Assessment and Future Directions of Nonlinear Model Predictive ...

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210 A.G. Wills and W.P. Heathto the point x(µ c ). A damped Newton method is used to generate the new pointx k+1 , and, we employ a long-step approach, so that aggressive reductions inµ k are allowed. This has the consequence of increasing theoretical complexitybounds but tends to be favoured for practical algorithms since these bounds areusually conservative.With this in mind, it remains to find an initial point x 0 and value µ 0 suchthat x 0 is close to x(µ 0 ). This is the subject of Section 3.1 which discusses aninitialisation algorithm.Further to this, however, is a minor technical issue. The algorithm presentedin this section requires that the objective function f is linear, but we are interestedin a more general choice of f (typically quadratic). Nevertheless, it isstraightforward to embed problem (P) into a slightly larger formulation whichdoes have a linear objective. In fact, we do this now using the epigraphic form [2].(A) :min t s.t. f(x) ≤ t, x ∈ G, (A µ): min(t,x) (t,x)1µt−ln(t−f(x))+F (x),where t ∈ R and −ln(t − f(x)) is assumed to be a ν f -self-concordant barrierfunction for f(x) ≤ t (e.g. in the case where f is quadratic then this assumptionis satisfied with ν f = 2). Note that (A µ ) has a unique minimiser, denoted by(t(µ),x(µ)). The following lemma shows that if (t(µ),x(µ)) minimises (A µ )thenx(µ) minimises (P µ ) and it therefore appears reasonable to solve problem (A µ )and obtain the minimiser for problem (P µ ) directly.Lemma 1. Let (t(µ),x(µ)) denote the unique minimiser of problem (A µ ) forsome µ>0, thenx(µ) also minimises problem (P µ ) for the same value of µ.3.1 Initialisation StageThe purpose of an initialisation stage is to generate a point (t 0 ,x 0 ) and a valueµ 0 such that (t 0 ,x 0 )iscloseto(t(µ 0 ),x(µ 0 )) (the minimiser of (A µ )), whichenables the main stage algorithm to progress as described above. The usualapproach is to minimise the barrier function F A (t, x) −ln(t − f(x)) + F (x),which in turn provides a point at the “centre” of the constraint set and close tothe central-path of (A) forµ large enough.However, by construction F A (t, x) is unbounded below as t →∞.Thissproblemis overcome, in the literature, by including a further barrier on the maximumvalue of t resulting in the combined barrier functionF B (t, x) − ln(R − t)+F A (x), F A (t, x) −ln(t − f(x)) + F (x). (2)The unique minimiser of the barrier function F B and the central path ofproblem (A) are related in the following important way.Lemma 2. Let (t ∗ ,x ∗ ) denote the unique minimiser of F B (t, x). Then(t ∗ ,x ∗ )coincides with a point on the central path of problem (A) identified by µ = R−t ∗ .Therefore, minimising the barrier function F B actually provides a point onthe central path of (A), which is precisely the goal of the initialisation stage.
Interior-Point Algorithms for Nonlinear Model Predictive Control 211Given x ∈ G o and f(x) 0 is pre-specified by the user.We are interested in the total number of Newton iterations taken in Step 4 inthe above algorithm. In the following proposition we bound this number usingresults from [7].Proposition 2. The total number of Newton iterations required in Algorithm3.2 is bounded from above byN T = ⌊log κ (µ c /µ 0 )⌋N (β)+N (λ ∗ )+⌈log 2 ((2 − 3 1/2 )/ɛ)⌉.
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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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Page 169 and 170: Numerical Methods for Efficient and
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Page 201 and 202: 196 V. Sakizlis et al.linear PWA fu
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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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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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