Model Predictive Control edited by Tao ZHENG SCIYO Model Predictive Control Edited by Tao ZHENG Published by Sciyo Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2010 Sciyo All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by Sciyo, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Jelena Marusic Technical Editor Sonja Mujacic Cover Designer Martina Sirotic Image Copyright Richard Griffin, 2010. Used under license from Shutterstock.com First published September 2010 Printed in India A free online edition of this book is available at www.sciyo.com Additional hard copies can be obtained from publication@sciyo.com Model Predictive Control, Edited by Tao ZHENG p. cm. ISBN 978-953-307-102-2 SCIYO.COM WHERE KNOWLEDGE IS FREE free online editions of Sciyo Books, Journals and Videos can be found at www.sciyo.com Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Preface VII Robust Model Predictive Control Design 1 Vojtech Veselý and Danica Rosinová Robust Adaptive Model Predictive Control of Nonlinear Systems 25 Darryl DeHaan and Martin Guay A new kind of nonlinear model predictive control algorithm enhanced by control lyapunov functions 59 Yuqing He and Jianda Han Robust Model Predictive Control Algorithms for Nonlinear Systems: an Input-to-State Stability Approach 87 D. M. Raimondo, D. Limon, T. Alamo and L. Magni Model predictive control of nonlinear processes 109 Author Name Approximate Model Predictive Control for Nonlinear Multivariable Systems 141 JonasWitt and HerbertWerner Multi-objective Nonlinear Model Predictive Control: Lexicographic Method 167 Tao ZHENG, Gang WU, Guang-Hong LIU and Qing LING Model Predictive Trajectory Control for High-Speed Rack Feeders 183 Harald Aschemann and Dominik Schindele Plasma stabilization system design on the base of model predictive control 199 Evgeny Veremey and Margarita Sotnikova Predictive Control of Tethered Satellite Systems 223 Paul Williams MPC in urban traffic management 251 Tamás Tettamanti, István Varga and Tamás Péni Contents VI Chapter 12 Chapter 13 Off-line model predictive control of dc-dc converter 269 Tadanao Zanma and Nobuhiro Asano Nonlinear Predictive Control of Semi-Active Landing Gear 283 Dongsu Wu, Hongbin Gu, Hui Liu Since Model Predictive Heuristic Control (MPHC), the earliest algorithm of Model Predictive Control (MPC), was proposed by French engineer Richalet and his colleagues in 1978, the explicit background of industrial application has made MPC develop rapidly to satisfy the increasing request from modern industry. Different from many other control algorithms, the research history of MPC is originated from application and then expanded to theoretical eld, while ordinary control algorithms often has applications after sufcient theoretical research. Nowadays, MPC is not just the name of one or some specic computer control algorithms, but the name of a specic thought in controller design, from which many kinds of computer control algorithms can be derived for different systems, linear or nonlinear, continuous or discrete, integrated or distributed. The basic characters of the thought of MPC can be summarized as the model used for prediction, the online optimization based on prediction and the feedback compensation for model mismatch, while there is no special demands on the form of model, the computational tool for online optimization and the form of feedback compensation. After three decades’ developing, the MPC theory for linear systems is now comparatively mature, so its applications can be found in almost every domain in modern engineering. While, MPC with robustness and MPC for nonlinear systems are still problems for scientists and engineers. Many efforts have been made to solve them, though there are some constructive results, they will remain as the focuses of MPC research for a period in the future. In rst part of this book, to present the recent theoretical improvements of MPC, Chapter 1 will introduce the Robust Model Predictive Control and Chapter 2 to Chapter 5 will introduce some typical methods to establish Nonlinear Model Predictive Control, with more complexity, MPC for multi-variable nonlinear systems will be proposed in Chapter 6 and Chapter 7. To give the readers an overview of MPC’s applications today, in second part of the book, Chapter 8 to Chapter 13 will introduce some successful examples, from plasma stabilization system to satellite system, from linear system to nonlinear system. They can not only help the readers understand the characters of MPC, but also give them the guidance for how to use MPC to solve practical problems. Authors of this book truly want to it to be helpful for