Polynomial fuzzy model based control systems

Studies in Systems, Decision and Control 64 Hak-Keung Lam Polynomial Fuzzy Model-Based Control Systems Stability Analysis and Control Synthesis Using Membership Function-Dependent Techniques Studies in Systems, Decision and Control Volume 64 Series editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland e-mail: kacprzyk@ibspan.waw.pl About this Series The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control- quickly, up to date and with a high quality The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output More information about this series at http://www.springer.com/series/13304 Hak-Keung Lam Polynomial Fuzzy Model-Based Control Systems Stability Analysis and Control Synthesis Using Membership Function-Dependent Techniques 123 Hak-Keung Lam Department of Informatics King’s College London London UK ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-319-34092-0 ISBN 978-3-319-34094-4 (eBook) DOI 10.1007/978-3-319-34094-4 Library of Congress Control Number: 2016941072 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland To my family, my beloved wife, Esther Wing See Chan, and my lovely daughter, Katrina Faye Lam, for standing by me Hak-Keung Lam Preface I have researched on fuzzy model-based control systems since 1995 as a Ph.D student During the past two decades, I have gained a rich experience and knowledge in the field, and have summarized the research achievements by a number of publications I have witnessed the development of the research in the field of fuzzy model-based control systems in the past 20 years In general, I divide it into four stages In this first stage, from early 1970s to late 1980s, fuzzy control has become well known as an intelligent control strategy for ill-defined and complex systems due to successful applications from household appliances to chemical plants Using fuzzy logic concept, human spirit can be captured by linguistic rules which can be realized by machines Fuzzy logic controller is thus able to incorporate human knowledge to control complex systems In the early applications, the fuzzy logic controller was designed heuristically without the need of the mathematical model of nonlinear systems Although good performance can be demonstrated by some application examples, essential issues such as system stability and robustness are not guaranteed which put the users and applications at risk In the second stage, from early 1990s to mid-2000s, thanks to the T–S fuzzy model fuzzy model-based control has become very popular and offered a systematic way for system analysis and control design Stability analysis has become a very promising research topic since then Fruitful analysis results have been reported in many articles Relaxation of stability conditions has drawn a great deal of attention from the researchers in the fuzzy control community As the stability analysis has not considered the membership functions in most of the work during this period, the stability analysis/conditions are named as membership function-independent stability analysis/conditions in my publications The third stage was from mid-2000s to late 2000s I have proposed the membership function-dependent stability analysis which is able to bring the information and characteristic of the membership functions into the stability conditions Consequently, it is named as membership function-dependent stability conditions in my publications As the membership functions are the nonlinearity of the nonlinear vii viii Preface system, it plays an important role to achieve more relaxed stability analysis results compared with the membership function-dependent stability analysis Furthermore, opposite to the concept of parallel distributed compensation, I have promoted the concept of partially/imperfectly matched premises that the number of rules and/or premise membership functions used in the fuzzy controller are different from those of the fuzzy model to achieve greater control design flexibility and lower control implementation complexity (to reduce the implementation costs) The fourth stage started from late 2000s The introduction of the polynomial fuzzy model takes the stability analysis and fuzzy control to another level using sum-of-squares approach instead of linear matrix inequality This book focuses on the work on the fourth stage which is the research on the stability analysis of polynomial