×

The explicit linear quadratic regulator for constrained systems. (English) Zbl 0999.93018

A technique to compute the explicit state-feedback solution of a discrete-time linear quadratic control problem subject to state and input constraints is presented. First the quadratic program (QP), which must be solved to determine the optimal control action, is derived. The original QP is viewed as a multi-parametric QP (mp-QP). The properties of an mp-QP are analysed, and an efficient algorithm to solve it is developed. It is shown that the closed form solution is piecewise affine and continuous for both the finite horizon problem (model predictive control, MPC) and the usual infinite time measure (constrained linear quadratic regulation). The controller can be implemented with substantially reduced on-line calculations preserving all performance and stability properties of MPC. The special on-line QP solvers are no longer required, only the evaluation of an explicitly defined piecewise linear function must be performed on-line. The proposed technique is attractive for a wide range of practical problems in which the computational complexity of on-line optimization is prohibitive.

MSC:

93B40 Computational methods in systems theory (MSC2010)
93B51 Design techniques (robust design, computer-aided design, etc.)
93C55 Discrete-time control/observation systems
49N10 Linear-quadratic optimal control problems
65Y20 Complexity and performance of numerical algorithms

Software:

cdd
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Acevedo, J.; Pistikopoulos, E.N., An algorithm for multiparametric mixed-integer linear programming problem, Operations research letters, 24, 139-148, (1999) · Zbl 0941.90057
[2] Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M., Nonlinear programming—theory and algorithms, (1993), Wiley New York · Zbl 0774.90075
[3] Bellman, R., Adaptive control processes—a guided tour, (1961), Princeton University Press Princeton, NJ · Zbl 0103.12901
[4] Bemporad, A., A predictive controller with artificial Lyapunov function for linear systems with input/state constraints, Automatica, 34, 10, 1255-1260, (1998) · Zbl 0938.93524
[5] Bemporad, A. (1998). Reducing conservativeness in predictive control of constrained systems with disturbances. Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL (pp. 1384-1391).
[6] Bemporad, A., Borrelli, F., & Morari, M. (2000). Explicit solution of LP-based model predictive control. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December. · Zbl 1364.93697
[7] Bemporad, A., Borrelli, F., & Morari, M. (2000). Piecewise linear optimal controllers for hybrid systems. Proceedings of the American Control Conference, Chicago, IL. · Zbl 1046.49019
[8] Bemporad, A.; Chisci, L.; Mosca, E., On the stabilizing property of SIORHC., Automatica, 30, 12, 2013-2015, (1994) · Zbl 0825.93418
[9] Bemporad, A., & Filippi, C. (2001). Suboptimal explicit MPC via approximate multiparametric quadratic programming. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida. · Zbl 1044.90080
[10] Bemporad, A.; Fukuda, K.; Torrisi, F.D., Convexity recognition of the union of polyhedra, Computational geometry, 18, 141-154, (2001) · Zbl 0976.68163
[11] Bemporad, A., & Morari, M. (1999). Robust model predictive control: A survey. In A. Garulli, A. Tesi, & A. Vicino (Eds.), Robustness in identification and control, Lecture Notes in Control and Information Sciences Vol. 245 (pp. 207-226). Berlin: Springer. · Zbl 0979.93518
[12] Bemporad, A., Torrisi, F. D., & Morari, M. (2000). Performance analysis of piecewise linear systems and model predictive control systems. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December. · Zbl 0939.93523
[13] Blanchini, F., Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions, IEEE transactions on automatic control, 39, 2, 428-433, (1994) · Zbl 0800.93754
[14] Borrelli, F., Baotic, M., Bemporad, A., & Morari, M. (2001). Efficient on-line computation of closed-form constrained optimal control laws. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida. · Zbl 1171.49026
[15] Borrelli, F., Bemporad, A., & Morari, M. (2000). A geometric algorithm for multi-parametric linear programming. Technical Report AUT00-06, Automatic Control Laboratory, ETH Zurich, Switzerland, February 2000. · Zbl 1061.90086
[16] Chiou, H. W., & Zafiriou, E. (1994). Frequency domain design of robustly stable constrained model predictive controllers. Proceedings of the American Control Conference, Vol. 3 (pp. 2852-2856).
[17] Chisci, L.; Zappa, G., Fast algorithm for a constrained infinite horizon LQ problem, International journal of control, 72, 11, 1020-1026, (1999) · Zbl 0942.49027
[18] Chmielewski, D.; Manousiouthakis, V., On constrained infinite-time linear quadratic optimal control, Systems and control letters, 29, 3, 121-130, (1996) · Zbl 0867.49025
