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A Polynomial Approach for Analysis and Optimal Control of Switched Nonlinear Systems

Abstract : In this dissertation, we investigate how convex semialgebraic geometry and global polynomial optimization can be used to analyze and to design switched nonlinear systems. To deal with stability analysis of switched nonlinear systems it is shown that the representation of the original switched problem into a continuous polynomial system allows us to use the dissipation inequality for polynomial systems. With this method, and from a theoretical point of view, we provide an alternative way to search for a common Lyapunov function for switched nonlinear systems. The main idea behind the proposed approach is to include in the system analysis the hidden constraints. We need to check the negative semidefinite of V with respect to the constrained set. In order to do that, we use the idea of penalization used in optimization theory with constraints. For that, we use a function λ(x, s), which can be interpreted as a penalization function or a Lagrange multiplier. This idea is based on some results for constrained control systems, where we can use the dissipation inequality concept using storage functions and supply rates. We then extend the results to a more general class of switched systems, those modeled by elementary and nested elementary functions. This class of functions is related to explicit symbolic derivatives, such as exponential, logarithm, power-law, trigonometric, and hyperbolic functions. For this aim, we transform, using a recasting process, the system obtained by the equivalent representation in a system with polynomial form, and then we use the results of the previous section for stability analysis. Besides stability analysis, optimal control problems for switched nonlinear systems are also investigated. We propose an alternative approach for solving effectively the optimal control problem for an autonomous nonlinear switched system based on the Generalized Maximum Principle (GMP). The essence of this method is the transformation of a nonlinear, non-convex optimal control problem, i.e., the switched system, into an equivalent optimal control problem with the linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computing. Consequently, we propose to convexify the state and control variables by means of the method of moments obtaining SDP programs. A generalization to solve the optimal control problem of nonlinear switched systems based on the recasting process is investigated then. Finally, we concentrate on the industrial application obtaining a piecewise-linear approximation of nonlinear cellular growth using orthonormal canonical piecewise linear functions, which is tested by a probing control strategy for the feed rate. We deal with the mammalian cells BHK (Baby Hamster Kidney) in bioreactor in batch, fed-batch, and continuous mode operation. Simulation results show that this piecewise linear approximation is well suited for modeling such nonlinear dynamics
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  • HAL Id : tel-02723110, version 1

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Eduardo Mojica-Nava. A Polynomial Approach for Analysis and Optimal Control of Switched Nonlinear Systems. Automatic. École Centrale de Nantes, 2009. English. ⟨tel-02723110⟩

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