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 SystemEdit

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

 

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

The DataEdit

  •   n,   l ,   m are respectively the state vector, the measured

output vector, and the output vector of interests

  •   p and   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 ProblemEdit

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

|| ||2 <  

An observer of the form

 

is to be designed, where L is the observer gain.
Defining the error state as
 
The break dynamics are found to be

 


For the system we introduce a full state observer in the following form:
    are the observation vector and the observer gain.
The transfer function for this case is
 
and thus the problem of   state observer design is to find L such that
|| 

The LMI: LMI for H2 Observer estimationEdit

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

  


  


 


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

 

ConclusionEdit

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  

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



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

 


  


 

When a minimal   is obtained, the minimal attenuation level is

 

ImplementationEdit

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

https://github.com/yashgvd/ygovada

Related LMIsEdit

H  State Observer Design
Discrete time H2 State Observer Design

External LinksEdit

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