In this paper, we state the positive real lemma and the strictly positive real lemma (KYP lemma) for non-minimal realization systems. First we show the positive real lemma for stabilizable and observable systems under only the constraint on the regularity of the systems, by using the generalized algebraic Riccati equation. Moreover we show that the solution of the Lyapunov equation in the posit...

Stability properties (uniform stability/uniform asymptotic stability) of nonlinear time-varying systems are explored using positive semi-definite time-varying Lyapunov candidates whose derivative along trajectories is either non-positive or negative semi-definite. Once these positive semi-definite time-varying Lyapunov candidates are available, conditional stability properties on some specific ...

Let Q(z, w) = − ∏n k=1[(z − ak)(w̄ − āk) − R]. The main result of the paper states that in the case when the nodes aj are situated at the vertices of a regular n-gon inscribed in the unit circle, the matrix Q(ai, aj) is positive definite if and only if R < ρn, where z = 2ρ 2 n − 1 is the smallest 6= −1 zero of the Jacobi polynomial Pn−2ν,−1 ν (z), ν = [n/2].