Traffic signal control on similarity logic reasoning

Fuzzy Sets and Systems 133 (2003) 109 – 131

www.elsevier.com/locate/fss

Tra c signal control on similarity logic reasoning Jarkko Niittym%akia , Esko Turunenb; ∗ b

a Helsinki University of Technology, P.O. Box 2100, FIN-02015 HUT, Finland Department of Mathematics, Tampere University of Technology, P.O. Box 692, FIN-33101 Tampere, Finland

Received 20 September 1999; received in revised form 20 December 2001; accepted 8 January 2002

Abstract The main intention in this study is to tie fuzzy reasoning to many-valued logic framework; a Lukasiewicz many-valued logic similarity based fuzzy control algorithm is introduced, and tested in three realistic tra c signal control systems. The results are compared to fuzzy control systems where the inference is based on standard Matlab Fuzzy Logic toobox’s Mamdani-style system. The compared tra c signal control modes are signalized pedestrian crossing and multi-phase signal control with phase selection. The statistical signi4cance in di5erences of expectations obtained by di5erent control schemes has been tested by two-sided approximate Students test on signi4cance level  = 0:01. Simulations made by HUTSIM tra c signal simulator show that the performances of the new control algorithm and that of standard Mamdani-style algorithm are almost equal. However, if tra c density is high then the new algorithm gives signi4cantly better statistical results. c 2002 Elsevier Science B.V. Moreover, a stronger mathematical approach o5ers a natural smoothness test.
All rights reserved. Keywords: Non-classical logics; Fuzzy control; Transportation; Tra c signal control

1. Introduction The aim of this study is two-fold. In the mathematical part we introduce a new algorithm to construct fuzzy IF–THEN inference systems. Our intention is to tie fuzzy reasoning to many-valued logic framework. The algorithm utilizes Lukasiewicz many-valued equivalence, and its design has a special feature that whenever the output would not be unique the 4nal decision should be left to human experts. This corresponds to a situation that sometimes occurs in mathematical modeling: a theoretical model leads to two or more solutions, say roots of equation of the second degree. However, a human expert knows that the solution cannot be negative. Thus, as much as possible the intelligence relies on a real controller, and technical defuzzi4cation methods are not needed. ∗

Corresponding author. E-mail address: [email protected] (E. Turunen).

c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 1 2 8 - 8

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In the application part we show how the algorithm can be used to solve various tra c signal control problems. In general, tra c signal control is used to maximize the e ciency of the existing tra c systems without new road constructions, maintain safe tra c Gows, minimize delays and holdups, and reduce air and noise pollution. At present, tra c signal control is based on tailor made solutions, and is poor at handling approaching tra c or cause and e5ect relationships. Based on recent research work [16, 5, 13] and a Special Issue of FSS (Vol. 116 (1), 2000), fuzzy control technology appears particularly well suited to tra c signal situation involving multiple approaches and vehicle movements. In previous studies (e.g. [17]), non-fuzzy and fuzzy control methods have been compared. Thus, one of the aims of this paper is to compare other fuzzy IF–THEN inference systems to the introduced Lukasiewicz many-valued equivalence based one. However, the above referred reports of various other fuzzy signal control systems are not described in su cient details, and therefore the comparison has been done between our system and a standard Matlab Fuzzy Logic Toolbox’s Mamdani-style system, where the defuzzi4cation method has been ‘least of maximum’ as it is most suggestive of our approach. The testbed for checking the performance of the proposed fuzzy control system and comparing it with other schemes is HUTSIM, a tra c Gow simulation package developed at the Laboratory of Transportation Engineering of the Helsinki University of Technology, Finland. Since being developed in 1989, HUTSIM has been used both for the real world application and also as a research tool. For the real world application it has been used to check the performance of the actual controller for di5erent hypothetical Gow conditions. For research purpose it has been used to test di5erent signal timing algorithms for given Gows of vehicles, transit vehicles and pedestrians at intersections of varying complexity. In HUTSIM di5erent control logics can be introduced as modules and under individual control schemes it simulates Gow microscopically and produces various measures of performance. This feature is particularly useful for this study because it allows us to compare the performance of di5erent control schemes for the same input. The user of HUTSIM can specify the locations of tra c detectors, pedestrian arrival patterns as well as the signal control scheme. The statistical signi4cance in di5erences of expectations  bigger and  smaller obtained by di5erent control schemes has been tested by two-sided approximate Students test on signi4cance level  = 0:01, null hypothesis H0 :  bigger =  smaller and H1 :  bigger =  smaller . 1.1. Bene7ts and disadvantages of fuzzy inference systems from a mathematical point of view Fuzzy logic has been introduced and successfully applied to a wide range of automatic control tasks. The main bene4t of fuzzy logic is the opportunity to model the ambiguity and the uncertainty of decision-making. Moreover, fuzzy logic has the ability to comprehend linguistic instructions and to generate control strategies based on priori communication. The point in utilizing fuzzy logic in control theory is to model control based on human expert knowledge, rather than to model the process itself. Indeed, fuzzy control has proven to be successful in problems where exact mathematical modeling is hard or impossible but an experienced human operator can control process. In general, fuzzy control is found to be superior in complex problems with multi-objective decisions. At present, there is a multitude of inference systems based on fuzzy technique. Most of them, however, su5er ill-de4ned foundations; even if they are mostly performing better that classical

