LMI for Mixed with desired pole location Controller for perturbed system case
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.
We consider the following state-space representation for a linear system:
- , 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.
- and are real valued matrix functions which represent the time varying parameters uncertainities.
Furthermore, the parameter uncertainties and are in the form of
- , and are known matrices of appropriate dimensions.
- is a matrix containing the uncertainty, which satisfies
We assume that all the four matrices of the plant,, , ,, ,,,, and are given.
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
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
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 locations[edit | edit source]
The optimization problem discussed above has a solution if there exist scalars two symmetric matrices and a matrix , satisfying
and are the weighting factors.
The calculated scalars and are the and norms of the system, respectively.
The controller is extracted as
A link to Matlab codes for this problem in the Github repository:
Mixed H2 Hinf with desired poles controller
LMIs in Control/Tools