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 $\gamma \in {ℝ}_{>0}$

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

The Data

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

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.

$\left[\begin{array}{cc}\gamma I& -C{A}^{-1}B+D\\ (-C{A}^{-1}B+D{)}^{\prime }& \gamma I\end{array}\right]$$\begin{array}{c}>0\end{array}$

OR

$\left[\begin{array}{cc}\gamma I& -B^T{A}^{-T}{C}^T+DT\\ (-B^T{A}^{-T}{C}^T+DT{)}^{\prime }& \gamma I\end{array}\right]$$\begin{array}{c}>0\end{array}$

Conclusion

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

• 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