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UBC Theses and Dissertations

On measuring closed-loop nonlinearity : a topological approach using the v-gap metric Tan, Guan Tien

Abstract

All chemical processes are inherently nonlinear. However, a nonlinear process does not necessarily require a nonlinear control since the feedback control itself has a certain degree of linearizing effect. This leads to an interesting and often subtle question: "When is a linear controller sufficient to control a nonlinear process?". This thesis aims to answer such a question and develop a systematic approach to quantify closed-loop nonlinearity from a controller design perspective. An immediate consequence arising from the quest of the answer for the above question leads to the two major contributions of this thesis. Firstly, a novel way to quantify closed-loop nonlinearity and a practical computational algorithm are developed. Secondly, the nonlinearity measure presented in this thesis can be used as an effective decision making tool when dealing with the question of choosing an appropriate strategy for a class of nonlinear plants that can be recast into a quasi-linear parameter varying (quasi-LPV) representation. An additional contribution of this thesis is the development of a pictorial approach that provides a better insight and understanding to the gap metric theory. In this thesis, the v-gap metric arising from the graph topology is used to quantify the "distance" of a nonlinear process and its linearized model in a closed-loop fashion. Since the v-gap metric is developed for linear time invariant (LTI) systems, the nonlinear system is recast into a quasi-LPV form. As a result, the largest v-gap induced by the closed-loop nonlinearity together with a time variation penalty (i.e. the developed nonlinearity measure) can be obtained. The theoretical optimal stability margin is subsequently computed and compared with the largest v-gap to check the severity of the closed-loop nonlinearity. If the developed nonlinearity measure is smaller than the theoretical optimal stability margin, then the closed-loop nonlinearity is manageable by an optimal single linear controller designed for the linear system. Otherwise, the optimal linear controller that results in a satisfactory robust stability in the linear system might show poor robust stability (or even destabilizes) the nonlinear system. Under this circumstance, a nonlinear control strategy might be needed. Since such an optimal controller might not be attainable in practice, a sub-optimal controller can be obtained via the [symbol is not included] loop shaping design procedure. Finally, the developed nonlinearity measure is applied in three different examples, a continuous stirred tank reactor control problem, an inverted cone tank control problem and a fictitious nonlinear plant control problem. Simulation results confirm the applicability and the reliability of the developed nonlinearity measure in tackling practical engineering control problems.

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