# LMIs in Control/pages/H2 Disk Region

Insensitive Disk Region Design with Minimum ${\displaystyle H_{2}}$ Gain

Apart from the design for the insensitive strip region with minimum ${\displaystyle H_{2}}$ 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 ${\displaystyle H_{2}}$ gain of the closed-loop system.

## The System

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

{\displaystyle {\begin{aligned}\rho x&=Ax+Bu\\y&=Cx\end{aligned}}}

where ${\displaystyle x\in \mathbb {R} ^{n}}$ , ${\displaystyle y\in \mathbb {R} ^{m}}$  and ${\displaystyle u\in \mathbb {R} ^{r}}$  are the system state, output, and the input vector respectively. ${\displaystyle \rho }$  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 ${\displaystyle \gamma _{0}}$  is required.

## The Optimization Problem

The problem of designing an ${\displaystyle H_{2}}$  optimal controller that results in the closed loop system insensitive to a certain disk region involves two sub-problems:

• Finding a control gain ${\displaystyle K}$  such that: ${\displaystyle \left\|K\right\|_{2}<\gamma }$ .
• 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 ${\displaystyle \gamma }$  such that above two hold.

## The LMI: ${\displaystyle H_{2}}$ Optimal Control Design for Insensitive Disk Region

The problem above has a solution if and only if the following optimization problem has a solution ${\displaystyle (K,\gamma )}$ :

{\displaystyle {\begin{aligned}{\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{aligned}}}

## Conclusion:

By using the design problem provided here, an optimal ${\displaystyle H_{2}}$  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: