Assessment and Future Directions of Nonlinear Model Predictive ...

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348 A. Alessandri, M. Baglietto, and G. Battistelliin [2] for the same class of hybrid systems. More specifically, following the linesof [2], we use a generalized least-squares approach that consists in minimizinga quadratic estimation cost function defined on a moving window including themost recent measurements. To account for the occurrences of switches, the proposedapproach relies on the estimation of both the discrete state and the continuousone. At every time step, first an estimate of the discrete state is determinedon the basis of the observations vector over the most recent time instants, thensuch an estimate is used to determine the quadratic loss function that has to beminimized in order to estimate the continuous state. The main novelties withrespect to the approach presented in [2] concern: i) the development of a newmethod for the estimation of the system mode, based on a minimum-distancecriterion, which turns out to be more computationally efficient; ii) the choice of asimpler quadratic loss function that allows one to derive a closed-form expressionfor the optimal estimate of the continuous state variables, thus avoiding the necessityof resorting to heavy on-line computations. Furthermore, it is importantto remark that the proposed estimation scheme can be always applied regardlessof the form of the sets to which the system and measurement noises belong.Likewise in [2], an explicit relationship is established between the observabilityand the convergence properties of the estimation error. More specifically, inthe presence of bounded noises and under suitable observability conditions, anexponentially converging sequence can be obtained that provides an upper boundon the estimation error for the continuous state vector (even when the systemdynamics is unstable). Simulation results show the effectiveness of the proposedapproach in comparison with the one proposed in [2]. As expected, the newestimation method leads to a great reduction in computation time at the priceof a small decay in performance. The proofs are omitted for the sake of brevity(the interested reader can contact the authors).Let us introduce some definitions that will be useful in the following. Given avector v , ‖v‖ denotes its Euclidean norm. For a generic time-varying vector v t ,let us define vt−Nt △=col(v t−N ,v t−N+1 ,...,v t ). Given a matrix M, wedenoteby σ(M) and ¯σ(M) its minimum and maximum singular values, respectively.Furthermore, M ⊤ is the matrix transpose of M and ‖M‖ =¯σ(M) isitsnorm.n∏△Given n square matrices M 1 ,M 2 ,...,M n , M i = M1 M 2 ···M n is the orderedproduct of such matrices.i=12 A Minimum-Distance Criterion for the Estimation ofthe Discrete StateLet us consider a class of switching discrete-time linear systems described byx t+1 = A(λ t ) x t + w ty t = C(λ t ) x t + v t(1)
Minimum-Distance Receding-Horizon State Estimation 349where t =0, 1,... is the time instant, x t ∈ R n is the continuous state vector (theinitial continuous state x 0 is unknown), λ t ∈L= △ {1, 2,...,L} is the systemmode or discrete state, w t ∈W⊂R n is the system noise vector, y t ∈ R m isthe vector of the measures, and v t ∈V⊂R m is the measurement noise vector.A(λ) andC(λ), λ∈L,are n × n and m × n matrices, respectively. We assumethe statistics of x 0 ,w t , and v t to be unknown as well as the law governing theevolution of the discrete state.In this section, a minimum-distance criterion is proposed for the estimation ofthe discrete state of system (1). More specifically, given the noisy observationsvector yt−Nt over a given time interval [t − N,t], such a criterion allows one to△estimate the switching pattern π t = λtt−N (or at least a portion of it [2]). Sincesystem (1) is time-invariant with respect to the extended state (x t ,λ t ), in thefollowing of this section, for the sake of simplicity and without loss of generality,we shall always consider the interval [0,N].If the evolution of the discrete state is completely unpredictable, the switching△pattern π N = λN0 can assume any value in the set L N+1 . However, in manypractical cases, the a-priori knowledge of the system may allow one to consider arestricted set of “admissible” switching patterns [4]. Think, for example, of thecase in which the discrete state is slowly varying, i.e., there exists a minimumnumber τ of steps between one switch and the following one. Of course, sucha-priori knowledge may make the task of estimating the discrete state from themeasures y0N considerably simpler. As a consequence, instead of considering allthe possible switching patterns belonging to L N+1 , we shall consider a restrictedset P N ⊆L N+1 of all the admissible switching patterns, i.e., of all the switchingpatterns consistent with the a-priori knowledge of the evolution of the discretestate.Let us consider a generic switching pattern π =col △ ( λ (0) ,...,λ (N)) and definethe matrices F (π) and H(π) as⎡⎤C(λ (0) )C(λ (1) )A(λ (0) )F (π) =△ .,⎢⎣N∏ ⎥C(λ (N) ) A(λ (N−i) )⎦H(π) =△ ⎢⎣⎡i=10 0 ··· 0C(λ (1) ) 0 ··· 0C(λ (2) )A(λ (1) ) C(λ (2) ) ··· 0..... ....N−1∏C(λ (N) )i=1.N−2∏⎥A(λ (N−i) ) C(λ (N) ) A(λ (N−i) ) ··· C(λ (N) ⎦)i=1⎤
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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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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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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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