# LMIs in Control/pages/reduced order state estimation

*WIP, Description in progress*

In this page, we investigate an LMI approach for the design of the problem of reduced-order observer design for the linear system.

## System SettingEdit

where are the state vector, the input vector, and the output vector, respectively. Without loss of generality, it is assumed that rank .

In the design of reduced-order state observers for linear systems, the following lemma performs a fundamental role.

## LemmaEdit

Given the linear system, and let be an arbitrarily chosen matrix which makes the matrix

nonsingular, then

.

Furthermore, let

, ,

then the matrix pair is detectable if and only if is detectable.

Let

, ,

then it follows from the relations in previous 3 equations that system is equivalent to

,

In the equivalent system, the substate vector is directly equal to the output of the original system. Thus to reconstruct the state of the original system, we suffice to get an estimate of the substate vector , namely, , from the earlier equivalent system. Once an estimate is obtained, an estimate of , that is, the state vector of original system, can be obtained as

.

## Problem FormulationEdit

For the equivalent continuous-time linear system, design a reduced-order state observer in the form of

such that for arbitrary control input , and arbitrary initial system values , there holds

.

## Solution/TheoremEdit

Problem has a solution if and only if one of the following two conditions holds:

1. There exist a symmetric positive definite matrix P and a matrix W satisfying

2. There exists a symmetric positive definite matrix P satisfying

In this case, a reduced-order state observer can be obtained as in problem with

where

with W and being a pair of feasible solutions to the first inequality condition or with being a solution to the second inequality condition.

*WIP, additional references to be added*

## External LinksEdit

A list of references documenting and validating the LMI.

- [1] - LMI in Control Systems Analysis, Design and Applications