강의노트 상태 변수 방정식과 시스템 응답

상태 변수 방정식과 시스템 응답

$u(t)=0$인 경우

(1)

$$\dot x(t) = Ax(t)$$

$$\begin{aligned} x(t) = e^{At} x(0) \end{aligned}$$

이면

$$e^{At} = I + At + \dfrac{A^2 t^2}{2} + \ldots + \dfrac{A^n t^n}{n!} + \ldots$$

$$\begin{aligned} \dfrac{de^{At}}{dt} &= A + A^2t + \ldots +\dfrac{ A^n t^{n-1}}{(n - 1)!} +  \ldots \\&= A (I + At + \dfrac{A^2t^2}{2} +\ldots ) = Ae^{At} \end{aligned}$$

$$\dot x(t) = Ae^{At} x(0) = Ax(t)$$

$$sX(s) - x(0) = AX(s)$$

$$X(s) = (sI - A)^{-1} x(0)$$

$x(t) = \mathcal{L}^{-1} [X(s)] =  \mathcal{L}^{-1}  [(sI - A)^{-1}]x(0)$ (1)과 비교하면

$e^{At} = \mathcal{L}^{-1}  [(sI - A)^{-1}] = \Phi(t)$→ 상태 천이 행렬

$$\mathcal{L}[e^{At}] = (sI - A)^{-1}$$

$$x(t) = e^{At} x(0)=\Phi(t)x(0)$$

상태 천이 행렬 성질

1. $\Phi(0) = e^{A0}=e^0=I$
2. $\Phi^{-1}(t)=(e^{At})^{-1}=e^{-At}= e^{A(-t)}=\Phi(-t)$
3. $\begin{aligned}\Phi(t_2-t_0)&=e^{A(t_2-t_0)}=e^{At_2}e^{-At_0}(e^{At_1}e^{-At_1})\\ &=(e^{At_2}e^{-At_1})(e^{At_1}e^{-At_0})=e^{A(t_2-t_1)}e^{A(t_1-t_0)} \\&=\Phi(t_2-t_1)\Phi(t_1-t_0) \end{aligned}$
4. $(\Phi(t))^k = \Phi(kt)$

$u(t) \neq 0$인 경우

$$\dot x(t) - Ax(t) = Bu(t)$$

$$e^{-At} [\dot x(t) - Ax(t)]=e^{-At}Bu(t)$$

$$\dfrac{d}{dt}[e^{-At}x(t)] = e^{-At}\dot x(t) - A e^{-At} x(t) = e^{-At}[\dot x(t) - Ax(t)]$$

$$\dfrac{d}{dt}[e^{-At}x(t)]= e^{-At}Bu(t)$$

$$e^{-At}x(t)-x(0)=\int_0 ^t  e^{-At}Bu(\tau) d\tau$$

$$\begin{aligned} x(t) &= e^{At}x(0) + \int_0 ^t  e^{A(t-\tau)}Bu(\tau) d\tau \\ &= \Phi(t)x(0)+\int_0 ^t  \Phi(t-\tau)Bu(\tau) d\tau \end{aligned}$$

• $t= t_0$ 인 경우

$$\begin{aligned} x(t_0 ) &= e^{At_0} x(0) + \int_0^{t_0}  e^{A (t_0 - \tau )} B u( \tau ) d \tau \\&= \Phi (t_0) x(0) + \int_0^{t_0}  \Phi (t_0-\tau ) B u( \tau ) d \tau \end{aligned}$$

$$\begin{aligned} x(t ) &= e^{At}e^{-At_0}e^{At_0} x(0) + \int_0^{t}  e^{A(t-\tau)}e^{A (t_0 - t_0)} B u( \tau ) d \tau \\ &= e^{A(t-t_0)}e^{At_0} x(0) + \int_0^{t_0}  e^{A(t-t_0)}e^{A (t_0 - \tau)} B u( \tau ) d \tau + \int_{t_0}^{t}  e^{A(t-\tau)} B u( \tau ) d \tau\\ &= e^{A(t-t_0)} \left( e^{At_0} x(0) + \int_0^{t_0} e^{A (t_0 - \tau)} B u( \tau ) d \tau \right)+ \int_{t_0}^{t}  e^{A(t-\tau)} B u( \tau ) d \tau\\ &= e^{A(t-t_0)} x(t_0) + \int_{t_0}^{t}  e^{A(t-\tau)} B u( \tau ) d \tau\\ &= \Phi (t-t_0) x(t_0) + \int_{t_0}^{t}  \Phi (t-\tau ) B u( \tau ) d \tau \\ &= x(t-t_0)\end{aligned}$$

$$\begin{aligned} y(t) & = Cx(t) + Du(t) \\ &= C e^{A(t-t_0)} x(t_0) + C\int_{t_0}^{t}  e^{A(t-\tau)} B u( \tau ) d \tau + Du(t) \\ &= C \Phi(t-t_0) x(t_0) + C\int_{t_0}^{t} \Phi(t-\tau) B u( \tau ) d \tau + Du(t) \end{aligned}$$