researchers and students, who are concerned about MPC, and further discussions on the contents of this book are warmly welcome. Preface VIII Finally, thanks to SCIYO and its ofcers for their efforts in the process of edition and publication, and thanks to all the people who have made contributes to this book. Editor Tao ZHENG University of Science and Technology of China Robust Model Predictive Control Design 1 Robust Model Predictive Control Design Vojtech Veselý and Danica Rosinová 0 Robust Model Predictive Control Design Vojtech Veselý and Danica Rosinová Institute for Control and Industrial Informatics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkoviˇcova 3, 81219 Bratislava Slovak Republic 1. Introduction Model predictive control (MPC) has attracted notable attention in control of dynamic systems and has gained the important role in control practice. The idea of MPC can be summarized as follows, (Camacho & Bordons, 2004), (Maciejovski, 2002), (Rossiter, 2003) : • Predict the future behavior of the process state/output over the finite time horizon. • Compute the future input signals on line at each step by minimizing a cost function under inequality constraints on the manipulated (control) and/or controlled variables. • Apply on the controlled plant only the first of vector control variable and repeat the previous step with new measured input/state/output variables. Therefore, the presence of the plant model is a necessary condition for the development of the predictive control. The success of MPC depends on the degree of precision of the plant model. In practice, modelling real plants inherently includes uncertainties that have to be considered in control design, that is control design procedure has to guarantee robustness properties such as stability and performance of closed-loop system in the whole uncertainty domain. Two typical description of uncertainty, state space polytope and bounded unstruc- tured uncertainty are extensively considered in the field of robust model predictive control. Most of the existing techniques for robust MPC assume measurable state, and apply plant state feedback or when the state estimator is utilized, output feedback is applied. Thus, the present state of robustness problem in MPC can be summarized as follows: Analysis of robustness properties of MPC. (Zafiriou & Marchal, 1991) have used the contraction properties of MPC to develop necessary- sufficient conditions for robust stability of MPC with input and output constraints for SISO systems and impulse response model. (Polak & Yang, 1993) have analyzed robust stability of MPC using a contraction constraint on the state. MPC with explicit uncertainty description. ( Zheng & Morari, 1993), have presented robust MPC schemes for SISO FIR plants, given un- certainty bounds on the impulse response coefficients. Some MPC consider additive type of uncertainty, (delaPena et al., 2005) or parametric (structured) type uncertainty using CARIMA model and linear matrix inequality, (Bouzouita et al., 2007). In (Lovas et al., 2007), for open- loop stable systems having input constraints the unstructured uncertainty is used. The robust stability can be established by choosing a large value for the control input weighting matrix R in the cost function. The authors proposed a new less conservative stability test for determin- ing a sufficiently large control penalty R using bilinear matrix inequality (BMI). In (Casavola 1 Model Predictive Control2 et al., 2004), robust constrained predictive control of uncertain norm-bounded linear systems is studied. The other technique- constrained tightening to design of robust MPC have been proposed in (Kuwata et al., 2007). The above approaches are based on the idea of increasing the robustness of the controller by tightening the constraints on the predicted states. The mixed H 2 /H ∞ control approach to design of MPC has been proposed by (Orukpe et al., 2007) . Robust constrained MPC using linear matrix inequality (LMI) has been proposed by (Kothare et al., 1996), where the polytopic model or structured feedback uncertainty model has been used. The main idea of (Kothare et al., 1996) is the use of infinite horizon control laws which guar- antee robust stability for state feedback. In (Ding et al., 2008) output feedback robust MPC for systems with both polytopic and bounded uncertainty with input/state constraints is pre- sented. Off-line, it calculates a sequence of output feedback laws based on the state estimators, by solving LMI optimization problem. On-line, at each sampling time, it chooses an appro- priate output feedback law from this sequence. Robust MPC