Assessment and Future Directions of Nonlinear Model Predictive ...

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138 T. Alamo et al.1515X 1X 2X 1X 210105500−5−5−10−10X 3−15−15 −10 −5 0 5 10 15−15 −10 −5 0 5 10 15Fig. 1. Sequence of outer bounds Ĉk leadingto an outer approximation of C ∞ robust control invariantFig. 2. Sequence of sets ˜C k leading to aset−15X 36 ConclusionsIn this paper, an algorithm that provides an outer approximation of C ∞ ispresented. This outer approximation has a number of practical and relevant applications.Based on the outer approximation of C ∞ , an algorithm that providesan inner approximation of C ∞ is given. This algorithm can be used to obtain arobust control invariant set for a piecewise affine system. An illustrative exampleis presented.References[1] Mayne, D.Q., Rawlings, J.B., Rao, C.V. and Scokaert, P.O.M. ”Constrained modelpredictive control: Stability and optimality”. Automatica, 36:789–814, (2000).[2] Limon, D., Alamo, T. and Camacho, E.F. ”Enlarging the domain of attraction ofMPC controllers”. Automatica, 41(4):629–635, (2005).[3] Heemels,W., De Schutter, B. and Bemporad, A., ”Equivalence of hybrid dynamicalmodels”. Automatica, 37:1085–1091, (2001).[4] Lazar, M., Heemels, W.P.M.H., Weiland, S. and Bemporad, A. ”Stabilizationconditions for model predictive control of constrained PWA systems”. Proceedingsof the 43rd IEEE Conference on Decision and Control. 4595-4600, (2004).[5] Johansson, M. and Rantzer, A., ”Computation of piecewise quadratic lyapunovfunctions for hybrid systems”. IEEE Transactions on Automatic Control,34(4):555–559, (1998).[6] Mignone, D., Ferrari-Trecate, G., and Morari, M. ”Stability and Stabilization ofPiecewise Affine and Hybrid Systems: An LMI Approach. Proceedings of the 39thIEEE Conference on Decision and Control. 504-509, (2000).[7] Rodrigues, Luis. ”Stability analysis of piecewise-affine systems using controlledinvariant sets”. Systems and Control Letters, 53(2):157-169, (2004).[8] Kerrigan, E.C. ” Robust Constraint Satisfaction: Invariant Sets and PredictiveControl”. PhD thesis, University of Cambridge, (2000).
On the Computation of Robust Control Invariant Sets 139[9] Grieder, P., Kvanisca, M., Baotic, M. and Morari, M. ”Stabilizing low complexityfeedback control of constrained piecewise affine systems”. Automatica, 41:1683–1694, (2005).[10] Rakovic, S.V., Grieder, P., Kvasnica, M., Mayne, D.Q. and Morari, M. ”Computationof invariant sets for piecewise affine discrete time systems subject tobounded disturbances”. Proceedings of the 43rd IEEE Conference on Decisionand Control. 1418-1423, (2004).[11] Bemporad, A., Fukuda, K. and Torrisi, F. ”Convexity recognition of the union ofpolyhedra”. Computational Geometry, 18, 141-154, (2001).[12] Alamo, T., Cepeda, A., Limon, D., Bravo, J.M., Fiacchini, M. and Camacho,E.F. ”On the computation of robust control invariant sets for piecewise affine systems”Workshop on Assessment and Future Directions on NMPC. Freudentadt-Lauterbad, Germany, August 26-30(2005).[13] Bemporad, A., Morari, M., Dua, V. and Pistikopoulos, E.N. ”The Explicit LinearQuadratic Regulator for Constrained Systems”. Automatica, 38(1), 3-20, (2002).[14] Blanchini F., ”Set invariance in control”. Automatica, 35:1747–1767, (1999).
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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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Page 123 and 124: ReferencesModel Predictive Control
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Page 169 and 170: Numerical Methods for Efficient and
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192 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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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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