Development of an optimal state transition graph for trajectory optimisation of dynamic systems by application of Dijkstra's algorithm

Abstract A technique is presented for modelling a dynamic system defined by differential-algebraic equations into a weighted directed graph of optimal state transitions. This aims to make it possible for the fixed final state optimal control problem of a dynamic system to be solved through shortest path graph search. The graph is generated by taking a defined dynamic system state space and modelling discrete states as vertices and the transitions between states as edges. The edge weights are optimized to represent the optimal transitions between their connected state-vertices, resulting in an optimal state transition graph. An optimal control solution for an optimal control problem can be determined by applying Dijkstra’s algorithm to this optimized graph. The graph is generated to have n-connections between the system states instead of n2-connections, allowing for a shortest path from an initial state to a specified final state to be determined at a feasible computational run-time.

[1]  S. A. Dadebo,et al.  Dynamic optimization of constrained chemical engineering problems using dynamic programming , 1995 .

[2]  A. Hindmarsh,et al.  Stiff ode slovers: a review of current and coming attractions , 1987 .

[3]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[4]  Alberto Bemporad,et al.  Dynamic programming for constrained optimal control of discrete-time linear hybrid systems , 2005, Autom..

[5]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[6]  R. Luus.  Optimal control by dynamic programming using systematic reduction in grid size , 1990 .

[7]  Z. Michalewicz,et al.  A modified genetic algorithm for optimal control problems , 1992 .

[8]  B. Bequette.  Nonlinear control of chemical processes: a review , 1991 .

[9]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[10]  N. Shimkin,et al.  Fast Graph-Search Algorithms for General-Aviation Flight Trajectory Generation , 2005 .

[11]  Lorenz T. Biegler,et al.  Nonlinear Waves in Integrable and Nonintegrable Systems , 2018 .

[12]  John N. Tsitsiklis,et al.  Implementation of efficient algorithms for globally optimal trajectories , 1998, IEEE Trans. Autom. Control..

[13]  Lorenz T. Biegler,et al.  Simultaneous dynamic optimization strategies: Recent advances and challenges , 2006, Comput. Chem. Eng..

[14]  Simant Ranjan Upreti,et al.  Optimal Control for Chemical Engineers , 2012 .

[15]  Rutherford Aris,et al.  An analysis of chemical reactor stability and control—I: The possibility of local control, with perfect or imperfect control mechanisms , 1958 .

[16]  Donald E. Kirk,et al.  Optimal control theory : an introduction , 1970 .

[17]  Rein Luus,et al.  Application of dynamic programming to final state constrained optimal control problems , 1991 .

[18]  A. Maidi,et al.  Optimal Control of Nonlinear Chemical Processes Using the Variational Iteration Method , 2012 .

[19]  Jens Vygen,et al.  A generalization of Dijkstra's shortest path algorithm with applications to VLSI routing , 2009, J. Discrete Algorithms.

[20]  O. Junge,et al.  A set oriented approach to global optimal control , 2004 .

[21]  Steven M. LaValle,et al.  Simplicial dijkstra and A* algorithms for optimal feedback planning , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[22]  Moo Ho Lee,et al.  Dynamic Optimization of a Continuous Polymer Reactor Using a Modified Differential Evolution Algorithm , 1999 .

[23]  D. Liu,et al.  Adaptive Dynamic Programming for Finite-Horizon Optimal Control of Discrete-Time Nonlinear Systems With $\varepsilon$-Error Bound , 2011, IEEE Transactions on Neural Networks.

[24]  Maarten Steinbuch,et al.  Implementation of Dynamic Programming for Optimal Control Problems With Continuous States , 2015, IEEE Transactions on Control Systems Technology.

[25]  R. Dooren.  A Chebyshev technique applied to a controlled nuclear reactor system , 1989 .

[26]  Leon Lapidus,et al.  Optimal control of engineering processes , 1967 .

[27]  Wolfgang Marquardt,et al.  Dynamic optimization using adaptive direct multiple shooting , 2014, Comput. Chem. Eng..

[28]  Oskar von Stryk,et al.  Direct and indirect methods for trajectory optimization , 1992, Ann. Oper. Res..

[29]  J. Tsitsiklis.  Efficient algorithms for globally optimal trajectories , 1995, IEEE Trans. Autom. Control..

[30]  G. Siouris,et al.  Optimum systems control , 1979, Proceedings of the IEEE.

[31]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[32]  D. E. Cormack,et al.  Multiplicity of solutions resulting from the use of variational methods in optimal control problems , 1972 .

[33]  Hans Bock,et al.  Time optimal trajectories of elbow robots by direct methods , 1989 .

[34]  R. Sargent.  Optimal control , 2000 .

[35]  R. Bellman.  The theory of dynamic programming , 1954 .

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

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

[38]  K. Teo,et al.  THE CONTROL PARAMETERIZATION METHOD FOR NONLINEAR OPTIMAL CONTROL: A SURVEY , 2013 .