Control predictivo basado en modelos borrosos. Reducción de la complejidad mediante el análisis de componentes principales funcionales

DOCTORAL THESIS Fuzzy Model Predictive Control. Complexity Reduction by Functional Principal Component Analysis Author: Juan Manuel Esca˜ no Gonz´ alez Supervisor: Dr. Carlos Bordons A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Departamento de Ingenier´ ıa de Sistemas y Autom´ atica July 2015 UNIVERSIDAD DE SEVILLA Abstract Fuzzy Model Predictive Control. Complexity Reduction by Functional Principal Component Analysis Juan Manuel Esca˜ no Gonz´ alez In Model-based Predictive Control, the controller runs a real-time optimisation to obtain the best solution for the control action. An optimisation problem is solved to identify the best control action that minimises a cost function related to the process predictions. Due to the computational load of the algorithms, predictive control subject to restrictions is not suitable to run on any hardware platform. Predictive control techniques have been well known in the process industry for decades. The application of advanced control techniques based on models is becoming increasingly attractive in other fields such as building automation, smart phones, wireless sensor networks, etc., as the hardware platforms have never been known to have high computing power. The main purpose of this thesis is to establish a methodology to reduce the computational complexity of applying nonlinear model based predictive control systems subject to constraints, using as a platform hardware systems with low computational power, allowing a realistic implementation based on industry standards. The methodology is based on applying the functional principal component analysis, providing a mathematically elegant approach to reduce the complexity of rule-based systems, like fuzzy and piece wise affine systems, allowing the reduction of the computational load on modelbased predictive control systems, subject or not subject to constraints. The idea of using fuzzy inference systems, in addition to allowing nonlinear or complex systems modelling, endows a formal structure which enables implementation of the aforementioned complexity reduction technique. This thesis, in addition to theoretical contributions, describes the work done with real plants on which tasks of modeling and fuzzy control have been carried out. One of the objectives to be covered for the period of research and development of the thesis has been training with fuzzy systems and their simplification and application to industrial systems. The thesis provides a practical knowledge framework, based on experience. UNIVERSIDAD DE SEVILLA Resumen de la tesis Control Predictivo basado en Modelos Borrosos. Reducci´ on de la complejidad mediante el An´ alisis de Componentes Principales Funcionales Juan Manuel Esca˜ no Gonz´ alez En el Control Predictivo basado en Modelo, el controlador ejecuta una optimizaci´ on en tiempo real para obtener la mejor soluci´ on para la acci´ on de control. Un problema de optimizaci´ on se resuelve para identificar la mejor acci´ on de control que minimiza una funci´ on de coste relacionada con las predicciones de proceso. Debido a la carga computacional de los algoritmos, el control predictivo sujeto a restricciones, no es adecuado para funcionar en cualquier plataforma de hardware. Las t´ ecnicas de control predictivo son bien conocidos en la industria de proceso durante d´ ecadas. Es cada vez m´ as atractiva la aplicaci´ on de t´ ecnicas de control avanzadas basadas en modelos a otros muchos campos tales como la automatizaci´ on de edificios, los tel´ efonos inteligentes, redes de sensores inal´ ambricos, etc., donde las plataformas de hardware nunca se han conocido por tener una elevada potencia de c´ alculo. El objetivo principal de esta tesis es establecer una metodolog´ ıa para reducir la complejidad de c´ alculo al aplicar control predictivo basado en modelos no lineales sujetos a restricciones, utilizando como plataforma, sistemas de hardware de baja potencia de c´ alculo, permitiendo una implementaci´ on basado en est´ andares de la industria. La metodolog´ ıa se basa en la aplicaci´ on del an´ alisis de componentes principales funcionales, proporcionando un enfoque matem´ aticamente elegante para reducir la complejidad de los sistemas basados en reglas, como los sistemas borrosos y los sistemas lineales a trozos. Lo que permite reducir la carga computacional en el control predictivo basado en modelos, sujetos o no a restricciones. La idea de utilizar sistemas de inferencia borrosos, adem´ as de permitir el modelado de sistemas no lineales o complejos, dota de una estructura formal que permite la implementaci´ on de la t´ ecnica de reducci´ on de la complejidad mencionada anteriormente. En esta tesis, adem´ as de las contribuciones te´ oricas, se describe el trabajo realizado con plantas reales en los que se han llevado a cabo tareas de modelado y control borroso. Uno de los objetivos a cubrir en el per´ ıodo de la investigaci´ on y el desarrollo de la tesis ha sido la experimentaci´ on con sistemas borrosos, su simplificaci´ on y aplicaci´ on a sistemas industriales. La tesis proporciona un marco de conocimiento pr´ actico, basado en la experiencia. Acknowledgements Firstly I would like to thank my supervisor Dr. Carlos Bordons for his dedication, support, patience and confidence to do this work. Also I would like to thank my family, especially my mother and my brother Dr. Carlos Esca˜ no who have been supporting this work (and everyone else) from the beginning until the last moment. For so many years combining this work with my working life, I have received support and help from many people who in one way or another, have left their mark on me. These include Antonio Nuevo, my friend and control and automation teacher, always available to help in good and bad times, Dr. Eduardo F. Camacho, whose support and help through my research career has been invaluable, Dr. Martin J. Hayes, who welcomed and looked after me, selflessly, in his University, my friend and teacher Dr. Carmine Luca Iandoli from whom I have learned the industrial reality of advanced control and my friend and brother Dr. Kritchai Witheephanich, with whom I have shared my research and he has been a support for me through good and bad moments. I especially want to thank my friends Dr. Niel Canty, Dr. Samira Roshany Yamchi, Jean-Michel Rubillon and Dr. Mairt´ ın O’Droma for their extensive revisions, unconditional help, advice and dedication. During my experience as a lecturer, I have dealt with many students, all of whom I have learned something from, but I must especially thank the future PhD Adolfo Juan S´ anchez del Pozo Fern´ andez for his invaluable support during this thesis. I also appreciate the support and time of my dear friends Maria Robledo, Elena Gonz´ alez, Samara Gonz´ alez and Cristina Mart´ ın. I would also like to thank my colleagues and friends in the Department of Systems and Control Engineering of the University of Seville especially Dr. Miguel ´ Angel Ridao, Dr. Fernando Dorado, Dr. David Mu˜ noz de la Pe˜ na, Dr. Jos´ e Mar´ ıa Maestre, Dr. Ignacio Alvarado, Dr. Daniel Lim´ on, Dr. Manuel Gil Ortega, Dr. Fernando Casta˜ no, Dra. Amparo N´ u˜ nez, Dr. Teodoro ´ Alamo and many others for their advice and contributions to this thesis. Also I also thank A´ oife Moloney and Louis Cronin for their assistance with corrections. I would finally like to thank Richard Linger and Dr. Dirk Pesch for the time that they have given me to finish this thesis at the most critical moment. iii Agradecimientos Primeramente me gustar´ ıa agradecer a mi director de tesis Dr. Carlos Bordons su dedicaci´ on, soporte, paciencia y confianza para poder realizar este trabajo. Tambi´ en en primer lugar de los agradecimientos est´ a mi familia, especialmente mi madre y mi hermano Dr. Carlos Esca˜ no quienes han estado apoyando este trabajo (y todos) desde el principio y hasta el ´ ultimo instante. Durante tantos a˜ nos compaginando este trabajo con mi vida laboral, he recibido el apoyo y ayuda de muchas personas que de una manera o de otra, han dejado huella en mi. Entre ellas est´ an Antonio Nuevo, mi amigo y maestro en control y automatizaci´ on, siempre disponible a ayudar en buenos y malos momentos, El Dr. Eduardo F. Camacho, cuyo soporte y ayuda en mi carrera investigadora ha sido inestimable, Dr. Martin J. Hayes, que me acogi´ o en su universidad desinteresadamente, mi amigo y maestro Dr. Carmine Luca Iandoli del cual he aprendido la realidad industrial del control avanzado y mi amigo y hermano Dr. Kritchai Witheephanich, con el que he compartido mi investigaci´ on con buenos y malos momentos y siempre ha sido un soporte para mi. Quiero agradecer especialmente a mis amigos Dr. Niel Canty, a la Dra. Samira Roshany- Yamchi, Jean Michel Rubillon y Dr. Mairt´ ın O’Droma por sus extensas revisiones, su incondicional ayuda, consejo y dedicaci´ on. Durante mi experiencia como profesor, he tratado con much´ ısimos alumnos, de todos he aprendido algo, pero debo de agradecer especialmente a mi amigo el futuro doctor Adolfo Juan S´ anchez del Pozo Fern´ andez su ayuda inestimable para esta tesis. Tambi´ en agradezco su apoyo y tiempo a a mis queridas Mar´ ıa Robledo, Elena Gonz´ alez, Samara Gonz´ alez y Cristina Mart´ ın. Quisiera tambi´ en agradecer a mis colegas y amigos del departamento de Ingenier´ ıa de Sistemas y Autom´ atica de la Universidad de Sevilla especialmente al Dr. Miguel ´ Angel Ridao, Dr. Fernando Dorado, Dr. David Mu˜ noz de la Pe˜ na, Dr. Jos´ e Mar´ ıa Maestre, Dr. Ignacio Alvarado, Dr. Daniel Lim´ on, Dr. Manuel Gil Ortega, Dr. Fernando Casta˜ no, Dra. Amparo N´ u˜ nez, Dr. Teodoro ´ Alamo y tantos otros, por sus consejos y contribuciones para esta tesis. Tambi´ en agradezco a A´ oife Moloney y Louis Cronin, por sus correcciones. Quisiera agradecer finalmente a Richard Linger y al Dr. Dirk Pesch el tiempo que me han concedido para terminar la tesis en el momento m´ as cr´ ıtico. iv Contents Abstract i Acknowledgements iii Contents v Acronyms viii Introduction 1 1 Model predictive control and its implementation 6 1.1 Model-based predictive control ...................... 6 1.2 Industrial implementation of MPC .................... 10 1.3 MPC for low-cost hardware: a practical implementation perspective . . 11 1.3.1 Ambulatory sensor network description ............. 12 1.3.2 Problem formulation ....................... 15 1.3.3 MPC ............................... 18 1.3.4 Robust MPC ........................... 19 1.3.5 Experimental results ....................... 21 1.4 Conclusion to the chapter ......................... 24 2 Implementation of fuzzy inference systems 25 2.1 Fuzzy logic ................................ 25 2.1.1 Mamdani fuzzy systems ..................... 27 2.1.2 Takagi-Sugeno fuzzy systems .................. 29 2.2 Industrial standardisation ......................... 30 2.3 Development of a FIS application for authentication gestures ...... 32 2.3.1 Result using the FME ...................... 34 2.4 Conclusion to the chapter ......................... 38 3 Fuzzy modelling techniques and applications 39 3.1 Fuzzy modelling ............................. 40 v Contents vi 3.1.1 Takagi-Sugeno models ...................... 42 3.1.2 Input selection .......................... 44 3.1.3 Fuzzy Time Series ........................ 46 3.2 Review of methods to build and train fuzzy systems ........... 48 3.2.1 Clustering methods ........................ 48 3.2.2 Back propagation algorithms ................... 49 3.2.3 Evolutionary algorithms ..................... 51 3.3 Development and validation of fuzzy models in real applications . . . . 53 3.3.1 Autoclave for food sterilization ................. 54 3.3.2 Gas Mixing Chamber ....................... 56 3.4 Conclusion of the chapter ......................... 61 4 Fuzzy control systems: practical implementation 63 4.1 Direct fuzzy controllers .......................... 64 4.1.1 Illustrative application: pneumatic levitation system ...... 68 4.1.2 Stability analysis of fuzzy control systems ............ 72 4.2 Fuzzy model-based control ........................ 73 4.3 Controllers with adaptive fuzzy parameters ............... 75 4.3.1 Air Separation Unit ........................ 75 4.4 Conclusion of the chapter ......................... 78 5 Fuzzy MPC for an industrial autoclave 80 5.1 Developement and implementation of a FMPC for an industrial autoclave 81 5.1.1 Neurofuzzy model of the temperature inside the autoclave . . . 82 5.1.2 Fuzzy generalised Predictive Control (FGPC) of the temperature 84 5.1.3 Experimental results ....................... 87 5.1.4 FLC code accomplishing IEC 61131-7 .............. 90 5.2 FMPC with constraints. Implementation issues ............. 91 5.3 Conclusion of the chapter ......................... 93 6 Complexity reduction in fuzzy systems using Functional Principal Component Analysis 94 6.1 Complexity reduction in fuzzy systems .................. 94 6.2 Principal Component Analysis ...................... 95 6.3 Functional Principal Component Analysis ................ 97 6.4 FPCA for fuzzy systems ......................... 99 6.5 Illustrative examples ...........................101 6.5.1 Pilot plant .............................101 6.5.2 Mechanical system ........................105 6.6 Conclusion of the chapter .........................108 7 FPCA to simplify MPC implementation 109 7.1 Dimensionality reduction of input variables space ............109 7.2 Application of FPCA to FMPC without constraints ...........113 7.3 FPCA applied to PWA systems ......................114 Contents vii 7.3.1 Example: distillation column ...................118 7.4 Simplified FMPC with constraints ....................121 7.5 Conclusion to the chapter .........................124 Conclusion and future works 125 Bibliography 127 Acronyms ANFIS Adaptive Neuro-Fuzzy Inference System ANN Adaptive Neural Networks AWG Additive White Gaussian BP Backpropagation CARIMA Controlled Auto-Regressive Integrated Moving Average CSMA/CA Carrier Sense Multiple Access/Collision Avoidance DCS Distributed Control System dm degree of membership DMC Dynamic Matrix Control EA Evolutionary algorithms EHAC Extended Horizon Adaptive Control eMMMPC explicit Min-Max MPC eMPC explicit nominal MPC EPSAC Extended Prediction Self-Adaptive Control FBD Functional Block Diagram FC Fuzzy Controllers FCL Fuzzy Control Language FGPC Fuzzy generalised Predictive Control FIC Fuzzy Incremental Controller FIE Fuzzy Inference Engine FIM Fuzzy Inference Machine FIS Fuzzy Inference System FL Fuzzy Logic FLF Fuzzy Lyapunov Function FME Fuzzy Matching Engine FMPC Fuzzy Model Predictive Control FNN Fuzzy Neural Network FPCA Functional Principal Component Analysis FPD Fuzzy Proportional Derivative GA Genetic Algorithms GPC Generalised Predictive Control HMI Human-Machine Interface IAE Integral of Absolute Error IIM-CSIC Instituto de Investigaciones Marinas Consejo Superior de Investigaciones Cientificas IMC Internal Model Control ITAE Integral of Time Absolute Error Introduction 5 how you can apply the same analysis of the previous chapter on piecewise linear systems. Practical applications of complexity reduction will be seen, laying the foundations of a methodology applicable to the industry. Last chapter will conclude the thesis and will propose further developments and research. Main contributions This thesis focuses on the following contributions: •A novel complexity reduction technique for fuzzy systems based on functional analysis has been study and proven experimentally. •Application of simplification technique to allow model predictive control in low capability hardware. •The same technique has been applied successfully for piece wise affine systems. •There has been an application of simplification technique to allow model predictive control subject to constraints in low capability hardware, permitting an adjusting parameter at runtime. •Transformation of inputs space for complex fuzzy models has been done in order to reduce dimensionality and therefore, complexity. •Experimental implementation of robust model predictive control (RMPC) subject to constraints in low capability hardware. A wireless sensor network has been used to implement a energy consumption control strategy in ambulatory environment. •Inputs selection for fuzzy modelling. Methods based on the application of component analysis and evolutionary algorithms have been carried out experimentally. •Real applications of fuzzy control systems have been developed using standard industrial languages. •Industrial application of fuzzy model predictive control (FMPC) has been developed and trialling. Chapter 1 Model predictive control and its implementation In this chapter, an overview of MPC is presented. It will not only review the formulation of various control designs that are most commonly used in industries, but practical examples will also be conducted during the investigation. The focus of this chapter is thus mainly on the practical implementation of predictive control. 1.1 Model-based predictive control MPC consists of a set of control strategies that began to be used in the industry since the 80s [19]. The idea behind such control strategies is the optimisation of an objective function, calculating the appropriate sequence of inputs over a prediction horizon based on a model of the plant. This calculation is repeated for each sampling instant to obtain the updated information of the plant. Depending on the structure of the objective function and the type of model being used, there are several well known strategies [18]. From 1973 [20],[18], it has been shown that one of the most popular algorithms is Dynamic Matrix Control (DMC), developed by Cutler and Ramaker [21]. The process model used in DMC is the step response of the control variable: y(k) = ∞ ∑ i=1 gi4u(k−i)(1.1) 6 Model predictive control and its implementation 7 The one-step ahead prediction calculated at instant kis: b y(k+1|k) = g14u(k)+ f(k+1),(1.2) where f(k+1)is the response of y(k+1)if 4u(k) = 0, i.e. the free response. The prediction over a horizon NPis: b y(k+1|k) = g14u(k)+ f(k+1) b y(k+2|k) = g24u(k)+ f(k+1) . . . b y(k+NP|k) = NP ∑ i=NP−NC+1 gi4u(k+NP−i)+ f(k+NP).(1.3) It can be written as a compact form, by=Gu +f,(1.4) where G=             g10... 0 g2g1... 0 . . .. . ..... . . gNCgNC−1... g1 . . .. . ..... . . gNPgNP−1... gNP−NC+1             (1.5) is the dynamic matrix. The objective of DMC is to obtain the minimum error between the reference w(k)and the control variable y(k). Therefore, the manipulated variables are calculated in order to minimize a functional such that: J(Np,Nc,λ) = Np ∑ j=1 δ(j)[b y(k+j|k)−w(k+j)]2+ Nc ∑ j=1 λ[∆u(k+j−1)]2,(1.6) where the control action u(k)is penalized by λ(known in the industry as move suppression). If there is no constraints, the sequence u={u(k)}can be calculated by computing 4J 4k=0, giving: u= (GTG+λI)−1GT(w−f).