View source for 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 <math>w(t)</math> to the estimate error is
prohibited to a desired level.

== System Setting == 
The system is following

<math>\dot{x}(t) = Ax(t) + B_1u(t) + B_2w(t), x(0) = x_0, </math>

<math>y(t) = C_1x(t) + D_1u(t) + D_2w(t), </math>

<math>z(t) = C_2x(t), </math>

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

<math> w \in \mathbb{R}^p</math> are the disturbance vector and control vector , respectively.

<math> A, \ B_1, \ B_2, \ C_1, \ C_2, \ D_1, \text{ and } D_2</math> are the system coefficient matrices of
appropriate dimensions.

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

<math>\dot{\hat{x}}=(A+LC_1)\hat{x}-Ly+(B_1+LD_1)u</math>

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

<math>\hat{z}(t) = C_2\hat{x}(t)</math>

which is desired to have as little affection as possible from the disturbance <math>w(t)</math>.

Using system dynamics, 

<math>\begin{align}\dot{x}(t) &= Ax(t) + B_1u(t) + B_2w(t) \\
&=Ax(t)+Ly-Ly+B_1u(t)+B_2w(t) \\ 
&=(A+LC_1)x(t)-Ly(t)+(B_1+LD_1)u(t)+(B_2+LD_2)w(t)\end{align}</math>

Denoting 

<math>\dot{e} = (A+LC_1)e+(B_2+LD_2)w</math>

<math>\tilde{z}(t)=C_2e</math>.

The transfer function of the system is clearly given by

<math> G_{\tilde{z}w}(s) = C_2(sI-A-LC_1)^{-1}(B_2+LD_2)</math>.

With the aforementioned preparation, the problems of <math>\mathcal{H}_\infty \text{ and } \mathcal{H}_2</math> state
observer designs can be stated as follows.

===Problem 1===
(<math>\mathcal{H}_\infty</math> state observers) Given system (9.22) and a positive scalar <math>\gamma</math> ,
find a matrix <math>L</math> such that

<math>||G_{\tilde{z}w}(s)||_\infty < \gamma</math>.


===Problem 2===
(<math>\mathcal{H}_2</math> state observers) Given system (9.22) and a positive scalar <math>\gamma</math> ,
find a matrix <math>L</math> such that

<math>||G_{\tilde{z}w}(s)||_2 < \gamma</math> 

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

<math>e(t) = x(t) - \hat{x}(t) \rightarrow 0, \text{ as } t\rightarrow \ \infty</math>

This states that <math>\hat{x}(t)</math> is an asymptotic estimate of <math>x (t)</math>.

== Solution/Theorem ==

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

===Theorem 1===
The <math>\mathcal{H}_\infty</math> state observers problem 1 has a solution
if and only if there exist a matrix <math>W</math> and a symmetric positive definite matrix <math>P</math>
such that

<math> \begin{bmatrix} A^TP+C_1^TW^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</math>

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

<math>L=P^{-1}W</math>

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 <math>\gamma</math> can be sought via the following optimization problem:

min <math>\gamma</math>

s.t. <math> \begin{align}P&>0 \\
\begin{bmatrix} A^TP+C_1^TW^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{align}</math>

===Theorem 2===
The <math>\mathcal{H}_2</math> 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

<math> \begin{bmatrix} XA+WC_1+(XA+WC_1)^T & XB_2+WD_2  \\ 
(XB_2+WD_2)^T & -I 
\end{bmatrix}<0</math>,

<math>\begin{bmatrix}-Q & C_2 \\ C_2^T & -X\end{bmatrix}<0</math>,

<math> \text{trace}(Q) < \gamma^2 </math>.

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

<math>L=X^{-1}W</math>.

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

<math>A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2<0</math>,

<math>\begin{bmatrix}-Z & (YB_2+VD_2)^T \\ YB_2+VD_2 & -Y\end{bmatrix}<0</math>,

trace<math>(Z)<\gamma^2</math>.

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

<math>L=Y^{-1}V</math>.

In applications, we are often concerned
with the problem of finding the minimal attenuation level <math>\gamma</math> . This problem can be
solved via the optimization

min <math>\rho</math>

s.t. <math> \begin{bmatrix} XA+WC_1+(XA+WC_1)^T & XB_2+WD_2  \\ 
(XB_2+WD_2)^T & -I 
\end{bmatrix}<0</math>,

<math>\begin{bmatrix}-Q & C_2 \\ C_2^T & -X\end{bmatrix}<0</math>,

<math> \text{trace}(Q) < \gamma^2 </math>,

or 

min <math>\rho</math>

<math>A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2<0</math>

<math>\begin{bmatrix}-Z & (YB_2+VD_2)^T \\ YB_2+VD_2 & -Y\end{bmatrix}<0</math>

trace<math>(Z)<\gamma^2</math>

When a minimal ρ is obtained, the minimal attenuation level is <math>\gamma=\sqrt{\rho}</math>.

''WIP, additional references to be added''
== External Links ==
A list of references documenting and validating the LMI.

== '''Return to Main Page:''' ==
{{bookcat}}