Absolute stability in linear delay-differential systems: ill-posedness and robustness

Abstract

of f~~ ( s ) in the closed right half plane {Re ( 8 ) 2 0). The system ($1 is said to be stable independent of delay (i.0.d.) if it is asymptotically stable for every /I E H . In this case we say that the system is Example 2.1: Consider the delay-differential system (*) s'(t) = James Louisell absolutely stable. -x ( t ) -~( t h ) -0.5:r(t -2h). the system used by Datko [41Abs~cl-Tbeaulhorconsidersmatrlxdelay-di~eranUaIsyste~which as an example,the while is simply the delay operator acting on scalar functions. The characteristic function is f h ( 8 ) = 8 + 1 + e-"' + 0.5~-'*'', and one can easily show that for each h E H, the function flE ( 8 ) has no zeros in (Re(8) 2 0). Thus the system (*) is stable i.0.d.Dalko,s example one considers the systems ( * a ) ,rr(t) = -r ( t ) -z ( t -/I.) -0 . 5 z ( t -(~+ E ) / I ) ,where E > 0 is a perturbation ape polynomial In several delay operators. Using a neceasary and sulRclent cdterion for stability independent or delay, or absolute anlhor shows thatsystemstabilltyforaUv~lupaofthedelay vector lyingin a sector wlll imply absolute stability. We then show that absolute stability is Ill-ped with respect to arbitrarily small perturbations Of the delay ratios if a cerlaln extended delay.diffen?ntlal system which Is formed fmm the original Is not also absolutely stable.

Dynamic Controllers Which Use Difference Approximates of Output Derivatives and Their Practical Stability

“…the closed-loop system becomes of neutral type. The instability caused by such a small delay in the loop has been studied for a various kind of systems (see e.g., Barman et al 1973;Datko 1988;Louisell 1995). Owing to the improved tool of Logemann et al (1996), we know that the neutral system is practically stable if à is Schur.…”

Stability Preserving Transition From Derivative Feedback to Its Difference Counterparts

“…We first give a simple example in which the derivative feedback stabilizes, whereas its difference approximation can not for all small Ì . Such ill-posedness may be compared to known instability caused by introducing small time-delays in the feedback loop (see e.g., Datko 1988;Louisell 1995;Longemann and Townley 1996).…”

Characterization of Delay‐Independent Stability and Delay Interference Phenomena

“…In [15] it is shown that delay-independent stability is equivalent to asymptotic stability for all delay values lying in a nontrivial sector in the delayparameter space, and to the robustness of stability of a ray, consisting of commensurate delay values, w.r.t. small perturbations of the direction (see [15] for precise formulations). Note that the latter two statements imply the existence of a stable ray, which is not subjected to the interference phenomenon.…”

“…A further example (scalar system including two delays) of delay interference can be found in [5]. This is also discussed in [15] and [22], where some characterizations regarding interference are given (a frequency-sweeping test combined with a matrix pencil condition in [22], a sector characterization in [15]). As we shall discuss at the end of Section 5, these results appear as corollaries of the general theory developed throughout the paper.…”

“…Due to the fragility of delay-independent stability properties (note that they involve a non-compact set of delay values), these issues are nontrivial and deserve special attention, as we shall see. They have, however, barely been treated in the literature, excepting some contributions on the interference phenomenon and related topics [16,17,5,15,22].…”