LMIs in Control/pages/Switched Systems Rise time

Rise time Specification for Switched Systems

This LMI lets you specify desired performance (rise time). Note that arbitrarily switching between stable systems can lead to instability whilst switching can be done between individually unstable systems to achieve stability.

The System

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Suppose we were given the switched system such that

 

where  ,  ,  , and   for any  .

  modes of operation

The Data

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

  • matrices  ,  
  • rise time ( )

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

The Optimization Problem

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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 using the following inequality constraints:

Rise Time:  

Assume that   is the complex pole location, then:

 

This then allows us to modify our inequality constraints as:

Rise Time:  


The LMI: An LMI for Rise time Specification

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Suppose there exists   and   such that

 


for  

Conclusion:

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The resulting controller can be recovered by

 .

Implementation

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

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A list of references documenting and validating the LMI.


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