LMIs in Control/pages/DT-SOFS

LMIs in Control/pages/DT-SOFS

The static output feedback (SOF) problem has been investigated and analyzed by many people and the literature concerning this topic is vast. In practicality, it is not always possible to have full access to the state vector and only a partial information through a measured output is available. This explains why this problem has challenged many researchers in control theory.
Here is a systematic approach for the SOF control design for discrete time linear systems.

The SystemEdit

Consider a discrete-time LTI system, with state-space realization  ,

 

  is the state,   is the measured output,   is the control input.

The DataEdit

  are the constant matrices of appropriate dimensions.

The Optimization ProblemEdit

The full state is not measurable and only partial information is available through   which can used for control purposes.

We have to find a static output feedback gain with respect to

 

where   is the output feedback gain such that the final closed loop system is asymptotically stable.

The LMI: LMI for Discrete-Time Static Output Feedback StabilizabilityEdit

The discrete time system considered is static output feedback stabilizable under any of the following equivalent necessary or sufficient conditions.

  • There exists a   and   where   such that
  


  • There exists a   and   where   such that
  

ConclusionEdit

If it is feasible we obtain a output feedback gain matrix   such that the closed loop system is asymptotically stable.
While implementing the optimization problem the following conditions are assumed to be satisfied

  • The Transfer matrix and its inverse are both analytical a s=0
  • The matrix   is non-singular
  • The triple   are reachable and observable.

ImplementationEdit

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

Related LMIsEdit

Continuous-time Static Output Feedback Stabilizability

External LinksEdit

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