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mathematical methods, they still contain black boxes, e.g. defuzzi4cation, which are very di cult to justify logically. For example, fuzzy IF–THEN rules, which are in the core of fuzzy inference systems, are often reported to be generalizations of classical Modus Ponens rule of inference, but literally this not the case; the relation between these rules and any known many-valued logic is complicated and arti4cial. Moreover, the performance of an expert system should be equivalent to that of human expert: it should give the same results that the expert gives, but warn when the control situation is so vague that an expert is not sure about the right action. The existing fuzzy expert systems very seldom ful4ll this latter condition. Therefore, the main theoretical aim of this study is to show a solid logical foundation of fuzzy IF–THEN rules. Many researches observe that fuzzy inference is based on similarity. Kosko [12], for example, writes ‘Fuzzy membership...represents similarities of objects to imprecisely de4ned properties’. Taking this remark seriously, we study systematically many-valued equivalence, i.e. fuzzy similarity. In literature there is, however, a multitude of approaches to similarity; it can be divided at least into the following four groups: (I) Similarity in two-valued logic setting. Excellent studies can be found e.g. in a book edited by Helman [10]. Niiniluoto, for example, writes in his paper ‘Analogy and Similarity in Scienti4c Reasoning’ ‘Rule for simple Analogy (RA) present ... it tells how to transfer knowledge from one source object a to a target object b. In the case of multiple analogy, we try to extract information about the target b from several sources a1 ; : : : ; an : : : : A real challenge ...is...that we have to extend our treatment from simple analogy to multiple analogy’. (II) Similarity measure of two fuzzy sets. Bandler and Kohout [1] (see also [29, 4]) de4ne and study the degree to which the fuzzy sets A and B are the same, or their degree of sameness. (III) Fuzzy similarity relation in a crisp set, i.e. the original notion by Zadeh [31] as a generalization of equivalence relation; a binary fuzzy relation that is reGexive, symmetric and weakly transitive. Later many other authors have developed Zadeh’s ideas, see e.g. [26, 8, 22, 3]. Dubois and Prade consider the same problem than Niiniluoto as they write in their paper ‘Similarity-Based Approximate Reasoning’ ...The evaluation of similarity between two multi-feature descriptions of objects may be specially of interest in analogical reasoning. If we assume that each feature is associated with an attribute domain equipped with similarity relation modelling approximate equality on this domain, the problem is then to aggregate the degrees of similarity between the objects pertaining to each feature into a global similarity index. This means that the resulting index should still have properties like re=exivity, symmetry and max-transitivity. ... . Moreover, we may think of a weighted aggregation if we consider that we are dealing with a fuzzy set of features having di>erent levels of importance’. (IV) Other approaches, such as Tversky’s paper ‘Features of Similarity’ [28], studies on CaseBased Reasoning, e.g. [23], and Li’s paper [14], where fuzzy inference systems are de4ned to have similarity if they ful4ll a certain equation between input and output values. We study many-valued similarity in the sense of (III). By means of a result presented in [3] and two new theorems, we solve the problem stated by Niiniluoto and Dubois and Prade above. It turns out that, starting from the Lukasiewicz well-de4ned many-valued logic, we are able to construct a method performing fuzzy reasoning such that the inference relies only on experts knowledge and on well-de4ned logical concepts. Therefore we do not need any arti4cial defuzzi4cation method (like Center of Gravity) to determine the 4nal output of the inference.

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Our basic observation is that any fuzzy set generates a fuzzy similarity, and that these similarities can be combined to a fuzzy relation which turns out to a fuzzy similarity, too. We call this induced fuzzy relation total fuzzy similarity. Fuzzy IF–THEN inference systems are, in fact, problems of choice: compare each IF-part of the rule base with an actual input value, 4nd the most similar case and 4re the corresponding THEN-part; if it is not unique, use a criteria given by an expert to proceed. Thus, the main distinction between our approach and other fuzzy inference systems is that only one IF–THEN rule determines the output. Based on the Lukasiewicz well-de4ned many-valued logic, we show how this method can be carried out formally. Thus, the aim of this paper is to give a mathematical approach to fuzzy reasoning, a method that is based on many-valued equivalence rather that many-valued implication. Our approach relies on the fundamental studies of HQajek [9], H%ohle [11], Mundici et al. [6], Pavelka [21]] and Turunen [27]. Besides, we discuss a kind of stability or smoothness of fuzzy control systems, a topic which is often neglected in this framework, and show its relation to many-valued equivalence. 1.2. Tra?c signals as a control system Since the inception of tra c signal control in the year 1913 in Cleveland, OH, USA, the technical and algorithmic developments are continually improving the safety, e ciency, and environmental impact of control [2]. Signal control is a necessary measure to maintain the quality and safety of tra c circulation. Further development of present signal control has great potential to reduce travel times, vehicle and accident costs, and vehicle emissions. The development of detection and computer technology has changed tra c signal control from 4xed-time open-loop regulation to adaptive feedback control. Present adaptive control methods, like the British MOVA, Swedish SOS (isolated signals) and British SCOOT (area-wide control), use mathematical optimization and simulation techniques to adjust the signal timing to the observed Guctuations of tra c Gow in real time. The optimization is done by changing the green time and cycle lengths of the signals. In area-wide control the o5sets between intersections are also changed. Several methods (see e.g. [15]) have been developed for determining the optimal cycle length and the minimum delay at an intersection but, based on uncertainty and rigid nature of tra c signal control, the global optimum is not possible to 4nd out. In adaptive tra c signal control the increase in Gexibility increases the number of overlapping green phases in the cycle, thus making the mathematical optimization very complicated and di cult. For that reason, the adaptive signal control in most cases is not based on precise optimization but on the green extension principle. In practice, uniformity is the principle followed in signal control for tra c safety reasons. This sets limitations to the cycle time and phase arrangements. Hence, tra c signal control in practice are based on tailor-made solutions and adjustments made by the tra c planners. The modern programmable signal controllers with a great number of adjustable parameters are well suited to this process. For good results, an experienced planner and 4ne-tuning in the 4eld is needed. As noted, fuzzy control has proven to be successful in problems where exact mathematical modeling is hard or impossible but an experienced human can control the process operator. Thus, tra c signal control in particular is a suitable task for fuzzy control. Indeed, one of the oldest examples of the potentials of fuzzy control is a simulation of tra c signal control in an inter-section of two

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one-way streets [20]. Even in this very simple case the fuzzy control was at least as good as the traditional adaptive control. In general, fuzzy control is found to be superior in complex problems with multi-objective decisions. In tra c signal control several tra c Gows compete from the same time and space, and di5erent priorities are often set to di5erent tra c Gows or vehicle groups. In addition, the optimization includes several simultaneous criteria, like the average and maximum vehicle and pedestrian delays, maximum queue lengths and percentage of stopped vehicles. So, it is very likely that fuzzy control is very competitive in complicated real intersections where the use of traditional optimization method is problematic. 1.2.1. Hypothesis and principles of fuzzy tra?c signal control Tra c signal control is used to maximize the e ciency of the existing tra c systems [16]. However, the e ciency of tra c system can even be fuzzy. By providing temporal separation of rights of way to approaching Gows, tra c signals exert a profound inGuence on the e ciency of tra c Gow. They can operate to the advantage or disadvantage of the vehicles or pedestrians, depend on how the rights of ways are allocated. Consequently, the proper application, design, installation, operation, and maintenance of tra c signals is critical to the orderly safe and e cient movement of tra c at intersections. In tra c signal control, we can 4nd some kind of uncertainties in many levels. The inputs of tra c signal control are inaccurate, and that means that we cannot handle the tra c of approaches exactly. The control possibilities are complicated, and handling these possibilities are an extremely complex task. Maximizing safety, minimizing environmental aspects and minimizing delays are some of the objectives of control, but it is di cult to handle them together in the traditional tra c signal control. The cause–consequence relationship is also not possible to explain in tra c signal control. These are typical features of fuzzy control. Fuzzy logic based controllers are designed to capture the key factors for controlling a process without requiring many detailed mathematical formulas. Due to this fact, they have many advantages in real time applications. The controllers have a simple computational structure, since they do not require many numerical calculations. The IF–THEN logic of their inference rules does not require much computational time. Also, the controllers can operate on a large range of inputs, since di5erent sets of control rules can be applied to them. If the system related knowledge is represented by simple fuzzy IF–THEN rules, a fuzzy-based controller can control the system with e ciency and ease. The main goal of tra c signal control is to ensure safety at signalized intersections by keeping conGict tra c Gows apart. The optimal performance of the signalized intersections is the combination of time value, environmental e5ects and tra c safety. Our goal is the optimal system, but we need to decide what attributes and weights will be used to judge optimality. The entire knowledge of the system designer about the process, tra c signal control in this case, to be controlled is stored as rules in the knowledge base. Thus the rules have a basic inGuence on the closed-loop behavior of the system and should therefore be acquired thoroughly. The development of rules is time consuming, and designers often have to translate process knowledge into appropriate rules. Sugeno and Nishida [24] mentioned four ways to derive fuzzy control rules: 1. operators experience, 2. control engineer’s knowledge,

