LMIs in Control/pages/Insensitive Disk Region Design

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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 System[edit]

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 Data[edit]

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

The Optimization Problem[edit]

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 Design[edit]

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 .

Implementation[edit]

  • Example Code - A GitHub link that contains code that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs[edit]

Links to other closely-related LMIs

External Links[edit]

A list of references documenting and validating the LMI.

Return to Main Page:[edit]