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

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

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

The Data

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

output vector, and the output vector of interests

• $w$ ∈ ${\begin{aligned}\mathbb {R} \end{aligned}}$^p and $u$ ∈ ${\begin{aligned}\mathbb {R} \end{aligned}}$^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

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

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

${\begin{aligned}{\dot {e}}=(A+LC_{1})e+(B_{2}+LD_{2})w\\{\bar {z}}(t)=C_{2}e\end{aligned}}$

For the system we introduce a full state observer in the following form:
${\dot {\hat {x}}}=(A+LC_{1}){\hat {x}}-Ly+(B_{1}+LD_{1})u$ ${\hat {x}},L$ are the observation vector and the observer gain.
The transfer function for this case is
$G_{zw}(s)=C_{2}(sI-A-LC_{1})^{-1}(B_{2}+LD_{2})$
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

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

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

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

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.

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

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

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

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_$\infty$ 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.