Assessment and Future Directions of Nonlinear Model Predictive ...

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Sampled-Data MPC for Nonlinear Time-Varying Systems 127[7] F. A. C. C. Fontes. A general framework to design stabilizing nonlinear modelpredictive controllers. Systems & Control Letters, 42:127–143, 2001.[8] F. A. C. C. Fontes. Discontinuous feedbacks, discontinuous optimal controls, andcontinuous-time model predictive control. International Journal of Robust andNonlinear Control, 13(3–4):191–209, 2003.[9] F. A. C. C. Fontes and L. Magni. Min-max model predictive control of nonlinearsystems using discontinuous feedbacks. IEEE Transactions on Automatic Control,48:1750–1755, 2003.[10] L. Grüne. Homogeneous state feedback stabilization of homogeneous systems.SIAM Journal of Control and Optimization, 98(4):1288–1308, 2000.[11] E. Gyurkovics. Receding horizon control via Bolza-time optimization. Systemsand Control Letters, 35:195–200, 1998.[12] E. Gyurkovics and A. M. Elaiw. Stabilization of smpled-data nonlinear systems byreceding horizon control via discrete-time aproximations. Automatica, 40:2017–2028, 2004.[13] L. Magni and R. Scattolini. Model predictive control of continuous-time nonlinearsystems with piecewise constant control. IEEE Transactions on AutomaticControl, 49:900–906, 2004.[14] L. Magni and R. Scattolini. Robustness and robust design of mpc for nonlineardiscrete-time systems. In Preprints of NMPC05 - International Workshop onAssessment and Future Directions of Nonlinear Model Predictive Control, pages31–46, IST, University of Stuttgart, August 26-30, 2005.[15] H. Michalska and R. B. Vinter. Nonlinear stabilization using discontinuousmoving-horizon control. IMA Journal of Mathematical Control and Information,11:321–340, 1994.[16] D. Nesić and A. Teel. A framework for stabilization of nonlinear sampled-datasystems based on their approximate discrete-time models. IEEE Transactions onAutomatic Control, 49:1103–1034, 2004.[17] J. E. Slotine and W. Li. Applied Nonlinear Control. Prentice Hall, New Jersey,1991.AppendixProof of Lemma 4At a certain sampling instant t i , we measure the current state of the plant x tiand we solve problem P(x ti ,T c ,T p ) obtaining as solution the feedback strategy¯k to which corresponds, in the worst disturbance scenario, the trajectory ¯x andcontrol ū. The value of differential game P(x ti ,T c ,T p )isgivenbyV ti (t i ,x ti )=t i+T∫ pt iL(¯x(s), ū(s))ds + W (¯x(t i + T p )). (13)Consider now the family of problems P(x t ,T c − (t − t i ),T p − (t − t i )) for t ∈[t i ,t i + δ). These problems start at different instants t, but all terminate at thesame instant t i + T p . Therefore in the worst disturbance scenario, by Bellman’sprinciple of optimality we have that
128 F.A.C.C. Fontes, L. Magni, and É. GyurkovicsV ti (t, ¯x(t)) =∫t i+T ptL(¯x(s), ū(s))ds + W (¯x(t i + T p )). (14)Suppose that the worst disturbance scenario did not occur and so, at time t, weare at state x ∗ (t) which is, in general, distinct from ¯x(t). Because such scenario ismore favorable, and by the assumption on the existence of value to the differentialgamewehavethatV ti (t, x ∗ (t)) ≤ V ti (t, ¯x(t)) for all t ∈ [t i ,t i + δ). (15)We may remove the subscript t i from the value function if we always choose thesubscript t i to be the sampling instant immediately before t, that is (recall that⌊t⌋ π =max i {t i ∈ π : t i ≤ t})For simplicity define the functionV (t, x) :=V ⌊t⌋π (t, x).V ∗ (t) =V (t, x ∗ (t)).We show that t ↦→ V ∗ (t) is decreasing in two situations:(i) on each interval [t i ,t i + δ)∫ tV ∗ (t) ≤ V ∗ (t i ) −(ii) from one interval to the othertherefore yielding the result.t iM(¯x(s)) ds for all t ∈ [t i ,t i + δ), and all i ≥ 0;V ∗ (t i + δ + ) ≤ V ∗ (t i + δ − ) for all i ≥ 0;(i) The first assertion is almost immediate from (15), (14) and (13).V ∗ (t) ≤ V ti (t, ¯x(t))= V ti (t i , ¯x(t i )) −∫ tt i∫ t= V ti (t i ,x ∗ (t i )) −∫ t≤ V ∗ (t i ) −for all t ∈ [t i ,t i + δ), and all i ≥ 0;t it iM(¯x(s)) dsL(¯x(s), ū(s)) dsL(¯x(s), ū(s)) ds
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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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Hybrid MPC: Open-Minded but Not Eas
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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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A Computationally Efficient Schedul
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The Potential of Interpolation for
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The Potential of Interpolation 693.
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The Potential of Interpolation 71Pr
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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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n p,2 ≤Interior-Point Algorithms
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Hard Constraints for Prioritized Ob
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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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Minimum-Distance Receding-Horizon S
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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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Integration of Economical Optimizat
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Controlling Distributed Hyperbolic
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444 K.R. Muske, A.E. Witmer, and R.
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Robust NMPC for a Benchmark Fed-Bat
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Real-Time Implementation of Nonline
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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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Hybrid NMPC Control of a Sugar Hous
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Application of the NEPSAC Nonlinear
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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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A New Real-Time Method for Nonlinea
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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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