# LMIs in Control/pages/Quadratic DStabilization

Continuous-Time D-Stability Controller

This LMI will let you place poles at a specific location based on system performance like rising time, settling time and percent overshoot, while also ensuring the stability of the system.

**The System**Edit

Suppose we were given the continuous-time system

whose stability was not known, and where , , , and for any .

Adding uncertainty to the system

**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. In order to do that, we have to first 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

Given the resulting controller , we can now determine that the pole locations of satisfies the inequality constraints , and for all

**Implementation**Edit

The implementation of this LMI requires Yalmip and Sedumi https://github.com/JalpeshBhadra/LMI/blob/master/quadraticDstabilization.m

**Related LMIs**Edit

- Continuous Time D-Stability Observer - Equivalent D-stability LMI for a continuous-time observer.

## External LinksEdit

- 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.