线性时变系统的能控能观性

对于线性时变系统

\dot{\mathbf{x}}(t)&=\mathbf{A}(t) \mathbf{x}(t)+ \mathbf{B}(t) \mathbf{u}(t) \\ \mathbf{y}(t)&=\mathbf{C}(t)\mathbf{x}(t)+\mathbf{D}(t) \mathbf{u}(t) \end{align}$$ $$\begin{gathered} \mathbf{x}(t)=\varPhi(t,t_{0})\mathbf{x}_{0}+\int _{t_{0}}^{T} \varPhi(t,\tau)\mathbf{B}(\tau)\mathbf{u}(\tau) \, d\tau \\ \mathbf{y}(t)=\mathbf{C}(t)\varPhi(t,t_{0})\mathbf{x}_{0}+\mathbf{C}(t)\int _{t_{0}}^{T} \varPhi(t,\tau)\mathbf{B}(\tau)\mathbf{u}(\tau) \, d\tau+\mathbf{D}(t) \mathbf{u}(t) \end{gathered}$$ ### 一、线性时变系统的可控性判据 #### 充分必要条件 [[Gram矩阵\|Gram矩阵]]非奇异 $$\begin{align} W_{c}(t_{0},t_{f})=\int _{t_{0}}^{t_{f}} \varPhi(t_{0},t)B(t)B^{T}(t) \varPhi^{T}(t_{0},t)\, dt \end{align}$$ #### 充分条件 $$\begin{gathered} B_{1}(t)=B(t)\quad B_{i}(t)=-A(t)B_{i-1}(t)+\dot{B}_{i-1}(t) \\ Q_{c}(t)=\begin{pmatrix} B_{1}(t),B_{2}(t),\cdots,B_{n}(t) \end{pmatrix} \end{gathered}$$ 在 $[0,t_{f}]$ 上状态完全能控： $$\begin{align} rank\;Q_{c}(t_{f})=n \end{align}$$ ### 二、线性时变系统的能观性 时变系统能观性：根据不能观测的定义，得到不能观测状态的数学表达式： $$\begin{align} C(t)\varPhi(t,t_{0})\mathbf{x}(t_{0}) \equiv 0,t\in\left[t_{0},t_{f}\right] \end{align}$$ #### 充分必要条件 $$\begin{align} W_{0}(t_{0},t_{f})= \int _{t_{0}}^{t_{f}} \varPhi^{T}(t_{0},t)C^{T}(t) C(t) \varPhi(t_{0},t)\, dt \end{align}$$ #### 充分条件 $C_{1}(t)=C(t),C_{i}(t)=C_{i-1}(t)A(t)+\dot{C}_{i-1}(t)$ $$\begin{align} R(t)=\begin{pmatrix}\; C_{1}(t),C_{2}(t),\cdots,C_{n}(t)\; \end{pmatrix}^{T} \end{align}$$