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Stable Fuzzy Control of Single-Input-Single-Output Systems. Exponential . Chapters may be used for the fuzzy system half of the course. A Course in Fuzzy Systems and Control - Ebook download as PDF File .pdf), Text File .txt) or read book online. Provides a comprehensive, self-tutorial course in fuzzy logic and its increasing role in control theory. The book answers key questions about.

By "handle," I mean "to model and minimize the effect of. The expanded FL, type-2 FL, is able to handle uncertainties because it can model them and minimize their effects. And, if all uncertainties disappear, type-2 FL reduces to type-1 FL, in much the same way that, if randomness disappears, probability reduces to determinism. Although many applications were found for type-1 FL, it is its application to rule-based systems that has most significantly demonstrated its importance as a powerful design methodology. A rule-based fuzzy logic system FLS is shown in Figure 1. Its fuzzifier, inference mechanism which is associated with rules, the heart of an FLS , and output processor involve operations on fuzzy sets that are characterized by membership functions.

X is the name of the linguistic variable. The terms "not. These atomic terms may be classified into three groups: Primary terms.

From numerical variable to linguistic variable. Our task now is to characterize hedges. This is the first step to incorporate human knowledge into engineering systems in a systematic and efficient manner. In our daily life. Linguistic Hedges 61 numerical variable linguistic variable Figure An atomic fuzzy proposition is a single statement. In this spirit. Let A be a fuzzy set in U. M and F denote the fuzzy sets "slow. Note that in a compound fuzzy proposition.

How to determine the membership functions of these fuzzy relations? For connective "and" use fuzzy intersections. A compound fuzzy proposition is a composition of atomic fuzzy propositions using the connectives "and.

For connective "or" use fuzzy unions. A is a fuzzy set defined in the physical domain of x.

V and A represent classical logic operations "not. V and A operators in 5. From Table 5. The fuzzy proposition 5. In classical propositional calculus.

Since there are a wide variety of fuzzy complement. We list some of them below. Dienes-Rescher Implication: If we replace the logic operators. By generalizing it to fuzzy propositions.

Giidel Implication: The Godel implication is a well-known implication for- mula in classical logic.. Y and max[l. Another question is: Are 5.

Since 0 I 1. For all x. When p and q are crisp propositions that is. PQL x. Lemma 5. This is an important question and we will discuss it in Chapters PFPz Y. In logic terms. THEN resistance is high. PQD x. PFPI x.

We now try to answer this question. So a question arises: Based on what criteria do we choose the combination of fuzzy complements. The fol- lowing lemma shows that the Zadeh implication is smaller than the Dienes-Rescher implication. THEN resistance is low. THEN y is large 5. Let xl be the speed of a car. We now consider some examples for the computation of Q D. If we use algebraic product for the t-norm in 5. Figure 5. Division of the domains of Zadeh and Mamdani implications.

A way to resolve this complexity is to use a single smooth function t o approximate the nonsmooth functions. IF x is large. If we use the Lukasiewicz implication 5. Suppose we know that x E U is somewhat inversely propositional to y E V.

To formulate this knowledge. Suppose we use to approximate the pslOw xl of 5. Now if we use Mamdani product implication 5. Godel and Mamdani implications. This is consistent with our earlier discussion that Dienes-Rescher. This paper is another piece of art and the reader is highly recommended to study it. Zadeh and Godel implications are global. Summary and Further Readings 71 For the Zadeh implication 5. The concept of linguistic variables and the characterization of hedges..

Linguistic variables were introduced in Zadeh's seminal paper Zadeh []. QZ and QG give full membership value to them. Properties and computation of these implications. The comprehensive three-part paper Zadeh [I summarized many concepts and principles associated with linguistic variables.

Show that Exercise 5. Combine these lin- guistic variables into a compound fuzzy proposition and determine its membership function. Give three examples of linguistic variables. Exercise 5. Let QL. Use basic fuzzy operators 3. Let Q be a fuzzy relation in U x U. Plot these membership functions. Q is called reflexive if pQ u. Consider some other linguistic hedges than those in Section 5.

Use m i n for the t-norm in 5. Show that if Q is reflexive. The fundamental truth table for conjunction V. By combining. In classical logic. The most commonly used complete set of primitives is negation-. Given n basic propositions p l. Chapter 6 Fuzzy Logic and Approximate Reasoning 6. This generalization allows us to perform approximate reasoning. Fuzzy logic generalizes classical two-value logic by allowing the truth values of a proposition to be any number in the interval [0. V and A in appropriate algebraic expressions.

