LMIs in Control/Matrix and LMI Properties and Tools/Negative Imaginary System DC Constraint

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 $$\bold G_1(s)$$ and an NI transfer matrix $$\bold G_2(s) = \bold C_2(s\bold 1 - \bold A_2)^{-1} \bold B_2 + \bold D_2$$. The condition λ̅ $$(\bold G_1(0)\bold G_2(0) < 1$$ is satisfied if and only if
 * $$\bold S^T(-\bold C_2 \bold A_2^{-1} \bold B_2 + \bold D_2) \bold S < \bold 1$$,

Conclusion
The above equation holds true if and only if $$\bold {S}\bold{S}^T = \bold G_1(0)$$.

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

Related LMIs

 * Negative Imaginary Lemma