- Email: [email protected]
Determining fuzzy membership functions with tabu search—an application to control
Fuzzy Sets and Systems 139 (2003) 209 – 225 www.elsevier.com/locate/fss
Determining fuzzy membership functions with tabu search—an application to control Aytekin Ba)gi*s∗ Department of Electronic Engineering, Erciyes University, Kayseri, 38039 Turkey Received 19 April 2002; received in revised form 4 October 2002; accepted 10 October 2002
Abstract This paper presents a new approach for the optimum determination of membership functions for a fuzzy logic controller based on the use of tabu search algorithm. To demonstrate the e0ciency of the suggested approach, a speci2c control problem—operation of spillway gates of reservoirs during 3oods is selected. Simulation results showed that the proposed approach could be employed as a simple and e4ective optimization method for achieving optimum determination of membership functions. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic controller; Membership functions; Tabu search algorithm; Reservoir control
1. Introduction Fuzzy logic controllers (FLCs) are intelligent control systems characterized by a set of linguistic statements based on expert knowledge or experience. If the mathematical model of a process is de2cient or complicated, or if the process is nonlinear or time dependent, or if it is di0cult or impossible to control the process with the conventional methods, then fuzzy logic control is the most suitable method for these types of problems [18,13]. A fuzzy controller is formed by input and output fuzzy sets assigned over the controller input and output variables, and a collection of fuzzy rules. These rules provide the necessary connection between the controller input and output fuzzy sets. In design of a fuzzy controller, a major di0culty is encountered in the production of optimal fuzzy rule base. Determination of the most appropriate membership functions consisting of the rule base makes signi2cant impacts on the 2nal performance of the controller. In order to get the best performance from FLC, the membership functions must be optimally determined. In general, de2nitions of the membership functions and production of the ∗
Fax: +90-352-437-57-84. E-mail address: [email protected] (A. Ba)gi*s).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 5 0 2 - X
210
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
rule base are acquired from a human expert using trial-and-error method. However, converting the experts’ know-how into fuzzy rules is di0cult, and often, results are incomplete, unnecessary and include con3icting knowledge, since operators and control engineers are not capable of speci2c details or cannot express all their knowledge including intuition and inspiration. Therefore, applying automatic techniques to obtain the optimum membership functions appears as an alternative. To improve behaviour of designed fuzzy controller it is necessary to use some optimization methods such as genetic algorithm (GA) [3,11,14,20,21], simulated annealing (SA) [5,12], tabu search (TS) [4,15]. Chang and Wu [3] introduced a GA based tuning method for symmetric membership functions in a fuzzy control system. Homaifar and McCormick [11] examined the applicability of GA’s in the simultaneous design of membership functions and rule sets for FLCs. They employed a GA procedure to determine only the base lengths of triangular fuzzy sets and not to location of the peaks. Surmann [20] applied the GA approach to optimize a fuzzy rule based system for charging high-power NiCd batteries. In [5], SA algorithm with an adaptation for continuous minimization by the simplex method is used to tune of fuzzy models, and the height of fuzzy terms is considered as a tuneable parameter within the modelling process. Denna et al. [4] present an approach for automatic de2nition of the fuzzy rules based on the use of TS algorithm. To determine the most appropriate rule base for the problem, they employed the reactive form of TS algorithm. Karaboga [15] described a method for constructing the fuzzy rule base. In the method, using the TS algorithm, search of the solution is based on the basis of the automatic learning of fuzzy rule table with the preselected membership functions. GA’s are search algorithms that use operations found in natural genetic to guide journey through a search space [17]. They work with a population of individuals or strings (chromosomes). Each string represents a feasible solution to the problem, and a population consists of a set of admissible solutions. GA’s use three basic operators to manipulate the genetic composition of a population: reproduction, crossover, and mutation. However, there are many decisions the user has to make such as the representation, population size, chromosome structure, mutation and crossover probability, recombination probability, and mating subset selection method. GA’s consider many points in the search space simultaneously, not a single point. Thus, they generate entire populations of points, test each point independently, and combine qualities from existing points to form a new population containing improved points. Furthermore, they might evaluate the same solutions several times since the GA’s use probabilistic rules to guide their search, not deterministic rules. Hence, a large amount of computation time for obtaining a near optimal solution can be required during search of the algorithm. In contrast, the evolution of a TS algorithm is carried on using a single individual. In this case, the computational resources required are much more limited, and evolution can proceed very rapidly. Because the TS is usually based on the use of two important factors (tabu list size and tabu conditions), organization of the algorithm has not a complex procedure. SA is a heuristic optimization technique that derives from statistical mechanics [4,17]. The algorithm consists of an iterative method with a single solution. Four basic ingredients are needed to apply SA in practice: a concise problem representation, a neighbourhood function, a transition mechanism, and a cooling schedule. As for the choice of the cooling schedule, there exist some general guidelines. However, no general rules that guide the choice of the other ingredients are known. This method uses a probabilistic transition rule. Therefore, each of iteration consists of randomly changing the current solution to create a new solution in the neighbourhood of the current solution. Since the steps in the SA algorithm are taken randomly, the search for good solution can be long,
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
211
and 2nal solution could be suboptimal especially in problems with enormous solution spaces. The performance of the algorithm based on the annealing scheduling, and, therefore, it has to be chosen very carefully. Furthermore, in the basic strategy of the SA, time is a critical factor [4]. If SA is too fast, the resulting solution could be unsatisfactory. In contrast, time is not a critical factor in TS algorithm. Although TS is also a form of iterative search, it can discover an optimal solution for the problem in a reasonable time. The biggest di4erence of the TS algorithm is the use of memory during the search process [4,8,9,17]. In the GA and SA, the memory cannot be e4ectively used as that in TS. They use memory to store a set of individuals, and the past history of the search does not in3uence the current move. On the other hand, TS employs an explicit memory to store historical information on the course of the search trajectory. Thus, memory is used to guide the selection of the next solutions. This property of the TS algorithm is a signi2cant superiority over other optimization algorithms mentioned above. In this paper, we present a simple and e4ective method for the optimum determination of the membership functions based on the use of TS algorithm. This method has been applied to fuzzy logic based reservoir operating system of dams during 3oods. The membership functions consisting of fuzzy rule base are optimized by using di4erent objective functions. The results obtained by means of the fuzzy control system are compared with those of conventional control methods.
2. Structure of a fuzzy logic controller A fuzzy controller allows us to use a control strategy expressed in the form of linguistic rules for the de2nition of an automatic control strategy. A typical fuzzy rule can be composed as IF A is Ai AND B is Bi OR C is Ci THEN D is Di ; where A, B, C, and D are fuzzy variables, Ai , Bi , Ci , and Di are fuzzy linguistic values, “AND” and “OR” are connectives of the rule. The advantage of using this linguistic representation is that it is very easy to change the de2nition of system by means of this description. A basic FLC can be decomposed into four basic components [18]. These are fuzzi2cation unit, knowledge base (rule base and data base), decision making unit (inference unit), and defuzzi2cation unit (Fig. 1). Fuzzi2cation unit transforms the measurement data into fuzzy sets. Knowledge base Knowledge Base data base
fuzzification unit
input
rule base
decision making unit
defuzzification unit output
Fig. 1. Con2guration of a basic fuzzy logic controller.
212
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
consists of a rule base and data base. This unit has knowledge about the physical system to be controlled. Decision making unit de2nes how the system should make inferences through the fuzzy rules contained in the rule base. Defuzzi2cation unit aggregates the outputs of all the rules that have 2red for the particular input fuzzy sets to produce a crisp output.
3. Tabu search algorithm TS algorithm was proposed by Glover as an intelligent optimization technique to overcome local optimality in solution processes for hard combinatorial optimization problems [8,9]. The search is based on neighbourhood mechanism. The neighbourhood of a solution is the set of all formations that can be arrived at by a move. The move is a process that transforms the search from the current solution to its neighbouring solution. The 3owchart of TS algorithm procedure is shown in Fig. 2 [17]. To avoid performing a move returning to a recently visited region, the reverses of last moves are forbidden. In other words, these moves are considered as “tabu”. They are recorded in a list called initial solution
create a set of neighbour solutions
evaluate the neighbour solutions
choose the best admissible solution
stopping criteria satisfied?
update tabu list
No
Yes final solution Fig. 2. Flowchart of a standard TS algorithm.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
213
“tabu list”. Therefore, in order to record the history of search, an explicit memory is employed. Tabu list is initialized empty, constructed in consecutive iterations of search and updated circularly in later iterations. A move that is not on the tabu list is called as admissible move. After making a move, if a solution produced by this move is better than all solutions found in prior iterations, then this solution is saved as the best solution. At each iteration of the optimization process, candidate solutions are checked with respect to “tabu conditions”, and hence, the next solution is determined by depending on evaluation values and tabu conditions. The re-generation of a solution previously obtained is avoided by using the tabu conditions (or restrictions) on the possible moves. The tabu conditions are usually based on two important factors: frequency memory and recency memory. Frequency memory keeps the knowledge of how often the same solutions have been made in the past. Recency memory prevents cycles of length less than or equal to a predetermined number of iterations. The tabu conditions used in this work are as follows: If the element k of solution vector does not satisfy the conditions (i) recency (k)¿r:M , (ii) frequency (k)¡f:avgfreq, then it is accepted as tabu and not used to create a neighbour. Here, r and f are recency and frequency factors, M is the number of elements in the solution vector, and avgfreq is the average frequency value. TS has another important element called aspiration mechanism. If a move on the tabu list leads to a solution with a value strictly better than the best obtained so far, it is possible to allow this move to get out of tabu list. This property is used to avoid removing very good moves from consideration and plays an essential role on the search process. In this work, if all available moves are classi2ed tabu, then a least tabu move is selected for new solution. The least tabu move means that this solution is changed less recently and frequently among them but it is still classi2ed as tabu. In this work, maximum number of iterations reached is selected as stopping criterion of the search process. 4. An application to the reservoir operation problem To demonstrate the e0ciency of TS algorithm, we illustrate the application of the approach to a nonlinear control problem—the operation of spillway gates of a reservoir during 3oods. In this paper, the use of a FLC to automatically realize the reservoir operation system of Adana Catalan dam in Turkey is proposed. To ensure that an optimal FLC is designed for the problem, the membership functions of fuzzy rules are optimized by using TS algorithm. The results of the operating system, which is controlled by the optimum fuzzy controller, are compared with the results of conventional ones. 4.1. Physical system to be controlled The reservoir operation system is a control system that manages the spillway gates in a dam to increase or decrease the amount of discharge water [22]. The most important task of this system is to keep the reservoir water level within a prescribed range in any condition by adjusting the opening of the spillway gates. Main variables of a reservoir operation system are in3ow rate (I; m3 =s), out3ow rate (O; m3 =s), reservoir capacity (S; 106 m3 ), minimum and maximum reservoir water surface elevation (Hmin and Hmax , m), actual water level (H; m), and spillway gate opening (d; m) (Fig. 3). The accumulation
214
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 3. Diagrammatic description of reservoir operation.
of storage in a reservoir depends on the di4erence between the rates of in3ow and out3ow. For the interval of Rt, this continuity relationship can be expressed as the following: RS(t) = [I (t) − O(t)]Rt; where RS is the storage accumulated or depleted during Rt, I (t)=O(t) is the average rate of in3ow=out3ow during Rt. Operation of the spillway gates of a reservoir during 3oods is a complex, nonlinear, nonstationary control process because: (a) System to be controlled is signi2cantly a4ected by hydrological conditions. The hydrological conditions have a nonlinear and nonstationary behaviour. (b) The spillway gate must be regulated according to probable in3ow hydrograph. Determination of the in3ow hydrograph can be extremely di0cult if not impossible. (c) Since the system complexity is high and correct prediction of the system behaviour is not easy, it is very di0cult to make a precise model to control the system. (d) Controlling the spillway gate of the dams are usually performed by experienced human operators. But, humans are emotional and forgetful, and may reach incorrect decisions especially under extreme conditions such as over3ows. The wrong decisions during a dangerous in3ow hydrograph may cause serious safety problems for nature and human life. Hence, owing to human factor, there is no guarantee of the safe and e0cient control of process. Some methods used to achieve the reservoir control have been presented in [1,2,10,16,19,23]. Beards presents a deterministic operation procedure [1]. Ozelkana et al. applied the dynamic programming technique for water reservoir management [16]. In [2], it is alleged that to build an intelligent reservoir operation system is impossible without information of optimal input and output hydrological patterns. The model given in [10] presents a set of operation rules with ten-stage for controlling the spillway gate opening. The approaches mentioned above are insu0cient and in3exible modelling methods with low performance. Currently the most common reservoir control strategy in Turkey is based on human operator decisions [7]. In this strategy, a constant amount of water is discharged depending on the reservoir level. The most signi2cant drawback of this approach is how to determine the constant amount of water that must be discharged for a given lake level. Another disadvantage of this strategy is that it does not consider the change rate of lake level to determine the water amount to be released. Therefore, there is no guarantee that the control strategy will have su0ciently good
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
215
Calculation h TS algorithm
dh
Fuzzy Logic Controller
Sensors
d inflow hydrograph
Reservoir, Dam, and Spillway Gates
lake level (H)
Fig. 4. Fuzzy controlled reservoir operating system.
performance for the present complex system with large number of input variables of unpredictable in3ow rates. Because of uncertain process information, system dynamics varying nonlinearly, and inaccurate mathematical model, conventional control techniques will also be insu0cient to provide a solution for an accurate and e0cient reservoir operation. 4.2. Fuzzy controller for the reservoir operation problem The main objectives of the reservoir operation considered herein can be given as the following: (1) The control system must be fast and reliable during any 3ood of any magnitude, which is not predictable beforehand. (2) The control method must drive the lake water surface level to desired minimum set point (Hmin = 118:6 m) in the shortest time during spillway over3ows [6]. (3) Reservoir water level must not be exceed the predetermined maximum set point (Hmax = 127 m) in any time [6]. (4) The rate of change of the released water must be as smooth as possible. The fuzzy control system proposed for the present problem is given in Fig. 4. In the fuzzy control system design, the selection of the controller structure involves the following choices: (a) Input and output variables: The input variables for the fuzzy controller are “lake level” (h) and “change in the lake level” (dh). The output of the controller is “gate opening” (d). Out3ow rate of the reservoir is adjusted by the gate opening, which is controlled by FLC. For h, dh, and d variables, the normalization intervals are selected as [118; 127], [−1; 1], and [0; 12], respectively. (b) Number and type of membership functions for variables: The membership functions used for the fuzzy values of fuzzy variables are selected based on human=expert experience. As shown in Fig. 5, the input variables h and dh are characterized by three fuzzy membership functions, and the output variable d is characterized by 2ve membership functions. All of the fuzzy values are represented by triangular membership functions for simplicity. But, fuzzy membership functions can have di4erent shapes and sizes depending on the designer’s preference or experience, and the system’s performance can also be signi2cantly improved.
216
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 5. Initial membership functions.
