Design of VLSI Systems - Fuzzy Logic Systems

# Chapter 9 FUZZY LOGIC SYSTEMS

## 1 Systems Considerations

The use of fuzzy logic is rapidly spreading in the realm of consumer products design in order to satisfy the following requirements: (1) to develop control systems with nonlinear characteristics and decision making systems for controllers, (2) to cope with an increasing number of sensors and exploit the larger quantity of information, (3) to reduce development time, (4) to reduce costs associated with incorporating the technology into the product. Fuzzy technology can satisfy these requirements for the following reasons.

Nonlinear characteristics are realized in fuzzy logic by partitioning the rule space, bu weighting the rules, and by the nonlinear membership function. Rule-based systems compute their output by combining results from different parts of the partition, each part being governed by separate rules. In fuzzy reasoning, the boundaries of these parts overlap, and the local results are combined by weighting them appropriately. That is why the output in a fuzzy system is a smooth, nonlinear function.

In decision-making systems, the target of modeling is not a control surface but the person whose decision-making is to be emulated. This kind of modeling is outside the realm of conventional control theory. Fuzzy reasoning can tackle this easily since it can handle qualitative knowledge (e.g. linguistic terms like “big” and “fast”, and rules of thumb) directly. In most applications to consumer products, fuzzy systems do not directly control the actuators, but determine the parameters to be used for control. For example, they may determine washing time in washing machines, or if it is the hand or the image that is shaking in a camcorder, or they compute which object is supposed to be the focus in an auto-focus system, or they determine the contrast optimal for watching television.
A fuzzy system encodes knowledge at two levels: knowledge which incorporates fuzzy heuristics, and the knowledge that defines the terms being used in the former level. Due to this separation of meaning, it is possible to directly encode linguistic rules and heuristics. This reduces the development time, since the expert’s knowledge can be directly built in.

Although the developed fuzzy system may have complex input-output characteristics, as long as the mapping is static during the operation of the device, the mapping can be discretized and implemented as a memory lookup on simple hardware. This further reduces the costs involved in incorporating the knowledge into the device.

## 2 Fuzzy Logic Based Control Background

### 2.1 Mamdani-Type Controllers

E.H. Mamdani is famous in the circle of fuzzy for the works on fuzzy logic he made in the 70s and that are still topical now. He has extended the application field of fuzzy logic theory to technical systems whereas most scientists thought that these applications were restricted to non-technical fields (such as human sciences, trade, jurisprudence, etc.). At first, he suggested that a control that can be done by an operator could be done as well by fuzzy logic after having translated the operator experience into qualitative linguistic terms ([Mam73]-[Mam77]). Mamdani's method gave then rise to many engineering applications, especially for industrial fuzzy control and command systems.

Mamdani introduced the fuzzification/inference/defuzzification scheme and used an inference strategy that is generally mentioned as the max-min method. This inference type is a way of linking input linguistic variables to output ones in accordance with the generalized modus ponens, using only the MIN and MAX functions (as T-norm and S-norm (or T-conorm) respectively). It allows to achieve approximate reasoning (or interpolative inference).

Let consider a set of inference rules in the guise of a fuzzy associative memory (FAM) represented in an inference matrix or table, and some fuzzy sets and respective membership functions that have been attributed to each variables (fuzzification). Figure 2.1 represents the case where 2 rules are activated involving 2 input variables (x & y) and one output variable (r). Let assume x be a measure of x(t) at time t, and y a measure of y(t) at the same time. Let now consider the fuzzy sets A1, A2, B0 and B1 which respective membership functions µA1(x), µA2(x), µB0(y) and µB1(y) take positive values for x and y. Affected inference rules are for example as follow:

```        if      (x = A1   AND    y = B0)        then    r = C0
else    if      (x = A2   AND    y = B1)        then    r = C1```
They also can be expressed as follow:
A statement like x=A1 is true at a degree µA1(x) and a rule is activated when the combination of all membership grades (or truth degrees) µi(k) in its condition part (premise) take a strictly positive value. Several rules may be activated simultaneously. The max-min method realises the AND operators of the different rule conditions by taking the respective maximum membership functions. The premises can also include some OR operators realised by taking the minimum of the membership functions but it is rarely the case in control systems. The implications (connective then) are realised by the truncation (or clipping) of the output sets. That consists in taking for each point the minimum value between the membership grades resulting from rules conditions (fig. 2.1: µA2 (x) and µB0 (y)) and the membership functions of the respective output fuzzy sets (fig. 2.1: µC1 (r) and µC0 (r)):
The rules are finally combined by using the connective else, acting then as the OR operator and interpreted as the maximum operation for each possible value of the output variable (r on fig. 2.1) according to the defined fuzzy sets (n sets):
It is then possible with the above defined operations to give an algorithm for fuzzy reasoning (in order to achieve a control action for example).
[Click to enlarge image]Figure-2.1: max-min method for 2 rules involving 2 input and 1 output variables

The max-min method finally requires a defuzzification stage which is generally performed using the centre of gravity method. It will give for the above example the real value r, using the membership function resulting from the max-min method:

### 2.2 Characteristics

There are several mathematical properties that make the MIN and MAX operators well appropriated (this is simple and efficient) for fuzzy inferences as notably described in [Dub80] and [God94]. There are however several other methods based on different ways to realise the OR & AND operators, in purpose to improve some mathematical properties or numerical implementation characteristics.

The max-prod method for example is similar to the max-min method, except that all implications in the rules (then operations) are realised by a product instead of a minimisation. The truth values of the rule conditions are used to multiply uniformly the corresponding output sets instead of clipping them at a certain level. This allow to keep the information that consist in the shape of these sets, and that is partly lost with the max-min method. The product is moreover simpler and faster to execute than the minimum operator in software implementations and allows some simplifications in numerical realizations of inferences (since it can deal with analytic expressions instead of comparing each couple of stored points).

Another common method is the sum-prod method that uses the arithmetical mean and the product to realise respectively all the OR & AND operators. Unlike the MAX operator which select only the maximums values, the sum takes into account all involved sets and conserves part of the information that contain their shapes.

These different methods are described and compared in [Büh94]. From this analysis emerges the fact that they lead to very similar input/output characteristics in the case of a single input variable, when used with triangular or trapezoid output sets. With several input variables, the max-min method produces non-linear characteristics with strong discontinuities, while the sum-prod method produces non-linear characteristics with smoother discontinuities. Nevertheless the choice of a specific method is mainly influenced by the way to implement it. This suggests the choice of the max-min method for hardware implementations, because the MAX and MIN operators are then the easiest to implement. These two operators are moreover the most robust to realize T- and S-norm according to the authors of [Ngu93]. Since they are the most reliable when membership grades have imprecise and noisy values they are well appropriate to fuzzy hardware with questionable accuracy. The max-min method is finally well suitable for fuzzy rule-based systems when there is no precise model. It generally leads in a simple way to consistent rules as it can be noticed in practical applications.

The choice of an inference method has nevertheless a great importance for one-rule inferences, to select for example one among several candidates or to choose an optimal solution (this case is frequent especially in non-technical fields). The operators are then to be chosen with care because they influence directly the criterion of evaluation and consequently the final decision.

