false" data-index="1" data-equation="h1、h2"><span></span><span></span></span>
5、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dot{\hat{x}}=(A-HC){\hat{x}}+Bu+Ly"><span></span><span></span></span>
验证
<span class="equation-text" contenteditable="false" data-index="0" data-equation="det[s-(A-HC)]=\alpha^*(s)"><span></span><span></span></span>
方案2
子主题
降维
1、判断{A,C}能观测,构造<span class="equation-text" contenteditable="false" data-index="0" data-equation="m=n-q(rankC=q)"><span></span><span></span></span>维降维观测器
2、构造n*n矩阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q=\left[ \begin{array} {c} C\\R \end{array} \right]"><span></span><span></span></span>
↔化为能控标准形
从选取Qk中选取r个线性无关的R列向量，再任取n-r与R线性无关的列向量组成P
3、<span class="equation-text" data-index="0" data-equation="\bar{A}=QAQ^{-1} =\left[ {\begin {array} {cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array} } \right]" contenteditable="false"><span></span><span></span></span>、<span class="equation-text" data-index="1" data-equation="\bar{B}=QB=\left[ \begin{array} {c} \bar{B}_1\\\bar{B} _2\end{array} \right]" contenteditable="false"><span></span><span></span></span> ，<span class="equation-text" contenteditable="false" data-index="2" data-equation="\bar{C}=CQ^{-1}=\left[ \begin{array} {cc} I_q&0\end{array} \right]"><span></span><span></span></span>其中A22是m*m维的矩阵
4、对<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_{22}^T、A_{12}^T"><span></span><span></span></span>进行极点配置得到k
5、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{L}=k^{T}"><span></span><span></span></span>
6、降维观测器<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dot z=(\bar{A}_{22}-\bar{L}\bar{A}_{12})z+[(\bar{A}_{22}-\bar{L}\bar{A}_{12})\bar{L}+(\bar{A}_{21}-\bar{L}\bar{A}_{11})]y+(\bar{B}_{2}-\bar{L}\bar{B}_{1})u"><span></span><span></span></span>
7、重构的状态<span class="equation-text" contenteditable="false" data-index="0" data-equation="\hat{x}=Q^{-1}\left[ \begin{array} {c} y\\z+\bar{L}y \end{array} \right]"><span></span><span></span></span>
验证
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{C}=CQ^{-1}=\left[ \begin{array} {cc} I_q&0\end{array} \right]"><span></span><span></span></span>
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