Since n propositions can assume 2n possible combinations of truth values. The new proposition is usually called a logic function. Table 6. If p is a proposition. Truth table for five operations that are frequently applied to propositions. Logic formulas are defined recursively as follows: The truth values 0 and 1 are logic formulas. The following logic formulas are tautologies: To prove 6.

The three most commonly used inference rules are: The only logic formulas are those defined by a - c. When the proposition represented by a logic formula is always true regardless of the truth values of the basic propositions participating in the formula.

Various forms of tautologies can be used for making deductive inferences. Example 6. If p and q are logic formulas. They are referred to as inference rules. The ultimate goal of fuzzy logic is t o provide foundations for approximate reasoning with imprecise propositions using fuzzy set theory as the principal tool. They are the fundamental principles in fuzzy logic.

To achieve this goal. Intuitive criteria relating Premise 1 and the Conclusion for given Premise 2 in generalized modus ponens. ELSE y is not B. Criterion p7 is interpreted as: Generalized M o d u s Ponens: Premise 1: We note that if a causal relation between "x is A" and "y is B" is not strong in Premise 2.

B' and B are fuzzy sets. Similar to the criteria in Table 6. B and B' are fuzzy sets. C and C' are fuzzy sets. Intuitive criteria relating Premise 1 and the Conclusion for given Premise 2 in generalized modus tollens. Other criteria can be justified in a similar manner.

By applying the hedge more or less to cancel the very. Criteria s2 is obtained from the following intuition: Intuitive criteria relating y is B' in Premise 2 and z is C' in the Conclusion in generalized hypothetical syllogism.

We have now shown the basic ideas of three fundamental principles in fuzzy logic: Going one step further in our chain of generalization.

Although these criteria are not absolutely correct. The next question is how to determine the membership functions of the fuzzy propositions in the conclusions given those in the premises.

Then we project I on V yielding the interval b. Figure 6. To find the interval b which is inferred from a and f x. They should be viewed as guidelines or soft constraints in designing specific infer- ences. The compositional rule of inference was proposed to answer this question.

Let us generalize the above procedure by assuming that a is an interval and f x is an interval-valued function as shown in Fig.

Inferring fuzzy set B' from fuzzy set A' and fuzzy relation Q. Inferring interval b from interval a and interval-valued function f x. The Compositional Rule of Inference 79 Figure 6.

Different implication principles give different fuzzy relations. In summary. In the literature. For generalized hypothetical syllogism. In Chapter 5. Our task is to determine the corresponding B'. Suppose we use min for the t-norm and Mamdani's product impli- cation 5.

These results show the properties of the implication rules. We consider the generalized modus ponens. Consider four cases of A': Properties of the Implication Rules 81 is C.

We now study some of these properties. UA XO. From 6. In this example. Y in the generalized modus ponens 6. To summarize. This approximate reasoning is truely approximate! Similar to Example 6.

Consider four cases of B': Properties of the Implication Rules 85 Finally. Similar to Examples 6. Zadeh [I and other papers of Zadeh in the s. The idea and applications of the compositional rule of inference. Basic inference rules Modus Ponens. A comprehensive treatment of many-valued logic was prepared by Rescher []. The generalizations of classical logic principles to fuzzy logic were proposed in Zadeh [].

Exercise 6. The compositional rule of inference also can be found in these papers of Zadeh. THEN y is B" is given. Modus Tollens. Using truth tables to prove the equivalence of propositions. Generalized Modus Tollens. Use the truth table method to prove that the following are tautologies: Determining the resulting membership functions from different implication rules and typical cases of premises.

For any two arbitrary propositions. Show exactly how they are restricted. A and B. Imposing such requirement means that pairs of truth values of A and B become restricted to a subset of [O. Repeat Exercise 6. Consider a fuzzy logic based on the standard operation min. With min as the t-norm and Mamdani minimum implication 5. Let U. Use min for the t-norm and Lukasiewicz implication 5. Use min for the t-norm and Dienes-Rescher implication 5.

Now given a fact "y is B'. Exercises 87 Exercise 6. Y in the generalized modus tollens 6. In Chapter 9. We will see how the fuzzy mathematical and logic principles we learned in Part I are used in the fuzzy systems. In Chapters 10 and In Chapter 7. In this part Chapters We will derive the compact mathematical formulas for different types of fuzzy systems and study their approximation properties. In Chapter 8. Figure 7. A multi-input-multi-output fuzzy system can be decomposed into a collection of multi-input-single- output fuzzy systems.

