# 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 ${\displaystyle (A,B,C,D)}$ and ${\displaystyle \gamma \in \mathbb {R} _{>0}}$

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

## The Data

${\displaystyle A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},D\in \mathbb {R} ^{p\times m}}$

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

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

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

OR

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

## Conclusion

The DC Gain of the continuous-time LTI system, whose state space realization is give by (${\displaystyle A,B,C,D}$), is
${\displaystyle K=D-CA^{-1}B}$

• Upon implementation we can see that the value of ${\displaystyle \gamma }$ obtained from the LMI approach and the value of ${\displaystyle K}$ 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: