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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 6, NOVEMBER 2001

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A Fuzzy Controller for an Aeroload Simulator Using Phase Plane Method Sang Yeal Lee and Hyung Suck Cho, Member, IEEE

Abstract—The objective of this paper is to realize a fuzzy logic controller for an aeroload simulator, which simulates the aerodynamic load that a missile is subjected to on flight. In this paper, we propose a new fuzzy logic controller using the phase plane, in order to improve the overall performance of the aeroload simulator system. The design procedure and method of this controller are easy and simple such that performance evaluation of the aeroload simulator is carried out in a phase plane mapped from a decision rule table. The effectiveness of this control scheme is verified by comparison with a proportional integral derivative (PID) and a fixed fuzzy control through a series of simulation studies. Index Terms—Aeroload simulator, design methodology, electrohydraulics, fuzzy control, phase plane, tuning.

I. INTRODUCTION

I

N ORDER to test the dynamic performance of a fin position servo system, an aeroload simulator to generate the aerodynamic load to which a missile is subjected to during flight is needed. The aerodynamic load is a highly nonlinear function that varies with the deflection angle of the control fin, the angle of attack, and Mach number, etc. Therefore, the aeroload simulator has to be designed as a torque-control servo system, which has fast and accurate dynamic performance to follow desired torque profiles. In the load simulation, the electro-hydraulic servo systems have been frequently used because of their high power, fast response characteristics, and good positioning capabilities. Hydraulic servo systems, however, have uncertainties, time-varying and nonlinear characteristics due to the flow-pressure relationship, oil leakage, oil temperature variation, and so on. Consequently, the conventional control approaches based on a linearized model near the operating point of interest may not guarantee satisfactory control performance for these load simulator systems. To solve such hydraulic servo problems some research efforts on adaptive control approaches have been made in recent years [1]–[4]. The control techniques provide satisfactory results over relatively much larger ranges of changes in the plant dynamics. But in the case of plants whose parameters are not completely known, or change rapidly over very large ranges, the adaptive control strategies require considerable computation time due to the complexity in their algorithms and often lead to instability.

To avoid such criticism, the fuzzy logic-based controllers have been actively researched and widely utilized for many industrial processes [5]–[21]. These fuzzy logic controllers show good results in the cases of controlling high-order nonlinear systems. Accordingly, many research efforts have adopted fuzzy logic for the control of hydraulic servo systems [22]–[26]. Even though fuzzy logic controllers often produce results superior to those of classical controllers as mentioned above, the control engineer has difficulty in accessing the fuzzy logic controller because of the following limitations: • The design of the fuzzy logic controller is not straightforward due to heuristics involved with control rules and membership functions. • The tuning of the parameters of the fuzzy logic controller is very complex. When the design of a fuzzy system is undertaken, one immediately faces many design parameters such as scaling factors, membership functions, and control rules. However, at present there are not many simple methods currently available for control engineers in designing the knowledge base and tuning of a fuzzy logic controller. Therefore, the designers have to devise a knowledge base by heuristic methods, employing experience and, accordingly, the parameters of a fuzzy control system are tuned repeatedly by trial and error method. This leads to a well-known fact that the design of a fuzzy logic controller is more difficult than the design of a conventional controller. In this paper, we propose a novel fuzzy logic controller, which can improve the transient performance and robustness of the aeroload simulator control system in response to the set point changes and disturbances. The uniqueness of this controller is that it is designed in a phase plane in a rather straightforward manner. To demonstrate effectiveness of this proposed controller, a series of numerical simulations and experiments are performed on an aeroload simulator for various conditions. The simulation and experimental results show that the proposed control scheme gives faster and more accurate responses, as compared with a proportional integral derivative (PID) control and a fuzzy control with initially fixed rules. II. DYNAMICS OF THE AEROLOAD SIMULATOR

Manuscript received September 30, 1999; revised October 30, 2000. Manuscript received in final form June 11, 2001. Recommended by Associate Editor D. W. Repperger. S. Y. Lee is with the Agency for Defense Development, Taejon 305-600, Korea (e-mail: [email protected]). H. S. Cho is with the Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea (e-mail: [email protected]). Publisher Item Identifier S 1063-6536(01)09474-X.

The schematic diagram of an aeroload simulator servo system is illustrated in Fig. 1. The simulator system is composed of a hydraulic power supply, an electro-hydraulic servo valve, a hydraulic motor, a torsion spring to combine with a fin position servo system, and a torque sensor. The torque of the hydraulic motor is controlled as follows: Once the voltage input corresponding to the torque input is

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 6, NOVEMBER 2001

applied to the fluid flowing in each chamber of the hydraulic motor, the following expression can be derived: (3) where volumetric displacement coefficient of the hydraulic motor; angular displacement of the hydraulic motor; total leakage coefficient of the hydraulic motor; total volume of the servo valve and the hydraulic motor; effective bulk modulus of oil. The final basic relation is the torque balance equation for the ) developed by the hydraulic hydraulic motor. The torque ( motor is given by

(4) where ,

Fig. 1. A schematic diagram of the aeroload simulator servo system.

transmitted to the servo controller, the input current is generated in proportion to the error between the voltage input and the voltage output from the torque sensor. Then, the valve spool position is controlled according to the input current applied to the torque motor of the servo valve. Depending on the spool position and the load conditions of the hydraulic motor, the rate as well as the direction of the flows supplied to each hydraulic motor chamber is determined. The motion of the hydraulic motor is controlled by these flows. At the same time, the hydraulic motor is influenced by an external disturbance generated from the fin position servo system. If it is necessary to represent servo valve dynamics through a wider frequency range, a second-order transfer function must be used. The relation between the servo valve spool position and the input current can be written as (1) where torque motor gain; natural frequency of the servo valve; damping ratio of the servo valve. as and the Defining the load pressure as , the relationship between load flow and the load flow for an ideal critical the load pressure center servo valve with a matched and symmetric orifice can be expressed as follows [27]: (2) represents the servo valve’s sizing factor where is the supply pressure. When the continuity equation is and

moment of inertia and the viscous damping coefficient of the hydraulic motor, respectively; torsion spring coefficient; external disturbance applied to the aeroload simulator; friction of the hydraulic motor. The torque output of the hydraulic motor system, then can be calculated by (5)

III. FUZZY LOGIC CONTROLLER USING PHASE PLANE The fuzzy logic controller adopted herein consists of hierarchically two levels. The lower level is the fuzzy logic controller designed with simplicity to control the torque output of the aeroload simulator system. The higher level is the self-compensating controller, which evaluates control performance in the phase plane and modifies the output of the fuzzy logic controller according to the evaluation. The overall structure of the fuzzy logic controller using phase plane is depicted in Fig. 2. A. Phase Plane Analysis Since the error and change in error of the output variable of a system are analyzed in a phase plane, the relationship between a decision rule table and the phase plane is analyzed here. The phase plane is assumed to be bounded and normalized by scaling factors. The generalized fuzzy logic controller consists of a set of linguistic conditional statements or rules as follows: IF is and is then is IF is and is then is .. . IF is and is then is

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Fig. 2. The structure of the fuzzy logic controller using phase plane.

.. . IF

is

and

is for

then and

is for (6)

