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

When designing controllers with insensitive region conditions, the aim is to place the closed-loop poles of the system in a particular region defined by its inner boundary. These regions are specified based on their insensitivity to perturbations to the system parameter matrices.

One type of such design is the Insensitive Strip Region Design. In this section, building upon that, optimization problems will be provided that ensure that the conditions for insensitive strip 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} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t) \end{align}$$

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$u(t)\in \R^r$$ are the system state, output, and the input vector respectively.

The Data
To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the strip region on the eigenvalue-space, two parameters $$ \gamma_1 $$ and $$ \gamma_2 $$ are required.

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


 * Finding a control gain $$ K $$ such that: $$ \left\| K \right\|_2 < \gamma $$.
 * The conditions for insensitive strip region design for the closed-loop system, as provided in the section Insensitive Strip 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 Strip Region
The problem above has a solution if and only if the following optimization problem has a solution $$ (K, \gamma) $$:

$$ \begin{align} \text{min   } & \gamma \\ \text{s.t.  } & \begin{bmatrix} - \gamma I & K \\ K^\top & - \gamma I \end{bmatrix} < 0 \\ & 2 \gamma_1 I < (A+BKC)^\top + (A+BKC) < 2 \gamma_2 I \end{align} $$

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_Strip_example.m

Related LMIs
Insensitive Strip Region Design