(1.7) Model predictive control and its implementation 8 Only the first control move u(1)is sent to the plant and the whole process is repeated for the next sampling instant. Thus, a continuous feedback of the control variable is obtained. The prediction horizon is always the same in each sample (common thing in predictive control strategies). The difference is, with respect to any other optimal control strategy, the MPC is a receding horizon optimal control. Another popular algorithm is the generalised Predictive Control (GPC) [22]. The output prediction in GPC is given by a CARIMA (Controlled Auto-Regressive Integrated Moving Average) model of the plant: A(z−1)y(t) = z−dB(z−1)u(t−1)+C(z−1)ε(t) ∆,(1.8) where u(t)and y(t)are the control and output sequence of the system, dthe dead time of the plant, ε(t)is a zero mean white noise, A(z−1) = 1+a1z−1+a2z−2+...+anaz−na B(z−1) = b0+b1z−1+b2z−2+...+bnbz−nb C(z−1) = 1+c1z−1+c2z−2+...+cncz−nc with ∆=1−z−1. For simplicity, in the following, C(z−1)is chosen to be 1. The sequence of future control signals is calculated such that it minimizes a multistage cost function defined by: J(N1,N2,Nu) = N2 ∑ J=N1 δ(j)[b y(t+j|t)−w(t+j)]2+ Nu ∑ j=1 λ(j)[∆u(t+j−1)]2,(1.9) where b y(t+j|t)is a j-step ahead prediction of the system output on data up to time t,N1and N2are the minimum and maximum prediction horizon, δ(j)and λ(j)are weighted sequences, and w(t+j)is the future reference trajectory. To solve the problem, we consider the following Diophantine equation: 1=∆ej(z−1)A(z−1)+ z−jfj(z−1),(1.10) where ej(z−1),fj(z−1)are polynomials uniquely defined. They can be obtained recursively in an easy way and they can be expressed as: ej(z−1) = ej,0+ej,1z−1+ej,2z−2+ ...+ej,j−1z−(j−1), fj(z−1) = fj,0+fj,1z−1+fj,2z−2+...+fj,nyz−ny, Model predictive control and its implementation 9 In order to obtain the predictive output, let us first define the following expression: gj(z−1) = ∆ej(z−1)B(z−1),(1.11) where gj(z−1) = gj,0+gj,1z−1+gj,2z−2+...+gj,j+Nu−1z−(j+Nu−1), and let us multiply equation (1.8) by ∆zjej(z−1)to get ∆zjej(z−1)A(z−1)y(k) = ∆zjej(z−1)B(z−1)u(k−d)+ ¯ ξ(k)(1.12) where ¯ ξ(k) = ∆zjej(z−1)ε(t) ∆.Therefore, by using (1.10) and (1.11), the following predictive output is obtained: ˆy(k+j|k) = fj(z−1)y(k)+ gj(z−1)∆u(k+j−d).(1.13) The horizon can be defined by N1=d+1, N2=d+Nand Nu=N. To solve the GPC problem, the set of control signals u= [u(t),u(t+1),...,u(t+N)]Thas to be obtained in order to optimize (1.9). As the cost function is quadratic, optimum can be easily obtained, assuming there are no constraints on the control signals, making the gradient of Jequal to zero. Considering δ(j)and λ(j)constants and grouping the terms of equation (1.13) which depend on the past input and output, into f, this leads to u= (GTG+λI)−1GT(w−f),(1.14) which is the same expression as (1.7). Here, G=gd,0gd+1,0... gNp,0T and w= [w(t+d+1)w(t+d+2)... w(t+d+N)]T. The control signal that is sent to the process is the first element of u, given by: ∆u(t) = K(w−f)(1.15) The previous control laws (1.15) and (1.7) are calculated when there is no any constraint is taken into account. The consideration of constraints allow the process to operate closer to constraints and optimal operating conditions and may reduce the number of constraint violations, hence reducing the number of costly emergency shutdowns. Model predictive control and its implementation 10 However, an online optimisation is needed. The formulation of such constrained optimal control problem can be expressed as, [18]: min unJ(u) = 1 2uTHu +bu +f0o s.t. Ru ≤r+Vz,(1.16) where H=2(GTG+λI) bT=2(f−w)TG f0= (f−w)T(f−w) with R,r,Vare parameters and signal bounds, and zis a vector composed of present and past signals (in case of state space representation, zis x(t)). To solve this problem, there are many available and reliable Quadratic Programming (QP) algorithms, e.g. Active Set, Feasible Direction, Pivoting methods, etc. They all use an iterative algorithm, which means that due to the computational burden they are not suitable for every hardware platform. In [23], the implementation aspects are presented and the restriction of the horizon on limited-resource hardware such as fixed-point arithmetics is addressed. Faster on line optimisation technique is described in [24,25]. 1.2 Industrial implementation of MPC MPC implementation has always been associated with commercial products in industry [20,26,27]. Companies like Adersa, Aspen Tech., Honeywell, Shell Global, Invensys, Continental Controls, DOT Products, Pavilion Technologies, etc., offer their platform externally. Most of them use an objective similar to (1.16). However, for fast process or large number of variables, QP may not be sufficient. Some of the commercial platform offer suboptimal approximated optimisation algorithms for achieving that purpose. In addition to the DMC and GPC (seen above), other algorithms are well known and widely used in industry: Model Algorithmic Control (MAC)[28], Predictive Functional Control (PFC)[29], Extended Prediction Self-Adaptive Control (EPSAC) [30], Extended Horizon Adaptive Control (EHAC) [31]. Although DCS are adequate for the implementation of MPC controllers, nowadays, Model predictive control and its implementation 11 there are PLCs ready to run MPC algorithms, like Siemens (unconstraint DMC)[32] and Schneider Electric (PFC)[33]. On the other hand, it is very important to offer the plant supervisors or instrumentation engineers the adjustment parameters of the controllers. As we have seen in the previous sections, these parameters are the prediction horizon and control and weights in the objective function. Several authors have proposed practical tuning methods based on real experience [34–36], [37,38] including some tuning techniques based on trial and error. A method based on principal component selection is used in [39,40]. [41] shows a simplification of a DMC tuning. Tuning method focusing on stability [42–45] and robustness [46,47] are also proposed. Techniques to evaluate the performance of the controllers are presented in [48–50]. A good review of the techniques is presented in [51]. From the point of view of the implementation, prediction and control horizons are not suitable for being used as setting parameters. They should be chosen, like the sample time, depending on the process dynamic. The weights used in the formulation of DMC and GPC modulate the error terms (δ) and control action (λ), respectively. The parameter that mostly impacts the robustness is the move suppression (λ)[34]. In practice, it is very important to give the plant supervisor and process instrumentation and control engineers the chance to adjust this parameter. 1.3 MPC for low-cost hardware: a practical implementation perspective For linearly constrained MPC problems of a low dimensional system, one can partially avoid the computational burden by precomputing the solution of the optimisation using multi-parametric quadratic programming (mpQP) [52–55]. This leads to an offline MPC. Examples can be found in [56,57], where a constrained MPC problem is developed using a low range PLC. Also in [58], an explicit MPC is applied in building controllers. Following this line, an application will be presented in this section regarding to MPC that is implementable on a hardware with low capacity such as Wireless Sensor nodes (motes). A constrained explicit generalised predictive control for ambulatory wireless sensor network power management will be described. Model predictive control and its implementation 12 1.3.1 Ambulatory sensor network description This application is framed within a context that control engineering can aid in the energy sensitive provision of vital biometric data, [59]. An ambulatory wireless sensor network (WSN) scenario is considered within a typical ambient healthcare setting illustrated by Figure 1.1[60–62]. This network consists of a set of low cost (in this particular case, Mote type), communicating sensor nodes featuring low-power radios. In each healthcare unit, there is a remote base station that is used to aggregate biometric data and to process the flow of information locally through wireless links. The base station is expected to relay the data wirelessly to another base station, e.g., from base station 2 (BS2) to base station 1 (BS1), when a patient moves to an adjoining area in an ambulatory fashion. It is anticipated that the sensor data will be routed from there via an Internet connection to some higher supervisory level within the network, [63,64]. The nodes Hospitalhealthcarenetwork Non-acutehealthcarenetwork Doctor Internet Central PC Monitoring and control Bodysensor BaseStation Wirelesscommunication Environmentalsensor HopStation Wiredcommunication x1 x1 x2 x2 x3 x3 CentralPC FIGURE 1.1: An ambulatory WSN in a general ambient healthcare environment. are deployed in either a static or mobile manner. The variability in wireless link quality for a static network deployment, e.g., in environmental, agricultural, or structural monitoring applications, has been shown (empirically) to place significant constraints on information throughput, [65,66]. The introduction of wearable devices, organized within an ambulatory setting is well known to exacerbate this problem, [67–69]. The communication link between any transceiver pair within such a network deployment is known to be time-varying, non-linear; and can exhibit large, rapid deviations that are dependent on placement and body movement, [70]. The resulting network must therefore be able to withstand complex radio dynamics such as fast and shadow fading, as well as relatively variable antenna orientation between transceiver pairs [71,72], that can attenuate the received signal power at the base station, frequently leading to packet Model predictive control and its implementation 13 errors (vital health status information to be lost) or even total breakdown in communication. The loss of data during wireless transfer becomes particularly more significant in this class of application than most other types of monitoring data. The primary performance requirement in the healthcare application space is to reliably achieve a target level, i.e., quality reference, on the received signal strength indicator (RSSI) metric, [73], so that a health care provider can be assured that the link quality and continuous link connectivity are sufficient to guarantee satisfactory levels of subject observation. Commercial sensor node platforms are now generally available that support wearable devices within a body sensor network setting. Examples include the MicaZ and Telos motes used in the CodeBlue project [74] based at Harvard University, that operate with low-power radios and are based on an adoption of the IEEE 802.15.4 standard, employing a carrier sense multiple access/collision avoidance (CSMA/CA) technique for data transmission. However, the CSMA/CA mechanism does not work perfectly due to the so called “hidden terminal” or multiple access interference (MAI) problem, [75]. This interference due to the existence of multiple simultaneously transmitting nodes can unnecessarily increase transmit power level and significantly degrade network capacity. In addition, the phenomena of uncertain fading channel and interference means that the question of dynamic energy management is a challenging component of any WSN that is constrained by finite battery resources. Certainly, transmission at lower power levels will compromise the quality of communication, and the desired quality of service (QoS) might not be met. An outage-based QoS constraint is considered wherein the received signal strength must be kept above a given threshold level so that no instances of potentially catastrophic outage or disconnection due to deep fading occur [76]. It is a particular objective of this work to highlight how dynamic control can aid the development of practicable radio power control strategies that provide an intelligent way of determining the optimal transmit power levels to be used by the sensor nodes on a network. The result will be an improvement in overall network lifetime and the maintenance of an acceptable received signal strength for all sensor nodes, while also preserving a satisfactory QoS for as many nodes as possible on the network. Furthermore, the use of commercial sensor node platforms that characterises this work means that only limited computational capacity and memory are available in the implementation stage of the radio power control law. The sensor nodes are programmed to send sensor data framed in an 802.15.4 format, [77–79]. Figure 1.2 illustrates the typical problem that has been considered. In this testbed, a star topology is adopted: a Tmote Sky sensor node is connected to a personal computer acting as the fixed base station or coordinator via the USB port, and there Model predictive control and its implementation 14 FIGURE 1.2: An experimental ambulatory test scenario containing two static and two mobile nodes. are four other Tmote Sky sensor nodes that are wirelessly connected to, (and randomly located around), the base station within 200 cm distance. Reconfigurable obstacles that highlight the practical phenomena influencing the loss of line-of-sight between the base station and the sensor nodes are integrated to form part of the testbed as illustrated in Figure 1.2. In addition, a selection of fully autonomous MIABOT Pro miniature mobile robots are used to provide a controlled, ambulatory dimension to the experiment. Each of the robots can be mounted with the sensor node in order to imitate various activities performed by a patient. The data flow within the WSN testbed is depicted in Figure 1.3. In this work, power control algorithms are directly implemented (via nesC) on the Tmote Sky sensor node platform running the TinyOS operating system, [80], that is identified a priori as the coordinator. We chose this environment, since the TinyOS and the Tmote Sky architectures are de facto benchmark standards for commercial and academic WSNs. An interface between Matlab and TinyOS has been established using stable bridging tools written in Java for data management purposes. Upon receipt of data packets from each sensor node at time sample t, the coordinator takes an RSSI measurement and then performs the power control strategy, resulting in an optimal power increment desired for updating the actual transmit power for the next sampling instant. After the hardware constraints (i.e., quantisation and power limitation) of the radio power amplifier are taken into account, the quantised power level is then transmitted over a fading channel to the corresponding connected sensor, where the RF output power level is adjusted accordingly. Model predictive control and its implementation 21 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 x u region #1 region #2 region #3 FIGURE 1.6: Min-max MPC-based radio power control law. ¯v∗(0) = −0.059∗¯x(k)−3.125 else if "2.236 −1#¯x(k)≤"−27.5 12.5#then ¯v∗(0) = −0.618∗¯x(k)−10 else {problem is infeasible} ¯v∗(0) = 0 end if It can be seen that when the optimal control law is saturated, the control correction input ¯vhas the same gain as the feedback law, i.e., K=0.618, but is of inverted sign, so that the applied input is therefore independent of the state of the system. As a result, the min-max MPC-based radio power control law ¯u∗(0)generates the profile illustrated in Fig. 1.6. 1.3.5 Experimental results The min-max MPC mpQP solution computed requires a memory usage of only 285 bytes from an available 48 Kbytes of memory provided within a typical Tmote Sky Model predictive control and its implementation 22 TABLE 1.1: Summary of test scenarios. Test Sensor nodes Motion of scenarios deployment the mobile robot(s) 13 static nodes, 1 robot — straight path 1 mobile node 22 static nodes, 1 robot — straight path 2 mobile nodes 1 robot — circular path 31 static node, 2 robots — straight path 3 mobile nodes 1 robot — circular path sensor node, [89]. There still exists space for various actual user level sensor node applications. Note that the number of regions depend on the complexity of the model and the prediction horizon. In theory, the possible number of regions can grow exponentially with the horizon dimension. However, for low order systems this number is much lower because of the state constraints, [90,91]. To benchmark the advantages of the proposed algorithm, the explicit min-max MPC- based radio power controller (denoted henceforth as eMMMPC) is compared in each scenario with the explicit nominal MPC approach seen before(denoted as eMPC), a power control that utilizes a balanced adaptive scheme (denoted as Adaptive 1) and has previously been shown to perform well in the WSN power control literature [68] as well as a power control strategy with adaptive step-size (denoted as Adaptive 2), taken from [92]. In order to benchmark the proposed control law with the other aforementioned designs, three test scenarios, in which the sensor nodes are deployed in both a static and mobile fashion, have been considered and the setup details are summarised in Table 1.1. Specifically, in scenarios 2 and 3 the motion of robot is configured to move continuously in a circular path about its initial position, along a short distance of 20 cm, so as to produce a high variation in the observed RSSI feedback signal. The wireless channel can hence be well described as exhibiting a Rayleigh fading distribution in this circumstance, [81]. For each scenario, the test has been iteratively performed for 4 runs with a duration of 140 sec and a sampling interval of 1 sec. For each run, different static nodes positions and mobile nodes trajectories (either straight or circular path) are defined. Note that, for consistency, the positions of static nodes and the trajectories of the mobile robots are not changed while an experiment is repeated for each control law. During each experiment, dynamic data, i.e., RSSI and the corresponding transmit power level of all nodes are recorded to analyze the system performance according to the following three criteria: Model predictive control and its implementation 23 •Power consumption: Pi(mW) = Ns ∑ k=1 pi(k),(1.32) where Nsis the total number of samples. •Outage probability: Poi(%) = Prob{¯ri<¯rth}.(1.33) •Standard deviation of the RSSI tracking error: σei(dBm) = 1 Ns Ns ∑ k=1 (¯rt−¯ri(k))21 2.(1.34) The mean values of Pi,Poi, and σeiof all nodes are calculated for each experiment of each controller. The averages of these results for each test scenario are the final performance metrics of each controller. 