Assessment and Future Directions of Nonlinear Model Predictive ...

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Sampled-Data MPC for Nonlinear Time-Varying Systems 125Theorem 1. Assume condition RSC is satisfied and that the differential gamesP(t, x t ,T c ,T p ) have a value for all x t ∈ X and t ≥ t 0 . Then, the robust MPCstrategy robustly stabilizes the system to the target set Θ.ProofThe proof starts by establishing a monotone decreasing property of the MPCValue function. Then the application of the generalized Barbalat’s Lemma yieldsthe robust stability result.Let V (t, x) be the value function of P(t, x, T c − (t − t i ),T p − (t − t i )) witht i = ⌊t⌋ π .Letalsoˆx be the concatenation of predicted trajectories ¯x for eachoptimization problem. That is for i ≥ 0ˆx(t) =¯x i (t) for all t ∈ [t i ,t i + δ) (12)where ¯x i is the trajectory of a solution to problem P(t i ,x ∗ (t i ),T c ,T p ). Note thatˆx coincides with x ∗ at all sampling instants t i ∈ π, but they are typically notidentical on [t i ,t i + δ), since they correspond to different disturbances.The following lemma establishes a monotone decreasing property of V .Lemma 4. ([9], Lemma 4.4) There exists an inter-sample time δ > 0 smallenough such that for any t ′
126 F.A.C.C. Fontes, L. Magni, and É. Gyurkovicsresults on existence of a minimizing solution to optimal control problems (e.g. [7,Proposition 2]). However, for implementation, using any optimization algorithm,the control functions need to be described by a finite number of parameters (theso called finite parameterizations of the control functions). The control can beparameterized as piecewise constant controls (e.g. [13]), polynomials or splinesdescribed by a finite number of coeficients, bang-bang controls (e.g. [9, 10]), etc.Note that we are not considering discretization of the model or the dynamicequation. The problems of discrete approximations are discussed in detail e.g.in [16] and [12].But, in the proof of stability, we just have to show at some point that theoptimal cost (the value function) is lower than the cost of using another admissiblecontrol. So, as long as the set of admissible control values U is constant forall time, an easy, but nevertheless important, corollary of the previous stabilityresults followsIf we consider the set of admissible control functions (including the auxiliarycontrol law) to be a finitely parameterizable set such that the set ofadmissible control values is constant for all time, then both the nominalstability and robust stability results here described remain valid.An example, is the use of discontinuous feedback control strategies of bang-bangtype, which can be described by a small number of parameters and so makethe problem computationally tractable. In bang-bang feedback strategies, thecontrols values of the strategy are only allowed to be at one of the extremesof its range. Many control problems of interest admit a bang-bang stabilizingcontrol. Fontes and Magni [9] describe the application of this parameterizationto a unicycle mobile robot subject to bounded disturbances.References[1] R. W. Brockett. Asymptotic stability and feedback stabilization. In R. W. Brockett,R. S. Millman, and H. S. Sussmann, editors, Differential Geometric ControlTheory, pages 181–191. Birkhouser, Boston, 1983.[2] H. Chen and F. Allgöwer. Nonlinear model predictive control schemes with guaranteedstability. In R. Berber and C. Kravaris, editors, Nonlinear Model BasedProcess Control. Kluwer, 1998.[3] H. Chen and F. Allgöwer. A quasi-infinite horizon nonlinear model predictivecontrol scheme with guaranteed stability. Automatica, 34(10):1205–1217, 1998.[4] F. H. Clarke. Nonsmooth analysis in control theory: a survey. European Journalof Control; Special issue: Fundamental Issues in Control, 7:145–159, 2001.[5] F. H. Clarke, Y. S. Ledyaev, E. D. Sontag, and A. I. Subbotin. Asymptoticcontrollability implies feedback stabilization. IEEE Transactions on AutomaticControl, 42(10):1394–1407, 1997.[6] R. Findeisen, L. Imsland, F. Allgower, and B. Foss. Towards a sampled-data theoryfor nonlinear model predictive control. In W. Kang, M. Xiao, and C. Borges,editors, New Trends in Nonlinear Dynamics and Control, and their applications,volume 295 of Lecture Notes in Control and Information Sciences, pages 295–311.Springer Verlag, Berlin, 2003.
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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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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 65De
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The Potential of Interpolation 67Al
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The Potential of Interpolation 693.
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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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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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New Extended Kalman Filter Algorith
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NLMPC: A Platform for Optimal Contr
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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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Integrating Fault Diagnosis with No
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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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A Two-Time-Scale Control Scheme for
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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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