We consider only the multi-input-single-output case. The partial rule 7. IF xl is A: It is the heart of the fuzzy system in the sense that all other components are used to implement these rules in a reasonable and efficient manner. We call the rules in the form of 7. Fuzzy Rule Base 91 inference engine.

The smaller the x. Let M be the number of rules in the fuzzy rule base. Lemma 7. Definition 7. For d. The fuzzy statement 7. That is.. Are there any conflicts among these rules? To answer these sorts of questions. In our fuzzy system framework. The preceding rule is in the form of 7.

Based on intuitive meaning of the logic operator "or. MI and L1 in Ul. If any rule in this group is missing. In order for a fuzzy rule base to be complete.. L1 with Sz. Define three fuzzy sets S1. This problem is called the curse of dimensionality and will be further discussed in Chapter From Example 7.

Fuzzy Rule Base 93 Example 7. An example of membership functions for a two-input fuzzy system. If the fuzzy rule base consists of only a single rule. We should first understand what a set of rules mean intuitively. For fuzzy rules. The first one views the rules as independent conditional statements. The second one views the rules as strongly coupled conditional statements such that the conditions of all the rules must be satisfied in order for the whole set of rules to have an impact.

Because any practical fuzzy rule base constitutes more than one rule. It is difficult to explain this concept in more detail at this point. If we adapt this view. There are two opposite arguments for what a set of rules should mean. There are two ways to infer with a set of rules: For nonfuzzy production rules.

If we accept this point of view. So the key question is how to perform this combination. J PQM x..

Fuzzy Inference Engine 95 to combine the rules. Let A' be an arbitrary fuzzy set in U and be the input to the fuzzy inference engine. We now show the details of these two schemes. The second view may look strange. If we accept the first view of a set of rules. From Chapter 5 we know that A. Let RU ' be a fuzzy relation in U x V. This combination is called the Godel combination. M Step 4: M according to any one of these i for implications. The combination can be taken either by union or by intersection.

Step 2: View All x.. Step 3: Determine The computational procedure of the individual-rule based inference is summa- rized as follows: Steps 1and 2: Same as the Steps 1and 2 for the composition based inference..

M according to 7. Step 4: For given input A'. PRU " x. B b either by union. For given input fuzzy set A' in U If these properties are desirable. Zadeh implication 5. We now show the detailed formulas of a number of fuzzy inference engines that are commonly used in fuzzy systems and fuzzy control. Fuzzy Inference Engine 97 7. In product inference engine.. Lukasiewicz implication 5. So a natural question is: M i n i m u m Inference Engine: In minimum inference engine.

The choice should result in a formula relating B' with A'. Intuitive appeal: The choice should make sense from an intuitive point of view. Godel implication 5. Mamdani inference or Godel inference. Special properties: Some choice may result in an inference engine that has special properties The product inference engine is unchanged if we replace "individual- rule based inference with union combination 7. If the fuzzy set A' is a fuzzy singleton.

Substituting 7. We now show some properties of the product and minimum inference engines. From 7. The product inference engine and the minimum inference engine are the most commonly used fuzzy inference engines in fuzzy systems and fuzzy control. The main advantage of them is their computational simplicity. Zadeh and Dienes-Rescher inference engines. Zadeh Inference Engine: In Zadeh inference engine. This may cause problems in implementation.

A disadvantage of the product and minimum inference engines is that if at some x E U the pA. Fuzzy Inference Engine 99 fuzzy inference engine can be greatly simplified if the input is a fuzzy singleton the most difficult computation in 7.

The following three fuzzy inference engines overcome this disadvantage. In Dienes-Rescher inference engine. Lukasiewicz Inference Engine: In Lukasiewicz inference engine.. We would like to plot the PBI Y obtained from the five fuzzy inference engines.

Zadeh and Dienes-Rescher inference engines are simplified to respectively. We now have proposed five fuzzy inference engines. Suppose that a fuzzy rule base consists of only one rule IF x1 i s A1 and..

From Figs. If A' is a fuzzy singleton as defined by 7. Example 7. Using the same arguments as in the proof of Lemma 7. Let B. Then from 7. Zadeh and Dienes-Rescher inference engines give very large member- ship values. Output membership functions using the Lukasiewicz. Zadeh and Dienes-Rescher inference engines are similar. Zadeh and Dienes-Rescher inference engines for the par. Fuzzy Inference Engine and minimum inference engines give very small membership values.

