Assessment and Future Directions of Nonlinear Model Predictive ...

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Discrete-Time Non-smooth Nonlinear MPC: Stability and Robustness 1017 ConclusionIn this paper we have presented an overview of stability and robustness theory fordiscrete-time nonlinear MPC while focusing on the application and the extensionof the classical results to discontinuous nonlinear systems. A stability theoremhas been developed, which unifies many of the previous results. An ISS resultfor discrete-time discontinuous nonlinear MPC has also been presented. A newMPC scheme with an ISS guarantee has been developed for a particular class ofdiscontinuous PWA systems.AcknowledgementsThe authors are grateful to the reviewers for their helpful comments. This researchwas supported by the Dutch Science Foundation (STW), Grant “ModelPredictive Control for Hybrid Systems” (DMR. 5675) and the European Communitythrough the Network of Excellence HYCON (contract number FP6-IST-511368).References[1] Kalman, R.E., Bertram, J.E.: Control system analysis and design via the secondmethod of Lyapunov, II: Discrete-time systems. Transactions of the ASME,Journal of Basic Engineering 82 (1960) 394–400[2] Keerthi, S.S., Gilbert, E.G.: Optimal, infinite horizon feedback laws for a generalclass of constrained discrete time systems: Stability and moving-horizon approximations.Journal of Optimization Theory and Applications 57 (1988) 265–293[3] Alamir, M., Bornard, G.: On the stability of receding horizon control of nonlineardiscrete-time systems. Systems and Control Letters 23 (1994) 291–296[4] Meadows, E.S., Henson, M.A., Eaton, J.W., Rawlings, J.B.: Receding horizoncontrol and discontinuous state feedback stabilization. International Journal ofControl 62 (1995) 1217–1229[5] Scokaert, P.O.M., Rawlings, J.B., Meadows, E.B.: Discrete-time stability withperturbations: Application to model predictive control. Automatica 33 (1997)463–470[6] Scokaert, P.O.M., Mayne, D.Q., Rawlings, J.B.: Suboptimal model predictivecontrol (feasibility implies stability). IEEE Transactions on Automatic Control44 (1999) 648–654[7] Magni, L., De Nicolao, G., Scattolini, R.: Output feedback receding-horizon controlof discrete-time nonlinear systems. In: 4th IFAC NOLCOS. Volume 2., Oxford,UK (1998) 422–427[8] Jiang, Z.P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems.Automatica 37 (2001) 857–869[9] Limon, D., Alamo, T., Camacho, E.F.: Input-to-state stable MPC for constraineddiscrete-time nonlinear systems with bounded additive uncertainties. In: 41stIEEE Conference on Decision and Control, Las Vegas, Nevada (2002) 4619–4624[10] Magni, L., Raimondo, D.M., Scattolini, R.: Regional input-to-state stability fornonlinear model predictive control. IEEE Transactions on Automatic Control 51(2006) 1548–1553
102 M. Lazar et al.[11] Grimm, G., Messina, M.J., Tuna, S.E., Teel, A.R.: Nominally robust model predictivecontrol with state contraints. In: 42nd IEEE Conference on Decision andControl, Maui, Hawaii (2003) 1413–1418[12] Grimm, G., Messina, M.J., Tuna, S.E., Teel, A.R.: Examples when nonlinearmodel predictive control is nonrobust. Automatica 40 (2004) 1729–1738[13] Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, andconstraints. Automatica 35 (1999) 407–427[14] Borrelli, F.: Constrained optimal control of linear and hybrid systems. Volume290 of Lecture Notes in Control and Information Sciences. Springer (2003)[15] Kerrigan, E.C., Mayne, D.Q.: Optimal control of constrained, piecewise affinesystems with bounded disturbances. In: 41st IEEE Conference on Decision andControl, Las Vegas, Nevada (2002) 1552–1557[16] Mayne, D.Q., Rakovic, S.V.: Model predictive control of constrained piecewiseaffine discrete-time systems. International Journal of Robust and Nonlinear Control13 (2003) 261–279[17] Lazar, M., Heemels, W.P.M.H., Weiland, S., Bemporad, A., Pastravanu, O.: Infinitynorms as Lyapunov functions for model predictive control of constrained PWAsystems. In: Hybrid Systems: Computation and Control. Volume 3414 of LectureNotes in Computer Science., Zürich, Switzerland, Springer Verlag (2005) 417–432[18] Grieder, P., Kvasnica, M., Baotic, M., Morari, M.: Stabilizing low complexityfeedback control of constrained piecewise affine systems. Automatica 41 (2005)1683–1694[19] Rakovic, S.V., Mayne, D.Q.: Robust model predictive