Appendix B
State Space Analysis
B.l State Space Form for Continuous-Time Systems
线性时不变（LTI）、连续、动态系统描述如下
\[\begin{aligned} \dot{\mathbf{x}}(t) = \mathbf{Ax}(t) + \mathbf{Bu}(t) & \qquad \text{state equation} \\ \mathbf{y}(t) = \mathbf{Cx}(t) + \mathbf{Du}(t) & \qquad \text{output equation} \end{aligned} \tag{B.1.1}\]初始条件\(\mathbf{x}(t=t_0) = \mathbf{x}(t_0)\)。
对上述方程组作Laplace变换，得
\[\begin{aligned} & \mathbf{X}(s) = [s\mathbf{I-A}]^{-1} [\mathbf{x}(t_0)+\mathbf{BU}(s)] \\ & \mathbf{Y}(s) = \mathbf{C}[s\mathbf{I-A}]^{-1} [\mathbf{x}(t_0)+\mathbf{BU}(s)] + \mathbf{DU}(s) \end{aligned} \tag{B.1.3}\]根据零初始条件\(\mathbf{x}(t_0)=0\)得传递函数\(\mathbf{G}(s)\)，得
\[\mathbf{G}(s) = \frac {\mathbf{Y}(s)} {\mathbf{U}(s)} = \mathbf{C} [s \mathbf{I} - \mathbf{A}]^{-1} \mathbf{B} + \mathbf{D}\]LTI系统的解为
\[\begin{aligned} & \mathbf{x}(t) = \pmb{\Phi}(t,t_0) \mathbf{x}(t_0) + \int_{t_0}^t {\pmb{\Phi}(t,\tau) \mathbf{Bu}(\tau)} d \tau \\ & \mathbf{y}(t) = \mathbf{C} \pmb{\Phi}(t,t_0) \mathbf{x}(t_0) + \mathbf{C} \int_{t_0}^t {\pmb{\Phi}(t,\tau) \mathbf{Bu}(\tau)} d \tau + \mathbf{Du}(t) \end{aligned} \tag{B.1.6}\]其中，\(\pmb{\Phi}(t,t_0)\)为系统的状态转移阵
\[\pmb{\Phi}(t,t_0) = e^{\mathbf{A}(t-t_0)} \tag{B.1.7}\]且有如下性质
\[\pmb{\Phi}(t_0,t_0) = \mathbf{I} , \quad \pmb{\Phi}(t_2,t_1)\pmb{\Phi}(t_1,t_0) = \pmb{\Phi}(t_2,t_0) \tag{B.1.8}\]线性时变（LTV）、连续、动态系统描述如下
\[\begin{aligned} \dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t) & \qquad \text{state equation} \\ \mathbf{y}(t) = \mathbf{C}(t) \mathbf{x}(t) + \mathbf{D}(t) \mathbf{u}(t) & \qquad \text{output equation} \end{aligned} \tag{B.1.5}\]初始条件\(\mathbf{x}(t=t_0) = \mathbf{x}(t_0)\)。
系统的解为
\[\begin{aligned} & \mathbf{x}(t) = \pmb{\Phi}(t,t_0) \mathbf{x}(t_0) + \int_{t_0}^t {\pmb{\Phi}(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau)} d \tau \\ & \mathbf{y}(t) = \mathbf{C}(t) \pmb{\Phi}(t,t_0) \mathbf{x}(t_0) + \mathbf{C}(t) \int_{t_0}^t {\pmb{\Phi}(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau)} d \tau + \mathbf{D}(t) \mathbf{u}(t) \end{aligned} \tag{B.1.9}\]其中，\(\pmb{\Phi}(t,t_0)\)仍为系统的状态转移阵，难以计算，也有性质(B.1.8)。但是，根据基本矩阵\(\mathbf{X}(t)\)满足
\[\dot{\mathbf{X}}(t) = \mathbf{A}(t) \mathbf{X}(t) \tag{B.1.10}\]可以写成
\[\pmb{\Phi}(t,t_0) = \mathbf{X}(t) \mathbf{X}^{-1}(t_0) \tag{B.1.11}\]B.2 Linear Matrix Equations
关于已知矩阵\(\mathbf{A}\)和\(\mathbf{Q}\)的未知矩阵\(\mathbf{P}\)的线性联立方程组写作
\[\mathbf{PA+A'P+Q} = 0 \tag{B.2.1}\]特别地，如果\(\mathbf{Q}\)正定，那么就存在一个唯一的正定阵\(\mathbf{P}\)满足方程，当且仅当\(\mathbf{A}\)是渐近稳定的或\(\lambda \{\mathbf{A}\}\)的实部\(（\text{Re}）\lt 0\)。(B.2.1)被称为Lyapunov equation，解为
\[\mathbf{P} = \int_0^{\infin} e^{\mathbf{A}'t} \mathbf{Q} e^{\mathbf{A}t} dt \tag{B.2.2}\]
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