离散线性系统能控性与能观性

离散线型系统

状态方程：
$$\begin{array}{l}{x}_{t+1}=A{x}_t+B{u}_t\end{array}$$
输出方程：
$$\begin{array}{l}{y}_{t+1}=C{x}_{t+1}+D{u}_t\end{array}$$
$A\in R^{n\times n},B\in R^{n\times m},C\in R^{o\times n},D\in R^{o\times m},$
$x\in R^n,u\in R^m,y\in R^o.$

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$x_t.$

---

x 1 = A x 0 + B u 0 , x 2 = A x 1 + B u 1 = A ( A x 0 + B u 0 ) + B u 1 = A 2 x 0 + A B u 0 + B u 1 , … x t = A t x 0 + [ A t − 1 B ⋯ A B B ] [ u 0 ⋮ u t − 1 ] \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} x1​x2​xt​​=Ax0​+Bu0​,=Ax1​+Bu1​=A(Ax0​+Bu