[19] Dua, V.; Pistikopoulos, E.N., Algorithms for the solution of multiparametric mixed-integer nonlinear optimization problems, Industrial engineering chemistry research, 38, 10, 3976-3987, (1999)
[20] Dua, V.; Pistikopoulos, E.N., An algorithm for the solution of multiparametric mixed integer linear programming problems, Annals of operations research, 99, 123-139, (2000) · Zbl 0990.90079
[21] Fiacco, A.V., Sensitivity analysis for nonlinear programming using penalty methods, Mathematical programming, 10, 3, 287-311, (1976) · Zbl 0357.90064
[22] Fiacco, A.V., Introduction to sensitivity and stability analysis in nonlinear programming, (1983), Academic Press London, UK · Zbl 0543.90075
[23] Filippi, C. (1997). On the geometry of optimal partition sets in multiparametric linear programming. Technical report 12, Department of Pure and Applied Mathematics, University of Padova, Italy, June.
[24] Fletcher, R. (1981). Practical methods of optimization; Vol. 2: Constrained optimization. New York: Wiley. · Zbl 0474.65043
[25] Fukuda, K. (1997). cdd/cdd+ reference manual, (0.61 (cdd) 0.75 (cdd+), ed.). Zurich, Switzerland: Institute for Operations Research.
[26] Gal, T., Postoptimal analyses, parametric programming, and related topics, (1995), de Gruyter Berlin
[27] Gilbert, E.G.; Tin Tan, K., Linear systems with state and control constraints: the theory and applications of maximal output admissible sets, IEEE transactions on automatic control, 36, 9, 1008-1020, (1991) · Zbl 0754.93030
[28] Gutman, P.O., A linear programming regulator applied to hydroelectric reservoir level control, Automatica, 22, 5, 533-541, (1986) · Zbl 0605.93040
[29] Gutman, P.O.; Cwikel, M., Admissible sets and feedback control for discrete-time linear dynamical systems with bounded control and states, IEEE transactions on automatic control, AC-31, 4, 373-376, (1986) · Zbl 0589.93048
[30] Johansen, T. A., Petersen, I., & Slupphaug, O. (2000). On explicit suboptimal LQR with state and input constraints. Proceedings of the 39th IEEE Conference on Decision and Control (pp. 662-667). Sydney, Australia, December.
[31] Keerthi, S.S.; Gilbert, E.G., Optimal infinite-horizon feedback control laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations, Journal of optical theory and applications, 57, 265-293, (1988) · Zbl 0622.93044
[32] Kerrigan, E. C., & Maciejowski, J. M. (2000). Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. Proceedings of the 39th IEEE Conference on Decision and Control.
[33] Kothare, M.V.; Campo, P.J.; Morari, M.; Nett, C.N., A unified framework for the study of anti-windup designs, Automatica, 30, 12, 1869-1883, (1994) · Zbl 0825.93312
[34] Kothare, M.V.; Morari, M., Multiplier theory for stability analysis of anti-windup control systems, Automatica, 35, 917-928, (1999) · Zbl 0935.93048
[35] Mulder, E. F., Kothare, M. V., & Morari, M. (1999). Multivariable anti-windup controller synthesis using iterative linear matrix inequalities. Proceedings of the European Control Conference. · Zbl 0996.93035
[36] Qin, S. J., & Badgwell, T. A. (1997). An overview of industrial model predictive control technology. In Chemical process control—V, Vol. 93, No. 316 (pp. 232-256). AIChe Symposium Series—American Institute of Chemical Engineers.
[37] Rawlings, J.B.; Muske, K.R., The stability of constrained receding-horizon control, IEEE transactions on automatic control, 38, 1512-1516, (1993) · Zbl 0790.93019
[38] Scokaert, P.O.M.; Rawlings, J.B., Constrained linear quadratic regulation, IEEE transactions on automatic control, 43, 8, 1163-1169, (1998) · Zbl 0957.93033
[39] Sontag, E.D., An algebraic approach to bounded controllability of linear systems, International journal of control, 39, 1, 181-188, (1984) · Zbl 0531.93013
[40] Sznaier, M., & Damborg, M. J. (1987) Suboptimal control of linear systems with state and control inequality constraints. Proceedings of the 26th IEEE Conference on Decision and Control, Vol. 1 (pp. 761-762).
[41] Tan, K. T. (1991). Maximal output admisible sets and the nonlinear control of linear discrete-time systems with state and control constraints. Ph.D. thesis, University of Michigan.
[42] Teel, A., & Kapoor, N. (1997). The \(L2\) anti-windup problem: its definition and solution. European Control Conference, Brussels, Belgium.
[43] Zafiriou, E., Robust model predictive control of processes with hard constraints, Computers and chemical engineering, 14, 4/5, 359-371, (1990)
[44] Zheng, A.; Kothare, M.V.; Morari, M., Anti-windup design for internal model control, International journal of control, 60, 5, 1015-1024, (1994) · Zbl 0825.93283
[45] Zheng, A.; Morari, M., Stability of model predictive control with mixed constraints, IEEE transactions on automatic control, 40, 1818-1823, (1995) · Zbl 0846.93075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.