[제어공학개론] Lec 07 - Simularity transform

📢Precaution

본 게시글은 서울대학교 심형보 교수님의 23-2 제어공학개론 수업 내용을 바탕으로 작성되었습니다.

State Transformation

state-space representation is not unique (state 좌표 변환의 관계)

let’s define a transform matrix :

\[x(t) = Tx(t)\]

$T$ invertible, invarient to time. (유사변환, simularity transformation)

define State space representation with $z$

\[\begin{aligned} \dot z & = T\dot x = TAx + TBu = TAT^{-1}z+TBu \\ y &=Cx+Du = CT^{-1}z + Du \\ \begin{bmatrix}\dot z \\ y\end{bmatrix} & =\begin{bmatrix}TAT^{-1} & TB \\ CT^{-1} & D\end{bmatrix}\begin{bmatrix}z \\ u\end{bmatrix} \end{aligned}\]

Transfer function (matrix form) are equivalent

\[\begin{aligned} CT^{-1} (sI-TAT^{-1})^{-1}TB+D &= C(sI-A)^{-1}B+D \\ T^{-1}(sI-TAT^{-1})T &= T^{-1}(sTT^{-1}-TAT^{-1})T \\ &= T^{-1}(T(sI-A)T^{-1})^{-1} T \\ &= (sI-A)^{-1} \end{aligned}\]

solution to $\dot x = Ax$

solution to matrix (state-space) with no input : $\dot x = Ax$

\[\begin{aligned} x (t ) &\in R^n \\ x(0) &= {\bf x_0} \text{(given)} \end{aligned}\]

Answer :

\[\begin{aligned} x(t) &= e^{At}x_0 \\ e^{At} &= (I+At+\frac{(At)^2}{2!}+\cdots) \end{aligned}\]

solution이라는 사실 proof : 대입

\[\begin{aligned} \dot x &= (0+A+A^2t+\frac{A^3t^2}{2!}+\cdots) x_0 \\ &= A(I+At+\frac{(At)^2}{2!}+\cdots)x_0 = Ax(t) \\ &t=0 \rightarrow X(0) = e^{A0}x_0 = x_0 \end{aligned}\]

solution to $\dot x = Ax+Bu$

solution to state-space with input

\[\dot x = Ax+Bu\]

Multiplying integrating factor (variation of constant formula) :

\[\begin{aligned} (e^{-At})\dot x -Ax &= e^{-At}Bu \\ \int_0^t\frac{d}{dt}(e^{-At}x(t))dt &= e^{-At}x(t)-e^{-A0}x(0)\\ & = \int_0^t e^{-A\tau} Bu(\tau)d\tau \\ x(t)-e^{At}x(0) &= \int_0^t e^{A(t-\tau)}B u(\tau)d\tau \end{aligned}\]

• Response to the initial condition : $e^{At}x_0$
• State transition matrix : $e^{A(t-\tau)}$

Matrixwise convolution. 외우는 방법은 summing all jumps from previous time $\tau \text{ to } t$

Characteristics of Matrix exponent

\[\begin{aligned} \left. e^{At} \right\vert_{t=0}&= I \\ e^{At_1} \times e^{At_2} &= e^{A(t_1+t_2)} \end{aligned}\]

$t_1$초만큼 solution이 이동한 것 + $t_2$초만큼 이동한 solution : $t_1+t_2$초만큼 이동한 sol.

이로부터,

\[\begin{aligned} e^{At} \times e^{-At} &= I \\ e^{At} \times e^{Bt} &\neq e^{(A+B)t} \end{aligned}\]

comparing solutions at Transfer function

ss2tf를 이용한 laplace transform of $X$와 $x(t)$의 비교

\[X(s) = (sI-A)^{-1} x(0) + (sI-A)^{-1} BU(s)\]

matrixwise convolution이 s-domain에서의 곱으로 표현된 것을 확인할 수 있음.

\[\int_0^t e^{A(t-\tau)} B u(\tau) d\tau \leftrightarrow (sI-A)^{-1} BU(s)\]

Laplace transform of state transition matrix : Taylor series of inverse of $sI-A$

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

calculation about $y(t)$

\[y(t) = Ce^{At}x_0 + C\int_0^t e^{A(t-\tau)}Bu(\tau) d\tau + Du(t)\]