# LMIs in Control/pages/DCGain

LMIs in Control/pages/DCGain
The continuous-time DC gain is the transfer function value at the frequency s = 0.

## The System

Consider a square continuous time Linear Time invariant system, with the state space realization $(A,B,C,D)$ and $\gamma$ {\begin{aligned}\mathbb {R} \end{aligned}} >0

{\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}} ## The Data

$A$ {\begin{aligned}\mathbb {R} \end{aligned}} nxn, $B$ {\begin{aligned}\mathbb {R} \end{aligned}} nxm , $C$ {\begin{aligned}\mathbb {R} \end{aligned}} pxn , $D$ {\begin{aligned}\mathbb {R} \end{aligned}} pxm

## The LMI: LMI for DC Gain of a Transfer Matrix

The transfer matrix is given by$G(s)=C(sI-A)^{-1}B+D$ The DC Gain of the system is strictly less than $\gamma$ if the following LMIs are satisfied.

${\begin{bmatrix}\gamma I&-CA^{-1}B+D\\(-CA^{-1}B+D)'&\gamma I\end{bmatrix}}$ {\begin{aligned}>0\end{aligned}} OR

${\begin{bmatrix}\gamma I&-B^{T}A^{-T}C^{T}+DT\\(-B^{T}A^{-T}C^{T}+DT)'&\gamma I\end{bmatrix}}$ {\begin{aligned}>0\end{aligned}} ## Conclusion

The DC Gain of the continuous-time LTI system, whose state space realization is give by ($A,B,C,D$ ), is
$K=D-CA^{-1}B$ • Upon implementation we can see that the value of $\gamma$ obtained from the LMI approach and the value of $K$ obtained from the above formula are the same

## Implementation

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