Table 1 Rule base Rule number
Rules
(1) (2) (3) (4) (5)
IF IF IF IF IF
h h h h h
is is is is is
h1 , h1 , h2 , h2 , h3 ,
AND AND AND AND AND
dh dh dh dh dh
is is is is is
dh1 , dh2 , dh2 , dh3 , dh3 ,
THEN THEN THEN THEN THEN
d d d d d
is is is is is
d1 d2 d3 d4 d5
(c) Rule base: The rules of the reservoir operation strategy are obtained from information gathered by engineers and experts informed about the dam and operator experience. The rule base of the FLC contains only 2ve rules, which are tabulated in Table 1. (d) Type of inference mechanism: The output of each rule is determined by Mamdani’s max–min inference method. (e) Defuzzi7cation method: Standard centre of area method is employed for defuzzi2cation process.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
217
Fig. 6. Parameterized membership function.
4.3. De7nition of the membership functions Determination of the FLC membership functions using an optimization algorithm takes place in four phases: 1. 2. 3. 4.
creation of a primary membership functions which are arbitrarily initialized, parameterization, de2nition of the rule base, and adjustment of the membership functions.
Number and type of the membership functions for variables is mentioned earlier. Each membership function is characterized by three numerical values. These numerical values represent the position of the membership function on the universe of discourse. In Fig. 6, parameterized fuzzy membership functions (A1 ; A2 ; : : : ; An ) are demonstrated. It can be seen from Fig. 6 that three numerical values are used in de2nition of a membership function. In any physical system with two inputs and one output, when the same membership functions are used for all variables of the system, a sample rule structure can be written as follows: Linguistic rule → IF input (1) is A2 , AND input (2) is A3 , THEN output is A2 . Parameterized rule → IF (a21 a22 a23 ), AND (a31 a32 a33 ), THEN (a21 a22 a23 ). Here, aij is a numerical parameter that characterizes the membership functions, and aij ∈ [0; 1] for h; d; aij ∈ [−1; 1] for dh (i = 1; 2; 3; j = 1; 2; 3). As to this de2nition, each rule consists of [(input number+output number)×3] parameters. Therefore, the number of parameters required for de2ning a fuzzy rule set is [rule number ×(input number + output number)×3]. Since the fuzzy controller with two inputs and one output is used in this work, each fuzzy rule is represented by nine parameters. Thus, 45 parameters are used for 2ve di4erent rules (Fig. 7). This value is also equal to the size of the tabu list. This de2nition of the membership functions for fuzzy rule base has three main advantages: (1) It is easy to understand and de2ne the rules. (2) It provides the evaluation of the all possible solution points for the universe of discourse since the membership functions have not symmetric structure. (3) The same control parameters can be obtained for di4erent membership functions, and hence, the number of membership functions can be reduced during optimization process.
218
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225 h1 = 0.00 h2 = 0.25 h3 = 0.50 dh1 = -1.00 dh2 = -0.50 dh3 = 0.00 d1 = 0.00 d2 = 0.00 d3 = 0.25 d4 = 0.50 d5 = 0.75
dh d h
dh1
dh2
dh3
h1
d1
d2
---
h2
---
d3
d4
h3
---
---
d5
0.00 0.00 0.50 0.00 0.00 0.50 0.25 0.50 0.75 0.25 0.50 0.75 0.50 1.00 1.00
-1.00 –1.00 0.00 -0.50 0.00 0.50 -0.50 0.00 0.50 0.00 1.00 1.00 0.00 1.00 1.00
0.00 0.00 0.25 0.00 0.25 0.50 0.25 0.50 0.75 0.50 0.75 1.00 0.75 1.00 1.00
h
dh
d
0.00 0.50 1.00 -1.00 0.00 1.00 0.00 0.25 0.50 0.75 1.00
0.50 0.75 1.00 0.00 0.50 1.00 0.25 0.50 0.75 1.00 1.00
Rule 1 Rule 2 Rule 3 Rule 4 Rule 5
Fig. 7. Parameter matrix used to represent the rule base.
Table 2 Objective functions used in this work Function (1)
F1 = (1=N )
(2)
F2 = (1=N )
(3)
F3 = F1 + F2
N t=1 N t=1
O(t) exp(|O(t) − O(t − 1)|)
The membership functions used play a crucial role in the 2nal performance of a fuzzy control system. Therefore, selection of the appropriate functions is an important design problem. So, in order to design an optimal FLC the proper membership functions are searched by using TS algorithm. In the design of fuzzy controller for this problem, two main factors are considered by TS algorithm: peak values of the out3ow hydrograph, and changes of these values. During the optimization process, these factors are minimized by TS for the better parameters. In the reservoir operation, three di4erent objective functions are considered (Table 2). In these equations, N is the number of samples, t is time, and O is the peak value of out3ow hydrograph. Neighbourhood values of the parameters used to represent the membership functions are found using the neighbourhood structure given in Table 3. In this structure based on binary code number of the neighbour solutions is B for a solution with B bits. Each neighbour is obtained by changing the only one bit of the solution. The other control parameters r, f, and iteration number were selected as 0.2, 2.0, 250, respectively.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
219
Table 3 Neighbourhood structure Solution Neighbour Neighbour Neighbour Neighbour Neighbour
(1) (2) (3) (4) (5)
1
0
1
1
0
0 1 1 1 1
0 1 0 0 0
1 1 0 1 1
1 1 1 0 1
0 0 0 0 1
Fig. 8. Performance of the FLC with initial membership functions: (a) 3ood versus time; (b) lake level versus time; (c) gate opening versus time curves.
4.4. Simulation results As mentioned earlier, initially the membership functions and rule base are de2ned intuitively. Performance of the FLC using this rule base is presented in Fig. 8. According to Fig. 8, performance of the control system is not certainly good. Time curve of the amount of water discharged is not smooth enough, and exhibits sudden discontinuities. The out3ow hydrograph and reservoir level has undesirable sudden changes due to unsystematic behaviour of gate opening. Spillway gate of the dam is not controlled e4ectively. Therefore, controlling the spillway gate by this operating system is dangerous for nature and human life.
220
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 9. Membership functions optimized by TS algorithm with (a) F1 , (b) F2 , and (c) F3 ..
To improve behaviour of operating system, membership functions were adjusted by using three di4erent objective functions F1 , F2 , and F3 . After the optimization process, the 2nal state of the membership functions for system variables was obtained as shown in Fig. 9. As to this 2gure, number of the membership functions is di4erent for each objective function. The reason for this is that the parameter matrix contains di4erent numerical values for de2ning the membership functions within the fuzzy rule base after the optimization processes (Table 4). FLCs optimized by these objective functions are represented with FLC(1), FLC(2), FLC(3), respectively. The overall performances of these controllers are given in Figs. 10 and 11. Signi2cant performance indexes for fuzzy controllers
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
221
Table 4 Parameter matrix determined by di4erent objective functions Parameter matrix Optimized by F1
0.52 0.25 0.10 0.00 0.25
0.86 0.52 0.10 0.08 0.52
1.00 0.74 0.55 0.26 0.74
−1:00 −0:87 −0:100 −0:87 −1:00
−1:00 0.60 0.03 0.60 −1:00
0.30 0.60 0.30 0.60 0.30
0.28 0.23 0.00 0.00 0.00
0.68 0.60 0.02 0.00 0.13
0.74 0.71 0.09 0.02 0.58
Optimized by F2
0.55 0.40 0.25 0.08 0.08
0.97 0.85 0.74 0.48 0.08
0.97 0.97 0.81 0.74 0.28
−1:00 −0:81 −0:70 −1:00 −0:56
−0:04 0.20 0.60 −1:00 1.00
0.30 0.60 0.94 0.30 1.00
0.30 0.05 0.13 0.00 0.00
0.68 0.13 0.30 0.05 0.00
0.68 0.30 0.50 0.13 0.02
Optimized by F3
0.55 0.48 0.25 0.08 0.00
0.86 0.86 0.52 0.21 0.08
1.00 1.00 0.71 0.61 0.29
−1:00 −1:00 −0:60 −1:00 −0:60
−1:00 −0:42 0.60 0.30 0.60
0.29 0.30 0.60 0.60 0.60
0.50 0.00 0.23 0.00 0.00
0.68 0.06 0.48 0.05 0.00
0.68 0.48 0.65 0.10 0.02
Fig. 10. Routing of out3ow hydrograph from Catalan dam by di4erent FLCs.
are listed in Table 5. In this table, FLC(0) indicates the fuzzy controller that is not optimized by TS algorithm. Figs. 10 and 11 and Table 5 show all the controllers tuned by TS were able to successfully operate the system according to main objectives. It should be noted that the best performance, on average value of the out3ow hydrograph (Oavg ) and maximum change of the out3ow hydrograph
222
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 11. Lake level versus time from Catalan dam by di4erent FLCs.
