Assessment and Future Directions of Nonlinear Model Predictive ...

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548 D. DeHaan and M. Guaythe parameterization, if desired, helps to maximize the efficiency of the parameterization.By allowing for stabilization to a general target set, a broad class ofcontrol problems can be addressed within the given framework.References[1] Diehl, M., Bock, H., and Schloder, J., “A real-time iteration scheme for nonlinearoptimization in optimal feedback control”, SIAM Journal of Control andOptimization, 43(5), 1714–1736 (2005).[2] Ohtsuka, T., “A continuation/GMRES method for fast computation of nonlinearreceding horizon control”, Automatica, 4(40), 563–574 (2004).[3] Cannon, M. and Kouvaritakis, B., “Continuous-time predictive control of constrainednon-linear systems”, in: F. Allgower and A. Zheng (eds.), Nonlinear ModelPredictive Control: Assessment and Future Directions for Research, Birkhauser-Verlag (2000).[4] Findeisen, R. and Allgöwer, F., “Stabilization using sampled-data open-loop feedback– a nonlinear model predictive control perspective”, in: Proc. IFAC Symposiumon Nonlinear Control Systems, 735–740 (2004).[5] DeHaan, D. and Guay, M., “A real-time framework for model predictive controlof continuous-time nonlinear systems”, in: Proc. IEEE Conf. on Decision andControl (2005), To Appear.[6] Mayne, D. Q., Rawlings, J. B., Rao, C. V., and Scokaert, P. O. M., “Constrainedmodel predictive control: Stability and optimality”, Automatica, 36, 789–814(2000).[7] Clarke, F., Ledyaev, Y., Sontag, E., and Subbotin, A., “Asymptotic controllabilityimplies feedback stabilization”, IEEE Trans. Automat. Contr., 42(10), 1394–1407(1997).[8] Nešić, D. and Teel, A., “A framework for stabilization of nonlinear sampled-datasystems based on their approximate discrete-time models”, IEEE Trans. Automat.Contr., 49(7), 1103–1122 (2004).[9] Wills, A. and Heath, W., “Barrier function based model predictive control”, Automatica,40, 1415–1422 (2004).[10] Grimm, G., Messina, M., Tuna, S., and Teel, A., “Examples when model predictivecontrol is non-robust”, Automatica, 40(10), 1729–1738 (2004).[11] Nesterov, Y. and Nemirovskii, A., Interior-Point Polynomial Algorithms in ConvexProgramming, SIAM, Philadelphia (1994).[12] Leis, J. and Kramer, M., “The simultaneous solution and sensitivity analysisof systems described by ordinary differential equations”, ACM Transactions onMathematical Sofware, 14(1), 45–60 (1988).[13] Krstic, M., Kanellakopoulos, I., and Kokotovic, P., Nonlinear and Adaptive ControlDesign, Wiley and Sons, New York (1995).[14] Goebel, R., Hespanha, J., Teel, A., Cia, C., and Sanfelice, R., “Hybrid systems:generalized solutions and robust stability”, in: Proc. IFAC Symposium on NonlinearControl Systems, 1–12 (2004).[15] Lygeros, J., Johansson, K., Simić, S., Zhang, J., and Sastry, S., “Dynamical propertiesof hybrid automata”, IEEE Trans. Automat. Contr., 48(1), 2–17 (2003).[16] Magni, L., De Nicolao, G., Magnani, L., and Scattolini, R., “A stabilizing modelbasedpredictive control for nonlinear systems”, Automatica, 37(9), 1351–1362(2001).
A New Real-Time Method for Nonlinear Model Predictive Control 549A Proof of Lemma 2It can be shown from (11a) that ∇ t J = −L a (x p ,u φ ) −〈∇ x J, f(x p ,u φ )〉. From(12a),dJdt = ∇ tJ + ∇ z J ż=−L a (x p ,u φ )− 〈 ∇ t θJ, Proj{k t αΓ t ∇ t θJ T ,Ξ} 〉 − 〈 ∇ θ J, Proj{k θ Γ θ ∇ θ J T , Θ N } 〉≤−γ L (‖x p ,u φ ‖ Σ)The conditions of the lemma guarantee that J(t 0 ,z 0 ) is bounded (although notuniformly), and the above ensures that J(t, z) ≤ J(t 0 ,z 0 ), for all t ∈ [t 0 ,t 1 ].Since all dynamics in (12a) are locally Lipschitz on the set Z = { z : (t θ ,θ) ∈Φ(t, x(t), P) }, continuity of the solution implies that the states can only exit Zby either 1) reaching the boundary A = cl{Z} \ Z (i.e. the set where x ∈ ∂X,u φ ∈ ∂U, orx p (t θ N ) ∈ ∂X f ), or 2) passing through the boundary B = Z\˚Z. Thefirst case is impossible given the decreasing nature of J and lim z→A J(t, z) =∞,while the second case is prevented by the parameter projection in (12a).B Proof of Lemma 3The first claim follows fromJ(t, z + )−J(t, z) ==δ(x p f )∫0t θ+∫Nt θ NL a (x p (τ,t,z + ,φ), u φ (τ,z + ,φ)) dτ + W a (x p+f) − W a (x p f )L(x κ (τ), φ(τ,κ(x p f )) + µ (B x (x κ (τ)) + B u (φ(τ,κ(x p f ))) )dτ()+ W (x p+f) − W (x p f )+µ B xf (x p+f) − B xf (x p f )≤ 0 (by (4) and (9))where x p f xp (t θ N ,t,z,φ), xp+f x p (t θ+N ,t,z+ ,φ), and x κ (·) is the solution toẋ κ = f(x κ ,φ(t, κ(x p f ))), x κ(0) = x p f. The second claim follows by the propertiesof κ(x) guaranteed by Assumption 2, since the portion of the x p (τ) andu φ (τ)trajectories defined on τ ∈ (t, t θ N ] are unaffected by (14).C Proof of Theorem 1Using the cost J(z π ) as an energy function (where J(z π ) ≡ J(s, x, t θ − s, θ) from(11a), with s arbitrary), the result follows from the Invariance principle in [15,Thm IV.1]. The conditions of [15, Thm IV.1] are guaranteed by Lemmas 2, 3,
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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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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 71Pr
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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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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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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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486 R. Lepore et al.In this study,
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488 R. Lepore et al.3 Control Strat
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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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