LMIs in Control/pages/SSFP

LMIs in Control/pages/SSFP
We are attempting to stabilizing The Static State-Feedback Problem

The SystemEdit

Consider a continuous time Linear Time invariant system

 

The DataEdit

  are known matrices

The Optimization ProblemEdit

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

 

and

 


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.

Problem 2:

 

where   and we find  
 

The Problem 1 is equivalent to Problem 2

ConclusionEdit

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

ImplementationEdit

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

https://github.com/yashgvd/ygovada

Related LMIs Edit

Hurwitz Stability

External LinksEdit

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