Table 5 Signi2cant performance indexes Out3ow rate (m3 =s)
FLC(0) FLC(1) FLC(2) FLC(3)
Gate opening (m)
Lake level (m)
Omax
Oavg
ROmax
dmax
Rdmax
hmax
5031.27 5797.26 6149.12 4528.29
2503.68 2483.43 2515.85 2508.26
1538.05 614.08 412.50 291.11
6.87 8.09 8.32 5.34
3.00 1.20 0.72 0.44
124.75 125.57 126.57 126.49
(ROmax ), came from the FLC(3) as shown in Table 5. Using FLC(3), maximum values of the out3ow hydrograph and spillway gate opening can be remarkably reduced. These characteristics are highly desirable in a reservoir operating system, and are important indications of power and e4ectiveness of optimum fuzzy control. In order to demonstrate the excellence of the operating system that employs the FLC(3), its performance is compared with the other conventional control methods (Fig. 12). These control methods are human-based manual control [7], and the operation method with ten-stage [10]. As can be seen from Fig. 12, the performance achieved by the proposed controller is apparently superior to that obtained from the other control methods. It is clear that the optimized fuzzy controller exhibits a more smooth out3ow hydrograph than the others. In addition, there are no undesirable sudden changes in the reservoir water level. When the fuzzy control approach is used for the reservoir operation, peak value of the out3ow hydrograph is also less than those of the other methods. This
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
223
Fig. 12. Performances of the di4erent control methods; (a) routing of out3ow hydrograph; (b) lake level versus time variation.
striking reduction in the out3ow rate is the most signi2cant advantage of fuzzy logic based control system. This means that less damage occurs due to this over3ow in the waterways and nature. As to simulation results, reservoir level is successfully kept within a predetermined lower and upper boundary. Furthermore, the overall control mechanism can be systematically carried out without needing a human operator. Moreover, the fuzzy logic based operating system ran as e0ciently as an expert operator does for the control. In the light of these observations, there is no doubt that the
224
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
fuzzy logic based control method designed by TS algorithm provides a more desirable and reliable control action in comparison with the human-based (conventional) control methods. 5. Conclusion In this paper, we present a new method for optimum determination of the membership functions in a fuzzy rule set based on the use of tabu search algorithm. In order to evaluate the e4ectiveness and robustness of the proposed method, we apply it to fuzzy logic control based reservoir operating system. The simulation results indicate that the presented approach works e4ectively, and provides a good relation between the objective function that optimizes the fuzzy controller and dynamic response of the system to be controlled. The performance of tabu search algorithm can be a4ected by adjusting the control parameters such as the initial solution, type of move, tabu conditions, stopping criterion, and hence, dynamic behaviour of the system under control can be signi2cantly improved. References [1] L.R. Beard, Flood control operation of reservoirs, J. Hydraul. Division, ASCE 89 (1) (1963) 1–23. [2] L. Chang, F. Chang, Intelligent control for modelling of real-time reservoir operation, Hydrol. Process. 15 (2001) 1621–1634. [3] C.H. Chang, Y.C. Wu, Genetic algorithm based tuning method for symmetric membership functions of fuzzy logic control system, Proc. IEEE/IAS Int. Conf. on Industrial Automation and Control Emerging Technologies, 1995, pp. 421– 428. [4] M. Denna, G. Mauri, A.M. Zanaboni, Learning fuzzy rules with tabu search-an application to control, IEEE Trans. Fuzzy Systems 7 (2) (1999) 295–318. [5] J.M. Garibaldi, E.C. Ifeachor, Application of simulated annealing fuzzy model tuning to umbilical cord acid-base interpretation, IEEE Trans. Fuzzy Systems 7 (1) (1999) 72–84. [6] General Directorate of State Hydraulic Works, Storage-area relationships and the design 3ood hydrographs of the Catalan dam and hydropowerplant, Project No. HD-003, Division of Dams and HPPs, Yucetepe, Ankara, Turkey, 1987. [7] General Directorate of State Hydraulic Works, Spillway rule curves of the Catalan dam and hydropowerplant, Project No. HD-004, Division of Dams and HPPs, Yucetepe, Ankara, Turkey, 1987. [8] F. Glover, Tabu search-part I, ORSA J. Comput. 3 (1) (1989) 190–206. [9] F. Glover, Tabu search-part II, ORSA J. Comput. 1 (2) (1990) 4–32. [10] T. Haktanir, O. Kisi, Ten-stage discrete 3ood routing for dams having gated spillways, J. Hydrologic Eng. ASCE 6 (1) (2001) 86–90. [11] A. Homaifar, E. McCormick, Simultaneous design of membership functions and rule sets for fuzzy controllers using genetic algorithms, IEEE Trans. Fuzzy Systems 3 (2) (1995) 129–139. [12] S. Isaka, A.V. Sebald, An optimization approach for fuzzy controller design, IEEE Trans. Systems Man Cybernet. 22 (6) (1992) 1469–1473. [13] M. Jamshidi, L. Zadeh, A. Titli, S. Boverie (Eds.), Application of Fuzzy Logic: Towards High Machine Intelligence Quotient Systems, Prentice-Hall, Engleewood Cli4s, NJ, 1997. [14] D. Karaboga, Design of fuzzy logic control systems using genetic algorithms, Thesis, University of Wales College of Cardi4, 1994. [15] D. Karaboga, Design of fuzzy logic controllers using tabu search algorithms, Biennial Conf. North American Fuzzy Inf. Processing Society, 1996, pp. 489 – 491. Y Galambosia, E.F. Gaucheranda, L. Ducksteina, Linear quadratic dynamic programming for water [16] E.C. Ozelkana, A. reservoir management, Appl. Math. Modelling 21 (9) (1997) 591–598.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
225
[17] D.T. Pham, D. Karaboga, Intelligent Optimisation Techniques: Genetic Algorithms, Tabu Search, Simulated Annealing, and Neural Networks, Springer, Berlin, 2000. [18] T.J. Ross, Fuzzy Logic With Engineering Applications, McGraw-Hill, New York, 1995. [19] S.P. Simonovic, Reservoir systems analysis: closing gap between theory and practice, J. Water Resour. Plann. Manage. ASCE 118 (3) (1992) 262–280. [20] H. Surmann, Genetic optimization of a fuzzy system for charging batteries, IEEE Trans. Indus. Electron. 43 (5) (1996) 541–548. [21] P. Thrift, Fuzzy logic synthesis with genetic algorithms, Proc. 4th Internat. Conf. on Genetic Algorithms, USA, 1991, pp. 509 –513. [22] S.L. Udall, Design of small dams, 1961. [23] T.A. Wurbs, Reservoir system simulation and optimisation models, J. Water Resour. Plann. Manage. ASCE 119 (4) (1993) 455–472.
