LMIs in Control/pages/Switched Systems Rise time

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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[edit | edit source]

Suppose we were given the switched system such that

where , , , and for any .

modes of operation

The Data[edit | edit source]

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[edit | edit source]

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[edit | edit source]

Suppose there exists and such that


Conclusion:[edit | edit source]

The resulting controller can be recovered by


Implementation[edit | edit source]

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

External Links[edit | edit source]

A list of references documenting and validating the LMI.

Related LMIs[edit | edit source]

Return to Main Page:[edit | edit source]