LMIs in Control/pages/Insensitive Strip Region Design

Insensitive Strip Region Design


Suppose if one were interested in robust stabilization 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. This would be accomplished with the help of 2 design problems: the insensitive strip region design and insensitive disk region design (see link below for the latter).


The SystemEdit

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

 

where  ,  , and   are the state, output and input vectors respectively. 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 scalars   and  . Then the output feedback control law   would be such that  , where:

 

Letting   being the solution to the above problem, then

 

where

 

The LMI: Insensitive Strip Region DesignEdit

Using the above info, we can simplify the problem by setting   to   for all practical applications. This then simplifies our problem and results in the following LMI:

 

Conclusion:Edit

If the resulting solution from the LMI above produces a negative  , then the output feedback controller   is Hurwitz-stable. Hoewever, if   is a really small positive number, then   must be negative for the controller to be Hurwitz-stable.

ImplementationEdit

  • Example Code - A GitHub link that contains code (titled "InsensitiveStripRegion.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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