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Receding Horizon Control for Free-Flight Path Optimization 569S2: Receive updated environment data from ATC agencies, set P(k) as the initialpoint to start flight path optimization, and then solve the following minimizationproblemminJ 2(k) (9)P (k+1|k),P (k+2|k),...,P (k+N|k)subject to available headings in Ω and unavailable regions, where P (k +i|k),i =1,...,N, is the end waypoint of ith sub-trajectory in an originalpotential flight path at kth step. Denote the optimal solution as [ ˆP (k +1|k), ˆP (k +2|k),..., ˆP (k + N)|k], and the associated shortcut-taken flightpath as [ ˆP (k +1|k), ˆP (k +2|k),..., ˆP (k + ceil(M(k))|k], where M(k) isthenumber of time slices in the shortcut-taken flight path, and ceil rounds M(k)to the nearest integer towards infinity.S3: When the aircraft arrives at P (k), set P (k +1)= ˆP f (k +1|k) and then flyalong the sub-trajectory determined by [P (k),P(k +1)].S4: If P (k + 1) is not the destination airport, let k = k + 1, and go to Step 2;otherwise, the algorithm stops.3.2 The Length of Receding Horizon and Terminal PenaltyThe choice of N, the horizon length, is important to design the proposed algorithm.The online computational time for each optimization is covered by anupper bound, which mainly depends on N and can be estimated through simulations.As long as the time-slice is larger than the upper bound, no matterhow long the entire flight distance is, the real-time properties of the proposedalgorithm are always guaranteed. Also, a properly chosen receding horizon canwork like a filter to remove unreliable information for the far future. A larger Nresults in heavier online computational burden, but if N is too small, the RHCalgorithm becomes “shortsighted”, and the performance significantly degrades.A properly chosen N should be a good trade-off on these factors which dependon the dynamics of the systems and the quality of the information.However, the nature of the receding horizon concept makes the proposed algorithmonly taking into account the cost within the receding horizon, whichimplies shortsightedness in some sense. The introduction of terminal penaltyW term (k) inJ 2 (k) can compensate for this shortsightedness. When applyingRHC in online FF path optimization, if no terminal penalty is used, very poorperformance even instability (in the sense that the aircraft fails to arrive at thedestination airport) is observed in [9]. Several choices of the terminal penaltyhave been proposed and investigated in [9]. Due to space limit, only one terminalpenalty is presented in this paper, which is defined asW term (k) =(β|θ 3 |/θ 4 +1)dis(P last (k),P DA )/v E (10)where θ 3 , θ 4 and β are illustrated in Fig.3. P SA ,P DA ,P prev (k) andP last (k)are the source airport, the destination airport, the second last waypoint in anoptional FF path, and the last waypoint in an optional FF path, respectively,
570 X.-B. Hu and W.-H. ChenFig. 3. Definition of a terminal penaltyand IW/OW stands for unavailable airspace regions located on/outside the waydirectly from P last (k) toP DA . From Fig.3, one can see that: θ 3 > 0meansthatthe heading of the last sub-trajectory in a potential flight path is over-turning,i.e., aircraft will turn unnecessarily far away from P DA ; θ 3 < 0 means underturning,i.e., aircraft will fly into IW unavailable airspace. |θ 3 |/θ 4 is used to assesshow much the over-turning or under-turning is when compared with θ 4 .Alargervalue of |θ 3 |/θ 4 means more excessive turning of the aircraft (either over-turningor under-turning), and will therefore lead to a heavier terminal penalty. β ≥ 0is a tuning coefficient, and β = 0 when there is no IW unavailable airspace.In order to evaluate the proposed RHC algorithm, the simulation system reportedin [3] is adopted to set up different FF environments, and the conventionaldynamic optimization based GA in [3], denoted as CDO, is also used for the comparativepurpose. The proposed RHC algorithm, denoted as RHC, modifies theonline optimizer in [3] by taking into account the concept of RHC, as discussedbefore. More details of the GA optimizer can be found in [3]. In the simulation,unless it is specifically pointed out, the horizon length is N = 6, and the terminalpenalty W term (k) defined in (10) is adopted for RHC. Six simulation cases aredefined in Tab. 1 with different degrees of complexity of the FF environment,where DD stands for the Direct Distance from the source airport to the destinationone, and UR for Unavailable Region. In Cases 1 to 3, the UR’s are static,while the UR’s vary in Cases 4 to 6; in other words, they may move, change insize, and/or disappear randomly. The comparative simulation focuses on onlinecomputational times (OCT’s) and performances, i.e., actual flight times (AFT’s)from the source airport to the destination one. Numerical results are given inTables 2 to 4, where 10 simulation runs are conducted under either RHC or CDOfor each static case, while 200 simulation runs are carried out for each dynamiccase. Firstly, RHC is compared with CDO in static cases, and simulation resultsare given in Table 2. One can see that CDO achieves the best performance,i.e., the least AFT’s, in all 3 cases. This is understandable because conventionaldynamic optimization strategy, by its nature, should be the best in terms of
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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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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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New Extended Kalman Filter Algorith
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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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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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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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