LMIs in Control/pages/SSFP
We are attempting to stabilizing The Static State-Feedback Problem
Consider a continuous time Linear Time invariant system
are known matrices
The Optimization ProblemEdit
The Problem's main aim is to find a feedback matrix such that the system
is stable Initially we find the matrix such that is Hurwitz.
The LMI: Static State Feedback ProblemEdit
This problem can now be formulated into an LMI as Problem 1:
From the above equation and we have to find K
The problem as we can see is bilinear in
- 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.
where and we find
The Problem 1 is equivalent to Problem 2
If the (A,B) are controllable, We can obtain a controller matrix that stabilizes the system.
A link to the Matlab code for a simple implementation of this problem in the Github repository:
Related LMIs Edit
-  - 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.
-  -Mathworks reference to DC Gain