Finally, one important aspect of Mamdani's method is that it is essentially heuristic and it can sometimes be very difficult to express an operator's or engineer's knowledge in terms of fuzzy sets and implications. Moreover such a knowledge is often incomplete and episodic rather than systematic. There is in fact no specific methodology for clearly deriving and analysing a rule base of a fuzzy inference system (there is consequently no exhaustive choice of optimal rule and set numbers, shapes, operators,...). Since their principle seems to be rather simple, fuzzy system includes a lot of parameters and can lead then to a great deal of different and complicated characteristics. Some problems may occur when the rules have to describe a process that is too complex or to deal with a high number of variables. It can be then very difficult to define a sufficient set of coherent rules, and the danger of having not enough or conflicting rules occurs. There are several methods used to optimise inference rules and fuzzy sets, that are more or less perfected and sometimes quite complicated (typical ones are gradient method, least squares method, simulated annealing, neural-networks, etc.). They give rise to adaptive fuzzy systems which parameters are suited to the conditions of a specific application. The high parametrical level of fuzzy systems makes automatic adaptation solutions rather difficult.

### 2.3 Application field

Mamdani's method is currently and effectively applied to process control, robotics and other expert systems. It is especially well appropriated to execute an operator's control or command action. It leads to good results that are often close to the operator's ones while dismissing the risk of human error. Thus it has been used successfully in the control of several plants, such as those in the chemical, cement or steel industries.

Mamdani-type control is simpler than most of standard ones and requires much shorter development cycle when linguistic rules can be easily expressed (because there is no need to develop, analyse and implement a mathematical model). It is even in several cases as or more efficient, especially when no precise model exists, for example when a process to control is governed by non-linear laws or includes bad-known parameters or disturbances. When the mathematical model contains some non-linear terms, they are linearized and simplified under the assumption of small error signals, whereas a non-linear fuzzy method often allows to control bigger ranges of error.

The non-linearity of Mamdani-type control can moreover have a favourable influence on transitory phenomenon. Consequently it can sometimes supplant classic control to provide fast responses. When increasing the response speed of conventional controllers the overshoot is also increasing (in position control for example). Generally fuzzy controllers with highly non-linear characteristics gives lower overshoot before setting time than conventional PID control, but small oscillations often remain after settlement. This oscillatory behaviour can become very difficult to restrain and fuzzy controllers are sometimes combined with conventional ones (in cascade configuration for example) to exploit the advantages of both. However fuzzy controllers can also have perfectly linear characteristics and replace standard comtrollers, which can sometimes be useful to provide some features of fuzzy controllers.

Fuzzy control is attractive in some cases where parameters are varying with time and easily expressed into linguistic terms. It is indeed easier to reassign one or several fuzzy rules rather than to calculate the mathematical equations of a new model necessary to adapt classic controllers. However these last ones have a smaller number of parameters to adjust, and these parameters can for example be issued from a fuzzy system which regulates them according to the features fluctuations of a controlled process.

### 2.4 Defuzzification Strategy

The common centre of gravity defuzzification method requires a quantity of calculation that is prohibitive for many real-time applications with software implementations. Its calculation can however be simplified when associated with the sum-prod method [Büh94]. The computation of the centre of gravity can take advantage of the high speed afforded by VLSI when integrated on an IC, which is however quite complex. Simplest defuzzification types are often used, as for example the mean of maximum or height method [Dri93],[Büh94],[God94].

### 2.5 Takagi-Sugeno's method

A Sugeno-type controller differs from a Mamdani-type controller in the nature of the inference rules that are generally called Takagi-Sugeno's rules in this case. Whereas Mamdani's method uses some inference rules that only deal with linguistic variables, the rules of Takagi- Sugeno directly lead to real values that are not membership functions but deterministic crisp values [Tak83],[Tak85]. This method only use fuzzy sets for the input variables, and there is no need of any defuzzification stage. Whereas the antecedents still consist in logical functions of linguistic variables, the output values are resulting from standard functions of the input variables (real crisp functions). In most cases, only linear functions are used. Some variables can appear in such a function and not in the corresponding condition or vice versa. Each function belongs to the consequence part of a rule and is considered when the respective condition has a positive truth degree. This degree results from the different membership factors of the input fuzzy sets that the rule condition deal with. The final output value is calculated as the weighted average of all the linear functions and the weights are the truth degrees of their respective rule conditions.
[Click to enlarge image]Figure-2.2: Takagi-Sugeno's method for 2 rules involving 2 input and 1 output variables

Control rules are deriving in the following way:

Let the antecedent part results from a specific fuzzification achieved according to human operator's or engineer's knowledge. That means some fuzzy sets and some logical functions making up premises are defined. The input space has been thus divided into a certain number of fuzzy subsets, i.e. a certain number of fuzzy rules. Then the setting-up of the consequence part of the rules requires a certain amount of numerical data (corresponding to input and output variables) coming from the operation which has to be modelled (fig. 2.3). The coefficients of each linear function are identified from the analysis of these data in order to minimized the difference between outputs of an original system and the ones of a model (fuzzy engine). This optimisation is achieved by the analysis of a weighted linear regression using a certain amount of input data (the weights are calculated as the truth degrees of the fuzzy conditions).

Finally the system identification is function of the consequence parameters (coefficients in the linear functions) as well as the premise parameters (input sets) and the fuzzy partition of the input space. The identification of this last has no solution since it is a combinatorial problem and is generally issued from an heuristic search. An iterative algorithm for parameters and structure identification can be used efficiently (fig. 2.4) [Sug85],[Tak85]. The resulting fuzzy model gives generally much better results than a statistical model [Tak83].

[Click to enlarge image]Figure-2.3: Takagi-Sugeno's method: identification of a fuzzy logic inference engine
[Click to enlarge image]Figure-2.4: Takagi-Sugeno's method: outline of an identification algorithm [Sug85],[Tak85]

### 2.6 Characteristics

It is not possible to derive efficient control rules by Mamdani's method when it is too difficult to translate exactly an human operator's or engineer's knowledge into linguistic terms. The adaptive Takagi-Sugeno's method is however efficient in such cases, provided that inference rules can be derived from the analysis of appropriate numerical data. One more advantage of this method occurs in some complicated cases where many different variables play a part in a process. It leads then more certainly to consistent rules than Mamdani's method.
When fuzzy reasoning is used to describe some processes, methods as Mamdani's one are not so powerful partly because of their nonlinearity. The use of linear relations in the consequences enable to easily deal with Takagi-Sugeno'rules as an efficient mathematical tool for fuzzy modelling. It is however not possible to cascade inference rules without performing again a fuzzification step of crisp output values.

Takagi-Sugeno'rules implementation is simplified because there is no need of a defuzzification stage. Output values are simply calculated from input ones, using a set of weighted linear functions. One needs however to be able to collect a certain amount of data (sometimes during a rather long period for time-varying processes) and then to find good coefficients with the least squares method. Measurements and data analysis often require a great deal of time, and it is furthermore not always possible to find satisfying coefficients. Takagi-Sugeno' method becomes actually difficult or even impossible to apply when a control has strongly non-linear characteristics (for example when there is a large hysteresis loop) or which is moreover changing with time.

Some method as the simulated annealing for example could be used to reduce the iterative sequence of tests achieving the coefficients and structure identification. The approximation of measured characteristic or of a calculated linear regression could also be efficiently achieved by using a neural network. The structure identification would then result from a supervised learning phase instead of a long and unwieldy iterative research.

### 2.7 Application field

The rules of Takagi-Sugeno are mainly used to realise process modelling. Fuzzy controllers are realised by regarding an operator's control action as a process to model, and thus by collecting a lot of data when the operator is proceeding. A typical application of Takagi- Sugeno'method is described in [Sug85], pp.125-138. Sugeno has experienced this method with good results in the case of an automated car parking into a garage. Fuzzy control rules are then derived from observation and modelling of a man's parking action. The method is well appropriated to this situation which is easier to observe than to express into linguistic terms. The arms and legs actions are indeed easier to measure than to verbalise, just like most gestures which have been acquired by heuristic experiences or which are instinctive, rather than issued from precise reasoning.

