Tractable stability analysis for systems containing repeated scalar slope‐restricted nonlinearities

Abstract

SUMMARYThis paper proposes an LMI-based approach for studying the stability of feedback interconnections of a finite dimensional LTI system and a nonlinear element that consists of several identical scalar nonlinearities that have restrictions on their sector and slope. The results are based on the integral quadratic constraint stability analysis framework and other recent results that give a sharp characterisation of stability multipliers for monotone, repeated scalar nonlinearities. Several examples show the effectiveness of the proposed approach; the lack of conservatism in the results is noteworthy.

Zames-Falb Multipliers: don't panic

“…The generalisations are not trivial and require further assumptions on the form of the nonlinearity to be made, but they do have IQC interpretations in much the same way as their scalar-valued counterparts [8]. The search for multipliers is somewhat more involved however: the generalisation of [35] to the multivariable case [37] is effectively too conservative and too computationally demanding 3 . The approaches given in [10,30] both have merit but require choices to be made which are not entirely systematic.…”

Full‐block multipliers for repeated, slope‐restricted scalar nonlinearities

“…Example For ν =0, the Zames‐Falb multiplier is static (and independent from ρ ). In (, Lemma 3), a similar multiplier (with the (unnecessary) additional constraint G = G ⊤ ) is, slightly misleadingly, introduced as a less conservative substitute for a circle criterion multiplier in certain cases. As a consequence of our exposition (see also ), this multiplier neither requires a separate proof for its validity nor does it serve as a replacement for circle criterion multipliers.…”

“…In a series of papers (see, e.g., [43,48,[59][60][61][62][63][64]), various authors tried to reduce several of the non-intrinsic drawbacks of this method, for example, by combining the obtained Zames-Falb multipliers with those corresponding to the Popov criterion or by allowing for higher order approximations. Still, for k-fold repeated nonlinearities, numerical tractability suffers because a non-convex search must be performed for a k-dimensional parameter [25,44,65]; also, the use of a common Lyapunov function for all L 1 -constraints and stability LMIs might cause additional conservatism for increasing values of k. Our approach avoids these troubles and is guaranteed to achieve no worse results, because it freely combines multiple pole Zames-Falb multipliers (each based on asymptotically exact parameterizations) with those from other criteria for repeated nonlinearities.…”

“…As one of the distinguishing features of the previously proposed parametrization, if compared for example with the one in (extended to the full‐block case in ), even for a fixed choice of ρ <0, it can be proven to be asymptotically exact.…”

Analysis of Systems With Slope Restricted Nonlinearities Using Externally Positive Zames–Falb Multipliers

“…As shown in [5], [6], this approach can be competitive, but sometimes it is conservative and numerically unreliable. The extensions of the Zames-Falb analysis tools to multivariable systems have been discussed in [23], [15] and tools for the search over the class of MIMO Zames-Falb multipliers have been proposed in [9], [29], [10]. It is probably fair to say that a complete solution to the problem originally considered by O'Shea some fifty years ago -that of finding a multiplier rendering a system with a slope restricted nonlinearity stable -is not yet available.…”