鲁鹏
北京理工大学宇航学院
2019.10.11
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 t 0 t_{0} t0, a i j ( t ) a_{ij}(t) aij(t)分段连续
微分方程(1)解为
x ( t ) = [ I + ∫ t 0 t A ( τ ) d τ + ∫ t 0 t A ( τ 1 ) ∫ t 0 t A ( τ 2 ) d τ 2 d τ 1 + ∫ t 0 t A ( τ 1 ) ∫ t 0 t A ( τ 2 ) ∫ t 0 t A ( τ 3 ) d τ 3 d τ 2 d τ 1 + ⋯ ] x ( t 0 ) \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 ) [ ∫ t 0 t A ( τ ) d τ ] = [ ∫ t 0 t A ( τ ) d τ ] A ( t ) ⟺ x ( t ) = exp [ ∫ t 0 t A ( τ ) d τ ] x ( t 0 ) 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)
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 t 0 t_{0} t0, a i j ( t ) a_{ij}(t) aij(t), b i j ( 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 × n n\times n n×n阶解矩阵 Φ ( t , t 0 ) \Phi(t,t_{0}) Φ(t,t0)
Φ ˙ ( t , t 0 ) = A ( t ) Φ ( t , t 0 ) , Φ ( t 0 , t 0 ) = I n (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)
设连续线性时变系统(2)的解为[3]
x ( t ) = Φ ( t , t 0 ) 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 , t 0 ) c ( t ) + Φ ( t , t 0 ) c ˙ ( t ) = A ( t ) Φ ( t , t 0 ) c ( t ) + Φ ( t , t 0 ) 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 , t 0 ) c ( t ) + Φ ( t , t 0 ) c ˙ ( t ) = A ( t ) Φ ( t , t 0 ) 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 , t 0 ) 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 ( t 0 ) = ∫ t 0 t Φ − 1 ( τ , t 0 ) B ( τ ) u ( τ ) d τ (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 = t 0 t = t_{0} t=t0时有 c ( t 0 ) = x ( t 0 ) \boldsymbol{c}(t_{0}) = \boldsymbol{x}(t_{0}) c(t0)=x(t0),所以
c ( t ) = x ( t 0 ) + ∫ t 0 t Φ − 1 ( τ , t 0 ) B ( τ ) u ( τ ) d τ (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 , t 0 ) c ( t ) = ⋯ = Φ ( t , t 0 ) x ( t 0 ) + ∫ t 0 t Φ ( t , τ ) B ( τ ) u ( τ ) d τ (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。
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