### 2.8 Sugeno-Type controllers

Sugeno's method is a special case of Takagi-Sugeno'one and its particularity lies in the fact that all the output membership functions are precise values. The consequence functions are then replaced by constants which are weighted by the condition truth degrees to give the final output values. The fuzzy sets forming these precise values are generally called singletons or sets of Sugeno (fig. 2.5). It can also be regarded as a particular case of Mamdani'method where straight and without thickness non-overlapping output sets are used.
[Click to enlarge image]Figure-2.5: Example of Singletons (Sets of Sugeno)

### 2.9 Characteristics

The main advantage of using singletons is the great simplicity of implementing the consequence part. It can be used with Mamdani'method to simplify considerably the defuzzification stage, whose task is reduced to the calculation of a weighted average with a restricted set of crisp values. An other defuzzification type called maxima method is sometimes used (cf.[Mam74bis],[Dri93]). In this case each output crisp set corresponds to a specific action, and only the one with the maximum weight is selected (or an action midway between two maximum values).

The use of singletons has no bad consequence on the output variable domain which can be the same as with triangular or trapezoid output sets when using the centre of gravity method. The nonlinearity of a controller characteristic can be modulated by the distribution of these sets (but not more by their shapes). The difficulty to describe some processes can however be then increased in comparison with the use of more complex sets in Mamdani'method or the use of linear functions in Takagi-Sugeno'method. For the latter, the restriction to constant functions in the consequence part doesn't allow any more to describe complicated relationships between input and output variables in a rather simple format. This method leads then sometimes to a worse minimisation of the output error between a real process and a fuzzy model, and structure identification (fuzzy partition of the input space) has to be much more improved to obtain satisfactory results in complicated cases.

## 3 Integrated Implementations of Fuzzy Logic Circuits

### 3.1 Fuzzy dedicated circuits

The recent advances in fuzzy logic theory has brought about rule-based algorithms to a growing field of applications. Several implementation approaches have been proposed during the last ten years, especially in Japan where fuzzy systems have known a real proliferation. Fuzzy control has proved it could lead to good performance with short design times in a wide variety of low complexity real world applications. Many specific application programs have been developed to implement fuzzy operations on standard digital computers, and most of processor manufacturers provide software environments to develop and simulate fuzzy applications on their available microcontrollers.

The design of fuzzy dedicated integrated circuits is however of great interest, because of the increasing number of fuzzy applications requiring highly parallel and high-speed fuzzy processing. They attempt to give a concrete expression to the idea of "fuzzy computers" (sometimes called computers of the sixth generation), which deal with analog values or some digital representation of them. Fuzzy processors are designed in a way to optimise fuzzy logic functions (cf.§3.2) as regards their implementation size and execution speed. Since practical systems often require a great number of rule evaluations per second, a huge amount of real- time data processing is necessary and the speed of fuzzy circuits is of prime importance. Their architectures are generally suited to the structure of approximate reasoning and decision- making algorithms and generally include three distinct parts, proceeding fuzzification, inference and defuzzification respectively. Fuzzy controllers are thus made on one hand of a knowledge base, which contains rules and membership functions outlines, as other configuration parameters of the system. On the other hand stands an inference mechanism (based on interpolative reasoning) interfacing a real process in a feedback loop configuration.

[Click to enlarge image]Figure-3.1: General structure of fuzzy logic dedicated processors for approximate reasoning

The structure of fuzzy dedicated circuits is characterized by the number and shape range of input and output variables, the number of rules they can evaluate simultaneously, the type(s) of inference(s) (size of premises, operators, consequences,...), the type(s) of defuzzification method(s), and so on. Their performance is evaluated according to their processing speed (that is the number of fuzzy logic inferences per second (FLIPS)), as to their precision (error and noise generation in analog circuits and number of bit representing fuzzy values in digital ones). Fast response is required for non-linear functions as MIN and MAX which output signals can be subject to sudden discontinuities. These functions are however piecewise linear and accuracy is also quite important.

Fuzzy chips are mainly used in expert systems as well as in control and command field to achieve real-time performance. They are however less efficient for applications that deal with a large amount of data, because of the limited I/O. It can thus be interesting to design several compatible circuits, identical or different. They can be for example dedicated to inference rules and defuzzification respectively. In this manner it is possible to connect them together in several different ways, using some of them for example to proceed parallel fuzzifications and inferences, and connecting them to a defuzzification circuit [Hir93]. With a large number of input variables it may be useful to split up the control system into several cascaded or superposed units, which aims at simplifying the inference engine [Büh94].

Fuzzy software is useful when an application can be modelled to simulate and calculate in advance a multidimensional response characteristic. They give the different parameters relative to an optimal characteristic in order to design or program a fuzzy dedicated circuit. Some of these parameters (fuzzy sets and inference rules) have just to be adjusted then in the real implementation. In some simple cases, the numeric response characteristic can be stored as a reference rule table in a multidimensional associative memory, and provides then response values for real-time proceeding without any more inference calculations. The stored values are provided by fuzzy software (when a model exist), by the measure of an operator's action or by an adaptive system with a retroactive learning scheme (which comes close to the principle of fuzzy dedicated neural networks)[Chan93].

When no model exists, a fuzzy processor has to be programmable because its parameters are established by iterative testing on the real implementation, not by simulation. This process requires a lot of tests and is time consuming, and may furthermore not necessarily reach an optimal solution. Since no simulation of the system's dynamic behaviour are achieved, the danger of unstable working states occurs in control applications. The configuration flexibility of integrated fuzzy system requires a great storage capacity (digital circuits) or tunable components (analog circuits) to deal with variable numbers and shapes of membership functions as variable numbers and sizes of rules. Since the shape of the membership functions is generally less important than their degree of overlap the horizontal position of sets should always be tunable, but often complicated shapes are not essential.

[Click to enlarge image]Figure-3.2: Fuzzy logic circuits

### 3.2 Basic fuzzy logic functions and their relations

The fuzzy operators presented below have been introduced to deal with fuzzy values and are relative to specific membership grades µ(x). A fuzzy operator  that deals with tho fuzzy sets A and B defines to a new membership function . The real variable (x) has been omitted in the expressions below.

Conventional operators as algebraic sum (+), algebraic difference (-), algebraic product (.) are commonly used in fuzzy reasoning systems. The first nine fuzzy operators are the more commonly used in hardware system design to implement fuzzy information processing. They have moreover the particularity that they all can be expressed by the only bounded difference () and algebraic sum (+) functions [Yam86],[Zhij90],[Sas90]. There has been several approaches using these two functions to design fuzzy units, which provides attractive perspectives for CAD automation and semicustom circuits. Other operators also allows to express any fuzzy formulae when combined together, as for example the bounded sum ( ) and the bounded product ( ) [Lem94]. These ones are associative and commutative, but are not distributive to each other. The non-distributiveness of bounded operators leads to long and complicated manipulations when used to solve fuzzy formulae, and it is quite not obvious to substite or eliminate some terms. Fuzzy formulae are unfortunately not able to be reduced as much and as simply than boolean formulae.

Some other fuzzy operators as symmetric difference (), drastic difference (), drastic sum ( ) or drastic product () are sometimes to resolve some fuzzy equations.

## 4 Digital Implementations of Fuzzy Logic Circuits

### 4.1 Digital approach

The VLSI digital implementation of Fuzzy Logic systems offers several advantages issued from the sound knowledge of digital circuit design and technology. Several mature CAD tools allow relatively easy design automation (synthesis & simulation) reducing consequently time and cost of development. The automatic regeneration of logic levels involves high noise immunity and low sensitivity to the variances of transistor characteristics. This provides accurate and reliable data and signal processing. Binary data can be easily stored and allows to realize programmable and multistage fuzzy processing. Complex representation of fuzzy vectors and parallel structures are however required to obtain accurate and fast processing. Digital implementations of common fuzzy operations leads unfortunately rapidly to complicated, enormous VLSI circuits. The density and speed of these circuits are nevertheless continually increasing according to technological advances, so that they will become more and more efficient to implement fuzzy logic systems.

