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FREQUENCY DOMAIN ANALYSIS OF FEEDBACK INTERCONNECTIONS OF STABLE SYSTEMS

Maya Gonzalez, Martin

[Thesis]. Manchester, UK: The University of Manchester; 2015.

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Abstract

The study of non-linear input-output maps can be summarized by three concepts: Gain, Positivity and Dissipativity. However, in order to make efficient use of these theorems it is necessary to use loop transformations and weightings, or so called ”multipliers”.The first problem this thesis studies is the feedback interconnection of a Linear Time Invariant system with a memoryless bounded and monotone non-linearity, or so called Absolute Stability problem, for which the test for stability is equivalent to show the existence of a Zames-Falb multiplier. The main advantage of this approach is that Zames–Falb multipliers can be specialized to recover important tools such as Circle criterion and the Popov criterion.Albeit Zames-Falb multipliers are an efficient way of describing non-linearities in frequency domain, the Fourier transform of the multiplier does not preserve the L1 norm. This problem has been addressed by two paradigms: mathematically complex multipliers with exact L1 norm and multipliers with mathematically tractable frequency domain properties but approximate L1 norm. However, this thesis exposes a third factor that leads to conservative results: causality of Zames-Falb multipliers. This thesis exposes the consequences of narrowing the search Zames-Falb multipliers to causal multipliers, and motivated by this argument, introduces an anticausal complementary method for the causal multiplier synthesis in [1].The second subject of this thesis is the feedback interconnection of two bounded systems. The interconnection of two arbitrary systems has been a well understood problem from the point of view of Dissipativity and Passivity. Nonetheless, frequency domain analysis is largely restricted for passive systems by the need of canonically factorizable multipliers, while Dissipativity mostly exploits constant multipliers.This thesis uses IQC to show the stability of the feedback interconnection of two non-linear systems by introducing an equivalent representation of the IQC Theorem, and then studies formally the conditions that the IQC multipliers need. The result of this analysis is then compared with Passivity and Dissipativity by a series of corollaries.