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 $\gamma \in \mathbb {R} _{>0}$

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

The Data

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

The LMI: LMI for DC Gain of a Transfer Matrix

The transfer matrix is given by$G(s)=C(sI-A)^{-1}B+D$
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:

https://github.com/yashgvd/LMI_wikibooks

External Links

• [1] - LMI in Control Systems Analysis, Design and Applications
• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• [2] -Mathworks reference to DC Gain