# LMIs in Control/pages/DCGain

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LMIs in Control/pages/DCGain

**The continuous-time DC gain is the transfer function value at the frequency s = 0.**

**The System**[edit]

Consider a square continuous time Linear Time invariant system, with the state space realization and **∈**_{>0}

**The Data**[edit]

∈ ** ^{nxn}**, ∈

**, ∈**

^{nxm}**, ∈**

^{pxn}

^{pxm}

** The LMI: LMI for DC Gain of a Transfer Matrix**[edit]

The transfer matrix is given by

The DC Gain of the system is strictly less than if the following LMIs are satisfied.

**OR**

**Conclusion**[edit]

The DC Gain of the continuous-time LTI system, whose state space realization
is give by (), is

**Upon implementation we can see that the value of obtained from the LMI approach and the value of obtained from the above formula are the same**

**Implementation**[edit]

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**[edit]

- [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