Adaptation de la methode multi-unites a l'optimisation sous contraintes en presence d'unites non identiques

Real-time optimization methods seek to keep a given process at its optimum operating conditions despite plant variations and external disturbances. This is achieved by identifying these variations using the available measurements, and thereby reacting to them. Among the different techniques available for real-time optimization, extremum-seeking methods are those where optimization is recast as a problem of controlling the gradient to zero.The multi-unit optimization is a recently proposed extremum-seeking method which provides faster convergence for slow dynamic processes. This method requires the presence of multiple identical units, with each of them operated at input values that differ by a pre-determined constant offset. The gradient is then estimated by finite differences between the outputs of the units. As the perturbation is not in the temporal domain, the convergence is faster.The assumption of having identical units is indeed very strong and may not be realizable in practice. This thesis first studies the effects of the differences between the static characteristics on the stability and convergence of the standard multi-unit optimization scheme. It is shown that the choice of the offset parameter is crucial to stability, while the equilibrium point could be quite far away from the real optimum. To avoid such a situation, correctors which compensate for the differences between the units have been proposed. Two types of adaptation are analyzed: a sequential approach where the multi-unit adaptation and the correction are done separately and a simultaneous approach were the two are performed together. Local stability of both approaches has been established. It was shown that the differences can indeed be corrected and both units would converge to their respective optima.Extremum-seeking methods have traditionally been used for unconstrained problems. Typically, constrained problems have been transformed to unconstrained ones using barrier functions. Such an approach results in a loss of performance due to a gap between the equilibrium point and the set of active constraints. On the other hand, control of the gradient, projected on the set active constraints, would allow the equilibrium point to be directly on the active constraints. However, the main challenge is in the identification of the set of active constraints in a continuous framework. Existing methods have a problem of "jamming", i.e., being stuck at a non-optimal set of constraints. A new jamming-free switching logic is developed and a rigorous proof is provided to show that the system in fact converges to the optimum. The multi-unit optimization method is then coupled with the idea of gradient projection on the set of active constraints.The principle difference between the various extremum-seeking methods lie in the way the gradient is estimated. Most of these schemes involve injecting a periodic temporal perturbation signal and several time-scale separations are necessary to isolate the effects of the system dynamics on the estimated gradient. Time-scale separation will not be an issue for processes with fast responses, e.g. electrical or mechanical systems, though, for slower processes such as the chemical or biological ones, the convergence time could be prohibitive.The last part of this thesis contains the experimental verification of the multi-unit optimization method for the maximum power point tracking of microbial fuel cells. Microbial fuel cells produce electricity from waste water though an electro-chemical reaction with bacteria acting as the catalyst. The sequential adaptation technique presented in this thesis is used to correct the difference between the cells. The results from multi-unit optimization are compared with two other traditional techniques that involve temporal perturbation, i.e., the perturbation/observation method and the steepest descent method. Also, different disturbances are introduced and the ability to track the optimum is observed. The experimental results confirm the main advantage of the multi-unit optimization method, i.e., a faster convergence to the optimum than methods using temporal perturbation. It also verifies the fact that differences between the units can be corrected leading each of them to their respective optima.