LMIs in Control/pages/hinfinityoptimalobserver

${\displaystyle H\infty }$-Optimal observers yield robust estimates of some or all internal plant states by processing measurement data. Robust observers are increasingly demanded in industry as they may provide state and parameter estimates for monitoring and diagnosis purposes even in the presence of large disturbances such as noise etc. It is there where Kalman filters may tend to fail. State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system. The goal of ${\displaystyle H_{\infty }}$ -optimal state estimation is to design an observer that minimizes the ${\displaystyle H_{\infty }}$ norm of the closed-loop transfer matrix from w to z.

The System

Consider the continuous-time generalized plant ${\displaystyle P}$ with state-space realization

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1}w,\\y&=C_{2}x+D_{21}w\\\end{aligned}}}

The Data

The matrices needed as input are ${\displaystyle A,B_{1},B_{2},C_{2},D_{21},D_{11}}$.

The Optimization Problem

The observer gain ${\displaystyle L}$ is to be designed such that the ${\displaystyle H\infty }$ of the transfer matrix from w to z, given by

{\displaystyle {\begin{aligned}T(s)=C_{1}(s1-(A-LC_{2}))^{-1}(B_{1}-LD_{21})+D_{11}\\\end{aligned}}}

is minimized. The form of the observer would be:

{\displaystyle {\begin{aligned}{\dot {\hat {x}}}=A{\hat {x}}+L(y-{\hat {y}}),\\{\hat {y}}=C_{2}{\hat {x}}\\\end{aligned}}}

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

The ${\displaystyle H\infty }$-optimal observer gain is synthesized by solving for ${\displaystyle P\in \mathbb {S} ^{n_{x}},G\in \mathbb {R} ^{n_{x}\times n_{y}}}$, and ${\displaystyle \gamma \in \mathbb {R} _{>0}}$ that minimize ${\displaystyle \zeta (\gamma )=\gamma }$ subject to ${\displaystyle P>0}$ and

{\displaystyle {\begin{aligned}{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}&&C_{1}\\\star &&-\gamma 1&&{D_{11}}^{T}\\\star &&\star &&-\gamma 1\end{bmatrix}}<0\\\end{aligned}}}

Conclusion:

The ${\displaystyle H_{\infty }}$ -optimal observer gain is recovered by ${\displaystyle L=P^{-1}G}$ and the ${\displaystyle H_{\infty }}$ norm of T(s) is ${\displaystyle \gamma }$.

Implementation

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