LMIs in Control/pages/Discrete-Time Mixed H2 HInf Optimal Observer

LMIs in Control/pages/Discrete-Time Mixed H2 HInf Optimal Observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

The SystemEdit

 

where   and is the state vector,   and is the state matrix,   and is the input matrix,   and is the exogenous input,   and is the output matrix,   and is the feedthrough matrix,   and is the output, and it is assumed that   is detectable.


 

The DataEdit

The matrices  .

The Optimization ProblemEdit

An observer of the form:

 

is to be designed, where   is the observer gain.

Defining the error state  , the error dynamics are found to be

 ,

and the performance output is defined as

 .

The observer gain   is to be designed to minimize the   norm of the closed loop transfer matrix   from the exogenous input   to the performance output   is less than  , where

 

The LMI: Discrete-Time Mixed H2-Hinf-Optimal ObserverEdit

The discrete-time mixed- -optimal observer gain is synthesized by solving for  ,  ,  , and   that minimize J  subject to  ,

 

where   refers to the trace of a matrix.

Conclusion:Edit

The mixed- -optimal observer gain is recovered by  , the   norm of   is less than  , and the   norm of   is less than  . This result gives us a matrix of observer gains   that allow us to optimally observe the states of the system indirectly as:

 

ImplementationEdit

This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m

Related LMIsEdit

Discrete-Time_Hinfinity-Optimal_Observer

Discrete-Time_H2-Optimal_Observer

External LinksEdit

This LMI comes from Ryan Caverly's text on LMI's (Section 5.3.2):

Other resources:

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