LMIs in Control/pages/Switched Systems Pole Placement

Pole Placement for Switched Systems

This LMI lets you provide specifications of the switched system closed loop poles. Note that arbitrarily switching between stable systems can lead to instability whilst switching can be done between individually unstable systems to achieve stability.



The SystemEdit

Suppose we were given the switched system such that

 

where  ,  ,  , and   for any  .

  modes of operation

The DataEdit

In order to properly define the acceptable region of the poles in the complex plane, we need the following pieces of data:

  • matrices  ,  
  • desired closed loop pole bound ( )

Having these pieces of information will now help us in formulating the optimization problem.

The Optimization ProblemEdit

Using the data given above, we can now define our optimization problem. We first have to define the acceptable region in the complex plane that the poles can lie on. Assume that   is the complex pole location, then:

 

The LMI: An LMI for Pole PlacementEdit

Suppose there exists   and   such that


 


for  

Conclusion:Edit

The resulting controller can be recovered by

 .

ImplementationEdit

The implementation of this LMI requires Yalmip and Sedumi /MOSEK [1]

External LinksEdit

A list of references documenting and validating the LMI.


Related LMIsEdit


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