Global stabilization of multiple integrators by a bounded feedback with constraints on its successive derivatives

In this paper, we address the global stabilization of chains of integrators by means of a bounded static feedback law whose p first time derivatives are bounded. Our construction is based on the technique of nested saturations introduced by Teel. We show that the control amplitude and the maximum value of its p first derivatives can be imposed below any prescribed values. Our results are illustrated by the stabilization of the third order integrator on the feedback and its first two derivatives

Global stabilization of linear systems with bounds on the feedback and its successive derivatives

We address the global stabilization of linear time-invariant (LTI) systems when the magnitude of the control input and its successive time derivatives, up to an order $p\in\mathbb N$, are bounded by prescribed values. We propose a static state feedback that solves this problem for any admissible LTI systems, namely for stabilizable systems whose internal dynamics has no eigenvalue with positive real part. This generalizes previous work done for single-input chains of integrators and rotating dynamics

Stability and stabilization of sampled-data control for lure systems

Este trabalho apresenta um novo método para a análise de estabilidade e estabilização de sistemas do tipo Lure com controle amostrado, sujeitos a amostragem aperiódica e não linearidades que são limitadas em setor e restritas em derivada, em ambos contextos global e regional. Assume-se que os estados da planta estão disponíveis para medição e que as não linearidades são conhecidas, o que leva a uma formulação mais geral do problema. Os estados são adquiridos por um controlador digital que atualiza a entrada de controle em instantes de tempo discretos e aperiódicos, mantendo-a constante entre dois instantes sucessivos de amostragem. A abordagem apresentada neste trabalho é baseada no uso de uma nova classe de looped-functionals e em uma função do tipo Lure generalizada, que leva a condições de estabilidade e estabilização que são escritas na forma de desigualdades matriciais lineares (LMIs) e quasi-LMIs, respectivamente. Com base nestas condições, problemas de otimização são formulados com o objetivo de computar o intervalo máximo entre amostragens ou os limites máximos do setor para os quais a estabilidade assintótica da origem do sistema de dados amostrados em malha fechada é garantida. No caso em que as condições de setor são válidas apenas localmente, a solução desses problemas também fornece uma estimativa da região de atração para as trajetórias em tempo contínuo do sistema em malha fechada. Como as condições de síntese são quasi-LMIs, um algoritmo de otimização por enxame de partículas é proposto para lidar com as não linearidades envolvidas nos problemas de otimização, que surgem do produto de algumas variáveis de decisão. Exemplos numéricos são apresentados ao longo do trabalho para destacar as potencialidades do método.This work presents a new method for stability analysis and stabilization of sampleddata controlled Lure systems, subject to aperiodic sampling and nonlinearities that are sector bounded and slope restricted, in both global and regional contexts. We assume that the states of the plant are available for measurement and that the nonlinearities are known, which leads to a more general formulation of the problem. The states are acquired by a digital controller which updates the control input at aperiodic discrete-time instants, keeping it constant between successive sampling instants. The approach here presented is based on the use of a new class of looped-functionals and a generalized Luretype function, which leads to stability and stabilization conditions that are written in the form of Linear Matrix Inequalities (LMIs) and quasi-LMIs, respectively. On this basis, optimization problems are formulated aiming to compute the maximal intersampling interval or the maximal sector bounds for which the asymptotic stability of the origin of the sampled-data closed-loop system is guaranteed. In the case where the sector conditions hold only locally, the solution of these problems also provide an estimate of the region of attraction for the continuous-time trajectories of the closed-loop system. As the synthesis conditions are quasi-LMIs, a Particle Swarm Optimization (PSO) algorithm is proposed to deal with the involved nonlinearities in the optimization problems, which arise from the product of some decision variables. Numerical examples are presented throughout the work to highlight the potentialities of the method

