ROBUST STATE FEEDBACK CONTROL OF UNCERTAIN POLYNOMIAL DISCRETE-TIME SYSTEMS: AN INTEGRAL ACTION APPROACH

his paper examines the problem of designing a nonlinear state feedback controller for polynomial discrete-time systems with parametric uncertainty. In general, this is a challenging controller design problem due to the fact that the relation between Lyapunov function and the control input is not jointly convex; hence, this problem cannot be solved by a semidenite programming (SDP). In this paper, a novel approach is proposed, where an integral action is incorporated into the controller design to convexify the controller design problem of polynomial discrete-time systems. Based on the sum of squares (SOS) approach, sufficient conditions for the existence of a nonlinear state feedback controller for polynomial discrete-time systems are given in terms of solvability of polynomial matrix inequalities, which can be solved by the recently developed SOS solver. Numerical examples are provided to demonstrate the validity of this integral action approach

Expensive control of long-time averages using sum of squares and Its application to a laminar wake flow

The paper presents a nonlinear state-feedback con- trol design approach for long-time average cost control, where the control effort is assumed to be expensive. The approach is based on sum-of-squares and semi-definite programming techniques. It is applicable to dynamical systems whose right-hand side is a polynomial function in the state variables and the controls. The key idea, first described but not implemented in (Chernyshenko et al. Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problem of optimizing a cost function involving long-time averages is replaced by an optimization of the upper bound of the same average. As such, controller design requires the simultaneous optimization of both the control law and a tunable function, similar to a Lyapunov function. The present paper introduces a method resolving the well-known inherent non-convexity of this kind of optimization. The method is based on the formal assumption that the control is expensive, from which it follows that the optimal control is small. The resulting asymptotic optimization problems are convex. The derivation of all the polynomial coefficients in the controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The proposed approach is applied to the problem of designing a full-information feedback controller that mitigates vortex shedding in the wake of a circular cylinder in the laminar regime via rotary oscillations. Control results on a reduced-order model of the actuated wake and in direct numerical simulation are reported

Contributions to nonlinear system modelling and controller synthesis via convex structures

Esta tesis discute diferentes metodologías de modelado para extraer mejores prestaciones o resultados de estabilidad que aquéllas que el modelado convencional basado en sector no-lineal de sistemas Takagi-Sugeno (también denominados cuasi-LPV) es capaz de producir. En efecto, incluso si las LMIs pueden probar distintas cotas de prestaciones o márgenes de estabilidad (tasa de decaimiento, $\mathcal H_\infty$, etc.) para sistemas politópicos, es bien conocido que las prestaciones probadas dependen del modelo elegido y, dado un sistema no-lineal, dicho modelo politópico no es único. Por tanto, se presentan exploraciones hacia cómo obtener el modelo que es menos perjudicial para la medida de prestaciones elegida. Como una última contribución, mejores resultados son obtenidos mediante la extensión del modelado politópico Takagi-Sugeno a un marco de inclusiones en diferencias cuasi-convexas con planificación de ganancia. En efecto, una versión sin planificación de ganancia fue propuesta por un equipo de investigadores de la Universidad de Sevilla (Fiaccini, Álamo, Camacho) para generalizar el modelado politópico, y esta tesis propone una version aún más general de algunos de dichos resultados que incorpora planificación de ganancia.This thesis discusses different modelling methodologies to eke out best performance/stability results than conventional sector-nonlinearity Takagi-Sugeno (also known as quasi-LPV) systems modelling techniques are able to yield. Indeed, even if LMIs can prove various performance and stability bounds (decay rate, $\mathcal H_\infty$, etc.) for polytopic systems, it is well known that the proven performance depends on the chosen model and, given a nonlinear dynamic systems, the polytopic embeddings available for it are not unique. Thus, explorations on how to obtain the model which is less deletereous for performance are presented. As a last contribution, extending the polytopic Takagi-Sugeno setup to a gain-scheduled quasi-convex difference inclusion framework allows to improve the results over the polytopic models. Indeed, the non-scheduled convex difference inclusion framework was proposed by a research team in University of Seville (Fiacchini, Alamo, Camacho) as a generalised modelling methodology which included the polytopic one; this thesis poses a further generalised gain-scheduled version of some of these results.Aquesta tesi discuteix diferents metodologies de modelatge per extreure millors prestacions o resultats d'estabilitat que aquelles que el modelatge convencional basat en sector no-lineal de sistemes Takagi-Sugeno (també anomenats quasi-LPV) és capaç de produir. En efecte, fins i tot si les LMIs poden provar diferents cotes de prestacions o marges d'estabilitat (taxa de decaïment, $\mathcal H_\infty$, etc.) per a sistemes politòpics, és ben conegut que les prestacions provades depenen del model triat i, donat un sistema no-lineal, el dit model politòpic no és únic. Per tant, es presenten exploracions cap a com obtenir el model que és menys perjudicial per a la mesura de prestacions triada. Com una darrera contribució, millors resultats són obtinguts mitjançant l'extensió del modelatge politòpic Takagi-Sugeno a un marc d'inclusions en diferències quasi-convexes amb planificació de guany. En efecte, una versió sense planificació de guany va ser proposada per un equip d'investigadors de la Universitat de Sevilla (Fiaccini, Álamo, Camacho) per a generalitzar el modelatge politòpic, i aquesta tesi proposa una versió més general d'alguns d'aquests resultats que incorpora planificació de guany.Robles Ruiz, R. (2018). Contributions to nonlinear system modelling and controller synthesis via convex structures [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/100848TESI