, and are the fuzzy variables of the error, the where , change in error and the control input, respectively. , , and are the corresponding fuzzy sets, respectively. The and are the number of fuzzy sets having the antecedent variables, in total. The respectively, and the number of rules is number of fuzzy sets having the consequent variable is and through are one of the fuzzy sets. Without loss of generality, we will adopt the fuzzy membership function as a triangular shape. We then define the peak value and the scaling factor, which are the parameters related to the chosen shape of membership function as follows. • Peak value: the value of the fuzzy variable at which the membership function is 1.0. The peak values of the error, , change in error, and control input are expressed by , and , respectively. • Scaling factor: maximum peak value, which defines the universe of discourse of the fuzzy variable. The scaling factor of the error, change in error, and control input are , , and , respectively. expressed by In general, since the inputs to fuzzy logic controller (FLC) are the error and the change in error, the behavior of these to a step input on a phase plane becomes the phase trajectory that indicates dynamic behavior of the system. Here, the phase plane is normalized as [ 1, 1] by scaling factors and partitioned by the peak values of the antecedent fuzzy sets. Then, the rule table rule nodes as is mapped to the phase plane composed of shown in Fig. 3. We will assign the number of these rule nodes as follow: The upper corner point of the left-hand side is and the lower corner of the right-hand side is . Then, the , ) and the memcoordinates of each rule node become ( bership value of the consequent part corresponding to this point . That is, intersecting points of the phase plane becomes become a point of zero fuzziness as a point which indicates the

Fig. 3.

The fuzzy rule table transformed to the phase plane.

peak values of the antecedent part and the consequent part fuzzy membership function of control rules. The phase plane, composed like in the above, contains not only the control rules but also the quantitative information about the membership function. The phase trajectory passes through the rule node on the phase plane at every sampling instant. Therefore, if a reference phase trajectory is given a priori, it is possible to design the FLC which makes the trajectory of the plant follow as closely as possible the desired one in the phase plane. B. Performance Measure A reference model is introduced in order to make the trajectory of a plant follow the desired one in the phase plane. The reference model that generates the desired trajectory as a linear time-invariant system can be expressed by (7)

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 6, NOVEMBER 2001

where model state vector; stable system matrix; reference input matrix; reference input vector; model output; model output matrix. For effective tuning of the fuzzy logic controller, the control performance of the plant is measured by the difference of phase trajectory between the reference model and the plant. and change in error As shown in Fig. 2, the error between the reference model output and the plant output are defined by (8) (9) represents the sampling period and the sample where number. The fuzzification is performed by the scale mapping from the range of actual values to the corresponding universe of discourse. The fuzzification procedure can be expressed by

(a)

(10) (11) where and

fuzzy singleton of the error and change in error between the model and the plant states, respectively; and scaling factors for the error and change in error, respectively; fuzzification procedure. In order to measure the control performance of plant, the magnitude and the direction of the error vector between the reference model output and the plant output on the phase plane are calculated by (12) (13) where and

Euclidean distance and the phase difference between the reference model and the plant output, respectively; and phase angle of the reference model and the plant on the phase plane, respectively; signum function. Since the control objective is to force the error vector to vanish, converges to a small the tuning algorithm is performed until . value C. Tuning Scheme 1) Firing Strength: Assume that there is samples delay in the plant. Also, the phase trajectories of the reference model and the plant are assumed to be as shown in Fig. 4(a). Assume that samples in the past, was the one that cona control action, tributed to the present system performance. The and would have been the error and change in

(b) Fig. 4. (a) Phase trajectories of a reference model and the plant on the phase plane. (b) Four rule nodes fired at (nT mT ) sample instant.

0

error at that time, and would have been the controller output. Consequently, the controller output that would rather than have been desired is , where is the input reinforcement. If a fuzzy singleton is assigned into the phase plane as composed in Fig. 3, four rule nodes are fired. The four rule nodes, , , and , fired at sample instant are presented in Fig. 4(b). Here, we define the firing strength of each rule node as how much the node affects the fuzzy inference output at that instant. Then, the firing strength can be calculated from the distance between the phase in the phase plane and each rule node. The state closest rule node can be considered to have the highest firing strength. Thus, we define the firing strength of each node as follows:

(14)

LEE AND CHO: A FUZZY CONTROLLER FOR AN AEROLOAD SIMULATOR

(15)

(16)

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is a linguistic fuzzy set of consequent part of the control rule. To determine the crisp value of the control input reinforcement from the corresponding fuzzy output reinforcement, the center of gravity method is used as a defuzzification scheme. ) is given by Then, the control input reinforcement (