Determining fuzzy membership functions with tabu search—an application to control Aytekin Ba)gi*s∗ Department of Electronic Engineering, Erciyes University, Kayseri, 38039 Turkey Received 19 April 2002; received in revised form 4 October 2002; accepted 10 October 2002
Abstract This paper presents a new approach for the optimum determination of membership functions for a fuzzy logic controller based on the use of tabu search algorithm. To demonstrate the e0ciency of the suggested approach, a speci2c control problem—operation of spillway gates of reservoirs during 3oods is selected. Simulation results showed that the proposed approach could be employed as a simple and e4ective optimization method for achieving optimum determination of membership functions. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic controller; Membership functions; Tabu search algorithm; Reservoir control
1. Introduction Fuzzy logic controllers (FLCs) are intelligent control systems characterized by a set of linguistic statements based on expert knowledge or experience. If the mathematical model of a process is de2cient or complicated, or if the process is nonlinear or time dependent, or if it is di0cult or impossible to control the process with the conventional methods, then fuzzy logic control is the most suitable method for these types of problems [18,13]. A fuzzy controller is formed by input and output fuzzy sets assigned over the controller input and output variables, and a collection of fuzzy rules. These rules provide the necessary connection between the controller input and output fuzzy sets. In design of a fuzzy controller, a major di0culty is encountered in the production of optimal fuzzy rule base. Determination of the most appropriate membership functions consisting of the rule base makes signi2cant impacts on the 2nal performance of the controller. In order to get the best performance from FLC, the membership functions must be optimally determined. In general, de2nitions of the membership functions and production of the ∗
Fax: +90-352-437-57-84. E-mail address: [email protected] (A. Ba)gi*s).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 5 0 2 - X
210
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
rule base are acquired from a human expert using trial-and-error method. However, converting the experts’ know-how into fuzzy rules is di0cult, and often, results are incomplete, unnecessary and include con3icting knowledge, since operators and control engineers are not capable of speci2c details or cannot express all their knowledge including intuition and inspiration. Therefore, applying automatic techniques to obtain the optimum membership functions appears as an alternative. To improve behaviour of designed fuzzy controller it is necessary to use some optimization methods such as genetic algorithm (GA) [3,11,14,20,21], simulated annealing (SA) [5,12], tabu search (TS) [4,15]. Chang and Wu [3] introduced a GA based tuning method for symmetric membership functions in a fuzzy control system. Homaifar and McCormick [11] examined the applicability of GA’s in the simultaneous design of membership functions and rule sets for FLCs. They employed a GA procedure to determine only the base lengths of triangular fuzzy sets and not to location of the peaks. Surmann [20] applied the GA approach to optimize a fuzzy rule based system for charging high-power NiCd batteries. In [5], SA algorithm with an adaptation for continuous minimization by the simplex method is used to tune of fuzzy models, and the height of fuzzy terms is considered as a tuneable parameter within the modelling process. Denna et al. [4] present an approach for automatic de2nition of the fuzzy rules based on the use of TS algorithm. To determine the most appropriate rule base for the problem, they employed the reactive form of TS algorithm. Karaboga [15] described a method for constructing the fuzzy rule base. In the method, using the TS algorithm, search of the solution is based on the basis of the automatic learning of fuzzy rule table with the preselected membership functions. GA’s are search algorithms that use operations found in natural genetic to guide journey through a search space [17]. They work with a population of individuals or strings (chromosomes). Each string represents a feasible solution to the problem, and a population consists of a set of admissible solutions. GA’s use three basic operators to manipulate the genetic composition of a population: reproduction, crossover, and mutation. However, there are many decisions the user has to make such as the representation, population size, chromosome structure, mutation and crossover probability, recombination probability, and mating subset selection method. GA’s consider many points in the search space simultaneously, not a single point. Thus, they generate entire populations of points, test each point independently, and combine qualities from existing points to form a new population containing improved points. Furthermore, they might evaluate the same solutions several times since the GA’s use probabilistic rules to guide their search, not deterministic rules. Hence, a large amount of computation time for obtaining a near optimal solution can be required during search of the algorithm. In contrast, the evolution of a TS algorithm is carried on using a single individual. In this case, the computational resources required are much more limited, and evolution can proceed very rapidly. Because the TS is usually based on the use of two important factors (tabu list size and tabu conditions), organization of the algorithm has not a complex procedure. SA is a heuristic optimization technique that derives from statistical mechanics [4,17]. The algorithm consists of an iterative method with a single solution. Four basic ingredients are needed to apply SA in practice: a concise problem representation, a neighbourhood function, a transition mechanism, and a cooling schedule. As for the choice of the cooling schedule, there exist some general guidelines. However, no general rules that guide the choice of the other ingredients are known. This method uses a probabilistic transition rule. Therefore, each of iteration consists of randomly changing the current solution to create a new solution in the neighbourhood of the current solution. Since the steps in the SA algorithm are taken randomly, the search for good solution can be long,
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
211
and 2nal solution could be suboptimal especially in problems with enormous solution spaces. The performance of the algorithm based on the annealing scheduling, and, therefore, it has to be chosen very carefully. Furthermore, in the basic strategy of the SA, time is a critical factor [4]. If SA is too fast, the resulting solution could be unsatisfactory. In contrast, time is not a critical factor in TS algorithm. Although TS is also a form of iterative search, it can discover an optimal solution for the problem in a reasonable time. The biggest di4erence of the TS algorithm is the use of memory during the search process [4,8,9,17]. In the GA and SA, the memory cannot be e4ectively used as that in TS. They use memory to store a set of individuals, and the past history of the search does not in3uence the current move. On the other hand, TS employs an explicit memory to store historical information on the course of the search trajectory. Thus, memory is used to guide the selection of the next solutions. This property of the TS algorithm is a signi2cant superiority over other optimization algorithms mentioned above. In this paper, we present a simple and e4ective method for the optimum determination of the membership functions based on the use of TS algorithm. This method has been applied to fuzzy logic based reservoir operating system of dams during 3oods. The membership functions consisting of fuzzy rule base are optimized by using di4erent objective functions. The results obtained by means of the fuzzy control system are compared with those of conventional control methods.
2. Structure of a fuzzy logic controller A fuzzy controller allows us to use a control strategy expressed in the form of linguistic rules for the de2nition of an automatic control strategy. A typical fuzzy rule can be composed as IF A is Ai AND B is Bi OR C is Ci THEN D is Di ; where A, B, C, and D are fuzzy variables, Ai , Bi , Ci , and Di are fuzzy linguistic values, “AND” and “OR” are connectives of the rule. The advantage of using this linguistic representation is that it is very easy to change the de2nition of system by means of this description. A basic FLC can be decomposed into four basic components [18]. These are fuzzi2cation unit, knowledge base (rule base and data base), decision making unit (inference unit), and defuzzi2cation unit (Fig. 1). Fuzzi2cation unit transforms the measurement data into fuzzy sets. Knowledge base Knowledge Base data base
fuzzification unit
input
rule base
decision making unit
defuzzification unit output
Fig. 1. Con2guration of a basic fuzzy logic controller.
212
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
consists of a rule base and data base. This unit has knowledge about the physical system to be controlled. Decision making unit de2nes how the system should make inferences through the fuzzy rules contained in the rule base. Defuzzi2cation unit aggregates the outputs of all the rules that have 2red for the particular input fuzzy sets to produce a crisp output.
