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Algebraic design of anisochronic controller for time delay systems. (English) Zbl 1044.93058

Stability conditions of linear systems with divergent argument are considered, where the characteristic quasipolynomial is \[ H(s)= s^n+ \sum^{n-1}_{i=0}\, \sum^h_{j=1} a_i s^i\exp(-\upsilon_{ij}s).\tag{1} \] Conditions analogous to Mikhajlov’s criterion for polynomials are given.
Proposition 1 (Stability criterion). A retarded quasipolynomial \(H(s)\) of the form (1) is stable if and only if
a) \(H(0)> 0\), a real positive number, and
b) the argument increment limit for \(\omega\to\infty\) is \[ \lim_{\omega\to\infty}\arg H(s)_{s= j\omega}= (j\omega)^n+ \sum^{n-1}_{i=0}\,\sum^h_{j=1} a_i(j\omega)^i \exp(-j\upsilon_{ij}\omega)= n{\pi\over 2}. \] Different particular examples of such quasipolynomials are given, too.

MSC:

93D20 Asymptotic stability in control theory
93C80 Frequency-response methods in control theory
93C23 Control/observation systems governed by functional-differential equations
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