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3. fuzzy modeling of the operator’s control actions, 4. fuzzy modeling of the process, Zimmermann [32] added three sources more, 5. crisp modeling of the process, 6. heuristic design rules, and 7. on-line adaptation of the rules. 2. Mathematical preliminaries Recall a binary operation  : [0; 1]2  [0; 1] is called t-norm if, for all elements x; y; z ∈ [0; 1], (i) if x6y, then x  z6y  z, (ii) x  y = y  x, (iii) x  1 = x, (iv) x  (y  z) = (x  y)  z, (v) x  0 = 0. In particular, continuous t-norms  and their residua → play a fundamental role in fuzzy logic. The most frequently used continuous t-norms in various fuzzy inference systems generate the following algebraic structures: G/odel algebra:
1 if x 6 y; x  y = min{x; y}; x → y = y otherwise: Product t-algebra: x  y = xy;


x→y=

1 if x 6 y; y otherwise: x

Lukasiewicz algebra: x  y = max{0; x + y − 1};


x→y=

1 if x 6 y; 1 − x + y otherwise:

These three examples are fundamental since, in a certain sense, they characterize all possible continuous t-norms (for details, see [7, 9]). They are the generators of all BL-algebras of the real unit interval, too; by 4xing a continuous t-norm we 4x a Basic Logic, a well-de4ned many-valued logic modeling mathematically fuzzy reasoning. The operations  and → are the algebraic counterparts of the logical connectives conjunction and implication, respectively. In particular, the complement x∗ of an element x ∈ [0; 1] de4ned by x∗ = x → 0, is the algebraic counterpart of negation, while many-valued equivalence is interpreted algebraically by bi-residuum de4ned, for all x; y ∈ [0; 1], via x ↔ y = min{(x → y); (y → x)}: In any BL-algebra, a bi-residuum ↔ has the following properties (cf. [27]) x ↔ x = 1;

(1)

x ↔ y = y ↔ x;

(2)

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(x ↔ y)  (y ↔ z) 6 x ↔ z;

(3)

x ↔ 1 = x:

(4)

It is easy to see that, in Lukasiewicz algebra, the ↔ bi-residuum is calculated via x ↔ y = 1 − max{x; y} + min{x; y}: Castro and Klawonn [3] among others set the following important Denition 1. Let A be a non-void set and  a continuous t-norm. Then a fuzzy similarity S on A is such a binary fuzzy relation that, for each x; y; z ∈ A, (i) Sx; x = 1 (everything is similar to itself), (ii) Sx; y = Sy; x (fuzzy similarity is symmetric), (iii) Sx; y  Sy; z 6Sx; z (fuzzy similarity is weakly transitive). Trivially, fuzzy similarity is a generalization of classical equivalence relation, thus called manyvalued equivalence, too. Recall a fuzzy set X is an ordered couple (A;  X ), where the reference set A is a non-void set and the membership function  X : A  [0; 1] tells the degree to which an element a ∈ A belongs to the fuzzy set X . Theorem 1. Any fuzzy set (A;  X ) on a reference set A generates a fuzzy similarity S on A, de7ned by S(x; y) = X (x) ↔ X (y);

where x; y are elements of A:

Moreover, if X (y) = 1

then S(x; y) = X (x):

Proof. By (1)–(4). Theorem 2. For each x; z ∈ A, condition (iii) is equivalent to (iv)



y∈A (Sx; y

 Sy; z )6Sx; z .

Proof. Assume (iii) holds. Let x; z ∈ A be 4xed elements. Since (iii) holds for any y ∈ A, (iv) holds by de4nition of lowest upper bound. Similarly (iv) implies (iii). Notice that the original de4nition of fuzzy similarity given by Zadeh in [31] is (i), (ii) and (iv). Moreover, it is worth noting that, in Lukasiewicz algebra, ‘the negation of equivalence is distance’. Indeed, for all a; b ∈ [0; 1]  (a ↔ b)∗ = 1 − [1 − max{a; b} + min{a; b}] = |a − b| = (a − b)2 the one-dimensional Euclidean distance between a and b. In [26], or independently in [27], the following result was proved.

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Theorem 3. Consider n Lukasiewicz valued fuzzy similarities Si ; i = 1; : : : ; n on a set X . Then Sx; y =

n 1  Si x; y n i=1

is a Lukasiewicz valued fuzzy similarity on X . More generally, the weighted mean Sx; y = where M = on X .

n

n 1  mi · Si x; y ; M i=1

i=1

mi and mi are natural numbers, is again a Lukasiewicz valued fuzzy similarity

Theorem 3 does not hold for other BL-algebras than Lukasiewicz algebra. Indeed, consider the following two fuzzy similarities S1 and S2 on a set {a; b; c} (with respect to any BL-algebra on the real unit interval!), de4ned by S1 a b c

a 1 1 0

b 1 1 0

c 0 0 1

and

S2 a b c

a 1 0 0

b 0 1 1

c 0 : 1 1

The combined fuzzy relation is not a fuzzy similarity on the set {a; b; c} if one uses G%odel algebra or Product t-algebra; in general, weak transitivity does hold. Indeed, for G%odel algebra  = ∧ (min) and 1 [S (a; b) 2 1

+ S2 (a; b)] ∧ 12 [S1 (b; c) + S2 (b; c)] =

1 2


0 = 12 [S1 (a; c) + S2 (a; c)]:

Similarly, for Product t-algebra  = · and 1 [S (a; b) 2 1

+ S2 (a; b)] · 12 [S1 (b; c) + S2 (b; c)] =

1 4


0 = 12 [S1 (a; c) + S2 (a; c)]:

2.1. Algorithm to construct fuzzy inference systems A fuzzy IF–THEN inference system is composed of an input universe of discourse X , the IF-parts, and an output universe of discourse Y , the THEN-parts. We assume there are n input variables and one output variable. The dynamics of S are characterized by a 4nite collection of IF–THEN rules; e.g. Rule 1 Rule 2 .. .