In this example..

Output membership functions using the prod- uct and minimum inference engines for the PA. Fuzzy Inference Engine Figure 7. Zadeh and Dienes-Rescher inference engines for the PA. Output membership function using the prod- uct inference engine for the case of two rules.

Exercise 7. If the third and sixth rules in 7. Use the Godel implication to propose a so-called Godel inference engine. Lee [I provided a very good survey on fuzzy rule bases and fuzzy inference engines. This paper gives intuitive analyses for various issues associated with fuzzy rule bases and fuzzy inference engines. Consider Example 7. B6 such that the set of the six rules 7. A mathematical analysis of fuzzy infer- ence engines.

Suppose that a fuzzy rule base consists of only one rule 7. The detailed formulas of the five specific fuzzy inference engines: Give an example of fuzzy sets B1.

Hellendoorn and Reinfrank []. The computational procedures for the composition based and individual-rule based inferences.

Plot the output membership functions. Chapter 8 Fuzzifiers and Defuzzifiers We learned from Chapter 7 that the fuzzy inference engine combines the rules in the fuzzy rule base into a mapping from fuzzy set A' in U to fuzzy set B' in V. Because in most applications the input and output of the fuzzy system are real- valued numbers.

Matsushita vacuum cleaners use microcontrollers running fuzzy algorithms to interrogate dust sensors and adjust suction power accordingly. Hitachi washing machines use fuzzy controllers to load-weight, fabric-mix, and dirt sensors and automatically set the wash cycle for the best use of power, water, and detergent. Canon developed an autofocusing camera that uses a charge-coupled device CCD to measure the clarity of the image in six regions of its field of view and use the information provided to determine if the image is in focus.

It also tracks the rate of change of lens movement during focusing, and controls its speed to prevent overshoot. The camera's fuzzy control system uses 12 inputs: 6 to obtain the current clarity data provided by the CCD and 6 to measure the rate of change of lens movement. The output is the position of the lens. The fuzzy control system uses 13 rules and requires 1.

An industrial air conditioner designed by Mitsubishi uses 25 heating rules and 25 cooling rules. A temperature sensor provides input, with control outputs fed to an inverter , a compressor valve, and a fan motor. Other applications investigated or implemented include: character and handwriting recognition; optical fuzzy systems; robots, including one for making Japanese flower arrangements; voice-controlled robot helicopters hovering is a "balancing act" rather similar to the inverted pendulum problem ; rehabilitation robotics to provide patient-specific solutions e.

Work on fuzzy systems is also proceeding in the United State and Europe, although on a less extensive scale than in Japan. The US Environmental Protection Agency has investigated fuzzy control for energy-efficient motors, and NASA has studied fuzzy control for automated space docking: simulations show that a fuzzy control system can greatly reduce fuel consumption.

Firms such as Boeing , General Motors , Allen-Bradley , Chrysler , Eaton , and Whirlpool have worked on fuzzy logic for use in low-power refrigerators, improved automotive transmissions, and energy-efficient electric motors.

In Maytag introduced an "intelligent" dishwasher based on a fuzzy controller and a "one-stop sensing module" that combines a thermistor , for temperature measurement; a conductivity sensor, to measure detergent level from the ions present in the wash; a turbidity sensor that measures scattered and transmitted light to measure the soiling of the wash; and a magnetostrictive sensor to read spin rate.

The system determines the optimum wash cycle for any load to obtain the best results with the least amount of energy, detergent, and water. It even adjusts for dried-on foods by tracking the last time the door was opened, and estimates the number of dishes by the number of times the door was opened. Research and development is also continuing on fuzzy applications in software, as opposed to firmware , design, including fuzzy expert systems and integration of fuzzy logic with neural-network and so-called adaptive " genetic " software systems, with the ultimate goal of building "self-learning" fuzzy-control systems.

Please improve the article by adding information on neglected viewpoints, or discuss the issue on the talk page. April The input variables in a fuzzy control system are in general mapped by sets of membership functions similar to this, known as "fuzzy sets". The process of converting a crisp input value to a fuzzy value is called "fuzzification". A control system may also have various types of switch , or "ON-OFF", inputs along with its analog inputs, and such switch inputs of course will always have a truth value equal to either 1 or 0, but the scheme can deal with them as simplified fuzzy functions that happen to be either one value or another.