### 4.2 Digital fuzzy circuits features

Digital fuzzy processors are generally designed for multipurpose applications in order to interest a maximum of potential customers. They should thus implement a great and various number of fuzzy operators, membership functions and inference rules. This make them rather efficient for a large range of applications, provided that appropriate programming is possible (which supposes an appropriate internal or external memory). Combined with an appropriate object-oriented programming environment, linguistic rules derived from an human expert can be directly translated into an implementation on a chip.

The first hardware approaches of implementing fuzzy logic inference engine were the digital circuits designed by Togai and Watanabe [Tog86],[Tog87],[Wat90]. ASIC's implementing specific architectures and specialized instructions (MIN & MAX) exhibits much enhanced processing speed relatively to regular microprocessors.

operators: Min, Max implementation

architectures: Fuzzy microcontroller (embedded systems with A/D & D/A converters, Fuzzy arithmetic logic unit (FALU) and specific memories, ...) versus fuzzy coprocessor; coprocessor. Sequential processing -> enhance speed by parallel, pipeline [Nak93], systolic, RISC, SIMD architectures

fuzzification strategy, defuzzification strategy

For applications where a restricted number of pins of a chip can be inappropriate (such as fuzzy databases (= ambiguous data or fuzzy relations between crisp data)), a fuzzy coprocessor is used rather than a stand-alone chip. When other conventionnal digital processing are required (flexible system)

### 4.3 Digital Fuzzy Processing

Analog fuzzy values should be converted into strictly binary signals before being processed by standard digital circuits. On one hand the analog input signals should be quantized through A/D converters, and on the other hand the membership functions should be quantized to obtain their digital representations. Fuzzy sets are then storable but in the guise of stair-functions. The combination of these two round-off effect can deteriorate fuzzy processing if the fuzzy values are not represented with a sufficient number of bit. There is however a trade-off between precision and size (or speed) since the latter is proportional to the number of bit.

The input are furthermore sampled at the frequency of digital circuits. Sampling effects: non-continuous (or pseudo-continuous) control [Büh94] § 7.3

### 4.4 Different approaches of digital IC's

Togai's FC110
fuzzy accelerator, coprocessor
Watanabe
Risc approach
• THOMSON's WARP (Weight Associative Rule Processor)
• Cell-to-cell mapping approach, automatic synthesis [Pag93]
SIEMENS's coprocessor (SAE81C99)
optimized memory organisation [Eich92]
Sasaki
SIMD and Logic-in-Memory structure [Sas93]
OMRON's FP-3000, FP-1000 & FP-5000

NeuraLogix' controllers

INFORM's FUZZY-166 Processor

### 4.6 Application fields and future trends

Control, expert systems, robots, image recognition, diagnosis, database (interface with digital circuits), information retrieval, ...

High prices must be reduced to make these processors more attractive.

Some less common hardware implementations such as pulse-width-modulation [Ung93] or sperconducting processor have also been implemented, and are not exclude from future developments according to technological evolution.

## 5 Analog Implementations of Fuzzy Logic Circuits

### 5.1 Analog fuzzy circuits features

Analog circuits present several advantages towards digital ones, especially regarding speed of processing, power dissipation and functional density. They can moreover perform continuous-time processing and have the particularity to be well compatible with sensors, actuators and all other analog signals. Therefore they are obviously indicated to deal with fuzzy values which are analog by nature. Some continuous representations of symbolic membership functions and some non-linear fuzzy operations can be easily synthesised by dealing with transistor characteristics. There is no need of A/D or D/A converters when implemented in a real system, provided that no specific digital signal processing is required. Analog circuits can then supplant digital controllers for some applications requiring low-cost, low-consumption, compact and high-speed stand-alone chips. They suffer nevertheless of the lack of reliable memory cells. They are consequently not well appropriate to pipeline structures and have very restricted programmability possibilities. Fortunately, the nature of fuzzy variable systems requires extensive parallelism which make analog circuits well appropriate to proceed high- speed numerous inferences and also limits the problem of error accumulation.

Analog controllers can achieve fuzzy real-time reasoning with a large amount of fuzzy implications, especially when no high-level accuracy is required. They are well suited to deal with vague and imprecise models, like for example some tasks that interface human senses (eye, tactile nerves, ear,...) or replace human reasoning (pattern recognition, reflex processes,...). Accuracy of fuzzy systems is actually not always so important since there is no thorough mathematical background usable to define precise and exhaustive fuzzy methods. Imprecise but adjustable analog devices are consequently suitable for a great number of cases (tunable membership functions are thus needed to optimize the performance). They are nevertheless much less flexible and adaptable than digital ones that are programmable, and they must be designed according to the structure of a specific application. A basic programmability is afforded when some analog external parameters can be adjusted or when some binary inputs allow the control of internal switches.

The choice of common fuzzy operators (§3.2) is not exhaustive, and other methods could just as well be implemented since there is no mathematical proof than some of them are really optimal. Thus some other operators can be chosen instead of the common MIN and MAX ones, in order to be efficiently performed with analog circuits. Some dense, fast and accurate hardware operators can give very good practical results, even if they are theoretically not optimal. A novel way of evaluating the condition part of fuzzy rules has thus been introduced in [Land93]. The degree of membership of an input vector to some fuzzy subspace is defined using a measure of distance between this vector and some central point in the subspace. MIN and MAX operators are not implemented any more and very dense and high-speed analog hardware may be realised (in current-mode).

Analog signals are represented either by voltages (voltage-mode, §5.2) or by currents (current-mode, §5.4). Stable and low-noise analog technologies (n-well CMOS, BiCMOS) must be used in order to design analog circuits having sufficient accuracy with wide frequency range. Reliable CAD tools for automatic design as fast verification and simulation tools are required for effective design.

### 5.2 Voltage-mode

#### 5.2.1 Yamakawa's approach

Voltage-mode is attractive because it makes easy to distribute a signal in various parts of a circuit. Non-linear operators such as the MIN, MAX and truncation ones are quite easy to implement in voltage-mode. Multiple-input MIN & MAX circuits constructed with bipolar transistors are represented in figure 5.1 and 5.2 and are called emitter coupled fuzzy logic gates. These basic non-linear gates present good characteristics and robustness [Yam93]. Such circuits are impractical with MOS transistors which cause anacceptable error associated with the transition regions in which multiple devices are active. CMOS multiple-input MIN & MAX circuits using gain-enhanced voltage followers based on differencial amplifiers are presented in [Fatt93] & [Tou94]. They are more complicated but have high frequency and accurate performance according to the authors.
[Click to enlarge image]Figure-5.1: MIN cell, voltage-mode [Yam88]

[Click to enlarge image]Figure-5.2: MAX cell,,voltage-mode [Yam88]

Yamakawa has designed a BiCMOS monolithic rule chip (TG004MC[Yam88], TG005MC[Hir93]) which architecture is shown on figure 5.3. It is constructed with about 600 transistors and 800 resistors. The response time of fuzzy inference is about 1µs (1 Mega- FLIPS). This chip implement one fuzzy inference including three variables in the antecedent and one variable in the consequent. The antecedent part is made of membership function circuits (fig. 5.4) providing for each variable seven possible fuzzy linguistic values (one of which is selected according to a voltage label VLAB). These values can have four possible shapes  assigned by an external binary signal, and the slopes can be changed by external resistors. A special label "not assigned" corresponding to a constant high level (+5V) can stand for constant membership functions which value is 1.

