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 SystemEdit

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 DataEdit

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

The Optimization ProblemEdit

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  

ImplementationEdit

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

Related LMIsEdit

Mixed H2 Hinf with desired pole location for perturbed system

External LinksEdit

Return to Main PageEdit

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