1 2 3 0 5 10 15 20 25 30 35 40 45 50 Outage probability (%) 1: One mobile, 2: Two mobiles, 3: Three mobiles eMMMPC eMPC Adaptive 1 Adaptive 2 (a) Average outage probability 1 2 3 0 20 40 60 80 100 120 140 160 RSSI tracking error (dBm) 1: One mobile, 2: Two mobiles, 3: Three mobiles eMMMPC eMPC Adaptive 1 Adaptive 2 (b) Average standard deviation of the RSSI tracking error 1 2 3 0 10 20 30 40 50 60 70 80 90 100 Power consumption (mW) 1: One mobile, 2: Two mobiles, 3: Three mobiles eMMMPC eMPC Adaptive 1 Adaptive 2 (c) Average power consumption FIGURE 1.7: Average values of outage probability, standard deviation of the RSSI tracking error, and power consumption for different radio power controllers of all sensor nodes. Figure 1.7 and Table 1.2 present the summary of performance evaluation in terms of the average power consumption ˜ P, the average outage probability ˜ Po, and the average standard deviation of the RSSI tracking error ˜ σeproduced for different test scenarios according to the performance criteria (1.32), (1.33), and (1.34), respectively. More detail can be seen in [62]. Model predictive control and its implementation 24 TABLE 1.2: Performance evaluation for different radio power controllers taking into account all sensor nodes. eMMMPC eMPC Adaptive 1 Adaptive 2 One mobile ˜ Po(%) 2.10 4.36 18.12 29.65 ˜ σe(dBm) 19.55 22.24 48.16 101.22 ˜ P(mW) 13.11 14.77 15.65 48.66 eMMMPC eMPC Adaptive 1 Adaptive 2 Two mobiles ˜ Po(%) 3.19 6.86 18.38 25.78 ˜ σe(dBm) 26.11 38.12 58.84 106.20 ˜ P(mW) 19.31 23.41 24.57 68.32 eMMMPC eMPC Adaptive 1 Adaptive 2 Three mobiles ˜ Po(%) 6.18 9.48 21.93 26.46 ˜ σe(dBm) 37.74 46.48 73.37 111.76 ˜ P(mW) 28.05 36.72 35.53 75.68 1.4 Conclusion to the chapter In this chapter, an introduction of MPC and its implementation have been presented. An overview of tuning techniques for MPC commonly used in the industrial process is also addressed. From a practical point of view, a RSSI-based explicit min-max MPC approach has been presented to address the radio power control problem encountered in ambulatory sensor networks. It has been shown that an explicit solution of the constrained min-max MPC problem can be computed for the WSN power control problem by solving an mpQP. The feasibility of the proposed design and its performance has been experimentally validated using a variety of test scenarios. The experimental results clearly show that the explicit min-max MPC-based power management strategy performing optimal radio power assignments exhibits good performance for this particular problem. Moreover, the algorithm has been implemented on a reduced functionality wireless sensor platform and the numerical overhead involved is relatively insignificant from an implementation perspective. However, although the strategy is suitable for industrial processes (process industry, manufacturing, etc.), lack of on-line tuning parameters to the designed controller is a weakness in terms of implementation. One possible solution is to design different PWA controllers for different settings parameters and linearly interpolating the longline action of the different controllers. The problem with this is an increase in complexity. In the following chapters, we will address this problem by using Fuzzy models and a complexity reduction technique. Chapter 2 Implementation of fuzzy inference systems This introductory chapter will give an overview about the foundation of fuzzy logic (FL) and its applications. Now days, FL is the paradigm of the Soft Computing, discipline of computer science which integrates a set of techniques that deal with approximation, imprecision, uncertainty and partial truth. (such as neural networks, evolutionary computation, support vector machines, etc.). Many engineering applications have been developed based on the use of fuzzy logic [93]. Since 1965, when L. Zadeh published his article about fuzzy sets, the count of publications containing the word “fuzzy” in title, as cited in INSPEC database is 171.420 and in MathSciNet database, 27.201 (Compiled on July 26, 2015). The total number of papers with “fuzzy” in title in Google Scholar is 2.310.000. There are 29 journals with ”fuzzy” in the title (and 21 with ”soft computing”). And there are 1255 patents issued with ”fuzzy logic” into the title and 2.314 with ”fuzzy control”, according to The Lens (http://www.lens.org). 2.1 Fuzzy logic FL is a logical system built on the basis of fuzzy sets, introduced by Lotfi A. Zadeh[93]. According to him, if X={x}is a space of objects (points) a fuzzy sets Ain Xis characterised by a function fA(x):x→[0,1], called membership function. In classic set theory the membership function can take only two possible values: {0,1}. FL is, therefore, a multivalued logic. Fuzzy and classic inference are to obtain a conclusion statement 25 Implementation of fuzzy inference systems 26 from another (premise) by applying inference rules, they only differ on belonging to sets of variables used in the antecedents and consequences. Fuzzy logic can handle information closer to the human way, ie, uncertain, vague or imprecise. A plant operator is able to control a system using logic, running perfectly control actions not defined by numbers, such as: close valve slightly,slightly slow down or set to a fairly high temperature. The operator controls with significance more than accurately. Modus Ponens inference rule is used in applications of logic in engineering because it preserves the cause-effect. Modus Ponens or direct reasoning can be summarized as follows: •Premise 1: xis A •Premise 2: IF xis A, THEN yis B •Consequent: yis B Modus Ponens is associated with the implication A→B. As for the theory of fuzzy sets, also the foundations of the theory of fuzzy logic depart and take the fundamental concepts of classical logic. In fuzzy logic, modus ponens extends to generalised modus ponens, where the antecedents (premise 1 and 2) and consequent are activated by a degree of membership ∈[0,1]. Generally, a fuzzy inference system (FIS) is structured as shown in fig 2.1. The fuzzification is determined by the degrees of membership of the inputs to the sets mentioned on the premises. In order to formulate mathematically the fuzzy inference system a mechanism has to be defined for the operators, implication, aggregation and defuzzification. There are two types of well known fuzzy systems: Operator Inputx Output Inputy Implication Defuzzification Aggregation IFxIS A1 ANDyISB1 THENzISC1 IFxIS A1 ANDyISB1 THENzISC1 IFxIS A1 ANDyISB1 THENzISC1 ... { S Fuzzification FIGURE 2.1: Fuzzy inference system structure Mamdani-type and Takagi-Sugeno type. Implementation of fuzzy inference systems 27 2.1.1 Mamdani fuzzy systems Mamdani System is a structure proposed by E. Mamdani [94] in 1975. A Mamdani fuzzy system consists of fuzzy antecedent and consequent, a rule set, a inference engine that processes the reasoning process using the rules set. The idea of this structure is to carry out an approximation of the human reasoning. A set of conditional rules (IF...THEN...) connects the inputs with the outputs. Those rules work with fuzzy predicates. Given the inputs (non-fuzzy), the fuzzification is carried out by the evaluation of prototypical memberships functions (fig. 2.2) The degree of membership (dm) is a value 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 1.2 Membership Grades (a) Trianguler MF 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 1.2 Membership Grades (b) Trapezoidal MF 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 Membership Grades (c) Gaussian MF 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 Membership Grades (d) Generalized Bell MF FIGURE 2.2: Common Membership functions between 0 and 1 assigned to a non-fuzzy number and gives it the degree of belonging to a particular fuzzy set represented by a membership function. In other words, Definition 2.1. Let Xbe an universal set, and a fuzzy subset Aof X. The membership function of Ais defined as: µA:X→[0,1] Implementation of fuzzy inference systems 28 the value µA(x)is the degree of belonging of xto the fuzzy set A. Once the fuzzification is made and the dm to every fuzzy system are calculated, they will decide the degree of activation of each rule. In case of multiple inputs, an operation between degrees of membership must be performed in order to obtain the dm of the antecedent. Multiplication or minimum of dm could be chosen if they are connected by AND operator, and summation or maximum of dm if OR operator. The result of the operation, will decide the degree of firing of the particular rule. A way to define the numerical firing of a rule is by using α-cuts. Definition 2.2. Let Xbe an universal set, and a fuzzy subset Aof X. Let α∈Rand α∈[0,1]. then, the α-cut of Ais defined as: α(A) = {x∈X|µA(x)≥α} If the activation degree of rule iis µiand the fuzzy consequent is the fuzzy set Ui(x), defined by the membership function µUi(x), then, if αiis the α-cut of Ui(x), with α=µi, the firing degree of such rule is defined by the function: Ui(x) = (µii f x ∈αi µUi(x)i f x /∈αi (2.1) Taking the Ui(x)function for each rule, all of them are combined into a single fuzzy set. Normally, set is formed by the union of the new membership functions of the outputs u(x) = SUi(x), so called aggregated fuzzy set. There are several methods of defuzzification to obtain the nonfuzzy real output. The most popular defuzzification method for Mamdani systems is the centroid [95]. The idea of this method is to get the centre of gravity of the aggregated fuzzy set. Let cibe the centre of the membership function Ui(x), the centre of gravity can be computed by: COG =∑ici·RiUi(x) ∑iRiUi(x)(2.2) The set of methods chosen to simulate the logic inference, is known as fuzzy inference engine. Implementation of fuzzy inference systems 29 2.1.2 Takagi-Sugeno fuzzy systems Takagi-Sugeno (TS) fuzzy systems [96] have been applied successfully in non-linear model based techniques [97] where the nonlinearity can be decomposed into multiple linear regions defined by each rule. In TS models, the system may be described by j rules by the following way: Rule Rj: IF x1is Ax1j,..., and xn(k)is Axnj, THEN: fj=g0j+g1jx1+...+gn jxn being xi,yjfor each rule, the inputs and outputs of the system respectively, and Axij is the fuzzy set respective to xi(k)on the rule j,gi∈R,fj(k)is the output of the model respective to the operating region associated to that rule. The structure of antecedents describes fuzzy regions in the inputs space, and the one of consequents presents nonfuzzy functions of the model inputs. These models may be formulated as an adaptive neuro-fuzzy inference system (ANFIS) [98]. In figure 2.3 an ANFIS is presented as an example with ninput variables, one output variable and five layers. The first layer is composed of membership functions of A11 D N N x1 Layer1 Layer2 Layer3 Layer4 Layer5 x ...x 1n P xn A1S An1 AnS x ...x 1n ... ... . . . . . . . . .. . .. . . . . . P FIGURE 2.3: Fuzzy neural network [99] each Ai j, defined by the membership degree µAi j :xi∈R7−→ µAi j (xi)∈R(2.3) The output of each node iis µAi j (xi), the membership degree of xi. For the definition of these membership functions, some standard types are used, like gaussian membership Implementation of fuzzy inference systems 30 functions (see figure 2.2). The second layer has nodes labelled with Πwhich implement fuzzy inference machine. Logical operation AND may be carried out by multiplication or minimum value for example, the output of each node jof this layer may be: ωj(x) = µA1j(x1)·µA2j(x2)·...·µAn j (xn)(2.4) Or ωj(x) = min{µA1j(x1),µA2j(x2),...,µAn j (xn)}(2.5) The third layer normalises the inference motor. The output of each node of this layer is: aj(x) = ωj(x) ∑N j=1ωj(x)(2.6) Where Nis the number of rules of the system. The fourth layer has adaptive nodes: aj(x)·fj(x) = aj(x)·(g0j+g1jx1+...+gn jxn)(2.7) Finally, the fifth layer is the defuzzification node. For TS systems, the output will be: N ∑ j=1 aj(x)·fj(x) = ∑N j=1ωj(x)·fj(x) ∑N j=1ωj(x)(2.8) We will call aj(x)antecedent functions and fj(x)consequent functions. The output of the fuzzy complete model may be described by f(x) = N ∑ j=1 aj(x)g0j+g1jx1+...+gn jxn(2.9) 2.2 Industrial standardisation Although the advanced control, in each and every one of his strategies, has been widely used in many industrial applications, only some paradigms are defined by industry standards. Certainly the standard ISO/FDIS 15746-1 [100], about Automation systems and integration -Integration of advanced process control and optimisation capabilities for Implementation of fuzzy inference systems 37 negative positive 0.0 0.2 0.4 0.6 0.8 Distribution of Positives and Negatives Score FIGURE 2.10: Distribution of positives and negatives Distribution of Scores for Positive Gestures Score Frequency 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 FIGURE 2.11: Distribution of scores for positives gestures •Scores for positive gestures were in the range 0.21−0.89 (mean=0.56, median=0.61) In this case, the threshold could safely be lowered to 0.5. However, there would still be no authenticated negatives but the number of rejected positives would have been reduced to 18% and authenticated positives would increase to 45%. Figures 2.11 and 2.12 show the distribution of scores for positive and negative gestures respectively. As expected, these are somewhat skewed to the left and right respectively. It should be noted that these statistics were compiled by manually examining gestures. The human operative would not know if a gesture was drawn from left to right or from right to left. However the system would always reject a gesture that was drawn in the opposite direction from its prototype. Consequently, some gestures labelled positive may, in fact, Implementation of fuzzy inference systems 38 Distribution of Scores for Negative Gestures Score Frequency 0.0 0.1 0.2 0.3 0.4 0 2 4 6 8 10 12 FIGURE 2.12: Distribution of scores for negatives gestures have been negative. This may account for the unexpectedly high prevalence of positive gestures with a low score (the spike to the left on figure 2.11). Therefore, the system may have performed somewhat better than these numbers suggest. 2.4 Conclusion to the chapter This chapter provides an overview of fuzzy inference systems. The industrial implementation through the FCL for PLC, shows that the use of fuzzy logic in the industrial world is well established, unlike other advanced control techniques. The standard provides manufacturers and users a common understanding well defined base, a means to integrate fuzzy control applications in the languages of Programmable Controllers according to the PLC standard languages, as well as the ability to exchange fuzzy control portable programs between different programming systems. A real application of the FIS has been described. It is a real example of commercial use of fuzzy techniques for authentication gestures in smart phones and tablets. The application is commercialised by the company Sensipassr. That application can be seen as a minor contribution for this thesis, which shows a practical implementation of a FIS for embedded systems. Chapter 3 Fuzzy modelling techniques and applications The use of models for prediction, simulation and control of systems is very common in engineering. There is vast literature regarding the modeling of dynamic systems. Obtaining a system model is not always an easy process. In many real cases, system complexity due to the number of variables in play, the random nature of the process, ignorance of the physics of the system, etc., makes it virtually impossible to obtain equations that reflect the behavior of the real system. At other times, an accurate model will consist of a large number of equations, which make the model impractical for use in control applications. In many cases, fuzzy inference systems can be the solution to the problem of modeling. Given that the behavior of systems can be described by rules captured by the experts and described with human language, fuzzy logic is an appropriate tool to to formulate that knowledge. Despite fuzzy models being highly successful in industry, even today, many scientists are reluctant to use fuzzy systems for modeling or control. This may be due to confusion caused by its name. Paraphrasing Professor Zadeh, the father of fuzzy logic, ”There are many misconceptions about fuzzy logic. Fuzzy logic is not fuzzy. Like traditional logical systems and probability theory, fuzzy logic is precise. However, there is an important difference. In fuzzy logic, the objects of discourse are allowed to be much more general and much more complex than the objects of discourse in traditional logical systems and probability theory.”[121]. The typical questions of the classical system identification community is: Why do we use a fuzzy inference system (FIS) for approximating continuous functions instead of more classical techniques, like regression, for performing this task? [122]. An answer 39 Fuzzy modelling techniques and applications 40 may be that a FIS forms a collection of fuzzy rules which can be extracted from experts knowledge, or from common sense. Moreover, each rule can represent a local model that is easily interpretable and analysable. This local type of representation enables improved approximation accuracy to be obtained [123]. One might object that Piece Wise Affine (PWA) models [124] already exist for that reason. However the transition from one region containing a linear model, to another region in which another linear model prevails, is naturally smooth and not sharp, as is the case with the PWA systems. This makes fuzzy systems more suitable than PWA models for describing nonlinear systems. The goal of fuzzy modeling is to obtain a set of rules that describe the dynamics of the system through experimental data. FIS are generic functional approximators, i.e., given a certain level of error, you can find a FIS that approximates any function with less than that fixed error. To do this, various techniques are used, some from the field of neural networks (NN) and also from other fields such as statistics, genetic algorithms, etc. This chapter will provide an introduction to the techniques of fuzzy modeling. In Section 3.1, the mathematical formulation of fuzzy models is presented, considering the problem of the choice of inputs and modeling error. In Section 3.2, the different groups will be most popular methods to determine the structure and parametrisation of these models. Finally, in section 3.3, several real applications made under this thesis will be presented. 