A small gain theorem for mixed deterministic and stochastic uncertainties and its application to networked stabilization

Abstract

Classical small gain theorem is a powerful tool in robust control theory to deal with the stability of closed-loop systems with uncertainty. When the uncertainty is deterministic norm-bounded nonlinearity or stochastic gain with known mean and variance, the theorem asserts that the robust stability, i.e. the simultaneous stability for all possible uncertainty, is true if and only if the corresponding small gain condition holds. The theorem is very useful in analyzing stability of uncertain systems, however its use is limited to the case when only one type of uncertainty exists. In recent years, the research on networked control systems (NCSs) has drawn considerable attention from the community. In particular, we are interested in the state feedback stabilization of a linear time-invariant (LTI) system over quantized fading channels. This leads to the introduction of both deterministic norm-bounded and stochastic wide-sense-stationary (WSS) multiplicative uncertainties into the closed-loop. Therefore to analyze the stability of the NCS, we need a small gain theorem being able to handle both uncertainty at the same time, which has not yet appeared in the literature. This thesis seeks for establishing such a small gain theorem such that both types of uncertainty are allowed to exist simultaneously in the closed-loop. In particular, we extend the l2-based input-output theory for deterministic systems to stochastic case. We first extend the definitions of l2 signals, spaces and norms to the stochastic case. Then they are used to induce the system stability and norms. The space structure, as well as the important properties of these definitions are discussed. Then the new small gain theorem is derived in the sense of these new definitions of stability and norms. A general sufficient condition is shown first. The condition holds for any given systems and uncertainty satisfying the small gain condition, but the stability is not in the robust sense. The theorems concerning robust stability are then established and the small gain condition is proved to be necessary and sufficient for robust stability. Both unstructured and structured cases are worked out. Finally the theorems are applied to the NCS problem which motivates the research. The quantization is modeled as a nonlinearity with norm bound and the fading is modeled as a multiplicative stochastic noise, which fits the setup of the new small gain theorems exactly. Hence the necessary and sufficient stabilizability conditions of the NCS are derived by using the theorems, both for single-input and multi-input cases. It is shown that the proposed NCS is stabilizable if and only if the channel capacity, as defined in this thesis instead of in classical communication theory, is strictly larger than the topological entropy of the given LTI plant. In summary, this thesis extends the l2 input-output theory to the stochastic setup and establishes a new small gain theorem under such a setup. The application of the theorem is then shown by using it to solve an NCS problem with mixed channel uncertainty. The theorem may be useful to more problems which is also the future research direction of the author.

Date of Award	2014
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology