LMIs in Control/pages/H2-OSE

LMIs in Control/pages/H2-OSE

The H2 norm of a stable system H is the root-mean-square of the impulse response of the system. The H2 norm measures the steady-state covariance (or power) of the output response to unit noise input. In this module, the goal of H₂ optimal state estimation is to design an observer that minimizes the H₂ norm of the closed loop transfer matrix

The System

Consider the continuous-time generalized plant P with state-space realization

$\begin{array}{ccc}\dot{x}(t)& =Ax(t)+B_{1}u(t)+B_{2}w(t)& x(0)=x_0\\ y(t)& =C_{1}x(t)+D_{1}u(t)+D_{2}w(t)\\ & z(t)=C_{2}x(t)\end{array}$

where it is assumed that (A,C₂) is detectable. An observer of the form

The Data

• $x$ ∈ $\begin{array}{c}ℝ\end{array}$ⁿ, $y$ ∈ $\begin{array}{c}ℝ\end{array}$^l , $z$ ∈ $\begin{array}{c}ℝ\end{array}$^m are respectively the state vector, the measured

output vector, and the output vector of interests

• $w$ ∈ $\begin{array}{c}ℝ\end{array}$^p and $u$ ∈ $\begin{array}{c}ℝ\end{array}$^r are the disturbance vector and the control vector,

respectively

• A, B₁, B₂, C₁, C₂, D₁, and D₂ are the system coefficient matrices of

appropriate dimensions

The Optimization Problem

Given the system and a positive scalar $\gamma$ we have to find the matrix L such that

||$G_{zw}(s)$||₂ < $\gamma$

An observer of the form

$\begin{array}{c}\dot{x}(t)=Ax(t)+(Ly-Ly)+B_{1}u(t)+B_{2}w(t)\\ \dot{x}(t)=(A+L{C}_1)x(t)-Ly+(B_1+L{D}_1)u(t)\\ +(B_2+L{D}_2)w(t)\end{array}$

is to be designed, where L is the observer gain.
Defining the error state as
$e=x-\hat{x}$
The break dynamics are found to be

$\begin{array}{c}\dot{e}=(A+L{C}_1)e+(B_2+L{D}_2)w\\ \overline{z}(t)=C_{2}e\end{array}$

For the system we introduce a full state observer in the following form:
$\dot{\hat{x}}=(A+L{C}_1)\hat{x}-Ly+(B_1+L{D}_1)u$ $\hat{x},L$ are the observation vector and the observer gain.
The transfer function for this case is
$G_{zw}(s)=C_2(sI-A-L{C}_1{)}^{-1}(B_2+L{D}_2)$
and thus the problem of $H_2$ state observer design is to find L such that
||$G_{zs}(s)|{|}_2<\gamma$

The LMI: LMI for H₂ Observer estimation

The H₂ state observer problem has a solution if and only if there exists a matrix $W$, a symmetric matrix $Q$ and a symmetric matrix $X$ such that

$\left[\begin{array}{cc}XA+W{C}_1+(XA+W{C}_1{)}^T& X{B}_2+W{D}_2\\ (X{B}_2+W{D}_2{)}^T& -I\end{array}\right]$$\begin{array}{c}<0\end{array}$

$\left[\begin{array}{cc}-Q& C_2\\ C_2^T& -X\end{array}\right]$$\begin{array}{c}<0\end{array}$

$\begin{array}{c}trace(Q)<{\gamma }^2\end{array}$

and from the solution of the above LMIs we can obtain the observer matrix as

$L=X^{-1}W$

Conclusion

Thus by formulation, we have converted the problem of H₂ state observer design into an LMI feasibility problem by optimizing the above LMIs. In application we are often concerned with the problem of finding the minimal attenuation level $\gamma$

On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 3 matrices as output, $X,WandQ$ and also $\rho$ which is used to calculate $\gamma$ which is the H₂ norm of the system.

There exists another set of LMIs which holds true for the same optimization problem as above.

$\begin{array}{c}{A}^{T}Y+C_1^T{V}^T+YA+V{C}_1+C_2^T{C}_2<0\\ \end{array}$

$\left[\begin{array}{cc}-Z& Y{B}_2+V{D}_2\\ (Y{B}_2+V{D}_2{)}^T& -Y\end{array}\right]$$\begin{array}{c}<0\end{array}$

$\begin{array}{c}trace(Z)<{\gamma }^2\end{array}$

When a minimal $\rho$ is obtained, the minimal attenuation level is

$\gamma =\sqrt{\rho }$

Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

Related LMIs

H_{$\mathrm{\infty }$} State Observer Design
Discrete time H₂ State Observer Design

External Links

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