controller design with one step ahead prediction is proposed in (Veselý & Rosinová , 2009). The survey of optimal and robust MPC design can be consulted in (Mayne et al., 2000). Some interesting results for nonlinear MPC are given in (Janík et al., 2008). In MPC approach generally, control algorithm requires solving constrained optimization prob- lem on-line (in each sampling period). Therefore on-line computation burden is significant and limits practical applicability of such algorithms to processes with relatively slow dynam- ics. In this chapter, a new MPC scheme for an uncertain polytopic system with constrained control is developed using model structure introduced in (Veselý et al., 2010). The main con- tribution of the first part of this chapter is that all the time demanding computations of output feedback gain matrices are realized off-line ( for constrained control and unconstrained control cases). The actual value of control variable is obtained through simple on-line computation of scalar parameter and respective convex combination of already computed matrix gains. The developed control design scheme employs quadratic Lyapunov stability to guarantee the ro- bustness and performance (guaranteed cost) over the whole uncertainty domain. The first part of the chapter is organized as follows. A problem formulation and preliminaries on a predictive output/state model as a polytopic system are given in the next section. In Section 1.2, the approach of robust output feedback predictive controller design using linear matrix inequality is presented. In Section 1.3, the input constraints are applied to LMI feasi- ble solution. Two examples illustrate the effectiveness of the proposed method in the Section 1.4. The second part of this chapter addresses the problem of designing a robust parameter dependent quadratically stabilizing output/state feedback model predictive control for linear polytopic systems without constraints using original sequential approach. For the closed-loop uncertain system the design procedure ensures stability, robustness properties and guaran- teed cost. Finally, conclusions on the obtained results are given. Hereafter, the following notational conventions will be adopted: given a symmetric matrix P = P T ∈ R n×n , the inequality P > 0(P ≥ 0) denotes matrix positive definiteness (semi- definiteness). Given two symmetric matrices P, Q, the inequality P > Q indicates that P − Q > 0. The notation x(t + k) will be used to define, at time t, k-steps ahead prediction of a system variable x from time t onwards under specified initial state and input scenario. I denotes the identity matrix of corresponding dimensions. 1.1 Problem formulation and preliminaries Let us start with uncertain plant model described by the following linear discrete-time uncer- tain system with polytopic uncertainty domain x (t + 1 ) = A(α )x(t) + B(α)u(t) (1) y (t) = Cx(t) where x(t) ∈ R n , u(t) ∈ R m , y(t) ∈ R l are state, control and output variables of the system, respectively; A (α), B(α) belong to the convex set S = {A(α) ∈ R n×n , B(α) ∈ R n×m } (2) {A(α) = N ∑ j=1 A j α j B(α) = N ∑ j=1 B j α j , α j ≥ 0 }, j = 1, 2 N, N ∑ j=1 α j = 1 Matrices A i , B i and C are known matrices with constant entries of corresponding dimensions. Simultaneously with (1) we consider the nominal model of system (1) in the form x (t + 1 ) = A o x(t) + B o u(t) y(t) = Cx(t) (3) where A o , B o are any constant matrices from the convex bounded domain S (2). The nominal model (3) will be used for prediction, while (1) is considered as real plant description provid- ing plant output. Therefore in the robust controller design we assume that for time t output y (t) is obtained from uncertain model (1), predicted outputs for time t + 1, t + N 2 will be obtained from model prediction, where the nominal model (3) is used. The predicted states and outputs of the system (1) for the instant t + k, k = 1, 2, N 2 are given by • k=1 x (t + 2 ) = A o x(t + 1) + B o u(t + 1) = A o A(α)x(t) + A o B(α)u(t) + B o u(t + 1) • k=2 x (t + 3 ) = A 2 o A(α)x(t) + A 2 o B(α)u(t) + A o B o u(t + 1) + B o u(t + 2) • for k x (t + k + 1) = A k o A(α)x(t) + A k o B(α)u(t) + k−1 ∑ i=0 A k−i−1 o B o u(t + 1 + i) (4) and corresponding output is y (t + k) = Cx(t + k) (5) Consider a set of k = 0, 1, 2, , N 2 state/output model predictions as follows z (t + 1 ) = A f (α)z(t) + B f (α)v(t), y f (t) = C f z(t) (6) where z (t) T = [x(t) T x(t + N 2 ) T ], v(t) T = [u(t) T u(t + N u ) T ] (7) y f (t) T = [y(t) T y(t + N 2 ) T ] and B f (α) =     B (α) 0 0 A o B(α) B o 0 0 A N 2 o B(α) A N 2 −1 o B o A N 2 −N u o B o     (8) [...]