Assessment and Future Directions of Nonlinear Model Predictive ...

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J(U, x(t)) =Computational Aspects of Approximate Explicit NMPC 183N−1∑k=0[x T t+k|t Qx t+k|t + u T t+kRu t+k]+ x T t+N|t Px t+N|t (9)Here, N is a finite horizon. From a stability point of view it is desirable to chooseδ in (6) as small as possible [15]. If the system is asymptotically stable (or prestabilized)and N is large, then it is more likely that the choice of a small δ willbe possible. The following assumptions are made:A1. P, Q, R ≻ 0.A2. y min < 0
184 A. Grancharova, T.A. Johansen, and P. Tøndelobtain this is to find a close-to-global solution at a point w 0 ∈ X by comparingthe local minima corresponding to several initial guesses and then to use thissolution as an initial guess at the neighbouring points w i ∈ X, i =1, 2, ..., l, i.e.to propagate the solution. This is described in the following procedure:Procedure 1. (close-to-global solution of mp-NLP)Consider any hyper-rectangle X 0 ⊆ X f with vertices Θ 0 = { }θ1,θ 0 2, 0 ..., θM 0 andcenter point w 0 . Consider also the hyper-rectangles X j 0 ⊂ X 0, j =1, 2, ..., N jwith vertices respectively Θ j =X 2 0{θ j 1 ,θj 2 , ..., θj M}, j =1, 2, ..., N j . Suppose X 1 0 ⊂⊂ ... ⊂ X Nj0 . For each of the hyper-rectangles X 0 and X j 0 ⊂ X 0, j =1, 2, ..., N j determine a set}of points that belongs to its facets and denote thisset Ψ j ={ψ j 1 ,ψj 2 , ..., ψj N Ψ, j = 0, 1, 2, ..., N j . Define the set of all points{ } {Nj⋃NjW = {w 0 ,w 1 ,w 2 , ..., w N1 }, where w i ∈ Θ j ⋃∪ Ψ}, j i =1, 2, ..., N 1 .Then:a). Determine a close-to-global solution of the NLP (10)–(11) at the center pointw 0 through the following minimization:U ∗ (w 0 )=argj=0j=0Ò minÓJ(U localUilocal ∈ U1 local ,U2 local ,...,UN localUi ,w 0 ) (13)where Uilocal , i =1, 2, ..., N NU correspond to local minima of the cost functionJ(U, w 0 ) obtained for a number of initial guesses Ui 0, i =1, 2, ..., N N U.b). Determine a close-to-global solution of the NLP (10)–(11) at the points w i ∈W , i =1, 2, ..., N 1 in the following way:1. Determine a close-to-global solution of the NLP (10)–(11) at the centerpoint w 0 by solving problem (13). Let i =1.2. Let W s = {w 0 ,w 1 ,w 2 , ..., w N2 }⊂W bethesubsetofpointsatwhichafeasible solution of the NLP (10)–(11) has been already determined.3. Find the point ˜w ∈ W s that is most close to the point w i , i.e. ˜w =arg min ‖w − w i‖. Let the solution at ˜w be U ∗ ( ˜w).w∈W s4. Solve the NLP (10)–(11) at the point w i with initial guess for the optimizationvariables set to U ∗ ( ˜w).5. If a solution of the NLP (10)–(11) at the point w i has been found, markw i as feasible and add it to the set W s . Otherwise, mark w i as infeasible.6. Let i = i +1.If i ≤ N 1 , go to step 2. Otherwise, terminate. □3.2 Computation of Feasible Approximate SolutionDefinition 1. Let X = {w 1 ,w 2 , ..., w L }⊂R n be a discrete set. A function U(x)is feasible on X if G(U(w i ),w i ) ≤ 0, i ∈{1, 2, ..., L}.We restrict our attention to a hyper-rectangle X ⊂ R n where we seek to approximatethe optimal solution U ∗ (x) to the mp-NLP (10)–(11). We require
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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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The Potential of Interpolation 71Pr
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The Potential of Interpolation 73Th
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The Potential of Interpolation 75de
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Techniques for Uniting Lyapunov-Bas
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94 M. Lazar et al.Lipschitz continu
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96 M. Lazar et al.let X T ⊆ X den
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98 M. Lazar et al.S 0 {j ∈S|0
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100 M. Lazar et al.Fig. 1. State-sp
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102 M. Lazar et al.[11] Grimm, G.,
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Model Predictive Control for Nonlin
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ReferencesModel Predictive Control
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116 F.A.C.C. Fontes, L. Magni, and
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Page 169 and 170: Numerical Methods for Efficient and
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Page 199 and 200: 194 V. Sakizlis et al.presented her
Page 201 and 202: 196 V. Sakizlis et al.linear PWA fu
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Page 205 and 206: 200 V. Sakizlis et al.Remark 1. For
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Page 209 and 210: 204 V. Sakizlis et al.g0.050.10.15x
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A Nonlinear Model Predictive Contro
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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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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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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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