Decidable Controller Synthesis for Classes of Linear Systems

A problem of great interest in the control of hybrid systems is the design of least restrictive controllers for reachability specifications. Controller design typically uses game theoretic methods which compute the region of the state space for which there exists a control such that for all disturbances, an unsafe set is not reached. In general, the computation of the controllers requires the steady state solution of a Hamilton-Jacobi-Isaacs partial differential equation which is very difficult to compute, if it exists. In this paper, we show that for classes of linear systems, the controller synthesis problem is decidable: There exists a computational algorithm which, after a finite number of steps, will exactly compute the least restrictive controller. This result is achieved by a very interesting interaction of results from mathematical logic and optimal control.

[1]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..

[2]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[3]  S. Shankar Sastry,et al.  Conflict resolution for air traffic management: a study in multiagent hybrid systems , 1998, IEEE Trans. Autom. Control..

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

[5]  Thomas Sturm,et al.  REDLOG: computer algebra meets computer logic , 1997, SIGS.

[6]  H. Wong-Toi,et al.  The synthesis of controllers for linear hybrid automata , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[7]  Joseph Lewin,et al.  Differential Games , 1994 .

[8]  S. Shankar Sastry,et al.  O-Minimal Hybrid Systems , 2000, Math. Control. Signals Syst..

[9]  Wolfgang Thomas,et al.  On the Synthesis of Strategies in Infinite Games , 1995, STACS.

[10]  Gerardo Lafferriere,et al.  A New Class of Decidable Hybrid Systems , 1999, HSCC.

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

[12]  John Lygeros,et al.  Verified hybrid controllers for automated vehicles , 1998, IEEE Trans. Autom. Control..

[13]  Panos J. Antsaklis,et al.  Hybrid Systems II , 1994, Lecture Notes in Computer Science.

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

[15]  George J. Pappas,et al.  Reachability computation for linear hybrid systems , 1999 .

[16]  Jim Alves-Foss,et al.  Higher Order Logic Theorem Proving and its Applications 8th International Workshop, Aspen Grove, Ut, Usa, September 11-14, 1995 : Proceedings , 1995 .

[17]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[18]  George J. Pappas,et al.  Con ict Resolution for Air Tra c Management: a Study in Multi-Agent Hybrid Systems , 1997 .

[19]  C. Tomlin,et al.  Controllers for Reachability Speciications for Hybrid Systems , 1999 .

[20]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

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

[22]  L. van den Dries,et al.  Tame Topology and O-minimal Structures , 1998 .

[23]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[24]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .