# LMIs in Control/pages/D-Stabilization of Switched Systems

**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.

**The System**Edit

Suppose we were given the switched system such that

where , , , and for any .

modes of operation

**The Data**Edit

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 Problem**Edit

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**:

**Settling Time**:

**Percent Overshoot**:

Assume that is the complex pole location, then:

This then allows us to modify our inequality constraints as:

**Rise Time**:

**Settling Time**:

**Percent Overshoot**:

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

for

**Conclusion:**Edit

The resulting controller can be recovered by

.

**Implementation**Edit

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

**External Links**Edit

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.