A truncation circuit is not more than a certain number n of 2-inputs MIN circuits connected to a n-inputs MAX circuit. A voltage level  as truncation factor is applied to one input of each MIN circuit, and membership grades of output fuzzy sets are applied to the second inputs. The output fuzzy sets are sampled in the consequent part by a membership function generator (MFG) in order to form fuzzy words of 25 elements which can be processed by the truncation circuit. These words are realised by a switch array controlled by a switch setting circuit [Gup88],[Yam93], and represent the consequent membership functions as analog voltages on 25 lines.

[Click to enlarge image]Figure-5.3: Architecture of the TG005MC rule chip [Yam93]

[Click to enlarge image]Figure-5.4: Principle of a basic membership function circuit [Hir85]

Yamakawa also designed an analog defuzzifier chip (TB005PL[Yam88], TB010PL[Hir93]) which implements the centre of gravity calculation for 25 values (fig. 5.5). Its architecture consist of an ordinary addition circuit in parallel with a weighted addition circuit, and followed by an analog divider. It is constructed with resistor arrays, OP amplifiers, an analog divider and capacitors in hybrid structure. The response time of defuzzification is about 5µs and is almost determined by that of the divider. The sum calculation is proceeded by a simple network of 25 identical resistors connected to a same node, which produce a current proportional to the desired sum. The weighted addition is proceeded in the same way, but with 25 different resistors: 0,R, R/2, R/3,..., R/24. For an emitter junction of a bipolar transistor, the base-to- emitter voltage is proportional to the logarithm of the emitter current (over the range where the effects of series and shunt resistors and reverse saturation current are negligible). Thus the division of the two currents (proportional to the sum and the weighted sum respectively) is implemented by the subtraction of two base-to-emitter voltages. Finally, the divider is followed by a level shifter with current-voltage conversion [Yam88bis].

Several rule chips can be connected to such a defuzzifier chip which calculate a deterministic value from the maximal membership function. Several defuzzifier chips can also be used to realise systems involving several conditions and several conclusions. The main weak point of such systems is that they generally lead to non-optimal cumbersome implementations.

OMRON has developed the above analog fuzzy chips, as an analog rudimentary fuzzy computer based on these two circuits: the FZ-5000. This last one consists of a multi-boards system, which each board includes four inference chips and one defuzzifier chip. Inference time is about 15µs, including defuzzification. Programming is done using switches and jumper pins, or by a specific software tool (FT-6000) using a personal computer.

[Click to enlarge image]Figure-5.5: Architecture of the TB005PL defuzzifier chip [Yam88bis]

Voltage-mode fuzzy circuit implies a large stored energy into the node parasitic capacitances (CV2/2) and speed is limited by charge delays of various capacitors. They are moreover penalized by a certain lack of precision because signals are sensitive to changes of supply voltages. This is especially significant when the voltage range is restricted in order to limit transistor functioning to a small parts of their characteristic, or when the electrical consumption should be limited. The problems mainly lie in the sizing of some components.

Several functions are very difficult to build in voltage-mode, and it is also true for some basic ones as the algebraic sum. The above described approach needs resistors to achieve additions and to convert voltages into currents. Integrated resistors are unfortunately inaccurate, cumbersome and involve significant parasitic capacitances. The truncation of the consequent and the defuzzification pose an important problem as regards the parallelism of the inference engine (especially when the number and size of output sets is large). This approach implies high-power dissipation and large chip area, and leads to high-costs implementations.

### 5.3 Mixed-mode

#### 5.3.1 Transconductance

An hybrid mode can be realized by implementing on a same chip both current-mode and voltage-mode operations. This allows to avoid some problems inherent to voltage-mode while taking advantage of some benefits of current-mode. For example, the sum of analog voltages is rather complex to realize, while it corresponds to simple wire connections in current-mode. This way to do has been used in the defuzzifier chip in §5.2.1. The difficulty is to obtain linear and accurate conversions to swap between voltage - and current - modes without loosing too much precision. Efficient transconductance elements (fig.5.6) should exhibit at the same time good linearity and good frequency response to deal with fast and non-linearly varying membership values. They should moreover occupy few place and consume few energy.
[Click to enlarge image]Figure-5.6: A linear tunable transconductance [Park86]

#### 5.3.2 OTA-based approach

Operational transconductance amplifiers (OTA) can be used as basic building blocks to design analog CMOS circuits for fuzzy logic. The use of OTAs, diodes and OTA-simulated resistors is sufficient to realize every useful fuzzy circuits. Fig.5.7 represents an OTA of the same type that the ones used in [Inou91]. The fuzzy grade interval [0,1] is represented by [0V,1V] in order to assure a linear operation of the OTAs. More effective OTA based on the transconductance element of fig.5.6 are also presented in [Park86]. It is well linear and efficient in a wide range of frequencies, but at the expense of bigger consumption and chip area.
[Click to enlarge image]Figure-5.7: Operational Transconductance Amplifier (OTA)

A high-speed bounded difference operation can easily be implemented by use of OTAs. For the circuit represented on fig.5.8, the relationship between the input voltages (VIN1 & VIN2) and the current of the diode satisfies the definition of the bounded difference. As the first OTA supplies a current proportional to the difference of input voltages, the bounded difference is realized for µ1 greater or equal to  µ2 when input voltages represent the membership functions µ1 and µ2. It is also realized for µ1 < µ2 since the diode can only pass the unidirectional output current of the OTA. The second OTA finally converts this current into a proportional output voltage (VOUT), in order to obtain a voltage-mode bounded difference characteristic. Its output current is indeed identical to the diode one, and is also proportional to the output voltage which is connected back to an input of the second OTA.

[Click to enlarge image]Figure-5.8: OTA-based bounded difference operator

This OTA-based bounded difference operator can be effectively used to synthesise all other fuzzy function according to the relationships described in §3.2. The algebraic sum used in these relationships is realised by simple wire connections before finally converting currents into voltages with OTAs. The fuzzy complement operation is realized when the positive input voltage (VIN1) is connected to the high-level (µ1 = 1) in the circuit of fig.5.8 (VOUT is then the complement of VIN2). Two-input MAX and MIN operators synthesis are represented on fig.9. Since µ1 = (µ1 µ2) in any case, MIN(µ12) = µ1 µ2) = µ1 - (µ1 µ2). The algebraic difference leads to a faster MIN operator synthesis than two cascaded bounded differences. A current equal to the negative value -(µ1 µ2) is obtained by permuting the two input voltages (VIN1 as µ1 & VIN2 as µ2) in a common bounded difference circuit operation which diode (D1) has been connected in reverse sense. The diode D2 is just useful if the output voltage has to be limited to positive values.

[Click to enlarge image]Figure-5.9: OTA-based MAX and MIN operators [Inou91]

Multiple-input MAX and MIN circuits are also described and analysed in [Inou91], where a fuzzy membership function circuit is also presented. They are made of two-stages OTA structures which provide high-speed signal processing. OTA-based realisations of other fuzzy functions are however often quite complicated and requires more OTAs stages, which radically deteriorates accuracy and speed of processing. Several OTA are actually necessary to synthesise some relationships of §3.2 since many voltage-current or current-voltage conversions are needed before performing bounded differences (which deals with voltages) and algebraic sums (which deals with currents). There is moreover quite difficult to proceed simplifications when these different functions are combined together and the principal optimisation consists of increasing parallelism of OTA stages. This OTA-based approach leads unfortunately rapidly to very big and not so accurate circuits for implementing fuzzy processing of complex inference systems.

