Assessment and Future Directions of Nonlinear Model Predictive ...

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Trajectory Control of Multiple Aircraft: An NMPC Approach 637Fig. 2. Test field for the case studies. Aircraft 1 (dotted) and 2 (solid).respectively. After some simplifying assumptions, the equations of motion arestated for each aircraft as:ẋ = v cos γ cos χ, ˙v = g(u 1 − sin γ),( )ẏ = v cos γ sin χ, ˙χ = − g u 2 sin u 3v cos γ,ḣ = v sin γ, ˙γ = − g v (u 2 cos u 3 +cosγ) ,where g is the standard acceleration. In order to produce flyable maneuvers, constraintsdefining the flight envelope and other restrictions modeling the performancecapabilities of the aircraft are added to the formulation. Using SI units,we have the air density, ρ = 1.222 exp (−h/9144.0) and bounds on velocity,v ≥ v S√9144.0/ρ, v 2 ≤ 2q max /ρ and control variables u j ∈ [u j min ,u j max ],j=1,...3. Here v S is the stall speed and q max is the maximum dynamic pressure.We now consider two aircraft flying in the test field presented in Figure 2,where the three small cylinders are pop-up obstacles; their presence is not knownapriori. Two missions are assigned to each aircraft: wp1→wp3→wp4 for aircraft1, and wp3→wp1→wp2 for aircraft 2. Using the proposed multi-stageNMPC approach, both aircraft are able to reach the assigned way-points, whileavoiding obstacles and (locally) minimizing the load factor terms. In Figure 3,notice that aircraft 1 reached the second way-point in 560 seconds, while aircraft2 reached the second way-point in 660 seconds. The optimization problemsolved at each NMPC horizon varies in size, since different information isadded and subtracted as the flight evolves. The largest NLP solved consists of1406 variables and 1342 constraints. The average CPU time required to solve theNMPC problem was 0.2236 seconds, and the maximum CPU time required was of0.8729 seconds. 11 SUN Java Workstation: dual AMD64-250 processors @ 2.4GHz with 16GB RAM,running Linux operating system.(7)
638 J.J. Arrieta–Camacho, L.T. Biegler, and D. SubramanianFig. 3. Optimal control actions for aircraft 1 (left) and 2 (right). Forward load factor(solid, inner axis) and vertical load factor (dashed, outer axis).5 Conclusions and Future WorkWe present a multi-stage, NMPC-based control strategy for the real time coordinationof multiple aircraft. The controller couples a finite state machine witha nonlinear model predictive controller. With the proposed methodology, it ispossible to coordinate several aircraft, such that they can perform several missionsin partially unknown environments. The main advantage of this controlleris its ability to consider new information as it becomes available and its abilityto define several modes of operation.A case study with two aircraft was presented. It is noticed that the CPU timesrequired to compute the control action are small compared to the physical timeof the implementation (3.7%, on average). Stability of the controller is achievedbased on properties of the dual mode NMPC controller and robustness can bepromoted by tuning certain parameters within the NMPC regulator. Althoughgood results can be obtained with the methodology presented, it is desirable toinvestigate more general conditions under which the controller is stable for theentire FSM, in the presence of disturbances, and also with known robustnessmargins.The NMPC controller could also be used to assist in the decision-makingprocess involved in the unmanned control of aerospace vehicles. We believe thatthe concept of combining the versatility of hybrid systems theory with the powerof large-scale optimal control can prove very useful in the design of advancedcontrol strategies for the efficient coordination of multiple aircraft.References[1] Allgöwer, F., T. A. Badgwell, J. S. Qin, J. B. Rawlings, and S. J. Wright, Nonlinearpredictive control and moving horizon estimation: An introductory overview, inAdvances in Control, Highlights of ECC99, P. M. Frank, ed., Springer, 1999, pp.391.[2] Arrieta-Camacho, J. J., L. T. Biegler and D. Subramanian, NMPC-Based Real-Time Coordination of Multiple Aircraft, submitted for publication (2005).
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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 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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State Estimation Analysed as Invers
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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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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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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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526 M. AlamirDefinition 4. The syst
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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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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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