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

Model predictive control techniques for hybrid systems

This paper describes the main issues encountered when applying model predictive control to hybrid processes. Hybrid model predictive control (HMPC) is a research field non-fully developed with many open challenges. The paper describes some of the techniques proposed by the research community to overcome the main problems encountered. Issues related to the stability and the solution of the optimization problem are also discussed. The paper ends by describing the results of a benchmark exercise in which several HMPC schemes were applied to a solar air conditioning plant.Ministerio de Eduación y Ciencia DPI2007-66718-C04-01Ministerio de Eduación y Ciencia DPI2008-0581

Robust nonlinear receding horizon control with constraint tightening: off line approximation and application to networked control system

2007/2008Nonlinear Receding Horizon (RH) control, also known as moving horizon control or nonlinear Model Predictive Control (MPC), refers to a class of algorithms that make explicit use of a nonlinear process model to optimize the plant behavior, by computing a sequence of future ma- nipulated variable adjustments. Usually the optimal control sequence is obtained by minimizing a multi-stage cost functional on the basis of open-loop predictions. The presence of uncertainty in the model used for the optimization raises the question of robustness, i.e., the maintenance of certain properties such as stability and performance in the presence of uncertainty. The need for guaranteeing the closed-loop stability in presence of uncertainties motivates the conception of robust nonlinear MPC, in which the perturbations are explicitly taken in account in the design of the controller. When the nature of the uncertainty is know, and it is assumed to be bounded in some compact set, the robust RH control can be determined, in a natural way, by solving a min–max optimal control problem, that is, the performance objective is optimized for the worst-case scenario. However, the use of min-max techniques is limited by the high computational burden required to solve the optimization problem. In the case of constrained system, a possibility to ensure the robust constraint satisfaction and the closed-loop stability without resorting to min-max optimization consists in imposing restricted (tightened) constraints on the the predicted trajectories during the optimization. In this framework, an MPC scheme with constraint tightening for discrete-time nonlinear systems affected by state-dependent and norm bounded uncertainties is proposed and discussed. A novel method to tighten the constraints relying on the nominal state prediction is described, leading to less conservative set contractions than in the existing approaches. Moreover, by imposing a stabilizing state constraint at the end of the control horizon (in place of the usual terminal one placed at the end of the prediction horizon), less stringent assumptions can be posed on the terminal region, while improving the robust stability properties of the MPC closed-loop system. The robust nonlinear MPC formulation with tightened constraints is then used to design off- line approximate feedback laws able to guarantee the practical stability of the closed-loop system. By using off-line approximations, the computational burden due to the on-line optimization is removed, thus allowing for the application of the MPC to systems with fast dynamics. In this framework, we will also address the problem of approximating possibly discontinuous feedback functions, thus overcoming the limitation of existent approximation scheme which assume the continuity of the RH control law (whereas this condition is not always verified in practice, due to both nonlinearities and constraints). Finally, the problem of stabilizing constrained systems with networked unreliable (and de- layed) feedback and command channels is also considered. In order to satisfy the control ob- jectives for this class of systems, also referenced to as Networked Control Systems (NCS’s), a control scheme based on the combined use of constraint tightening MPC with a delay compen- sation strategy will be proposed and analyzed. The stability properties of all the aforementioned MPC schemes are characterized by using the regional Input-to-State Stability (ISS) tool. The ISS approach allows to analyze the depen- dence of state trajectories of nonlinear systems on the magnitude of inputs, which can represent control variables or disturbances. Typically, in MPC the ISS property is characterized in terms of Lyapunov functions, both for historical and practical reasons, since the optimal finite horizon cost of the optimization problem can be easily used for this task. Note that, in order to study the ISS property of MPC closed-loop systems, global results are in general not useful because, due to the presence of state and input constraints, it is impossible to establish global bounds for the multi-stage cost used as Lyapunov function. On the other hand local results do not allow to analyze the properties of the predictive control law in terms of its region of attraction. There- fore, regional ISS results have to employed for MPC controlled systems. Moreover, in the case of NCS, the resulting control strategy yields to a time-varying closed-loop system, whose stability properties can be analyzed using a novel regional ISS characterization in terms of time-varying Lyapunov functions.XXI Ciclo198

Robustness of Prediction Based Delay Compensation for Nonlinear Systems

Control of systems where the information between the controller, actuator, and sensor can be lost or delayed can be challenging with respect to stability and performance. One way to overcome the resulting problems is the use of prediction based compensation schemes. Instead of a single input, a sequence of (predicted) future controls is submitted and implemented at the actuator. If suitable, so-called prediction consistent compensation and control schemes, such as certain predictive control approaches, are used, stability of the closed loop in the presence of delays and packet losses can be guaranteed. In this paper, we show that control schemes employing prediction based delay compensation approaches do posses inherent robustness properties. Specifically, if the nominal closed loop system without delay compensation is ISS with respect to perturbation and measurement errors, then the closed loop system employing prediction based delay compensation techniques is robustly stable. We analyze the influence of the prediction horizon on the robustness gains and illustrate the results in simulation.Comment: 6 pages, 3 figure

Advanced control designs for output tracking of hydrostatic transmissions

The work addresses simple but efficient model descriptions in a combination with advanced control and estimation approaches to achieve an accurate tracking of the desired trajectories. The proposed control designs are capable of fully exploiting the wide operation range of HSTs within the system configuration limits. A new trajectory planning scheme for the output tracking that uses both the primary and secondary control inputs was developed. Simple models or even purely data-driven models are envisaged and deployed to develop several advanced control approaches for HST systems