Assessment and Future Directions of Nonlinear Model Predictive ...

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30 S.E. Tuna et al.21.521.5µ=51 → 2µ=101 → 21(x1 (0),x 2(0)) 0(x 1(T),x 2(T)) 2π1µ=201 → 20.50.5µ=31 → 2x 20−0.5−1−1.5−2(x 1(0),x 2(0)) 2π(x 1(T),x 2(T)) 0x 1−1 0 1 2 3 4 5 6 7x 20−0.5−1−1.5µ=202 → 1µ=32 → 1µ=102 → 1µ=52 → 1µ=31 → 2µ=32 → 1µ=52 → 1−20 1 2 3 4 5 6x 1(a) Vulnerability of standard MPC tomeasurement noise: trajectories, denotedby ∗, starting close to the thick line approachdifferent equilibrium points.(b) MPC with logic: switching linesfrom q =1→ 2andq =2→ 1 for variousµ.Fig. 3. Swing-up with standard MPC and MPC with memorylines. The margin is independent of the sampling time T as long as NT remainsconstant, N being the horizon for MPC with logic. The design of an MPC withmemory controller and the extension to the case of swinging up the pendulumon a cart follows directly, but due to space limitations we do not include themhere.5.2 Obstacle Avoidance with Constant Horizontal VelocityConsider a vehicle moving on the plane x + = x + δ, y + = y + uδ where δ>0and u ∈ {−1, 1} (note that this system can be thought of as sampling thesystem ẋ =1,ẏ = u). Suppose that the goal for the vehicle is to avoid hittingan obstacle defined by a block of unit height centered about the horizontalaxis at x = 0 (i.e. the vehicle must leave the region y ∈ [−0.5, 0.5]before x = 0). We design a controller using MPC with logic. Let q ∈{1, 2},l 1 ([x, y] T ,u)=l 2 ([x, −y] T ,u)=exp(y), and g(·) = 0. Since the costs are invarianton x and symmetric about the x axis, the decision lines defined by µturn out to be horizontal lines. Let the spacing between these lines be s(µ). Inthis case, s(µ) =ln(µ), since VN 1 ([x, y]T )=µVN 2 ([x, y]T )wheny = ln(µ)2andVN 2 ([x, y]T )=µVN 1 ([x, y]T )wheny = − ln(µ)2for any N.Note that when µ = 1 (or s(µ) = 0) MPC with logic is equivalent tothe standard MPC algorithm implemented using the stage cost l([x, y] T ,u)=min{l 1 ([x, y] T ,u),l 2 ([x, y] T ,u)}. Asµ is increased, the spacing s(µ) increases.Table 1 shows the average number of switches and the total number of crashes for50, 000 runs of the system. The initial conditions are set to be x(0) = −1.5 andy(0) normally distributed (though kept within (−1, 1)) around y = 0. The noiseis uniformly distributed in [−0.8, 0.8]. The key variables of comparison are thespacing of the decision lines s(µ) and the sampling time δ. With the increased
Hybrid MPC: Open-Minded but Not Easily Swayed 31Table 1. Simulations of system with differing decision line spacing and sampling timefor uniformly distributed measurement noise e ∈ [−0.8, 0.8]. Each datum is generatedby 50,000 runs starting at x(0) = −1.5 andy(0) normally distributed constrained to(−1, 1). “TC” is total number of crashes and “AS” is average number of switches.δ 0.1 0.06 0.03 0.01 0.006 0.003 0.001s(µ) TC AS TC AS TC AS TC AS TC AS TC AS TC AS0.00 5110 2.61 5444 4.36 5791 8.73 5927 26.1 5922 43.8 6107 88.3 6125 2630.25 3248 1.75 3716 2.88 3862 5.72 4197 17.2 4334 28.9 4375 57.3 4144 1690.50 1575 1.13 1875 1.76 2253 3.38 2549 10.0 2529 16.7 2625 33.2 2638 1010.75 428 0.70 609 1.01 863 1.80 1102 5.00 1140 8.26 1155 16.4 1189 48.61.00 47 0.46 51 0.57 110 0.86 241 2.05 274 3.21 276 6.14 298 17.71.25 1 0.34 2 0.39 1 0.48 6 0.73 9 0.94 8 1.55 17 3.921.50 0 0.25 0 0.29 0 0.35 0 0.42 0 0.45 0 0.48 0 0.54spacing for a given sampling time, there are fewer crashes, as expected, and thetrajectories contain fewer switches. The number of switches can be thought of asa measure of the sensitivity to measurement noise. As the sampling time is decreased,the system also becomes more sensitive to measurement noise due to thesmaller movements of the system making it difficult to escape the neighborhoodof the horizontal axis.For this system, a crash-free bound on the measurement noise (that solelydepends on µ) can be calculated as follows.Claim. Suppose the MPC with logic controller is implemented with the costfunctions l 1 ,l 2 . If the buffer gain µ>1, the measurement ( noise ) is bounded bys(µ)2 , and the horizontal component of the state x0, and constrain the horizontaldisplacement to z ∈ [−1, 1]. The goal of the vehicle is to reach some target whileavoiding the obstacle whose goal is to reach the vehicle. Both of the agents areconsidered as single points in R 2 and run MPC to achieve their goals as follows.The stage cost of the vehicle puts a high penalty on the current location of the
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:
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Page 34 and 35: 22 S.E. Tuna et al.this says that i
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Page 40 and 41: 28 S.E. Tuna et al.where ᾱ := max
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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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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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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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Controlling Distributed Hyperbolic
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444 K.R. Muske, A.E. Witmer, and R.
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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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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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566 X.-B. Hu and W.-H. Chen2 Online
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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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