LMIs in Control/pages/H stabilization

$$\mathbb{H}_{(\alpha,\beta)}$$-Stabilization

There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of $$\mathbb{H}_{(\alpha,\beta)}$$-Stabilization, a form of $$\mathbb{D}$$-Stabilization where the real components are located on the left-half of the complex plane.

The System
For this particular problem, suppose that we were given a linear system in the form of:



\begin{align} \dot x&=Ax+Bu,\\ \end{align}$$

where $$x\in\mathbb{R}^n$$ and $$u\in\mathbb{R}^r$$. Then the LMI for determining the $$\mathbb{H}_{(\alpha,\beta)}$$-stabilization case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 2 matrices: $$A\text{ and }B$$.

The Optimization Problem
Suppose - for the linear system given above - we were asked to design a state-feedback control law where $$u=Kx$$ such that the closed-loop system:

\begin{align} \dot x&=(A+BK)x\\ \end{align}$$ is $$\mathbb{H}_{(\alpha,\beta)}$$ stable, then the system would be stabilized as follows.

The LMI: $$\mathbb{H}_{(\alpha,\beta)}$$-Stabilization
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix $$W$$ and a symmetric matrix $$P>0$$ that satisfy the following:



\begin{align} \begin{cases} AP+P{A^T}+BW+{W^T}{B^T}+2{\alpha}P &<0\\ -AP-P{A^T}-BW-{W^T}{B^T}-2{\beta}P &<0 \end{cases} \end{align}$$

Conclusion:
Given the resulting controller matrix $$K=WX^{-1}$$, it can be observed that the matrix is $$\mathbb{H}_{(\alpha,\beta)}$$-stable.

Implementation

 * Example Code - A GitHub link that contains code (titled "HStability.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * ../D stabilization/ - Equivalent LMI for $$\mathbb{D}_{(q,r)}$$-stabilization.