[제어공학개론] Lec 11 - Observability

📢Precaution

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

Definition of Observability

• Definition:

Model, $u[0,T]$(a.k.a. trajectory of input signal), $y[0,T]$(also history of output)만을 이용하여 $x(0)$를 알아낼 수 있다.

Examples of unobservable cases :

\[\begin{aligned}\dot x_1 &= -x_1 \\\dot x_2 &= x_1+x_2 \\y &= x_1 \end{aligned}\]

$x_2$ is not observable state

Another example :

\[\begin{aligned} \dot x_1 &= x_1 \\ \dot x_2 &= x_2 \\ y &= x_1+x_2 \end{aligned}\]

Applying Solution :

\[y = e^t x_1(0) + e^T x_2(0)\]

Unable to distinguish $x_1, x_2$

Observability matrix

Derivation (naive)

let $B=0$ , $D=0$

\[\begin{aligned} \dot x &= Ax\\ y &= Cx \end{aligned}\]

differenating output $y$ (one time to $(n-1)$ times)

\[\begin{aligned} \dot y &= CAx \\ \ddot y &= CA^2x \\ y^{(n-1)} &= CA^{n-1}x \end{aligned}\]

LHS is measurable (output을 미분하는 것은 그다지 권장되지 않으나 수학적으로는 계산가능)

\[\begin{bmatrix}y \\ \dot y \\ \ddot y \\ \vdots \\ y^{(n-1)}\end{bmatrix} = \begin{bmatrix}C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1}\end{bmatrix} x\]

우변의 $x$와 곱해진 matrix가 invertible하다면, $x$를 알 수 있음.

• Observability matrix $O$

\[\text{system or A, C is observable if }\begin{bmatrix}C \\ CA \\ CA^2\\ \vdots \\ CA^{n-1}\end{bmatrix} \text{ has full column rank}\]

• Controllability matrix와 duality를 확인할 수 있음.

Derivation (exact) :

\[\begin{aligned} \dot x &= Ax+Bu \\ y &=Cx+Du \\ y-Du\text{(known)} &= Cx \\ \dot y - D\dot u &= C\dot x = C(Ax+Bu) \\ \dot y - D\dot u - CBu &= CAx \end{aligned}\]

with same method, LHS is always measurable while RHS has form $=C^{k}x, k=0,\cdots,n-1$

if observability matrix $O$ has full column rank, $x$ is measurable w/ using pseudo-inverse of $O$

Observability Gramian

Consider the solution of $y$

\[y(t) = Ce^{At}x(0) + C\int_0^T e^{A(t-\tau)}Bu(\tau) d\tau + Du(t)\]

while $y, C, D, u$ are known,

\[\begin{aligned}\bar y(t) Ce^{At}x(0) \text{ if } \bar y(t) &= y(t)-C\int_0^T e^{A(t-\tau)}Bu(\tau) d\tau - Du(t)\\ \int_0^T e^{A^Tt}C^T\bar y(t)dt &= \int_0^T e^{A^Tt}C^TCe^{At}dt\cdot x(0) \\ \text{Observability Gramian : }& \int_0^T e^{A^Tt}C^TCe^{At}dt \\ \text{if } G(T)>0,& \text{system is observable} \end{aligned}\]

PBH test of observability

Definition of PBH test in observability

\[\begin{aligned}rank \begin{bmatrix}\lambda I - A \\ C\end{bmatrix} = n, &\forall \lambda \in \text{e.v. of } A \\ \nexists v\in \Re^n-\{ {\bf 0}\} \text{ s.t. } &\lambda v=Av, Cv = {\bf 0}\end{aligned}\]

if $x(0) = v$, $\dot x = Ax$, $y=Cx \equiv = 0$

즉 sensing은 안되는데 $v$방향으로 내부 값은 계속 변하는 unobservable state가 생김

Controllability에서의 증명의 dual이 Observability에서 성립하므로, Controllability쪽만 다룸.

동일한 방법으로 Observable state, nonobservable state를 구분할 수 있음.