Fuzzy Gain Scheduling and Tuning of Multivariable Fuzzy Control Methods of Fuzzy Computing in Control Systems

Transcription

1 Tampereen teknillinen korkeakoulu Julkaisuja 293 Tampere University of Technology Publications 293 Pauli Viljamaa Fuzzy Gain Scheduling and Tuning of Multivariable Fuzzy Control Methods of Fuzzy Computing in Control Systems Tampere 22

2 Tampereen teknillinen korkeakoulu PL Tampere Tampere University of Technology P. O. B. 527 FIN-3311 Tampere Finland ISBN ISSN

3 Tampereen teknillinen korkeakoulu Julkaisuja 293 Tampere University of Technology Publications 293 Pauli Viljamaa Fuzzy Gain Scheduling and Tuning of Multivariable Fuzzy Control Methods of Fuzzy Computing in Control Systems Thesis for the degree of Doctor of Technology to be presented with due permission for public examination and criticism in Auditorium S1, at Tampere University of Technology, on the 2th of June 2, at 12 o clock noon. Tampere 22

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5 Abstract The thesis deals with fuzzy computing in control systems and particularly in multivariable control and gain scheduling. It addresses research problems as to how to apply methods of fuzzy set theory efficiently in control tasks, and how to tune a fuzzy controller. The problem is solved by analyzing properties of fuzzy sets and fuzzy systems, selecting a structure for the fuzzy system, and packaging the system in a compact parameterized presentation. The thesis has two main results. It proposes a multivariable fuzzy controller aimed to control a stable, two-input two-output system. Also a method to automate a rule base updating in a fuzzy gain scheduling scheme is developed. In the thesis, fuzzy logic is not considered as a logic, but as a computing method. Fuzzy systems are considered from two points of view. First the system has linguistic representation with linguistic variables and fuzzy if-then rules. On the other hand, fuzzy logic is used as a numerical method to create a nonlinear mapping between inputs and outputs. Some special properties of fuzzy systems are discussed in more detail. Transparency and locality of design parameters of the fuzzy system are important properties and it is not worthwhile to lose them with careless design of the system. Therefore a structure and a parameterization is proposed for fuzzy computing systems. Interpolation behaviour of different implementations in fuzzy reasoning is discussed. It is shown that under certain assumptions the fuzzy system is piece-wise multi-linear which ensures well-behaved interpolation. The assumptions are less restrictive than presented in earlier literature. Practical advice for the design and tuning of fuzzy controllers are given. Input fuzzy sets should be placed according the nonlinearity needed. Each rule should have an individual fuzzy set as a consequence that clustering of the rules can be avoided and the controller can be tuned locally. A systematic method to tune a multivariable fuzzy controller for an unknown two-input two-output process is developed. The method groups the rules for two rule bases which are tuned with different objectives. Thus, a good control performance is obtained and also the total number of the rules can be kept small. The design is based on a scheme of dominant rules. The consequences of the rules are designed for a situation where the rule is dominant. Conventional control and fuzzy computing is combined together in the scheme of fuzzy gain scheduling. Tuning parameters of the controller are stored in a fuzzy rule base beforehand, and during control the fuzzy system gives suitable parameter values for the controller. In addition to the process output, the reference signal is also used as a scheduling variable in order to predict near future operation conditions. A method is proposed to keep the rule base up-to-date. When the controller is retuned i

6 ii Abstract in some operation conditions, fresh tuning is added to the rule base. But in order to keep the number of the rules small a rule must be removed. The rule to be removed is selected by the principle of the slightest detriment, so that the removal deteriorates the approximation accuracy the least. The algorithm has the advantage that all data needed are stored in the rule base.

7 Acknowledgement The work has been carried out in the Automation and Control Institute of Tampere University of Technology. I would like to express my gratitude to Prof. Heikki N. Koivo, the advisor of my thesis, for his guidance and support during the course of this thesis. I am also grateful to Prof. Jan Jantzen (Technical University of Denmark) and Dr. Tech. Enso Ikonen (University of Oulu) for providing their comments and recommendations as the reviewers of the thesis. I am grateful to my friends and colleagues in the Automation and Control Institute for providing the inspiring environment in which the work has been carried out. It is my pleasure to thank Prof. Pentti Lautala and students in his seminar course for reading an unfinished version of the manuscript. I also wish to thank many people for co-operation in research projects during which many of the results of this thesis have been created. The text of the manuscript was revised by James Rowland. The research was funded by the Finnish Academy and the Tekes Technology Development Centre. Financial support was also provided by the Jenny and Antti Wihuri Foundation, the Neles Foundation, the Walter Ahlström Foundation, the Paavo Sirén Foundation, and the Ulla Tuominen Foundation, which are gratefully acknowledged. Most of all, I thank my wife, Lea, and Laura, Iiris, Olli, Juho and Alisa for their support and patience. Lempäälä, May 2 Pauli Viljamaa iii

