Observability of a system with affine polytopic parametric uncertainty

Question

I have the following system with affine parametric uncertainty: $$x(k+1)=A_{\theta} \ x(k) + B \ u(k)$$ $$y(k)=C_{\theta} \ x(k) + D \ u(k)$$

where $$A_{\theta} = A_0 + \sum\limits_{i=1}^{n_p} [\theta]_i A_i $$ $$C_{\theta} = C_0 + \sum\limits_{i=1}^{n_p} [\theta]_i C_i $$ and $\theta \in \Theta$: a convex polytope in $\mathbb{R^{n_p}}$; $A_i,C_i$ are constant matrices.

Is there a way to check the observability condition on $(C,A)$ for all values of $\theta$ in $\Theta$?

Any hints would be appreciated, thanks.

Edit_1:

• $\theta$ is considered to be constant, and it exists inside the known set $\Theta$.
• The observer is to estimate the state variable $x$, not $\theta$.
• I know how to check the observability of a simple LTI system, using the rank of the observability matrix, or using Hautus Lemma, but how can we check the rank of a matrix that is a function of $\theta$?
• I was thinking of turning the problem into an optimization problem since $\Theta$ is convex, but I am not sure how can I do that...

Answer 1

There are many ways, some being mathematically easier but more restrictive/conservative. Clearly, it is difficult to ensure that the rank condition holds because of the presence of the parameters.

However, if the pair $(A(\theta),C(\theta))$ is observable or, at least detectable, then there exists a parameter-dependent matrix $L(\theta)$ such that $A(\theta)-L(\theta)C(\theta)$ is Schur stable for all $\theta\in\Theta$. The difficulty here is that $L(\theta)$ is not necessarily affine in $\theta$ (in fact, it does not need to) and it is unclear how it could be computed efficiently.

On top of that, the Schur stability of $A(\theta)-L(\theta)C(\theta)$ for all $\theta\in\Theta$ is only a necessary condition for the stability of the associated parameter-varying system $\dot{x}=(A(\theta)-L(\theta)C(\theta))x$. But if the parameters vary sufficiently slowly, then stability of the system also holds. This may be difficult to quantify.

One way is to overcome that is to consider a constant observer gain and try to show that $$(A(\theta)-LC(\theta))^TP(A(\theta)-LC(\theta))-P$$ is negative definite for some positive definite $P$ and for all $\theta\in\Theta$. This can be exactly reformulated as the following Linear Matrix Inequality (LMI) problem

$$\begin{bmatrix} -P & (PA(\theta)-UC(\theta))^T\\ PA(\theta)-UC(\theta) & -P \end{bmatrix}\prec0,$$

where the inequality sign means that the expression is negative definite and the two decision variables are $P\succ0$ and $U$. Again that must hold for all $\theta\in\Theta$.

Assume that $\Theta$ is the unit-simplex (any convex polytope can be transformed to the unit-simplex), then by exploiting the linearity of the conditions in $\theta$, this expression can be rewritten as

$$\begin{bmatrix} -P & (PA_0-UC_0+PA_i-UC_i)^T\\ PA_0-UC_0+PA_i-UC_i & -P \end{bmatrix}\prec0,$$

which must now holds for some $P\succ0$, $U$, and all $i=1,\ldots,n_p$. When this holds, the observer gain is given by $L=P^{-1}U$. Both conditions imply that $A(\theta)-LC(\theta)$ is Schur stable for all $\theta\in\Theta$.

This has several drawbacks

1. We use a constant observer gain and so, this restricts the structure of the observer. If $\theta$ is not known there is nothing we can really do.
2. We use a constant matrix $P$, which results in a "quadratic stability" result. This could be refined using a parameter-dependent matrix $P(\theta)$ which as the same structure as the system. In that case, a possible condition (assuming constant parameters) is

$$\begin{bmatrix} -P(\theta)+X(A(\theta)-LC(\theta))+(A(\theta)-LC(\theta))^TX^T & -X+A(A(\theta)-LC(\theta))^TX^T\\ -X^T+X(A(\theta)-LC(\theta)) & P(\theta)-X-X^T \end{bmatrix}\prec0,$$

where $X$ is a new decision matrix (slack-variable). It can be shown using the elimination lemma, Finsler's lemma or projection lemma, that the above inequality implies that

$$ (A(\theta)-LC(\theta))^TP(\theta)(A(\theta)-LC(\theta))-P(\theta)\prec0, $$ which implies that $A(\theta)-LC(\theta)$ is Schur stable for all $\theta\in\Theta$.

Assuming now that $P$ is affine in $\theta$ and that $\Theta$ is the unit simplex, we get the equivalent condition

$$\begin{bmatrix} -P_0-P_i+(X\bar A_i-U\bar C_i)+(X\bar A_i-U\bar C_i)^T & -X+(X\bar A_i-U\bar C_i)^T\\ -X^T+X\bar A_i-U\bar C_i & (P_0+P_i)-X-X^T \end{bmatrix}\prec0,$$

for some $P_i\succ0,X,U$ and for all $i=1,\ldots,n_p$, where $\bar A_i:=A_0+A_i$, $\bar C_i:=C_0+C_i$. When this holds, the observer gain is given by $L=P^{-1}U$.

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I am posting already this and will improve it later. I am hungry.