Assessment and Future Directions of Nonlinear Model Predictive ...

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160 T. Raff, C. Ebenbauer, and F. Allgöwer16151412x 110x 210860 50 100 150 200 250Time50 50 100 150 200 250TimeFig. 1. Closed loop trajectory x 1:Passivity-based MPC (gray) and MPCwithout guaranteed stability (black)Fig. 2. Closed loop trajectory x 2:Passivity-based MPC (gray) and MPCwithout guaranteed stability (black)x 33530252015105x 425201510500 50 100 150 200 250Time00 50 100 150Time200 250Fig. 3. Closed loop trajectory x 3:Passivity-based MPC (gray) and MPCwithout guaranteed stability (black)Fig. 4. Closed loop trajectory x 4:Passivity-based MPC (gray) and MPCwithout guaranteed stability (black)300300250250200200u 1150u 2150100100505000 50 100 150 200 250Time00 50 100 150 200 250TimeFig. 5. Control input u 1: Passivity-basedMPC (gray) and and MPC without guaranteedstability (black)Fig. 6. Control input u 2: Passivity-basedMPC (gray) and and MPC without guaranteedstability (black)
Nonlinear Model Predictive Control: A Passivity-Based Approach 161[Ṡ(x) ≤ − a 3√2gx3 + (1 − γ ] [2)u 2 x 3 + − a 4√2gx4 + (1 − γ ]1)u 1 x 4 ,A 3 A 3 A 4 A 1≤ x 3 u 2 + x 4 u 1 .Consequentely, y =[x 3 ,x 4 ] T is a passive output for the quadruple tank systemwith the storage function S(x) = 1 2 x2 3 + 1 2 x2 4. Since the quadruple tank system iszero state detectable with respect to the output y =[x 3 ,x 4 ] T , the passivity-basednonlinear model predictive control (12) can be used to asymptotically stabilizethe quadruple tank system. In the following, the control task is to stabilize the thequadruple tank system at the equilibrium point x s =[14cm 14 cm 15.3 cm20.3 cm] T . The steady state control input u s at the equilibrium point x s is u s =[43.2 mls38.2 mls ]T . The performance index (5) was chosen as 1000([x 1 − x 1s ] 2 +[x 2 − x 2s ] 2 )+(u 1 − u 1s ) 2 +(u 2 − u 2s ) 2 . Furthermore, the nonlinear model predictivecontrol schemes were implemented with a prediction horizon T =60sand a sampling time δ =1s in the nonlinear model predictive control toolbox[13]. Figure 1 to Figure 6 show the dynamic behavior with the initial conditionx 0 =[5cm 5.6 cm 4 cm 4.5 cm] T of the passivity-based nonlinear model predictivecontroller (12) and the model predictive controller (9), i.e., a model predictivecontroller without guaranteed closed loop stability. As it can be seen fromthe figures, the closed loop system is unstable with the nonlinear model predictivecontroller (9). This instability is not a new fact in model predictive control [1] andhas motivated the development of model predictive control schemes with guaranteedclosed loop stability. In contrast to the model predictive controller (9), thepassivity-based nonlinear model predictive controller (12) asymptotically stabilizesthe quadruple system. Hence, this examples illustrates nicely that the proposedapproach achieves closed loop stability while improving the performance.6 ConclusionsIn this paper a nonlinear model predictive control scheme based on the conceptof passivity was developed. It was shown that by using a specific passivitybasedstate constraint closed loop stability is guaranteed. The basic idea ofthe passivity-based nonlinear model predictive control scheme is to unify optimalcontrol, passivity and nonlinear model predictive control based on theirrelationships. Since passivity and stability are closely related, the proposed approachcan be seen as an alternative to the control Lyapunov function basednonlinear model predictive control scheme [16]. Finally, the passivity-based nonlinearmodel predictive control scheme was applied to control a quadruple tanksystem in order to demonstrate its applicability.References[1] R. R. Bitmead, M. Gevers, I. R. Petersen, and R. J. Kaye, “Monotonicity andStabilizability Properties of Solutions of the Riccati Difference Equation: Propositions,Lemmas, Theorems, Fallacious Conjectures and Counterexamples”, Systemsand Control Letters , pages 309-315, (1985).
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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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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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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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