3. Tabu search algorithm TS algorithm was proposed by Glover as an intelligent optimization technique to overcome local optimality in solution processes for hard combinatorial optimization problems [8,9]. The search is based on neighbourhood mechanism. The neighbourhood of a solution is the set of all formations that can be arrived at by a move. The move is a process that transforms the search from the current solution to its neighbouring solution. The 3owchart of TS algorithm procedure is shown in Fig. 2 [17]. To avoid performing a move returning to a recently visited region, the reverses of last moves are forbidden. In other words, these moves are considered as “tabu”. They are recorded in a list called initial solution
create a set of neighbour solutions
evaluate the neighbour solutions
choose the best admissible solution
stopping criteria satisfied?
update tabu list
No
Yes final solution Fig. 2. Flowchart of a standard TS algorithm.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
213
“tabu list”. Therefore, in order to record the history of search, an explicit memory is employed. Tabu list is initialized empty, constructed in consecutive iterations of search and updated circularly in later iterations. A move that is not on the tabu list is called as admissible move. After making a move, if a solution produced by this move is better than all solutions found in prior iterations, then this solution is saved as the best solution. At each iteration of the optimization process, candidate solutions are checked with respect to “tabu conditions”, and hence, the next solution is determined by depending on evaluation values and tabu conditions. The re-generation of a solution previously obtained is avoided by using the tabu conditions (or restrictions) on the possible moves. The tabu conditions are usually based on two important factors: frequency memory and recency memory. Frequency memory keeps the knowledge of how often the same solutions have been made in the past. Recency memory prevents cycles of length less than or equal to a predetermined number of iterations. The tabu conditions used in this work are as follows: If the element k of solution vector does not satisfy the conditions (i) recency (k)¿r:M , (ii) frequency (k)¡f:avgfreq, then it is accepted as tabu and not used to create a neighbour. Here, r and f are recency and frequency factors, M is the number of elements in the solution vector, and avgfreq is the average frequency value. TS has another important element called aspiration mechanism. If a move on the tabu list leads to a solution with a value strictly better than the best obtained so far, it is possible to allow this move to get out of tabu list. This property is used to avoid removing very good moves from consideration and plays an essential role on the search process. In this work, if all available moves are classi2ed tabu, then a least tabu move is selected for new solution. The least tabu move means that this solution is changed less recently and frequently among them but it is still classi2ed as tabu. In this work, maximum number of iterations reached is selected as stopping criterion of the search process. 4. An application to the reservoir operation problem To demonstrate the e0ciency of TS algorithm, we illustrate the application of the approach to a nonlinear control problem—the operation of spillway gates of a reservoir during 3oods. In this paper, the use of a FLC to automatically realize the reservoir operation system of Adana Catalan dam in Turkey is proposed. To ensure that an optimal FLC is designed for the problem, the membership functions of fuzzy rules are optimized by using TS algorithm. The results of the operating system, which is controlled by the optimum fuzzy controller, are compared with the results of conventional ones. 4.1. Physical system to be controlled The reservoir operation system is a control system that manages the spillway gates in a dam to increase or decrease the amount of discharge water [22]. The most important task of this system is to keep the reservoir water level within a prescribed range in any condition by adjusting the opening of the spillway gates. Main variables of a reservoir operation system are in3ow rate (I; m3 =s), out3ow rate (O; m3 =s), reservoir capacity (S; 106 m3 ), minimum and maximum reservoir water surface elevation (Hmin and Hmax , m), actual water level (H; m), and spillway gate opening (d; m) (Fig. 3). The accumulation
214
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 3. Diagrammatic description of reservoir operation.
of storage in a reservoir depends on the di4erence between the rates of in3ow and out3ow. For the interval of Rt, this continuity relationship can be expressed as the following: RS(t) = [I (t) − O(t)]Rt; where RS is the storage accumulated or depleted during Rt, I (t)=O(t) is the average rate of in3ow=out3ow during Rt. Operation of the spillway gates of a reservoir during 3oods is a complex, nonlinear, nonstationary control process because: (a) System to be controlled is signi2cantly a4ected by hydrological conditions. The hydrological conditions have a nonlinear and nonstationary behaviour. (b) The spillway gate must be regulated according to probable in3ow hydrograph. Determination of the in3ow hydrograph can be extremely di0cult if not impossible. (c) Since the system complexity is high and correct prediction of the system behaviour is not easy, it is very di0cult to make a precise model to control the system. (d) Controlling the spillway gate of the dams are usually performed by experienced human operators. But, humans are emotional and forgetful, and may reach incorrect decisions especially under extreme conditions such as over3ows. The wrong decisions during a dangerous in3ow hydrograph may cause serious safety problems for nature and human life. Hence, owing to human factor, there is no guarantee of the safe and e0cient control of process. Some methods used to achieve the reservoir control have been presented in [1,2,10,16,19,23]. Beards presents a deterministic operation procedure [1]. Ozelkana et al. applied the dynamic programming technique for water reservoir management [16]. In [2], it is alleged that to build an intelligent reservoir operation system is impossible without information of optimal input and output hydrological patterns. The model given in [10] presents a set of operation rules with ten-stage for controlling the spillway gate opening. The approaches mentioned above are insu0cient and in3exible modelling methods with low performance. Currently the most common reservoir control strategy in Turkey is based on human operator decisions [7]. In this strategy, a constant amount of water is discharged depending on the reservoir level. The most signi2cant drawback of this approach is how to determine the constant amount of water that must be discharged for a given lake level. Another disadvantage of this strategy is that it does not consider the change rate of lake level to determine the water amount to be released. Therefore, there is no guarantee that the control strategy will have su0ciently good
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
215
Calculation h TS algorithm
dh
Fuzzy Logic Controller
Sensors
d inflow hydrograph
Reservoir, Dam, and Spillway Gates
lake level (H)
Fig. 4. Fuzzy controlled reservoir operating system.
performance for the present complex system with large number of input variables of unpredictable in3ow rates. Because of uncertain process information, system dynamics varying nonlinearly, and inaccurate mathematical model, conventional control techniques will also be insu0cient to provide a solution for an accurate and e0cient reservoir operation. 4.2. Fuzzy controller for the reservoir operation problem The main objectives of the reservoir operation considered herein can be given as the following: (1) The control system must be fast and reliable during any 3ood of any magnitude, which is not predictable beforehand. (2) The control method must drive the lake water surface level to desired minimum set point (Hmin = 118:6 m) in the shortest time during spillway over3ows [6]. (3) Reservoir water level must not be exceed the predetermined maximum set point (Hmax = 127 m) in any time [6]. (4) The rate of change of the released water must be as smooth as possible. The fuzzy control system proposed for the present problem is given in Fig. 4. In the fuzzy control system design, the selection of the controller structure involves the following choices: (a) Input and output variables: The input variables for the fuzzy controller are “lake level” (h) and “change in the lake level” (dh). The output of the controller is “gate opening” (d). Out3ow rate of the reservoir is adjusted by the gate opening, which is controlled by FLC. For h, dh, and d variables, the normalization intervals are selected as [118; 127], [−1; 1], and [0; 12], respectively. (b) Number and type of membership functions for variables: The membership functions used for the fuzzy values of fuzzy variables are selected based on human=expert experience. As shown in Fig. 5, the input variables h and dh are characterized by three fuzzy membership functions, and the output variable d is characterized by 2ve membership functions. All of the fuzzy values are represented by triangular membership functions for simplicity. But, fuzzy membership functions can have di4erent shapes and sizes depending on the designer’s preference or experience, and the system’s performance can also be signi2cantly improved.
216
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 5. Initial membership functions.
