Optimal control with regular objectives using an abstraction-refinement approach

This paper presents a method to synthesize a sequence of control inputs for a discrete-time piecewise linear system equipped with a cost function, such that the controlled system behavior satisfies a finite-word linear-time temporal objective with minimal cost. An abstract finite state weighted transition system is constructed, such that the cost of the optimal control on the abstract system provides an upper bound on the cost of the optimal control for the original system. Specifically, the abstract system is constructed from finite partitions of the state and input spaces by solving optimization problems, and a sequence of suboptimal controllers is obtained by considering a sequence of uniformly refined partitions. Furthermore, the costs achieved by the sequence of suboptimal controllers converge to the optimal cost for the piecewise linear system. The abstraction refinement algorithm is implemented in the tool OptCAR. The feasibility of this approach is illustrated on an example, by constructing automatically, sub-optimal controllers with improving optimal costs.

[1]  Jun-ichi Imura,et al.  Discrete Abstractions of Nonlinear Systems Based on Error Propagation Analysis , 2012, IEEE Transactions on Automatic Control.

[2]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[3]  Amir Pnueli,et al.  Symbolic Controller Synthesis for Discrete and Timed Systems , 1994, Hybrid Systems.

[4]  George J. Pappas,et al.  Optimal Paths in Weighted Timed Automata , 2001, HSCC.

[5]  T. Lindvall.  ON A ROUTING PROBLEM , 2004, Probability in the Engineering and Informational Sciences.

[6]  P. Ramadge,et al.  Supervisory control of a class of discrete event processes , 1987 .

[7]  Panos J. Antsaklis,et al.  Optimal Control of Switched Hybrid Systems: A Brief Survey , 2013 .

[8]  C. Seatzu,et al.  Optimal Control of Discrete-Time Hybrid Automata under Safety and Liveness Constraints , 2005, Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, 2005..

[9]  Thomas A. Henzinger,et al.  Alternating Refinement Relations , 1998, CONCUR.

[10]  Patricia Bouyer,et al.  Weighted O-Minimal Hybrid Systems Are More Decidable Than Weighted Timed Automata! , 2007, LFCS.

[11]  Ricardo G. Sanfelice,et al.  Optimal control of Mixed Logical Dynamical systems with Linear Temporal Logic specifications , 2008, 2008 47th IEEE Conference on Decision and Control.

[12]  Calin Belta,et al.  Temporal logic model predictive control , 2015, Autom..

[13]  Calin Belta,et al.  Temporal Logic Control of Discrete-Time Piecewise Affine Systems , 2012, IEEE Transactions on Automatic Control.

[14]  Ufuk Topcu,et al.  Receding Horizon Temporal Logic Planning , 2012, IEEE Transactions on Automatic Control.

[15]  Alberto Bemporad,et al.  Piecewise linear optimal controllers for hybrid systems , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[16]  Leong Yoke Peng,et al.  Optimal control with regular objectives using an abstraction-refinement approach , 2016 .

[17]  Pierre Wolper,et al.  An Automata-Theoretic Approach to Automatic Program Verification (Preliminary Report) , 1986, LICS.

[18]  Ufuk Topcu,et al.  Optimal control of non-deterministic systems for a computationally efficient fragment of temporal logic , 2013, 52nd IEEE Conference on Decision and Control.

[19]  Alonzo Church,et al.  Logic, arithmetic, and automata , 1962 .

[20]  Edmund M. Clarke,et al.  Design and Synthesis of Synchronization Skeletons Using Branching Time Temporal Logic , 2008, 25 Years of Model Checking.

[21]  Bo Lincoln,et al.  LQR optimization of linear system switching , 2002, IEEE Trans. Autom. Control..

[22]  Ufuk Topcu,et al.  Synthesis of Reactive Switching Protocols From Temporal Logic Specifications , 2013, IEEE Transactions on Automatic Control.

[23]  Alessandro Abate,et al.  Efficient suboptimal solutions of switched LQR problems , 2009, 2009 American Control Conference.

[24]  Roberto Bagnara,et al.  The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems , 2006, Sci. Comput. Program..

[25]  A. Rantzer,et al.  Piecewise linear quadratic optimal control , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[26]  G. Dullerud,et al.  A Course in Robust Control Theory: A Convex Approach , 2005 .

[27]  Jörg Raisch,et al.  Discrete approximation and supervisory control of continuous systems , 1998, IEEE Trans. Autom. Control..

[28]  Thomas A. Henzinger,et al.  Rectangular Hybrid Games , 1999, CONCUR.

[29]  Patricia Bouyer,et al.  O-Minimal Hybrid Reachability Games , 2009, Log. Methods Comput. Sci..

[30]  Hadas Kress-Gazit,et al.  Guaranteeing reactive high-level behaviors for robots with complex dynamics , 2013, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[31]  Gunther Reissig,et al.  Abstraction-based solution of optimal stopping problems under uncertainty , 2013, 52nd IEEE Conference on Decision and Control.

[32]  P. Tabuada,et al.  Symbolic approximate time-optimal control , 2010, Syst. Control. Lett..

[33]  Mahesh Viswanathan,et al.  Specifications for decidable hybrid games , 2011, Theor. Comput. Sci..

[34]  Antoine Girard,et al.  Controller synthesis for safety and reachability via approximate bisimulation , 2010, Autom..

[35]  Pierre Wolper,et al.  Synthesis of Communicating Processes from Temporal Logic Specifications , 1981, TOPL.

[36]  Calin Belta,et al.  Temporal Logic Planning and Control of Robotic Swarms by Hierarchical Abstractions , 2007, IEEE Transactions on Robotics.