مرکز منطقه ای اطلاع رساني علوم و فناوري - A maximum modulus theorem for spectral radius and absolutely stable amplifiers

Abstract :

Stability problems invariably impose constraints on eigenvalues and the spectral radius, Spr $A$ , of a matrix $A$ emerges as an important concept. Unfortunately, the spectral radius of a matrix does not qualify as a norm. Nevertheless, with the aid of the Lyapunov lemma we prove the following: let $A(z) \\equiv A(z_1, z_2, \\cdots , z_k)$ denote a rational matrix in the $k$ independent variables $z_{i}, i \\rightarrow k$ , which is analytic in the closed unit polydisc, $\\bar{D}^k(1) \\equiv {z: |z_1| \\leq 1,|z_2| \\leq 1, \\cdots ,|z_k| \\leq 1}$ . Let $\\partial \\bar{D}^k(1) \\equiv {z:|z_1|=1,|z_2|=1,\\cdots , |z_k|=1}$ denote the distinguished boundary of $\\bar{D}^{k}(1)$ . Then, Spr $A(z)\\leq 1$ for all $z\\in|\\bar{D}^k(1)$ if and only if Spr $A(z) \\leq 1$ for all $z \\in \\partial \\bar{D}^k(1)$ . In addition to pointing out several obvious generalizations, we also employ the above theorem to give a rigorous proof of the long accepted conjecture that the absolute stability of a $k$ -port amplifier can always be tested by closing its $k$ -ports on $k$ uncoupled pure reactances. Lastly, we present an entirely new justification of the well-known fact that a reciprocal $k$ -port amplifier is absolutely stable if and only if it is strictly passive.