Assessment and Future Directions of Nonlinear Model Predictive ...

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260 M. Cannon, P. Couchmann, and B. Kouvaritakisx(k +1)=Φ(k)x(k), Φ(k) = ¯Φ +L∑q i (k)Φ i , q(k) ∼N(0,I) (11)(where ¯Φ = A + ¯BK and Φ i = B i K), then confidence ellipsoids for q can beused to determine conditions on the vertices x j , j =1,...,M of Ω so that Ωis invariant with a given probability. Specifically, the condition vi T Φx j ≤ 1isequivalent to x T j [ΦT 1 v i ··· Φ T L v i]q ≤ 1 − x T ¯Φ j T v i , and, since ‖q‖ 2 is distributedas χ 2 with L degrees of freedom, it follows that vi T Φx j ≤ 1 with probability p Ω ifi=1r Ω∥ ∥ xTj[ ΦT1 v i ··· Φ T L v i] ∥∥2 ≤ 1 − x T j ¯Φv i (12)where r Ω satisfies Pr(χ 2 (L)
MPC for Stochastic Systems 261form based on the solution of a pair of Lyapunov equations, and shows that theLyapunov-like property:E k[L(k + N +1|k +1)+l(k + N|k +1)]≤ L(k + N|k) (15)holds whenever predictions at time k + 1 are generated by the sequenceu(k +1)={u(k +1|k),...,u(k + N − 1|k),Kx(k + N|k +1)} (16)where u(k) ={u(k|k),u(k +1|k),...,u(k + N − 1|k)} is the predicted inputsequence at time k and u(k) =u(k|k).To simplify notation, let x δ = x − ¯x, where¯x(k + j|k) =E k x(k + j|k), anddefine[Z 1 (k + j|k) =E k x(k + j|k)x T (k + j|k) ] ,[Z 2 (k + j|k) =E k xδ (k + j|k)x T δ (k + j|k)] .Lemma 3. If the terminal penalty in (4) is defined byL(k + N|k) =Tr ( Z 1 (k + N|k)S 1)+(κ21 − 1)Tr ( Z 2 (k + N|k)S 2)(17)where S 1 = S T 1 ≻ 0 and S 2 = S T 2 ≻ 0 are the solutions of the Lyapunov equations¯Φ T S 1 ¯Φ + L ∑i=1Φ T i¯Φ T S 2 ¯Φ + c1 c T 1 = S 2(18a)(S1 +(κ 2 1 − 1)S )2 Φi + c 1 c T 1 = S 1 (18b)then L(k+N|k) is the cost-to-go: L(k+N|k) = ∑ ∞j=Nl(k+j|k) for the closed-loopsystem formed by (3) under the terminal control law u(k + j|k) =Kx(k + j|k).Proof. With u(k + j|k) =Kx(k + j|k), it is easy to show that, for all j ≥ N:Z 1 (k + j +1|k) = ¯ΦZ 1 (k + j|k) ¯Φ T +Z 2 (k + j +1|k) = ¯ΦZ 2 (k + j|k) ¯Φ T +L∑Φ i Z 1 (k + j|k)Φ T ii=1L∑Φ i Z 2 (k + j|k)Φ T i .Using these expressions and (18a,b) to evaluate L(k + j +1|k), we obtaini=1(19a)(19b)L(k + j +1|k)+l(k + j|k) =L(k + j|k), (20)which can be summed over all j ≥ N to giveL(k + N|k) − lim L(k + j|k) = ∑∞ l(k + j|k),j→∞but x(k +1)=Φx(k) is necessarily mean-square stable [16] in order that thereexist positive definite solutions to (18a,b), and it follows that lim j→∞ L(k +j|k) =0.j=N
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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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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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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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