... Continuous model has been converted to discrete time one with sampling time of 0.1s, the nominal model turns to (1) where   9996 0383 013 1 −.0322  −.0056 9647 7446 00 01   Ao =    002 −.0097 9543 0 00 01 −.0005 0978 1 10 Model Predictive Control Fig 1 Dynamic behavior of controlled system for unconstrained case for u(t)  10 02 018 3  C = 0586  0029  00 01  −.0 615  Bo =  − .11 33 −.0057 and model. .. are A1u  0  0  = 0 0 B1u  0 0.0005 0 0 0  −0.02 =  −0 .12 0 1 0 0 1 0 0.0 017 0.00 01 0 0 0 0 7.74  0 0   0  0  0 .12 0 .1  −3  10 0  0 For the case when number of uncertainty is p = 1, the number of vertices is N = 2 p = 2, the matrices (2) are calculated as in example 1 Note that nominal model Ao is unstable Consider N2 = Nu = 1, = 20000 and weighting matrices Q0 = Q1 = 1I, R0 = R1 = I... obtained from (12 ) for F = F1 ; compute the actual feedback gain matrix from (29) and respective constrained control vector from (12 ) All on-line computations follow general MPC scheme, i.e the first part of computed control vector u(t) is applied on real controlled plant and the other part of control vector is used for model prediction Robust Model Predictive Control Design 9 1. 4 EXAMPLES Two examples... proposed control algorithm (29) for output feedback gain matrix F The input constraint case is studied, in each case maximal value of u(t) is checked; stability is assessed using spectral radius of closed-loop system matrix First example serves as a benchmark The model of double integrator turns to (1) where Ao = Bo = 1 1 0 1 1 0 ,C = 0 1 and uncertainty matrices are A1u = 0. 01 0.02 B1u = 0. 01 0.03 0.0 01. .. can be derived − − − − P 1 ≤ Yk 1 ( P − Yk )Yk 1 − Yk 1 = lin(− P 1 ) (19 ) where Yk , k = 1, 2, in iteration process Yk = P We can recast bilinear matrix inequality (18 ) to the linear matrix inequality (LMI) using linearization (19 ) The following LMI is obtained for quadratic stability   −P + Q C T F T A Ti + C T F T B Ti f f f f   (20) FC f − R 1 0   ≤ 0 i = 1, 2, N 1 ) A f i + B f i FC f 0... responses for in the paper proposed control algorithm (29) and (32) are in Fig.6 Maximal values of control variables are about u1max = 0.75 < 1, u2max = 0.6 < 1 Input constraint conditions were applied only for plant control variable u(t) Both examples show that using tuning parameter θ the demanded input Robust Model Predictive Control Design 11 Fig 2 Dynamic behavior of controlled system for constrained... A f j αj j =1 B f (α) = N ∑ B f j αj j =1 We can conclude that if the LMIs (20) are feasible with respect to ∗ I > P = P T > 0 and matrix F then the closed-loop system with control algorithm (12 ) is quadratically stable with 6 Model Predictive Control guaranteed cost (17 ) Note that due to control horizon strategy only the first m rows of matrix F are used for real plant control, the other part of matrix... , Nu are output and control prediction horizons of model predictive control, respectively Note that for output/state prediction in (6) one needs to put A(α) = Ao , B(α) = Bo Matrices dimensions are A f (α) ∈ Rn( N2 +1) ×n( N2 +1) , B f (α) ∈ Rn( N2 +1) ×m( Nu +1) and C f ∈ Rl ( N2 +1) ×n( N2 +1) Consider the cost function associated with the system (6) in the form J= ∞ ∑ J (t) (10 ) t =0 where Nu N2... given by • k =1 x (t + 2) = Ao x (t + 1) + Bo u(t + 1) = Ao A(α) x (t) + Ao B(α)u(t) + Bo u(t + 1) • k=2 x (t + 3) = A2 A(α) x (t) + A2 B(α)u(t) + Ao Bo u(t + 1) + Bo u(t + 2) o o • for k x ( t + k + 1) = A k A ( α ) x ( t ) + A k B ( α ) u ( t ) + o o and corresponding output is k 1 ∑ Ak−i 1 Bo u(t + 1 + i) o (4) i =0 (5) y(t + k) = Cx (t + k) Consider a set of k = 0, 1, 2, , N2 state/output model predictions... αj (2) N N j =1 j =1 B(α) = j =1 ∑ Bj α j , α j ≥ 0}, j = 1, 2 N, ∑ α j = 1 Matrices Ai , Bi and C are known matrices with constant entries of corresponding dimensions Simultaneously with (1) we consider the nominal model of system (1) in the form x (t + 1) = Ao x (t) + Bo u(t) (3) y(t) = Cx (t) where Ao , Bo are any constant matrices from the convex bounded domain S (2) The nominal model (3) will . . B o =     .00 01 .10 02 −.0 615 . 018 3 − .11 33 .0586 −.0057 .0029     C =  1 0 0 0 0 1 0 7.74  and model uncertainty matrices are A 1u =     0 0 0 0 0 0.0005 0.0 017 0 0 0 0.00 01 0 0 0 0 0     B 1u =     0. proposed method. Robust Model Predictive Control Design 11 Fig. 1. Dynamic behavior of controlled system for unconstrained case for u(t) . B o =     .00 01 .10 02 −.0 615 . 018 3 − .11 33 .0586 −.0057. from predictive model (44). Substituting control algorithm (36) to (33) we obtain x (t + 1 ) = D 1 (j)x(t) (38) where D 1 (j) = A j + B j K 1 (j) K 1 (j) = (I − F 12 CB j ) 1 (F 11 C + F 12 CA j ),