DocumentCode :
1183122
Title :
A maximum modulus theorem for spectral radius and absolutely stable amplifiers
Author :
Youla, D.C.
Volume :
27
Issue :
12
fYear :
1980
fDate :
12/1/1980 12:00:00 AM
Firstpage :
1274
Lastpage :
1276
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.
Keywords :
Automatic testing; Integrated circuit fabrication; MOS integrated circuits; Pattern recognition; Circuits and systems; Matrices;
fLanguage :
English
Journal_Title :
Circuits and Systems, IEEE Transactions on
Publisher :
ieee
ISSN :
0098-4094
Type :
jour
DOI :
10.1109/TCS.1980.1084757
Filename :
1084757
Link To Document :
بازگشت