User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Negative Imaginary 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{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: 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 $$P$$ ∈ $$\begin{align}\mathbb{S}\end{align}$$n ,where $$P \geq 0$$, such that,

\begin{bmatrix} A^TP + PA & PB-A^TC^T \\ B^TP-CA & -(CB+B^TC^T) \end{bmatrix}$$$$\begin{align} \leq 0\end{align}$$


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

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

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