LMIs in Control/pages/mixh2hinfdesiredpole4perturbed

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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.

The System[edit | edit source]

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.
  • and are real valued matrix functions which represent the time varying parameters uncertainities.

Furthermore, the parameter uncertainties and are in the form of where

  • , and are known matrices of appropriate dimensions.
  • is a matrix containing the uncertainty, which satisfies

The Data[edit | edit source]

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

The Optimization Problem[edit | edit source]

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 locations[edit | edit source]

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

min
s.t

where

and are the weighting factors.

Conclusion:[edit | edit source]

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

Implementation[edit | edit source]

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

Related LMIs[edit | edit source]

Mixed H2 Hinf with desired poles controller

External Links[edit | edit source]

Return to Main Page[edit | edit source]

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