(24) (17) and represent the firing strength of the where , to the error and change in error , rerule node spectively. As we can see in Fig. 4(b), the firing strength to the and is identical, and the firing error of rule node and is strength to the change in error of rule node also identical. Therefore, the combined firing strength of each rule node can be obtained from the compositional operation of the firing strength about the error of (14) or (15) and the firing strength about the change in error of (16) or (17). Here, we , define the combined firing strength of four rule nodes as , , . Then the combined firing strength deof each node can be represented by the algebraic product fined as -norm [28] as follows: (18) (19) (20) (21) Here, since the -norm is a 2-valued function, mapping to , the value of firing strength is . When the phase trajectory is on a rule node, the value of combined firing strength is one, and those of remaining three rule nodes are zero. Meanwhile, when the phase trajectory is on the side of rule cell, only two rule nodes on both sides are fired and the combined firing strengths of remaining two rule nodes are zero as well. 2) Output Reinforcement: Various methods to determine the output reinforcement are available in order to improve the current control performance. In this work we adopt a method to obtain the output reinforcement by using the inference result of FLC. This method is a very simple way to increase execution speed, thereby, reducing computational burden. With four can be rule nodes fired, the fuzzy output reinforcement calculated from the following equations by using Mamdani’s – operation: for

for (22) (23)

represents the contribution factor of th rule calcuwhere lated in the fuzzy logic controller of the lower level and represents the contribution reinforcement factor of th rule. , is calculated from (18)–(21). The combined firing strength,

is defined by the final output reinforcement, it can If the be computed by the self-compensating mechanism in the higher level (25) is the speed factor to accelerate the reference model where following speed, and the constant auxiliary parameter to be is the self-compensating tuned by the designer. In (25), factor defined by the following equation and the use of (12) and (13) yields (26) is the Euclidean distance between the reference Since model and the plant output, this time-variant weighting has to be multiplied to calculate the output reinforcement at each sample is the direction of the error vector between instant. The the model and the plant output on the phase plane, and the sign of the output reinforcement is determined from it. The resulting control input to the plant is given by (27) is the output scaling factor and the same as the one for where represents the fuzzy logic controller of the lower level. The the crisp value of the control input calculated in the fuzzy logic controller of the lower level. may lead to poor or unstable It is noted that the tracking performance in the case of choosing an extremely large speed factor. But it is experimentally observed that if we always choose a proper value for the speed factor, the leads to an improved phase plane behavior and the stability of the closed-loop system is guaranteed with respect to any reasonable disturbances. This observation is also obtained from the simulation results. An extensive stability study needs to be made for more general cases. IV. FUZZY CONTROLLER DESIGN USING PHASE PLANE FOR THE AEROLOAD SIMULATOR In order to design the fuzzy logic controller using phase plane (FLCPP), the FLC of the lower level is designed with a simplias the control fied method. The error and change in error input variables for the FLC are defined by following equations: (28) (29)

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(a)

(b) Fig. 5. (a) The fuzzy membership functions of the error (E ), the change in error (CE ) and the control input (U ) of the aeroload simulator. (b) The rule table and the membership function of the aeroload simulator servo system transformed to the phase plane.

where and represent the desired reference input and the torque output of the aeroload simulator system, respectively. and The control inputs for the fuzzy controller, , are converted to the fuzzy values, and , by the fuzzification process of (10) and (11), respectively. Each control variable should be decomposed into a set of fuzzy regions. These regions are given unique names, called labels, within the domain of the variables. In the case of the aeroload simulator system, the input and output space are partitioned by fuzzy sets of seven regions. Therefore, the labels used are positive large (PL), positive medium (PM), positive small (PS), zero (ZE), negative small (NS), negative medium (NM), and negative large (NL). The membership functions of the fuzzy variables are adopted as a triangular shape. The membership function plays an important role in determining the control action prescribed and the performance of the system. Therefore, it is important to design the peak value of the membership function properly, but it is very difficult. Here, we propose a method to select the peak

value of the membership function for servo system by our experience as follows:

