# 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 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:

and

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]

- Disk Region Design - LMI for Disk Region Design with minimal gain
- Insensitive Strip Region Design - Equivalent LMI for Insensitive Strip Region Design

## External Links[edit| edit source]

A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.