Rule k

IF x is A1 and y is B1 and z is C1 IF x is A2 and y is B2 and z is C2 IF x is Ak and y is Bk and z is Ck

THEN w is D1 , THEN w is D2 , .. . THEN w is Dk ;

where A1 ; : : : ; Dk are fuzzy sets of height 1, that is, in each fuzzy set there is at least one element that obtains the membership degree 1. Generally, the output fuzzy sets D1 ; : : : ; Dk should have members that obtain all the same values ∈ [0; 1] the input fuzzy sets A1 ; : : : ; Ck do, however, the outputs can

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be crisp actions, too. All these fuzzy sets are to be speci4ed by the fuzzy control engineer. We avoid disjunction between the rules by allowing some of the output fuzzy sets Di and Dj ; i = j; be possibly equal. Thus, a 4xed THEN-part can be followed by various IF-parts. Some of the input fuzzy sets may be equal, too (e.g. Bi = Bj for some i = j). However, the rule base should be consistent; a 4xed IF-part precedes a 4xed THEN-part. Moreover, the rule base can be incomplete; if an expert is not able to de4ne the THEN-part of some combination ‘IF x is Ai and y is Bi and z is Ci ’ then the rule can be skipped. This is, in fact, a useful feature all fuzzy inference systems have. In fact, it is an implicit mechanism for interpolation. Now we are in the position to formulate an algorithm a fuzzy control engineer has to perform to construct a total fuzzy similarity based inference system. Step 1. Create the dynamics of S, i.e. de4ne the IF–THEN rules, give the shapes of the input fuzzy sets (e.g. A1 ; : : : ; Ck ) and the shapes of the output fuzzy sets (e.g. D1 ; : : : ; Dk ). Step 2. Give weights to various parts of the input fuzzy sets (e.g. to Ai .s, Bi .s and Ci .s) to emphasize the mutual importance of the corresponding input variables. Step 3. Put the IF–THEN rules in a linear order with respect to their mutual importance, or give some criteria on how this can be done when necessary; i.e. give a criteria on how to distinguish inputs causing equal degree of total fuzzy similarity in di5erent IF-parts. Step 4. For each THEN-part i, give a criteria on how to distinguish outputs with equal degree on membership (e.g. w0 and v0 such that  Di (w0 ) =  Di (v0 ); w0 = v0 ). A general framework for the inference system is now ready. Assume then that we have actual input values, e.g. (x 0 ; y 0 ; z 0 ). The corresponding output value w0 is found in the following way. Step 5. Consider each IF-part of the rule base as a crisp case, and compare the actual input values separately with each IF-part, in other words, count total fuzzy similarities between the actual inputs and each IF-part of the rule base; by the above Theorems, this is equivalent to counting weighted means, e.g. m1 A1 (x0 ) + m2 B1 (y0 ) + m3 C1 (z0 ) = Similarity(actual; Rule 1) m1 A2 (x0 ) + m2 B2 (y0 ) + m3 C2 (z0 ) = Similarity(actual; Rule 2) .. .. . . m1 Ak (x0 ) + m2 Bk (y0 ) + m3 Ck (z0 ) = Similarity(actual; Rule k) where m1 ; m2 and m3 are the weights given in Step 2. Step 6. Fire an output value w0 such that Di (w0 ) = Similarity(actual; Rule i) corresponding to the maximal total fuzzy similarity Similarity(actual,Rule i), if such Rule i is not unique, use the mutual order given in Step 3, and if there are several such output values w0 utilize the criteria given in Step 4. Notice that, in Step 5, we have introduced a new way to count fuzzy conjunction. Indeed, if A and B are fuzzy set and for an element x we have membership degrees  A (x) = a;  B (x) = b where a; b ∈ [0; 1], then  A and B (x) = (m1 a + m2 b)=(m1 + m2 ). Moreover, Step 6 can be view as an instance of Generalized Modus Ponens in the sense of Pavelka’s well-de4ned fuzzy logic

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(cf. [21]), i.e. RGMP :

; ( imp !) !

;

a; b ab

;

where  corresponds to the IF-part, ! corresponds to the THEN-part, a is the degree of maximal total fuzzy similarity and b = 1. In general, this gives a many-valued logic based theoretical justi4cation to Step 6. In case the outputs are concrete actions that can either be taken or not we de4ne ‘take the action only in case a ∈ [c; 1], where c is a suitable value’. Of course, there is no universal guarantee that the above presented schema of similarity-based algorithm would always perform better than other fuzzy IF–THEN inferences systems do. However, we can specify our algorithm by putting extra demands, for example, in some cases the degree of total fuzzy similarity of the best alternative should be greater than some 4xed value  ∈ [0; 1]; in particular, if the rule base is incomplete. Sometimes all the alternatives possessing the highest fuzzy similarity should be indicated, or the di5erence between the best candidate and second one should be larger than a 4xed value ! ∈ [0; 1]. All this depends on an expert’s choice. 2.1.1. Fuzzy input–crisp output The simplest—and a bit modi4ed—case to utilize the algorithm is to make choices between discrete alternatives, say n routes between two cities. We need only one IF-THEN-rule; the IF-part corresponds to the criteria (expressed by fuzzy sets) the choice should meet, and the THEN-part simply reads: ‘select’. The route possessing the highest total fuzzy similarity to the IF-part will be chosen. Of course, in this case Steps 3 and 4 is empty. In the application part of this paper we will present in details two more complicated inference systems such that the outputs are discrete actions, namely tra c signals ‘green’ and ‘red’. We will see that in such case Step 4 will be empty. 2.1.2. Fuzzy input–fuzzy output The most complicated are such control systems that inputs as well as outputs are fuzzy; in the application part of this study we will give an example on how to proceed. 2.1.3. Smoothness of fuzzy control Any fuzzy IF–THEN inference systems generates a mapping " : X  Y , called control map [11], which determine the real performance of the system, the relation between inputs and outputs. In our algorithm the control map is such that "(x) = y, where Di (y) = Similarity(x; Rule i), the maximal total fuzzy similarity. Due to Steps 3 and 4, the corresponding Rule i and the output value y are unique. Intuitively, a control systems is smooth if small changes in inputs cause small changes in the corresponding outputs. This idea can be formalized by using total fuzzy similarity. Without loss of generality we may assume that the input and output universes X and Y are scaled, i.e. X = [0; 1]n and Y = [0; 1]. Now de4ne, for any x = (x1 ; : : : ; xn ); x = (x1 ; : : : ; xn ) ∈ X , n 1  (xi ↔ xi ); Sim(x; x ) = n i=1