fuzzy model-based control systems where the concept of partially/imperfectly matched premises and membership function-dependent analysis are considered I would like to summarize my recent achievements on this topic which present the latest research outcomes including findings, observations, concepts, ideas, research directions, stability analysis techniques, and control methodologies The membership function-dependent analysis offers a new research direction for the fuzzy model-based control systems by taking into account the characteristic and information of the membership functions (related to the nonlinearity of the plant) in the stability analysis Membership function-dependent stability conditions are far more relaxed compared with some state-of-the-art membership function-independent stability conditions It is more effective to deal with nonlinear control problems as membership function-dependent approach considers the dedicated nonlinear system on hand rather than a family of nonlinear systems tackled in the membership function-independent approach Through this book, I would like to promote the membership function-dependent analysis to be a new research direction and hope to see that it becomes a popular technique to deal with the stability analysis problem for fuzzy model-based control systems The content of this book is mainly at the research level presenting the most recent and advanced research results, which aims to promote the research of polynomial fuzzy model-based control systems, provide theoretical support, and point a research direction to postgraduate students and fellow researchers The introduction and preliminary parts of the book provides an overview of the topics and technical materials are presented in a very detailed manner Numerical examples are provided in each chapter to verify the analysis results, demonstrate the effectiveness of the proposed polynomial fuzzy control schemes, and explain the design procedure This book is comprehensively written with detailed derivation steps and mathematical details to enhance the reading experience, in particular, for readers without extensive knowledge on the topics It is thus also recommended to undergraduate students with control background who are interested in polynomial fuzzy model-based control systems This book has four parts consisting of ten chapters The first part Introduction and Preliminaries provides the overview and technical background of the fuzzy model-based control systems offering fundamental knowledge and mathematical support for the subsequent parts The second part Stability Analysis Techniques Preface ix presents the latest techniques based on the membership function-dependent stability analysis for polynomial fuzzy model-based control systems The third part Advanced Control Methodologies extends the stability analysis techniques to more challenging control problems The fourth part Advanced Lyapunov Functions introduces more effective Lyapunov functions for stability analysis and polynomial fuzzy control strategy for the control of nonlinear plants The content of each chapter is briefly introduced below Part I Introduction and Preliminaries • Chapter gives a general overview of the fuzzy model-based control which covers the background, literature review, development of the field, fuzzy models, fuzzy control methodologies, stability analysis approaches, and control problems • Chapter provides the technical and mathematical background for the fuzzy model-based control which offers the equations of the fuzzy model and closed-loop systems, definition of variables, published stability conditions in terms of linear matrix inequalities, and sum of squares (SOS) These materials are essential for the work in the subsequent chapters Part II Stability Analysis Techniques • Chapter investigates the stability of polynomial fuzzy model-based control systems by treating the membership functions and system states as symbolic variables The information of membership functions is considered in the stability analysis and brought to the SOS-based stability conditions Techniques are proposed to introduce slack matrix variables carrying the information of membership functions to the SOS-based stability conditions without increasing much the computational demand • Chapter investigates the stability of polynomial fuzzy model-based control systems by bringing the approximated membership functions