Cost-to-travel functions: A new perspective on optimal and model predictive control

Abstract This paper concerns a class of functions, named cost-to-travel functions, which find applications in model-based control. For a given (potentially nonlinear) control system, the cost-to-travel function associates with any given start and end point in the state space and any given travel duration the minimum economic cost of the associated point-to-point motion. Cost-to-travel functions are a generalization of cost-to-go functions, which are often used in the context of dynamic programming as well as model predictive control. We discuss the properties of cost-to-travel functions, their relations to existing concepts in control such as dissipativity, but also a variety of control-theoretic applications of this function class. In particular, we discuss how cost-to-travel functions can be used to analyze the properties of economic model predictive control with return constraints.

[1]  Boris Houska.  Enforcing asymptotic orbital stability of economic model predictive control , 2015, Autom..

[2]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[3]  M. Connor.  Calculus of Variations and Optimal Control Theory , 1967 .

[4]  Lars Grüne,et al.  Economic receding horizon control without terminal constraints , 2013, Autom..

[5]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[6]  Frank Allgöwer,et al.  On the role of dissipativity in economic model predictive control , 2015 .

[7]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[8]  E. Blum,et al.  The Mathematical Theory of Optimal Processes. , 1963 .

[9]  Moritz Diehl,et al.  An auto-generated real-time iteration algorithm for nonlinear MPC in the microsecond range , 2011, Autom..

[10]  David Angeli,et al.  On Average Performance and Stability of Economic Model Predictive Control , 2012, IEEE Transactions on Automatic Control.

[11]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[12]  Viorel Barbu,et al.  Periodic optimal control in Hilbert space , 1996 .

[13]  A. F. Filippov.  On Certain Questions in the Theory of Optimal Control , 1962 .

[14]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[15]  David Angeli,et al.  Receding horizon cost optimization for overly constrained nonlinear plants , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[16]  Frank Allgöwer,et al.  Economic and Distributed Model Predictive Control: Recent Developments in Optimization-Based Control , 2017 .

[17]  Lars Grne,et al.  Nonlinear Model Predictive Control: Theory and Algorithms , 2011 .

[18]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[19]  Lars Grüne,et al.  Economic model predictive control without terminal constraints for optimal periodic behavior , 2015, Autom..

[20]  David Angeli,et al.  On Necessity and Robustness of Dissipativity in Economic Model Predictive Control , 2015, IEEE Transactions on Automatic Control.

[21]  Stephen J. Wright.  Partitioned Dynamic Programming for Optimal Control , 1991, SIAM J. Optim..

[22]  P. Tseng.  Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization , 2001 .

[23]  James B. Rawlings,et al.  Postface to “ Model Predictive Control : Theory and Design ” , 2012 .

[24]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[25]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[26]  F. Allgöwer,et al.  A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability , 1997 .

[27]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[28]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[29]  Stephen J. Wright.  Coordinate descent algorithms , 2015, Mathematical Programming.

[30]  L. Grüne,et al.  Nonlinear Model Predictive Control : Theory and Algorithms. 2nd Edition , 2011 .

[31]  P. Moylan,et al.  Dissipative Dynamical Systems: Basic Input-Output and State Properties , 1980 .