3.1 Fuzzy modelling Most real applications require mathematical models as functional assignments, corresponding actual inputs to outputs, where the aim is to approximate a function y=f(x) in a limited area (compact) of input space x= (x1,x2,...,xn). In this section, the functional representation of fuzzy models will be described. As we mention in 2.1, there are two kind of Fuzzy Inference Systems, called Mamdani and Takagi-Sugeno fuzzy systems. Given a Mamdani system by rules of the form: Rule Rj: IF x1is A1jAND x2is A2j, AND,..., AND xnis An j, THEN: yjis Bj Where Ai j is a membership function (MF) of the input xiand Bjthe consequent of the output yj. Using the product for the AND operator, the minimum for the implication, the union for the aggregation and µi j(x)being the degree of membership of the input xi Fuzzy modelling techniques and applications 41 to the fuzzy set Ai j and νj(y)represents the degree of membership of the output yjto Bj. To obtain the output, we can use the centroid defuzzifier: y=RYµ(y)·y·dy RYµ(y)·dy (3.1) Where µ(y) = [ j"min( n ∏ i=1 µi j(x)!,νj(y))# Once the rule base is set, the problem of approximation is reduced to find the parameters that define each µi j(x)and νj(y), which usually are chosen as prototypical functions. Examples are: •Triangular m(x;a,b,c) = maxminx−a b−a,c−x c−b,0 Where a,b,c(with a<b<c) determine the xcoordinates of the three corners of the triangular MF •Trapezoidal m(x;a,b,c,d) = maxminx−a b−a,1,d−x d−c,0 a<b≤c<ddetermine the xcoordinates of the four corners of the trapezoidal MF •Gaussian m(x;c,σ) = e −1 2x−c σ2 Where cis the MFs centre and σdetermines the MFs width •Generalised bell m(x;a,b,c) = 1 1+ x−c a 2b cdetermines the centre of the corresponding membership function; ais the half width; and b 2acontrols the slopes at the crossover points. Fuzzy modelling techniques and applications 42 3.1.1 Takagi-Sugeno models Many methods can be found in the literature for the identification of a fuzzy model. One of the most popular methods is the formulation of a Takagi-Sugeno Fuzzy system as a NN [98], also called Adaptive Neuro Fuzzy Inference System (ANFIS). One of the classical ways to model complex systems experimentally is by using artificial NN. Models based on NN[125] are relatively easy to design, they often impose initial assumptions (as a functional dependency), simulation responses are quick. It is true that operators and engineers have a good insight into the operation of complex process, achieved over many years of experience. However a problem often arises when this knowledge is incomplete and imprecise, making the formulation of accurate mathematical equations very difficult. In general, all the training methods used in artificial NN can be transferred to the field of fuzzy systems [98], so called NeuroFuzzy Systems, which is an attempt to combine the learning ability of NN to the handling of uncertain information of the fuzzy inference system (FIS). The rule basis of an FIS can be based on expert knowledge, however, there are methods based on techniques that do not require a priori information about its structure. FIS used for modeling, can handle nonlinear processes well, and provide knowledge of the system that is impossible with the use of NN. The models based on ANFIS combine the advantage of adaptive neural networks (ANN), such as the ability to learn and adapt, and fuzzy logic, i.e. knowledge based on rules and management uncertainty and significance of knowledge. Unlike ANNs based systems, ANFIS can incorporate a priori knowledge in order to improve the model. In the neurofuzzy model proposed by Takagi-Sugeno (TS)[96], the structure of antecedent describes fuzzy regions in the inputs space, and the one of consequent presents non-fuzzy functions of the model inputs. Recurrent Fuzzy Neural Networks (RFNN) have demonstrated better results at identifying all the dynamics of nonlinear systems. They are systems which have the same advantages as recurrent neural networks [126, 127]. RFNN are also named Fuzzy Dynamical Systems (see figure 3.1) and extend the application domain of FNN to temporal problems. Feedback enables dynamics to be captured and updated. If we use recurrent functions with NARMAX structure (Nonlinear Auto Regressive Moving Average with eXogenous input), of the kind: ˆy(k+1) = f(y(k),...,y(k−m),u(k),...,u(k−n)), where u,yare the inputs and outputs of the system, each rule of the system may be described by Rj: IF x1(k)is F1j,..., and xn(k)is Fn j, Fuzzy modelling techniques and applications 43 Outputs Inputs i FIGURE 3.1: Dynamical Neurofuzzy System THEN: yj(k) = aj(z−1)y(k−1)+ bj(z−1)u(k−d)+ξ(k) Where aj(z−1) = a1j+a2jz−1+...+anyjz−(ny−1)and bj(z−1) = b0j+b1jz−1+b2jz−2+ ...+bnujz−nu X(k)=[x1(k)x2(k)...xn(k)]Tis the inputs vector of the neurofuzzy system in the instant k,Fi j is the fuzzy set respective to xi(k)on the rule j,yj(k)is the output of the model respective to the operating region associated to the rule. If µi j(k)is the membership degree of xj(k)in the fuzzy set Fi j and the number of implications or rules is L, the Recurrent Fuzzy Neural Network (RFNN) complete model is described by y(k) = L ∑ j=1 wj(k)aj(z−1)y(k−1)+ bj(z−1)u(k−d)+ξ(k)(3.2) Where wj(k) = ¯ µj(k) ∑L j=1¯ µj(k),¯ µj(k) = n ∏ i=1 µi j(k) and ξ(k)is a white noise sequence with zero mean. Rewriting equation (3.2) as ¯a(z−1)y(k) = ¯ b(z−1)u(k−d)+ ξ(k)(3.3) Where dis the delay and ¯a(z−1) = 1−¯a1z−1−¯a2z−2−...−¯anyz−ny(3.4) ¯ b(z−1) = 1−¯ b1z−1−¯ b2z−2−...−¯ bnuz−nu(3.5) ¯ai= L ∑ j=1 wj(k)ai jz−i(3.6) ¯ bi= L ∑ j=1 wj(k)bi jz−i(3.7) Fuzzy modelling techniques and applications 44 TS systems are computationally more efficient than Mamdani systems and work well with optimisation and adaptive techniques, which makes them very attractive in control problems, particularly for dynamic non linear systems [128]. 3.1.2 Input selection When establishing the structure ANFIS structure, one problem is to determine the membership functions. A large number will increase modelling accuracy, resulting in numerous management rules. This problem of the explosion of rules as linguistic terms, can be overcome with the application of clustering methods [129] seeking classification of data into subsets. In addition to the proper choice of rules and membership functions via a methodology, the choice of the input variables is important. In a simple application, consisting of a few variables, it is easier to choose the inputs, observing causality. When the system is composed of many variables, coupled with each other and hardly visible to the naked eye, a method can be to choose the maximum number of variables in the input FIS, and let any method from multivariate analysis decide the appropriate subsets, including the input. On the other hand, in many applications, it is important to catch the dynamics of a system and predict it for a variable time horizon ahead. A model with previous samples of the output as inputs, may be useless if it’s very sensitive to input errors, i.e., due to modelling error, a recurrent scheme may yield poor results if used in model-based control strategies with a large prediction horizon. For example, in figures 3.2,3.3,3.4 a comparison between a dynamical neurofuzzy model and the real data, in function of prediction horizon (one, three and five-step ahead) is presented. The mean absolute error |e|is used as a measurement of validation. While there is a huge literature on the modeling of fuzzy systems and its application to engineering, the literature reports few instances of a systematic methodology to obtain good models. Most of the contributions in this field refer to the methods for setting the parameters and the structure of fuzzy systems, but there is little reference made to the proper selection of inputs. In [130], a method based on testing models with different input selection is applied. The procedure for input variable selection is systematised as: 1. Evaluate the performance of the initial model with all candidate input variables in the model. 2. For each remaining input variable, evaluate the performance of the initial model with this variable temporarily removed. Fuzzy modelling techniques and applications 45 0 200 400 600 800 1000 1200 1400 1600 1800 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 model real data FIGURE 3.2: One-step ahead model, |e|=0.0142 0 200 400 600 800 1000 1200 1400 1600 1800 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 model real data FIGURE 3.3: Three-step ahead model, |e|=0.0284 3. Permanently remove the variable associated with the best model performance obtained in step 2. Record the resultant reduced variable set and the associated model performance. 4. If there are still variables remaining in the model, go back to step 2 to eliminate another variable. Otherwise go to step 5. 5. Choose the best variable set from the sets recorded in step 3. Fuzzy modelling techniques and applications 46 0 200 400 600 800 1000 1200 1400 1600 1800 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 model real data FIGURE 3.4: five-step ahead model, |e|=0.0412 The idea in [131] is to find the most relevant inputs by successively removing inputs from the initial input data set and checking whether the reduced data set is still consistent. Yen et al.[132] proposed using principal component analysis (PCA), a statistical analysis technique, to reduce the number of inputs for fuzzy models. Principal components are eigenvectors of the input variables’ covariance matrix; the first principal component corresponds to a linear combination of the input variables that produces maximum variance in the input value (i.e., maximum input excitation) [133]. After a normalisation of the variables, the application of PCA will produce a new set of uncorrelated variables. However, it’s important to select variables with any correlation with the output before PCA, because the selection is based on variability in the input value, not based on whether the input actually affects the output. An input variable with large variance may be completely unrelated to the output. 3.1.3 Fuzzy Time Series The main objective of time series analysis is the construction of mathematical models based on known past instances of a variable in order to predict future values of the same. There are dynamic systems that exhibit repetitive dynamics in time or have a profile that appears regularly. When a system is complex enough to define the causal variables of dynamic effects, we have to resort to the study of time series to predict Fuzzy modelling techniques and applications 53 Thus, in the following string: 101100101, the samples xk−1,xk−3,xk−6,xk−7,xk−9will be chosen as inputs for the FIS. GA can use the binary strings as individuals to evaluate an objective function, based on the model error between a FIS generated using that particular input structure and the series. In figure 3.8 a Fuzzy time series is compared with the Mackey-Glass equation. After the application of GA, the input set of the FIS is xk−4,xk−7,xk−17,xk−19,xk−20, being ˆxk the fuzzy time serie output, the Root Mean Squared Error (RMSE) one step ahead is: RMSE =sN ∑ i (xi−ˆxi)2 N=3.6131·10−4(3.18) 0 1000 2000 3000 4000 5000 6000 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 k x(k) Mackey−Glass time serie Mackey−Glass equation Fuzzy Time serie FIGURE 3.8: Mackey-Glass Fuzzy Time Serie 3.3 Development and validation of fuzzy models in real applications In this section, three real applications carried out during the research work of the thesis are presented. The first is an application of modelling using Takagi-Sugeno models, based on linear models, which were determined from experimental data. The second one is also a TS model but differs by changing the input space for the subspace of the principal components (Principal Component Analysis). The third case study is an application of fuzzy time series for electrical demand prediction in a building. Fuzzy modelling techniques and applications 54 3.3.1 Autoclave for food sterilization Here an example of a fuzzy model being applied to a steam autoclave for the sterilization of food is given. This unit is located in Spain, in the IIM-CSIC (Instituto de Investigaciones Marinas- Consejo Superior de Investigaciones Cient´ ıficas). The purpose of sterilization is to eliminate health risks by the thermal destruction of microorganisms, and to achieve product stability over long storage periods. The treatment is carried out at an elevated temperature (>100oC) so that FIGURE 3.9: Autoclave for food sterilization (IIM-CSIC, Vigo, SPAIN) the process is relatively short and, thus, having a minimal effect on factors of quality and nutrient retention. The unit uses a stream of saturated steam to heat the enclosed product. It has three inputs (steam, air and water) and two outputs (draining and purging). The sterilisation process is carried out in three stages: venting, heating and cooling. In the first stage, saturated steam passes through the vessel in order to evacuate the air from the system. Once the pressure into the vessel is equal to the saturated steam, the second stage starts. During the heating cycle, a fine control strategy is needed in order to track the reference and reject disturbances. After a predefined time, the third stage starts, cooling with water and controlling the pressure by injection of air. [167] In order to identify the process in the heating stage, a series of steps, of opening of the steam valve sequentially, have been done, for experimentally obtaining the temperature inside. Given the same increase in step, different temperature increases are obtained, as Figure 3.10 shows. Fuzzy modelling techniques and applications 55 0 2 4 6 8 10 12 14 16 0 200 400 600 800 1000 1200 1400 Steam flow control valve (%) 102 104 106 108 110 112 114 116 118 120 122 0 200 400 600 800 1000 1200 1400 AutoclaveTemperature (ºC) FIGURE 3.10: Autoclave experimental open loop test The system can be described by a set of ordinary differential equations derived from mass and energy balances. The following assumptions are made in the derivation of the model equations of this unit: •The steam, air and vapor-air mixture are considered ideal gases. •The unit is heated homogeneously, i.e. the temperature is the same in all points of the autoclave. •Liquid water and steam are considered in balance throughout the process. Using Mass and Energy balance equations [168], is obtained: [mv(Cpv −Rv)+ma(Cpa −Ra)+mwCpw]dT dt =Fi vCpv(Ti v−T)+RvT+Fi aCpa(Ti a−T)+ RaT]+Fi wCpw(Ti w−T)−FpT[xvRv+xaRa]−λΨ−(Qrad +Qconv)−Qcarc −Qsol (3.19) Where mis the mass, Cpthe specific heat, Fis the flow rate, xymass fraction of component y,Ry is the relationship between the universal gas constant (R) and the molecular weight of y,λlatent heat of water, Ψwater flow that is transferred between liquid and vapor phases, Qr,Qc,Qh,Qs, radiant, convection, through the casing, and through the solid heat, respectively, Tthe temperature. The subscripts v,a,windicate vapor, air and water, respectively and the superscript i indicates ”input”. Equation 3.19 describes a nonlinear system and the nonlinear behaviour is evident in figure 3.10. If we linearise at every equilibrium point, we will have different linear systems. In order to simplify the control problem and taking into account the nonlinear behaviour, a FIS can be made using the results of the experiment for identification. A set of rules formed by linear systems as a consequents can be set by the operating point variable values for the antecedents. For example, consider that T(z)represents the autoclave temperature (in discrete time), and Us(z)represents the positions of the steam valve, and G(z) = T(z) U(z), we can formulate the rule-base of the FIS as: Fuzzy modelling techniques and applications 56 IF T(z)is 108oCTHEN G1(z) = 0.09259z+0.09259 z−0.9259 IF T(z)is 112.5oCTHEN G2(z) = 0.07121z+0.07121 z−0.9394 IF T(z)is 116oCTHEN G3(z) = 0.06515z+0.06515 z−0.9394 IF T(z)is 118oCTHEN G4(z) = 0.05405z+0.05405 z−0.9459 IF T(z)is 120.5oCTHEN G5(z) = 0.03919z+0.03919 z−0.9459 IF T(z)is 121.5oCTHEN G6(z) = 0.00303z+0.00303 z−0.9394 Where for the variable T(z), a set of memberships have been chosen as shown in figure 3.11. Figure 3.12 shows the validation of the Fuzzy model vs the real data. 105 107 109 111 113 115 117 119 121 123 0 0.2 0.4 0.6 0.8 1 Temperature(celsius) Degreeofmembership mf1 mf2 mf3 mf4 mf5 mf6 FIGURE 3.11: Membership functions of the temperature of Autoclave 3.3.2 Gas Mixing Chamber As an application of PCA for Fuzzy modelling, a gas mixing chamber is proposed. The process is a part of Atlantic Copper Smelter facilities in Huelva (Spain), whose anual production is around three hundred thousand tons of copper [3]. This plant includes a Flash Furnace and four Pierce-Smith converters, two of them blowing simultaneously. The three gas streams generated in these processes are mixed in the mixing chamber and sent to three acid plants operating in parallel (see figure 3.14). It is very important to maintain the gas pressure in the mixing chamber at a desired value, always bellow ambient pressure in order to avoid gas losses to the atmosphere. That pressure depends on other variables of the production line and it is very difficult to get an accurate prediction of it. On one hand, the causes of the pressure oscillations are hard to detect. Moreover, since there are different control systems in the copper smelter and the acid plant, no clock synchronization is possible, so there are considerable uncertainties when Fuzzy modelling techniques and applications 57 1000 2000 3000 4000 5000 6000 7000 8000 9000 102 104 106 108 110 112 114 116 118 120 122 Time(seconds) Temperature(celsius) Temperaturesteps Realtemperature FuzzyModel FIGURE 3.12: Real data and model output for the autoclave FIGURE 3.13: General view of the copper smelter [3] Fuzzy modelling techniques and applications 58 pressure Gassesoutputlines Acidplants PT chamber Mixing FIGURE 3.14: Gas mixing. Acid plants trying to measure cause-effect delays. A suitable model for one step ahead prediction has been developed in [3]. Other models have been made in [169,170]. All of them are prediction models, because the actual pressure value at time k PMC(k)is used to predict PMC(k+1). Taking into account that the converters operate on a batch mode, while those of the flash furnace and acid plants are continuous, extremely high disturbances both in flow and SO2concentration occur at the acid plants inlet due to the converters’ operating schedule. The existing control strategy, based on independent single loop PID controllers, is not able to cope with those disturbances [3]. Advanced control schemes should be applied, but it would be interesting to have a model suitable for simulation. In this case, it is difficult to derive a precise mathematical model, based on first principles. Besides, the computation of the solution of models obtained through this methodology may require a large computational effort making them useless for real time tasks, such as control or optimisation. Neurofuzzy modeling, which permits an easy way to derive successful models, is a good alternative which can be employed to overcome such limitations [171–174]. After a preliminary study based on some experiments with steps on the variables, the evolution of the pressure in the mixing chamber (PMC), is influenced by others that are divided into two groups: control signals and disturbances. In table 3.1 a brief description of the considered variables is given, whereas in figure 3.15 a scheme depicting each of them is presented. Figure 3.17 presents the scheme followed to obtain the neurofuzzy model. The principal component analysis obtains new set of variables which are linear combination of the original input variables. Let X= [x1x2...xn]a set of variables. If the principal components are y1,y2,...,ypWe can write yi=ai1x1+ai2x2+...