Constrained Control of Nonlinear Systems

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1 Diss. ETH No Constrained Control of Nonlinear Systems A dissertation submitted to the Swiss Federal Institute of Technology (ETH) Zurich for the degree of Doctor of Technical Sciences presented by Vesna Nevistic Dipl. Engineer of Electronics, Military Technical Academy, Zagreb Magister of Sciences in Electrical Engineering, Zagreb University born June 20, 1965 citizen of Croatia accepted on the recommendation of Prof. Dr. M. Morari, examiner Prof. Dr. D. Bonvin, co-examiner January, 1997

2 Summary One of the most important practical and theoretical open problems in control is the design and analysis of feedback controllers for systems with constraints Constraints on the actuators and often also on the outputs are present in all real systems and can lead to performance deterioration and even instability if not properly accounted for in the design procedure Constraints render a control problem nonlmear even when the plant itself is linear Over the years many different approaches have been proposed for the analysis and design of control systems subject to constraints They have ranged from simple ad hoc techniques like anti-wmdup to the solution of appropriately defined optimal control problems and their on-line implementation This dissertation includes contributions on a wide range of problems related to the control of constrained linear and nonlinear systems First, m order to put the work in perspective, the optimal control approach to determining the feedback law for gen eral nonlinear systems is reviewed The classical variational technique is contrasted with the Hamilton-Jacobi-Bellman approach in terms of open vs closed loop local vs global, and necessary vs sufficient characterization of optimahty While these mentioned tools can lead to useful benchmark results their applicability to real-world problems is severely limited by numerical and dimensionality issues Model Predictive Control (MPC) constitutes a practical alternative Different formu lations of MPC have been employed m industry for about two decades with varying degrees of success For these implementations stability can often be determined by trial and error only, except in the special case of a linear plant with a quadratic objec tive there are no efficient computational procedures, robustness can only be assessed via case studies, any comparison with other nonlinear control techniques is highly sub jective depending on the designers effort and experience all these topics are reported in this dissertation Significant developments on It had become customary to guarantee stability for MPC by either extending the out put horizon to infinity or by imposing a terminal (end) constraint For constrained problems with a finite horizon and no end constraint, essentially no workable stability analysis techniques were available Here, based on a Lyapunov approach, sufficient conditions for stability are derived In the unconstrained case, evaluation of these conditions requires the computation of some eigenvalues M\ other similar conditions reported in the literature are shown to be special cases of this general result More importantly, these conditions also yield bounds on the infinite horizon performance v

3 of this finite honzon controller For the constrained case, the solution of a max-min problem is necessary to obtain a sufficient condition for stability Simulation experi ence suggests that this max-min problem is well behaved A performance bound is established similarly as for unconstrained systems For the full execution of MPC for nonlinear systems ("Nonlinear MPC ), a dynamic optimization problem must be solved on line at each time step, which is rarely a prac tical proposition Here, two alternate approaches are derived which require much less numerical effort and where the performance is essentially indistinguishable from that obtained with standard nonlinear MPC One technique combines feedback lineariza tion with MPC For open-loop stable nonlinear systems with constraints it guarantees global stability The only other method with equally strong properties is full nonlinear MPC, but it achieves these characteristics at the cost of much more involved on-line computations The other proposed technique approximates the nonlinear plant model with a linear time varying model, obtained from a linearization of the system along the predicted trajectory of the nonlinear system it is widely applicable Unlike MPC with feedback linearization In the last decade, it has become customary to assess closed loop performance and, in particular, robust performance via the induced 2-norm For linear systems with a wide range of uncertainty descriptions, a so-called fi problem results For general non linear systems a robust performance test can be formulated in the form of a two-point boundary-value problem which is solvable via an efficient numerical method This technique is used here to compare the robust performance of several MPC variants Finally, a systematic technique, the converse constrained Hamilton-Jacobi-Bellman approach, is developed to generate examples of nonlinear constrained systems, for which the optimal feedback control law is known analytically By comparing various control techniques in terms of their closeness to this standard, an objective evaluation of the relative merits becomes possible MPC is found to be a most effective strategy for nonlinear constrained problems - was investigated equal to or better than any other technique which

