Assessment and Future Directions of Nonlinear Model Predictive ...

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166 H.G. Bock et al.in a straightforward way. Given suitable choices of the objective function andthe final state constraint (3f), stability of the nominal NMPC dynamics can beproven [6, 7, 16].The present article is concerned with efficient ways to calculate the feedbackcontrol ū(x 0 ) or a suitable approximation in real-time while the considered processmoves on.2.3 Direct Multiple Shooting for DAEOur approaches to the online solution of the optimal control problem (3a)–(3f)– the real-time iteration schemes – are based on the direct multiple shootingmethod [5] for DAE models [15], which is briefly reviewed in this section.Parameterization of the Infinite Optimization ProblemThe parameterization of the infinite optimization problem consists of two steps.For a suitable partition of the time horizon [0,T p ]intoN subintervals [t i ,t i+1 ],,0=t 0 0 is non-increasing on t ∈ [t i ,t i+1 ].Consistency of the algebraic states is ensured by adding consistency conditions(7c) for each multiple shooting node at t i in the overall NLP defined in the nextsection. For more details on the relaxation of the DAE the reader is referred,e.g. to [4, 15, 18].Since the trajectories x i (t) andz i (t) ontheinterval[t i ,t i+1 ] are functionsof the initial values s i := (s x i ,sz i ) and control parameters u i only, they will bereferred to as x i (t; s i ,u i )andz i (t; s i ,u i ) in the following. The integral part ofthe cost function is evaluated on each interval independently:
L i (s i ,u i ):=Numerical Methods for Efficient and Fast NMPC 167t∫i+1t iL(x i (t),z i (t),φ i (t, u i )) dt. (6)Note that up to now the multiple shooting parameterization does not involveany discretization of differential operators f,g, but is exact.Structured Nonlinear Programming ProblemThe parameterization of problem (3a)–(3f) using multiple shooting and a suitablecontrol representation leads to the following structured nonlinear programming(NLP) problem :N−1∑minu,si=0L i (s i ,u i ) + E(s x N ) (7a)subject to s x 0 = x 0, (7b)0=g(s x i ,s z i ,φ i (t i ,u i )), i =0, 1,...N−1, (7c)s x i+1 = x i(t i+1 ; s i ,u i ), i =0, 1,...N−1, (7d)r(s x N ) ≥ 0,(7e)with initial condition (7b), consistency conditions (7c), continuity conditions(7d), and terminal constraint (7e). Additional control and path constraints aresupposed to be imposed pointwise for a suitable discretization (at n i points τ ijon each interval, τ ij ∈ [t i ,t i+1 ), j =0,...,n i − 1)h(x i (τ ij ; s i ,u i ),z i (τ ij ; s i ,u i ),u i ) ≥ 0, j =0,...,n i −1, i =0,...N−1. (7f)The NLP (7a)–(7e) can be summarized asP (x 0 ):{bx0 (w)=0min a(w) subject towc(w)≥0,(8)where w contains all the multiple shooting state variables and controls:w =(s x 0,s z 0,u 0 ,s x 1,s z 1,u 1 ,...,u N−1 ,s x N ) ∈ R nw .The function a(w) is the objective (7a), the vector valued equation b x0 (w) =0summarizes all equalities from (7b)-(7d), and the vector valued c(w) ≥ 0containsthe inequality constraints (7f) and (7e).It is important to note that the initial condition (7b) is a linear constraintamong the equality constraints, with the varying parameter x 0 entering linearlyonly in this constraint, so that⎡s x 0 − x ⎤⎡ ⎤0−I nxg(s x 0,s z 0,φ 0 (t 0 ,u 0 ))b x0 (w) =⎢ s⎣1 − x 0(t 1 ; s x 0 ,sz 0 ,u 0) ⎥⎦ = b 00(w)+Lx 0 with L :=⎢ 0 ⎥⎣ ⎦ . (9)..
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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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Page 121 and 122: Model Predictive Control for Nonlin
Page 123 and 124: ReferencesModel Predictive Control
Page 125 and 126: 116 F.A.C.C. Fontes, L. Magni, 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
Page 203 and 204: 198 V. Sakizlis et al.non-linear sy
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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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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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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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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