LMIs in Control/pages/Insensitive Strip Region Design

From Wikibooks, open books for an open world
Jump to navigation Jump to search

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 System[edit | edit source]

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 Data[edit | edit source]

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

The Optimization Problem[edit | edit source]

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


The LMI: Insensitive Strip Region Design[edit | edit source]

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 | edit source]

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.

Implementation[edit | edit source]

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

Related LMIs[edit | edit source]

External Links[edit | edit source]

A list of references documenting and validating the LMI.

Return to Main Page:[edit | edit source]