8 iv Acknowledgement

9 Table of Contents List of Publications Notational Conventions and Abbreviations ix xi 1 Introduction Historical background Fuzzy computing Fuzzy control Fuzzy gain scheduling Objectives of thesis Contributions of thesis Outline of thesis Fuzzy Computing Membership functions Properties of fuzzy sets Properties of sets of membership functions Relation to basis functions Fuzzy system Completeness Transparency Locality Normality Reasoning mechanism Similarity between Mamdani and Sugeno reasonings Min-max composition versus product-sum composition Partial solution to the problem of min-max composition Piece-wise multi-linear fuzzy system Summary Fuzzy Control Fuzzy controller Design of the fuzzy controller Selection of input and output variables Selection of rule base Selection of fuzzy parameters Parameterization of the fuzzy controller Sufficient structure of the fuzzy controller v

10 vi Table of Contents Proposed parameterization Tuning of the fuzzy controller Rules of the thumb for tuning Tuning based on linear controller Tuning based on other methods Feasibility of the fuzzy control Possibilities of the fuzzy control Requirements for the platform An industrial example Summary Multivariable Fuzzy Control Problem statement Structure of the multivariable fuzzy controller Tuning of the multivariable fuzzy controller Simulation example Fuzzy control of pilot head-box process Tuning Results Limitations of tuning method Summary Fuzzy Gain Scheduling Introduction Gain scheduling Fuzzy performance measure Fuzzy tuning Related work Gain scheduling with fuzzy logic Simulation example Updating of rule base Updating of piece-wise linear approximation Numerical example Combination of gain scheduling and updating method Limitations of the updating method Limitations of scheduling concept Summary Conclusions 77 7 Summary of Publications Contents of publications Author s contribution to publications References 83

11 Table of Contents vii A Basics of Fuzzy Sets 89 A.1 Fuzzy sets A.2 T-norm and T-conorm A.3 Fuzzy reasoning A.3.1 Mamdani fuzzy rules A.3.2 Sugeno fuzzy rules A.4 Defuzzification A.5 Definitions for fuzzy sets A.6 Definitions for sets of membership functions A.7 Definitions for fuzzy systems B Analysis of Fuzzy System 95 B.1 Normality of fuzzy system B.2 Linear fuzzy system B.3 Piece-wise multi-linear fuzzy system C Experimental Data 99 Publications 11

12 viii Table of Contents

13 List of Publications This thesis consists of the following publications: 1. Viljamaa, P., Raitamäki, J., Neittaanmäki, P. & Koivo, H. N Basis functions in soft computing a survey. Proceedings of the World Automation Congress (WAC 96), Intelligent Automation and Control. Montpellier, France, May 28 3, WAC, Vol. 4, pp Raitamäki, J., Viljamaa, P., Koivo, H. N. & Neittaanmäki, P Basis functions in soft computing and in finite element method. Fourth European Congress on Fuzzy and Intelligent Technologies EUFIT 96. Aachen, Germany, September 2 5, ELITE-foundation, Vol. 1, pp Viljamaa, P. & Koivo, H. N Comparison of minimum and product operators in fuzzy systems with more than two inputs. IASTED Int. Conf. on Control and Applications CA 98. Honolulu, HI, USA, August 12 14, IASTED, Vol. 1, pp Viljamaa, P., Peltonen, H. & Koivo, H. N. 2. Target value computing in paper machine grade changes by fuzzy system. The 9th IEEE Int. Conf. on Fuzzy Systems FUZZ-IEEE 2. San Antonio, TX, USA, May 7 1, IEEE, Vol. 1, pp Viljamaa, P. & Koivo, H. N Tuning of a multivariable fuzzy logic controller. Intelligent Automation and Soft Computing. Vol. 1, No. 1, pp Viljamaa, P. & Koivo, H. N Tuning of a multivariable PI-like fuzzy logic controller. Proceedings of the 3rd IEEE Int. Conf. on Fuzzy Systems FUZZ- IEEE 94. Orlando, FL, USA, June 26 July 2, IEEE, Vol. 1, pp Viljamaa, P. & Koivo, H. N Multivariable fuzzy logic controller of pilot head-box process. The 13th World Congress of IFAC. San Francisco, CA, USA, June 3 July 5, IFAC, Vol. N, pp Viljamaa, P. & Koivo, H. N Fuzzy logic in PID gain scheduling. Third European Congress on Fuzzy and Intelligent Technologies EUFIT 95. Aachen, Germany, August 28 31, ELITE-foundation, Vol. 2, pp ix

14 x List of Publications

15 Notational Conventions and Abbreviations a T, A T transpose of vector a or matrix A A (i) the ith row of the matrix A (i,j) the element of the matrix in the ith row and in the jth column t time k discrete time h sampling interval z 1 delay operator u(k) control signal at time k y(k) process output at time k y(t) process output at time t y r (k) reference signal for process output at time k e(k) = y r (k) y(k), error between reference and measurement e(k) = e(k) e(k 1), change in error Y (s) Laplace transform of y(t) G(s) transfer function of plant n x dimension of x (x R nx ) m x number of fuzzy sets for x X i the ith fuzzy set of x µ X i(x) membership function of fuzzy set X i p x R mx, cores of normalized membership functions for x δ x (j) i degree of membership of the ith input to its jth fuzzy set ψ (i) truth value of the ith rule θ (i) in parameters of input membership functions of the ith input θ rules rule base index θ (i) out places of output singletons for the ith rule ẋ = dx dt x steady state of signal x x = x x, deviation from steady state K p proportional gain T i integral time K i integral gain T d derivative time α, β fine tuning parameters F ( ) fuzzy system xi