into the SOS-based stability conditions Various approximation methods of membership functions are reviewed and their characteristics are discussed Using the Taylor series expansion, the original membership functions are represented by approximated membership functions which are a weighted sum of local polynomials in a favorable form for stability analysis SOS-based stability conditions are obtained which guarantee the system stability if the fuzzy model-based control system is stable at all chosen Taylor series expansion points • Chapter investigates the stability of general polynomial fuzzy model-based control systems In Chaps and 4, a constraint that the polynomial Lyapunov function matrix is allowed to be dependent on some state variables determined by the structure of the input matrices is required to obtain convex stability x Preface conditions In this chapter, this constraint is removed and a two-step procedure is proposed to search for a feasible solution to the SOS-based stability conditions Consequently, the stability analysis results can be applied to a wider range of polynomial fuzzy model-based control systems Part III Advanced Control Methodologies • Chapter considers a regulation problem for polynomial fuzzy model-based control systems An output-feedback polynomial fuzzy controller is employed to drive the system outputs to reach a desired level SOS-based stability conditions for the three cases (perfectly, partially and imperfectly matched premises) are obtained, which are facilitated by considering different information of membership functions, to determine the system stability and synthesize the controller With the support of Barbalat’s Lemma, it is guaranteed that a stable output-feedback polynomial fuzzy controller will produce no steady state error • Chapter considers a tracking problem for polynomial fuzzy model-based control systems An output-feedback polynomial fuzzy controller is employed to drive the system outputs to follow a reference trajectory SOS-based stability conditions are obtained to determine the system stability and synthesize the controller where the tracking performance satisfies an H1 performance index governing the tracking error • Chapter considers a sampled data output-feedback polynomial fuzzy model-based control system which is formed by a nonlinear plant represented by the polynomial fuzzy model and a sampled data output-feedback polynomial fuzzy controller connected in a closed loop SOS-based stability analysis considering the effect due to sampling and zero-order-hold activities is performed using the input-delay method SOS-based stability conditions are obtained to determine the system stability and synthesize the controller Part IV Advanced Lyapunov Functions • Chapter proposes a switching polynomial Lyapunov function candidate, which consists of a number of local sub-Lyapunov function candidates, for the stability analysis of polynomial fuzzy model-based control systems where switching is dependent on the system states When the system state vector falls into the pre-defined local operating domain, the corresponding local sub-Lyapunov function candidate is employed to take care of the system stability Corresponding to each local sub-Lyapunov function candidate, a local polynomial fuzzy controller is employed for the control of the nonlinear plant resulting in a switching polynomial fuzzy control strategy A favorable form of switching polynomial Lyapunov function candidate is proposed to make sure 282 10 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems … Table 10.14 G jd1 (x2 ) given by Theorem 10.1 corresponding to the point ‘ ’ at a = 200 and b = 31 in Fig 10.1b for Example 10.1 with δ = 0.25 Sub-domain d1 Feedback gains G jd1 (x2 ) Φ1 G11 (x2 ) = −1288.9232 7.9005 G21 (x2 ) = −82.8418 1.6457 Φ2 G12 (x2 ) = −1288.9232 7.9005 G22 (x2 ) = −82.8418 1.6457 Φ3 G13 (x2 ) = −1288.9232 7.9005 G23 (x2 ) = −82.8418 1.6457 Φ4 G14 (x2 ) = −1288.9232 7.9005 G24 (x2 ) = −82.8418 1.6457 Φ5 G15 (x2 ) = −1288.9232 7.9005 G25 (x2 ) = −82.8418 1.6457 10 x2(t) −2 −4 −6 −8 −10 −10 −8 −6 −4 −2 10 x1(t ) Fig 10.2 Phase portrait of x1 (t) and x2 (t) for Example 10.1 corresponding to the point ‘◦’ at a = 200 and b = 38 in Fig 10.1a, where ‘◦’ indicate the initial conditions 10.4 Simulation Examples 283 10 x2(t ) −2 −4 −6 −8 −10 −10 −8 −6 −4 −2 10 x1(t ) Fig 10.3 Phase portrait of x1 (t) and x2 (t) for Example 10.1 