线性系统理论

可控标准型

可观标准型

自身性

求导

求逆

定义法

特征值法

多项式法

拉氏变换法

<span class="equation-text" contenteditable="false" data-index="0" data-equation="x(t)=\int_{t_0}^{t}&nbsp;e^{A(t-\tau)}Bu(\tau)\,&nbsp;\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="x(t)=\int_{0}^{t} e^{A(t-\tau)}Bu(\tau)\, \mathrm{d}\tau+ e^{At}x_0"><span></span><span></span></span>

<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>

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="\phi(t,t_0)=\psi(t)*\psi^{-1}(t_0)"><span></span><span></span></span>

状态转移矩阵的判断

<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>

判据

能控性指数<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mu"><span></span><span></span></span>

多输入系统

判据

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>

2、变换矩阵P<br>

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>

4、状态<span class="equation-text" contenteditable="false" data-index="0" data-equation="\bar{x}"><span></span><span></span></span>

对偶系数矩阵

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

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>

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>

一致稳定：与时间无关

渐进稳定：对终点有要求

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

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)&nbsp;\to\infty"><span></span><span></span></span>

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

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)&nbsp;\to\infty"><span></span><span></span></span>

充分条件

充要条件：<br>特征值判据

系统完全能控

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 &amp; 0 &amp; 0\\ \alpha_2 &amp; 1 &amp; 0 \\ \alpha_1 &amp; \alpha_2 &amp; 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>

方案1

方案2

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>

3、<span class="equation-text" data-index="0" data-equation="\bar{A}=QAQ^{-1} =\left[ {\begin {array} {cc} A_{11} &amp; A_{12} \\ A_{21} &amp; 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&amp;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>