(30) In the above equation the peak value of the membership functions nearby zero need to be adjusted, depending upon the characteristics of the system to be controlled. In the aeroload and of the output membership simulator system here functions are made doubled the calculated value for faster response. The fuzzy membership functions of the aeroload simulator system are depicted in Fig. 5(a). In order to generate the rule table reasonably, it requires the knowledge or experience of experts about the skill of aeroload simulator control. However, we constitute the rule table of the aeroload simulator system with simplicity as shown in Table I, only. with the consequent part depending on the error

LEE AND CHO: A FUZZY CONTROLLER FOR AN AEROLOAD SIMULATOR

TABLE I THE FUZZY RULE TABLE OF THE AEROLOAD SIMULATOR SYSTEM (7

2 7)

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tion matrix follows:

, and the command vector

of (7) become as

(34)

To obtain the output value from the linguistic control rules and the fuzzy membership functions, fuzzy inference must be performed. In this work we introduce a simplified inference method, firing only four rule nodes selected by the input point on the phase plane illustrated as in Fig. 3. In the case of using a min–max operator introduced by Mamdani, most widely used of the FLC is obtained in fuzzy inference, the fuzzy output by

for

for

(31) (32)

represents the contribution factor of th rule and where is a linguistic fuzzy set of the consequent part of the fuzzy control rule. The defuzzification process must be performed to determine of the control input from the corresponding the crisp value fuzzy value. The center of gravity method of (24) is used as a to the hydraulic defuzzification scheme. The control input servo valve of the aeroload simulator system is calculated by the following equation: (33) The control rule table shown in Table I is mapped to a phase plane by using the peak value of fuzzy membership functions depicted in Fig. 5(a). Positioning the peak values of consequent part on rule nodes, the phase plane of the aeroload simulator system becomes as shown in Fig. 5(b). The phase plane expressed like this includes all the quantitative information of membership functions as well as control rules. The reference model can be chosen as a typical second-order transfer function in order to make the specification of control performance easy. There are ITAE transfer functions and Bessel transfer functions that represent the response characteristics of a control system. In case of ITAE, the rise time is faster than that of Bessel, but there exists overshoot. In case of Bessel, the rise time is slower than that of ITAE, but there is almost no overshoot. In the case of a typical second-order transfer function, the , the system matrix , the command distribustate vector

and represent the model torque output and the dewhere represent the damping sired torque input, respectively. and coefficient and the natural frequency of the reference model, respectively. The reference model for the aeroload simulator system is chosen as the second-order Bessel transfer function. The damping coefficient of the second-order Bessel transfer function is 0.8660 [29]. Since the required bandwidth of the aeroload simulator system is 50 Hz, the natural frequency of the reference model is chosen as 50 Hz. , and change in error, , between The error, and the plant output the reference model output are defined by (8) and (9), respectively. They are converted to and , by fuzzification process fuzzy values, of (10) and (11), respectively. In order to know how much the rule node affects the fuzzy inference output, the firing strength of each rule node must be calculated on the phase plane. Since the delay of the aeroload simulator system can be negligible and the sampling period of digital controller is very small (2 ms), we can ignore the delay effect. Consequently, the controller output that would . The have been desired is assumed to be firing strength of the four rule nodes is obtained by (14)–(17). The combined firing strength of each node is computed by the algebraic product defined by (18)–(21). In order to improve the current control performance, the output reinforcement is calculated. The fuzzy output reinforcement is inferred from (22) and (23). The crisp value of the control input reinforcement is obtained by (24). The final output reinforcement of the self-compensating mechanism is computed by (25) and (26). Then, the actual control input of the aeroload simulator system is obtained by using (27). V. SIMULATION AND EXPERIMENTAL RESULTS To test the performance of the proposed controller as applied to the aeroload simulator system, a series of simulations and experiments are performed for various conditions. The simulation conditions include the cases in which the operating point changes, external disturbances are applied, and the speed factor of the controller varies. The experimental conditions include the cases in which the operating point changes are applied, and the speed factor of the controller varies. The characteristic values of the aeroload simulator system are shown in Table II and documented in detail in [30]. As a preliminary test, step inputs of different magnitude are input to the system and the responses are compared with those of the PID and fuzzy controller with initially fixed rules. and ) for the error and The input scaling factors ( change in error of the fuzzy logic controller are chosen to be 1/5 and 1/5000, respectively. The output scaling factor ( )