+ainxn(3.20) and if Y= [y1y2...yp],Y=AX The idea is to find Ato maximise the variance of Y, subject to <aik,ajk >=1 if i=jand <aik,ajk >=0 if i6=j. This analysis will be studied in more detail in chapter 6. Fuzzy modelling techniques and applications 59 FlashFurnace AcidPlant1 Scrubber Converter1 Converter2 ST ST ST ZT FT PT PT AcidPlant3 AcidPlant2 FT FT FT FT DilutionValve DilutionValve DilutionValve FanSpeed FanSpeed MixingChamberPressure MC FIGURE 3.15: Process and manipulated variables TABLE 3.1: List of variables Description Units Type Pressure in mixing chamber mbar Output Flow to plant 1 kNm3/h Manipulated Flow to plant 2 kNm3/h Manipulated Flow to plant 3 kNm3/h Manipulated Dilution flow to plant 1 Nm3/h Manipulated Dilution flow to plant 2 Nm3/h Manipulated Dilution flow to plant 3 Nm3/h Manipulated Reference for flash furnace feeding Ton/h Disturbance Flow control valve for flash furnace % Disturbance Fan speed in flash furnace rpm Disturbance Reference for fan speed line 1 rpm Disturbance Reference for fan speed line 2 rpm Disturbance Fuzzy modelling techniques and applications 60 The performance of the model can be seen in figure 3.16, where it is validated using a different real data set from the process. 0 200 400 600 800 1000 1200 1400 1600 −6 −4 −2 0 seconds Pressure (mbar) Real Data Model 0 200 400 600 800 1000 1200 1400 1600 −2 −1 0 1 seconds Error (mbar) FIGURE 3.16: Validation of the model used in [170] An improvement of that model is proposed using a major number of inputs, including squares of variables, to also provide a non linear dependence for each rule. A PCA has been used both in a model used in [170] and the one proposed here. In the first, the analysis is concerned with determining uncorrelated variables. In the second, further simplification is achieved in the FIS. The addition of inputs does not complicate the model when PCA is applied. Figure 3.17 presents FIS PCA Z-1 . . . Z-1 . . . . . . 24inputvariablestoPCA 11 inputvariables 21variables afterPCA PMC FIGURE 3.17: Model Scheme used in [170] Fuzzy modelling techniques and applications 61 the scheme followed to obtain the neurofuzzy model. The inputs are the variables presented in table 3.1, including pressure in mixing chamber, their squares and the previous samples of all of them. To carry out the PCA, data have been used for approximately three hours of operation, sampled every 2 seconds. The first 7 components involve 99% of the variability of the data. Using these new 7 uncorrelated variables as inputs to the fuzzy system, and the next sampling pressure as output, an ANFIS is designed, using Subtractive clustering technique [172]. The performance of the model can be seen in figure 3.18, where it is validated using a real data set from the process. It is important to note that PMC(t−1)is generated using an independent model output, that is, the model is not fed with measured data [29]. Looking at the figures, it is evident that the error does not grow indefinitely, this fact makes the models appropriate to simulate the process. In figure 3.16, the mean error is 0.4099 mbar, while for the new model proposed, the mean error is 0.1469 mbar, obtaining a significant improvement of 64%. 0 200 400 600 800 1000 1200 1400 1600 −6 −4 −2 0 seconds Pressure (mbar) Real Data Model 0 200 400 600 800 1000 1200 1400 1600 −1 −0.5 0 0.5 1 seconds Error (mbar) FIGURE 3.18: Validation of the proposed model used in [170] 3.4 Conclusion of the chapter In this chapter, an overview of various fuzzy modeling techniques has been described. After a general description, the experience of several years of work with fuzzy models was presented. There have been two examples of modeling real industrial systems. The contributions to the Fuzzy modelling techniques and applications 62 modeling of fuzzy systems, include using PCA to reduce the input space, allowing the modeling of complex systems. Accurate independent model outputs were obtained. In relation to the main objective of the thesis, this chapter helps to define an effective methodology for modeling nonlinear and complex systems. Since MPC relies on accurate model predictions, the performance and accuracy of the independent model outputs demonstrated in this chapter, ensures that these models are suitable for predictive control. Fuzzy control systems: practical implementation 69 -1 -0.5 00.5 1 -1 0 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 e d t d e dt du FIGURE 4.9: Control surface of a fuzzy incremental direct controller PC PLC Ethernet 0-10V position Motor camera Variable-speed Driver FIGURE 4.10: Pneumatic Levitation System [180] linked to it through Ethernet using the MODBUS TCP/IP protocol. Finally, the ball position is computed by means of a camera connected to the PC, which has an appropriate image processing software available [181]. As noted in [180], there is a dependence of the air flow velocity, vand the distance to the fan output, h. Although a wide zone where the velocity can be considered to be proportional to h−1exists, if h is large enough (in this application, about 50 cm) the flow velocity decreases drastically, making the system dynamics very complex. Moreover, taking into account that other effects (such as lateral and rotational motions, lack of sphericity, rugosity, etc...) have not been considered, it is easy to realise that the behaviour of this system is subject to a nonlinear dynamics. Presumably a linear controller shall have adequate performance, where the system shows a linear behavior. In order to compare, three control strategies have been implemented. The first controller is a classical PI, which provided a good trade-off between tracking and disturbance rejection as can be observed in fig. 4.11. A model-based H∞ Fuzzy control systems: practical implementation 70 controller was also tried out (Fig. 4.12). Although a better behavior was achieved for tracking experiments, this controller provided poor performance for disturbance rejection. This fact may be explained taking into account the uncertainty of the system dominant mode, which is too oscillatory. The direct fuzzy control test can be seen in (Fig. 4.13). Despite having only a few rules, this controller has slightly improved performance over that attained by the PI controller. The advantage of applying a direct fuzzy controller in this case, is clearly seen in the table 4.1, where the Integral of Time Absolute Error (ITAE) index of the time responses are exposed. FIGURE 4.11: Experimental results with the PID controller TABLE 4.1: ITAE index for each controller Controller Tracking Disturbance rejection PID 19.42 10.88 H∞17.54 16.12 Fuzzy 18.66 9.72 This controller has been implemented on a industrial PLC following the standard IEC61131-7 [11], (see 2.2). The code of the fuzzy block, according with this standard is: FUNCTION_BLOCK fuzzy1 VAR_INPUT e REAL; (* RANGE(-1 .. 1) *) de REAL; (* RANGE(-1 .. 1) *) Fuzzy control systems: practical implementation 71 FIGURE 4.12: Experimental results with the H∞controller FIGURE 4.13: Experimental results with the fuzzy controller Fuzzy control systems: practical implementation 72 END_VAR VAR_OUTPUT T_0 REAL; (* RANGE(0.7460 .. 0.9928) *) END_VAR FUZZIFY e TERM e_m1 := (-1.0, 1) (1.0, 0); TERM e_m2 := (-1.0, 0) (1.0, 1); END_FUZZIFY FUZZIFY de TERM de_m1 := (-1.0, 1) (1.0, 0); TERM de_m2 := (-1.0, 0) (1.0, 1); END_FUZZIFY DEFUZZIFY du TERM du_m1 := -1.0 ; TERM du_m2 := 0.0 ; TERM du_m3 := 1.0 ; METHOD: MoM; END_DEFUZZIFY RULEBLOCK No1 AND:MIN; ACCU:MAX; RULE 0: IF (e IS e_m1) AND (de IS de_m1) THEN (du IS du_m1); RULE 1: IF (e IS e_m1) AND (de IS de_m2) THEN (du IS du_m2); RULE 2: IF (e IS e_m2) AND (de IS de_m1) THEN (du IS du_m2); RULE 3: IF (e IS e_m2) AND (de IS de_m1) THEN (du IS du_m3); END_RULEBLOCK END_FUNCTION_BLOCK Where the inputs eand de have to be previously multiplied by the constants GE and GCE respectively and the output du needs to be multiplied by GCU before being integrated. 4.1.2 Stability analysis of fuzzy control systems Lliterature about studies of stability of fuzzy control systems abound, e.g. for direct controllers: [182–192], and for fuzzy model-based control systems: [193–204]. In [205] and [206] a design algorithm is proposed in order to obtain stable fuzzy controllers. Li-Xin Wang [207,208], raised an interesting scheme based on adding a supervisory control for fuzzy controller. Figure 4.14 shows the scheme proposed. The control action is composed by u=uc+uswith ucbeing the fuzzy controller action and usthe Fuzzy control systems: practical implementation 73 Plant Fuzzy controller Supervisory control - ++ + Adaptive law rey u uC uS FIGURE 4.14: Scheme of adaptive supervisory control[208] supervisory controller action. The idea of this control scheme is to compute a term us based on the structure of the controller and apply it to avoid violating stability limits established by a Lyapunov function. Within the limits of stability us=0. 4.2 Fuzzy model-based control One of the well known techniques that provided a procedure to design a controller from a TS fuzzy model is the Parallel Distributed Compensation (PDC). In [209], Kang and Sugeno design a way to control a system using a TS model. In [210–212] a procedure was developed, taking into account the stability of the system in the design. Using the state space formulation, the ith rules of a TS fuzzy model will be [213]: IF z1(t)is Mi1and ... zp(t)is Mip THEN (x(t+1) = Aix(t) +Biu(t) y(t) = Cix(t) Where zi(t)are the input premises of the system. They can be state variables or function of the inputs or disturbances. Mi j are fuzzy sets defined for the variables zj(t). The consequent is a linear system with outputs vector y(t)and state variables vector x(t) If for each rule we design a linear fuzzy controller such that IF z1(t)is Mi1and ... zp(t)is Mip THEN u(t) = −Fix(t)(4.7) Fuzzy control systems: practical implementation 74 The global state vector will be x(t+1) = r ∑ i=1 ai(z(t)){Aix(t)+Biu(t)}(4.8) The overall fuzzy controller will be u(t) = −∑r i=1wi(z(t))Fix(t) ∑r i=1wi(z(t)) =− r ∑ i=1 ai(z(t))Fix(t)(4.9) substituting 4.9 into 4.8, x(t+1) = r ∑ i=1 r ∑ j=1 ai(z(t))aj(z(t)){{Ai−BiFj}x(t)(4.10) According to [214] and [210] we can state the following sufficient stability condition [210]: Theorem 4.1. The equilibrium of a fuzzy system is globally asymptotically stable if there exists a common positive definite matrix Psuch that for all subsystem i, AT iPAi−P<0 (4.11) Finding a common Pis a LMI problem. From equation 4.7 and theorem 4.1, the following theorem can be formulate [211,215]: Theorem 4.2. The equilibrium of a fuzzy control system is globally asymptotically stable if there exists a common positive definite matrix P such that the following two conditions are satisfied: {Ai−BiFi}TP{Ai−BiFi}−P<0,i=1,2,...,r(4.12) GT i jPGi j −P<0,i<j≤r,s.t.ai∩aj6=φ(4.13) The control design problem is to select Fi(i=1,2,...,r)such that conditions 4.12 and 4.13 are satisfied. As seen in Chapter 1.1, the current paradigm of model-based control is the predictive control. The use of fuzzy techniques in predictive control was proposed for the first time by [216]. In the literature there are two approaches to conceive using fuzzy inference systems on predictive control. One is based on the use of fuzzy optimisation to solve the problem of predictive control, an approach that transparently translates objectives Fuzzy control systems: practical implementation 75 and constraints to predictive control by fuzzy multicriteria decision making [217–219]. Following this line, [220] a stable model-based fuzzy predictive control based on fuzzy dynamic programming is proposed. Other approach is the use of fuzzy systems as a prediction model for any MPC strategy, [221–225]. The use of fuzzy modeling for predictive control is justified by the complexity of the system to be modeled, either because the number of variables and their interactions, as the nonlinearities or hybrid nature [226]. In chapter 5a FMPC application will be studied in more detail. 4.3 Controllers with adaptive fuzzy parameters The FIS are often used to modify controller parameters, following an adaptive control scheme. In [227–230], a fuzzy system is used to adapt online the parameters of a PID. This strategy has also been used to adjust the MPC parameters setting [231–234]. For example, in [231], an evaporator process has been regulated by an MPC with a fuzzy parameter adaptive system. In that work the idea was to establish a set of rules changing the adjustments of the MPC parameters as a function of the degree of bound violation. That reactive change gives the system a better performance as the fuzzy system will provoke changes before the bounds are violated. 4.3.1 Air Separation Unit Another application done for this thesis is an adjustment of the move suppression in a MPC for one Air Separation Unit. The cryogenic process is comprised of unit operations that compress, purify, and separate the air feed into the required gaseous and liquid oxygen, nitrogen, and argon product flows 1[235] The argon column gets its feed as an intermediate vapor stream from the low pressure column which is mostly returned to the low pressure column as liquid condensed by that crude oxygen stream. Clearly there is significant interaction (Fig.4.16) when almost any of the manipulated variables are adjusted or a disturbance affects one of the column controlled variables.1 In this process, one of the important variables is the mid point purity (MPP). This is the gaseous oxygen plus argon product leaving the low pressure column. In a continuous operation process, it is important to keep this value as stable as possible. Fuzzy control systems: practical implementation 76 FIGURE 4.15: Simplified process for the cryogenic production of oxygen and argon 1 FIGURE 4.16: Portion of the dynamic matrix for a cryogenic air separation plant 1 Fuzzy control systems: practical implementation 77 There are several disturbances that affect this variable: input air to the high pressure column and the reflux from the high to the low pressure column. When the molecular sieve changes, a disturbance affects the MPP. A fuzzy adaptive system has been designed to modulate the move suppression in the MPC of the MPP control depending on the molecular sieve operating stage, making the controller less aggressive during that disturbance. An air separation plant supplying gaseous products is either connected to a large pipeline system supplying multiple customers or it may be directly supplying product to a single customer. Regardless, the product must be supplied to the customer when it is needed, so the plant has to respond rapidly to the changing product demand. Given the long time to steady state, it is possible that the plant may rarely ever settle into a steady state condition. This requires that the control strategy handle both the dynamic and steady state effects in order to achieve efficient operation. At the same time, the energy intensive nature of cryogenic liquid production often requires changing production rates to take advantage of variable power pricing or supply chain demands.1 A setting of a controller that allows a sufficiently aggressive action to changes in production can lead to a greater sensitivity to the disturbance caused by changes sieves. The designed fuzzy system takes into account the variation in demand, as measured by total oxygen flow out of the plant. When the system is experiencing changes in production, the rules cause a more aggressive driver, manipulating the main flow of air compressor that affects the MPP, to keep control variable at the set-point. Decreasing the variation in demand the fuzzy supervisor increases the move suppression, causing smoother control action, which will be not excited by the periodic disturbances created by changing sieves. The action of this adaptive system provides better performance of the MPC, achieving a reduction of about 10% in the standard deviation of the MPP regarding its set-point. Figure 4.17 shows the performance of this control scheme (b). Comparing with a MPC with fix move suppression (a), the controller reacts smoothly with sieves changes (periodic disturbances in the incoming air), but with more aggressive changes with the oxygen demand. Although the change in the demand for oxygen is lower in the graph (a), an improved performance compared to MPP control with fuzzy supervisor (b) can be seen. Fuzzy control systems: practical implementation 78 0 200 400 600 800 1000 0.8 0.85 0.9 0.95 1 1.05 MPP (Without Fuzzy S.) (a) 0 200 400 600 800 1000 0.75 0.8 0.85 0.9 0.95 1 Inlet air (a) 0 200 400 600 800 1000 0.8 0.85 0.9 0.95 1 1.05 MPP (With Fuzzy S.) (b) 0 200 400 600 800 1000 0.75 0.8 0.85 0.9 0.95 1 Inlet air (b) FIGURE 4.17: MPP control without (a) and with fuzzy supervisory system (b). Normalised variables 4.4 Conclusion of the chapter In this chapter, the main fuzzy control techniques have been summarised . Fuzzy control is a large field of study where many engineers have designed different control strategies [175]. There are excellent books and articles related to fuzzy control Systems [175, 236], even surveys related to specific fields of applications [237,238]. In addition to the summary, the major contribution of the chapter is the actual implementation of control strategies in real plants. A comparison of other drivers with a direct fuzzy controller in air levitation plant has been made and an application of a supervisory fuzzy control system in a real chemical plant in operation has been applied. In the next chapter we will focus on a specific application of fuzzy model-based predictive control. Fuzzy MPC for an industrial autoclave 85 RealdataRealdata Model Temperature(ºC) Time(s) FIGURE 5.3: Validation of neurofuzzy model of the industrial autoclave neurofuzzy models [225,254–259]. Perhaps, the main issue in non-linear Predictive Control is how to obtain the optimiser solution, which consists in a non-convex problem and its resolution includes a high computational cost to solve in real time. During the last few years, several techniques have arisen to avoid the problems associated to determine the solution of the non-convex optimisation problem [18]. Moreover, closed loop stability must be guaranteed. Two strategies based on the neurofuzzy model obtained for the autoclave are applied here. It is seen in chapter 2, the nonlinear model of the Takagi-Sugeno system, has a set of linear models equal to the number of rules. In this case, the linear models equations are: T1(t) = 0.385u(t−1)−0.385u(t−2)+0.125T(t−3)+0.606T(t−2)+1.732T(t−1) (5.2) T2(t) = 0.035u(t−1)−0.035u(t−2)−0.308T(t−3)+ 1.580T(t−2)+ 2.273T(t−1) (5.3) where Ti(t)is the temperature of autoclave for the local model i, and u(t−j)is the position of steam valve at jprevious samples. A strategy proposed in [252,257,259,260] involves calculating as many GPC controllers as linear models obtained in the neurofuzzy model, such that the controller output be (3.2): u(k) = L ∑ j=1 wj(k)uj(k)(5.4) Fuzzy MPC for an industrial autoclave 86 where Lis the number of linear models and wjiss defined as wj(k) = ¯ µj(k) ∑N j=1¯ µj(k),¯ µj(k) = n ∏ i=1 µi j(k)(5.5) This procedure will be referred to as the FGPC1. The advantage of this technique is an easy and fast implementation and can be applied on a simple PLC. The main disadvantage of this strategy is that global optimum is not always found. However a local optimum is guarantied for each rule or implication [260]. Another non-linear strategy based on neurofuzzy models (FGPC2) is proposed in [254]. A recurrent neurofuzzy model could be rewritten as a Linear Time Variant (LTV): ¯a(z−1)y(k) = ¯ b(z−1)u(k−d)+ ξ(k)(5.6) Where dis a transport delay, ξ(k)is a white noise sequence with null average and: ¯a(z−1) = 1−¯a1z−1−¯a2z−2−...