LMIs in Control/Click here to continue/Observer synthesis/Full-order Hinf state Observer

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

The System

Given a state-space representation of a linear system

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

• $x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{l},zin\mathbb {R} ^{m}$ 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 .

${\begin{aligned}\ {\dot {\hat {x}}}=(A+LC_{1}){\hat {x}}-Ly+(B_{1}+LD_{1})u\\\end{aligned}}$

(1)

The estimate of interested output is

${\begin{aligned}{\hat {z}}(t)=C_{2}{\hat {x}}(t)\\\end{aligned}}$

(2)

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

${\begin{aligned}\|G_{{\tilde {z}}w}(s)\|_{\infty }=\gamma \\\end{aligned}}$

(3)

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

${\begin{aligned}{\begin{bmatrix}A^{T}P+C_{1}^{T}W^{T}+PA+WC_{1}&PB_{2}+WD_{2}&C_{2}^{T}\\(PB_{2}+WD_{2})^{T}&-\gamma I&0\\C_{2}&0&-\gamma I\end{bmatrix}}\\\end{aligned}}$

(4)

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

${\begin{aligned}L=P^{-1}W\\\end{aligned}}$

(5)

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.

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