DocumentCode :
2999638
Title :
On necessary and sufficient conditions for stability for n-input n-output convolution feedback systems with a finite number of unstable poles
Author :
Callier, F.M. ; Desoer, C.A.
Author_Institution :
University of California, Berkeley, California
fYear :
1972
fDate :
13-15 Dec. 1972
Firstpage :
127
Lastpage :
127
Abstract :
We consider n-input n-output convolution feedback systems characterized by y = G * e and e = u - y, where the open-loop transfer function ?? contains a finite number of unstable poles and the expansion ??a(s) + ??i=0 ?? Gi exp(-sti). The latter is such that, i) ??a is the Laplace transform of L1 functions, ii) t0 = 0, iii) the delay-times ti, for i = 1, 2, ..., are positive and iv) the coefficient matrices Gi form an absolutely convergent series, i.e. ??i=0 ?? |Gi| < ??. Thus the subsystem represented by G is an open-loop unstable distributed parameter system. One theorem is obtained. It gives necessary and sufficient conditions for stability. A basic device is the following: the sum of the principal parts of the Laurent expansions of ?? at the unstable poles is factored as a ratio of two right coprime polynomial matrices. There are two necessary and sufficient conditions, the first is the usual infimum one, i.e. infRe s ?? 0 |det[I+??(s)]| > 0, and the second is required to prevent the closed-loop transfer function from being unbounded in some small neighborhood of each open-loop unstable pole. The latter condition has a nice interpretation in terms of McMillan degree theory. The modification of the theorem for the discrete-time case is immediate. The same is true if, instead of unity feedback, constant nonunity feedback represented by a nonsingular matrix is used. This allows us, through the use of the small gain theorem or the contraction mapping theorem, to consider non-linear systems, derived from the original one by introducing a non-linear time-varying gain in the forward loop.
Keywords :
Convolution; Delay; Distributed parameter systems; Feedback; Laplace equations; Polynomials; Stability; Sufficient conditions; Time varying systems; Transfer functions;
fLanguage :
English
Publisher :
ieee
Conference_Titel :
Decision and Control, 1972 and 11th Symposium on Adaptive Processes. Proceedings of the 1972 IEEE Conference on
Conference_Location :
New Orleans, Louisiana, USA
Type :
conf
DOI :
10.1109/CDC.1972.268961
Filename :
4044884
Link To Document :
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