−¯anyz−ny ¯ b(z−1) = ¯ b1z−1+¯ b2z−2+...+¯ bnuz−nu ¯ai= L ∑ j=1 wj(k)ai jz−i ¯ bi= L ∑ j=1 wj(k)bi jz−i The cost function defined in (1.9) can be expressed like: J(k) = (Fy(k)+G∆u(k)+Λ−ΦW)T(Fy(k)+G∆u(k)+Λ−ΦW)+(λ(z−1)∆u(k))2 (5.7) Where: F=fd(z−1)fd+1(z−1)... fNp(z−1)T, Λ="d+Nu−1 ∑ ρ=1 gd,ρ∆u(k−ρ) d+Nu ∑ ρ=1 gd+1,ρ∆u(k−ρ)... Np+Nu−1 ∑ ρ=1 gNp,ρ∆u(k−ρ)#T Φ=diag{δdδd+1... δNp} W= [w(k+d)w(k+d+1)... w(k+Np)]T Fuzzy MPC for an industrial autoclave 87 The prediction horizon was chosen as Np=3 and, to reduce computational cost, the control horizon was reduced to Nu=1. This leads to: GT(Fy(k)+Λ−ΦW)+(GTG+λ(z−1)λ0)∆u∗(k) = 0 (5.8) To be able to simplify the control law, λ2 0=λ>0 is chosen. Values λ1,λ2,...,λNpare adjusted such that they meet: ∆u∗(k) = GT(ΦW−Fy(k)) GTG+λ.(5.9) A stability study of this control strategy applied to neurofuzzy model was made in [254]. 5.1.3 Experimental results The pilot plant described in the introduction will be now be used a case study to illustrate the performance of the different controllers derived in this work. Using equal prediction and control horizons, based on a simulation tuning process, the values chosen for the prediction and control horizons were Np=Nu=4 and the weighting parameter of the control action λ=0.6, a GPC controller was obtained with the following law: u(k) = u(k−1)−12.01y(k−1)+21.68y(k−2)−12.85y(k−3)+2.30y(k−4) +0.44w(k+1)+0.27w(k+2)+0.14w(k+3)+0.04w(k+4)(5.10) Where a soft approximation of the future reference trajectory has been used with α=0.7 w(t+k) = αw(t+k−1)+(1−α)r(t+k),k=1,...,Np(5.11) Figure 5.4 shows the closed loop response of the retort temperature (black line) with this controller. Four different steps have been introduced in the set point (green line). As illustrated in the figure 5.4, the controller is not only slow but it rarely reaches a good approximation to the set point. As mentioned in the introduction, good controller performance is required in this type of process. In order to improve the closed loop response, the FGPC controllers have been Fuzzy MPC for an industrial autoclave 88 Temperature(ºC) Time(s) FIGURE 5.4: Performance of GPC implemented. To carry out FGPC controllers proposed here, first of all it is necessary to change gaussian membership functions (see figure 5.2) to new triangular membership function defined by IEC61131-7. In figure 5.5, the new equivalent functions are shown. To have the output given in 5.4, FCL code is given in 5.1.4. 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 In1 In2 In3 In4 In5 In6 FIGURE 5.5: Equivalent triangular membership functions Fuzzy MPC for an industrial autoclave 89 The results for FGPC controller are represented in Figure 5.6. It is clear that this con- Temperature(ºC) Time(s) FIGURE 5.6: Comparison between Predictive Control strategies Non-linear vs linear trol scheme is able to approach the retort temperature (black line) to the set point (green line) better than the GPC (turquoise line). In a third test, the FGPC2 technique has also been applied to the pilot plant under the same conditions than the GPC and FGPC1. The black line in figure 5.7 corresponds Temperature(ºC) Time(s) FIGURE 5.7: Comparison between strategies with the closed loop (FGPC2) temperature response. On initial examination, it seems (although difficult to assertain) that it improves the FGPC1 results (orange line). In order to obtain a more quantitative comparison between these two methods, the Integral of Absolute Error (IAE) (as a measurement of the tracking error) has been computed and presented in Table 5.1. The IAE for the FGPC2 is about a 20% lower than the IAE for the FGPC1 asserting that its performance is better. A previous control strategy using a PI-type controller parameterized by means of the Internal Model Control (IMC) technique [261] has been presented in [167]. It is important to highlight that the FGPC2 presents a similar performance to the PI, but with Fuzzy MPC for an industrial autoclave 90 GPC FGPC1 FGPC2 1.8688·1048.2837·1036.8766·103 TABLE 5.1: IAE comparison a saving of approximately a 70% in the energy consumption computed by using the index: Et=Zt us(t)dt. 5.1.4 FLC code accomplishing IEC 61131-7 As we saw in Chapter 2, the IEC61131-7 standard defines the language of fuzzy control for PLCs. The us1 and us1 variables, for FGPC1 strategy, must be previously calculated as the result of two GPC. Subsequently, the following code implements the fuzzy controller: FUNCTION_BLOCK fmpc VAR_INPUT us REAL; (* RANGE(0 .. 1) *) Ts REAL; (* RANGE(0.7 .. 1) *) ub REAL; (* RANGE(0 .. 1) *) ud REAL; (* RANGE(0 .. 1) *) T_2 REAL; (* RANGE(0.4 .. 1) *) T_1 REAL; (* RANGE(0.4 .. 1) *) END_VAR VAR_OUTPUT T_0 REAL; (* RANGE(0.7460 .. 0.9928) *) END_VAR FUZZIFY us TERM us_m1 := (-0.3558, 0) (0.6463, 1) (1.6480, 0) ; TERM us_m2 := (-0.3949, 0) (0.6016, 1) (1.5980, 0) ; END_FUZZIFY FUZZIFY Ts Ts_m1 := (0.5273, 0) (0.6774, 1) (0.8275, 0) ; Ts_m2 := (0.4507, 0) (0.9000, 1) (1.3490, 0) ; END_FUZZIFY FUZZIFY ub TERM us_m1 := (-0.3038, 0) (0.6954, 1) (1.6950, 0) ; TERM us_m2 := (-0.9986, 0) (-0.0005, 1) (0.9975, 0) ; END_FUZZIFY Fuzzy MPC for an industrial autoclave 91 FUZZIFY ud TERM us_m1 := (-0.5639, 0) (0.4354, 1) (1.4350, 0) ; TERM us_m2 := (-0.5664, 0) (0.4314, 1) (1.4290, 0) ; END_FUZZIFY FUZZIFY T_2 T2_m1 := (0.3081, 0) (0.9347, 1) (1.5610, 0) ; T2_m2 := (0.4255, 0) (0.9885, 1) (1.5520, 0) ; END_FUZZIFY FUZZIFY T_1 T1_m1 := (0.3003, 0) (0.9335, 1) (1.5670, 0) ; T1_m2 := (0.4376, 0) (0.9917, 1) (1.5460, 0) ; END_FUZZIFY DEFUZZIFY T0 T0_m1 := us1 ; T0_m2 := us2 ; METHOD: MoM; END_DEFUZZIFY RULEBLOCK first AND:MIN; ACCU:MAX; RULE 0: IF (us IS us_m1) AND (Ts IS Ts_m1) AND (ub IS ub_m1) AND (ud IS ud_m1) AND (T_2 IS T_2_m1) AND (T_1 IS T1_m1) THEN (T_0 IS T0_m1); RULE 1: IF (us IS us_m2) AND (Ts IS Ts_m2) AND (ub IS ub_m2) AND (ud IS ud_m2) AND (T_2 IS T_2_m2) AND (T_1 IS T1_m2) THEN (T_0 IS T0_m2); END_RULEBLOCK END_FUNCTION_BLOCK 5.2 FMPC with constraints. Implementation issues One of the problems of NMPC when considering constraints is the heavy computational burden associated with its implementation. On the other hand, it is also difficult to analyze the stability of the closed-loop system [262]. In [263] a Takagi-Sugeno (TS) fuzzy model predictive controller, based on piecewise Lyapunov function (PLF), has been proposed to solve the above problems with FMPC. However this method may lead to a conservative performance of the controller. [264] presents an alternative way to reduce the conservatism using fuzzy Lyapunov function (FLF) which only needs to find an independent positive definite matrix for each submodel. More recently, [262,265] have presented an extended-fuzzy Lyapunov function to improve the results. Notwithstanding the foregoing, the computing load of these methods prevents the implementation in Fuzzy MPC for an industrial autoclave 92 low-cost or industrial hardware as had been described in Chapter 1. There are some applications where the NN are adjusted to imitate the controller [266–268]. Once the NN is trained from the MPC controller, the amount of computation required is very small and can be compared to the online computation of the fast implementation methods. In this thesis, we propose a FIS for the same purpose. The use of Fuzzy system will permit the inclusion of adjusting parameters as another input variable. Let Fλibe a FIS obtained from a NMPC tuned by λi, composed by Nirules Rjas was shown in 3.2: Rj: IF x1(k)is F1j,..., and xn(k)is Fn j, THEN: yj(k) = aj(z−1)y(k−1)+ bj(z−1)u(k−d)+ξ(k) Figure 5.8 shows the procedure to adjust Fλifrom the NMPC. Changing parameter λito reach several control performance modes, e.g. aggresive, moderate, slow, we will obtain as many fuzzy systems as λichosen. The union of fuzzy systems {Fλ1,Fλ2,...,Fλp}will result another fuzzy system with a new input variable λiwith N=N1+N2+... +Np rules: Rj: IF x1(k)is F1j,..., and xn(k)is Fn j,and λjis Λr j THEN: yj(k) = aj(z−1)y(k−1)+ bj(z−1)u(k−d)+ξ(k)(5.12) MPC controller Fuzzy Inference System + - Training r,x e uC uf l FIGURE 5.8: Scheme to obtain a fast FIS from another controller Fuzzy MPC for an industrial autoclave 93 5.3 Conclusion of the chapter In this chapter FNMPC has been presented as an alternative to NMPC, saving computational burden. Two different schemes have been implemented in a real industrial plant, using low computational cost hardware with FCL. The results have been compared with a linear GPC, obtaining better performance with no significant programming effort and computational resources. NMPC subject to constraint has also been introduced, and a new technique to carry out a fast implementation providing an adjusting parameter has been proposed. Both the complexity of the system and the requirement of a higher precision in the parameter setting can cause an explosion of rules in the fuzzy model. Reasonably managing a FIS for use in control, involves applying a rules reduction technique. In the following chapters we will introduce a novel technique for complexity reduction in fuzzy systems based on their structure. Chapter 6 Complexity reduction in fuzzy systems using Functional Principal Component Analysis In this chapter a novel technique to reduce complexity in fuzzy models will be described. The aim of such reduction is the suitability to control systems which run on low capability hardware platforms. Firstly, in section 6.1, the state of the art in complexity reduction of fuzzy systems will be explored. Secondly in sections 6.2 and 6.3 the Functional Principal Component Analysis (FPCA) will be described. Section 6.4 will show the main contribution of this chapter: the application of FPCA to reduce complexity of a fuzzy inference system structures. Section 6.5 will describe example applications before drawing conclusions on this chapter. 6.1 Complexity reduction in fuzzy systems The ability to build fuzzy logic applications for control problems has been hindered by the well-known problem of combinatorial rules explosion, causing complexity in modeling. The existence of redundant rules may also cause performance degradation of the FIS [123]. There has been a increased interest in the issues of complexity of fuzzy systems over recent years. There are a remarkable number of methods aimed at reducing the complexity of fuzzy systems. Most of them are based on systematic and heuristic methods 94 Complexity reduction in fuzzy systems using FPCA 101 6.5 Illustrative examples 6.5.1 Pilot plant To illustrate the ideas developed in this chapter we will examine two examples. The first one is a pilot plant site in the Department of System Engineering and Automatic Control of the University of Seville 6.2. The plant is used to emulate exothermic chemical reactions based on temperature changes. It has previously been used as a benchmark for control by researchers [283]. The main elements of the pilot plant are the reactor, the heat exchanger, the cooling jacket and the valve to manipulate the flow rate through the cooling jacket 6.3. FIGURE 6.2: Pilot plant of the Department of System Engineering and Automatic Control of University of Seville The emulated chemical reaction represents a refinement process. At the same flow at the inlet of the reactor outlet and a constant volume, the model of the chemical reaction Complexity reduction in fuzzy systems using FPCA 102 FIGURE 6.3: Diagram of the pilot plant TABLE 6.1: Parameters and variables of the mathematical model of the pilot plant Parameter Value Unit Cp(Specific heat capacity) 4.18 KJ/K·Kg ∆H(Molar reaction heat) −105.57 KJ/mol V(Volume of the reactor content) 25 l M(Mass of the reactor content) 25 Kg CA,in (Reactan concentration in the feed) 1.2mol/l Fj(Cooling jacket flow rate) 0.05 l/s k0(Constant) 1.265x1017 l/mol E/R(Constant) 13550 K can be defined as: dT dt =−Fj V(Tj,in −Tj,out)+ (−∆H)V M·Cp k0e−E/(RT)C2 A(6.30) dCA dt =Ff V(CA,in −CA)−k0e−E/(RT )C2 A(6.31) Where Tis the temperature into the reactor, Tj,in the inlet temperature of the cooling jacket fluid, CAdenotes the reactant concentration in the reactor. The other variables and parameters can be seen in 6.1 Using real data, the system can be modeled by a Neurofuzzy System. Being a discrete model, in order to capture the dynamics of the system, two input variables have been chosen: The valve previous position V(k−1) Complexity reduction in fuzzy systems using FPCA 103 and the previous sampled temperature T(k−1). The output is the actual temperature T(k). Figure 6.4 shows the membership functions. Using a training learning method, we obtain the following rules: 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 ºC Temperature low high 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 % Valve low high FIGURE 6.4: Membership functions for the Pilot plant Fuzzy model •if T(k−1)is LOW and V(k−1)is LOW then T(k) = 1.0045T(k−1)+0.0005V(k−1)−0.0898 •if T(k−1)is LOW and V(k−1)is HIGH then T(k) = 1.0006T(k−1)+0.0005V(k−1)−0.0571 •if T(k−1)is HIGH and V(k−1)is LOW then T(k) = 1.0037T(k−1)−0.0005V(k−1)−0.3197 •if T(k−1)is HIGH and V(k−1)is HIGH then T(k) = 1.0002T(k−1)−0.0020V(k−1)−0.0158 To perform the FPCA on the model, we proceed as follows: FPCA algorithm: (6.5.1) 1. Calculation of Wfrom eq.7.22 Complexity reduction in fuzzy systems using FPCA 104 2. W1/2obtained through Cholewsky decomposition 3. Solving the eigenvalue problem 6.24 4. Getting b=W−1/2u The new system is: ˜ g(x) =     −0.04154 0.00001 0.00531    ·ξ(x)(6.32) and ξ(x) = a(x)T·      −0.01020 −0.00998 −0.01149 −0.00976       (6.33) A comparison between systems is shown in figure 6.5. 0 2000 4000 6000 8000 10000 0 20 40 60 80 100 120 Realdata Fuzzymodel SimplifiedFuzzymodel FIGURE 6.5: Pilot plant: Comparison between original, Fuzzy and simplified Fuzzy system Complexity reduction in fuzzy systems using FPCA 105 6.5.2 Mechanical system Another interesting example is the mechanical system shown in figure 6.6. It could be a simple manipulator with only one joint. The system is moved by an electrical motor which provides a torque Tuin order to move a bar an angle θ. If we consider all the mass (m) concentrated at the end of the bar, the equation that describes the system is: m¨ θl2+B˙ θ+mgsinθ=Tu(6.34) For simulation the parameters will be: g=9.8m/s2,l=1m,B=1Kgm2/s,m=1Kg. As we can observe in figure 6.6 the system has a non-linearity due to sinθ. Linearizing q Tu m·g l B FIGURE 6.6: Mechanical system around an equilibrium point, we could model the system as ¨ θ=−a˙ θ−bθ+Tu(6.35) Where a,bare parameters depending on the operating point (θ0). It is a second order linear system. In order to build a Fuzzy system, we use four variables in discrete mode, to get the dynamics of a 2nd order system: Tu(k−2),Tu(k−1),θ(k−2),θ(k−1). Providing data sets for training and checking, the FIS obtained is defined by the membership function depicted in figure 6.7. Taking small steps to the input (torque), we can model the response as second order system 6.35, different for each operating point determined Complexity reduction in fuzzy systems using FPCA 106 -10 -5 0 5 10 0 0.2 0.4 0.6 0.8 1 N·m Tu(k-2) Neg Zero Pos -10 -5 0 5 10 0 0.2 0.4 0.6 0.8 1 N·m Tu(k-1) Neg Zero Pos -100 -50 0 50 100 0 0.2 0.4 0.6 0.8 1 Position(º) Angle(k-2) Neg Zero Pos -100 -50 0 50 100 0 0.2 0.4 0.6 0.8 1 Position(º) Angle(k-1) Neg Zero Pos FIGURE 6.7: Membership functions for the mechanical system FIS by the position of the mechanism. Doing this in nine areas, we have nine linear systems: θ1(k) = 0.0037T(k−1)+0.0467T(k−2)−0.9705θ(k−1)+1.9705θ(k−2) θ2(k) = −0.0016T(k−1)+0.0525T(k−2)−0.9704θ(k−1)+1.9645θ(k−2) θ3(k) = −0.0001T(k−1)+0.0508T(k−2)−0.9704θ(k−1)+1.9628θ(k−2) θ4(k) = −0.0003T(k−1)+0.0511T(k−2)−0.9704θ(k−1)+1.9621θ(k−2) θ5(k) = −0.0003T(k−1)+0.0508T(k−2)−0.9705θ(k−1)+1.9619θ(k−2) θ6(k) = −0.0003T(k−1)+0.0510T(k−2)−0.9704θ(k−1)+1.9621θ(k−2) θ7(k) = −0.0002T(k−1)+0.0509T(k−2)−0.9704θ(k−1)+1.9629θ(k−2) θ8(k) = −0.0008T(k−1)+0.0515T(k−2)−0.9704θ(k−1)+1.9650θ(k−2) θ9(k) = 0.00065T(k−1)+0.0501T(k−2)−0.9704θ(k−1)+1.9705θ(k−2) where θi(k)is the angle variation for the local model i, and T(k−j)is the variation of the applied torque. establishing rules with variable angle as antecedent, we can model Complexity reduction in fuzzy systems using FPCA 107 the mechanical system with minimum error, as is shown in fig 6.8, obtaining a RMSE of ±0.1064◦over 3334 samples. In order to simplify the system applying a FPCA, the 0 10 20 30 40 50 60 70 80 90 100 -10 -5 0 5 10 Torque 010 20 30 40 50 60 70 80 90 100 -40 -20 0 20 40 60 Angle realangle FM FIGURE 6.8: Validation of the fuzzy model for the mechanical system procedure seen in 6.5.1 will be used. It is observed that the first eigenvalue contents almost all the variability of the system. Thus, the new simplified system will have just one rule and its structure is given by: ˜ g(x) =          0 0.0198 −0.3788 0.7669 −0.0034          ·ξ(x)(6.36) And ξ(x) = a(x)T·                    0.0425 0.0433 0.0433 0.0433 0.0433 0.0433 0.0434 0.0435 0.0444                    (6.37) Complexity reduction in fuzzy systems using FPCA 108 In the figure 6.9 we can distinguish differences between the fuzzy and simplified fuzzy models. However, with an RMSE of ±2.1692◦over 3334 samples, it is reasonable to use the simplified model for control or simulation. 0 10 20 30 40 50 60 70 80 90 100 -10 -5 0 5 10 Torque 0 10 20 30 40 50 60 70 80 90 100 -40 -20 0 20 40 60 Angle realangle FM SFM FIGURE 6.9: Mechanical system: Comparison between original, Fuzzy and simplified Fuzzy system 6.6 Conclusion of the chapter In this chapter a new analytic technique has been presented in order to reduce the complexity of TS fuzzy systems. The main contribution of the thesis is the application of FPCA to the Hilbert space of functions defined by the rule base. Defining a covariance operator, the technique permits to reduce the space to a subspace where the operator’s eigenvalues are bigger. There have been two examples where the technique has been applied successfully. The problem with this technique is the lack of interpretability in the fuzzy system, due to the algebraic combination before aggregation. If we focus on the use in predictive control, this is not a problem. Even predictive controllers with explicit solutions that could be implemented in industrial devices according to IEC61131-3 [7] and IEC61131-7 [11], there may be the possibility of non-prototypical membership functions or algebraic operations after fuzzification. Chapter 7 FPCA to simplify MPC implementation Multivariate Statistics is used in control engineering for many years [284]. Singular Value Decomposition (SVD) techniques such as PCA has been used in control engineering for sensor fault detection [285], variable decoupling [286] and modelling [287,288]. Dimensionality reduction [289] is the main feature that takes advantage of these techniques. In this chapter, PCA and FPCA techniques studied in Chapter 6will be used to simplify the control implementation. The first section of this chapter will show a PCA application for input space reduction, simplifying the control scheme. In section 7.2 the technique of the chapter 6will be applied to simplify a FMPC and make it implementable in nonlinear systems with low capability hardware. The same technique will be applied in section 7.3 and 7.4, in order to reduce complexity of PWA systems and implement FMPC with constraints. 