Efficient evaluation of piecewise control laws defined over a large number of polyhedra

We consider the class of piecewise state feedback control laws applied to discrete-time systems, motivated by recent work on the computation of closed-form MPC controllers. The on-line evaluation of such a control law requires the determination of the state space region in which the measured state lies, in order to decide which `piece' of the piecewise control law to apply. This procedure is called the point location problem, and the rate at which it can be solved determines the minimal sampling time of the system. In this paper we present a novel and computationally efficient search tree algorithm utilizing the concept of bounding boxes and interval trees that significantly improves this point-location search for piecewise control laws defined over a large number of (possibly overlapping) polyhedra. Furthermore, the required off-line preprocessing is low and so the approach can be applied to very complex controllers. The algorithm is compared with existing methods in the literature and its effectiveness is demonstrated for large examples.

[1]  Eduardo Sontag.  Nonlinear regulation: The piecewise linear approach , 1981 .

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[4]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[5]  Arjan van der Schaft,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[6]  Johannes Schumacher,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[7]  Alberto Bemporad,et al.  Control of systems integrating logic, dynamics, and constraints , 1999, Autom..

[8]  W. Heemels.  Linear complementarity systems : a study in hybrid dynamics , 1999 .

[9]  Chun-Liang Lin,et al.  Constrained optimal control for wheeled cars , 2000, 2009 4th International Conference on Autonomous Robots and Agents.

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

[11]  Bart De Schutter,et al.  Model predictive control for max-plus-linear discrete event systems , 2001, Autom..

[12]  P. Giblin.  Computational geometry: algorithms and applications (2nd edn.), by M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf. Pp. 367. £20.50. 2000. ISBN 3 540 65620 0 (Springer-Verlag). , 2001, The Mathematical Gazette.

[13]  Bart De Schutter,et al.  Equivalence of hybrid dynamical models , 2001, Autom..

[14]  Thomas H. Cormen,et al.  Introduction to algorithms [2nd ed.] , 2001 .

[15]  Jan M. Maciejowski,et al.  Predictive control : with constraints , 2002 .

[16]  D. Mayne,et al.  Optimal control of constrained, piecewise affine systems with bounded disturbances , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[17]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[18]  M. Baotic,et al.  Infinite time optimal control of hybrid systems with a linear performance index , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[19]  Manfred Morari,et al.  A clustering technique for the identification of piecewise affine systems , 2001, Autom..

[20]  Manfred Morari,et al.  Efficient mode enumeration of compositional hybrid systems , 2003, Int. J. Control.

[21]  D. Mayne,et al.  Model predictive control of constrained piecewise affine discrete‐time systems , 2003 .

[22]  Alberto Bemporad,et al.  Evaluation of piecewise affine control via binary search tree , 2003, Autom..

[23]  Mikael Johansson,et al.  Piecewise linear control systems - a computational approach , 2002, Lecture notes in control and information sciences.

[24]  Mato Baotic,et al.  Multi-Parametric Toolbox (MPT) , 2004, HSCC.

[25]  M. Morari,et al.  Optimal complexity reduction of piecewise affine models based on hyperplane arrangements , 2004, Proceedings of the 2004 American Control Conference.

[26]  Alberto Bemporad,et al.  Identification of piecewise affine systems via mixed-integer programming , 2004, Autom..

[27]  Jack Snoeyink.  Point Location , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[28]  Alberto Bemporad,et al.  Inner and outer approximations of polytopes using boxes , 2004, Comput. Geom..

[29]  C. Jones,et al.  A LOGARITHMIC-TIME SOLUTION TO THE POINT LOCATION PROBLEM FOR CLOSED-FORM LINEAR MPC , 2005 .

[30]  Francesco Borrelli,et al.  Constrained Optimal Control of Linear and Hybrid Systems , 2003, IEEE Transactions on Automatic Control.

[31]  Alberto Bemporad,et al.  Efficient On-Line Computation of Constrained Optimal Control , 2008, SIAM J. Control. Optim..