Assessment and Future Directions of Nonlinear Model Predictive ...

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112 L. Grüne, D. Nešić, and J. Pannek5 ExampleHere we present a model of a synchronous generator taken from [2]ẋ 1 = x 2 , ẋ 2 = −b 1 x 3 sin x 1 − b 2 x 2 + P, ẋ 3 = b 3 cos x 1 − b 4 x 3 + E + u. (16)We use the parameter b 1 =34.29, b 2 =0.0, b 3 =0.149, b 4 =0.3341, P =28.222221.81.51.61.51.41.211122210−1−20.2Emulation0.7MPC1.210−1−20.2Emulation0.7MPC1.210−1−20.20.7Reference1.2Fig. 1. Improvement by MPC control over emulation for a 1 =0.45, T =0.1 (left),T =0.5 (middle) and reference solution using continuous–time feedback (right)and E =0.2405, as well as the continuous–time feedback law u(x) =a 1 ((x 1 −x ∗ 1 )b 4 + x 2 ) with feedback gain a 1 > 0, whose purpose is to enlarge the domainof attraction of the locally stable equilibrium x ∗ ≈ (1.12, 0.0, 0.914) (note thatthis equilibrium is locally asymptotically stable also for u ≡ 0). As initial valuewe used the vector x 0 =(0.5, 0.0, 2.0) and generated results for T =0.1, T =0.5and a 1 =0.45.One can see that the fast dynamics of the problem require small samplingperiods to maintain stability using the emulated feedback law. The MPC controlon the other hand not only stabilizes the equilibrium even for rather large T butalso keeps the sampled–data solution close to the reference.6 ConclusionWe proposed an unconstrained model predictive algorithm for the sampled–data implementation of continuous–time stabilizing feedback laws. Stability andinverse optimality results were briefly revisited and numerical issues were discussed.Compared to direct MPC approaches without using continuous–timefeedbacks, advantages of our method are that the sampled–data solutions inheritthe performance properties of the continuous–time controller and that theknowledge of the continuous–time controller helps to reduce the computationalcost of the numerical optimization. Future research will include a systematicstudy about how this knowledge can be used in a numerically efficient way andan extension of our approach to dynamic continuous–time controllers.
ReferencesModel Predictive Control for Nonlinear Sampled-Data Systems 113[1] Grimm G, Messina M J, Teel A R, Tuna S (2005) Model predictive control: forwant of a local control Lyapunov function, all is not lost. In: IEEE Trans. Automat.Contr. 50: 546–558[2] Grüne L (2001) Subdivision Techniques for the Computation of Domains of Attractionsand Reachable Sets. In: Proceedings of NOLCOS 2001: 762–767[3] Gyurkovics E, Elaiw A M (2004) Stabilization of sampled–data nonlinear systemsby receding horizon control via discrete–time approximations. In: Automatica 40:2017–2028[4] Hairer E, Nørsett S P, Wanner G (1993) Solving Ordinary Differential EquationsI. Nonstiff Problems. 2nd Ed., Springer–Verlag[5] Jadbabaie A, Hauser J, Yu J (2001) Unconstrained receding horizon control ofnonlinear systems. In: IEEE Trans. Automat. Contr. 46: 776–783[6] Jadbabaie A, Hauser J (2005) On the stability of receding horizon control with ageneral terminal cost. In: IEEE Trans. Automat. Contr. 50: 674–678[7] Malanowski K, Büskens Ch, Maurer H (1997) Convergence of approximations tononlinear optimal control problems. In Fiacco, A. (ed.) Mathematical Programmingwith Data Perturbations. Marcel Dekker[8] Laila D S, Nešić D, Teel A R (2002) Open and closed loop dissipation inequalitiesunder sampling and controller emulation. In: Europ. J. Control 18: 109–125[9] Monaco S, Normand–Cyrot D (2001) Issues on nonlinear digital control. In: Europ.J. Control 7: 160–178[10] Nešić D, Teel A R, Sontag E D (1999) Formulas relating KL stability estimatesof discrete–time and sampled–data nonlinear systems. In: Syst. Contr. Lett. 38:49–60[11] Nešić D, Teel A R (2004) A framework for stabilization of nonlinear sampled–data systems based on their approximate discrete–time models. In: IEEE Trans.Automat. Contr. 49: 1103-1122[12] Nešić D, Teel A R, Kokotovic P V (1999) Sufficient conditions for stabilization ofsampled-data nonlinear systems via discrete-time approximations. In: Sys. Contr.Lett. 38: 259–270[13] Nešić D,Grüne L (2005) Lyapunov based continuous–time nonlinear controllerredesign for sampled–data implementation. In: Automatica 41: 1143–1156[14] Nešić D,Grüne L (2006) A receding horizon control approach to sampled–dataimplementation of continuous–time controllers. In: Syst. Contr. Lett. 55: 546–558
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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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L i (s i ,u i ):=Numerical Methods
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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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Distributed Model Predictive Contro
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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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