Given " mappings " of input variables into membership functions and truth values , the microcontroller then makes decisions for what action to take, based on a set of "rules", each of the form: IF brake temperature IS warm AND speed IS not very fast THEN brake pressure IS slightly decreased. In this example, the two input variables are "brake temperature" and "speed" that have values defined as fuzzy sets.

The output variable, "brake pressure" is also defined by a fuzzy set that can have values like "static" or "slightly increased" or "slightly decreased" etc. This rule by itself is very puzzling since it looks like it could be used without bothering with fuzzy logic, but remember that the decision is based on a set of rules: All the rules that apply are invoked, using the membership functions and truth values obtained from the inputs, to determine the result of the rule.

This result in turn will be mapped into a membership function and truth value controlling the output variable. These results are combined to give a specific "crisp" answer, the actual brake pressure, a procedure known as " defuzzification ".

This combination of fuzzy operations and rule-based " inference " describes a "fuzzy expert system". Traditional control systems are based on mathematical models in which the control system is described using one or more differential equations that define the system response to its inputs.

Such systems are often implemented as "PID controllers" proportional-integral-derivative controllers.

They are the products of decades of development and theoretical analysis, and are highly effective. If PID and other traditional control systems are so well-developed, why bother with fuzzy control? It has some advantages. In many cases, the mathematical model of the control process may not exist, or may be too "expensive" in terms of computer processing power and memory, and a system based on empirical rules may be more effective.

For the block diagram of a modified and expanded three-level LSE control system, see Figure Figure Three levels of LSE traffic control system [ 22 ]. The main change consisted of the expansion of the lowest level by a module, which predicted a value of the input variable using the fuzzy Neural network.

The authors moved the fuzzy inference system from the highest level to the second middle level. According to the authors a system assembled in this way solves the issues of stochastic traffic systems, such as an intersection with LSE, much better. Kuo and Lin proposed in their publication [ 23 ] a new approach for determination of an LSE signal plan based on fuzzy logic block diagram of the process, see Figure Figure Block diagram of the control process graphic representation of system process [ 23 ].

The combination of main input variables detected during the cycle average speed, congestion factor, and vehicle position further triggered the defined rules.

The defuzzification process resulted in the appropriate setting of the LSE signal plan. From the system design it is obvious that the authors expanded the number of input variables by input variables reflecting the geometry of the intersection and the vehicle position determined based on the distance from the intersection stop line. Murat and Gedizlioglu [ 24 ] developed a fuzzy control system for an isolated intersection, which differed from the above mentioned systems by the use of two fuzzy control systems.

Each of the fuzzy systems contained a different database block diagram of the LSE control system see Figure Jacques et al. In practice they tested three different fuzzy control systems the difference was in the application of three different methods of defuzzification MOM, COG, and SOM on an isolated intersection for three different loading levels low, medium, and high.

Expert publications dedicated to the application of fuzzy algorithms in the LSE control system after the year do not bring entirely new ideas; they mostly focus either on improvement of algorithms of other authors or on certain expansion of the control systems.

One of the other articles dealing with the application of the fuzzy algorithm in the LSE control system in combination with the genetic algorithm is the thesis by Chiou and Lan [ 25 ].

The genetic algorithm should select a suitable fuzzy rule and set the fitness function Figure Figure Illustration of the fuzzy rule and fitness function selection using the genetic algorithm [ 25 ]. In the article [ 58 ] Zeng et al.

The authors Zhang et al. Their expert publication described the proposal of a two-level fuzzy control system for an overloaded transportation network. The transportation area in question had compact central areas in which the intensities reached high values with a threat of a possible occurrence of traffic congestions.

The first goal was to minimize the delay of vehicles and the second goal was to prevent occurrence of traffic congestions. The first level of the fuzzy algorithm assessed the traffic situation in the network in question and the second level of the algorithm was to control the LSE on each intersection.

The peripheral intersections were used to regulate the number of vehicles approaching the inner area by altering the vehicle direction different selection of phase sequence. The result was the decrease in the probability of overloading the transportation network.

The authors Hu et al. The proposed control system contains an evolution algorithm, which generates the optimal fuzzy rule base. The evolution algorithm working with real measured data was applied to an isolated four-arm intersection containing lanes for straight direction and right turns. Left turns were solved only in one intersection arm as a separate bypass. The authors defined the fitness function performance function as an average time loss of all vehicles in.

