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

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}$$$$\begin{align} > 0\end{align}$$ OR

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

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