LMIs in Control/pages/NI-Lemma

LMIs in Control/pages/NI-Lemma Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".

The System

Consider a square continuous time Linear Time invariant system, with the state space realization $(A,B,C,D)$

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

The Data

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

The LMI: LMI for Negative Imaginary Lemma

According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions

• There exists a $P$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$ⁿ,where $P\geq 0$, such that,

${\begin{bmatrix}A^{T}P+PA&PB-A^{T}C^{T}\\B^{T}P-CA&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$

• There exists a $Q$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$ⁿ,where $Q\geq 0$, such that,

${\begin{bmatrix}QA^{T}+AQ&B-QA^{T}C^{T}\\B^{T}-CAQ&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$

Conclusion

The system is strictly negative if det($A$) $\neq$ 0 and either of the above LMIs are feasible with resulting $P>0$ or $Q>0$

Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

Related LMIs

Positive Real Lemma

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.
• Xiong, Junlin & Petersen, Ian & Lanzon, Alexander. (2010). A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems.