LMIs in Control/Discrete-Time Systems/Discrete-Time H2-Optimal Observer

LMIs in Control/Discrete-Time Systems/Discrete-Time H2-Optimal Observer

The System

$\begin{array}{cc}{x}_{k+1}& =A_d{x}_k+B_{d1}{w}_k,\\ y_k& =C_{c2}{x}_k+D_{d21}{w}_k\\ \end{array}$

Where it is assumed that :$(A_d,C_{d2})$ is detectable.

The Data

The matrices $A_d,B_{d1},C_{cd2},C_{cd1},D_{d21}$.

The Optimization Problem

An observer of the form:

$\begin{array}{cc}{\hat{x}}_{k+1}& =A_d{\hat{x}}_k+L_d(y_k-{\hat{y}}_k),\\ {\hat{y}}_k& =C_{d2}{\hat{x}}_k\\ \end{array}$

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

Defining the error state $e_k=x_k-{\hat{x}}_k$, the error dynamics are found to be

$e_{k+1}=(A_d-L_d{C}_{d2}{e}_k+(B_{d1}-L_d{D}_{d21})w_k$,

and the performance output is defined as

$z_k=C_{d1}{e}_k$.

The observer gain $L_d$ is to be designed such that the $H_2$ of the transfer matrix from $w_k$ to $z_k$, given by

$L_d\begin{array}{cc}T(z)& =C_{d1}(z1-(A_d-L_d{C}_{d2}){)}^{-1}(B_{d1}-L_d{D}_{d21}),\\ \end{array}$

is minimized.

The LMI: Discrete-Time H2-Optimal Observer

The discrete-time $H_2$-optimal observer gain is synthesized by solving for $P\in S^{n_x}$, $Z\in S^{n_z}$, $G_d\in R^{n_x\times n_y}$, and $v\in R_{>0}$ that minimize $J(v)=v$ subject to $P>0,Z>0$,

$\begin{array}{cc}\left[\begin{array}{ccc}P& P{A}_d-G_d{C}_{d2}& P{B}_{d1}-G_d{D}_{d21}\\ *& P& 0\\ *& *& 1\end{array}\right]& >0,\\ \left[\begin{array}{cc}Z& P{C}_{d1}\\ *& P\end{array}\right]& >0,\\ \mathrm{tr}Z<v\end{array}$

Conclusion:

The $H_2$-optimal observer gain is recovered by $L_d=P^{-1}{G}_d$ and the $H_2$ norm of $T(z)$ is $\mu =\sqrt{v}$.

Implementation

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/Discrete_Time_H2_Optimal_Observer_LMIs_Wikibook_Example.m

Related LMIs

Discrete-Time Mixed H2-H∞-Optimal Observer//

Discrete-Time H∞-Optimal Observer//

External Links

A list of references documenting and validating the LMI.

• 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.