16 xii Notational Conventions and Abbreviations w y, w yr weighting factors (x f, y f ) new (fresh) observation µ thr threshold truth value M Maximum number of fuzzy sets J (i) cost of the ith solution i index of Pareto optimal solution T T-norm S T-conorm FEM FLC IMC MFC MIMO PID RBF SISO T-norm Finite Element Method Fuzzy Logic Controller Internal Model Control Multivariable Fuzzy Controller Multi-Input Multi-Output Proportional, Integral and Derivative control, also PI and PD Radial Basis Function Single Input Single Output Triangular norm, also in T-conorm S M B N P Small Medium Big Negative Positive

17 Chapter 1 Introduction Due to the continuously developing automation systems and more demanding control performance requirements, conventional control methods are not always adequate. On the other hand, practical control problems are usually imprecise. The inputoutput relations of the system may be uncertain and they can be changed by unknown external disturbances. New schemes are needed to solve such problems. One such an approach is to utilize fuzzy control. Fuzzy control is based on fuzzy logic, which provides an efficient method to handle inexact information as a basis of reasoning. With fuzzy logic it is possible to convert knowledge, which is expressed in an uncertain form, to an exact algorithm. In fuzzy control, the controller can be represented with linguistic if-then rules. The interpretation of the controller is fuzzy but the controller is processing exact input-data and is producing exact output-data in a deterministic way. Fuzzy controllers are not the sole alternative to apply fuzzy computing in control systems. Automation systems have many hierarchical control loops and supervisory blocks which are not feedback controllers. Many of them can be managed by utilizing fuzzy rules. Those fuzzy systems can be considered rather as fuzzy computing units than fuzzy controllers. Traditionally those fuzzy systems are called fuzzy logic systems, which is a slightly misleading term. The systems include computing rather than logic. Therefore, the term fuzzy computing is used throughout this thesis. Here, fuzzy computing does not mean computing with fuzzy numbers (fuzzy calculus) but computing based on an algorithm that can be interpreted as fuzzy if-then rules. 1.1 Historical background Since the introduction of the theory of fuzzy sets by L. A. Zadeh in 1965 [8], and the industrial application of the first fuzzy controller by E. H. Mamdani in 1974 [46], fuzzy systems have obtained a major role in engineering systems and consumer products in the 198s and 199s. New theoretical results [14, 36] and new applications [21, 77] are presented continuously. A reason for this significant role is that fuzzy computing provides a flexible and powerful alternative to construct controllers, supervisory blocks, computing units and compensation systems in different application areas [14]. With fuzzy sets very nonlinear control actions can be formed easily. The transparency of fuzzy rules and 1

18 2 Chapter 1 Introduction the locality of parameters are helpful in the design and the maintenance of the systems [2]. Therefore, preliminary results can be obtained within a short development period. However, fuzzy control does have some weaknesses. One is that fuzzy control is still lacking generally accepted theoretical design tools. This nonexistence of a systematic procedure for the design of a fuzzy controller is stated in [34]. Although preliminary results are achieved easily, further improvements need a lot of work if any systematic tuning methods are not applied. Especially when the number of inputs increase, the maintenance of the multi-dimensional rule base is time-consuming. 1.2 Fuzzy computing Due to the active research in the field of fuzzy computing, many methodologies exist for the implementation of fuzzy systems, e.g., in [14]. A good example of this is the triangular norms (T-norms) and conorms (S-norms) which are utilized to implement fuzzy and and or connectives. A number of different implementations are available and comprehensive list of them can be found, e.g., in [17]. A designer of a fuzzy system can be confused by the many design choices that the fuzzy set theory provides. Fundamental comparisons and suggestions for appropriate implementations are needed. T-norm and S-norm implementations are compared in [16, 1, 2]. Gupta and Qi [16] compared simulated step responses of a closed loop control system. They used the same PI-type controller structure having two inputs and one output for each simulation. However, they did not retune the scaling factors between the comparisons. It is quite questionable to compare the implementations with respect to the step responses, at least the controllers should be retuned carefully before the comparison. Chen et al. [1] compared the interpolation properties of different implementations. This is a better approach. However, they considered only two input fuzzy systems as is also done in [2]. It is well-known that a fuzzy system produces linear interpolation between the rules, if the input membership functions are linear, output membership functions are selected appropriately, and fuzzy and connectives are implemented by product operator. It is also well-known that minimum operator generates nonlinearities to the output surface in the case of two inputs [2]. The differences to the linear surface are minor and it is usually assumed that it does not matter. But this is true only for two-input fuzzy systems. Such interpolation comparisons are not presented in the case that the fuzzy system has more than two inputs. It seems to be so that the performance of the fuzzy system is very poor with minimum operator when the system has several inputs. The performance can even be unacceptable as is shown in this thesis. The topic is important because the performance may be really poor and due to the fact that many commercial software tools provide minimum operator as the only alternative for T-norm. In the design of fuzzy systems different optimization methods have an increasingly important role. Fuzzy systems can be designed to match certain input-output data set. Efficient algorithms are adopted from the field of optimization theory. Fuzzy systems can be optimized by a linear least-squares algorithm, different Gauss-Newton algorithms, or pure gradient algorithms [37]. The selection of algorithm depends on

