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32 S.E. Tuna et al.obstacle and gradually vanishes at the target. The stage cost of the obstaclevanishes whenever x 1 = z and is positive elsewhere.Applying standard MPC with the vehicle and the obstacle initially alignedvertically at zero horizontal position, the obstacle is able to prevent the vehiclefrom converging to its target. Suppose the vehicle decides to move in the increasingx 1 direction to avoid the obstacle from the right. The obstacle will followthe vehicle with one step of delay. At some point, it will become necessary forthe vehicle to change its course since the optimal path, now that the obstaclehas moved, is now to the the left of the obstacle. Hence the vehicle can get stuckpossibly as shown in Fig. 4(a).Using the MPC with memory approach described in Section 4.1, the problemcan be resolved. Using the same stage cost, M =5,andµ =1.4, the vehicleavoids the obstacle. The sequence in memory is effectively used when the obstacleis at (2, 3): the vehicle stays within his initial course of passing the obstacle fromthe right as shown in Fig. 4(b). Similar results were obtained with MPC withlogic using two symmetric respect to x 1 stage cost functions that allow the vehicleavoid the obstacle from the left and from the right, respectively, but the resultsare omitted because of space limitations.Note that the moving obstacle can be thought of as noise for the measurementof the vehicle’s distance to a static obstacle. Since the displacement of theobstacle has magnitude equal to one, the “measurement noise” for the vehicle israther large.33.52.52k=6,...k=5,7,...k=0,4,... k=1,3,... k=2,...32.52k=0 k=1 k=2x 2(v)1.510.5k=6,7,8,...k=0,1,5,...k=2,3,4,...x 2(v)1.510.5k=0,1,7,... k=2,3,4,5,6k=30−0.50−0.5−1k=6,7,...k=5k=4−1−1.5−3 −2 −1 0 1 2 3x 1(v) ,z (o)(a) Standard MPC: target acquisitionfailed. Vehicle’s vertical position is alwaysequal to two. The thick ellipse correspondsto the bump created by the obstaclein the vehicle’s stage cost.−3 −2 −1 0 1 2 3x 1(v) ,z (o)(b) MPC with memory: target acquisitionsuccessful. Vehicle goes around theobstacle to the right. The ellipse centeredat (1, 1) denotes the bump in the vehicle’sstage cost when the obstacle moves.Fig. 4. Comparison of standard MPC and MPC with memory using identical stagecost and N = 6. The vehicle is represented by ∇, the obstacle by ◦, and the target by× (the origin). The discrete time is denoted by k and the level sets of the stage costfor the vehicle are plotted for the initial condition.
Hybrid MPC: Open-Minded but Not Easily Swayed 33References[1] A. Bemporad, F. Borrelli, and M. Morari. Optimal controllers for hybrid systems:Stability and piecewise linear explicit form. Proc. IEEE Conf. on Decision andControl, Vol. 2, pp. 1810–1815, (2000).[2] F.H. Clarke, S. Yu, S. Ledyaev, L. Rifford, and R.J. Stern. Feedback stabilizationand Lyapunov functions. SIAM Journal on Control and Optimization, Vol. 39,pp. 25–48, (2000).[3] P. Collins. A trajectory-space approach to hybrid systems. Proc.oftheInternationalSymposium on Mathematical Theory of Networks and Systems, (2004).[4] J-M. Coron and L. Rosier. A relation between continuous time-varying and discontinuousfeedback stabilization.Journal of Mathematical Systems, Estimation,and Control, Vol. 4, pp. 67–84, (1994).[5] R. Goebel, J.P. Hespanha, A.R. Teel, C. Cai, and R.G. Sanfelice. Hybrid systems:Generalized solutions and robust stability.Proc. NOLCOS, Vol. 1, pp. 1–12,(2004).[6] R. Goebel and A.R. Teel. Results on solution sets to hybrid systems with applicationsto stability theory. Proc. American Control Conf., Vol. 1, pp. 557–562,(2005).[7] R. Goebel and A.R. Teel. Solutions to hybrid inclusions via set and graphicalconvergence. Automatica, Vol. 42, pp. 573–587, (2006).[8] G. Grimm, M.J. Messina, S.E. Tuna, and A.R. Teel. Examples when nonlinearmodel predictive control is nonrobust. Automatica, Vol. 40, pp. 1729–1738, (2004).[9] G. Grimm, M.J. Messina, S.E. Tuna, and A.R. Teel. Model predictive control:for want of a local control Lyapunov function, all is not lost. IEEE Trans. Auto.Control, Vol. 50, pp. 546–558, (2005).[10] W.P.M.H Heemels, B. De Schutter, and A. Bemporad. Equivalence of hybriddynamical models. Automatica, Vol. 37, pp.1085–1091, (2001).[11] Y. Kuwata and J. How. Receding horizon implementation of MILP for vehicleguidance. In Proc. American Control Conf., pp. 2684–2685, (2005).[12] T. Lapp and L. Singh. Model predictive control based trajectory optimizationfor nap-of-the-earth (NOE) flight including obstacle avoidance. Proc. AmericanControl Conf., Vol. 1, pp. 891–896, (2004).[13] J. Lygeros, K.H. Johansson, S.N. Simić, J. Zhang, and S.S. Sastry. Dynamicalproperties of hybrid automata. IEEE Trans. Auto. Control, Vol. 48, pp. 2–17,(2003).[14] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained modelpredictive control: Stability and optimality. Automatica, Vol. 36, pp. 789–814,(2000).[15] M.J. Messina, S.E. Tuna, and A.R. Teel. Discrete-time certainty equivalence outputfeedback: allowing discontinuous control laws including those from modelpredictive control. Automatica, Vol. 41, pp. 617–628, (2005).[16] P. Mhaskar, N.H. El-Farrah, and P.D. Christofides. Robust hybrid predictive controlof nonlinear systems. Automatica, Vol. 41, pp. 209–217, (2005).[17] H. Michalska and D.Q. Mayne. Robust receding horizon control of constrainednonlinear systems. IEEE Trans. Auto. Control, Vol. 38, pp. 1623–1633, (1993).[18] P. Ogren and N.E. Leonard. A convergent dynamic window approach to obstacleavoidance. IEEE Trans. on Robotics, Vol. 21, pp. 188–195, (2005).
Page 2 and 3: Lecture Notesin Control and Informa
Page 4 and 5: Series Advisory BoardF. Allgöwer,
Page 7 and 8: ContentsFoundations and History of
Page 9 and 10: ContentsIXRobustness, Robust Design
Page 11 and 12: ContentsXIHybrid NMPC Control of a
Page 13 and 14: Nonlinear Model Predictive Control:
Page 29 and 30: Hybrid MPC: Open-Minded but Not Eas
Page 31 and 32: Hybrid MPC: Open-Minded but Not Eas
Page 34 and 35: 22 S.E. Tuna et al.this says that i
Page 36 and 37: 24 S.E. Tuna et al.In particular, f
Page 38 and 39: 26 S.E. Tuna et al.W N (f(x, ¯κ N
Page 40 and 41: 28 S.E. Tuna et al.where ᾱ := max
Page 42 and 43: 30 S.E. Tuna et al.21.521.5µ=51
Page 46 and 47: 34 S.E. Tuna et al.[19] C. Prieur.
Page 48 and 49: 36 É. Gyurkovics and A.M. Elaiw[8]
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Page 55 and 56: Conditions for MPC Based Stabilizat
Page 61 and 62: A Computationally Efficient Schedul
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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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Nonlinear Model Predictive Control:
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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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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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