Robust control and estimation for positive systems

Resumen

This thesis firstly presents systematic procedures to design robust constrained controllers for uncertain delayed positive systems. These approaches are based on resolution of optimization problems, expressed as Linear Programming or Linear Matrix Inequalities. Thus, they can be efficiently solved by available LP or LMI solvers. Furthermore, the proposed methodologies are extended to constrained design problems. These techniques have been validated using examples from the literature, giving satisfactory results. More precisely, the main contributions of the first part of this work are the proposal of an approach to solve the problem of designing state feedback controllers for positive delayed systems, which can be subject to interval uncertainties. We have also presented the robust ¿-stability notion that guarantees a specified decay rate in the presence of possible uncertainties on the system. Moreover, necessary and sufficient conditions are provided for the stabilization of uncertain positive systems with multiple delays by means of state feedback laws, that can be selected to have memory or not. Finally, we have demonstrated how the proposed approach can also be extended to delayed positive systems with constrained controls. Secondly, this thesis presented new results in the field of robust state estimation through the design of interval observers. These interval observers belong to a specific class of estimators called guaranteed set estimation methods. The strength of these methods is that they provide a region of the state space where the unknown variables are sure to lie. Our contribution is mainly focused on the estimation of the state variables of uncertain systems, which can be affected by noisy measurements. In this thesis, the theoretical developments improving interval estimations have been assessed, demonstrating their efficiency through various examples with significant uncertainties. More precisely, we first have considered the design of positive interval observers for uncertain linear positive systems. It was shown that the existence of robust interval positive observers can be expressed in terms of LP conditions, so they can be easily computed. In addition, we have shown that tight interval positive observers can be derived by minimizing an adequate objective function. Finally, we have presented a new approach to design interval observers for a class of nonlinear positive systems. These results are illustrated by examples from biology. In addition, we have presented a set membership approach dedicated to robust estimation of general uncertain continuous-time systems. We have considered a kind of interval observers based on the notion of dilatation functions, which leads to guaranteed interval estimates. The advantage of this approach is its robustness with respect to both model parameter uncertainties and noise measurements. Moreover, all the proposed conditions are also checkable conditions in terms of Linear Programming. Illustrative examples have shown the performance of the proposed approach. In addition, we have treated the problem of generating interval observers for a class of monotone systems. Finally, we have proposed some applications: routers in computer networks, electrical systems, biological systems, and chaotic systems.