线性时变系统的能控能观性

对于线性时变系统

\begin{aligned} \dot{x} (t) & = A (t) x (t) + B (t) u (t) \\ y (t) & = C (t) x (t) + D (t) u (t) \end{aligned}

\begin{matrix} x (t) = Φ (t, t_{0}) x_{0} + \int_{t_{0}}^{T} Φ (t, τ) B (τ) u (τ) d τ \\ y (t) = C (t) Φ (t, t_{0}) x_{0} + C (t) \int_{t_{0}}^{T} Φ (t, τ) B (τ) u (τ) d τ + D (t) u (t) \end{matrix}

\begin{array}{r} W_{c} (t_{0}, t_{f}) = \int_{t_{0}}^{t_{f}} Φ (t_{0}, t) B (t) B^{T} (t) Φ^{T} (t_{0}, t) d t \end{array}

\begin{matrix} B_{1} (t) = B (t) B_{i} (t) = - A (t) B_{i - 1} (t) + {\dot{B}}_{i - 1} (t) \\ Q_{c} (t) = (\begin{matrix} B_{1} (t), B_{2} (t), \dots, B_{n} (t) \end{matrix}) \end{matrix}

在 $[0, t_{f}]$ 上状态完全能控：

\begin{array}{r} r a n k Q_{c} (t_{f}) = n \end{array}

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{array}{r} C (t) Φ (t, t_{0}) x (t_{0}) \equiv 0, t \in [t_{0}, t_{f}] \end{array}

\begin{array}{r} W_{0} (t_{0}, t_{f}) = \int_{t_{0}}^{t_{f}} Φ^{T} (t_{0}, t) C^{T} (t) C (t) Φ (t_{0}, t) d t \end{array}

$C_{1} (t) = C (t), C_{i} (t) = C_{i - 1} (t) A (t) + {\dot{C}}_{i - 1} (t)$

\begin{array}{r} R (t) = {(\begin{array}{c} C_{1} (t), C_{2} (t), \dots, C_{n} (t) \end{array})}^{T} \end{array}