Assessment and Future Directions of Nonlinear Model Predictive ...

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46 É. Gyurkovics and A.M. Elaiw[17] Ito K, Kunisch K (2002) Asymptotic properties of receding horizon optimal controlproblems. SIAM J. Control Optim. 40:1585–1610[18] Jadbabaie A, Hauser J (2001) Unconstrained receding horizon control of nonlinearsystems. IEEE Trans. Automat. Control 46:776–783[19] Jadbabaie A, Hauser J (2005) On the stability of receding horizon control with ageneral terminal cost. IEEE Trans. Automat. Control 50:674–678[20] Keerthi S S, Gilbert E G (1985) An existence theorem for discrete-time infinitehorizonoptimal control problems. IEEE Trans. Automat. Control 30:907–909[21] Kwon W H, Han S H, Ahn Ch K (2004) Advances in nonlinear predictive control:A survey on stability and optimality. Int J Control Automation Systems 2:15–22[22] Magni L, Scattolini R (2004) Model predictive control of continuous-time nonlinearsystems with piecewise constant control. IEEE Trans. Automat. Control 49:900–906[23] Mayne D Q, Michalska H (1990) Receding horizon control of nonlinear systems.IEEE Trans. Automat. Control 35:814–824[24] Mayne D Q, Rawlings J B, Rao C V, Scokaert P O M (2000) Constrained modelpredictive control: Stability and optimality. Automatica 36:789–814[25] Nešić D, Teel A R (2004) A framework for stabilization of nonlinear sampled-datasystems based on their approximate discrete-time models. IEEE Trans. Automat.Control 49:1103–1122[26] Nešić D,TeelAR,Kokotović P V (1999) Sufficient conditions for stabilization ofsampled-data nonlinear systems via discrete-time approximation. Systems ControlLett. 38:259–270[27] Nešić D, Teel A R, Sontag E D (1999) Formulas relating KL stability estimates ofdiscrete-time and sampled-data nonlinear systems. Systems Control Lett. 38:49–60[28] Polushin I G, Marquez H J (2004) Multirate versions of sampled-data stabilizationof nonlinear systems. Automatica 40:1035–1041[29] Parisini T, Sanguineti M, Zoppoli R (1998) Nonlinear stabilization by recedinghorizonneural regulators. Int. J. Control 70:341–362AppendixProof. (Proof of Theorem 1) To obtain the properties of function VNA we shallsubsequently introduce several notations. Let ρ 1 > 0 be such that B ρ1 ⊂G η ,τ(s) ={ min {T, s/(2Mf (2∆ 0 ,∆ ∗ 2 ))} , if 0 ≤ s ≤ ∆ 0,T, if ∆ 0 0begivenby
Conditions for MPC Based Stabilization of Sampled-Data Nonlinear Systems 47Assumption A2 with ∆ ′ = ∆ ∗ 1 and ∆ ′′ = ∆ ∗ 2. A straightforward computationshows that for any x ∈X ∆ ∗1, u ∈ U ∆ ∗2we have ∥ ∥φ E (t, x, u) ∥ ≤ 2∆ ∗ 1 ,ift ∈ [0,T],and lT E(x, u) ≥ τ(x)ϕ 1 (‖x‖ /2) + T ϕ 1 (‖u‖). Let 0 < h ∗ 1 ≤ h∗ 0 be such thatτ(r 0 )ϕ 1 (r 0 /2) /2 ≥ Tγ(h ∗ 1), where γ is defined by Assumption A2 (ii). Then forall h ∈ (0,h ∗ 1 ]lT,h(x, A u) ≥ σ 1 (‖x‖)+T ϕ 1 (‖u‖),if x ∈X ∆ ∗1\B r0 and u ∈ U ∆ ∗2. Therefore for any x ∈ Ω ∆0 problem PT,h A (N,x) hasan optimal solution u ∗ (x), function VN A is continuous in its domain, V N A(0) = 0and VN A (x) > 0ifx ≠0.BeingN fixed, from Assumption A2 and Lemma 1 itfollows that there exists a 0 0 be arbitrary, let d = σ 1 (σ2 −1 (σ 1(r))/2), let r 0 = σ2 −1 (d)/2 andletδ = σ 1 (r 0 )/2. Let h, δ V , L V be defined by Theorem 1 according to this r 0 andδ, andleth ′ = h. The proof is based on the following claim:Claim A. Let k ∈{1, 2,...,l} be arbitrary and let d be defined above. If forj ≥ 1 ξj−1 E ∈ Γ max(h ′ ), ζj−1 A ∈ Γ max(h ′ ), and there exists a ε 1 ∈Ksuch that∥ ξEj−1 − ζj−1A ∥ ≤ ε1 (h), if 0
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Lecture Notesin Control and Informa
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Series Advisory BoardF. Allgöwer,
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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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On the Computation of Robust Contro
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142 M. Srinivasarao, S.C. Patwardha
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Nonlinear Model Predictive Control:
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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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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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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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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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