Table 1 Rule base Rule number
Rules
(1) (2) (3) (4) (5)
IF IF IF IF IF
h h h h h
is is is is is
h1 , h1 , h2 , h2 , h3 ,
AND AND AND AND AND
dh dh dh dh dh
is is is is is
dh1 , dh2 , dh2 , dh3 , dh3 ,
THEN THEN THEN THEN THEN
d d d d d
is is is is is
d1 d2 d3 d4 d5
(c) Rule base: The rules of the reservoir operation strategy are obtained from information gathered by engineers and experts informed about the dam and operator experience. The rule base of the FLC contains only 2ve rules, which are tabulated in Table 1. (d) Type of inference mechanism: The output of each rule is determined by Mamdani’s max–min inference method. (e) Defuzzi7cation method: Standard centre of area method is employed for defuzzi2cation process.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
217
Fig. 6. Parameterized membership function.
4.3. De7nition of the membership functions Determination of the FLC membership functions using an optimization algorithm takes place in four phases: 1. 2. 3. 4.
creation of a primary membership functions which are arbitrarily initialized, parameterization, de2nition of the rule base, and adjustment of the membership functions.
Number and type of the membership functions for variables is mentioned earlier. Each membership function is characterized by three numerical values. These numerical values represent the position of the membership function on the universe of discourse. In Fig. 6, parameterized fuzzy membership functions (A1 ; A2 ; : : : ; An ) are demonstrated. It can be seen from Fig. 6 that three numerical values are used in de2nition of a membership function. In any physical system with two inputs and one output, when the same membership functions are used for all variables of the system, a sample rule structure can be written as follows: Linguistic rule → IF input (1) is A2 , AND input (2) is A3 , THEN output is A2 . Parameterized rule → IF (a21 a22 a23 ), AND (a31 a32 a33 ), THEN (a21 a22 a23 ). Here, aij is a numerical parameter that characterizes the membership functions, and aij ∈ [0; 1] for h; d; aij ∈ [−1; 1] for dh (i = 1; 2; 3; j = 1; 2; 3). As to this de2nition, each rule consists of [(input number+output number)×3] parameters. Therefore, the number of parameters required for de2ning a fuzzy rule set is [rule number ×(input number + output number)×3]. Since the fuzzy controller with two inputs and one output is used in this work, each fuzzy rule is represented by nine parameters. Thus, 45 parameters are used for 2ve di4erent rules (Fig. 7). This value is also equal to the size of the tabu list. This de2nition of the membership functions for fuzzy rule base has three main advantages: (1) It is easy to understand and de2ne the rules. (2) It provides the evaluation of the all possible solution points for the universe of discourse since the membership functions have not symmetric structure. (3) The same control parameters can be obtained for di4erent membership functions, and hence, the number of membership functions can be reduced during optimization process.
218
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225 h1 = 0.00 h2 = 0.25 h3 = 0.50 dh1 = -1.00 dh2 = -0.50 dh3 = 0.00 d1 = 0.00 d2 = 0.00 d3 = 0.25 d4 = 0.50 d5 = 0.75
dh d h
dh1
dh2
dh3
h1
d1
d2
---
h2
---
d3
d4
h3
---
---
d5
0.00 0.00 0.50 0.00 0.00 0.50 0.25 0.50 0.75 0.25 0.50 0.75 0.50 1.00 1.00
-1.00 –1.00 0.00 -0.50 0.00 0.50 -0.50 0.00 0.50 0.00 1.00 1.00 0.00 1.00 1.00
0.00 0.00 0.25 0.00 0.25 0.50 0.25 0.50 0.75 0.50 0.75 1.00 0.75 1.00 1.00
h
dh
d
0.00 0.50 1.00 -1.00 0.00 1.00 0.00 0.25 0.50 0.75 1.00
0.50 0.75 1.00 0.00 0.50 1.00 0.25 0.50 0.75 1.00 1.00
Rule 1 Rule 2 Rule 3 Rule 4 Rule 5
Fig. 7. Parameter matrix used to represent the rule base.
Table 2 Objective functions used in this work Function (1)
F1 = (1=N )
(2)
F2 = (1=N )
(3)
F3 = F1 + F2
N t=1 N t=1
O(t) exp(|O(t) − O(t − 1)|)
The membership functions used play a crucial role in the 2nal performance of a fuzzy control system. Therefore, selection of the appropriate functions is an important design problem. So, in order to design an optimal FLC the proper membership functions are searched by using TS algorithm. In the design of fuzzy controller for this problem, two main factors are considered by TS algorithm: peak values of the out3ow hydrograph, and changes of these values. During the optimization process, these factors are minimized by TS for the better parameters. In the reservoir operation, three di4erent objective functions are considered (Table 2). In these equations, N is the number of samples, t is time, and O is the peak value of out3ow hydrograph. Neighbourhood values of the parameters used to represent the membership functions are found using the neighbourhood structure given in Table 3. In this structure based on binary code number of the neighbour solutions is B for a solution with B bits. Each neighbour is obtained by changing the only one bit of the solution. The other control parameters r, f, and iteration number were selected as 0.2, 2.0, 250, respectively.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
219
Table 3 Neighbourhood structure Solution Neighbour Neighbour Neighbour Neighbour Neighbour
(1) (2) (3) (4) (5)
1
0
1
1
0
0 1 1 1 1
0 1 0 0 0
1 1 0 1 1
1 1 1 0 1
0 0 0 0 1
Fig. 8. Performance of the FLC with initial membership functions: (a) 3ood versus time; (b) lake level versus time; (c) gate opening versus time curves.
4.4. Simulation results As mentioned earlier, initially the membership functions and rule base are de2ned intuitively. Performance of the FLC using this rule base is presented in Fig. 8. According to Fig. 8, performance of the control system is not certainly good. Time curve of the amount of water discharged is not smooth enough, and exhibits sudden discontinuities. The out3ow hydrograph and reservoir level has undesirable sudden changes due to unsystematic behaviour of gate opening. Spillway gate of the dam is not controlled e4ectively. Therefore, controlling the spillway gate by this operating system is dangerous for nature and human life.
220
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 9. Membership functions optimized by TS algorithm with (a) F1 , (b) F2 , and (c) F3 ..
To improve behaviour of operating system, membership functions were adjusted by using three di4erent objective functions F1 , F2 , and F3 . After the optimization process, the 2nal state of the membership functions for system variables was obtained as shown in Fig. 9. As to this 2gure, number of the membership functions is di4erent for each objective function. The reason for this is that the parameter matrix contains di4erent numerical values for de2ning the membership functions within the fuzzy rule base after the optimization processes (Table 4). FLCs optimized by these objective functions are represented with FLC(1), FLC(2), FLC(3), respectively. The overall performances of these controllers are given in Figs. 10 and 11. Signi2cant performance indexes for fuzzy controllers
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
221
Table 4 Parameter matrix determined by di4erent objective functions Parameter matrix Optimized by F1
0.52 0.25 0.10 0.00 0.25
0.86 0.52 0.10 0.08 0.52
1.00 0.74 0.55 0.26 0.74
−1:00 −0:87 −0:100 −0:87 −1:00
−1:00 0.60 0.03 0.60 −1:00
0.30 0.60 0.30 0.60 0.30
0.28 0.23 0.00 0.00 0.00
0.68 0.60 0.02 0.00 0.13
0.74 0.71 0.09 0.02 0.58
Optimized by F2
0.55 0.40 0.25 0.08 0.08
0.97 0.85 0.74 0.48 0.08
0.97 0.97 0.81 0.74 0.28
−1:00 −0:81 −0:70 −1:00 −0:56
−0:04 0.20 0.60 −1:00 1.00
0.30 0.60 0.94 0.30 1.00
0.30 0.05 0.13 0.00 0.00
0.68 0.13 0.30 0.05 0.00
0.68 0.30 0.50 0.13 0.02
Optimized by F3
0.55 0.48 0.25 0.08 0.00
0.86 0.86 0.52 0.21 0.08
1.00 1.00 0.71 0.61 0.29
−1:00 −1:00 −0:60 −1:00 −0:60
−1:00 −0:42 0.60 0.30 0.60
0.29 0.30 0.60 0.60 0.60
0.50 0.00 0.23 0.00 0.00
0.68 0.06 0.48 0.05 0.00
0.68 0.48 0.65 0.10 0.02
Fig. 10. Routing of out3ow hydrograph from Catalan dam by di4erent FLCs.
are listed in Table 5. In this table, FLC(0) indicates the fuzzy controller that is not optimized by TS algorithm. Figs. 10 and 11 and Table 5 show all the controllers tuned by TS were able to successfully operate the system according to main objectives. It should be noted that the best performance, on average value of the out3ow hydrograph (Oavg ) and maximum change of the out3ow hydrograph
222
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
Fig. 11. Lake level versus time from Catalan dam by di4erent FLCs.
