LMIs in Control/pages/Optimal Output Feedback H2 LMI

Optimal Output Feedback LMIEdit

Similar to state feedback, output feedback is necessary when information about the output is not known. Often techniques such as Kalman filtering are implemented to tackle this problem. The method below, however, does not use a filtering technique and instead uses a combination of LMI constraints to perform the output feedback as well as find the minimal bound on the   norm of the system.   is often done using more classical tools such as Riccati equations. More recently LMI techniques have been created to solve problems such as full state feedback or output feedback as seen below.

The SystemEdit

The system is represented using the 9-matrix notation shown below.

 

where   is the state,   is the regulated output,   is the sensed output,   is the exogenous input, and   is the actuator input, at any  .

The DataEdit

 ,  ,  ,  ,  ,  ,  ,  ,   are known.

The LMI: Optimal Output Feedback Control LMIEdit

The following are equivalent.

1) There exists a   such that  

2) There exists  ,  ,  ,  ,  ,  ,   such that

 
 
 
 

Conclusion:Edit

The above LMI determines the the upper bound   on the H2 norm. In addition to this the controller   can also be recovered.

 
 
 
 

where,

 

for any full-rank   and   such that

 .

ImplementationEdit

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/OF_H2.m

Related LMIsEdit

Optimal Output Feedback Hinf


External LinksEdit

A list of references documenting and validating the LMI.


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