Assessment and Future Directions of Nonlinear Model Predictive ...

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Discrete-Time Non-smooth Nonlinear MPC: Stability and Robustness 103due to x ∗ i+1|k ∈ Ω j i+1∼L i+1µ , it follows by Lemma 2 of [9] that x i|k+1 ∈ Ω ji+1 ∼L i µ ⊂ X i for i =1,...,N − 2. From x N−1|k+1 = x ∗ N|k + ∏ N−1p=1 A j pw k it followsthat F (x N−1|k+1 ) − F (x ∗ N|k ) ≤ ξηN−1 µ, which implies that F (x N−1|k+1 ) ≤θ 1 + ξη N−1 µ ≤ θ due to x ∗ N|k ∈ X T = F θ1 and µ ≤ θ−θ1ξη. Hence, x N −1 N−1|k+1 ∈F θ ⊂ X U ∩ (Ω j ∗ ∼L N−1µ ) ⊂ X U ∩ X N−1 so that h(x N−1|k+1 ) ∈ U and x N|k+1 ∈F θ1 = X T . Thus, the sequence of inputs u k+1 is feasible at time k +1 andProblem 1 with ŨN (x k ) instead of U N (x k ) remains feasible. Moreover, fromg(x, h(x)) ∈ F θ1 for all x ∈ F θ and F θ1 ⊂ F θ it follows that F θ1 is a positivelyinvariant set for system (7a) in closed-loop with u k = h(x k ), k ∈ Z + . Then, sinceF θ1 ⊂ F θ ⊆ (Ω j ∗∼L N−1µ ) ∩ X U ⊂ X i ∩ X U for all i =1,...,N − 1and X T = F θ1 , the sequence of control inputs (h(x 0|k ),...,h(x N−1|k )) is feasiblewith respect to Problem 1 (with ŨN (x k ) instead of U N (x k )) for all x 0|k ˜x k ∈F θ1 . Therefore, X T = F θ1 ⊆ ˜X f (N).(ii) The result of part (i) implies that ˜X f (N) isaRPIsetforsystem(7b)inclosed-loop with the MPC control (4) and disturbances in B µ .Moreover,since0 ∈ int(X T ), we have that 0 ∈ int(˜X f (N)). The choice of the terminal cost andof the stage cost ensures that there exist a, b > 0, α 1 (s) as and α 2 (s) bssuch that α 1 (‖x‖) ≤ ṼMPC(x) ≤ α 2 (‖x‖) for all x ∈ ˜X f (N). Let ˜x k+1 denote thesolution of (7b) in closed-loop with u MPC (·) obtained as indicated in part (i) ofthe proof and let x ∗ 0|k ˜x k. Due to full-column rank of Q there exists γ>0suchthat ‖Qx‖ ≥γ‖x‖ for all x. Then, by optimality, property (P1), x N−1|k+1 ∈ F θand from inequality (5) it follows that:Ṽ (˜x k+1 ) − Ṽ (˜x k) ≤ J(˜x k+1 , u k+1 ) − J(˜x k , u ∗ k)=−L(x ∗ 0|k ,u∗ 0|k )+F (x N|k+1)+[−F (x N−1|k+1 )+F (x N−1|k+1 )] − F (x ∗ N|k )+L(x N−1|k+1,h(x N−1|k+1 ))+N−2∑i=0[L(x i|k+1 , u k+1 (i +1))− L(x ∗ i+1|k ,u∗ i+1|k ) ]≤−L(x ∗ 0|k ,u∗ 0|k )+F (x N|k+1) − F (x N−1|k+1 )+L(x N−1|k+1 ,h(x N−1|k+1 ))()N−2∑+ ξη N−1 + ‖Q‖ η p ‖w k ‖p=0(5)≤−‖Qx ∗ 0|k ‖ + σ(‖w k‖) ≤−α 3 (‖˜x k ‖)+σ(‖w k ‖),with σ(s) (ξη N−1 + ‖Q‖ ∑ N−2p=0 ηp )s and α 3 (s) γs. Thus, it follows thatṼ MPC (·) satisfies the hypothesis of Theorem 3. Hence, the closed-loop system(7b)-(4) is ISS for initial conditions in ˜X f (N) and disturbance inputs in B µ . ✷
Model Predictive Control for NonlinearSampled-Data SystemsL. Grüne 1 ,D.Nešić 2 , and J. Pannek 31 Mathematical Institute, University of Bayreuthlars.gruene@uni-bayreuth.de2 EEE Department, University of Melbourne, Australiad.nesic@ee.mu.oz.au3 Mathematical Institute, University of Bayreuthjuergen.pannek@uni-bayreuth.deSummary. The topic of this paper is a new model predictive control (MPC) approachfor the sampled–data implementation of continuous–time stabilizing feedback laws. Thegiven continuous–time feedback controller is used to generate a reference trajectorywhich we track numerically using a sampled-data controller via an MPC strategy. Hereour goal is to minimize the mismatch between the reference solution and the trajectoryunder control. We summarize the necessary theoretical results, discuss several aspectsof the numerical implemenation and illustrate the algorithm by an example.1 IntroductionInstead of designing a static state feedback with sampling and zero order hold bydesigning a continuous–time controller which is stabilizing an equilibrium anddiscretizing this controller ignoring sampling errors which leads to drawbacks instability, see [5, 8], our approach is to use a continuous–time feedback and toanticipate and minimize the sampling errors by model predictive control (MPC)with the goal of allowing for large sampling periods without loosing performanceand stability of the sampled–data closed loop. Therefore we consider two systems,the first to be controlled by the given continuous–time feedback which willgive us a reference trajectory, and a second one which we are going to controlusing piecewise constant functions to construct an optimal control problem byintroducing a cost functional to measure and minimize the mismatch betweenboth solutions within a time interval.In order to calculate a feedback instead of a time dependent control functionand to avoid the difficulties of solving a Hamilton-Jacobi-Bellman equation foran infinite horizon problem we reduce the infinite time interval to a finite one byintroducing a positive semidefinite function as cost–to–go. To re–gain the infinitecontrol sequence we make use of a receding horizon technique. For this approachwe will show stability and (sub-)optimality of the solution under certain standardassumptions.We will also show how to implement an algorithm to solve this process of iterativelygenerating and solving optimal control problems. The latter one is doneR. Findeisen et al. (Eds.): Assessment and Future Directions, LNCIS 358, pp. 105–113, 2007.springerlink.com c○ Springer-Verlag Berlin Heidelberg 2007
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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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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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Numerical Methods for Efficient and
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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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MPC for Stochastic SystemsMark Cann
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MPC for Stochastic Systems 263( )κ
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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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294 H. Chen et al.[CSA97] H.Chen,C.
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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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310 H. Arellano-Garcia et al.RECIPE
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384 K. Naidoo et al.polymer control
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392 K. Naidoo et al.Fig. 4. Bounded
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398 K. Naidoo et al.[3] N.M.C. Oliv
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406 R. Franke and J. Doppelhamer[3]
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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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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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608 W.B. Dunbar and S. Desaand sequ
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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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