# 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

{\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:

## Related LMIs

Hurwitz Stability