Table 5 Signi2cant performance indexes Out3ow rate (m3 =s)
FLC(0) FLC(1) FLC(2) FLC(3)
Gate opening (m)
Lake level (m)
Omax
Oavg
ROmax
dmax
Rdmax
hmax
5031.27 5797.26 6149.12 4528.29
2503.68 2483.43 2515.85 2508.26
1538.05 614.08 412.50 291.11
6.87 8.09 8.32 5.34
3.00 1.20 0.72 0.44
124.75 125.57 126.57 126.49
(ROmax ), came from the FLC(3) as shown in Table 5. Using FLC(3), maximum values of the out3ow hydrograph and spillway gate opening can be remarkably reduced. These characteristics are highly desirable in a reservoir operating system, and are important indications of power and e4ectiveness of optimum fuzzy control. In order to demonstrate the excellence of the operating system that employs the FLC(3), its performance is compared with the other conventional control methods (Fig. 12). These control methods are human-based manual control [7], and the operation method with ten-stage [10]. As can be seen from Fig. 12, the performance achieved by the proposed controller is apparently superior to that obtained from the other control methods. It is clear that the optimized fuzzy controller exhibits a more smooth out3ow hydrograph than the others. In addition, there are no undesirable sudden changes in the reservoir water level. When the fuzzy control approach is used for the reservoir operation, peak value of the out3ow hydrograph is also less than those of the other methods. This
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
223
Fig. 12. Performances of the di4erent control methods; (a) routing of out3ow hydrograph; (b) lake level versus time variation.
striking reduction in the out3ow rate is the most signi2cant advantage of fuzzy logic based control system. This means that less damage occurs due to this over3ow in the waterways and nature. As to simulation results, reservoir level is successfully kept within a predetermined lower and upper boundary. Furthermore, the overall control mechanism can be systematically carried out without needing a human operator. Moreover, the fuzzy logic based operating system ran as e0ciently as an expert operator does for the control. In the light of these observations, there is no doubt that the
224
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
fuzzy logic based control method designed by TS algorithm provides a more desirable and reliable control action in comparison with the human-based (conventional) control methods. 5. Conclusion In this paper, we present a new method for optimum determination of the membership functions in a fuzzy rule set based on the use of tabu search algorithm. In order to evaluate the e4ectiveness and robustness of the proposed method, we apply it to fuzzy logic control based reservoir operating system. The simulation results indicate that the presented approach works e4ectively, and provides a good relation between the objective function that optimizes the fuzzy controller and dynamic response of the system to be controlled. The performance of tabu search algorithm can be a4ected by adjusting the control parameters such as the initial solution, type of move, tabu conditions, stopping criterion, and hence, dynamic behaviour of the system under control can be signi2cantly improved. References [1] L.R. Beard, Flood control operation of reservoirs, J. Hydraul. Division, ASCE 89 (1) (1963) 1–23. [2] L. Chang, F. Chang, Intelligent control for modelling of real-time reservoir operation, Hydrol. Process. 15 (2001) 1621–1634. [3] C.H. Chang, Y.C. Wu, Genetic algorithm based tuning method for symmetric membership functions of fuzzy logic control system, Proc. IEEE/IAS Int. Conf. on Industrial Automation and Control Emerging Technologies, 1995, pp. 421– 428. [4] M. Denna, G. Mauri, A.M. Zanaboni, Learning fuzzy rules with tabu search-an application to control, IEEE Trans. Fuzzy Systems 7 (2) (1999) 295–318. [5] J.M. Garibaldi, E.C. Ifeachor, Application of simulated annealing fuzzy model tuning to umbilical cord acid-base interpretation, IEEE Trans. Fuzzy Systems 7 (1) (1999) 72–84. [6] General Directorate of State Hydraulic Works, Storage-area relationships and the design 3ood hydrographs of the Catalan dam and hydropowerplant, Project No. HD-003, Division of Dams and HPPs, Yucetepe, Ankara, Turkey, 1987. [7] General Directorate of State Hydraulic Works, Spillway rule curves of the Catalan dam and hydropowerplant, Project No. HD-004, Division of Dams and HPPs, Yucetepe, Ankara, Turkey, 1987. [8] F. Glover, Tabu search-part I, ORSA J. Comput. 3 (1) (1989) 190–206. [9] F. Glover, Tabu search-part II, ORSA J. Comput. 1 (2) (1990) 4–32. [10] T. Haktanir, O. Kisi, Ten-stage discrete 3ood routing for dams having gated spillways, J. Hydrologic Eng. ASCE 6 (1) (2001) 86–90. [11] A. Homaifar, E. McCormick, Simultaneous design of membership functions and rule sets for fuzzy controllers using genetic algorithms, IEEE Trans. Fuzzy Systems 3 (2) (1995) 129–139. [12] S. Isaka, A.V. Sebald, An optimization approach for fuzzy controller design, IEEE Trans. Systems Man Cybernet. 22 (6) (1992) 1469–1473. [13] M. Jamshidi, L. Zadeh, A. Titli, S. Boverie (Eds.), Application of Fuzzy Logic: Towards High Machine Intelligence Quotient Systems, Prentice-Hall, Engleewood Cli4s, NJ, 1997. [14] D. Karaboga, Design of fuzzy logic control systems using genetic algorithms, Thesis, University of Wales College of Cardi4, 1994. [15] D. Karaboga, Design of fuzzy logic controllers using tabu search algorithms, Biennial Conf. North American Fuzzy Inf. Processing Society, 1996, pp. 489 – 491. Y Galambosia, E.F. Gaucheranda, L. Ducksteina, Linear quadratic dynamic programming for water [16] E.C. Ozelkana, A. reservoir management, Appl. Math. Modelling 21 (9) (1997) 591–598.
A. Ba$gis% / Fuzzy Sets and Systems 139 (2003) 209 – 225
225
[17] D.T. Pham, D. Karaboga, Intelligent Optimisation Techniques: Genetic Algorithms, Tabu Search, Simulated Annealing, and Neural Networks, Springer, Berlin, 2000. [18] T.J. Ross, Fuzzy Logic With Engineering Applications, McGraw-Hill, New York, 1995. [19] S.P. Simonovic, Reservoir systems analysis: closing gap between theory and practice, J. Water Resour. Plann. Manage. ASCE 118 (3) (1992) 262–280. [20] H. Surmann, Genetic optimization of a fuzzy system for charging batteries, IEEE Trans. Indus. Electron. 43 (5) (1996) 541–548. [21] P. Thrift, Fuzzy logic synthesis with genetic algorithms, Proc. 4th Internat. Conf. on Genetic Algorithms, USA, 1991, pp. 509 –513. [22] S.L. Udall, Design of small dams, 1961. [23] T.A. Wurbs, Reservoir system simulation and optimisation models, J. Water Resour. Plann. Manage. ASCE 119 (4) (1993) 455–472.