Ellipsoidal Techniques for Reachability Under State Constraints

The paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with time-varying coefficients, bounds on the controls, and constraints on the state. The scheme provides tight external approximations by ellipsoid-valued tubes. The tubes touch the reach tubes from the outside at each point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The result is an exact parametric representation of reach tubes through families of external ellipsoidal tubes. The parameters that characterize the approximating ellipsoids are solutions of ordinary differential equations with coefficients given partly in explicit analytical form and partly through the solution of a recursive optimization problem. The scheme combines the calculation of external approximations of infinite sums and intersections of ellipsoids, and suggests an approach to calculate reach sets of hybrid systems.

[1]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[2]  A. I. Subbotin.  Generalized Solutions of First Order PDEs , 1995 .

[3]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[4]  Pravin Varaiya,et al.  On Ellipsoidal Techniques for Reachability Analysis. Part I: External Approximations , 2002, Optim. Methods Softw..

[5]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[6]  Hlawka.  Theory of the integral , 1939 .

[7]  H. Hermes,et al.  Foundations of optimal control theory , 1968 .

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

[9]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[10]  Pravin Varaiya,et al.  Epsilon-Approximation of Differential Inclusions , 1996, Hybrid Systems.

[11]  Andrei I. Subbotin,et al.  Generalized solutions of first-order PDEs - the dynamical optimization perspective , 1994, Systems and control.

[12]  A. Kurzhanski,et al.  On the Theory of Trajectory Tubes — A Mathematical Formalism for Uncertain Dynamics, Viability and Control , 1993 .

[13]  P. Varaiya,et al.  Dynamic Optimization for Reachability Problems , 2001 .

[14]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[15]  F. Chernousko.  State Estimation for Dynamic Systems , 1993 .

[16]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[17]  Pravin Varaiya,et al.  On Reachability Under Uncertainty , 2002, SIAM J. Control. Optim..

[18]  A. B. Kurzhanskii,et al.  On control problems with restricted coordinates: PMM vol. 33, n≗4, 1969, pp. 705–719 , 1969 .

[19]  L. Schwartz.  Théorie des distributions , 1966 .

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

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

[22]  Pravin Varaiya,et al.  Reach Set Computation Using Optimal Control , 2000 .

[23]  P. Varaiya,et al.  Ellipsoidal techniques for reachability analysis: internal approximation , 2000 .

[24]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[25]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

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

[27]  Alexander B. Kurzhanski,et al.  Ellipsoidal Techniques for Reachability Problems Under Nonellipsoidal Constraints , 2001 .