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which is a total fuzzy similarity in X . In a same manner set, for any y; y ∈ Y , Sim(y; y ) = y ↔ y : Denition 2. Assume ; !61 are real numbers close to 1. A fuzzy control system is called (; !)smooth if ∀x; x ∈ X :  6 Sim(x; x )

implies ! 6 Sim("(x); "(x ))

and, in particular, -smooth if ∀x; x ∈ X :  6 Sim(x; x )

implies  6 Sim("(x); "(x )):

Since x ↔ y = 1 if, and only if x = y [27], we reason that any fuzzy inference system is 1smooth. Generally the (; !)-smoothness of a fuzzy inference system, however, depends on fuzzy control engineer’s choice of rule base, shapes of fuzzy sets, etc. Roughly speaking, in a smooth system  and ! are close to each other.

3. Signalized isolated pedestrian crossing: fuzzy input–crisp output The 4rst application of the Algorithm controls a signalized isolated pedestrian crossing (see Picture 1). Normally, the signals of isolated pedestrian crossings in Finland are working as tra c actuated, and the rest phase is vehicle green (important safety aspect). Two detectors are located per each lane, one at the stop line and the other 60 m from the stop line. Pedestrian green time is constant (10 s) or even actuated (6–14 s) in speci4c conditions if children or elderly people are numerous. The main goal of fuzzy control is to give pedestrians an opportunity to cross the street safely, and with minimum waiting time, but also that the risk of rear-end collisions is minimized (minimize the number of approaching vehicles at the termination moment). It is also important that control does not encourage pedestrians to cross the street during the pedestrian red phase. Controlling the timing of a tra c signal means making the following evaluation constantly: either to terminate the current phase and to change it to the next phase, or to continue the current phase. In other words, a controller incrementally evaluates these two options and takes the most appropriate option. This means the output is the decision about the termination (T ) or the extension (E) of vehicle signal group (crisp value). The input parameters in use are ◦ pedestrian waiting time in seconds, PWT, corresponding to three fuzzy sets; short, long, very long, ◦ maximum number of approaching vehicles=lane, NoAV, corresponding to three fuzzy sets none, some, many, ◦ discharging queue indicator, gap between vehicles at stop line GAP, corresponding two fuzzy sets; low, high. For the corresponding fuzzy sets, see Picture 2.

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Picture 1. Layout of a signalized isolated pedestrian crossing.

Normal 1

Membership Grade

0. 8 0. 6 0. 4 0. 2 0 0

5

10

15

20

25

30

35

Pedestrian Wait Time (sec) Short

Long

Very long

Membership Functions re : Discharge Gap (S) 1

0. 8

0. 8

Membership Grade

Membership Grade

Membership Functions re : ApproachVehicle (A) 1

0. 6 0. 4 0. 2 0

0. 6 0. 4 0. 2 0

0

1

2

3

4

5

6

1

1 .5

Approaching Vehicles (veh) Few

Some

2

2.5

Discharge Gap (sec) Many Large

Picture 2. Fuzzy sets for pedestrian signals.

Small

3

J. Niittym/aki, E. Turunen / Fuzzy Sets and Systems 133 (2003) 109 – 131

121

Picture 3. Two decision tables for pedestrian signals.

In this fuzzy IF–THEN inference system, general rule formulation is the following: IF

PWT is short=long=verylong AND NoAV is none=some=many AND GAP is low=high THEN Terminate=Extend

The rule base is complete, indeed, total number of rules is 18 (= 3 × 3 × 2). There are 9 rules for the extension and 9 rules for the termination decisions. The rules are shown in Two decision tables for Pedestrian Signals (see Picture 3.) We have constructed a complete rule-base, however, looking at Picture 3, one can easily reduce the number of rules to 11. We have now settled Steps 1 and 2 of the Algorithm. According to an experienced tra c signal designer, in 4fty–4fty situation the decision is Extension. This corresponding to Step 3 of the Algorithm. Clearly, Step 4 is empty. Steps 5 and 6 is straightforward.

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To illustrate the performance of the control system, assume a pedestrian has been waiting 13 s, there are 2 vehicles approaching and their gap is 1:5s. Such a situation is the most similar to the case PVT is long,  long (13) = 0:6, NoAV is some,  some (2) = 1, and GAP is small,  small (1:5) = 0:8. The degree of total fuzzy similarity to this IF-part is 0.8, and the corresponding THEN-part is ‘Extend vehicles green signal’. 3.1. Simulation results A previous version of this control system was a Matlab Fuzzy Logic Toolbox’s Mamdani style fuzzy inference machine. Simulations made by HUTSIM showed [17] that even the previous version performed better or at least as well as traditional isolated pedestrian signals do. Therefore the performance of the Lukasiewicz based control system was compared to this Mamdani style inference machine. 1 In Table 1, average pedestrian delays obtained by Mamdani-style Control Algorithm, noted by Fuz, and Lukasiewicz Control Algorithms, noted by Fsim(1,1,1) and Fsim(3,2,1), are shown. In Table 2, the corresponding average vehicle delays are shown. The performances of the three control systems Mamdani-style, fuzzy similarity based with weights 1,1,1 and fuzzy similarity based with weights 3,2,1 were compared with respect to the two measures: average pedestrian and vehicle delays for three pedestrian volumes: 15 pedestrian per hour, 50 pedestrian per hour and 150 pedestrian per hour. The results on Tables 1 and 2 are based on a simulation by HUTSIM of more than 90 h. Besides, a statistical hypothesis ‘The average delays are equal in each case’ was tested by approximate t-test on the risk level  = 0:01. According to the results on Table 1, there is no di5erence if pedestrian volume is low (15 ped=h). The pedestrian delays seems to be slightly smaller if control strategy is Mamdani. However, this di5erence is not statistically signi4cant. If tra c volume is high (see Table 2) and, especially, if pedestrian volume is high too, then the results of fuzzy similarity are better. This di5erence has also statistical signi4cance. Obviously, the Similarity Algorithm gives larger capacity, but Mamdani-style Algorithm is more pedestrian friendly. 4. Multi-phase vehicle control For the other two applications of the Algorithm, consider a T-junction in Picture 4. The fuzzy rules used in our project work at three levels: the tra c situation level, the phase and sequence level, and the green extension level. The most common tra c signal control mode ‘Two-phase Control’ was discussed Niittym%aki and Pursula [18]. In this paper we concentrate on multi-phase control by presenting the algorithms and results of the second and third level. Tra c Gow on the main street (phase A) is from two to ten times more intensive than tra c Gow from the other direction. Normally the phase order is A–B–C–A, however, if there is low request, i.e. very few or no vehicles in the next phase B or C, then this phase can be skipped. Thus, the order can be e.g. A–C–A–B–C or A–B–A–B–C. The 4rst task is to determine the right phase order; fuzzy phase selector decides the next signal group. The second task is the exact timing and length of the current green phase. 1

The rule base and the fuzzy sets were same.