For the block diagram of the LSE control system containing a block for evaluation of the performance function, see Figure Figure The block diagram of the genetic fuzzy generator database [ 26 ]. Hu et al. Together with the other authors they proposed a hierarchic fuzzy control system, which was applied to a real four-arm intersection containing fourteen lanes, two pedestrian crossings, and 7 phases. It is obvious from Figure 19 that the hierarchic control system contained six levels of control subprocesses subcontrollers.

Each of these individual control elements contained two inputs and one output. Levels 1 to 5 used two identical input parameters representing column lengths. The consolidated column length from the previous layer was the output. The last sixth level used the consolidated column length of previous levels and the column length of the selected phase as an input parameter.

The output of the sixth level was the duration of the green light of the selected phase. Figure Diagram of the hierarchic fuzzy control system [ 27 ]. Yang et al. The proposed control system was a two-level one and contained three modules with fuzzy logic.

The first level was represented by two modules with fuzzy logic, which estimated the actual duration of the green light and the subsequent red light phase based on the intensity of the traffic flow of the given phase. The second level contained one fuzzy module, which decided on the extension of the actual green phase.

This proposed control system shows, according to the study conclusions, better control results than the previously used one-level LSE fuzzy control systems single stage fuzzy logic controller SSFLC. Figure HFLC architecture hierarchical fuzzy logic controller [ 28 ]. Authors Cheng and Yang proposed in their thesis [ 60 ] another LSE control system containing a fuzzy-genetic algorithm. They used fuzzy clustering analysis to create the knowledge base, which was created based on identified intensities.

They separated the rules of the knowledge base into two sets, fixed and variable rules. The genetic algorithm was used for setting the set of variable rules during the LSE control and was also part of the process of determination of optimal cycle duration. The proposed control system provided again better results than the fixed signal plan or dynamic LSE control. The fitness function was optimized using the Bellman-Zdeh principle.

According to the authors it was a very effective method due to the fact that the result led to the so-called Paret optimal solution ideal balanced state. The fitness functions were further optimized through a genetic algorithm. For the block diagram of the transportation system, see Figure The authors verified the proposed system on a case study of the Marylebone Road, Baker Street intersection in London.

Figure Block diagram of the transportation system [ 29 ]. Li and Zhang proposed in their article [ 30 ] a multiphase fuzzy control system, which is comprised of two parts: the first part composed of the so-called fuzzy green extension controller FGEC , and the second part provided the change of phases fuzzy phase change controller FPCC again using the fuzzy logic block diagram, see Figure Figure Block diagram of the fuzzy control system [ 30 ].

Zarandi and Rezapour [ 31 ] chose a different approach to multilevel fuzzy LSE control. The fuzzy control system contained a fuzzy system of phase selection and a fuzzy system for green extension; see Figure The system either extended the current running phase or selected a new phase diagram. Figure Block diagram of the multilevel LSE fuzzy control system [ 31 ]. The publication of Rhung et al.

Figure Block diagram of the control system [ 32 ]. Wen et al. The authors claim that this system Figure 25 can adapt to fluctuating traffic load intensity, vehicle direction and prevent oversaturation of the intersection. The proposed control system was tested at a four-arm intersection and was compared with control system using the fixed signal plan and with fully dynamic LSE control system. Chen et al.

The traffic in China can be characterized by big amount of pedestrians and cyclists, who slow down the traffic flow. Niittymaki and Kikuchi [ 41 ] deal in their paper with specifications of pedestrian crossings at intersections, where the pedestrians prevail. Using the fuzzy logic the traffic control system became the so-called pedestrian friendly.

In the article [ 33 ] the authors proposed a hybrid LSE control system combining fuzzy logic, learning automata, and CPN coloured Petri nets. The system was based on a learning automata block diagram, see Figure 26 , where input data changed its internal settings and chose selected output action.

Fuzzy coloured Petri net should predict optimized future system conditions. Figure Relationship between learning automata and their environment [ 33 ]. In the recent study [ 62 ] concern was devoted to exploitation of neural networks for better prediction of traffic behaviour.

However, the use of fuzzy logic for control of LSE is not limited to four-arm or T intersections; it can be used to control rotary intersection, as suggested in the article [ 63 ] by Gong et al. Lu et al. The roundabouts are very popular in the USA, but it is almost impossible for pedestrians, especially the handicapped ones, to judge the safe space between the vehicles.