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TABLE II THE CHARACTERISTIC VALUES OF THE AEROLOAD SIMULATOR SYSTEM

of the fuzzy logic controller is chosen to be 10. The speed factor of the FLCPP was chosen as 20 by the tuning method was chosen as 0.001. The gains on the trial and error basis. , , of the PID controller were tuned as which is obtained by a trial and error method. A. The Influence of Set Point Change Fig. 6 shows the simulation results of the step response of various controllers PID, fixed fuzzy, and FLCPP when set points are changed. Fig. 7 shows the experimental results of the step response of the fixed fuzzy and FLCPP when set points are changed. These controllers were tuned at 500 lb/in (56.5 Nm), which is assumed the main operating point of the aeroload simulator. To see an ability to adapt the operating point, the reference input torque is changed to 100 lb/in (11.3 Nm). As shown in Figs. 6(a) and 7(a), for large values of the reference input the FLCPP gives faster response, compared with the PID and the fixed fuzzy controller. For small reference input, the FLCPP and the fixed fuzzy controller show nearly similar response characteristics as shown in the Figs. 6(b) and 7(b), while the PID controller shows rather slower response. B. The Influence of External Disturbances Fig. 8 exhibits the simulation results of the response characteristics of the three controllers in case when an external disturbance of 5 step is applied to the aeroload simulator system at s. The response time of the PID, the fixed fuzzy the time and the FLCPP, during the output regains within 2% of the reference input, is 34 ms, 27 ms, and 17 ms, respectively. This indicates that the robustness against the external disturbance of the FLCPP is superior to those of the PID and the fixed fuzzy.

C. The Influence of the Speed Factor Variation Fig. 9 exhibits the simulation results and experimental ones of the step response characteristics of the FLCPP according to the variation of the speed factor. As the speed factor increases, the step response becomes faster. However, the speed factor larger than 30 gives some oscillatory motion, and much larger speed factor causes the aeroload simulator servo system to be unstable. VI. CONCLUSION In this paper, we proposed a new fuzzy controller design method using phase plane analysis. To demonstrate effectiveness of this type of controller, a series of numerical simulations and experiments were performed on the aeroload simulator system. The simulation and experimental results show that the fuzzy control scheme designed using the proposed method gives faster and more accurate responses, compared with the PID and the fixed fuzzy controller in the cases when the operating point is changed and external disturbances are applied. The proposed method has the following advantages: 1) Since the control rule table can be selected as a very simple one from the start and the initial membership functions can be easily determined by the (30) proposed, there will be reduced burden to have an expert’s knowledge about the plant to be controlled. 2) Since this control scheme utilizes a phase plane to which the control rules and membership functions are quantitatively transformed, simultaneous adjustments of them are accomplished by the self-compensating mechanism at each sampling instant. If the speed factor is determined, the fuzzy logic controller is automatically tuned without repetitive adjustment of many parameters of the fuzzy logic controller. This makes it very easy to tune a fuzzy logic controller to work properly.

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(a)

(b) Fig. 6. Step responses of PID, fixed rule fuzzy and FLCPP: simulation. (PID: K = 10. FLCPP:  = 20).

61=5000, G

6

= 4, K = 2, K

= 0:001. Fixed rule fuzzy: G

=

61=5, G

(a)

(b) Fig. 7. Step responses of fixed rule fuzzy and FLCPP: experiment. (Fixed rule fuzzy: G

= 61=5, G = 61=5000, G = 610. FLCPP:  = 20).

=

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(a)

(b) Fig. 8.

The influence of an external disturbance of PID, fixed fuzzy, and FLCPP: simulation.

(a)

(b) Fig. 9. The influence of the speed factor variation of FLCPP.