19 1.3 Fuzzy control 3 the parameters optimized, necessity to preserve a priori parameter knowledge, etc. For example, when a fuzzy system with many input variables is used to supervise or control a system, the number of tuning parameters increases rapidly. This is not a big problem if the design is performed in different stages. Firstly the input parameters are selected based on a priori information about the system. This can be done because they have only a minor effect on the final performance of the fuzzy system. The input parameters form only a basis for the system behaviour. After that suitable input-output data pairs are generated using the operator s or expert s knowledge. The output parameters can be obtained from the data using least-squares algorithm. 1.3 Fuzzy control Fuzzy controllers can be very versatile. Tools of fuzzy logic provide many possibilities [14]. Therefore, it is common that a beginner tries many alternatives and searches for improvements to the control performance by altering the shape of membership functions and the number of fuzzy sets and rules. However, the result can be a deep bog where an acceptable solution is difficult to find. Usually, a very simple structure of the fuzzy controller can result in good performance with minimum effort of design and tuning. Several different techniques have been tried to solve the tuning problems of fuzzy controllers. Self-organizing controllers create the working rule base based on the meta rules [15]. However, the method needs some kind of model of the system and the scaling factors must still be tuned by the operator. The fuzzy controller can also be designed with neuro-fuzzy techniques. An approach where effective optimization technique is applied to automate the tuning of the fuzzy controller in the feedback control loop has been proposed in [44]. The disadvantage of these methods is that a priori knowledge about the system is not easy to use. The most widely used controller in industrial applications is PID-controller (proportional-integral-derivative). It is easy to tune and it has good disturbance attenuation properties. A disadvantage of the PID controller is that it is linear and cannot successfully control a plant, which has strong nonlinearities. In fuzzy control [14], PD-type and PI-type fuzzy controllers are the best-known counterparts of the PID controller. They are used to achieve better performance with nonlinear processes. Good experiences have been obtained especially with the PD-type fuzzy controllers in servo applications [15, 43]. However, the standard fuzzy controller, which has the error and the change in the error as inputs and the control signal or its change as an output, cannot react to changes in the operating point. The fuzzy controller needs more information to compensate nonlinearities when the operation conditions change. When the number of the inputs of the fuzzy system is increased, the dimension of the rule base also increases. Thus, the maintenance of the rule base is more time-consuming. Another disadvantage of fuzzy controllers is the lack of systematic, effective and useful design methods, which can use a priori knowledge of the plant dynamics. Deficiencies of the PID controller and the fuzzy controller can be solved by combining them together.

20 4 Chapter 1 Introduction 1.4 Fuzzy gain scheduling Conventional gain scheduling can be seen as a part of a wide range of adaptive control [6], although gain scheduling cannot take into account unexpected changes in the process dynamics. It is not adaptive in that sense. A common idea of conventional gain scheduling [31, 32, 18] is that a nonlinear model of the process is known. The conventional gain scheduling scheme is developed by linearizing the model at several operating points and designing linear controllers for the models. Depending on the method used in the design the controllers can be very sophisticated and may not even be of the same order. The approach considered in this thesis has a different background. It is a controller driven approach when the methods mentioned above are model driven approaches. The model driven approach has the advantage that, e.g., robustness of gain scheduling system can be considered [18]. Fuzzy systems are applied in gain scheduling in many ways. A fuzzy system can be used to tune the PID controller on-line if the response of the closed loop system is not acceptable [84]. In this approach, the fuzzy system can change the controller parameters even if the dynamics is not changed. This is not desirable. Fuzzy gain scheduling can mean a fuzzy controller with Sugeno type rules. Thus, the consequence of each rule is interpreted as a local controller [53, 23]. Or, fuzzy gain scheduler operates like its conventional counterparts, e.g., in [54, 4, 69]. In this thesis, the fuzzy system is used to schedule off-line tuned parameters based on the system operating point. This is closer to the conventional gain scheduling [6] than the method proposed in [84]. The fuzzy computing is based on the plant output, the reference signal and a table of the off-line tuned PID parameters. Thus, different dynamics of the system can be taken into account in different operating conditions. However, the dimension of the rule base can be kept small, which helps the maintenance. Although many results of fuzzy gain scheduling have been published (see above) maintenance questions have received no attention. When the dynamics of the controlled system change due to, e.g., wearing and soiling, some rules become obsolete and must be updated. The rule base can also be constructed on-demand, i.e., new rules are added if needed. This might result in unnecessarily many similar rules and sometimes they must be pruned. This thesis will give some answers to the maintenance. 1.5 Objectives of thesis The objective of this thesis is to simplify the current fuzzy control methods and to find the methods which are most useful in applications. The research problem studied is: How to tune a fuzzy controller? The problem is important from industrial point of view. An unmeasurable amount of human resources are wasted in testing if different implementations of fuzzy computing could improve the control performance. If a certain approach can be selected and justified, those resources could be used for the key point, the tuning. The control problem where fuzzy systems are applied is typically such that the process to be controlled is continuous and the controller is discrete. The control is performed in a closed loop or open loop as a supervisor, and is implemented by fuzzy logic either

