LMIs in Control/Stability Analysis/Continuous Time/Optimization Over Affine Family of Linear Systems

Optimization over an Affine Family of Linear Systems edit

Presented in this page is a general framework for optimizing various convex functionals for a system which depends affinely, or linearly, on a parameter using linear matrix inequalities. The optimization problem presented on this page generalizes an LMI which can be applied to various problems within linear systems and control. Some examples of these applications are finding the   and   norms, entropy, dissipativity, and the Hankel norm of an affinely parametric system.

The System edit

Consider a family of linear systems
 
 
with state space realization   where   and   depend affinely on the parameter  .

We assume   is stable and   is controllable.

The transfer function,   depends affinely on  .


The Data edit

The transfer function  , and system matrices  ,  ,  ,   are known.   represents the convex functionals,   and   represent some auxiliary variables dependent on the problem being solved.

The LMI:Generalized Optimization for Affine Linear Systems edit

Several control theory problems, mentioned earlier, take the following form:
minimize  
subject to  

Problems of this nature can be formulated as an LMI by representing   as an LMI in   and possibly   such that  

Thus, the general optimization problem to be applied to an affine family of linear systems is as follows:
minimize  
subject to  

Conclusion: edit

The LMI for this generalized optimization problem may be extended to various convex functionals for affine parametric systems. For extensions of this LMI, see the related LMIs section.

Implementation edit

Implementation of LMI's of this form require Yalmip and a linear solver such as Sedumi or SDPT3.

  Norm for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

Entropy Bond for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

LMI can be applied to other extensions in stability and controller analysis. Please see the related LMI pages in the section below.

Related LMIs edit

  Norm for Affine Parametric Systems

Entropy Bond for Affine Parametric Systems

Dissipativity for Affine Parametric Systems

for Affine Parametric Systems


External Links edit


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LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control