LEE AND CHO: A FUZZY CONTROLLER FOR AN AEROLOAD SIMULATOR

3) Since the results of fuzzy inference to obtain the output reinforcement are utilized, the execution speed is fast and, thus, the application of the algorithm can be made wider than that of the conventional fuzzy controllers. REFERENCES [1] R. B. Keller and J. Chen, “A high performance adaptive controller for nonlinear hydraulic servo system,” in ASME Winter Annu. Meet., Boston, MA, 1983, 83-WA/DSC-17. [2] H. J. Park and H. S. Cho, “An adaptive control of nonlinear time varying hydraulic servo systems,” in Proc. Amer. Contr. Conf., vol. 3, Pittsburgh, PA, 1989, pp. 1894–1898. [3] K. Takahasi, “Application of the model reference adaptive control technique to an electro-hydraulic servo system,” in Proc. Int. Conf. Fluid Power Transmission Contr., 1985, pp. 68–87. [4] J. S. Yun and H. S. Cho, “Adaptive model following control of electrohydraulic velocity control systems subjected to unknown disturbances,” Proc. Inst. Elect. Eng., pt. D, vol. 135, no. 2, pp. 149–156, 1988. [5] K. S. Boo and H. S. Cho, “A self-organizing fuzzy control of weld pool size in GMA welding processes,” Contr. Eng. Practice, vol. 2, no. 6, pp. 1007–1018, 1994. [6] S. Daley and K. F. Gill, “A design study of a self-organizing fuzzy logic controller,” Proc. Inst. Mech. Eng., vol. 200, no. C1, pp. 59–69, 1986. [7] P. J. King and E. H. Mamdani, “The application of fuzzy control systems to industrial processes,” Automatica, vol. 13, no. 3, pp. 235–242, 1977. [8] C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller—Part I,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 404–435, 1990. [9] Y. F. Li and C. C. Lau, “Development of fuzzy algorithms for servo system,” IEEE Contr. Syst. Mag., vol. 9, no. 3, pp. 65–72, 1989. [10] M. Maeda and S. Murakami, “A self-tuning fuzzy controller,” Fuzzy Sets Syst., vol. 51, pp. 29–40, 1992. [11] E. H. Mamdani, “Application of fuzzy algorithms for control of simple dynamic plant,” Proc. Inst. Elect. Eng., vol. 121, no. 12, pp. 1585–1588, 1974. [12] E. H. Mamdani and S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man–Machine Studies, vol. 7, no. 12, pp. 1–13, 1975. [13] T. J. Procyk and E. H. Mamdani, “A linguistic self-organizing process controller,” Automatica, vol. 15, pp. 15–30, 1979. [14] D. A. Rutherford and G. C. Bloore, “The implementation of fuzzy algorithms for control,” Proc. IEEE, vol. 64, no. 4, pp. 572–573, 1976. [15] S. Sao, “Fuzzy self-organizing controller and its application for dynamic processes,” Fuzzy Sets Syst., vol. 26, pp. 151–164, 1988. [16] R. Tanscheit and E. M. Scharf, “Experiments with the use of a rule-based self-organizing controller for robotics applications,” Fuzzy Sets Syst., vol. 26, pp. 195–214, 1988. [17] B. S. Zhang and J. M. Edmunds, “Self-organizing fuzzy logic controller,” Proc. Inst. Elect. Eng, pt. D, vol. 139, no. 5, pp. 460–464, 1992. [18] C. S. Hsu, “A theory of cell-to-cell mapping dynamical systems,” ASME J. Appl. Mechanics, vol. 47, pp. 931–939, 1980. [19] C. S. Hsu and R. S. Guttalu, “An unravelling algorithm for global analysis of dynamical systems: An application of cell-to-cell mappings,” ASME J. Appl. Mechanics, vol. 47, pp. 940–948, 1980. [20] Y. Y. Chen and T. C. Tsao, “A description of the dynamical behavior of fuzzy systems,” IEEE Trans. Syst., Man, Cybern., vol. 19, pp. 745–755, 1989. [21] B. H. Tongue and K. Gu, “Interpolated cell mapping for dynamical systems,” ASME J. Appl. Mechanics, vol. 55, pp. 461–466, 1988. [22] P. C. Chen and M. C. Shin, “An experimental study on the position control of a hydraulic cylinder using a fuzzy logic controller,” JSME Int. J., ser. III, vol. 34, no. 4, pp. 481–489, 1991.