21 1.6 Contributions of thesis 5 totally or partially. The tuning problem is that of how to adjust the parameters of the fuzzy controller so that the specifications of the control performance can be satisfied. Usually the adjustments have to be done with incomplete process knowledge, but some is always known. The tuning method should be capable of utilizing that partial knowledge. The methods to solve the research problem are to analyze different properties which fuzzy sets and fuzzy systems have and to decide which are important in the proposed case, to select the structure of the fuzzy system so that it has these important properties, to package the fuzzy system in a compact and powerful presentation so that it can be analyzed and applied conveniently. The results obtained in this thesis are a method to design and tune a multivariable fuzzy controller, a way to combine fuzzy computing with PID controller as a gain scheduler with a rule base updating algorithm. 1.6 Contributions of thesis Chapter 2 has two main contributions. Section reports the discovery of the poor performance of the min-max composition if the fuzzy system has more than two inputs (Viljamaa and Koivo, 1998, Publication 3). The analysis of the problem leads to a theorem of the multi-linearity of the fuzzy system (Section 2.3.4) with less restrictive assumptions than presented earlier. In addition, Section notices a unified framework of soft computing methods like radial basis function networks and B-splines (Viljamaa et al., 1996, Publication 1), and similarities with basis functions of finite element method and membership functions (Raitamäki et al., 1996, Publication 2). In Chapter 3 proposals are made for the structure (Viljamaa and Koivo, 1995, [76]) (Section 3.3.1) and the parameterization (Viljamaa and Koivo, 1998, Publication 3) (Section 3.3.2) of the fuzzy controller. The same structure is widely used but this thesis gives more solid arguments for it and the parameterization relies on it. The examples shown in this thesis are solved and presented using the structure and the parameterization. In Section 3.8 the structure and the parameterization are applied in an industrial paper machine (Viljamaa et al., 2, Publication 4). A structure and a tuning method of the multivariable fuzzy controller are developed (Viljamaa and Koivo, 1993, 1994, 1995, Publications 5 6) in Chapter 4. The controller is for a two-input two-output multivariable process. The rule base is divided to two parts. The tuning method utilizes experimental knowledge obtained from the process. The controller is applied to control a laboratory scaled paper-machine head-box (Viljamaa and Koivo, 1996, Publication 7). The most important contribution of this thesis is the controller driven approach for fuzzy gain scheduling (Viljamaa and Koivo, 1995, Publication 8) (Section 5.2) with the method to automate the updating of the scheduling rule base (Section 5.3). The method is integrated with a PID controller utilizing the concept of Pareto optimality. The author is the main contributor to the publications mentioned above.

22 6 Chapter 1 Introduction 1.7 Outline of thesis This thesis consists of the text part and the original publications. The publications are included to show the time perspective of the research. The results obtained in the publications are presented in a consistent and unified form in the text part which is organized to have methods in Chapters 2 3 and results in Chapter 4 5. The developments done after publication of the articles are shown in the following outline. Fuzzy computing related questions are discussed in Chapter 2. They include properties of membership functions and their effect on the behaviour of the fuzzy system. The chapter introduces similarities of membership functions to other basis functions utilized in function approximation (Publications 1 and 2). It gives some advice how to preserve transparency of a fuzzy system. A comparison of two possible implementations for fuzzy and connective is presented as well as in Publication 3. The chapter concludes with the result that multi-linearity of a fuzzy system ensures that the interpolation introduced by fuzzy computing is well-behaved. The results presented in the chapter are applicable both in fuzzy modelling and in fuzzy control. The chapter includes mainly unpublished material except in Sections 2.1.3, and Design and tuning of fuzzy controllers are treated in Chapter 3. The chapter deals with more control specific items than the previous chapter. A parametric representation of a fuzzy controller shown in Publication 3 is proposed in order to make the design and tuning easier. Also feasibility and possibilities of fuzzy control are discussed. In Publication 4, an example to apply fuzzy computing in Finnish process industry is shown. The chapter includes mainly unpublished material except that the parameterization of Section has been briefly described in Publication 3 and the example of Section 3.8 has been shown in Publication 4. Chapter 4 deals with the multivariable fuzzy control of Publications 5 7. It presents a new idea to classify the rules depending on their function and to compose two rule bases. A new tuning method is proposed and the controller is applied to control a pilot head-box process. The tuning methods have not been altered after the publication. Combinations of fuzzy systems and conventional controllers are discussed in Chapter 5. Scheduling of the controller parameters by a fuzzy system presented in Publication 8 is selected for more detailed treatment. A controller driven approach of gain scheduling is utilized and a new scheme to automate the updating of the rule base is proposed. The material in Sections 5.3 and 5.4 has not been published. A summary of the publications is given in Chapter 7. The author s contributions to the publications are also explained there. The theory of fuzzy sets is quite young and also a popular research area. Because of this, the terminology and the notational conventions are not well established. It is stated that many poorly written publications occur in the field [2]. Many publications are so fuzzy that the results cannot be checked. To avoid the problem, an introduction to fuzzy logic is included in Appendix A. It introduces only the very basics needed in this thesis and is not a comprehensive study to the theory.

