Assessment and Future Directions of Nonlinear Model Predictive ...

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Sampled-Data Model Predictive Control forNonlinear Time-Varying Systems: Stability andRobustness ⋆Fernando A.C.C. Fontes 1 2, Lalo Magni ,andÉva Gyurkovics31 Officina Mathematica, Departamento de Matemática para a Ciência e Tecnologia,Universidade do Minho, 4800-058 Guimarães, Portugalffontes@mct.uminho.pt2 Dipartimento di Informatica e Sistimistica, Università degli Studi di Pavia, viaFerrata 1, 27100 Pavia, Italylalo.magni@unipv.it3 Budapest University of Technology and Economics, Institute of Mathematics,Budapest H-1521, Hungarygye@math.bme.huSummary. We describe here a sampled-data Model Predictive Control frameworkthat uses continuous-time models but the sampling of the actual state of the plant aswell as the computation of the control laws, are carried out at discrete instants of time.This framework can address a very large class of systems, nonlinear, time-varying, andnonholonomic.As in many others sampled-data Model Predictive Control schemes, Barbalat’slemma has an important role in the proof of nominal stability results. It is arguedthat the generalization of Barbalat’s lemma, described here, can have also a similarrole in the proof of robust stability results, allowing also to address a very generalclass of nonlinear, time-varying, nonholonomic systems, subject to disturbances. Thepossibility of the framework to accommodate discontinuous feedbacks is essential toachieve both nominal stability and robust stability for such general classes of systems.1 IntroductionMany Model Predictive Control (MPC) schemes described in the literature usecontinuous-time models and sample the state of the plant at discrete instants oftime. See e.g. [3, 7, 9, 13] and also [6]. There are many advantages in consideringa continuous-time model for the plant. Nevertheless, any implementable MPCscheme can only measure the state and solve an optimization problem at discreteinstants of time.In all the references cited above, Barbalat’s lemma, or a modification of it, isused as an important step to prove stability of the MPC schemes. (Barbalat’s⋆ The financial support from MURST Project “New techniques for the identificationand adaptive control of industrial systems”, from FCT ProjectPOCTI/MAT/61842/2004, and from the Hungarian National Science Foundationfor Scientific Research grant no. T037491 is gratefully acknowledged.R. Findeisen et al. (Eds.): Assessment and Future Directions, LNCIS 358, pp. 115–129, 2007.springerlink.com c○ Springer-Verlag Berlin Heidelberg 2007
116 F.A.C.C. Fontes, L. Magni, and É. Gyurkovicslemma is a well-known and powerful tool to deduce asymptotic stability of nonlinearsystems, especially time-varying systems, using Lyapunov-like approaches;see e.g. [17] for a discussion and applications). To show that an MPC strategyis stabilizing (in the nominal case), it is shown that if certain design parameters(objective function, terminal set, etc.) are conveniently selected, then the valuefunction is monotone decreasing. Then, applying Barbalat’s lemma, attractivenessof the trajectory of the nominal model can be established (i.e. x(t) → 0as t →∞). This stability property can be deduced for a very general class ofnonlinear systems: including time-varying systems, nonholonomic systems, systemsallowing discontinuous feedbacks, etc. If, in addition, the value functionpossesses some continuity properties, then Lyapunov stability (i.e. the trajectorystays arbitrarily close to the origin provided it starts close enough to theorigin) can also be guaranteed (see e.g. [11]). However, this last property mightnot be possible to achieve for certain classes of systems, for example a car-likevehicle (see [8] for a discussion of this problem and this example).A similar approach can be used to deduce robust stability of MPC for systemsallowing uncertainty. After establishing monotone decrease of the valuefunction, we would want to guarantee that the state trajectory asymptoticallyapproaches some set containing the origin. But, a difficulty encountered is thatthe predicted trajectory only coincides with the resulting trajectory at specificsampling instants. The robust stability properties can be obtained, as we show,using a generalized version of Barbalat’s lemma. These robust stability resultsare also valid for a very general class of nonlinear time-varying systems allowingdiscontinuous feedbacks.The optimal control problems to be solved within the MPC strategy are hereformulated with very general admissible sets of controls (say, measurable controlfunctions) making it easier to guarantee, in theoretical terms, the existence ofsolution. However, some form of finite parameterization of the control functionsis required/desirable to solve on-line the optimization problems. It can be shownthat the stability or robustness results here described remain valid when theoptimization is carried out over a finite parameterization of the controls, such aspiecewise constant controls (as in [13]) or as bang-bang discontinuous feedbacks(as in [9]).2 A Sampled-Data MPC FrameworkWe shall consider a nonlinear plant with input and state constraints, where theevolution of the state after time t 0 is predicted by the following model.ẋ(s) =f(s, x(s),u(s)) a.e. s≥ t 0 , (1a)x(t 0 )=x t0 ∈ X 0 , (1b)x(s) ∈ X ⊂ IR n for all s ≥ t 0 , (1c)u(s) ∈ U a.e. s≥ t 0 . (1d)The data of this model comprise a set X 0 ⊂ IR n containing all possible initialstates at the initial time t 0 , a vector x t0 that is the state of the plant measured
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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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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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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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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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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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