corresponding to the point ‘×’ at a = 200 and b = 34 in Fig 10.1a, where ‘◦’ indicate the initial conditions 10 x2(t ) −2 −4 −6 −8 −10 −10 −5 10 x1(t ) Fig 10.4 Phase portrait of x1 (t) and x2 (t) for Example 10.1 corresponding to the point ‘ ’ at a = 200 and b = 30 in Fig 10.1a, where ‘◦’ indicate the initial conditions 284 10 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems … 10 x2(t ) −2 −4 −6 −8 −10 −10 −8 −6 −4 −2 10 x1(t ) Fig 10.5 Phase portrait of x1 (t) and x2 (t) for Example 10.1 corresponding to the point ‘◦’ at a = 200 and b = 41 in Fig 10.1b, where ‘◦’ indicate the initial conditions 10 x2(t) −2 −4 −6 −8 −10 −10 −8 −6 −4 −2 10 x1(t ) Fig 10.6 Phase portrait of x1 (t) and x2 (t) for Example 10.1 corresponding to the point ‘×’ at a = 200 and b = 36 in Fig 10.1b, where ‘◦’ indicate the initial conditions 10.4 Simulation Examples 285 10 x2(t ) −2 −4 −6 −8 −10 −10 −5 10 x1(t ) Fig 10.7 Phase portrait of x1 (t) and x2 (t) for Example 10.1 corresponding to the point ‘ ’ at a = 200 and b = 31 in Fig 10.1b, where ‘◦’ indicate the initial conditions Table 10.15 Δh ikd1 and Δh ikd1 with δ = 0.5 for Example 10.2 Δh ikd1 h 211 h 311 h 112 h 122 h 212 h 222 h 312 h 322 = −2.1737 × 10−3 , = −8.2124 × 10−6 , = −3.1118 × 10−3 , = −3.8515 × 10−4 , = −1.8971 × 10−3 , = −1.8971 × 10−3 , = −3.8515 × 10−4 , = −3.1118 × 10−3 , h 123 = −8.2124 × 10−6 , h 223 = −2.1737 × 10−3 ; the rest are of zero value Δh ikd1 h 111 = 2.1819 × 10−3 , h 112 = 1.9076 × 10−3 , h 122 = 1.6964 × 10−3 , h 212 = 3.2187 × 10−3 , h 222 = 3.2187 × 10−3 , h 312 = 1.6964 × 10−3 , h 322 = 1.9076 × 10−3 , h 323 = 2.1819 × 10−3 ; the rest are of zero value 286 10 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems … Table 10.16 Δh ikd1 and Δh ikd1 with δ = 0.25 for Example 10.2 Δh ikd1 h 211 h 311 h 112 h 122 h 212 h 222 h 312 h 322 = −5.8288 × 10−4 , = −2.3133 × 10−6 , = −7.9791 × 10−4 , = −9.8877 × 10−5 , = −4.7444 × 10−4 , = −4.7444 × 10−4 , = −9.8877 × 10−5 , = −7.9791 × 10−4 , Δh ikd1 h 111 = 5.8519 × 10−4 , h 112 = 4.7733 × 10−4 , h 122 = 4.3279 × 10−4 , h 212 = 8.2172 × 10−4 , h 222 = 8.2172 × 10−4 , h 312 = 4.3279 × 10−4 , h 322 = 4.7733 × 10−4 , h 323 = 5.8519 × 10−4 ; the rest are of zero value h 123 = −2.3133 × 10−6 , h 223 = −5.8288 × 10−4 ; the rest are of zero value The results demonstrate more or less the same trend as observed in Example 10.1 that the stability regions given by δ = 0.25 are larger than those of δ = 0.5 due to approximation error given by δ = 0.25 is smaller Also, the largest size of stability region is obtained by allowing both Nkd1 (x2 ) and Xk (x2 ) being different for all d1 and k However, the same size of stability region is achieved either with Xk (x2 ) being the same for all k or both Nkd1 and Xk (x2 ) being the same for all d1 and k Comparing the result with Example 10.1, larger size of stability regions are obtained with the 3-rule fuzzy polynomial Lyapunov function for both cases of δ = 0.5 and δ = 0.25 indicating that more sub-Lyapunov functions candidates used in the FPLF has a greater potential to produce more relaxed analysis results Example 10.3 (3-Rule Switching Polynomial Fuzzy Conroller and 3-Rule Fuzzy Polynomial Lyapunov Function) Similar to the case in Example 10.2, we consider the case that both switching polynomial fuzzy controller and FPLF share the same number of rules and premise membership functions where the number of rules is increased from to to investigate how it influences the size of the stability regions for the three scenarios The same 3-rule PFLF in Example 10.1 is employed The premise membership functions of switching polynomial fuzzy controller are the same as those of the PFLF to control the nonlinear plant According to the chosen membership functions, Δh ikd1 and Δh ikd1 are found numerically and given in Table 10.17 for δ = 0.5 and Table 10.18 for δ = 0.25 10.4 Simulation Examples (a) 287 41 39 37 b 35 33 31 29 27 25 100 110 120 130 140 150 160 170 180 190 200 160 170 180 190 200 a (b) 41 39 37 b 35 33 31 29 27 25 100 110 120 130 140 150 a Fig 10.8 Stability regions given by the SOS-based stability conditions in Theorem 10.2 for Example 10.2 ‘◦’: stability region for Nkd1 (x2 ) and Xk (x2 ) being allowed to be all different for all d1 and k ‘×’: stability region for Nkd1 (x2 ) being allowed to be all different for all d1 and Xk (x2 ) being the same for all k ‘ ’: stability region for Nkd1 (x2 ) being the same for