# LMIs in Control/pages/SSFP

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

## The System

Consider a continuous time Linear Time invariant system

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

## The Data

${\displaystyle A,B}$  are known matrices

## The Optimization Problem

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

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

and

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

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

## The LMI: Static State Feedback Problem

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

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

From the above equation ${\displaystyle X>0}$  and we have to find K

The problem as we can see is bilinear in ${\displaystyle K,X}$

• 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:

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

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

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:

## Related LMIs

Hurwitz Stability