4 MPC) Kurzfassung Der Entwurf und die Analyse von Regelungen fur Systeme mit Beschrankungen gehort zu den praktisch und theoretisch wichtigsten offenen Problemen in der Regelungstechnik. Beschrankungen der Stellgrosse und oft auch von einzelnen Ausgangen finden sich in alien wirklichen Systemen. Wenn diese Beschrankungen nicht explizite im En twurf berucksichtigt werden, konnen sie nicht nur eine Beeintrachtigung der Regelgiite mit sich bringen, sondern sogar zur tnstabilitat fiihren. Beschrankungen machen ein Regelproblem auch dann nichtlinear wenn die Strecke selbst linear ist. In den letzten Jahren wurden viele Ansatze zur Analyse und Synthese von Reglern fur Systeme mit Beschrankungen vorgestellt. Sie reichen von einfachen ad hoc Methoden wie "antiwindup" bis zur Losung von entsprechend definierten optimalen Regelproblemen und ihrer on-line Implementierung. Diese Dissertation beinhaltet eine Reihe von Beitragen im Zusammenhang mit der Regelung von linearen und nichtlinearen Systemen mit Beschrankungen. Zuerst, um einen entsprechenden Rahmen fur diese Arbeit zu setzen, wird besprochen, wie das Ruckfuhrungsgesetz fur allgemeine nichtlineare Systeme iiber die Losung eines opti malen Regelungproblems bestimmt werden kann. Die klassische Variaritionenrechnung wird mit der Methode von Hamilton, Jacobi und Bellman verglichen und die Unterschiede in Bezug auf Steuerung vs. Regelung, und lokale vs. globale, sowie notwendige vs. hinreichende Charakterisierung des Optimums werden aufgezeigt. Wahrend die erwahnten Werkzeuge niitzlich sind um gewisse Standards zu setzen, ist ihre Anwendbarkeit auf realistische Falle durch numerische Probleme sehr eingeschrankt. Die pradiktive Regelung ("Model Predictive Control" - stellt eine praktische Alternative dar. Verschiedene Varianten von MPC wurden in den letzten zwei Jahrzehnten in der Industrie mit unterschiedlichem Erfolg angewandt. Bei diesen Implementierungen kann Stabilitat oft nur durch "trial and error" bestimmt werden; ausser fur lineare Strecken mit quadratischen Zielfunktionen existieren keine effiziente Berechnungsmethoden; die Robustheit muss iiber Fallstudien untersucht werden; alle Vergleiche mit anderen nichtlinearen Reglern sind rein subjektiv und hangen voll von der Erfahrung des Anwenders ab. In dieser Dissertation werden signifikante Beitrage zu alien diesen Themen gebracht. Es war iiblich geworden, die Stabilitat von MPC Reglern durch Vorschreiben eines unendlichen Horizonts oder einer Endbedingung zu erzwingen. Fur Probleme mit Beschrankungen und einem endlichen Horizont ohne Endbedingung gab es keine brauchvn

5 baren Methoden zur Stabilitatsanalyse. Hier werden hinreichende Stabilitatsbedingungen basierend auf einem Lyapunov Ansatz hergeleitet. Im Fall ohne Beschrankungen ist fiir die Untersuchung dieser Bedingungen die Losung von Eigenwertproblemen notwendig. Es wird gezeigt, dass alle anderen ahnlichen Bedingungen in der Literatur Spezialfalle dieses allgemeinen Ergebnisses sind. Zusatzlich erlaubt die Losung dieses Problems auch, einen Grenzwert fiir die Regelgiite fiber einen unendlichen Horizont zu bestimmen, die mit diesem Regler, der fiber einen endlichen Horizont berechnet wird, erreicht werden kann. Im Falle mit Beschrankungen ergeben sich die hinreichenden Stabilitatsbedingungen aus der Losung eines max-min Problems. Erfahrungen mit Simulationen zeigen, dass diese Optimierungsaufgabe mit Standardalgorithmen fiir nichtlineare Programmierung losbar ist. Ein Grenzwert fur die Regelgiite kann in ahnlicher Weise wie fiir Systeme ohne Beschrankungen bestimmt werden. Fiir die voile Implementierung von MPC auf nichtlinearen Systemen ("Nonlinear MPC") muss man on-line ein dynamisches Optimierungsproblem losen, was in der Praxis nur selten realisierbar ist. In dieser Arbeit werden zwei Alternativen vorgeschlagen, die viel weniger numerischen Aufwand erfordern, fur die jedoch die Regelgiite kaum von der mit "Nonlinear MPC" erreichbaren unterscheidbar ist. Die erste Methode kombiniert "feedback linearization" mit MPC. Fiir stabile nichtlineare Systeme mit Beschrankungen garantiert diese Technik globale Stabilitat. Die einzige andere Meth ode mit ahnlich starken Eigenschaften ist nichtlineares MPC, das jedoch mit einem viel grosseren on-line Rechenaufwand verbunden ist. Die zweite Methode approximiert das nichtlineare Streckenmodell mit einem linearen zeitvarianten Modell, das man von der Linearisierung des Systems entlang der pradiktierten Trajektorie des nichtlinearen Systems erhalt. ode breit anwendbar. Im Gegensatz zur MPC mit "feedback linearization" ist diese Meth Im letzten Jahrzehnt ist es ublich geworden, die Regelgiite und insbesondere die robuste Regelgiite ("robust performance") durch eine induzierte 2-Norm zu messen. Fiir lineare Systeme mit einer breiten Auswahl von Unsicherheitsbeschreibungen erhalt man ein sogenanntes fj, Problem. Fiir allgemeine nichtlineare Systeme kann man einen "robust performance" Test als "two-point boundary-value problem" formulieren, das effizient gelost werden kann. Diese Methode wurde hier verwendet, um die robuste Regelgiite von verschiedenen MPC Varianten zu vergleichen. Schliesslich wird eine systematische Methode ("converse constrained Hamilton-Jacobi- Bellman approach") hergeleitet, um Beispiele nichtlinearer Systeme mit Beschrankun gen zu generieren, fiir die das optimale Regelgesetz analytisch bestimmt werden kann. Durch den Vergleich verschiedener Methoden nach ihrer Nahe zu diesem Standard, kann man ihre relativen Vorteile objektiv beurteilen. Hier zeigte es sich, dass MPC eine effektive Strategic fiir nichtlineare Probleme mit - Beschrankungen darstellt gleich gut oder besser als alle anderen Methoden, welche untersucht wurden.

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