LMIs in Control/pages/Mixed H2 HInf Optimal Observer

The goal of mixed $$ H_2-H\infin $$-optimal state estimation is to design an observer that minimizes the $$ H_2 $$ norm of the closed-loop transfer matrix from $$ w_1 $$ to $$ z_1 $$, while ensuring that the $$ H\infin $$ norm of the closed-loop transfer matrix from $$ w_2 $$ to $$ z_2 $$ is below a specified bound.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization



\begin{align} \dot x&=Ax+B_{1,1}w_1 +B_{1,2}w_2,\\ y&=C_2x+D_{21,1}w_1+D_{21,1}w_2\\ \end{align}$$

where it is assumed that $$ (A, C_2) $$ is detectable.

The Data
The matrices needed as input are $$ A,B_1,B_2,C_2,D_{21},D_{11} $$.

The Optimization Problem
The observer gain L is to be designed to minimize the $$ H_2 $$ norm of the closed-loop transfer matrix $$ T_{11}(s) $$ from the exogenous input $$ w_1 $$ to the performance output $$ z_1 $$ while ensuring the $$ H\infin $$ norm of the closed-loop transfer matrix $$ T_{22}(s) $$ from the exogenous input $$ w_2 $$ to the performance output $$ z_2 $$ is less than $$ \gamma_d $$, where



\begin{align} T_{11}(s) = C_{1,1}(s1-(A-LC_2))^{-1}(B_{1,1}-LD_{21,1})\\ T_{22}(s) = C_{1,2}(s1-(A-LC_2))^{-1}(B_{1,2}-LD_{21,2})+D_{11,22} \end{align} $$

is minimized. The form of the observer would be:



\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

is to be designed, where $$ L \in \mathbb{R}^{n_x \times n_y} $$ is the observer gain.

The LMI: $$ H_\infin $$ Optimal Observer
The mixed $$H_2 -H\infin $$-optimal observer gain is synthesized by solving for $$ P \in \mathbb{S} ^{n_x}, G \in \mathbb{R} ^{n_x \times n_y} $$, and $$ \nu \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\nu) = \nu $$ subject to $$ P>0, Z>0 $$,



\begin{align} \begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21}\\ \star && -1 \end{bmatrix} <0\\

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21} && C_1\\ \star && -\gamma1 && {D_{11}}^T\\ \star && \star && -\gamma1 \end{bmatrix} <0\\ \begin{bmatrix} P && C{_{1,1}}^T \\ \star && Z\end{bmatrix} >0\\ trZ < \nu \end{align}$$

Conclusion:
The mixed $$ H_2- H_\infin$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$, the $$ H_2 $$ norm of $$ T_{11}(s) $$ is less than $$ \mu = \sqrt {\nu} $$ and the $$ H_\infin $$ norm of T(s) is less than $$ \gamma_d$$.

Implementation
Link to the MATLAB code designing $$ H_2- H\infin $$- Optimal Observer

Code $ H_2-H\infin $ Optimal Observer

Related LMIs

 * $ H_2 $ Optimal observer
 * $ H\infin $ Optimal observer