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

文章目录

线性时变系统状态方程的解
- 线性时变系统齐次状态方程的解
- 线性时变系统的状态转移矩阵
- 线性时变系统非齐次状态方程的解
- 参考文献

随着学习的深入，线性定常系统（状态方程一般形式为 x˙(t)=Ax(t)+Bu(t)\dot{\boldsymbol{x}}(t)=Ax(t)+Bu(t)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 x˙(t)=A(t)x(t)+B(t)u(t)\dot{\boldsymbol{x}}(t)=A(t)x(t)+B(t)u(t)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：
x˙(t)=A(t)x(t)(1)\dot{\boldsymbol{x}}(t) = A(t)\boldsymbol{x}(t) \tag{1} x˙(t)=A(t)x(t)(1)

初始时刻为t0t_{0}t0，aij(t)a_{ij}(t)aij(t)分段连续

微分方程（1）解为
x(t)=[I+∫t0tA(τ)dτ+∫t0tA(τ1)∫t0tA(τ2)dτ2dτ1+∫t0tA(τ1)∫t0tA(τ2)∫t0tA(τ3)dτ3dτ2dτ1+⋯]x(t0)\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} x(t)=[I+∫t0tA(τ)dτ+∫t0tA(τ1)∫t0tA(τ2)dτ2dτ1+∫t0tA(τ1)∫t0tA(τ2)∫t0tA(τ3)dτ3dτ2dτ1+⋯]x(t0)

特殊情况：
A(t)[∫t0tA(τ)dτ]=[∫t0tA(τ)dτ]A(t)⟺x(t)=exp⁡[∫t0tA(τ)dτ]x(t0)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}) A(t)[∫t0tA(τ)dτ]=[∫t0tA(τ)dτ]A(t)⟺x(t)=exp[∫t0tA(τ)dτ]x(t0)

线性时变系统的状态转移矩阵

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

初始时刻为t0t_{0}t0，aij(t)a_{ij}(t)aij(t)，bij(t)b_{ij}(t)bij(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×nn\times nn×n阶解矩阵Φ(t,t0)\Phi(t,t_{0})Φ(t,t0)，如果如下初值问题（Initial Value Problem， IVP）无法解析求解可以使用数值积分求解
Φ˙(t,t0)=A(t)Φ(t,t0),Φ(t0,t0)=In(3)\dot{\Phi}(t,t_{0}) = A(t)\Phi(t,t_{0}), \quad \Phi(t_{0},t_{0}) = I_{n} \tag{3} Φ˙(t,t0)=A(t)Φ(t,t0),Φ(t0,t0)=In(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]
x(t)=Φ(t,t0)c(t)(4)\boldsymbol{x}(t)=\Phi(t,t_{0})\boldsymbol{c}(t) \tag{4} x(t)=Φ(t,t0)c(t)(4)

其中c(t)\boldsymbol{c(}t)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。
x˙(t)=Φ˙(t,t0)c(t)+Φ(t,t0)c˙(t)=A(t)Φ(t,t0)c(t)+Φ(t,t0)c˙(t)(5)\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} x˙(t)==Φ˙(t,t0)c(t)+Φ(t,t0)c˙(t)A(t)Φ(t,t0)c(t)+Φ(t,t0)c˙(t)(5)

将等式（2）带入等式（5）可得
A(t)Φ(t,t0)c(t)+Φ(t,t0)c˙(t)=A(t)Φ(t,t0)c(t)+B(t)u(t)(6)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} A(t)Φ(t,t0)c(t)+Φ(t,t0)c˙(t)=A(t)Φ(t,t0)c(t)+B(t)u(t)(6)

c˙(t)=Φ−1(t,t0)B(t)u(t)(7)\dot{\boldsymbol{c}}(t) = \Phi^{-1}(t,t_{0})B(t)\boldsymbol{u}(t) \tag{7} c˙(t)=Φ−1(t,t0)B(t)u(t)(7)

等式（7）两端积分得：
c(t)−c(t0)=∫t0tΦ−1(τ,t0)B(τ)u(τ)dτ(8)\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} c(t)−c(t0)=∫t0tΦ−1(τ,t0)B(τ)u(τ)dτ(8)

由等式（4）可知t=t0t = t_{0}t=t0时有c(t0)=x(t0)\boldsymbol{c}(t_{0}) = \boldsymbol{x}(t_{0})c(t0)=x(t0)，所以
c(t)=x(t0)+∫t0tΦ−1(τ,t0)B(τ)u(τ)dτ(9)\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} c(t)=x(t0)+∫t0tΦ−1(τ,t0)B(τ)u(τ)dτ(9)

x(t)=Φ(t,t0)c(t)=⋯=Φ(t,t0)x(t0)+∫t0tΦ(t,τ)B(τ)u(τ)dτ(10)\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} x(t)===Φ(t,t0)c(t)⋯Φ(t,t0)x(t0)+∫t0tΦ(t,τ)B(τ)u(τ)dτ(10)

类比上述推导过程，非线性时变系统x˙(t)=A(t)x(t)+B(t)u(t)+z(t)\dot{\boldsymbol{x}}(t) = A(t)\boldsymbol{x}(t) + B(t)\boldsymbol{u}(t) + \boldsymbol{z}(t)x˙(t)=A(t)x(t)+B(t)u(t)+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