LMIs in Control/KYP Lemmas/Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

The System


\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0 \end{align}$$ where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

The Data
The matrices $$ A,B,C,D $$ are known.

The LMI: The Positive Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad G \; \text{is passive, i.e.} \; \left\langle u, Gu \right\rangle _{L_2} \ge 0 \; (\hat G(s)+\hat G(s)^*\ge 0) $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB-C^T \\B^TX-C & -D^T-D\end{bmatrix}\le 0

$$

Conclusion:
The Positive Real Lemma can be used to determine if the system $$G$$ is passive. Note from the (1,1) block of the LMI we know that $$A$$ is Hurwitz.

Implementation
A link to CodeOcean or other online implementation of the LMI (in progress)

Related LMIs
KYP Lemma (Bounded Real Lemma)