control of contrained piecewiseaffine discrete time systems. In: 6th IFAC NOLCOS, Stuttgart, Germany(2004)[20] Lazar, M.: Model predictive control of hybrid systems: Stability and robustness.PhD thesis, Eindhoven University of Technology, The Netherlands (2006)[21] Kvasnica, M., Grieder, P., Baotic, M., Morari, M.: Multi ParametricToolbox (MPT). In: Hybrid Systems: Computation and Control. LectureNotes in Computer Science, Volume 2993, Pennsylvania, Philadelphia,USA, Springer Verlag (2004) 448–462 Toolbox available for download athttp://control.ee.ethz.ch/∼mpt.A Proof of Theorem 3Let (x ∗ 1|k ,...,x∗ N|k ) denote the state sequence obtained from initial state x 0|k ˜x k and by applying the input sequence u ∗ k to (7a). Let (x 1|k+1,...,x N|k+1 )denote the state sequence obtained from the initial state x 0|k+1 ˜x k+1 = x k+1 +w k = x ∗ 1|k + w k and by applying the input sequence u k+1 (u ∗ 1|k ,...,u∗ N−1|k ,h(x N−1|k+1 )) to (7a).(i) The constraints in (8) are such that: (P1) (x i|k+1 ,x ∗ i+1|k ) ∈ Ω j i+1× Ω ji+1 ,j i+1 ∈S, for all i =0,...,N−2 and, ‖x i|k+1 −x ∗ i+1|k ‖≤ηi µ for i =0,...,N−1.This is due to the fact that x 0|k+1 = x ∗ 1|k +w k, x i|k+1 = x ∗ i+1|k +∏ ip=1 A j pw k fori =1,...,N − 1and‖ ∏ ip=1 A j pw k ‖≤η i µ, which yields ∏ ip=1 A j pw k ∈L i+1µ .Pick the indices j i+1 ∈Ssuch that x ∗ i+1|k ∈ Ω j i+1for all i =1,...,N− 2. Then,
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Lecture Notesin Control and Informa
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Series Advisory BoardF. Allgöwer,
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ContentsFoundations and History of
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ContentsIXRobustness, Robust Design
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ContentsXIHybrid NMPC Control of a
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Nonlinear Model Predictive Control:
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22 S.E. Tuna et al.this says that i
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24 S.E. Tuna et al.In particular, f
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26 S.E. Tuna et al.W N (f(x, ¯κ N
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28 S.E. Tuna et al.where ᾱ := max
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30 S.E. Tuna et al.21.521.5µ=51
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32 S.E. Tuna et al.obstacle and gra
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34 S.E. Tuna et al.[19] C. Prieur.
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36 É. Gyurkovics and A.M. Elaiw[8]
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38 É. Gyurkovics and A.M. ElaiwWe
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40 É. Gyurkovics and A.M. ElaiwWe
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Conditions for MPC Based Stabilizat
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Numerical Methods for Efficient and
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182 A. Grancharova, T.A. Johansen,
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194 V. Sakizlis et al.presented her
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196 V. Sakizlis et al.linear PWA fu
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198 V. Sakizlis et al.non-linear sy
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200 V. Sakizlis et al.Remark 1. For
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202 V. Sakizlis et al.One should no
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204 V. Sakizlis et al.g0.050.10.15x
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Interior-Point Algorithms for Nonli
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A Nonlinear Model Predictive Contro
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Robustness and Robust Design of MPC
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MPC for Stochastic SystemsMark Cann
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y i (k) =∑n um=1MPC for Stochasti
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MPC for Stochastic Systems 259we ha
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MPC for Stochastic Systems 261form
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MPC for Stochastic Systems 263( )κ
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MPC for Stochastic Systems 265Fig.