J. Niittym/aki, E. Turunen / Fuzzy Sets and Systems 133 (2003) 109 – 131

123

Table 1 Average pedestrian delays and standard deviations Ped.volume

Veh.volume

Average pedestrian delay (s)

SD (s)

Fuz

Fsim(1,1,1)

Fsim(3,2,1)

Fuz

Fsim(1,1,1)

Fsim(3,2,1) 2,19 −1 −3; 42 4,07 5,95 5,27 6,45 5,3 7,08 8,66 7,35 3,13 2,36 2,36 3,42 2,49 4,6 4,86 5,51 7,67 3,74 1,34 1,28 1,28 1,69 1,63 2,86 2,4 2,37 4,89 2,6 1,97 2,05

15 15

200 400

4,5 8,1

4,5 8,1

4,5 8,1

2,25 3,42

2,19 3,42

15 15 15 15 15 15 15 15 50 50 50 50 50 50 50 50 50 50 150 150 150 150 150 150 150 150 150 150 150 150

600 800 1000 1200 1400 1600 1800 2000 200 400 600 800 1000 1200 1400 1600 1800 2000 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

7,7 10,4 9,6 10,4 13,5 12,7 14,8 16,5 5,8 6,1 7,2 8,6 9,3 11,3 10,7 13,6 14,5 14,3 5,5 6,7 8 8,8 8,6 9,6 12,2 12,3 12,4 14,4 14,7 17,6

7,7 10,4 9,7 10,4 14,2 13 16,5 17,6 5,8 6,1 7,2 8,4 9,7 11,2 12,7 14,8 16,2 15,6 5,5 6,7 7,8 8,8 9,2 10,2 13,7 14 14,5 17,3 17,4 17,4

7,8 10,6 10 10,7 15,2 14,6 17 18,9 5,9 6,2 7,3 8,4 9,4 11,7 13,2 14,9 16,4 16,1 5,5 7 8 9,1 9,4 10,3 13,9 13,7 15,7 16,2 17,2 18,6

3,3 5,89 5,59 6,03 5,11 5,73 8,7 7,71 1,55 2,38 2,36 3,78 2,49 4,56 3,23 8,08 6,37 3,73 1,34 1,46 1,28 1,65 2,87 2,65 2,91 4,54 3,49 2,12 2,87 3,31

4,07 5,89 5,27 6,03 4,7 5,68 7,16 6,32 3,13 2,38 2,36 3,42 2,31 4,56 3,69 4,73 6,07 3,47 1,34 1,46 1,1 1,65 3,2 2,19 2,09 1,43 2,06 1,72 2,41 2,71

4.1. Phase control: fuzzy input–crisp output The goal is to determine the right phase order; after A either B or C. The basic principle is that phase B can be skipped if there is no request or if total waiting time of vehicles V(B) in phase B is low, and similarly, phase C can be skipped if there is no request or if total waiting time of vehicles V(C) in phase C is low. Thus, after phase B the next phase is C or A, and after phase C

124

J. Niittym/aki, E. Turunen / Fuzzy Sets and Systems 133 (2003) 109 – 131

Table 2 Average vehicle delays and standard deviations Ped.volume

Veh.volume

15 15 15 15 15 15 15 15 15 15 50 50 50 50 50 50 50 50 50 50 150 150 150 150 150 150 150 150 150 150 150 150

200 400 600 800 1000 1200 1400 1600 1800 2000 200 400 600 800 1000 1200 1400 1600 1800 2000 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

Average vehicle delay (s)

SD (s)

Fuz

Fsim(1,1,1)

Fsim(3,2,1)

Fuz

Fsim(1,1,1)

Fsim(3,2,1)

1 1,1 1 1,6 2,1 1,8 2,4 2,5 2,9 3,5 1,8 2,5 2,9 3,1 3,7 3,6 4,3 4,2 5,9 6,5 3,7 5,4 5,1 5,6 6,4 7,3 7,4 8,6 10,4 15,2 21,7 180,3

1 1,1 1 1,6 2,1 1,8 2,3 2,5 2,8 3,4 1,8 2,5 2,9 3 3,7 3,5 4 4,4 5,5 5,7 3,7 5,3 4,8 5,6 6,1 6,9 7 8 10,2 10 11,3 118,3

1 1 1 1,6 2,1 1,8 2,3 2,4 2,8 3,5 1,7 2,5 2,9 3 3,6 3,5 4,1 4,2 5,2 5,8 3,7 5,3 4,8 5,5 5,9 6,9 7 7,8 9,1 9,4 12,7 64,1

1,11 0,64 1,05 0,65 1 1 0,9 0,9 1,21 1,26 1,17 1,32 1,06 1,2 1,34 0,91 1,79 1,72 2,34 2,05 1,05 1,05 1,59 1,21 0,86 1,23 1,37 2,38 2,6 4,7 6,23 60,4

1,06 0,57 0,47 0,87 1,14 1,14 1,03 0,71 1,15 1,44 1,07 3,38 2,36 1,24 2,31 1,1 1,95 1,67 2,34 2,04 1,85 1,75 1,01 0,94 1,34 1,53 1,34 2,38 1,63 4,47 3,42 82,8

1,06 0,57 0,47 0,87 1,14 1,14 1,03 0,69 1,24 1,38 1,07 1,11 2,36 1,24 2,49 1,14 1,71 1,71 2,04 1,33 1,85 0,95 0,95 0,88 1,48 1,65 1,18 1,87 167 1,51 3,32 50,3

the next phase is A. In details, the dynamics of the inference is the following. After phase A, IF IF IF IF

V(B) V(B) V(B) V(B)

is is is is

high medium low less than low

AND AND AND AND

V(C) V(C) V(C) V(C)

is is is is

any over saturated more than medium more than medium

THEN THEN THEN THEN

phase phase phase phase

is is is is

B C C C

J. Niittym/aki, E. Turunen / Fuzzy Sets and Systems 133 (2003) 109 – 131

125

Picture 4. Layout and three phases of a T-junction.