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[23] S. Y. Lee, Y. J. Park, and H. S. Cho, “A neuro-fuzzy control of an electrohydraulic fin position servo system,” in Proc. ASME Int. Mechanical Eng. Conf. Exposition, vol. FPST-3, Atlanta, GA, Nov. 17–22, 1996, pp. 101–106. [24] , “A self-tuning fuzzy control of an electro-hydraulic fin position servo system,” in Proc. ASME Int. Mechanical Eng. Conf. Exposition, vol. FPST-5, Anaheim, CA, Nov. 15–20, 1998, pp. 99–106. [25] E. C. Yeh and T. Y. Lin, “A fuzzy control scheme for hydraulic cylinder servo with flow compensation,” in Int. Fuzzy Eng. Symp., 1991, pp. 661–671. [26] T. Zhao and T. Virvalo, “Theoretical and experimental analysis on fuzzy control of a hydraulic position servo,” in Proc. Int. Conf. Fluid Power Transmission Contr., 1993, pp. 251–255. [27] H. E. Merritt, Hydraulic Control Systems. New York: Wiley, 1976. [28] W. Pedrycz, Fuzzy Control and Fuzzy Systems. Taunton, Somerset, U.K.: Research Studies, 1989, pp. 1–11. [29] G. F. Franklin, J. D. Powell, and A. Emani-Naeini, Feedback Control of Dynamic Systems. Reading, MA: Addison-Wesley, 1986, pp. 336–339. [30] S. Y. Lee, “Fuzzy controller design using phase plane with its application to electro-hydraulic servo system,” Ph.D. dissertation, Korea Advanced Inst. Sci. Technol., Dept. Mech. Eng., 1999.

Sang Yeal Lee was born in Taegu, Korea, in 1951. He received the B.Eng. degree in mechanical engineering from Han-yang University of Korea, Seoul, in 1974 and the M.S. and Ph.D. degrees in mechanical engineering from Korea Advanced Institute of Science and Technology (KAIST), Taejon, in 1986 and 1999, respectively. He is currently with the Agency for Defense Development (ADD), Korea. His present research interests include intelligent control, robust control, and servo system control.

Hyung Suck Cho (M’94) received the B.S. degree from Seoul National University, Seoul, Korea, in 1971, the M.S. degree from Northwestern University, Evanston, IL, in 1973 and the Ph.D. degree from the University of California, Berkeley, in 1977. From 1977 to 1978, he was a Postdoctoral Fellow with the Department of Mechanical Engineering, University of California, Berkeley. Since 1978, he has been a Professor with Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejon. His research interests include environment perception and recognition for mobile robots, machine vision and pattern classification, and application of artificial intelligence/machine intelligence. Dr. Cho serves on editorial boards of Journal of Robotic Systems, Robotica, Control Engineering Practice (IFAC), Journal of Advanced Robotics, and Journal of Engineering Manufacture (PIME). In 1998, he served as a Guest Editor of a special issue on “Intelligent Robotic Assembly” for Robotica. He general chaired/cochaired several conferences, including SPIE-Opto Mechatronic Systems in 2000, IEEE/RSJ IROS in 1999, International Workshop on Mechatronics Technology in 1999, IFAC Workshop on Intelligent Manufacturing in 1997, and two symposiums of ASME Winter Annual Meeting in 1991 and 1993. He has been endowed with a POSCO Professorship from Pohang Steel Company, Korea since 1995. In 1998, he was awarded Thatcher Bros Prize from Institution of Mechanical Engineers, U.K.