Development of efficient algorithms for model predictive control of fast systems

Die nichtlineare modellprädiktive Regelung (NMPC) ist ein vielversprechender Regelungsalgorithmus, der auf der Echtzeitlüsung eines nichtlinearen dynamischen Optimie- rungsproblems basiert. Nichtlineare Modellgleichungen wie auch die Steuerungs- und Zustandsbeschränkungen werden als Gleichungs- bzw. Ungleichungsbeschränkungen des Optimalsteuerungsproblems behandelt. Jedoch wurde die NMPC wegen des recht hohen Rechenaufwandes bisher meist auf relativ langsame Prozesse angewendet. Daher bildet die Rechenzeit bei Anwendung der NMPC auf schnelle Prozesse einen gewissen Engpass wie z. B. bei mechanischen und/oder elektrischen Prozessen. In dieser Arbeit wird eine neue Lüsungsstrategie für dynamische Optimierungsprobleme vorgeschlagen, wie sie in NMPC auftreten, die auch auf sog. schnelle Systeme anwendbar ist. Diese Strategie kombiniert Mehrschieß -Verfahrens mit der Methode der Kollokation auf finiten Elementen. Mittels Mehrschieß -Verfahren wird das nichtlineare dynamische Optimierungsproblem in ein hochdimensionales statisches Optimierungsproblem (nonlinear program problem, NLP) überführt, wobei Diskretisierungs- und Parametrisierungstechniken zum Einsatz kommen. Um das NLP-Problem zu lüsen, müssen die Zustandswerte und ihre Gradienten am Ende jedes Diskretisierung-Intervalles berechnet werden. In dieser Arbeit wird die Methode der Kollokation auf finiten Elementen benutzt, um diese Aufgabe zu lüsen. Dadurch lassen sich die Zustandsgrüß en und ihre Gradienten am Ende jedes Diskretisierungs-Intervalls auch mit groß er Genauigkeit berechnen. Im Ergebnis künnen die Vorteile beider Methoden (Mehrschieß -Verfahren und Kollokations-Methoden) ausgenutzt werden und die Rechenzeit lässt sich deutlich reduzieren. Wegen des komplexen Optimierungsproblems ist es im Allgemeinen schwierig, eine Stabilitätsanalyse für das zugehürige NMPC durchzuführen. In dieser Arbeit wird eine neue Formulierung des Optimalsteuerungsproblems vorgeschlagen, durch die die Stabilität des NMPC gesichert werden kann. Diese Strategie besteht aus den folgenden drei Eigenschaften. Zunächst wird ein Hilfszustand über eine lineare Zustandsgleichung in das Optimierungsproblem eingeführt. Die Zustandsgleichungen werden durch Hilfszustände ergänzt, die man in Form von Ungleichungsnebenbedingungen einführt. Wenn die Hilfszustände stabil sind, lässt sich damit die Stabilität des Gesamtsystems sichern. Die Eigenwerte der Hilfszustände werden so gewählt, dass das Optimalsteuerungsproblem lüsbar ist. Dazu benutzt man die Eigenwerte als Optimierungsvariable. Damit lassen sich die Stabilitätseigenschaften in einem stationären Punkt des Systemmodells untersuchen. Leistungsfähigkeit und Effektivität des vorgeschlagenen Algorithmus werden an Hand von Fallstudien belegt. Die Bibliothek Numerische Algorithmus Group (NAG), Mark 8, wird eingesetzt, um die linearen und nichtlinearen Gleichungen, die aus der Kollokation resultieren, zu lüsen. Weiterhin wird zur Lüsung des NLP-Problems der Lüser IPOPT für C/C++- Umgebung eingesetzt. Insbesondere wird der vorgeschlagene Algorithmus zur Steuerung einer Verladebrücke im Labor des Institutes für Automatisierungs- und Systemtechnik angewendet.Nonlinear model predictive control (NMPC) has been considered as a promising control algorithm which is based on a real-time solution of a nonlinear dynamic optimization problem. Nonlinear model equations and controls as well as state restrictions are treated as equality and inequality constraints of the optimal control problem. However, NMPC has been applied mostly in relatively slow processes until now, due to its high computational expense. Therefore, computation time needed for the solution of NMPC leads to a bottleneck in its application to fast systems such as mechanical and/or electrical processes. In this dissertation, a new solution strategy to efficiently solve NMPC problems is proposed so that it can be applied to fast systems. This strategy combines the multiple shooting method with the collocation on finite elements method. The multiple shooting method is used for transforming the nonlinear optimal control problem into nonlinear program (NLP) problem using discretization and parametrization techniques. To solve this NLP problem the values of state variables and their gradients at the end of each shooting need to be computed. We use collocation on finite elements to carry out this task, thus, a high precision approximation of the state variables and their sensitivities in each shoot are achieved. As a result, the advantages of both the multiple shooting and the collocation method can be employed and therefore the computation efficiency can be considerably enhanced. Due to the nonlinear and complex optimal control problem formulation, in general, it is difficult to analyze the stability properties of NMPC systems. In this dissertation we propose a new formulation of the optimal control problem to ensure the stability of the NMPC problems. It consists the following three features. First, we introduce auxiliary states and linear state equations into the finite horizon dynamic optimization problem. Second, we enforce system states to be contracted with respect to the auxiliary state variables by adding inequality constraints. Thus, the stability features of the system states will conform to the stability properties of the auxiliary states, i.e. the system states will be stable, if the auxiliary states are stable. Third, the eigenvalues of the linear state equations introduced will be determined to stabilize the auxiliary states and at the same time make the optimal control problem feasible. This is achieved by considering the eigenvalues as optimization variables in the optimal control problem. Moreover, features of this formulation are analyzed at the stationary point of the system model. To show the effectiveness and performance of the proposed algorithm and the new optimal control problem formulation we present a set of NMPC case studies. We use the numerical algorithm group (NAG) library Mark 8 to solve numerically linear and nonlinear systems that resulted from the collocation on finite elements to compute the states and sensitivities, in addition, the interior point optimizer (IPOPT) and in C/C++ environment. Furthermore, to show more applicability, the proposed algorithm is applied to control a laboratory loading bridge