#### 5.3.2 Current-mode multiple-input MIN and MAX circuits

Multiple-input MIN and MAX circuits can be realized with current mirrors composed of a standard MOS transistor and a lateral MOS transistor in bipolar operation [Tou94]. They are suitable for current-mode CMOS circuits and are based on the same principle than the voltage- mode circuits of fig. 5.1 & 5.2. Input currents are converted into gate-sources voltages which are applied to the base terminals of bipolar transistors connected as voltage followers. The voltages are processed according to MIN or MAX operation before being converted into output current. Precision depends on the symmetry of the structures based on MOS-bipolar mirrors followed by the inverse bipolar-MOS mirror (whose bipolar transistor also compensates the DC level shift of approximatively 0.7V).
[Click to enlarge image]Figure-5.10: Current-mode 3-input MIN circuit [Tou94]

### 5.4 Current-mode

#### 5.4.1 Current-mode circuits

Current-mode circuits do not need resistors and can achieve summation and subtraction in the most simple way, just by wire connections. This leads to simple and intuitive configurations, which exhibits high speed and great functional density. They are used more and more, especially for systems requiring a high level of interconnectivity (neural networks for example). High speed is provided when capacitive nodes are not subject to great voltage fluctuations. Current-mode circuits can also exhibit advantages as low power dissipation and low supply voltage, as good insensitivity to the fluctuation of the latter. Since they have a single fan-out, current repeatability is of prime importance and the distribution of signals requires multiple-output current mirrors.

#### 5.4.2 Current mirrors

A basic realisation of multiple-output CMOS current mirror is shown on fig.5.11. This circuit is however not suitable for synthesising accurate functions since each output current is slightly modulated by the output voltage throughout the Early conductance. The output current should be independent of the output voltage, which is obtained by reducing the output conductance as for the three common mirrors shown on fig.5.12. The drain voltage of the transistor which imposes the current (drain voltage of T2 on fig.5.12) is then independent of the output voltage of the circuit. Multiple-output cascade mirrors are often used but Wilson ones are preferable for low power applications because they require a single polarization voltage (VG(T1) on fig.5.12.a) instead of the two superposed voltages VG(T1) & VG(T3) on fig.5.12.b). The Mod-Wilson mirror is obtained by adding a transistor to the Wilson mirror (T4 on fig.5.12.c) to improve its symmetry. This mirror provides good accuracy and the input current is well reproduced with perfectly matched identical transistors. The precision of all these mirrors depends on their output resistance (which must be as high as possible) and on the matching of their transistors. The quality of current reproducibility is very important in cascade configurations to limit error accumulation. Dynamic mirrors can be used to obtain a greater accuracy, but at the expense of a clocking scheme which considerably increases their size and complexity.
[Click to enlarge image]Figure-5.11: Basic n-output CMOS current mirror and symbolic representation
[Click to enlarge image]Figure-5.12: NMOS current mirrors

All the above mentioned mirrors can be constructed in standard bipolar technology as well as in MOS one.

Multiple-output current mirrors can be realized by compact bipolar circuits when using multicollector transistors, whose accuracy is however poorer towards structures with single- collector transistors. Bipolar transistors produce two type of significant errors, due to the base current and to the reverse mode current. The latter causes the saturation of one collector to affect the other collectors, and the circuits should be designed so that no collector in the multiple-output current mirrors may be saturable. These errors do not appear in multiple-output MOS current mirrors since their input-output paths are separated and their drain currents are independent of each other. The design of cascade MOS structures is then much easier than the one of bipolar structures whose stages are interdependent. It is generally preferred, all the more since it is compatible with standard CMOS fabrication processes and efficient design tools.

The mismatch between two identical transistors is however smaller for bipolar than for MOS ones, since it does not depend on the collector current level and since they are affected by a much lesser influence of surface effects. All MOS transistors must work in saturation mode in order to reduce the mismatch effects. Bipolar current mirrors are then more appropriate to work with low voltages and are more precise and fast with low currents (the speed of MOS ones decreases at low currents because of their intrinsic capacitances).

#### 5.4.3 Fuzzy operator synthesis

Current mirrors can be used as building blocks to synthesise fuzzy logic operations and relevant processing.

In this way, nine basic fuzzy functions (cf. §3.2:  ) can be easily implemented on monolithic IC's with standard CMOS [Zhij90],[Yam86] or bipolar technologies [Yam87]. These current-mode basic logic cells exhibit good linearity which can not be easily achieved in voltage-mode, and lead to fuzzy integrated systems which are globally smaller than in voltage-mode.

[Click to enlarge image]Figure-5.13: CMOS and bipolar implementation of the bounded difference operation [Yam86],[Yam87]

A bounded difference circuit can be obtained by the combination of a current mirror and a diode (fig.5.13). The diode can easily be realized in the CMOS circuit either by a single FET which gate and drain are connected together (fig.5.13) or by a current mirror (fig.5.14). The first solution involves inevitable voltage drops due to the channel resistance and can influence the normal logic function of the circuits. Nevertheless the diode can be omitted in cascade connection of such circuits because the input current mirror of the following stage also achieves its task.

[Click to enlarge image]Figure-5.14: Two different implementations of the bounded difference operation [Zhij90]

Current mirrors are subdivided into two complementary types whether their transistors are n- or p-channel MOS FET (NPN or PNP in bipolar technology). The directions of input and output currents depend on the type of the respective components (input mirror and output diode in the bounded difference circuit). Thus there are four different configurations of input and output current directions (two of which are shown on fig.5.14 and are suitable for cascade connections). To each configuration corresponds a complementary one which is obtained by substituting p-channel current mirrors to n-channel ones and vice versa. This is convenient for designing circuits using such fuzzy logic units as basic bricks without worrying about specific current directions between neighbouring bricks.

The circuits of fig.5.13 and 5.14 realize the bounded difference operation on two values of membership functions µx & µy represented by the two currents I1 and I2 respectively. They also realize the complement operation on µy (represented by I2) when I1 has the maximum value (that represents a membership grade equal to 1). The bounded product is realized when I1 represents 1 and I2 represents the sum µx + µy. As the bounded difference and the algebraic sum are sufficient to realize all other fuzzy functions (according to the relations described in §3.2), fuzzy circuits can be designed only by specifying connections between bounded difference subcircuits. Multiple-output current mirrors are also required to distribute current to several logic units (fig.5.15). Such basic cells are attractive prospects for developing CAD tools and semicustom IC's (that are arrays of logic cells adaptable to various specifications according to the wire configurations). This leads however to solutions which are generally not optimal as regard to the number of transistors.

Multiple-input MAX and MIN circuits are proposed in [Sas90] which aim at avoiding error accumulation and increasing speed relatively to binary tree realizations based on two-input MIN and MAX subcircuits. The operation of these circuits can be formulated using simultaneous bounded difference equations. A simpler multiple-input MAX circuit is described in [Batu94].

[Click to enlarge image]Figure-5.15: CMOS multiple fan-out circuit [Yam86]

Three "primitive" operators have been introduced to obtain more elementary basic cells than the bounded difference [Lem93],[Lem93bis],[Lem94]. These operators are defined in the following way :

All fuzzy functions can be formulated as an additive combination of these primitive operators since the bounded difference can be expressed in the following way:

These operators are used to reduce the complexity of electrical realization of fuzzy functions, since they lead to simple relationship between transistor-level circuits and symbolic representation of fuzzy formulae. They are actually realisable in a most simple way by using current mirrors (fig.5.16) and exhibit special properties that help to obtain reduced forms of compositional expressions. Some of these properties are as following:
Several relations are rather complicated when expressed from bounded difference equations and should be reduced with the help of the above properties. This will at the same time decrease the number of cascade current mirror stages necessary to implement them. It is also important to favour parallel operations in rule evaluations so that error accumulation will be reduced and processing speed increased.
