Assessment and Future Directions of Nonlinear Model Predictive ...

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Towards the Design of Parametric Model Predictive Controllers 203and ψ g (x(t f )) = −x(t f ) + 1. The last expression represents that at final timethe beam should be positioned at a point where x(t f ) ≥ 1. We also assume thatt 0 = 0. Although, the problem has already been solved for a fixed initial pointx 0 =[0 0] T (as for example in [3]), here the optimal solution is derived for thefirst time as a function of the initial point coordinates.The problem is first dealt in discrete-time as described in Section 2. Thecontinuous-time system is turn into a discrete-time system assuming a samplingtime ∆t such that [t 0 ,t f ] is divided in three equally spaced time intervals of ∆ti.e. [t 0 ,t f ]=3∆t. The discrete-time problemmin ∆tγ kx k+1 = x k +(2gy k ) −1/2 cosγ k ∆t , k =0, 1, 2y k+1 = y k +(2gy k ) −1/2 sinγ k ∆t , k =0, 1, 2y k − 0.5x k − 1 ≤ 0 ,k=0, 1, 2, 3x 3 ≥ 1is then solved by transforming the above problem in (5) and solving the mp-LP problem to acquire the PWA solution. The continuous-time is solved nextfollowing Algorithm 3.1. The results for both the discrete-time case and thecontinuous-time case together with a simulation for x 0 =[0 0] T are shownin Figure 1 and 2. The straight line in both diagrams represents the boundaryof the linear constraint y − 0.5 − 1 ≤ 0. There are three control laws for thecontinuous-time case, depending in which region the initial state is. The controllaw in the unconstrained region (Figure 2.) is obtained by solving the followingsystem of algebraic equalities with respect to c 1 and γ0=c 2 1 − arccos(c 1√ y) − xc2 √ √1 + c 1 y sin arccos c1 yγ = −1/2(2g) 1/2 √c 1 t + arccos c 1 yIn the constrained region the control law is obtained as following. First, thefollowing system of equalities is solvedx(τ ′′ ) − (2g) 1 22c 1τ ′′ + 12c 2 1sin (2g) 1 2 c1 (t f − τ ′′ )=1− (2g) 1 22c 1t f = arccos c √ 11 y +0.5(2g) 2 c 1 τ ′′,x(τ ′′ )=0.1989gτ ′′2 +(2g 1 √0.5(2g) 1 2 )0.896 y0 τ ′′ + x 02 c 1y(τ ′′ )=(0.222(2g) 1 2 τ ′′ + √ y 0) 2, 0.46 = 0.5(2g)12 c1 (t f − τ ′′ )which is a system of five equations with five unknowns t f ,τ ′′ ,c 1 ,x(τ ′′ ),y(τ ′′ ).Then, the control to be applied is given asIf t ≤ τ ′′ then γ =0.46 = arctan(0.5) Else If t ≥ τ ′′ then γ =0.5(2g) 1 2 c 1 (t f − t)As it can be observed from Fig. 1 the approximating, discrete-time, PWAsolution is not the optimal one comparing to the optimal solution as it is givent f
204 V. Sakizlis et al.g0.050.10.15x0 0.2 0.4 0.6 0.8 10y0.20.250.30.350.40.450.5Fig. 1. Discrete-time Brachistochrone Problemg0.050.10.150.20 0.2 0.4 0.6 0.8 10Constrained regionxy0.250.30.35UnconstrainedRegion0.40.45Infeasible Region0.5Fig. 2. Continuous-time Brachistochrone Problemin [3], due to approximation error. On the other hand, the multi-parametricoptimal control law illustrated in Fig. 2, is the optimal solution for each initialcondition contained in the constrained and unconstrained regions.5 ConclusionsIn this paper the discrete-time MPC problem as well as the continuous-timeoptimal control problem were examined. A method was presented for obtaininga linear, PWA approximation to the discrete-time MPC problem where the
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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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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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Page 169 and 170: Numerical Methods for Efficient and
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Page 201 and 202: 196 V. Sakizlis et al.linear PWA fu
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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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