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624 S.V. Raković and D.Q. Maynez 0 N (x)⊕R ⊆ Z f ⊕R = X f ⊆ T. Similarly, U(X 0 i (·),µ0 i (·)) ⊆ U for all i ∈ N N−1because v 0 i (x) ∈ V yields that v0 i (x) ⊕ U ν ⊆ V ⊕ U ν ⊆ U for all i ∈ N N−1 .Finally, by Proposition 2 it follows that {Ay + Bµ 0 i (y; x)+w | (y, w) ∈ X 0 i (x) ×W} ⊆X 0 i+1 (x) for all i ∈ N N−1. Clearly, an analogous observation holds forany arbitrary couple (v,z) ∈V N (x) given any arbitrary x ∈X N .We consider the following implicit robust model predictive control law κ 0 N (·)yielded by the solution of P S N (x):κ 0 N (x) v 0 0(x)+ν(x − z 0 (x)) (33)We establish some relevant properties of the proposed controller κ 0 N (·) byexploitingthe results reported in [9].Proposition 3. (i) For all x ∈R, V 0 N (x) =0, z0 (x) =0, v 0 (x) ={0, 0,...,0}and κ 0 N (x) =ν(x). (ii) Let x ∈X N and let (v 0 (x),z 0 (x)) be defined by (28),then for all x + ∈ Ax + Bκ 0 N (x) ⊕ W there exists (v(x+ ),z(x + )) ∈V N (x + ) andV 0 N (x + ) ≤ V 0 N (x) − l(z 0 (x),v 0 0(x)). (34)The main stability result follows from Theorem 1 in [9] (definition of robustexponential stability of a set can be found in [9, 14]):Theorem 2. The set R is robustly exponentially stable for controlled uncertainsystem x + = Ax + Bκ 0 N (x)+w, w ∈ W. The region of attraction is X N .The proposed controller κ 0 N (·) results in a set sequence {X0 0 (x(i))}, where:X 0 0(x(i)) = z 0 (x(i)) ⊕R, i ∈ N (35)and z 0 (x(i)) → 0 exponentially as i →∞. The actual trajectory x(·) {x(i)},where x(i) is the solution of x + = Ax+Bκ 0 N (x)+w at time i ∈ N, correspondingto a particular realization of an infinite admissible disturbance sequence w(·) {w i }, satisfies x(i) ∈ X0 0 (x(i)), ∀i ∈ N. Proposition 3 implies that X0 0 (x(i)) ⊆X N , ∀i ∈ N and Theorem 2 implies that X0 0 (x(i)) →R(where R⊆T) asi →∞exponentially in the Hausdorff metric.4.2 Illustrative ExampleOur illustrative example is a double integrator:[ ] [ ]1 1 1x + = x + u + w (36)0 1 1with w ∈ W { w ∈ R 2 : |w| ∞ ≤ 0.2 }, x ∈ X {x ∈ R 2 ||x| ∞ ≤ 20, x 1 ≤1.85, x 2 ≤ 2}, u ∈ U {u ||u| ≤2} and T { x ∈ R 2 : |x| ∞ ≤ 3 }∩X ,where x i is the i th coordinate of a vector x. The cost function is defined by (26)with Q = 100I, R = 100; the terminal cost V f (x) is the value function (1/2)x ′ P f xfor the optimal unconstrained problem for the nominal system. The horizon isN = 8. The tube cross-section R is constructed by using methods of [15, 16].
Robust Model Predictive Control for Obstacle Avoidance 625The sequence of the sets {X i }, i =0, 1,...,8, where X i is the domain of Vi 0(·)and the terminal set X f = Z f ⊕Rwhere Z f satisfies Assumption 3.2 and is themaximal positively invariant set [1] for system z + =(A+BK)z under the tighterconstraints T f = T ⊖Rand V = U ⊖ U ν where K is unconstrained DLQRcontroller for (A, B, Q, R), is shown in Figure 1 together with obstacles O 1 andO 2 .x 20x 2055X fTZ f−5O 1−5O 2−15−15X(a)−15Target Sets and−5Obstacles0x 15X 7 X 0x 1(b)−15Controllability−5Sets {X0i}5Fig. 1. Sets {X i} 8 i=0, T, Z f , X f and Obstacles O 1 and O 2x 22z0(x(5)) 0 ⊕R0R−4X 0 0 (x(0)) = z 0 0(x(0)) ⊕Rx(0)−8−8−402x 1Fig. 2. RMPC Tube TrajectoryARMPCtube{z0 0(x(i)) ⊕R} for initial state x 0 =(−4.3, −5.3) ′ is shown inFigure 2 for a sequence of random admissible disturbances. The dash-dot line isthe actual trajectory {x(i)} due to the disturbance realization while the dottedline is the sequence {z0 0 (x(i))} of optimal initial states for corresponding nominalsystem.
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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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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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Conditions for MPC Based Stabilizat
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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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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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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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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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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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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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534 M. AlamirFig. 3. Closed loop be
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