7.1 Dimensionality reduction of input variables space In a Multiple Inputs Single Output (MISO) system with n inputs, the values that acquire the inputs can be considered as vectors in a space of dimension n. The control problem is to determine the sequence of vectors that produce the desired output. The application of PCA may reduce the inputs space dimension, simplifying the control problem. If the new input space has just 1 dimension, the system become Single Input Single Output 109 FPCA to Simplify MPC implementation 110 (SISO), reducing the complexity of the control problem, using only one manipulated variable In the plant showed in section 3.3.2, after applying a PCA over the input variables, the first principal component accounts by itself for almost 80% of the information. The idea is to use this component as a new virtual input of the system keeping the pressure as the controlled output (see Figure 7.1). The control action obtained using this technique is then projected into the original axis in order to obtain all the real manipulable variables needed to operate the plant. Therefore, only a SISO controller needs to be adjusted in order to control this new system. The manipulated variable is ξ=w0PMC +hw1w2w3w4w5w6i·u+hw7w8w9w10 w11i·d(7.1) Where wiare the coordinates of the principal component, and the vectors u,dare the manipulated variables and measurable disturbances, respectively. Then, having a measure of the disturbances and the current value of the pressure PMC, the virtual manipulated variable ξcan be transformed in the new set of real control actions, uT=ξ−w0PMC −dTwdwT u wuwT u (7.2) Where wd=hw7w8w9w10 w11iTand wu=hw1w2w3w4w5w6iT PMC Manipulated Disturbances { C RPlant -- - ++ wd w0 wu T wu wu T FIGURE 7.1: Control scheme Tuning a PI controller experimentally over the new virtual SISO system and testing in simulation using the neurofuzzy model described in section 3.3.2, using real data from the plant for disturbances, a set point changes are applied to the pressure in the mixing chamber, and the controller is able to follow the new operating point and reject FPCA to Simplify MPC implementation 117 The covariance functions of ˜ g(x), are: Cov[˜ g(x),˜ g(s)] = ∆(x)Tcov(R)∆(s)(7.19) Supposing that the eigen functions are: Γ(x) =        γ1(x) γ2(x) . . . γh(x)        =∆(x)T·b(7.20) Thus, taking in account (7.19): ZX 0 Cov[˜ g(x),˜ g(s)]·γ(s)ds =ZX 0 ∆(x)Tcov(R)∆(s)·∆(s)T·bds =∆(x)Tcov(R)·W·b cov(R)·W·b=λ·b(7.21) Where: W=ZX 0 ∆(s)·∆(s)Tds (7.22) Being γ(x)orthogonal, then hγi(x),γj(x)i=bT i·W·bj=0. As it was seen in 6.24: W1 2·cov(R)·W1 2·p=λ·p(7.23) A symmetric eigenvalue problem now remains to be solved. Since only one region will be active in each state vector, equation 7.20 will give only one of the values within b parameter, γi(x)∈ {bi1,bi,2,...,biN}(7.24) It may be that there are repeated bi j values, in such case, the regions associated to those values can be merged, reducing the number of regions, i.e. If bi j =bik with j6=k, then γi(x) = bi1δ1(x)+...+bi j(δi(x)+δj(x))+...+biNδN(x), being: δi(x)+δj(x) = (1i f x∈ΘiSΘj 0else (7.25) FPCA to Simplify MPC implementation 118 7.3.1 Example: distillation column To illustrate the performance of a reduced PWA system, a high purity distillation column, like the example in section 4.3.1 will be given. The distillation process typically works around an operating point, being identifiable by a linear model. The example will be carried out using the model shown in [294], where the linear model in an operational point is defined as: "˙x1 ˙x2#="−0.0133 0 0−0.0133 #"x1 x2#+"0.0117 0.0115 0.0144 0.0146 #"u1 u2#, "y1 y2#="x1 x2#(7.26) Where y1and y2are the top and bottom product compositions, respectively, and the inputs, u1and u2, are the reflux flow rate and the boil-up, respectively, as is shown in the Figure 7.6 Considering the reference tracking problem, i.e., the problem of driving FIGURE 7.6: Distillation Column [295] the output (Product composition) yto track a given reference signal r∈Rpby adjusting the control inputs (reflux and boiler flow rates) uunder the control input and control increment constraints. For the current x, the constrained MPC solves the following optimisation problem such that the optimal control increment ∆uis found at each sampling FPCA to Simplify MPC implementation 119 instant, min unJ(u,r,y(k))o s.t. umin ≤u(k+j|k)≤umax,j=0,...,Nu−1, x(k+j+1|k) = Ax(k+j|k)+Bu(k+j),j≥0, y(k+j|k) = Cx(k+j|k)+ Du(k+j),j≥0, u(k+j) = u(k+j−1)+∆u(k+j),j≥0,(7.27) where the cost function to be minimised is given by: J(u,r,y(k)) = Ny−1 ∑ j=0y(k+j|k)−r(k)TQ y(k+j|k)−r(k) + Nu−1 ∑ j=0 ∆u(k+j)TR∆u(k+j),(7.28) and u,[∆u(k)T,...,∆u(k+Nu−1)T]T,x(k+j|k)is the predicted state at time step k, Nyand Nuare the prediction and control horizons, and Q≥0, R>0. The above MPC optimisation problem (7.27) can be described in the standard QP form, [18], and as shown in [53] such an MPC QP problem can be transformed into: min unJ(u,θ(k)) = 1 2uTHu+θ(k)TFTuo s.t. Gu≤W+Sθ(k),(7.29) where uis defined as in (7.28), and θis the vector of parameters defined as: θ(k) = [x(k),u(k−1),r(k)]T The MPC mpQP problem (7.29) is solved explicitly, off-line, for all the feasible values of θof interest, resulting in the solution u∗(θ), which is a continuous piecewise-affine function defined over a polyhedral partition in the θ-space represented as: ∆u(k) = f(θ(k)) =              K1θ(k)+k1,if θ(k)∈Θ1 K2θ(k)+k2,if θ(k)∈Θ2 . . . KNre j θ(k)+kNre j ,if θ(k)∈ΘNre j , (7.30) FPCA to Simplify MPC implementation 120 with a polyhedral partition P={Θ1,...,ΘNre j }, where the polyhedral sets are represented by linear inequalities (hyperplanes), Θi={θ(k)|Liθ(k)≤li},i=1,...,Nre j.(7.31) Here, Kiand kiare the control gain and offset for each region respectively, and Nre j is the number of regions. Consequently, the on-line temperature control algorithm is reduced to a look-up table: the region associated with the current state θis first determined, and then the optimal control law valid for that region is applied. The tuning parameters used for deriving the explicit MPC controller are as follows: Ny=20, Nu=3, Q=R=I, with a sampling time of 10 min. The control input constraints are given as −2≤u≤2 and −1≤u≤2, respectively. The number of regions obtained for the control law is 140. Applying the FPCA above to this application, with the same tuning, just 5 consequents are obtained for the first manipulate variable and 6 for the second, ˜g1(x) =               0.0177 −0.1947 0.1562 0.0036 0.0096 −0.0207 0.2275 −0.1822 0.0075 −0.0103 −0.1298 0.8182 0.4014 −0.0068 −0.0012 −0.0003 0.0035 −0.0027 −0.0007 −0.0004 −0.0302 0.3351 −0.2730 −0.0245 0.0031 0.0336 −0.3704 0.3005 −0.0195 −0.0085 1.6627 0.2191 0.0795 −0.0067 −0.0013               ·∆(x) ˜g2(x) =               −0.0271 0.1330 −0.1683 −0.0030 0.0074 0.0045 −0.0322 0.1575 −0.1992 0.0083 0.0092 0.0027 −0.0006 0.0027 −0.0033 −0.0029 −0.0003 0.0032 −0.4149 0.8620 0.2987 0.0018 0.0062 0.0035 0.0455 −0.2331 0.2968 0.0203 0.0060 0.0038 0.0504 −0.2562 0.3258 −0.0136 0.0088 0.0033 1.3315 0.4020 0.1321 0.0018 0.0062 0.0035               ·∆(x) Associating regions with the same parameter as it has been seen in 7.25, the arithmetic operations of antecedents are reduced from 140 to 68. In Figure 7.7 it can be observed that the proposed transformation has a similar performance as the system. FPCA to Simplify MPC implementation 121 time (min) 0 200 400 600 800 Composition (%) -0.5 0 0.5 1 1.5 output 1 time (min) 0 200 400 600 800 Composition (%) -0.5 0 0.5 1 1.5 output 2 time (min) 0 200 400 600 800 reflux (l/s) -2 -1 0 1 2input 1 time (min) 0 200 400 600 800 boil-up (l/s) -2 -1 0 1 2input 2 FIGURE 7.7: Controllers performance for distillation column 7.4 Simplified FMPC with constraints The idea indicated in 1.4 about the lack of parameter setting in the MPC with explicit solution and constraints, can be addressed by the application of a fuzzy controller having the control actions of the various PWA designed for different parameter values as inputs, and setting the parameter itself. Figure 7.8 shows the scheme of this structure. This scheme requires more memory capacity and programming time. If greater accu- PWA1 PWA2 PWA3 l FIS u FIGURE 7.8: FMPC with explicit solution subject to contraints FPCA to Simplify MPC implementation 122 racy is desired, there should be more PWA controller programmed into the system. For example, in the previous application, three different PWA for three values of lambda are designed, giving the number of regions and the performance shown in the figure 7.9 Once the FPCA is applied, a huge reduction in the number of consequents can be 0 200 400 600 800 1000 -0.5 0 0.5 1 1.5 time(min) Composition(%) output1 0 200 400 600 800 1000 -0.5 0 0.5 1 1.5 time(min) Composition(%) output2 lambda=1 lambda=5 lambda=10 Reference 0 200 400 600 800 1000 -2 -1 0 1 2 time(min) reflux(l/s) input1 0 200 400 600 800 1000 -2 -1 0 1 2 time(min) reflux(l/s) input2 FIGURE 7.9: PWA controllers depending on λ observed (see table 7.1), and following 7.25, an association of regions can be done, reducing the antecedents between 45 and 51%. Following the structure of the system proposed above (see Figure 7.8), the parameter λ(move suppression) can be changed in order to give more or less aggressiveness to the system. Figure 7.10 shows the performance of the controller when λtakes different values between those which are in the table 7.1. FPCA to Simplify MPC implementation 123 TABLE 7.1: Controllers structure using FPCA parameter Variable Rules number Reduced consequents number/ reduction(%) Reduced antecedents number/ reduction(%) λ=1 u1 140 5 (96%) 68 (51%) λ=1 u2 140 6 (95%) 71 (49%) λ=5 u1 162 7 (95.6%) 89 (45%) λ=5 u2 162 5 (97%) 87 (46%) λ=10 u1 161 5 (97%) 88 (45%) λ=10 u2 161 6 (96%) 87 (46%) 0 200 400 600 800 1000 -0.5 0 0.5 1 1.5 time(min) Composition(%) output1 0 200 400 600 800 1000 -0.5 0 0.5 1 1.5 time(min) Composition(%) output2 lambda=2 lambda=4 lambda=8 Reference 0 200 400 600 800 1000 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 time(min) reflux(l/s) input1 0 200 400 600 800 1000 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 time(min) reflux(l/s) input2 FIGURE 7.10: FMPC for distillation column depending on lambda FPCA to Simplify MPC implementation 124 7.5 Conclusion to the chapter In this chapter three methods to reduce complexity have been seen. Firstly, reducing the dimension of the space of manipulated and disturbance variables to one, has permitted to reduce a MISO system to SISO. The second way is the application of the technique studied in chapter 6, reducing the number of rules of a fuzzy system and designing GPC controllers from the new consequents. An example has been a position control of a mechanism, reducing complexity just to one rule, designing one GPC to control the nonlinear system. The third way is the application of the FPCA to a PWA system to reduce its complexity. This last way permits to reduce complexity of the explicit solution of the optimisation problem subject to constraints, using an adjusting parameter. In order to use MPC with constraints on a system, we have to consider several issues: The computational load, which is determined by the number of operations performed. The use of on-line optimisation can become limited by the speed of the microprocessor in computing with a high number of variables. The explicit calculation of the optimisation solution solves this problem but a memory shortage may sometime limit its implementation, especially when is necessary to have a fitting parameter. Furthermore, in the actual implementation of a system, the time spent in programming, which is not negligible, must be taken into account, considering the lack of tools to automatically generate the appropriate code from the design and choice of industrial PLC and embedded systems. In this chapter a combination of the reduction technique showed in Chapter 6applied on PWA systems and, together with the use of a fuzzy system, has succeeded in designing a MPC scheme with constraints and adjustment parameter, with drastically reducing the number of regions of consequents, improving programming time (down 96%) without excessive increase in the computational load. The technique applied to the PWA also allows a new merge of regions (not necessarily adjacent), opening the door to future research on reducing their complexity. Conclusion and future work In this chapter, some very general concluding remarks and future work is presented. This thesis has focused on developing a new methodology for complexity reduction in rule based systems. The main contribution of the thesis is in Chapter 6, i.e. the application of Functional Principal Components Analysis to the Hilbert space of functions defined in a rule base. Defining a covariance operator, the technique permits the reduction of the space to a subspace where the operator’s eigenvalues are bigger. It has been published in [296]. A second contribution, shown in Chapter 7, is the application of the technique described in Chapter 6, to permit the structure reduction of the controller internally, either reducing the number of rules of a fuzzy system and designing GPC controllers from the new consequents, or reducing the rules of a PWA controllers which are the application of model predictive control, subject to constraints with explicit solution. As examples, a position control of a mechanism, reducing complexity just to one rule, designing one GPC to control the nonlinear system and a high distillation column, applying a linear GPC subject to constraints, with explicit solution of the optimisation problem, using a tuning parameter. Other derived contribution from the PCA application the reduction of a MISO system to SISO, which simplifies the complexity of the control system, by reducing the dimension of the space of the manipulated and disturbance variables to one, as is shown in section 7.1. Many practical applications of MPC implementation in low capability hardware and fuzzy systems which can be viewed as minor contributions of this thesis. The general fuzzy modelling methodology requires a considerable amount of empirical knowledge and experience [297]. The practical implementation of MPC and fuzzy systems, has resulted in several publications and the acquisition of experience needed to model and control systems in industrially feasible manner. A practical contribution of RSSI-based explicit GPC and a min-max MPC approach has 125 Conclusion and future work 126 been presented to address the radio power control problem encountered in ambulatory sensor networks. The results have been published already in [60], [61] and [62]. It has been shown that an explicit solution of the constrained min-max MPC problem can be computed for the WSN power control problem by solving an mpQP. The feasibility of the proposed design and its performance has been experimentally validated. A basic real contribution of a FIS has been a real software application for commercial purposes using fuzzy techniques. The application belongs to the company Sensipassr and two articles have been published in [102] and [103]. A contribution of fuzzy modelling been presented in section 3.2.3, applying GA to chose the input field structure for a fuzzy model, using the Mackey-Glass chaotic series as an example of prediction. Others contributions have been several real application of modelling published in [298] and [?] for an industrial autoclave and [169], [299] and [300] for an industrial gas mixing chamber. An application of a supervisory fuzzy control system in a real chemical plant in operation has been applied. A comparison of other strategies with a direct fuzzy controller in an air levitation plant has been made and published in [301] and [180]. In Chapter 5, two different schemes have been implemented in a real industrial plant, using low computational cost hardware with FCL. The results have been compared with a linear GPC, obtaining better performance with no significant programming effort and computational resources. The results have been published in [298] and [?]. Future works The application of the reduction technique presented in Chapter 6may result in a lack of interpretability in the fuzzy system, due to the algebraic combination before aggregation. If we focus on the use in predictive control, this is not a problem. Even predictive controllers with explicit solutions that could be implemented in industrial devices according to IEC61131-3 [7] and IEC61131-7 [11], there may be the possibility of non-prototypical membership functions or algebraic operations after fuzzification. Therefore, a future work may be to focus to obtain interpretable antecedents using a proper transformation of the input space, getting a simpler way to implement the reduction technique. The technique applied to the PWA also allows a new merge of regions (not necessarily adjacent), opening the door to future research on reducing their complexity when an explosion of rules arises due the accuracy of modelling. Bibliography 133 [57] Michal Kvasnica Miroslav Fikar Ivan Rauov´ a, Richard Valo. Real-time model predictive control of a fan heater via plc. In International Conference on Process Control (18th: 2011 Tatransk´ a Lomnica, Slovakia), pages 388–393, 2011. ISBN 978-80-227-3517-9. [58] K. Witheephanich, J.M. Esca˜ no, A.J. Gallego, and E.F. Camacho. Pressurized watertemperature control of a fresnel collector field-type solar cooling system using explicitmodel predictive control. Asian Conference onPower and Energy Systems. April 2013,Phuket, Thailand., 2013. [59] M. Walsh, M. Hayes, and J. Nelson. Robust performance for an energy sensitive wireless body area network - an anti-windup approach. International Journal of Control, 82(1):59–73, January 2009. ISSN 0020-7179. doi: 10.1080/00207170801983109. URL http://dx.doi.org/10.1080/ 00207170801983109. [60] K. Witheephanich, J.M. Esca˜ no, and M.J. Hayes. Ambulatory wireless sensor network power management using constrained explicit generalised predictive control. International Journal of Control, 84(8):1450–1466, 2011. doi: 10.1080/00207179.2011.605909. URL http://dx.doi.org/10.1080/ 00207179.2011.605909. [61] K. Witheephanich, J.M. Esca˜ no, and M.J. Hayes. Explicitly constrained generalised predictive control strategies for power management in ambulatory wireless sensor network systems. In American Control Conference (ACC), 2010, pages 1856–1861. IEEE, 2010. [62] K. Witheephanich, J.M. Esca˜ no, D. Mu˜ noz de la Pe˜ na, and M.J. Hayes. A minmax model predictive control approach to robust power management in ambulatory wireless sensor networks. Systems Journal, IEEE, 8(4):1060–1073, Dec 2014. ISSN 1932-8184. doi: 10.1109/JSYST.2013.2271388. [63] Sarangapani. Wireless Ad Hoc and Sensor Networks Protocols Performance and Control. CRC Press, 2007. [64] Jennifer Yick, Biswanath Mukherjee, and Dipak Ghosal. Wireless sensor network survey. Comput. Netw., 52(12):2292–2330, August 2008. ISSN 1389-1286. doi: 10.1016/j.comnet.2008.04.002. URL http://dx.doi.org/10.1016/j. comnet.2008.04.002. Bibliography 134 [65] Gang Zhou, Tian He, Sudha Krishnamurthy, and John A. Stankovic. Impact of radio irregularity on wireless sensor networks. In Proceedings of the 2Nd International Conference on Mobile Systems, Applications, and Services, MobiSys ’04, pages 125–138, New York, NY, USA, 2004. ACM. ISBN 1-58113-793-1. doi: 10.1145/990064.990081. URL http://doi.acm.org/10.1145/990064. 