The corresponding membership functions are shown in Picture 5. Corresponding to Step 3 of the Algorithm, if the maximal total similarity is not unique, the phase with the longest waiting time will be 4red, or in the worst case, the next phase will not be skipped. The performance of the fuzzy phase control is now straightforward; for example, after phase A, if there are 7 vehicles in phase B and 3 vehicles in phase C, then the next phase will be B. 4.2. Green ending control: fuzzy input–fuzzy output The 4rst two applications of the Algorithm are relatively simple. Indeed, the output is just a discrete action. To test what happens in such a situation that the output is a fuzzy variable we constructed a green ending control for the above T-junction. Depending on circumstances, the extension varies from 0 to 12 s. The main goal is to maximize the tra c capacity by minimizing inter-green times. The basic principle is that signal group can be kept in green while no disadvantages to other Gows occur. This is also called the method to use extra green. The main decision is the right termination moment of the green, the moment when the green of the 4rst signal group of phase A can be terminated, so that the 4rst signal group of phase B or C can be started. Secondly, the decision will be checked when the last signal group of phase A is ready to be terminated. The rule base is more or less a good guess by the authors, and the rules are as follows: Rule01 Rule02 Rule03 Rule04 Rule05

IF IF IF IF IF

A A A A A

is is is is is

Zero a Few MT a Few MT Medium None

AND AND AND AND AND

Q Q Q Q Q

is is is is is

Any Value LT Medium Any Value Any Value None

THEN THEN THEN THEN THEN

E1 E1 E1 E1 E1

is is is is is

RULE SET 1. First extension after 65 seconds green signal.

Zero Short Medium Long None

126

J. Niittym/aki, E. Turunen / Fuzzy Sets and Systems 133 (2003) 109 – 131 V(B)

Membership functions

1 low

0. 8

med

0. 6

high

0. 4

less than low

0. 2 0 0

2

4

6

8

10

12

14

16

Total wait time V(B) [10 sec] V(C)

Membership functions

1 0. 8 0. 6

over saturated

0. 4

more than media

0. 2

any

0 0

2

4

6

8

10

12

14

16

Total wait time V(C) [10 sec]

Picture 5. Membership functions in fuzzy phase control.

Rule06 Rule07 Rule08 Rule09 Rule10

IF IF IF IF IF

A A A A A

is is is is is

Zero a Few Medium Many None

AND AND AND AND AND

Q Q Q Q Q

is is is is is

Any Value LT Medium Any Value Any Value None

THEN THEN THEN THEN THEN

E2 E2 E2 E2 E2

is is is is is

Zero Short Medium Long None

RULE SET 2. E1 seconds after the 4rst extension. RULE11 RULE12 RULE13 RULE14 RULE15

IF IF IF IF IF

A A A A A

is is is is is

Zero a Few Medium Many None

AND AND AND AND AND

Q Q Q Q Q

is is is is is

Any Value LT Medium LT Medium LT Medium None

THEN THEN THEN THEN THEN

E3 E3 E3 E3 E3

RULE SET 3. E2 seconds after the second extension.

is is is is is

Zero Short Medium Long None

J. Niittym/aki, E. Turunen / Fuzzy Sets and Systems 133 (2003) 109 – 131

RULE16 RULE17 RULE18 RULE19 RULE20

IF IF IF IF IF

A A A A A

is is is is is

Zero MT a Few Medium Many Any Value

AND AND AND AND AND

Q Q Q Q Q

is is is is is

Any Value LT Medium LT Medium LT a Few Too Long

THEN THEN THEN THEN THEN

E4 E4 E4 E4 E4

is is is is is

127

Zero Short Medium Long Zero

RULE SET 4. E3 seconds after the third extension. RULE21 RULE22 RULE23 RULE24 RULE25

IF IF IF IF IF

A A A A A

is is is is is

Zero MT a Few Medium Many Any Value

AND AND AND AND AND

Q Q Q Q Q

is is is is is

Any Value a Few LT a Few LT a Few Too Long

THEN THEN THEN THEN THEN

E5 E5 E5 E5 E5

is is is is is

Zero Short Medium Long Zero

RULE SET 5. E4 seconds after the fourth extension. The corresponding fuzzy sets are shown in Picture 6. Again, corresponding to Steps 3 and 4 of the Algorithm, we need an extra rule. To minimize the risk of too long extensions for phase A, we decide to give the shortest possible extension in case total fuzzy similarity or the output values are not unique. Moreover, corresponding to Step 6—here fuzziness appears in outputs—if we have an input such that the largest total fuzzy similarity is obtained e.g. at Rule 24 and at a degree a there, then such an extension ‘Long’ is 4red that its membership degree is the largest number which is 6a, and which is simultaneously the shortest such an extension (in fact, this approach is known as least of maximum defuzzi4cation approach). This guarantees the soundness of inference in the sense of Pavelka [21]. A smoothness analysis shows that this rule base and the fuzzy sets do not o5er a particular smooth control. Indeed, in the worst case an extra approaching vehicle causes 6 s longer extension, thus only (0,95)(0.67)—smoothness for total fuzzy similarity based control. A Mamdani-style control system is even more unsmooth, namely (0.95)(0.583)-smooth. The performance could be improved, however, by writing down a 11 × 16 control map between the actual inputs and outputs, and simply removing the unstable parts of the control. 4.3. Simulation results The performances of the two control systems Mamdani-style and fuzzy similarity-style (denoted by Mamdani (without Phase Selector) and Mamdani PS (with Phase Selector), and tot.sim and tot.sim PS, respectively) were compared with respect to average vehicle delays on various vehicle densities and vehicle ratio (main=minor street volume). Moreover, a statistical hypothesis ‘The average delays are equal in each case’ was tested by approximate t-test on the risk level  = 0:01. Simulation results done by HUTSIM (see Tables 3–5) indicate that the fuzzy phase selector (PS) seems to improve the control performance in both cases, i.e. comparison between Mamdani and Mamdani PS and comparison between tot.sim and tot.sim PS; in some cases this improvement has statistical signi4cance. However, with larger minor street share of total volume, the phase selector is unable to skip phases and, because of that, no time saving is accomplished. With lower main street

128

J. Niittym/aki, E. Turunen / Fuzzy Sets and Systems 133 (2003) 109 – 131 A Membership functions

1 none zero a few med many mt a few mt med

0.8 0. 0.6 0. 0.4 0. 0.2 0. 0 0

2

4

6

8

10

12

Number of approaching vehi vehicl cles es Q

Membership functions

1 0.8 0. 0. 0.6

none a few too long any value ltt medium

0. 0.4 0.2 0. 0 0

2

4

6

8

10

12

14

16

Number of vehicles in queque Ei

zero

1 Membership functions

short 0. 0.8

medium long

0. 0.6 0. 0.4 0. 0.2 0 0

2

4

6

8

10

12

14

Extension length [sec]

