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

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

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
 * ST(-C2A2-1B2 + D2)S < 1,

Conclusion
The above equation holds true if and only if S ST = 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