all d1 and Xk (x2 ) being the same for all k a Stability regions with δ = 0.5 b Stability regions with δ = 0.25 288 10 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems … Table 10.17 Δh ikd1 and Δh ikd1 with δ = 0.5 for Example 10.3 Δh ikd1 h 211 h 311 h 212 h 222 h 312 h 322 h 113 h 123 h 133 h 213 h 223 h 233 h 313 h 323 h 333 = −1.6123 × 10−4 , = −4.0928 × 10−7 , = −1.3515 × 10−3 , = −1.8228 × 10−3 , = −1.1575 × 10−5 , = −1.0708 × 10−5 , = −2.3915 × 10−3 , = −9.9444 × 10−4 , = −1.8989 × 10−4 , = −1.3209 × 10−3 , = −7.9546 × 10−4 , = −1.3209 × 10−3 , = −1.8989 × 10−4 , = −9.9444 × 10−4 , = −2.3915 × 10−3 , Δh ikd1 h 111 = 1.6164 × 10−4 , h 112 = 1.3591 × 10−3 , h 122 = 1.8335 × 10−3 , h 113 = 4.4561 × 10−4 , h 123 = 8.0760 × 10−4 , h 133 = 1.3348 × 10−3 , h 213 = 2.4483 × 10−3 , h 223 = 1.0819 × 10−3 , h 233 = 2.4483 × 10−3 , h 313 = 1.3348 × 10−3 , h 323 = 8.0760 × 10−4 , h 333 = 4.4561 × 10−4 , h 324 = 1.8335 × 10−3 , h 334 = 1.3591 × 10−3 , h 335 = 1.6164 × 10−4 ; the rest are of zero value h 124 = −1.0708 × 10−5 , h 134 = −1.1575 × 10−5 , h 224 = −1.8228 × 10−3 , h 234 = −1.3515 × 10−3 , h 135 = −4.0928 × 10−7 , h 235 = −1.6123 × 10−4 ; the rest are of zero value As the number of rules and premise membership functions of both switching fuzzy polynomial controller and FPLF are the same, the SOS-based stability conditions in Theorem 10.2 are employed to determine the system stability for the three scenarios subject to δ = 0.5 and δ = 0.25 With the same settings as in Examples 10.1 and 10.2, the stability regions for the three scenarios are found and shown in Fig 10.9a for δ = 0.5 and Fig 10.9b for δ = 0.25 It can be seen from these two figures that larger size of stability regions are obtained with δ = 0.25 Also, the largest size of stability region is obtained by allowing both Nkd1 (x2 ) and Xk (x2 ) being different for 10.4 Simulation Examples 289 Table 10.18 Δh ikd1 and Δh ikd1 with δ = 0.25 for Example 10.3 Δh ikd1 h 211 h 311 h 212 h 222 h 312 h 322 h 113 h 123 h 133 h 213 h 223 h 233 h 313 h 323 h 333 = −4.5305 × 10−5 , = −1.1530 × 10−7 , = −3.3632 × 10−4 , = −4.7481 × 10−4 , = −3.1991 × 10−6 , = −3.0762 × 10−6 , = −6.0943 × 10−4 , = −2.4949 × 10−4 , = −4.7950 × 10−5 , = −3.2877 × 10−4 , = −2.1932 × 10−4 , = −3.2877 × 10−4 , = −4.7950 × 10−5 , = −2.4949 × 10−4 , = −6.0943 × 10−4 , Δh ikd1 h 111 = 4.5421 × 10−5 , h 112 = 3.3802 × 10−4 , h 122 = 4.7789 × 10−4 , h 113 = 1.6755 × 10−4 , h 123 = 2.2199 × 10−4 , h 133 = 3.3220 × 10−4 , h 213 = 6.2493 × 10−4 , h 223 = 2.7403 × 10−4 , h 233 = 6.2493 × 10−4 , h 313 = 3.3220 × 10−4 , h 323 = 2.2199 × 10−4 , h 333 = 1.6755 × 10−4 , h 324 = 4.7789 × 10−4 , h 334 = 3.3802 × 10−4 , h 335 = 4.5421 × 10−5 ; the rest are of zero value h 124 = −3.0762 × 10−6 , h 134 = −3.1991 × 10−6 , h 224 = −4.7481 × 10−4 , h 234 = −3.3632 × 10−4 , h 135 = −1.1530 × 10−7 , h 235 = −4.5305 × 10−5 ; the rest are of zero value all d1 and k However, the same size of stability region is achieved either with Xk (x2 ) being the same for all k or both Nkd1 and Xk (x2 ) being the same for all d1 and k Comparing with Example 10.2, it can be seen from Figs 10.8 and 10.9 that the size of stability regions for all scenarios are all larger than that in Example 10.2 at the cost of a larger number of rules in both switching polynomial fuzzy controller and FPLF Comparing with Fig 10.1a and b in Example 10.1, slightly larger sizes of stability regions are obtained at the cost of a larger number of rules being used in switching polynomial fuzzy controller 290 10 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems … (a) 41 39 37 b 35 33 31 29 27 25 100 110 120 130 140 150 160 170 180 190 200 160 170 180 190 200 a (b) 41 39 37 b 35 33 31 29 27 25 100 110 120 130 140 150 a Fig 10.9 Stability regions given by the SOS-based stability conditions in Theorem 10.2 for Example 10.3 ‘◦’: stability region for Nkd1 (x2 ) and Xk (x2 ) being allowed to be all different for all d1 and k ‘×’: stability region for Nkd1 (x2 ) being allowed to be all different for all d1 and Xk (x2 ) being the same for all k ‘ ’: stability region for Nkd1 (x2 ) being the same for all d1 and Xk (x2 ) being the same for all k a Stability regions with δ = 0.5 b Stability regions with δ = 0.25 10.5 Conclusion 291 10.5 Conclusion A switching polynomial fuzzy controller has been proposed to stabilize the nonlinear plant represented by the polynomial fuzzy model To investigate