Picture 6. Membership functions for fuzzy green ending control.

volumes, the phase selector control results to equal or slightly lower delays compared to the fuzzy control with 4xed phase order. Comparison between Mamdani and tot.sim show that there is no signi4cant di5erence between the fuzzy control methods operating without phase selector. With low volumes, the total similarity based algorithm seems to give somewhat bigger delays than the Mamdani method, but the di5erence has no statistical signi4cance, and it vanishes when the total volume increases. However, the total similarity algorithm (tot.sim. PS) 4ts better to the phase selection process than the Mamdani method

13 12.5 12.3 12.3

12.3 12.3 12.1 12.1

1.366 1.473 1.390 1.390

12.7 12.2 12.6 12.6

2.239 2.079 2.345 2.345

Tot.sim Mamdani Tot.sim PS Mamdani PS

12.8 11.6 11.9 11.9

400 Average SD delay

Veh=h 200 Veh. density ratio Average SD 10=5 delay 12.5 12.4 12.6 12.6

1.488 1.970 1.556 1.556

600 Average SD delay

1.957 1.874 1.579 1.579

600 Average SD delay

Table 5 T-junction; average vehicles delays and standard deviations

1.363 1.893 1.123 1.123

12.5 13.1 11.4 11.4

3.351 2.559 2.826 2.826

Tot.sim Mamdani Tot.sim PS Mamdani PS

12.7 10.3 10.8 10.8

400 Average SD delay

Veh=h 200 Veh. density ratio Average SD 10=2 delay

1.562 1.182 1.291 1.291

600 Average SD delay

Table 4 T-junction; average vehicles delays and standard deviations

2.061 1.941 1.893 1.893

14 12.2 11.8 11.8

1.921 2.406 2.447 2.447

Tot.sim Mamdani Tot.sim PS Mamdani PS

13.4 12.1 12 12

400 Average SD delay

Veh=h 200 Veh. density ratio Average SD 10=1 delay

Table 3 T-junction; average vehicles delays and standard deviations

1.661 2.197 2.046 1.937

1.651 1.612 2.259 2.259

13.5 14.2 13.9 13.9

1.328 1.790 1.461 1.461

800 Average SD delay

13.4 13.8 13.5 13.5

800 Average SD delay

13 13.2 11.7 12.4

800 Average SD delay 1.405 1.216 1.952 1.741

1.307 1.364 1.536 1.536

14.2 15 14.2 14.8

1.172 2.035 1.364 1.483

1000 Average SD delay

13.9 13.9 14.4 14.4

1000 Average SD delay

14.4 13.6 13.9 14.3

1000 Average SD delay 1.653 1.135 1.825 1.544

2.029 1.590 2.137 1.914

16.2 17 16.4 17.1

1.911 1.905 1.800 1.323

1200 Average SD delay

15.3 15.7 14.5 15

1200 Average SD delay

14.7 14.9 13.2 14.8

1200 Average SD delay 1.766 2.051 2.398 2.536

16.5 16.9 17.4 16.8

1.552 2.015 2.014 2.035

2.694 2.255 2.952 3.123

22.5 20.6 16.1 19.8

4.960 3.296 3.049 3.124

1600 Average SD delay

18.3 17.4 13.8 18.3

1600 Average SD delay

24.4 30.2 22.1 23.7

6.223 10.845 3.567 4.024

1600 Average SD delay

1.299 2.160 1.516 1.564

1400 Average SD delay

16.4 17.3 14.6 17

1400 Average SD delay

15.2 16.1 13.3 15.9

1400 Average SD delay

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(Mamdani PS). Indeed, the total similarity-based phase selector gives clearly lowest delays with high main street volumes and, by vehicle density 1600 vehicles=h, and this di5erence has statistical signi4cance, too. The Mamdani-selector’s performance remains at the same level as the 4xed order fuzzy controllers. 5. Conclusion Fuzzy control algorithms suit well to various tra c and transportation problems as was shown e.g. in [17, 20, 25]. In particular, this holds true in tra c signal control except, of course, pathological tra c jams or situations where there are very few vehicles in circulation; there 4rst-in-4rst-out is the only reasonable control strategy. In this paper we introduced a new fuzzy IF–THEN control algorithm based on Lukasiewicz equivalence. The algorithm is seeking for the most similar IF-part to the actual input value, and the corresponding THEN-part is then 4red. Thus, the main distinction between our approach and other fuzzy inference systems is that only one IF–THEN rule determines the output. Three realistic tra c signal control systems were constructed by means of the Algorithm and a simulation model tested their performance. Similar simulations were made to classical Mamdani style fuzzy inference system, too. The results with respect to average vehicle and pedestrian delay or average vehicle delay were in most cases almost equal on fuzzy similarity based control and on the Mamdani control systems. However, if tra c density is high then the similarity algorithm gives statistical signi4cant better results than Mamdani control. We conclude that similarity-based inference systems o5er a competitive method for control in tra c signal design. Acknowledgements The authors are grateful to the anonymous referees for their valuable comments, which lead to considerable improvement of this paper. References [1] W. Bandler, L. Kohout, Fuzzy power sets and fuzzy implication operations, Fuzzy Sets and Systems 4 (1980) 13–30. [2] M.G.H. Bell, Future directions in tra c signal control, Transport. Res. 26 (1992) 303–313. [3] J.L. Castro, F. Klawonn, Similarity in fuzzy reasoning, Mathware and Soft Comput. 2 (1995), 197–228, Using Fuzzy Logic, Proc. FUZZ-IEEE’99, pp. 1371–1376. [4] S.-M. Chen, Measures of similarity between vague sets, Fuzzy Sets and Systems 74 (1995) 217–223. [5] S. Chiu, S. Chand, Adaptive tra c signal control using fuzzy logic, Proc. FUZZ-IEEE’99, pp. 1371–1376. [6] R. Cignoli, M.L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 1999. [7] R. Cignoli, F. Esteva, L. Godo, A. Torres, Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft Computing 4 (2) (2002) 106–112. [8] D. Dubois, H. Prade, Similarity-based approximate reasoning, in: J.M. Zurada, R.J. Marks II, X.C.J. Robinson (Eds.), Computational Intelligence Imitating Life, Proc. IEEE Symp., Orlando, FL, June 27–July 1, 1994, IEEE Press, New York, pp. 69 –80.

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