LMIs in Control/pages/Insensitive Disk Region Design

Insensitive Disk Region Design


Similar to the insensitive strip region design problem, insensitive disk region design is another way with which robust stabilization can be achieved where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices.


The SystemEdit

Suppose we consider the following linear system that needs to be controlled:

 

where  ,  , and   are the state, output and input vectors respectively, and   represents the differential operator (in the continuous-time case) or one-step shift forward operator (i.e.,  ) (in the discrete-time case). Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.

The DataEdit

Prior to obtaining the LMI, we need the following matrices:  ,  , and  .

The Optimization ProblemEdit

Consider the above linear system as well as 2 positive scalars   and  . Then the output feedback control law   would be designed such that:

 

Recalling the definition, we have:

 

and

 

Letting   being the solution to the above problem, then

 

The LMI: Insensitive Strip Region DesignEdit

Using the above info, we can convert the given problem into an LMI, which - after using Schur compliment Lemma - results in the following:

 

Conclusion:Edit

For Schur stabilization, we can choose to solve the problem with  . Schur stability is achieved when  . Alternately, if   is greater than (but very close to) 1, then Schur stability is also achieved when  .

ImplementationEdit

  • Example Code - A GitHub link that contains code (titled "InsensitiveDiskRegion.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIsEdit

External LinksEdit

A list of references documenting and validating the LMI.

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