the closed-loop system stability, a FPLF consisting of a number of sub-Lyapunov candidates combined by state-dependent PLMFs has been proposed Each sub-Lyapunov candidate is responsible for an operating sub-domain in the stability analysis The PLMF has demonstrated a nice property that its derivative with respect to system states is a constant to make easy the stability analysis instead of using an estimated bounds in the existing work MFD stability analysis using PLMFs has been employed to investigate the system stability using the SOS-based approach SOS-based stability conditions have been derived to determine the system stability and polynomial feedback gains As the polynomial fuzzy model, switching polynomial fuzzy controller and the FPLF can have their own number of rules and set of premise membership functions, it offers the greatest design flexibility with the potential of reducing the implementation costs of controllers when smaller number of rules and/or simple form of membership functions are considered Simulation examples have been given to verify the effectiveness of the MFD stability analysis results using FPLF with imperfectly matched rule number and premise membership functions as well as the merits of the switching polynomial fuzzy control strategy References Wang, H.O., Tanaka, K., Griffin, 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function, 17 Fuzzy model, 41 Fuzzy model-based control system (FMB control system), 47 Fuzzy polynomial Lyapunov function, 260 H H∞ performance index, 176 I Imperfectly matched premises, 103, 120, 128, 156, 164 Integral action, 139 Integral gains, 139 Interval Type-2 polynomial fuzzy models, Interval Type-2 T–S fuzzy models, L Linear matrix inequality (LMI), 12 LMI-based stability conditions, 49–53 Lyapunov functions, 12, 14 Lyapunov’s direct method, 48 Lyapunov’s second method, 48 Lyapunov stability theory, 48 M Mamdani-type fuzzy controller, Membership function approximation, 20 Membership function boundary, 19 Membership function-dependent (MFD), 12, 17, 18 Membership function-independent (MFI), 12, 17 Membership function regional information, 19 Model reference adaptive control, 11 Monomial, 40 O Observer-based fuzzy controllers, Output-feedback fuzzy controllers, Output-feedback polynomial fuzzy controller, 178 P Parallel distributed compensation (PDC), Partially matched premises, 103, 114, 127, 152, 161 Perfectly matched premises, 103, 106, 125, 142, 159 © Springer International Publishing Switzerland 2016 H Lam, Polynomial Fuzzy Model-Based Control Systems, Studies in Systems, Decision and Control 64, DOI 10.1007/978-3-319-34094-4 295 296 Piecewise linear membership function, 20, 21, 260, 264, 267 Piecewise Lyapunov function, 16 Polynomial, 40 Polynomial fuzzy controller, 46 Polynomial fuzzy model, 7, 43 Polynomial fuzzy model-based control system (PFMB control system), 47 Polynomial Lyapunov function, 15 Polynomial membership function, 86 R Reference model, 177 Regulation, 22 S Sampled-data fuzzy controllers, 10 Sector nonlinearity, 42 Semi-definite program, 40 Simulation example, 54 SOS-based stability conditions, 55 SOSTOOLS, 40 S-procedure, 19, 62 Stability conditions, 49–53, 55 Stabilization, 22 Staircase membership function, 20, 21, 63 State-feedback fuzzy controller, 8, 45 Index Sum-of-squares (SOS), 12 Switching Lyapunov function, 16 Switching polynomial fuzzy controller, 230, 263 Switching polynomial Lyapunov function, 227 Switching/sliding-mode fuzzy controllers, 10 T Taylor series membership function (TSMF), 88 Tracking, 23 T-S fuzzy model, 41 T-S switching fuzzy models, T-S time-delay fuzzy models, Two-step procedure, 111, 145 Type-1 T-S/polynomial fuzzy models, Type-2 polynomial fuzzy models, Type-2 T-S fuzzy models, U Universe of discourse, 3, Z Zero order hold (ZOH), 10 ... are fuzzy models namely T–S fuzzy model, T–S polynomial fuzzy model, type-2 T–S fuzzy model and type-2 T–S polynomial fuzzy model 1.2.1.1 Type-1 T-S /Polynomial Fuzzy Models The (type-1) T–S fuzzy. .. discussed Combining various fuzzy models and fuzzy controllers, a wide range of fuzzy model- based control systems are formed Stability analysis of the fuzzy model- based control systems subject to various... time-delay T–S fuzzy model is reduced to the original T–S fuzzy model 1.2.1.2 Type-2 T–S /Polynomial Fuzzy Models The above mentioned T–S fuzzy model and polynomial fuzzy model are with type-1 fuzzy sets