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MPC for Stochastic Systems 267⎡
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NMPC for Complex Stochastic Systems
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284 H. Chen et al.of the moving hor
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286 H. Chen et al.Hence, at each sa
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288 H. Chen et al.control inputs in
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290 H. Chen et al.Corollary 1. The
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292 H. Chen et al.c A[mol/l]10−10
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294 H. Chen et al.[CSA97] H.Chen,C.
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296 L. Xie, P. Li, and G. Woznyand
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298 L. Xie, P. Li, and G. WoznyDeta
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300 L. Xie, P. Li, and G. Wozny3.2
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302 L. Xie, P. Li, and G. WoznyFig.
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304 L. Xie, P. Li, and G. Wozny[3]
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306 H. Arellano-Garcia et al.applic
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308 H. Arellano-Garcia et al.Fig. 1
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310 H. Arellano-Garcia et al.RECIPE
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312 H. Arellano-Garcia et al.optimi
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314 H. Arellano-Garcia et al.The re
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Interval Arithmetic in Robust Nonli
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Optimal Online Control of Dynamical
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State Estimation Analysed as Invers
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384 K. Naidoo et al.polymer control
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386 K. Naidoo et al.are two key qua
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388 K. Naidoo et al.1. It greatly s
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390 K. Naidoo et al.However, with t
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392 K. Naidoo et al.Fig. 4. Bounded
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394 K. Naidoo et al.1. Maintain pro
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396 K. Naidoo et al.7 Commissioning
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398 K. Naidoo et al.[3] N.M.C. Oliv
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400 R. Franke and J. DoppelhamerImp
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402 R. Franke and J. DoppelhamerFig
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404 R. Franke and J. DoppelhamerFig
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406 R. Franke and J. Doppelhamer[3]
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408 B.A. Foss and T.S. Scheicritica
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410 B.A. Foss and T.S. ScheiFig. 2.
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412 B.A. Foss and T.S. Scheidiscuss
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414 B.A. Foss and T.S. ScheiOther a
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416 B.A. Foss and T.S. ScheiIn the
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Non-linear Model Predictive Control
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NMPC of the Hashimoto Simulated Mov
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486 R. Lepore et al.In this study,
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488 R. Lepore et al.3 Control Strat
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490 R. Lepore et al.400.80.6|x−x
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492 R. Lepore et al.0.950.9w P;30.8
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524 M. Alamirmin V (x u(·,x),T) un
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526 M. AlamirDefinition 4. The syst
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528 M. Alamir♭V is radially unbou
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530 M. AlamirBut according to the s
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532 M. AlamirFig. 2. Stabilization
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534 M. AlamirFig. 3. Closed loop be
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566 X.-B. Hu and W.-H. Chen2 Online
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568 X.-B. Hu and W.-H. Chentime alo
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570 X.-B. Hu and W.-H. ChenFig. 3.
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572 X.-B. Hu and W.-H. ChenTable 4.
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574 M. Fujita et al. CameraCameraTa
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576 M. Fujita et al.J(u, t) =∫t+T
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578 M. Fujita et al.ξ 2[rad/s]6420
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580 M. Fujita et al.Acknowledgement
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582 A. Casavola, D. Famularo, and G
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4. c(t) ∈Cfor all t ∈ Z + ;5. T
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608 W.B. Dunbar and S. Desaand sequ
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610 W.B. Dunbar and S. Desademand r
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612 W.B. Dunbar and S. Desabacklog
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614 W.B. Dunbar and S. Desacasescas
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Robust Model Predictive Control for
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630 J.J. Arrieta-Camacho, L.T. Bieg
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Author IndexAlamir, Mazen 523Alamo,
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Lecture Notes in Control and Inform
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