LMIs in Control/Observer Synthesis/Continuous Time/Full-Order H-infinity State Observer

In this section, we design full order H-$$ \infty $$ state observer.

The System
Given a state-space representation of a linear system

\begin{align} \ \dot x(t) = Ax(t) + B_{1}u(t) + B_{2}w(t), x(0) = x_{0} \\ \ y(t) = C_{1}x(t) + D_{1}u(t) + D_{2}w(t) \\ \ z(t) = C_{2}x(t)\\ \end{align}$$


 * $$ x \in \mathbb{R}^{n}, y \in \mathbb{R}^{l}, z in \mathbb{R}^{m} $$ are the state vector, measured output vector and output vectors of interest.
 * $$ w \in \mathbb{R}^{p}, u \in \mathbb{R}^{r}, $$ are the disturbance vector and control vector respectively.

The Data
$$ A,B_{1},B_{2},C_{1},C_{2},D_{1},D_{2}$$ are system matrices

Definition
For the system, a full order state observer of the form of equation (1) is introduced and the estimate of interested output is given by.

The estimate of interested output is

Given the system and a positive scalar $$ \gamma $$, L is found such that

LMI Condition
The $$ H_{\infty}$$ state observers problem has a solution if and only if there exists a symmetric positive definite matrix $$ P $$ and a matrix $$ W $$ satisfying the below LMI

When such a pair of matrics is found, the solution is

Implementation
This implementation requires Yalmip and Mosek.
 * https://github.com/ShenoyVaradaraya/LMI--master/blob/main/hinf_obs.m

Conclusion
Thus, an $$H_{\infty} $$ state observer is designed such that the output vectors of interest are accurately estimated.