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 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  ,  
  • rise time ( )

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 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 SpecificationEdit

Suppose there exists   and   such that




The resulting controller can be recovered by



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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