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

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

The System

Given a state-space representation of a linear system

$\begin{array}{c}\dot{x}(t)=Ax(t)+B_{1}u(t)+B_{2}w(t),x(0)=x_0\\ y(t)=C_{1}x(t)+D_{1}u(t)+D_{2}w(t)\\ z(t)=C_{2}x(t)\\ \end{array}$

• $x\in {ℝ}^n,y\in {ℝ}^l,zin{ℝ}^m$ are the state vector, measured output vector and output vectors of interest.
• $w\in {ℝ}^p,u\in {ℝ}^r,$ 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 . Template:NumBlk

The estimate of interested output is Template:NumBlk

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

LMI Condition

The $H_{\mathrm{\infty }}$ 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 Template:NumBlk

Implementation

This implementation requires Yalmip and Mosek.

• https://github.com/ShenoyVaradaraya/LMI--master/blob/main/hinf_obs.m

Conclusion

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

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013