线性时变系统状态方程的解

鲁鹏
北京理工大学宇航学院
2019.10.11

随着学习的深入，线性定常系统（状态方程一般形式为$\dot{\boldsymbol{x}}(t)=Ax(t)+Bu(t)$）已经无法满足我的需求了。线性时变系统（状态方程一般形式为$\dot{\boldsymbol{x}}(t)=A(t)x(t)+B(t)u(t)$）更具一般性。非线性时变系统也可以通过泰勒展开简化为线性时变系统[1]。本文将总结线性时变系统的状态方程的解，主要参考了东北大学课件[2]。

线性时变系统齐次状态方程的解

系统描述：
$\dot{\boldsymbol{x}}(t) = A(t)\boldsymbol{x}(t) \tag{1}$

初始时刻为$t_{0}$，$a_{ij}(t)$分段连续

微分方程（1）解为
$\begin{gathered} \boldsymbol{x}(t) = \bigg[ I + \int^{t}_{t_{0}}A(\tau)d\tau + \int^{t}_{t_{0}}A(\tau_{1})\int^{t}_{t_{0}}A(\tau_{2})d\tau_{2}d\tau_{1} + \\\int^{t}_{t_{0}} A(\tau_{1}) \int^{t}_{t_{0}} A(\tau_{2}) \int^{t}_{t_{0}} A(\tau_{3}) d\tau_{3} d\tau_{2} d \tau_{1} + \cdots \bigg]\boldsymbol{x}(t_{0}) \end{gathered}$

特殊情况：
$A(t)\left[\int^{t}_{t_{0}}A(\tau)d\tau \right] = \left[\int^{t}_{t_{0}}A(\tau)d\tau \right]A(t) \\ \iff \boldsymbol{x}(t) = \exp\left[\int^{t}_{t_{0}}A(\tau)d\tau \right]\boldsymbol{x}(t_{0})$

线性时变系统的状态转移矩阵

对于连续线性时变系统
$\dot{\boldsymbol{x}}(t) = A(t)\boldsymbol{x}(t) + B(t)\boldsymbol{u}(t)\tag{2}$

初始时刻为$t_{0}$，$a_{ij}(t)$，$b_{ij}(t)$分段连续。

系统的状态转移矩阵是如下矩阵微分方程和初始条件的$n\times n$阶解矩阵$\Phi(t,t_{0})$
$\dot{\Phi}(t,t_{0}) = A(t)\Phi(t,t_{0}), \quad \Phi(t_{0},t_{0}) = I_{n} \tag{3}$

线性时变系统非齐次状态方程的解

设连续线性时变系统(2)的解为[3]
$\boldsymbol{x}(t)=\Phi(t,t_{0})\boldsymbol{c}(t) \tag{4}$

其中$\boldsymbol{c(}t)$为待定量，类比高等数学中的常数易变法（A method of constant variation）理解。
$\begin{aligned} \dot{\boldsymbol{x}}(t) =& \dot{\Phi}(t,t_{0})\boldsymbol{c}(t) + \Phi(t,t_{0})\dot{\boldsymbol{c}}(t) \\ =& A(t)\Phi(t,t_{0})\boldsymbol{c}(t) + \Phi(t,t_{0})\dot{\boldsymbol{c}}(t) \end{aligned} \tag{5}$

将等式（2）带入等式（5）可得
$A(t)\Phi(t,t_{0})\boldsymbol{c}(t) + \Phi(t,t_{0})\dot{\boldsymbol{c}}(t) = A(t)\Phi(t,t_{0})\boldsymbol{c}(t) + B(t)\boldsymbol{u}(t) \tag{6}$

$\dot{\boldsymbol{c}}(t) = \Phi^{-1}(t,t_{0})B(t)\boldsymbol{u}(t) \tag{7}$

等式（7）两端积分得：
$\boldsymbol{c}(t) - \boldsymbol{c}(t_{0}) = \int^{t}_{t_{0}}\Phi^{-1}(\tau,t_{0})B(\tau)\boldsymbol{u}(\tau)d\tau \tag{8}$

由等式（4）可知$t = t_{0}$时有$\boldsymbol{c}(t_{0}) = \boldsymbol{x}(t_{0})$，所以
$\boldsymbol{c}(t) = \boldsymbol{x}(t_{0}) + \int^{t}_{t_{0}}\Phi^{-1}(\tau,t_{0})B(\tau)\boldsymbol{u}(\tau)d\tau \tag{9}$

$\begin{aligned} \boldsymbol{x}(t) =& \Phi(t,t_{0})\boldsymbol{c}(t) \\ =& \cdots\\ =& \Phi(t,t_{0})\boldsymbol{x}(t_{0}) + \int^{t}_{t_{0}}\Phi(t,\tau)B(\tau)\boldsymbol{u}(\tau)d\tau \end{aligned} \tag{10}$

类比上述推导过程，非线性时变系统$\dot{\boldsymbol{x}}(t) = A(t)\boldsymbol{x}(t) + B(t)\boldsymbol{u}(t) + \boldsymbol{z}(t)$的解很好得到。

参考文献

[1] Szmuk M , Acikmese B . 2018 AIAA Guidance, Navigation, and Control Conference[C]. Successive Convexification for 6-DoF Mars Rocket Powered Landing with Free-Final-Time.

[2] 第三章 状态方程的解ppt（东北大学）

[3] 麻省理工学院公开课：微分方程