Abstract
A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality is proposed in this paper. The addressed approach is based on the fact that zonotopes are more flexible for representing sets than boxes in interval analysis. Then the solver of set inversion via interval analysis is extended to set inversion via zonotope geometry by introducing the novel idea of bisecting zonotopes. The main feature of the extended solver of set inversion is the bisection and the evolution of a zonotope rather than a box. Thus the shape of admissible domains for set inversion can be broadened from boxes to zonotopes and the wrapping effect can be reduced as well by using the zonotope evolution instead of the interval evolution. Combined with global optimization via interval analysis, the extended solver of set inversion via zonotope geometry is further applied to design control invariant sets for constrained nonlinear discrete-time systems in a numerical way. Finally, the numerical design of a control invariant set and its application to the terminal control of the dual-mode model predictive control are fulfilled on a benchmark Continuous-Stirred Tank Reactor example.
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References
Blanchini F.: Set invariance in control. Automatica J. IFAC 35(11), 1747–1767 (1999)
Rakovic S.V., Kerrigan E.C., Kouramas K.I., Mayne D.Q.: Invariant approximations of the minimal robust positively invariant set. IEEE Trans. Automat. Control 50(3), 406–410 (2005)
Mayne D.Q., Rawlings J.B., Rao C.V., Scokaert P.O.M.: Constrained model predictive control: stability and optimality. Automatica J. IFAC 26(6), 789–814 (2000)
Kerrigan, E.C.: Robust constraint satisfaction: invariant sets and predictive control. Ph.D. thesis, University of Cambridge (2000)
Cannon M., Deshmukh V., Kouvaritakis B.: Nonlinear model predictive control with polytopic invariant sets. Automatica J. IFAC 39(8), 1487–1494 (2003)
Bacic M., Cannon M., Kouvaritakis B.: Invariant sets for feedback linearisation based nonlinear predictive control. IEE Proc. Control Theory Appl. 152(3), 259–265 (2005)
Michalska H., Mayne D.Q.: Robust receding horizon control of constrained nonlinear systems. IEEE Trans. Automat. Control 38(11), 1623–1633 (1993)
Chen H., Allgower F.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica J. IFAC 34(10), 1205–1218 (1998)
Kothare M.V., Balakrishnan V., Morari M.: Robust constrained model predictive control using Linear Matrix Inequalities. Automatica J. IFAC 32(10), 1361–1379 (1996)
Magni L., De Nicolao G., Magnani L., Scattolini R.: A stabilizing model-based predictive control algorithm for nonlinear systems. Automatica 37(9), 1351–1362 (2001)
Jaulin L., Kieffer M., Didrit O., Walter E.: Applied Interval Analysis: with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, London (2001)
Jaulin L., Walter E.: Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica J. IFAC 29(4), 1053–1064 (1993)
Walter E., Jaulin L.: Guaranteed characterization of stability domains via set inversion. IEEE Trans. Automat. Control 39(4), 886–889 (1994)
Meizel D., Leveque O., Jaulin L., Walter E.: Initial localization by set inversion. IEEE Trans. Robotics Automat. 18(6), 966–971 (2002)
Kühn W.: Rigorously computed orbits of dynamical systems without the wrapping effect. Computing 61(1), 47–67 (1998)
Hansen E.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992)
Alamo T., Bravo J.M., Camacho E.F.: Guaranteed state estimation by zonotopes. Automatica J. IFAC 41(6), 1035–1043 (2005)
Combastel, C.: A state bounding observer for uncertain non-linear continuous-time systems based on zonotopes. In: IEEE ECC-CDC, Seville, Spain (2005)
Stancu, A., Puig, V., Cuguero, P., Quevedo, J.: Benchmarking on approaches to interval observation applied to robust fault detection. In: European Control Conference, Cambridge, UK (2003)
Ong C.J., Gilbert E.G.: The minimal disturbance invariant set: Outer approximations via its partial sums. Automatica J. IFAC 42(9), 1563–1568 (2006)
Fukuda K.: From the zonotope construction to the Minkowski addition of convex polytopes. J. Symb. Comput. 38(4), 1261–1272 (2004)
Kvasnica, M., Grieder, P., Baotic, M.: Multi-Parametric Toolbox (MPT), http://control.ee.ethz.ch/~mpt/. Cited July 2007
Ingimundarson, A., Bravo, J. M., Puig, V., Alamo, T.: Robust fault diagnosis using parallelotope-based set-membership consistency tests. In: IEEE ECC-CDC, Seville, Spain (2005)
Lasserre J.B.: An analytical expression and an algorithm for the volume of a convex polyhedron in R n. J. Optim. Theory Appl. 39(3), 363–377 (1983)
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Wan, J., Vehi, J. & Luo, N. A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality. J Glob Optim 44, 395–407 (2009). https://doi.org/10.1007/s10898-008-9334-6
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DOI: https://doi.org/10.1007/s10898-008-9334-6