Predictive Control for Polynomial Systems Subject to State and Input Constraints

Abstract We present a nonlinear model predictive control (NMPC) approach to design optimal feedback controllers for the class of continuous-time input affine polynomial systems subject to state and input constraints. The corresponding feedback law is obtained via the solution of an efficient to solve convex optimization problem subject to sum of squares (SOS) constraints. To serve the conflicting requirements of performance and computational costs in various applications, the approach comes in three different control schemes which differ in how many times the (updated) optimization problem has to be solved, and whether this is done on- or offline. Each of the proposed control schemes minimizes an upper bound on the cost functional while guaranteeing closed-loop stability and satisfaction of input and state constraints. Finally, simulations of an example system show the applicability and the effectiveness of the proposed controllers. Zusammenfassung Wir präsentieren einen rückführungsbasierten Ansatz zur Nichtlinearen Modelprädiktiven Regelung zeitkontinuierlicher, eingangsaffiner Polynomialsysteme unter Zustands- und Eingangsbeschränkungen. Das zugehörige Rückführgesetz wird durch die Lösung eines effizient zu lösenden konvexen Optimierungsproblem bestimmt, dessen Nebenbedingungen mit Hilfe der Summe-von-Quadraten Methode hergeleitet werden. Um den widersprüchlichen Anforderungen von Regelgüte und Rechenaufwand in verschiedenen Anwendungen Rechnung zu tragen, erlaubt der Ansatz drei verschiedene Regelstrategien, welche sich darin unterscheiden, wie oft das (aktualisierte) Optimierungsproblem gelöst werden muss, und ob dies on- oder offline geschieht. Jede der vorgeschlagenen Regelstrategien minimiert eine obere Grenze eines Kostenfunktionals und garantiert gleichzeitig sowohl die Stabilität des geschlossenen Kreises als auch die Einhaltung der Zustands- und Eingangsbeschränkungen. Schließlich, wird in Simulation anhand eines Beispielsystems die Anwendbarkeit und Effektivität der vorgeschlagenen Regler veranschaulicht.

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