# LMIs in Control/pages/full order Hinf H2 state observers

WIP, Description in progress

In this section, we treat the problem of designing a full-order state observer for system such that the effect of the disturbance ${\displaystyle w(t)}$ to the estimate error is prohibited to a desired level.

## System Setting

The system is following

${\displaystyle {\dot {x}}(t)=Ax(t)+B_{1}u(t)+B_{2}w(t),x(0)=x_{0},}$

${\displaystyle y(t)=C_{1}x(t)+D_{1}u(t)+D_{2}w(t),}$

${\displaystyle z(t)=C_{2}x(t),}$

where ${\displaystyle x\in \mathbb {R} ^{n},\ y\in \mathbb {R} ^{l},\ z\in \mathbb {R} ^{m}}$  are respectively the state vector, the measured output vector, and the output vector of interests.

${\displaystyle w\in \mathbb {R} ^{p}}$  are the disturbance vector and control vector , respectively.

${\displaystyle A,\ B_{1},\ B_{2},\ C_{1},\ C_{2},\ D_{1},{\text{ and }}D_{2}}$  are the system coefficient matrices of appropriate dimensions.

## Problem Formulation

For the system, we introduce a full-order state observer in the following form:

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

where ${\displaystyle {\hat {x}}}$  is the state observation vector and ${\displaystyle L\in \mathbb {R} ^{n\times m}}$  is the observer gain. Obviously, the estimate of the interested output is given by

${\displaystyle {\hat {z}}(t)=C_{2}{\hat {x}}(t)}$

which is desired to have as little affection as possible from the disturbance ${\displaystyle w(t)}$ .

Using system dynamics,

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}u(t)+B_{2}w(t)\\&=Ax(t)+Ly-Ly+B_{1}u(t)+B_{2}w(t)\\&=(A+LC_{1})x(t)-Ly(t)+(B_{1}+LD_{1})u(t)+(B_{2}+LD_{2})w(t)\end{aligned}}}

Denoting

${\displaystyle {\dot {e}}=(A+LC_{1})e+(B_{2}+LD_{2})w}$

${\displaystyle {\tilde {z}}(t)=C_{2}e}$ .

The transfer function of the system is clearly given by

${\displaystyle G_{{\tilde {z}}w}(s)=C_{2}(sI-A-LC_{1})^{-1}(B_{2}+LD_{2})}$ .

With the aforementioned preparation, the problems of ${\displaystyle {\mathcal {H}}_{\infty }{\text{ and }}{\mathcal {H}}_{2}}$  state observer designs can be stated as follows.

### Problem 1

(${\displaystyle {\mathcal {H}}_{\infty }}$  state observers) Given system (9.22) and a positive scalar ${\displaystyle \gamma }$  , find a matrix ${\displaystyle L}$  such that

${\displaystyle ||G_{{\tilde {z}}w}(s)||_{\infty }<\gamma }$ .

### Problem 2

(${\displaystyle {\mathcal {H}}_{2}}$  state observers) Given system (9.22) and a positive scalar ${\displaystyle \gamma }$  , find a matrix ${\displaystyle L}$  such that

${\displaystyle ||G_{{\tilde {z}}w}(s)||_{2}<\gamma }$

As a consequence of the requirements in the previous problems, the error system is asymptotically stable, and hence we have

${\displaystyle e(t)=x(t)-{\hat {x}}(t)\rightarrow 0,{\text{ as }}t\rightarrow \ \infty }$

This states that ${\displaystyle {\hat {x}}(t)}$  is an asymptotic estimate of ${\displaystyle x(t)}$ .

## Solution/Theorem

Regarding the solution to the problem of H∞ state observers design, we have the following theorem.

### Theorem 1

The ${\displaystyle {\mathcal {H}}_{\infty }}$  state observers problem 1 has a solution if and only if there exist a matrix ${\displaystyle W}$  and a symmetric positive definite matrix ${\displaystyle P}$  such that

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

When such a pair of matrices W and P are found, a solution to the problem is given as

${\displaystyle L=P^{-1}W}$

With a prescribed attenuation level, the problem of H∞ state observers design is turned into an LMI feasibility problem in the form problem stated before. The problem with a minimal attenuation level ${\displaystyle \gamma }$  can be sought via the following optimization problem:

min ${\displaystyle \gamma }$

s.t. {\displaystyle {\begin{aligned}P&>0\\{\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}}&<0\end{aligned}}}

### Theorem 2

The ${\displaystyle {\mathcal {H}}_{2}}$  state observers problem 2 has a solution the following 2 conclusions hold.

1.It has a solution if and only if there exists a matrix W, a symmetric matrix Q, and a symmetric matrix X such that

${\displaystyle {\begin{bmatrix}XA+WC_{1}+(XA+WC_{1})^{T}&XB_{2}+WD_{2}\\(XB_{2}+WD_{2})^{T}&-I\end{bmatrix}}<0}$ ,

${\displaystyle {\begin{bmatrix}-Q&C_{2}\\C_{2}^{T}&-X\end{bmatrix}}<0}$ ,

${\displaystyle {\text{trace}}(Q)<\gamma ^{2}}$ .

When such a triple of matrices are obtained, a solution to the problem is given as

${\displaystyle L=X^{-1}W}$ .

2. It has a solution if and only if there exists a matrix V, a symmetric matrix Z, and a symmetric matrix Y such that

${\displaystyle A^{T}Y+C_{1}^{T}V^{T}+YA+VC_{1}+C_{2}^{T}C_{2}<0}$ ,

${\displaystyle {\begin{bmatrix}-Z&(YB_{2}+VD_{2})^{T}\\YB_{2}+VD_{2}&-Y\end{bmatrix}}<0}$ ,

trace${\displaystyle (Z)<\gamma ^{2}}$ .

When such a triple of matrices are obtained, a solution to the problem is given as

${\displaystyle L=Y^{-1}V}$ .

In applications, we are often concerned with the problem of finding the minimal attenuation level ${\displaystyle \gamma }$  . This problem can be solved via the optimization

min ${\displaystyle \rho }$

s.t. ${\displaystyle {\begin{bmatrix}XA+WC_{1}+(XA+WC_{1})^{T}&XB_{2}+WD_{2}\\(XB_{2}+WD_{2})^{T}&-I\end{bmatrix}}<0}$ ,

${\displaystyle {\begin{bmatrix}-Q&C_{2}\\C_{2}^{T}&-X\end{bmatrix}}<0}$ ,

${\displaystyle {\text{trace}}(Q)<\gamma ^{2}}$ ,

or

min ${\displaystyle \rho }$

${\displaystyle A^{T}Y+C_{1}^{T}V^{T}+YA+VC_{1}+C_{2}^{T}C_{2}<0}$

${\displaystyle {\begin{bmatrix}-Z&(YB_{2}+VD_{2})^{T}\\YB_{2}+VD_{2}&-Y\end{bmatrix}}<0}$

trace${\displaystyle (Z)<\gamma ^{2}}$

When a minimal ρ is obtained, the minimal attenuation level is ${\displaystyle \gamma ={\sqrt {\rho }}}$ .