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 to the estimate error is prohibited to a desired level.

System SettingEdit

The system is following

 

 

 

where   are respectively the state vector, the measured output vector, and the output vector of interests.

  are the disturbance vector and control vector , respectively.

  are the system coefficient matrices of appropriate dimensions.

Problem FormulationEdit

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

 

where   is the state observation vector and   is the observer gain. Obviously, the estimate of the interested output is given by

 

which is desired to have as little affection as possible from the disturbance  .

Using system dynamics,

 

Denoting

 

 .

The transfer function of the system is clearly given by

 .

With the aforementioned preparation, the problems of   state observer designs can be stated as follows.

Problem 1Edit

(  state observers) Given system (9.22) and a positive scalar   , find a matrix   such that

 .


Problem 2Edit

(  state observers) Given system (9.22) and a positive scalar   , find a matrix   such that

 

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

 

This states that   is an asymptotic estimate of  .

Solution/TheoremEdit

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

Theorem 1Edit

The   state observers problem 1 has a solution if and only if there exist a matrix   and a symmetric positive definite matrix   such that

 

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

 

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   can be sought via the following optimization problem:

min  

s.t.  

Theorem 2Edit

The   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

 ,

 ,

 .

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

 .

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

 ,

 ,

trace .

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

 .

In applications, we are often concerned with the problem of finding the minimal attenuation level   . This problem can be solved via the optimization

min  

s.t.  ,

 ,

 ,

or

min  

 

 

trace 

When a minimal ρ is obtained, the minimal attenuation level is  .

WIP, additional references to be added

External LinksEdit

A list of references documenting and validating the LMI.

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