990081. [66] Wei Zhuang, Xi Chen, Jindong Tan, and Aiguo Song. An empirical analysis for evaluating the link quality of robotic sensor networks. In Wireless Communications Signal Processing, 2009. WCSP 2009. International Conference on, pages 1–5, Nov 2009. doi: 10.1109/WCSP.2009.5371588. [67] Guang-Zhong Yang. Body Sensor Networks. Computer Science. Springer London, 2014. ISBN 978-1-4471-6373-2. URL http://www.springer.com/ computer/hci/book/978-1-4471-6373-2. [68] Shuo Xiao, A. Dhamdhere, V. Sivaraman, and A. Burdett. Transmission power control in body area sensor networks for healthcare monitoring. Selected Areas in Communications, IEEE Journal on, 27(1):37–48, January 2009. ISSN 0733- 8716. doi: 10.1109/JSAC.2009.090105. [69] Hande Alemdar and Cem Ersoy. Wireless sensor networks for healthcare: A survey. Computer Networks, 54(15):2688 – 2710, 2010. ISSN 1389- 1286. doi: http://dx.doi.org/10.1016/j.comnet.2010.05.003. URL http://www. sciencedirect.com/science/article/pii/S1389128610001398. [70] Theodore Rappaport. Wireless Communications: Principles and Practice. Prentice Hall PTR, Upper Saddle River, NJ, USA, 2nd edition, 2001. ISBN 0130422320. [71] Dimitrios Lymberopoulos, Quentin Lindsey, and Andreas Savvides. An empirical characterization of radio signal strength variability in 3-d ieee 802.15.4 networks using monopole antennas. In Kay R¨ omer, Holger Karl, and Friedemann Mattern, editors, Wireless Sensor Networks, volume 3868 of Lecture Notes in Computer Science, pages 326–341. Springer Berlin Heidelberg, 2006. ISBN 978-3-540-32158-3. doi: 10.1007/11669463 24. URL http://dx.doi.org/ 10.1007/11669463_24. [72] Emiliano Miluzzo, Xiao Zheng, Krist´ of Fodor, and Andrew T. Campbell. Radio characterization of 802.15.4 and its impact on the design of mobile sensor Bibliography 135 networks. In In Proc. 5th European Conf. on Wireless Sensor Networks (EWSN ’08), pp 171–188, 2008. [73] Kannan Srinivasan and Philip Levis. Rssi is under appreciated. In In Proceedings of the Third Workshop on Embedded Networked Sensors (EmNets), 2006. [74] Victor Shnayder, Bor-rong Chen, Konrad Lorincz, Thaddeus R. F. Fulford Jones, and Matt Welsh. Sensor networks for medical care. In Proceedings of the 3rd International Conference on Embedded Networked Sensor Systems, SenSys ’05, pages 314–314, New York, NY, USA, 2005. ACM. ISBN 1-59593-054-X. doi: 10.1145/1098918.1098979. URL http://doi.acm.org/10.1145/1098918. 1098979. [75] F.A. Tobagi and L. Kleinrock. Packet switching in radio channels: Part ii–the hidden terminal problem in carrier sense multiple-access and the busy-tone solution. Communications, IEEE Transactions on, 23(12):1417–1433, Dec 1975. ISSN 0090-6778. doi: 10.1109/TCOM.1975.1092767. [76] Sunil Kandukuri and Stephen Boyd. Optimal power control in interferencelimited fading wireless channels with outage-probability specifications. IEEE Transactions on Wireless Communications, 1(1):46–55, 2002. URL http: //dblp.uni-trier.de/db/journals/twc/twc1.html#KandukuriB02. [77] Approved draft revision for ieee standard for information technologytelecommunications and information exchange between systems-local and metropolitan area networks-specific requirements-part 15.4b: Wireless medium access control (mac) and physical layer (phy) specifications for low rate wireless personal area networks (wpans) (amendment of ieee std 802.15.4-2003). IEEE Std P802.15.4/D6, pages –, 2006. [78] Draft amendment to ieee standard for information technologytelecommunications and information exchange between systems-part 15.4:wireless medium access control (mac) and physical layer (phy) specifications for low-rate wireless personal area networks (lr-wpans): Amendment to add alternate phy (amendment of ieee std 802.15.4). IEEE Std P802.15.4a/D5, pages –, 2006. Bibliography 136 [79] Approved draft amendment to ieee standard for information technologytelecommunications and information exchange between systems-part 15.4:wireless medium access control (mac) and physical layer (phy) specifications for lowrate wireless personal area networks (lr-wpans): Amendment to add alternate phy (amendment of ieee std 802.15.4). IEEE Approved Std P802.15.4a/D7, Jan 2007, pages –, 2007. [80] P. Levis, S. Madden, J. Polastre, R. Szewczyk, K. Whitehouse, A. Woo, D. Gay, J. Hill, M. Welsh, E. Brewer, and D. Culler. Tinyos: An operating system for sensor networks. In Werner Weber, JanM. Rabaey, and Emile Aarts, editors, Ambient Intelligence, pages 115–148. Springer Berlin Heidelberg, 2005. ISBN 978-3-540-23867-6. doi: 10.1007/3-540-27139-2 7. URL http://dx.doi. org/10.1007/3-540-27139-2_7. [81] B. Zurita Ares, P.G. Park, C. Fischione, A. Speranzon, and K.H. Johansson. On power control for wireless sensor networks: System model, middleware component and experimental evaluation. In European Control Conference, 2007. [82] Pan Gun Park, Carlo Fischione, and Karl Henrik Johansson. Experimental evaluation of power control algorithms for wireless sensor networks. In Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008, 2008. [83] S.M. Mahdi Alavi, Michael J. Walsh, and Martin J. Hayes. Robust distributed active power control technique for {IEEE}802.15.4 wireless sensor networks — a quantitative feedback theory approach. Control Engineering Practice, 17(7):805 – 814, 2009. ISSN 0967-0661. doi: http://dx.doi.org/10.1016/j. conengprac.2009.02.001. URL http://www.sciencedirect.com/science/ article/pii/S0967066109000161. [84] CC2420 datasheet, rev. b edition, 2007. URL http://www.ti.com/lit/ds/ symlink/cc2420.pdf. Datasheet. [85] Basil Kouvaritakis, J. Anthony Rossiter, and Jan Schuurmans. Efficient robust predictive control. IEEE Transactions on Automatic Control, 45(8):1545–1549, 2000. [86] T. Alamo, D. Mu˜ noz de la Pena, D. Limon, and E.F. Camacho. Constrained min-max predictive control: modifications of the objective function leading to Bibliography 137 polynomial complexity. Automatic Control, IEEE Transactions on, 50(5):710– 714, May 2005. ISSN 0018-9286. doi: 10.1109/TAC.2005.847039. [87] H.S. Witsenhausen. A minimax control problem for sampled linear systems. Automatic Control, IEEE Transactions on, 13(1):5–21, Feb 1968. ISSN 0018- 9286. doi: 10.1109/TAC.1968.1098788. [88] D.M. de la Pe˜ na, T. Alamo, D.R. Ramirez, and E.F. Camacho. Min-max model predictive control as a quadratic program. Control Theory Applications, IET, 1 (1):328–333, January 2007. ISSN 1751-8644. doi: 10.1049/iet-cta:20060016. [89] Moteiv. Tmote Sky Datasheet, 2006. URL http://www.eecs.harvard.edu/ ~konrad/projects/shimmer/references/tmote-sky-datasheet.pdf. [90] D. Mu˜ noz de la Pe˜ na, D.R. Ram´ ırez, E.F. Camacho, and T. Alamo. Application of an explicit min-max {MPC}to a scaled laboratory process. Control Engineering Practice, 13(12):1463 – 1471, 2005. ISSN 0967-0661. doi: http://dx.doi. org/10.1016/j.conengprac.2004.12.008. URL http://www.sciencedirect. com/science/article/pii/S0967066105000158. Special Section on Power Plants and Power Systems Control Special Section on Power Plants and Power Systems Control. [91] D. Mu˜ noz de la Pe˜ na, D.R. Ram´ ırez, E.F. Camacho, and T. Alamo. Explicit solution of min–max {MPC}with additive uncertainties and quadratic criterion. Systems & Control Letters, 55(4):266 – 274, 2006. ISSN 0167- 6911. doi: http://dx.doi.org/10.1016/j.sysconle.2005.08.006. URL http:// www.sciencedirect.com/science/article/pii/S0167691105001325. [92] H.J. Su and E. Geraniotis. Adaptive closed-loop power control with quantized feedback and loop filtering. Wireless Communications, IEEE Transactions on, 1 (1):76–86, Jan 2002. ISSN 1536-1276. doi: 10.1109/7693.975447. [93] L.A. Zadeh. Fuzzy sets. Information and Control, 8(3):338 – 353, 1965. ISSN 0019-9958. doi: http://dx.doi.org/10.1016/S0019-9958(65) 90241-X. URL http://www.sciencedirect.com/science/article/pii/ S001999586590241X. [94] E.H. Mamdani and S. Assilian. An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies, 7(1):1 – 13, 1975. ISSN 0020-7373. doi: http://dx.doi.org/10.1016/S0020-7373(75) Bibliography 138 80002-2. URL http://www.sciencedirect.com/science/article/pii/ S0020737375800022. [95] Werner Van Leekwijck and Etienne E. Kerre. Defuzzification: criteria and classification. Fuzzy Sets and Systems, 108(2):159 – 178, 1999. ISSN 0165- 0114. doi: http://dx.doi.org/10.1016/S0165-0114(97)00337-0. URL http: //www.sciencedirect.com/science/article/pii/S0165011497003370. [96] Tomohiro Takagi and Michio Sugeno. Fuzzy Identification of Systems and Its Applications to Modeling and Control. IEEE Transactions on Systems, Man, and Cybernetics, 15(1):116–132, February 1985. URL http://www.hi.cs.meiji. ac.jp/~{}takagi/paper/TS-MODEL.tar.gz. [97] K. Kiriakidis. Fuzzy model-based control of complex plants. Fuzzy Systems, IEEE Transactions on, 6(4):517 –529, nov 1998. ISSN 1063-6706. doi: 10. 1109/91.728444. [98] J.S.R. Jang. Anfis: adaptive-network-based fuzzy inference system. Systems, Man and Cybernetics, IEEE Transactions on, 23(3):665–685, May 1993. ISSN 0018-9472. doi: 10.1109/21.256541. [99] J.S.R. Jang, C.T. Sun, and E. Mizutani. Neuro-fuzzy and soft computing: a computational approach to learning and machine intelligence. MATLAB curriculum series. ISBN 9780132610667. [100] International Organization for Standardization (ISO). Automation systems and integration -Integration of advanced process control and optimization capabilities for manufacturing systems -Part 1: Framework and functional model. ISO, 2013. [101] Schneider Electric. Fuzzy control library v1.2. Technical Report 33004219.02, Schneider Electric SA, 35 rue Joseph Monier 92500 Rueil Malmaison - France, February 2009. URL http://download.schneider-electric.com/files? p_File_Id=27511099&p_File_Name=33004219_K01_000_02.pdf. [102] J.M. Esca˜ no, J.C. Stockdale, J. Liang, and A. Vakaloudis. Fuzzy matching engine for non-textual authentication: A case study. In Science and Information Conference (SAI), 2014, pages 615–619, Aug 2014. doi: 10.1109/SAI.2014.6918250. [103] J. Stockdale, A. Vakaloudis, J.M. Esca˜ no, J. Liang, and B. Cahill. A fuzzy system for three-factor, non-textual authentication. In Kohei Arai, Supriya Bibliography 139 Kapoor, and Rahul Bhatia, editors, Intelligent Systems in Science and Information 2014, volume 591 of Studies in Computational Intelligence, pages 125–137. Springer International Publishing, 2015. ISBN 978-3-319-14653- 9. doi: 10.1007/978-3-319-14654-6 8. URL http://dx.doi.org/10.1007/ 978-3-319-14654-6_8. [104] Z. Chi, H. Yan, and T. Pham. Fuzzy Algorithms: With Applications to Image Processing and Pattern Recognition. Advances in fuzzy systems - applications and theory. ISBN 9789810226978. [105] Ferdinando Di Martino and Salvatore Sessa. Image matching by using fuzzy transforms. 2013. doi: 10.1155/2013/760704. [106] Irina Perfilieva. Fuzzy transforms. In JamesF. Peters, Andrzej Skowron, Didier Dubois, JerzyW. Grzymała-Busse, Masahiro Inuiguchi, and Lech Polkowski, editors, Transactions on Rough Sets II, volume 3135 of Lecture Notes in Computer Science, pages 63–81. Springer Berlin Heidelberg, 2005. ISBN 978-3- 540-23990-1. doi: 10.1007/978-3-540-27778-1 5. URL http://dx.doi.org/ 10.1007/978-3-540-27778-1_5. [107] I. Bloch. Fuzzy relative position between objects in image processing: a morphological approach. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 21(7):657–664, Jul 1999. ISSN 0162-8828. doi: 10.1109/34.777378. [108] Isabelle Bloch and Anca Ralescu. Directional relative position between objects in image processing: a comparison between fuzzy approaches. Pattern Recognition, 36(7):1563 – 1582, 2003. ISSN 0031-3203. doi: http://dx.doi.org/10.1016/ S0031-3203(02)00263-7. URL http://www.sciencedirect.com/science/ article/pii/S0031320302002637. [109] Q. Tan and M. Akimoto. Fuzzy matching for robot localization. In Intelligent Robots and Systems ’96, IROS 96, Proceedings of the 1996 IEEE/RSJ International Conference on, volume 3, pages 1165–1170 vol.3, Nov 1996. doi: 10.1109/IROS.1996.568966. [110] Haiyuan Wu, Qian Chen, and M. Yachida. Face detection from color images using a fuzzy pattern matching method. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 21(6):557–563, Jun 1999. ISSN 0162-8828. doi: 10.1109/34.771326. Bibliography 140 [111] Tian Jinwen, Huang Jianzhong, Liu Jian, and Li Dchua. Image matching based on fuzzy information. In Signal Processing, 1996., 3rd International Conference on, volume 2, pages 946–949 vol.2, Oct 1996. doi: 10.1109/ICSIGP.1996. 566246. [112] Esko Ukkonen. Algorithms for approximate string matching. Information and Control, 64(1–3):100 – 118, 1985. ISSN 0019-9958. doi: http://dx.doi.org/ 10.1016/S0019-9958(85)80046-2. URL http://www.sciencedirect.com/ science/article/pii/S0019995885800462. International Conference on Foundations of Computation Theory. [113] Alberto Apostolico and Zvi Galil, editors. Pattern matching algorithms. Oxford University Press, New York, 1997. ISBN 0-19-511367-5. URL http://opac. inria.fr/record=b1092702. [114] Surajit Chaudhuri, Kris Ganjam, Venkatesh Ganti, and Rajeev Motwani. Robust and efficient fuzzy match for online data cleaning. In In SIGMOD, pages 313– 324, 2003. [115] Adnan A.Y. Mustafa. Fuzzy shape matching with boundary signatures. Pattern Recognition Letters, 23(12):1473 – 1482, 2002. ISSN 0167-8655. doi: http://dx.doi.org/10.1016/S0167-8655(02)00107-1. URL http://www. sciencedirect.com/science/article/pii/S0167865502001071. [116] Zhao kang Li, Liang jie Xu, Juan Fang, Qun jie Peng, and Min Wang. Research on the surrounding traffic flow of railway station based on license plate recognition and fuzzy matching. In Business Management and Electronic Information (BMEI), 2011 International Conference on, volume 4, pages 873–876, May 2011. doi: 10.1109/ICBMEI.2011.5914243. [117] P.T. Schultz and R.A. Sartini. Method and system for multi-factor biometric authentication. [118] Ritesh Vyas and Gourav Garg. Face recognition using feature extraction and neuro-fuzzy techniques. International Journal of Electronics and Computer Science Engineering, 1(4):2048–2056, 2012. ISSN 2277-1956. [119] Itseez. Opencv, open source computer vision, 2015. URL http://opencv. org/. Bibliography 141 [120] B. Bhanu and V. Govindaraju. Multibiometrics for Human Identification. ISBN 9781139501804. [121] Asli Celikyilmaz and I. Burhan Trksen. Modeling Uncertainty with Fuzzy Logic: With Recent Theory and Applications. Springer Publishing Company, Incorporated, 1st edition, 2009. ISBN 3540899235, 9783540899235. [122] Andras Bardossy and Lucien Duckstein. Fuzzy Rule-Based Modeling with Applications to Geophysical, Biological, and Engineering Systems. CRC Press, Inc., Boca Raton, FL, USA, 1995. ISBN 0849378338. [123] Oscar Cord´ on, Francisco Herrera, Frank Hoffmann, Luis Magdalena, O Cordon, F Herrera, and F Hoffmann. Genetic fuzzy systems. World Scientific Publishing Company Singapore, 2001. [124] Niel Canty. A new algorithm for piecewise affine system identification and its application to a MEMS switch. PhD thesis, Cork Institute of Technology, Rossa Avenue, Bishopstown. Cork, 2014. [125] Warren S. McCulloch and Walter Pitts. A logical calculus of the ideas immanent in nervous activity. The bulletin of mathematical biophysics, 5(4):115–133, 1943. ISSN 0007-4985. doi: 10.1007/BF02478259. URL http://dx.doi.org/10. 1007/BF02478259. [126] Chao-Chee Ku and Kwang Y Lee. Diagonal recurrent neural networks for dynamic systems control. Neural Networks, IEEE Transactions on, 6(1):144–156, 1995. [127] Ching-Hung Lee and Ching-Cheng Teng. Identification and control of dynamic systems using recurrent fuzzy neural networks. Fuzzy Systems, IEEE Transactions on, 8(4):349–366, 2000. [128] Arshdeep Kaur and Amrit Kaur. Comparison of mamdani-type and sugeno-type fuzzy inference systems for air conditioning system. International Journal of Soft Computing and Engineering (IJSCE), 2(2), May 2012. ISSN 2231-2307. [129] S. Chiu. Fuzzy Model Identification based on cluster estimation. Journal of Intelligent Fuzzy Systems, 2:267–278, 1994. [130] Stephen L Chiu. Selecting input variables for fuzzy models. Journal of Intelligent and Fuzzy Systems, 4(4):243–256, 1996. Bibliography 142 [131] R. Sindelar and R. Babuska. Input selection for nonlinear regression models. Fuzzy Systems, IEEE Transactions on, 12(5):688–696, Oct 2004. ISSN 1063- 6706. doi: 10.1109/TFUZZ.2004.834810. [132] J. Yen, Haojin Wang, and J. Liao. A method for automatic generation of a fuzzy model. In Industrial Fuzzy Control and Intelligent Systems, 1993., IFIS ’93., Third International Conference on, pages 88–92, Dec 1993. doi: 10.1109/IFIS. 1993.324208. [133] T.W. Anderson. An Introduction to Multivariate Statistical Analysis. Wiley Series in Probability and Statistics. ISBN 9780471360919. [134] Michael C. Mackey, Leon Glass, et al. Oscillation and chaos in physiological control systems. Science, 197(4300):287–289, 1977. [135] A. S. Weigend and N. A. Gershenfeld, editors. Time series prediction: Forecasting the future and understanding the past, 1994. URL http://adsabs. harvard.edu/cgi-bin/nph-bib_query?bibcode=1994tspf.conf.....W. [136] Pentti Saikkonen. Introduction to modern time series analysis by gebhard kirchg¨ assner, j¨ urgen wolters. International Statistical Review, 76(1):151–152, 2008. ISSN 1751-5823. doi: 10.1111/j.1751-5823.2007.00039 13.x. URL http://dx.doi.org/10.1111/j.1751-5823.2007.00039_13.x. [137] Richard McCleary, Richard A. Hay, Errol E. Meidinger, and David McDowall. Applied time series analysis for the social sciences. Sage Publications Beverly Hills, CA, 1980. [138] Scott L. Zeger, Rafael Irizarry, and Roger D. Peng. On time series analysis of public health and biomedical data. Annu. Rev. Public Health, 27:57–79, 2006. [139] Qiang Song and Brad S. Chissom. Forecasting enrollments with fuzzy time series—part i. Fuzzy Sets Syst., 54(1):1–9, February 1993. ISSN 0165- 0114. doi: 10.1016/0165-0114(93)90355-L. URL http://dx.doi.org/10. 1016/0165-0114(93)90355-L. [140] Qiang Song and Brad S. Chissom. Fuzzy time series and its models. Fuzzy Sets Syst., 54(3):269–277, March 1993. ISSN 0165-0114. doi: 10.1016/0165-0114(93)90372-O. URL http://dx.doi.org/10.1016/ 0165-0114(93)90372-O.