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Insensitive Disk Region Design with Minimum $$ H_2 $$ Gain

Apart from the design for the insensitive strip region with minimum $ H_2 $ gain, another type of such design is the Insensitive Disk Region Design. In this section, optimization problems will be provided that ensure that the conditions for insensitive disk region design are satisfied with some bounds on the $$ H_2 $$ gain of the closed-loop system.

The System
A state-space representation of a linear system as given below:



\begin{align} \rho x&=Ax+Bu\\ y&=Cx \end{align}$$

where $$x\in \R^n$$, $$y\in \R^m$$ and $$u\in \R^r$$ are the system state, output, and the input vector respectively. $$ \rho $$ represents the differential operation for continuous time systems, or the one-step shift forward operator for discrete time case.

The Data
To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the disk region on the eigenvalue-space, its radius $$ \gamma_0 $$ is required.

The Optimization Problem
The problem of designing an $$ H_2 $$ optimal controller that results in the closed loop system insensitive to a certain disk region involves two sub-problems:


 * Finding a control gain $$ K $$ such that: $$ \left\| K \right\|_2 < \gamma $$.
 * The conditions for insensitive disk region design for the closed-loop system, as provided in the section Insensitive Disk Region Design are fulfilled.
 * The optimization goal is to minimize $$ \gamma $$ such that above two hold.

The LMI: $$ H_2 $$ Optimal Control Design for Insensitive Disk Region
The problem above has a solution if and only if the following optimization problem has a solution $$ (K, \gamma) $$:

$$ \begin{align} \text{min   } & \gamma \\ \text{s.t.  } & \begin{bmatrix} - \gamma I & K \\ K^\top & - \gamma I \end{bmatrix} < 0 \\ & \begin{bmatrix} -\gamma_0 I & A+ BKC + qI \\ (A+ BKC + qI)^\top & -\gamma_0 I \end{bmatrix} < 0 \end{align} $$

Conclusion:
By using the design problem provided here, an optimal $$ H_2 $$ controller is designed to make the closed-loop system robust to perturbations in the system matrices.

Implementation
To solve the optimization problem with LMI presented here, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/H2_Disk_example.m

Related LMIs
Insensitive Disk Region Design