23 Chapter 2 Fuzzy Computing Fuzzy logic is a method which can be utilized to transform an inexact knowledge into the form of a computer algorithm. In applications it is applied by defining a fuzzy system with linguistic variables and with a set of if-then rules. The fuzzy system can be interpreted in a fuzzy way, but the computing algorithm itself is a deterministic system which processes exact input data and produces exact output data. Fuzzy logic offers only a notation to transfer heuristic knowledge in the form of linguistic rules to this mathematical algorithm. Zadeh announced the term soft computing in 1994 [81]. According to him, soft computing is a collection of methodologies including fuzzy logic, neuro-computing, and probabilistic reasoning. He stated that the methodologies are complementary, not competitive. Recently increasing interest is concerned with the unifying of the soft computing methodologies [65, 59]. Each approach above has parallel theoretical development, and the similarities are evident. If the unified theory of the soft computing could be developed it would have a positive impact on the field. The chapter deals with cross-directional transformation of the theory also by adopting definitions from the other fields and by making suggestions about useful methods of the other approaches. The chapter is entitled Fuzzy Computing instead of fuzzy logic system or fuzzy logic control. Firstly, although the thesis is aimed at developing control theory and applications, there exist applications which cannot be called controllers even in the field of control. The chapter is focused on general aspects of fuzzy systems. Secondly, although Zadeh used the term fuzzy logic in connection of soft computing [81], a question arises if fuzzy systems could be considered rather fuzzy computing systems than fuzzy logic systems. Where is the logic in a parameterized nonlinear input-output mapping? In this thesis fuzzy systems are considered from two points of view. The system is interpreted to be fuzzy with linguistic representation. On the other hand, fuzzy logic is used as a method to form nonlinear function between inputs and outputs. Depending on the situation, the view-point changes. In this chapter, properties of the nonlinear function generated by fuzzy logic are discussed. This chapter gives a practical meaning for definitions well-known in the field of fuzzy set theory. The properties are considered in modelling and control point of view in Sections 2.1 and 2.2. In order to put more focus on the contribution the definitions are presented in Appendix A. Section reports a poor performance of 7

24 8 Chapter 2 Fuzzy Computing implementation of T-norm by minimum operator. The analysis of the problem leads to the theorem of multi-linearity of a fuzzy system in Section Membership functions Membership functions form the basis for fuzzy computing. The shape, the overlapping, peak values, and their continuity properties determine how the fuzzy system can be designed and how it behaves. Thus it is worthwhile to consider the membership functions in more detail. Membership functions have a number of properties according to which they can be classified to different categories. Some of them are useful also from a practical point of view. Those properties are related to the membership function but also to the set of membership functions. Here the set means membership functions which are defined for a certain input variable and the membership functions together have some properties. Some of the properties are characteristics for fuzzy sets but there are also connections to function approximation theory and to other methodologies. There is an increasing interest to unify the theory of basis functions and membership functions of fuzzy sets [65] Properties of fuzzy sets Properties of single fuzzy sets are considered in this section. A practical meaning is given for a support, a normality, a core, a convexity, a degree, a locality and a globality of a fuzzy set. The output of the fuzzy system is usually composed of consequences of many rules. A rule affects the output on the support (Definition A.5) of its input fuzzy sets. Elsewhere the rule has no influence. When the fuzzy sets of a rule are normal (Definition A.6), the rule can achieve full truth. If full truth is obtained and each input has fuzzy sets which are normal as a set (see Definition A.9), the final output is produced by the rule alone. Usually there is no reason to define an abnormal fuzzy set. Usually the core of a fuzzy set (Definition A.7) includes only a point. It is useful to parameterize the membership functions with their cores. The support of the fuzzy set could range up to the core of the neighboring fuzzy sets. Thus, the membership functions can be parameterized exactly with the cores only. Fuzzy sets are often defined to be convex (Definition A.8). In modeling and control there is usually no reason to define non-convex fuzzy sets. Non-convexity can produce peculiar interpolation between rules. Definition 2.1. A membership function µ X (x) is said to have degree n, if it has continuous (n 1)th derivative. The membership function of degree n belongs to continuity class C n 1 (R). This definition is not common in the field of fuzzy theory. It can be found in books regarding B-splines, e.g., [27]. If fuzzy systems are optimized with respect to a cost function, the optimization method usually assumes membership functions of degree 2 or 3. If the input membership functions are fixed before the optimization which is quite a common approach,

