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

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

The System

${\begin{aligned}x_{k+1}&=A_{d}x_{k}+B_{d1}w_{k},\\y_{k}&=C_{c2}x_{k}+D_{d21}w_{k}\\\end{aligned}}$

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:

${\begin{aligned}{\hat {x}}_{k+1}&=A_{d}{\hat {x}}_{k}+L_{d}(y_{k}-{\hat {y}}_{k}),\\{\hat {y}}_{k}&=C_{d2}{\hat {x}}_{k}\\\end{aligned}}$

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

$L_{d}{\begin{aligned}T(z)&=C_{d1}(z1-(A_{d}-L_{d}C_{d2}))^{-1}(B_{d1}-L_{d}D_{d21}),\\\end{aligned}}$

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 $P>0,Z>0$,

${\begin{aligned}{\begin{bmatrix}P&PA_{d}-G_{d}C_{d2}&PB_{d1}-G_{d}D_{d21}\\*&P&0\\*&*&1\end{bmatrix}}&>0,\\{\begin{bmatrix}Z&PC_{d1}\\*&P\end{bmatrix}}&>0,\\\operatorname {tr} Z<v\end{aligned}}$

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.