[Click to enlarge image]Figure-5.16: CMOS implementation of primitive operators [Lem94]

As an example of such a process, the MIN operation can be implemented with two cascade current mirrors (fig.5.17), after having carried out a certain number of simplifications into its mathematical formulation:

[Click to enlarge image]Figure-5.17: Synthesis of the MIN function based on primitive operators [Lem94]

There are various solutions to realize non-linear analog membership function generator in current mode but they are directly influenced by their physical features and often exhibit bad temperature behaviour. Consequently it is quite attractive to design circuitry achieving piecewise-linear membership functions [Kett93],[Yam88-3]. The representation of complicated membership functions can also be subdivided into an appropriate composition of elementary logic subcircuits. The fuzzification of a crisp variable x thought membership functions with triangular shapes can be expressed, for example, in terms of the primitive operators , P and N as follows [Lem94]:

The input value i determines the horizontal position of the triangular functions and designates in this manner one of linguistics labels (negative high, negative low, zero,...).

#### 5.4.4 Mixed-mode fuzzifier

As current-mode circuits are restricted to single fan-out, multiple current mirrors are required to share out signals among several operational blocks. Voltage-mode inputs are thus preferable for fuzzy hardware systems since they must be distributed to the membership function circuits of many rule blocks. Current-mode signals are appreciated afterwards, because of the advantages provided by current-mode processing. Tunable voltage-input current-mode membership function circuits are consequently useful building blocks to proceed fuzzification with current-mode analog hardware [Chen92],[Sas92],[Ishi92]. They can also be used with the above described OTA-based approach. Such circuits can be realized with an OTA including variable source resistors which consist of integrated voltage-controlled resistors and which can change the OTA transconductance characteristic [Chen92]. This achieves triangular membership functions with variable heights, widths, slopes and horizontal positions. Such membership function circuits are suitable for realizing high-speed and small-size systems. Nevertheless their sizing become difficult as their complexity increase and their characteristics are affected by the variations of physical parameters (mismatches, temperature influence,...).

#### 5.4.5 Defuzzification strategy and normalization circuit

When achieved with the centre-of-gravity method, the defuzzification consists of a weighted average requiring a sum, a multiplication (weighting) and a division. The sum operation consists of wire connections while the multiplication can be realized by scaling current by means of asymmetrical current mirrors. The analog division is still the most complex operation and requires rather great chip area and processing time. The defuzzification operation is consequently often simplified in hardware implementations.

However the required division can be easily and rapidly performed by current-mode normalization circuits [Ishi92]. For the circuit of figure 5.18 there are the following relations:

[Click to enlarge image]Figure-5.18: 3-input normalization circuit [Ishi92]

The normalization circuit can directly be extended to more than three inputs. The sum of all output currents is normalized to I0 and so each output current (representing a membership grade) is divided by this sum (sum of all membership grades). There is no more need of dividing the weighted sum of these currents. The weight of each current can be realized in the normalization circuit by the W/L ration of the respective output transistor. Good precision is obtained when such a circuit is implemented by means of bipolar transistors or bipolar operated MOS. When using saturated MOS transistors operating in weak inversion with VS=0 the circuit is very inaccurate and variable with the temperature, due to the large variation of VT0 from device to device.

### 5.5 Singleton consequences and normalization loop

A main disadvantage of analog circuits concerns the defuzzification stage (and especially the analog divider) in terms of size, accuracy and processing speed. The quantity and complexity of calculations are however greatly decreased with the use of singleton consequences. These ones are more appropriate for hardware implementations than complicated output sets that must consist of a limited number of discrete analog values. It can moreover be observed thae singleton consequences facilitate a linear interpolation between inference results of different control rules [Yam89],[Yam93].

Yamakawa has searched to eliminate the analog division in a voltage-mode singleton- consequent controller by using grade-controllable membership function circuits (which are obtained by modifying ordinary membership function circuits [Yam89],[Yam93]). These can be tuned in such a way that the sum of all membership grades is equal to the unity (or constant), and that the weighted sum doesn't need any more division. The regulation consists of shifting down and up each membership function characteristic according to the variations of the sum of membership grades, through a negative feedback loop.

The current-mode rule block with voltage interface (and OTA-based membership function circuits) described in [Sas92] also includes a normalization locked loop which abolishes the division operation of the weighted average in a similar way (fig.5.19). The sum of membership grades () is regulated by using its fluctuations as modulation factor (Vm) of the membership function circuits throughout a negative feedback loop. This regulation operation is attractive since it is faster than the analog division in fuzzy hardware implementation. Nevertheless the implementation of such circuits requires complicated units and connections, and leads to difficult sizing and testing. The normalization solution described in §5.4.4 is finally far simpler for current- and mixed-mode circuits.

[Click to enlarge image]Figure-5.19: Principle of a normalization locked loop [Sas92]

### 5.6 Fuzzy analog memories

The main weak point of analog circuits is the luck of reliable analog memory modules. Since there is no accurate and lasting way to store analog fuzzy values, analog fuzzy computers exhibit bad features concerning programmability and multistage sequential processing.

Temporary memory elements have however been proposed to keep signals stable within a sampling period. This allowss to design fuzzy inference engines with pipelined structures and consequently enhanced speed [Zhij90bis]. Such basic memory cells are however not suitable for implementing complicated sequential algebra in the same way that digital circuits do thanks to the use of binary flip-flop and register circuits. An analog value can however be stored in a more lasting way when represented by a voltage, by mean of a capacitor. This last component is the core of sample-and-hold circuits which are basic cells of anolog memory elements.

A voltage-mode fuzzy flip-flop has been proposed as an extension form of binary J-K flip- flop [Gup88]. It is based on fuzzy negation, T-norm and S-norm which are respectively restricted to complementation, MIN and MAX operations in order to make easier its hardware implementation. The characteristics of fuzzy flip-flop based on other operations such as algebraic product and sum, bounded product and sum, and drastic product and sum are also reported in [Hir89]. Its structure is described by the following set and reset equations which generate the same state values than a digital J-K flip-flop in the case of boolean input and state values.

The fuzzy flip-flop can be implemented with a combinatorial part synthesising the above equation, and two sample-and-hold circuits driven by two control pulse circuits with opposite phases (fig.5.20). Present output Q(t) is memorized by two sample-and-hold circuits since the information is needed in the next state.

[Click to enlarge image]Figure-5.20: Fuzzy J-K flip-flop: circuit block diagram [Gup88]

Current-mode continuous valued memory elements can also be realized in the same way by using voltage-controlled current sources in both sample-and-hold circuits to store a control voltage representing the sampled current [Balt93]. The capacitor of a sample-and-hold circuit is thus charged according to the control voltage of a first current source. In the "hold mode" (after half a clock cycle), this stored voltage sets the output current through a second current source in the image of the input one.

Storage capacitors are designed according to a trade-off between high speed and small silicon area on one hand, and sufficient accuracy on the other hand (their capacitance should be high relatively to parasitic ones). High accuracy is however difficult to achieve since it is affected by the imprecision of integrated capacitors, the mismatch in couples of current sources, and overall the charge injection in parasitic capacitances. This last source of sampling error is however reduced with the master-slave configuration of the sample-and-hold circuits, since the injected charges are opposite in sign (the first order sampling errors are thus cancelled).

## 6 Mixed Digital/Analog Implementations of Fuzzy Systems

Fuzzy logic systems lend themselves well to analog integration, except for some control and reconfiguration structures. Several switches are thus often integrated on analog circuits and commanded by digital inputs.

