# LMIs in Control/pages/mixedhinfh2desiredpole

**LMI for Mixed with desired pole location Controller**

The mixed output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the controller, the channel is used to improve the robustness of the design while the channel guarantees good performance of the system and additional constraint is used to place poles at desired location.

**The System**Edit

We consider the following state-space representation for a linear system:

where

- , are the state vector and the output vectors, respectively

- , are the disturbance vector and the control vector

- , , , , , , , and are the system coefficient matrices of appropriate dimensions

**The Data**Edit

We assume that all the four matrices of the plant, , , , , , , , and are given.

**The Optimization Problem**Edit

For the system with the following feedback law:

The closed loop system can be obtained as:

the transfer function matrices are and

Thus the performance and the performance requirements for the system are, respectiverly

and

.
For the performance of the system response, we introduce the closed-loop eigenvalue
location requirement. Let

It is a region on the complex plane, which can be used to restrain the closed-loop
eigenvalue locations.
Hence a state feedback control law is designed such that,

- The performance and the performance are satisfied.
- The closed-loop eigenvalues are all located in , that is,

.

**The LMI:** LMI for mixed / with desired Pole locationsEdit

The optimization problem discussed above has a solution if there exist two symmetric matrices and a matrix , satisfying

min

s.t

where and are the weighting factors.

**Conclusion:**Edit

The calculated scalars and are the and norms of the system, respectively. The controller is extracted as

**Implementation**Edit

A link to Matlab codes for this problem in the Github repository:

**Related LMIs**Edit

Mixed H2 Hinf with desired pole location for perturbed system

**External Links**Edit

- 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.