D-Stability Controller for Switched Systems
This LMI lets you specify desired performance metrics like rising time, settling time and percent overshoot. Note that arbitrarily switching between stable systems can lead to instability whilst switching can be done between individually unstable systems to achieve stability.
Suppose we were given the switched system such that
where , , , and for any .
modes of operation
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 ( )
- settling time ( )
- percent overshoot ( )
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:
Assume that is the complex pole location, then:
This then allows us to modify our inequality constraints as:
which not only allows us to map the relationship between complex pole locations and inequality constraints but it also now allows us to easily formulate our LMIs for this problem.
The LMI: An LMI for Quadratic D-StabilizationEdit
Suppose there exists and such that
The resulting controller can be recovered by
The implementation of this LMI requires Yalmip and Sedumi /MOSEK
A list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.