Assessment and Future Directions of Nonlinear Model Predictive ...

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272 J.M. Maciejowski, A. Lecchini Visintini, and J. LygerosThe quality of the approximation is characterised by the following results,wherePerf max|ɛ =supE X perf(ω, x) subject to Pr{x ∉ X f }
NMPC for Complex Stochastic Systems 2731. The sequence of random variables (Ω,U J ) n should be a Markov chain. Thatis, for each n its probability distribution should depend only on the distributionof (Ω,U J ) n−1 .2. Furthermore, it should be a homogeneous Markov chain, namely it shouldhave an equilibrium distribution, independent of n. This is so that the chainconverges to its equilibrium distribution as n →∞. (In [17] this requirementis relaxed.)3. When the Markov chain is at its equilibrium distribution, the marginal distributionof Ω should be (proportional to) [E X u(ω, x)] J .IfJ is large enough,this distribution will be concentrated close to the optimal value(s) of ω, andmost random samples generated from it will be close to the optimal solution.This is achieved by the following algorithm. We start with a distribution g(ω),which is known as the instrumental (or proposal) distribution and is (almost)freely chosen by the user.Algorithm 2 (MCMC Algorithm)initialization:Extract a sample ω(0) of the random variable Ω(0) fromthe distribution g(ω).Extract J independent samples x j (0) of the random variable X j (0) fromthe (unknown) distribution p Ω(0) (x), by running the simulator Jtimes.Compute U J (0) = ∏ Jj=1 u(ω(0),x j(0)).Set k =0.repeatExtract a sample ˜ω of the random variable ˜Ω ∼ g(ω).Extract J independent samples ˜x j of the random variable ˜X j ∼ p ˜Ω(x),j =1,...,J, by running the simulator J times.Compute ŨJ = ∏ J{ j=1 u(˜ω, ˜x j)}ŨSet ρ =min 1, J g(ω(k))u J (k) g(˜ω)⎧⎪⎨ [˜ω, ŨJ] with probability ρSet [Ω(k +1),U J (k +1)]=⎪⎩[ω(k), U J (k)] with probability 1 − ρSet k = k +1.until TrueIn the initialization step the state [ω(0),U J (0)] is always accepted. In subsequentsteps the new extraction [˜ω, ŨJ] is accepted with probability ρ otherwiseit is rejected and the previous state of the Markov chain [ω(k), u J (k)] is maintained.In practice, the algorithm is executed until a certain number of extractions(say 1000) have been accepted. Because we are interested in the equilibriumdistribution of the Markov chain, the first few (say 10%) of the accepted statesare discarded to allow the chain to reach its equilibrium distribution (“burn inperiod”).
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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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On the Computation of Robust Contro
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142 M. Srinivasarao, S.C. Patwardha
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Numerical Methods for Efficient and
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L i (s i ,u i ):=Numerical Methods
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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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Hard Constraints for Prioritized Ob
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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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