25 2.1 Membership functions 9 PSfrag replacements 1 µ(x) convex local global normal abnormal non-convex core support x Figure 2.1: Properties of fuzzy sets. the degree of them does not naturally matter. However, it does not mean that input membership functions of degree 1 cannot be optimized [38]. A certain continuity class of the output function is easiest to obtain with input membership functions of the same continuity class. Definition 2.2. A fuzzy set X is local, if its membership function µ X (x) has a compact support. Then supp(x) is bounded and closed subset of R. Definition 2.3. A fuzzy set X is global, if its membership function µ X (x) has an infinite support, i.e., supp(x) = R. Whether input fuzzy sets are local or global affects the design and maintenance of the system. With global fuzzy sets the rules affect the output everywhere and local changes to the behaviour of the system cannot be made. Additionally no rule can compose the output individually anywhere. Definitions 2.2 and 2.3 differ from definitions for basis functions presented in [65], where definitions are related to the optimization tasks. In fuzzy computing the linguistic representation and user-friendliness are more important and therefore Definitions 2.2 and 2.3 are more appropriate for fuzzy computing. The properties of fuzzy sets are illustrated in Fig Properties of sets of membership functions The discussion in the previous section characterizes a membership function. Usually several fuzzy sets are defined for an input or an output variable. These fuzzy sets form a family of sets. The family can have some properties. A practical meaning is given for a normality, a completeness and an efficient area of a set of membership functions. The definition of the normality of a set of membership functions (Definition A.9) is distinct from the definition of the normality of a membership function (Definition A.6). The normality of a set of membership functions cannot be obtained with normal membership functions if they are also global. The antecedent of a fuzzy rule can be considered as a multivariable membership function which is constructed by T-norms and T-conorms according to the fuzzy connectives and the membership functions introduced in the rule. With an appropriate reasoning method and selections of T-norm and T-conorm the normality of the membership functions of each input implicates the normality of the multivariable mem-

26 1 Chapter 2 Fuzzy Computing bership functions of the rules. Thus the division expressed in the center-of-gravity defuzzification method (A.16) is not needed. If the sets of input membership functions are normal, they are also complete (Definition A.1). It is important that the input membership functions are complete, when center-of-gravity defuzzification method (A.16) is utilized. However, the centerof-gravity method assumes the completeness of the rule base, i.e., for all x R nx (the system has n x inputs) an active rule exists (implicating nonzero divider). That cannot be achieved, if the sets of the input membership functions are not complete. Definition 2.4. Let the membership functions µ X i(x) be piece-wise differentiable. An efficient area of membership functions µ X i(x) is a crisp set eff({x i }) = {x R i, j : d dx µ X i(x) µ Xj (x) > i j}. (2.1) When x eff({x i }), the membership functions µ X i(x) transform the changes in x to the reasoning mechanism. Therefore the area is called efficient. Otherwise the output keeps a constant value even if the input changes and the consequences of the rules let us assume a change in the output. The properties of sets of membership functions are illustrated in Fig Relation to basis functions Different function approximation schemes utilize basis functions from which the approximating function is formed, e.g., with a linear combination. They are called by different names like activation functions, wavelets or B-splines depending on the scheme. Membership functions can be considered to be a class of basis functions. Since the basis functions are widely used in approximation theory and in the solution of partial differential equations it seems possible to develop a unified theory of basis functions [65, 59]. In this section, a short summary of possible joint schemes is given. B-splines B-splines provide bases for certain spline spaces, and therefore all other spline functions can be obtained by forming linear combinations of them [27]. B-splines of degree are the starting point for a recursive definition of all the higher-degree B-splines. They can be evaluated, integrated and differentiated with a stable and efficient recursive algorithm. B-splines have compact support, i.e., they are local, and they form a normal base, i.e., the set of B-splines is normal. B-splines of degree 1 are triangle-shaped functions. They are set-normal like standard triangle-shaped membership functions. Multivariable B-splines are obtained by tensor product of single-variable B-splines. The set-normality is preserved by the tensor product. Therefore the B-spline system is analogous to a fuzzy system. Thus, theoretical results obtained for B-splines can be transfered directly to fuzzy computing. B-splines of higher degrees are not normal by themselves, although they are set-normal. However, they produce smoother output functions.

