LMIs in Control/pages/Conic Sector Lemma

Conic Sector Lemma

For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.

The SystemEdit

Consider a square, contiuous-time linear time-invariant (LTI) system,  , with minimal state-space relization  , where   and  . The state-space representation is:

 

where  ,   and   are the system state, output, and the input vector respectively.

The DataEdit

The system coefficient matrices   are required. Optionally, the parameters to define a cone, either in the form of   where   or a radius   and ceter  .

The Feasibility LMIEdit

The system   is inside the given cone   if the following is feasible:

 

The above LMI can be used to also determine the cone parameters by setting   as a variable along with the condition  , and use the bisection method to find  .

If the given cone is represented by a center   and radius  , then the following feasibility problem can be evaluated to check if   is inside the given cone:

 

In order to also find the cone parameters, substituting   as a decision variable with additional constraint   and then solving for   via the bisection method will give the cone in which the system   resides if the problem is feasible.

Conclusion:Edit

The aforementioned LMIs can be utilized to either check if   is in the specified cone or not, or can be used to check the stability of   by finding if a feasible cone can be obtained that encloses  . An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:

 .

ImplementationEdit

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Conic_sector_example.m

Related LMIsEdit

Exterior Conic Sector Lemma.

KYP Lemma

External LinksEdit

A list of references documenting and validating the LMI.

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