离散线性系统能控性与能观性

离散线型系统

状态方程：
$\begin{array}{ll} x_{t+1} = Ax_t+Bu_t \end{array}$
输出方程：
$\begin{array}{ll} y_{t+1} = Cx_{t+1}+Du_{t} \end{array}$
$A\in R^{n\times n}, B\in R^{n\times m}, C\in R^{o\times n},D\in R^{o\times m},$
$x\in R^n, u\in R^m, y\in R^o.$

能控性

能控性：可以通过控制输入 $u$，使得系统状态从任意初始状态 $x_0$到达任意终止状态 $x_t.$

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$\begin{array}{ll} x_1 &= Ax_0 + Bu_0, \\ x_2 &= Ax_1 + Bu_1 = A(Ax_0 + Bu_0) + Bu_1 = A^2x_0 + ABu_0 + Bu_1,\\ &\ldots \\ x_t &= A^tx_0 + \left[ \begin{array}{llll} A^{t-1}B &\cdots &AB &B \end{array} \right] \left[ \begin{array}{c} u_0\\ \vdots\\ u_{t-1} \end{array} \right] \end{array}$

要使 $x_t$ 到达任意状态，必须有 $C \triangleq \left[ \begin{array}{llll} A^{t-1}B &\cdots &AB &B \end{array} \right]$行满秩.

有哈密顿凯莱定理可知
$rank(\left[ \begin{array}{llll} A^{t-1}B &\cdots &AB &B \end{array} \right])= rank(\left[ \begin{array}{llll} A^{n-1}B &\cdots &AB &B \end{array} \right])$
所以能控性充要条件为：
$rank(\left[ \begin{array}{llll} A^{n-1}B &\cdots &AB &B \end{array} \right]) =n$

能观性

能观性：系统的当前状态 $x_0$ 可以由有限时间[t_0, t]内的输入输出完全确定.

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$\begin{array}{ll} y_0 &= Cx_0 + Du_0, \\ & \vdots \\ y_t &= Cx_t + Du_t\\ &=C\left( A^tx_0 + \left[ \begin{array}{llll} A^{t-1}B &\cdots &AB &B \end{array} \right] \left[ \begin{array}{c} u_0\\ \vdots\\ u_{t-1} \end{array} \right]\right) +D u_t \end{array}$
即
$\begin{array}{ll} y_0 - Du_0&= Cx_0 , \\ & \vdots \\ y_t - \left[ \begin{array}{llll} CA^{t-1}B &\cdots &CAB &CB \end{array} \right] \left[ \begin{array}{c} u_0\\ \vdots\\ u_{t-1} \end{array} \right] -D u_t&= CA^tx_0 \end{array}$
即
$\left[ \begin{array}{c} y_0 - Du_0\\ \vdots \\ y_t - CA^{t-1}Bu_0 -\cdots -CABu_{t-2} -CBu_{t-1} -D u_t& \end{array} \right]= \left[ \begin{array}{c} C\\ \vdots \\ CA^t \end{array} \right]x_0$
上式的左端已知，为唯一确定出 $x_0$，必须有 $\left[ \begin{array}{c} C\\ \vdots \\ CA^t \end{array} \right]$ 列满秩.

由哈密顿凯莱定理， $t\geq n-1$时，
$rank(\left[ \begin{array}{c} C\\ \vdots \\ CA^t \end{array} \right])= rank(\left[ \begin{array}{c} C\\ \vdots \\ CA^{n-1} \end{array} \right])$
所以系统能观的充要条件为： $rank(\left[ \begin{array}{c} C\\ \vdots \\ CA^{n-1} \end{array} \right])=n$