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 | edit source]

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 | 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 positive scalars and . Then the output feedback control law would be designed such that:

Recalling the definition, we have:


Letting being the solution to the above problem, then

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

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 | edit source]

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 | edit source]

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