It can actually be attractive to increase the complementarity between digital and analog features and to merge them into a single mixed chip, in order to improve the weak points of both [Fatt93],[Zhij90bis]. A fuzzy knowledge base can be programmed in a digital memory which consists of dedicated locations and stores a variable number of parameters characterizing membership function shapes and inference rules notably. Highly-parallel and non-sequential analog processing is then afforded provided that D/A converters are used. A/D converters can also be used when a digital computation of the centre-of-gravity is desired.

The VLSI Design group of SGS-THOMSON has also undertaken the design of an hybrid controller implemented by means of a mixed analog/digital technology [Pag93bis],[Pag93-3]. It consists of a digital storage and distribution unit followed by an analog inference core. The membership grades are converted into analog values by an internal D/A. This system does not need expensive A/D and D/A converters in comparison with digital controllers. Its high speed should be suitable for very demanding real-time requirements with a limited number of rules.

## 7 CAD Automation for Fuzzy Logic Circuits Design

### 7.1 CAD strategy for digital circuits

• define specific cells to implement fuzzy operations = standard cells approach with usual CAD environments
• THOMSON's approach [Pag93]

### 7.2 CAD strategy for analog circuits

Design automation of analog fuzzy blocks provides a standard-cell approach allowing to build a fast and reliable design strategy (very similar to a digital one). The use of p- and n- channel static current mirrors as building blocks is well appropriate to create a design automation framework for generating the layout of fuzzy units. Such a development environment for current-mode CMOS fuzzy logic circuits has been created from a standard graphical tool and a specific silicon compiler [Lem93],[Lem94]. The graphical tool provides a logical simulation of fuzzy algorithms and helps to the design of fuzzy system architectures. The silicon compiler generates from the mathematical expression of a given fuzzy algorithm its corresponding layout. The system is based on a three-level hierarchy, consisting of current mirror circuits as generic cells for building elementary logical blocks (MIN, MAX, ,...), which ones can finally be assembled into sophisticated fuzzy units. The use of the three primitive operators , P and N (cf.§5.4.3) leads to analytical expressions which are suitable to describe a fuzzy units at all levels, from the current mirrors to high level fuzzy algorithms. The aim of this methodology is to cancel redundant fuzzy function elements which exist in the trivial implementation of the fuzzy functions. The silicon compiler should also replace each terms of the mathematical equations by its electrical counterpart. It tries then to reduce and place each electrical block in such an optimized way that wire lengths and IC area are minimised. The interconnections of these wires determine the vertical current flow between p- and n- mirrors from one stage to another. Certain configurations which cause functional problems should be forbidden and trivial connections of p- and n- mirrors involving static consumption should be avoided (case where serial p- and n- mirrors of a branch are driving a static current between the high and low levels). The automatic cell generator produces then the final layout as physical representation of the fuzzy algorithms. As an application of this environment, the prototype fuzzy controller FLC#001 has been designed in standard 1.2 µm CMOS technology [Lem94]. This low power and small size circuit achieves 10 MegaFLIPS and is quite efficient for real- time control applications.

As the design issued of such a strategy is close to fuzzy algorithms, it makes easy the test of resulting circuits. Internal currents can however not be measured without adding supplementary output current mirrors which increase the circuit size. Gate voltages can be measured but they just indicate an imprecise estimation of the transistor currents.

[Click to enlarge image]Figure-7.1: Structure of a silicon compiler [Lem94]

## 8 Neural Networks Implementing Fuzzy Systems

Artificial neural networks and fuzzy logic system exhibits some similar or complementary features that suggest to merge these two techniques. Both have a certain ability to work in natural analog environment and to deal with uncertainty of real-world problems. Neural networks seem appropriate for implementing fuzzy logic since in principle they should provide massively parallel and highly parametric computation, achieve highly non-linear operations, do not require any model and are concerned with approximating input-output relationships. They have moreover the ability of achieving interpolate reasoning similar to fuzzy inference. This means that once a neural network has been trained for a representative set of data, it can interpolate and produce answers for the cases not present in the training data set (the training is a supervised learning which consists of supplying inputs and demanding the specific output in order to adjust the weights in the connections between neurons in different layers). The neural output characteristics (threshold functions with sigmoid characteristics) are able to synthesise complicated membership functions. Finally the sum of products operation that neural networks realize is close to the MIN-MAX and weighting operations that occur in approximate reasoning. One should however bear it in mind that the implementation of neural networks is still complicated and actually not really easier than the direct implementation of fuzzy logic engines. Thus hardware realizations of neural networks are not so performant as regard parralelism, computation speed, functional density, and so on.

The fusion of neural networks and fuzzy logic do not only concerns their similarities, but also mutual compensations between their different features. Thus neural networks can offer a solution in fuzzy systems to the problems of structure identification (number and size of rules making up a suitable fuzzy partition of the feature space) and parameter identification (number and characteristics of membership functions). They are actually able to replace in some cases the tedious and sometimes hazardous identification scheme by a supervised learning. The knowledge acquisition by self-organizing networks can be used to realize adaptive fuzzy systems, which are quite attractive when linguistic rules cannot be easily derived or when a great deal of rules are required. However there is no guaranty of convergence in the learning scheme. The necessary number of neurons is moreover unpredictable and the acquisition of new knowledge is fairly difficult without beginning a new learning scheme.

There is several manner to combine neural networks and fuzzy logic, which differ according to the authors. The first idea consists of using the high flexibility of neural networks produced by learning, in order to provide automatic design and fine tuning of the membership functions used in fuzzy control. Such an adaptive neuro-fuzzy controller adjusts its performances automatically by accumulating experience in a learning phase. National Semiconductor Corp. has developed the Neufuz embedded system that provides a front-end neural networks suitable for fuzzy logic technology. A first layer performs fuzzification, a second creates the rule base and a third, a single neuron, does rule evaluation and defuzzification. Neural networks involve a great deal of computations and hardware investments which is prohibitive for many real-time applications. So they emulate and optimise in this case a fuzzy logic system rather than implement directly the final application. The National Semiconductor solution includes then the capability to generate assembly code for a strictly fuzzy logic solution. Recently Neuralogix has put on the market the NLX230 fuzzy microcontroller which consists of a VLSI digital fuzzy logic engine based on the min-max method. It includes a neural network implementing a high-speed minimum comparator block connected with 16 parallel fuzzifier on one side and a maximum comparator on the other side. An other approach aims at solving the problems of consequence identification and of defuzzification by using the Takagi-Sugeno's method which lends itself good to implementation framework based on adaptive neural networks [Jan92].

Instead of using neurons to implement some part of fuzzy systems, the structure of approximate reasoning can be applied to neural networks. The aim of this approach is to improve neural network frameworks by bringing some advantages of fuzzy logic. The latter allows an explicit representation of the knowledge and has a logic structure that allows to handle high-order processing easier. In pattern recognition for example, the learning scheme is efficient to acquire a knowledge about reference objects. As for the structure of approximate reasoning rules, it can give information on the knowledge distribution inside the whole network. It is then easier to find out the internal part that cause poor performance by analysis of error according to the rule structure [Tak90bis].

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Hardware Implementation of Fuzzy Systems
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Digital Implementations
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Analog Implementation: Voltage-Mode
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Analog Implementation: Mixed-Mode
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Analog Implementation: Current-Mode
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• [Sas92] M. Sasaki, N. Ishikawa, F. Ueno & T. Inoue, Current-Mode Analog Fuzzy Hardware with Voltage Input Interface and Normalization Locked Loop, Second IEEE Int. Conf. on Fuzzy Systems, pp.451-457, San Diego, CA, USA, March 1992
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Mixed Analog-Digital Implementation
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Implementation on FPGA
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Technological Aspects
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Neural Networks and Adaptive Fuzzy Systems
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This chapter edited by D. Mlynek