Stabilization of cascaded nonlinear systems under sampling and delays

Over the last decades, the methodologies of dynamical systems and control theory have been playing an increasingly relevant role in a lot of situations of practical interest. Though, a lot of theoretical problem still remain unsolved. Among all, the ones concerning stability and stabilization are of paramount importance. In order to stabilize a physical (or not) system, it is necessary to acquire and interpret heterogeneous information on its behavior in order to correctly intervene on it. In general, those information are not available through a continuous flow but are provided in a synchronous or asynchronous way. This issue has to be unavoidably taken into account for the design of the control action. In a very natural way, all those heterogeneities define an hybrid system characterized by both continuous and discrete dynamics. This thesis is contextualized in this framework and aimed at proposing new methodologies for the stabilization of sampled-data nonlinear systems with focus toward the stabilization of cascade dynamics. In doing so, we shall propose a small number of tools for constructing sampled-data feedback laws stabilizing the origin of sampled-data nonlinear systems admitting cascade interconnection representations. To this end, we shall investigate on the effect of sampling on the properties of the continuous-time system while enhancing design procedures requiring no extra assumptions over the sampled-data equivalent model. Finally, we shall show the way sampling positively affects nonlinear retarded dynamics affected by a fixed and known time-delay over the input signal by enforcing on the implicit cascade representation the sampling process induces onto the retarded system

Applications of robust optimal control to decision making in the presence of uncertainty

This thesis is concerned with robustness of decision making in financial economics. Feedback control models developed in engineering are applied to three separate though linked problems in order to examine the role and impact of robustness in the creation and application of decision rules. Three problems are examined using robust optimal control techniques to evaluate the impact of robustness and stability in financial economic models. The first problem examines the use of linear models of robust optimal control in the pricing of castastrophe based derivatives and finds its relative performance to be superior to the popular jump diffusion and stochastic volatility models in the pricing of these emerging instruments. The novelty of the approach arises from the examination of the impact of robustness and stability of the pricing solution. The second problem involves robustness and stability of hedging. An alternative method of creating hedging rules is developed. The method is based on robust control Lyapunov functions that are simple, robust and stable in operation, yet in practice are not so conservative that they eliminate all trading gains. The third problem involves the development of robust control policies for managing risk, using non-linear robust optimal control techniques to provide clear evidence of superior performance of robust models when compared with existing VAR and EVT approaches to risk management. The novelty in the approach arises from the development of a simple and powerful risk management metric