LMIs in Control/Matrix and LMI Properties and Tools/Generalized Negative Imaginary Lemma

Introduction

These 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.

The System

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

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

The Data

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

The LMI

Consider an NI transfer matrix G₁(s) and an SNI transfer matrix G₂(s) = C₂(s1 - A₂)^-1 B₂ + D₂. The condition λ̅ (G₁(0)G₂(0) < 1 is satisfied if and only if

S^T(-C₂A₂^-1B₂ + D₂)S < 1,

Conclusion

The above equation holds true if and only if S S^T = G₁(0).

Implementation

This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.

Related LMIs

• Negative Imaginary Lemma