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Chance Constrained Nonlinear Model Predictive Control 301Li et al. [3] studied the linear case which only concerns the volume constraintsand the outlet flow rate u only affects the mean value of the output V . In contrast,for the nonlinear model in (16), u affects both the mean and covariance of thedistribution of the outlet concentration C.With the assumption that the feed flow rate q(k + i) and feed concentrationC 0 (k + i) follow a positive truncated normal distribution, namely, the low truncatingpoint a 1 in (4) is positive, the following monotone relation can be foundq(k + i) ↑⇒ V (k + i +1|k) ↑C 0 (k + i) ↑⇒ C(k + i +1|k) ↑Thus the chance constraints in (17) can be transformed into(18)P{V min ≤ V (k + i|k) ≤ V max ,i=1,...,P}⇒P{qk+i−1 min ≤ q(k + i − 1) ≤ qmaxk+i−1 ,i=1,...,P}(19)P{C min ≤ C(k + i|k) ≤ C max ,i=1,...,P}min⇒P{C 0 (k+i−1) ≤ C max0(k + i − 1) ≤ C 0 (k+i−1) ,i=1,...,P}Therefore (11) and (12) can be used to compute P{·} and its gradient.The proposed CNMPC controller is applied to the mixing process. The initialvalues of tank volume and product concentration are V (0) = 160 l and C(0) =50g/l, respectively. The inlet flow q(k) and concentration C 0 (k) areassumedto be multivariate truncated normal sequences with the truncating intervals of[0, 20] and [46, 56]. The mean profiles of q(k) andC 0 (k) within a period of 20minutesareshowninFig.1and2.Ineachtimeinterval,theyhavethestanddeviation values of 0.70 and 1.0. In addition, both q(k) andC 0 (k) at differentintervals are assumed to be independent. The dashed lines in Fig.1 and 2 are 10realizations of the disturbance from random samples and it is shown that theuncertainty is considerable. The prediction and control horizon of CNMPC isfixed at P = M = 5 and the lower and upper bounds of the output variables, Vand C, are [130, 170] and [49, 51], respectively. The probability level of α 1 andα 2 are both given as 0.9. The control results are illustrated in Fig.3 to Fig.6.In Fig.3, it can be seen that the control variable u(k) is more flat than the inletand thus the disturbance to the downstream unit is thus decreased. As shown inFig. 1. Inlet flow disturbance profile
302 L. Xie, P. Li, and G. WoznyFig. 2. Inlet concentration disturbance profileFig. 3. Inlet q(k) and outlet flow u(k)Fig. 4. Tank volume V (k)Fig.4 and Fig.5, the tank volume V (k) and outlet product concentration C(k)arestrictly restricted in the predefined bounds. In addition, oscillations also occurin the controlled variables profiles, which means that the controller takes theadvantage of the freedom available to keep the control action as flat as possible.The feasibility analysis of production concentration chance constraint is alsoperformed and the maximum reachable possibility of P{C min ≤ C(k + i|k) ≤C max } in each interval is depicted in Fig.6. It can be seen that the maximumreachable probabilities are all greater than the predefined value (α 2 =0.90), whichimplies that the corresponding CNMPC problem is feasible. Note that at the 4 thand 13 th minute when the concentration approaches its limits, the maximumprobability reaches its minimum value.
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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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Minimum-Distance Receding-Horizon S
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New Extended Kalman Filter Algorith
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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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Distributed Model Predictive Contro
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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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