27 2.1 Membership functions 11 1 µ(x) 1 µ(x) PSfrag replacements PSfrag replacements efficient area efficient area x x (a) Normal. (b) Abnormal. 1 µ(x) 1 µ(x) PSfrag replacements PSfrag replacements efficient area efficient area x x (c) Complete. (d) Incomplete. 1 µ(x) PSfrag replacements efficient area efficient area x (e) Efficient area. Figure 2.2: Properties of a set of membership functions. Basis functions in FEM Finite Element Method (FEM) can be used to approximate the solution of partial differential equations. It utilizes also basis functions, but they are of very special forms, e.g., they are triangular in the input space. A connection between soft computing methods and FEM were first reported by Raitamäki et al., 1996, [59]. A common

28 12 Chapter 2 Fuzzy Computing property of the basis functions is the locality, i.e., they have a compact support. In the field of FEM, the basis functions are important, because they determine the rate of convergence and the properties of approximating function [59]. Therefore a strong theory of basis functions exists in the field. It could be fruitful to transfer the theoretical results to fuzzy computing. Radial basis functions A Radial Basis Function (RBF) network can be used to interpolate or to approximate a (multivariable) function based on numerical input-output data. It includes radially symmetric basis functions which are usually Gaussian functions. The output of the RBF network is a weighted sum of n basis functions f(x) = n w i b i (r i (x)), (2.2) i=1 where r i ( ) is the distance, b i ( ) is the basis function, and w i is the weight [29]. The output function can also be biased or normalized. The distance function can be any Euclidean norm (r i (x)) 2 = (x c i ) T Λ T i Λ i (x c i ), (2.3) where the weight matrix Λ i can be diagonal or non-diagonal. The parameters of the network include the weights w i, and the centers c i and the widths Λ i of the basis functions. The parameters are obtained by fitting the mapping (2.2) to the set of data points by numerical optimization. If a fuzzy system utilizes Gaussian membership functions of equal width in each, only and connectives in the complete rule base, and singletons for the output are used, it can be expressed as f(x) = n i=1 w i exp( x ci 2 n i=1 δ 2 ) exp( x ci 2 δ 2 ), (2.4) where w i is the place of the singleton related to the ith rule and n is the number of the rules. Similarity to an RBF network with Gaussian basis functions is obvious. The denominator of (2.4) is the only difference. The denominator can be connected with the weights w i as is done in [26]. The other alternative is to see it as a normalization of the basis functions [22, 59]. In order to achieve a functional equivalence between the methods, a weighted sum must be utilized as a defuzzification algorithm or the RBF network must include normalization [22]. When the equivalence is obtained under such minor restrictions, the theoretical results achieved with the other approach can be applied straightforwardly for the other. The results include optimization algorithms like orthogonal least squares [11, 78, 19], and universal approximation properties [78, 12, 8]. 2.2 Fuzzy system A fuzzy system is a knowledge-based system which utilizes fuzzy if-then rules and fuzzy logic in order to obtain the output of the system. When the system is considered as a fuzzy block, the computing algorithm can be divided to three parts: fuzzification, reasoning and defuzzification [54, 23, 14, 28, 85].

29 2.2 Fuzzy system 13 Fuzzy sets of the inputs are defined by the membership functions [2]. The sets can be labeled by adjectives which represent the meaning of the sets. The membership function gives the grade of the membership which tells how well the current input value belongs to the fuzzy set. The part of the algorithm where the grades are calculated is usually called fuzzification [2]. After fuzzification the computing handles only the grades and the exact input values are ignored. The above is a practical definition of fuzzification. In fuzzy set theory, however, it is argued that the input itself can be a fuzzy set [2]. Or if the input is a crisp value, it can be fuzzified regarding the uncertainty related to it, e.g., by bell-shaped membership functions [2]. Thus the input value is transformed to a fuzzy set before any processing with actual input membership functions. This is a theoretical definition of fuzzification, which is not utilized in this thesis. The reasoning is performed based on the if-then rules and the grades calculated in the fuzzification [2]. In the design stage, different input fuzzy sets are combined together with fuzzy connectives, and a certain area of the input space can be detected, where only one rule is active. Selecting suitable values for the outputs in the situation and choosing them as consequences of the rule, the fuzzy system can be constructed element-by-element. Mamdani reasoning usually produces a fuzzy set as a consequence [2]. It must be converted to an exact value before it can be used. This part is called defuzzification. Sugeno reasoning does not need defuzzification [2] Completeness A fuzzy system is operating based on fuzzy if-then rules. In practice it is important that the rule base has a rule for each possible situation. This property is called completeness of the rule base [2] (Definition A.11). Definition A.11 is related to linguistic completeness. In semantic level it has a rule for each possible situation. However, in practical applications numerical completeness is more important. The center-of-gravity defuzzification (A.16) assumes that at least one rule is always fired. Commonly, certain input regions of the input domain are not of interest and therefore are not defined at all [14] Transparency A strength of the fuzzy computing is transparency of the system. The behaviour of the system is expressed in the form of the membership functions and the fuzzy if-then rules. This facilitates the validation and correction by experts, and provides a way to communicate with users [64]. However, the transparency can be lost with careless design of the system. Thus it is important to analyze the transparency in more detail and how to include it to the fuzzy system. Definition 2.5. A fuzzy system is transparent, if it gives what it implies in the rules. The definition does not sound mathematical but in that way the transparency is interpreted in this thesis. When a rule obtains full truth, the output of the transparent system really is that which the rule says. The transparency is illustrated with some examples: Example 2.1. A polynomial approximation of the form f(x) = 6.x 3 +.5x 2 4.x + 1. (2.5)