LMIs in Control/pages/Mixed H2 HInf Optimal Observer

The goal of mixed $H_2-H\mathrm{\infty }$-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 $H\mathrm{\infty }$ 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

$\begin{array}{cc}\dot{x}& =Ax+B_{1,1}{w}_1+B_{1,2}{w}_2,\\ y& =C_{2}x+D_{21,1}{w}_1+D_{21,1}{w}_2\\ \end{array}$

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 $H\mathrm{\infty }$ 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

$\begin{array}{c}{T}_{11}(s)=C_{1,1}(s1-(A-L{C}_2){)}^{-1}(B_{1,1}-L{D}_{21,1})\\ T_{22}(s)=C_{1,2}(s1-(A-L{C}_2){)}^{-1}(B_{1,2}-L{D}_{21,2})+D_{11,22}\end{array}$

is minimized. The form of the observer would be:

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

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

The LMI: $H_{\mathrm{\infty }}$ Optimal Observer

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

$\begin{array}{c}\left[\begin{array}{ccc}PA+A^{T}P-G{C}_2-{C_2}^T{G}^T& & P{B}_1-G{D}_{21}\\ \star & & -1\end{array}\right]<0\\ \left[\begin{array}{ccccc}PA+A^{T}P-G{C}_2-{C_2}^T{G}^T& & P{B}_1-G{D}_{21}& & C_1\\ \star & & -\gamma 1& & {D_{11}}^T\\ \star & & \star & & -\gamma 1\end{array}\right]<0\\ \left[\begin{array}{ccc}P& & C{{}_{1,1}}^T\\ \star & & Z\end{array}\right]>0\\ trZ<\nu \end{array}$

Conclusion:

The mixed $H_2-H_{\mathrm{\infty }}$ -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 $H_{\mathrm{\infty }}$ norm of T(s) is less than ${\gamma }_d$.

Implementation

Link to the MATLAB code designing $H_2-H\mathrm{\infty }$- Optimal Observer

Code $H_2-H\mathrm{\infty }$ Optimal Observer

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Related LMIs

• $H_2$ Optimal observer
• $H\mathrm{\infty }$ Optimal observer