LMIs in Control/Click here to continue/Controller synthesis/Static-State Feedback Problem

LMIs in Control/Click here to continue/Controller synthesis/Static-State Feedback Problem
We are attempting to stabilizing The Static State-Feedback Problem

The System

Consider a continuous time Linear Time invariant system

${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\\end{aligned}}$

The Data

$A,B$ are known matrices

The Optimization Problem

The Problem's main aim is to find a feedback matrix such that the system

${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\\end{aligned}}$

and

${\begin{aligned}u(t)=Kx(t)\\\end{aligned}}$

is stable Initially we find the $K$ matrix such that $(A+BK)$ is Hurwitz.

The LMI: Static State Feedback Problem

This problem can now be formulated into an LMI as Problem 1:

$X(A+BK)+(A+BK)^{T}X<0$

From the above equation $X>0$ and we have to find K

The problem as we can see is bilinear in $K,X$

• The bilinear in X and K is a common paradigm
• Bilinear optimization is not Convex. To Convexify the problem, we use a change of variables.

Problem 2:

$AP+BZ+PA^{T}+Z^{T}B^{T}<0$

where $P>0$ and we find $Z$
$K=ZP^{-1}$

The Problem 1 is equivalent to Problem 2

Conclusion

If the (A,B) are controllable, We can obtain a controller matrix that stabilizes the system.

Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

Related LMIs

Hurwitz Stability

External Links

• [1] - LMI in Control Systems Analysis, Design and Applications
• 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.
• [2] -Mathworks reference to DC Gain