Global stabilization of a Korteweg-de Vries equation with saturating distributed control

This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different types of saturated controls are considered. The well-posedness is proven applying a Banach fixed point theorem, using some estimates of this equation and some properties of the saturation function. The proof of the asymptotic stability of the closed-loop system is separated in two cases: i) when the control acts on all the domain, a Lyapunov function together with a sector condition describing the saturating input is used to conclude on the stability, ii) when the control is localized, we argue by contradiction. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation. 1. Introduction. In recent decades, a great effort has been made to take into account input saturations in control designs (see e.g [39], [15] or more recently [17]). In most applications, actuators are limited due to some physical constraints and the control input has to be bounded. Neglecting the amplitude actuator limitation can be source of undesirable and catastrophic behaviors for the closed-loop system. The standard method to analyze the stability with such nonlinear controls follows a two steps design. First the design is carried out without taking into account the saturation. In a second step, a nonlinear analysis of the closed-loop system is made when adding the saturation. In this way, we often get local stabilization results. Tackling this particular nonlinearity in the case of finite dimensional systems is already a difficult problem. However, nowadays, numerous techniques are available (see e.g. [39, 41, 37]) and such systems can be analyzed with an appropriate Lyapunov function and a sector condition of the saturation map, as introduced in [39]. In the literature, there are few papers studying this topic in the infinite dimensional case. Among them, we can cite [18], [29], where a wave equation equipped with a saturated distributed actuator is studied, and [12], where a coupled PDE/ODE system modeling a switched power converter with a transmission line is considered. Due to some restrictions on the system, a saturated feedback has to be designed in the latter paper. There exist also some papers using the nonlinear semigroup theory and focusing on abstract systems ([20],[34],[36]). Let us note that in [36], [34] and [20], the study of a priori bounded controller is tackled using abstract nonlinear theory. To be more specific, for bounded ([36],[34]) and unbounded ([34]) control operators, some conditions are derived to deduce, from the asymptotic stability of an infinite-dimensional linear system in abstract form, the asymptotic stability when closing the loop with saturating controller. These articles use the nonlinear semigroup theory (see e.g. [24] or [1]). The Korteweg-de Vries equation (KdV for short)Comment: arXiv admin note: text overlap with arXiv:1609.0144

Performance analysis of switching systems

Performance analysis is an important aspect in the design of dynamic (control) systems. Without a proper analysis of the behavior of a system, it is impossible to guarantee that a certain design satisfies the system’s requirements. For linear time-invariant systems, accurate performance analyses are relatively easy to make and as a result also many linear (controller) design methods have appeared in the past. For nonlinear systems, on the other hand, such accurate performance analyses and controller design methods are in general not available. A main reason hereof is that nonlinear systems, as opposed to linear time-invariant systems, can have multiple steady-state solutions. Due to the coexistence of multiple steady-state solutions, it is much harder to define an accurate performance index. Some nonlinear systems, i.e. the so-called convergent nonlinear systems, however, are characterized by a unique steady-state solution. This steady-state solution may depend on the system’s input signals (e.g. reference signals), but is independent of the initial conditions of the system. In the past, the notion of convergent systems has already been proven to be very useful in the performance analysis of nonlinear systems with inputs. In this thesis, new results in the field of performance analysis of nonlinear systems with inputs are presented, based on the notion of convergent systems. One part of the thesis is concerned with the question "how to analyse the performance for a convergent system?" Since the behavior of a convergent system is independent of the initial conditions (after some transient time), simulation can be used to find the unique steady-state solution that corresponds to a certain input signal, but this can be very time-consuming. In this thesis, a computationally more efficient approach is presented to estimate the steady-state performance of harmonically forced Lur’e systems, in terms of nonlinear frequency response functions (nFRFs). This approach is based on the method of harmonic linearization. It provides both a linear approximation of the nFRF and an upper bound on the error between this linear approximation and the true nFRF. It is shown in several examples that the approximation of the nFRF is accurate, and that it provides more detailed information on the considered system than the often used ‘L2 gain’ performance index. An additional observation that is made, is that the method of harmonic linearization can sometimes be ‘misleading’ for Lur’e systems with a saturation-like nonlinearity: for the case that the harmonic balance equation has a unique solution, it is shown that the corresponding nonlinear system can have multiple distinct steady-state solutions. Another part of the thesis is concerned with the question "under what conditions is a system with inputs guaranteed to be convergent?" In particular two types of systems were investigated: switched linear systems and Lur’e systems with a saturation nonlinearity and marginally stable linear part. For the switched linear systems, it is assumed that the dynamics of all the separate linear modes are given. For this setting, it was investigated if it is possible to find a switching rule (which defines when to switch between the available modes) such that the closed-loop system is convergent. Both a state-based, an observer-based, and a time-based switching rule are presented that guarantee a convergent system, assuming some conditions on the linear dynamics are met. The second type of systems that are discussed, are Lur’e systems with a saturation nonlinearity and marginally stable linear part. For this type of systems, the goal was to find sufficient conditions under which the closed-loop system is convergent. Because of the marginally stable linear part, however, a quadratically convergent system cannot be obtained. Instead, sufficient conditions are discussed that guarantee uniform convergency of the system. The obtained theory is shown to be also applicable to a class of anti-windup systems with a marginally stable plant. For this class of systems, the results of the convergency-based performance analysis are compared with the analysis results of existing anti-windup methods. It is shown that the convergency-based performance analysis can in some cases provide more detailed information on the steady-state behavior of the system. The results of uniform convergency for anti-windup systems are shown to be also applicable in the field of production and inventory control of discrete-event manufacturing systems. Since a manufacturing machine has a certain production capacity and cannot produce at a negative rate, it can be seen as an integrator plant (input: production rate, output: amount of finished products) preceded by a saturation function. For this marginally stable plant, a controller was constructed in such a way that the closed-loop system is uniformly convergent. The controller was also implemented in the discrete-event domain and the results from discrete-event simulations were compared with those of continuous-time simulations. Similarly, the controller was also applied for the production and inventory control of a line of four manufacturing machines. For both the single machine and the line of four machines, the resulting controlled discrete-event systems are shown to have the desired tracking properties. Besides these theoretical and numerical results, also experimental results are presented in this thesis. By means of an electromechanical construction, several experimental results were obtained, and used to validate the theoretical results for both the switched linear systems and the anti-windup systems

Process operating mode monitoring : switching online the right controller

This paper presents a structure which deals with process operating mode monitoring and allows the control law reconfiguration by switching online the right controller. After a short review of the advances in switching based control systems during the last decade, we introduce our approach based on the definition of operating modes of a plant. The control reconfiguration strategy is achieved by online selection of an adequate controller, in a case of active accommodation. The main contribution lies in settling up the design steps of the multicontroller structure and its accurate integration in the operating mode detection and accommodation loop. Simulation results show the effectiveness of the operating mode detection and accommodation (OMDA) structure for which the design steps propose a method to study the asymptotic stability, switching performances improvement, and the tuning of the multimodel based detector

Integrated chaos generators

This paper surveys the different design issues, from mathematical model to silicon, involved on the design of integrated circuits for the generation of chaotic behavior.Comisión Interministerial de Ciencia y Tecnología 1FD97-1611(TIC)European Commission ESPRIT 3110