Model Reference Adaptive Controller Design Based on Fuzzy Inference System

Transcription

1 Journal of Information & Computational Science 8: 9 (2011) Available at Model Reference Adaptive Controller Design Based on Fuzzy Inference System Zheng Li School of Electrical Engineering and Information Science, Hebei University of Science and Technology Shijiazhuang , China Abstract This paper describes the model reference adaptive fuzzy controller design and its application on automatic gauge control system. The controller produces the error of the closed loop control system response and the actual system output for the desired system by reference model, instead of ordinary adaptive mechanism. The analysis of dynamic performance for traditional PID controller and fuzzy adaptive controller is performed in detail with simulation software. Simulation results show that the system is with strong adaptive ability and can adapt to the wide range of changes of the controlled object. This study provides the primary theoretical guide for the design, optimization and control research of the automatic gauge control system. Keywords: Model Reference; Fuzzy Logical Control; Adaptive System; AGC; Self-tuning PID Control 1 Introduction Generally, fuzzy control method is fit for systems with complex mathematical models. In addition, to determine the fuzzy rules, it is generally the same parameters, so the system does not have parameters or the parameters are relatively small. If the system uncertain parameters or the system with large process parameters is difficult to get good results, to compensate for these shortcomings, the adaptive control scheme proposed may be fuzzy control method. In 1950 s, Professor Whithei presents the Model Reference Adaptive System (MRAS), which is currently a set of matured theory and design method of adaptive control system. MRAS can play a better role for the control of many industry control objects with the environment and parameters of controlled object change. However, there are more complex adaptive mechanisms, large amount of design work and hard for computer implementation and other difficulties [1]. Since Ichikawa put forward the innovative design of model reference adaptive fuzzy control, many scholars have Project supported by the Natural Science Foundation of Hebei Province of China (No. E ), the Foundation of Hebei Educational Committee under Grant No. Z , the Scientific Research Start Foundation of Doctoral Scholars of Hebei University of Science and Technology under Grant No. QD Corresponding author. address: lzhfgd@163.com (Zheng Li) / Copyright 2011 Binary Information Press September 2011

2 1684 Z. Li / Journal of Information & Computational Science 8: 9 (2011) made progresses on the application of fuzzy theory to design model reference adaptive system [2, 3]. Iron and steel industry is always in the process of industrialization based on the status of the industry, and the production of steel strip steel is a very important factor. With the rapid development of modern industrial technology, the thickness of the strip have become increasingly demanding precision thickness control system (Automatic Gauge Control System, referred to as the AGC system) as an integral part of automatic control of rolling mill. Strip thickness accuracy as a measure of material quality is one important aspect, and is general concern issue for domestic and international metallurgical industry. Use of Automatic Gauge Control (AGC) to improve the accuracy of strip thickness is currently a trend for accuracy improvement, whose core is to control the uniformity of the strip [4-7]. Strip mill automatic gauge control system has become a high-precision and necessary technology for strip rolling, with improving the finished production rate, saving raw materials and to facilitate the operator. The system has the advantages of fast response, high control precision, good reliability, complete system protection functions and fault diagnosis abilities. Intelligent control technology applications in the rolling mill process has become an inevitable trend, currently the technology is still in continuous improvement. Due to many problems of traditional AGC systems, the most obvious problem is that the control accuracy, and intelligent control technology as fuzzy control and neural network control system can improve the accuracy of AGC to improve the control effect of the rolling process [8]. Fuzzy inference system can take advantage of already acquired knowledge and experience of experts to determine the criteria of roll gap adjustment for the AGC system with high accuracy regulation requirements. This paper analyzes the mill thickness control based on the hydraulic AGC system through the establishment of a mathematical model of the system s static and dynamic characteristics that are analyzed, and a fuzzy adaptive control strategy is adopted to achieve satisfactory results. The aim of this study is to develop and apply a model reference adaptive controller based on the fuzzy inference system for industrial rolling mill gauge control, also with a detailed comparison with traditional PID controller. The work results can have maximum capabilities for further control system design and control strategy research of this kind of industry applications. 2 Adaptive Fuzzy Control Theory 2.1 Model Reference Adaptive Control System The basic system comprises the reference model, controlled object, feedback controller and adaptive controller. The reference model is an ideal model and its output y m (t) directly denotes the required dynamic response. The adaptive regulation process of the controller parameters is described as follows: when the input value r(t) is set to the controller, it is also simultaneously added to the reference model input; at the initial stage, since the origin parameters of controlled object are unknown, the controlled parameters are not determined causing the output response y(t) not in accordance with y m (t) and e(t) is produced. When e(t) is introduced into the adaptive regulation loop, through the calculation by adaptive laws and then proper dynamic signal of changing the controller parameters is derived to make the y(t) get approaching to y m (t), i.e. e(t) 0 with adaptive process ceased.

3 Z. Li / Journal of Information & Computational Science 8: 9 (2011) Fuzzy Adaptive Model Reference Control System The structure of Reference Model Based Fuzzy Adaptive System (FMRAS) can be shown in Fig. 1. In the control system, the fuzzy adaptive controllers together with the controlled object constitute the closed-loop system with adjustable parameters [9, 10]. The controller uses the indirect control method, which is firstly modeling the controlled object by the fuzzy logic system, and then producing the desired control action. The fuzzy logic system can get approaching the controlled object by regulating the adjustable parameters, so that the output of the system under certain conditions can track the reference model output for any precision. Reference model y m e,e* r Controller u Controlled object y P Adaptive fuzzy mechanism Fig. 1: Illustration of MRAS 2.3 Control Algorithm Design Considering nonlinear discrete system where g(t), are unknown nonlinear functions, and y(t + 1) = g(t) + u(t) (1) g(t) = g(y(t), y(t 1),, y(t n 1 + 1), u(t), u(t 1),, u(t n 2 + 1)) (2) = h(y(t), y(t 1),, y(t m 1 + 1), u(t), u(t 1),, u(t m 2 + 1)) (3) where k n 1, n 2, m 1, m 2 ; u(t) R and y(t) R are the input and output of the system respectively. Suppose g(t) and can be measured and estimated by small samples, with 0(t = 0, 1, 2, ). The control task is to make the output of controlled system tracking a given bounded reference signal y m (t + 1) with the constraints that all the signals are bounded. So the control purpose is to derive a feedback control signal u(t) and adaptive law of an adjustable vector W (t) for 1. In all variables W (t) and u(t) uniformly bounded sense, the system output error e(t + 1) = y m (t + 1) y(t + 1) is as small as possible; 2. Under certain conditions, the adjustable system is with global asymptotic stability.

4 1686 Z. Li / Journal of Information & Computational Science 8: 9 (2011) For the system as depicted in (1), if g(t) and are known, using the control law u c (t) = 1 [ g(t) + y m(t + 1)pe(t)] (4) where p < 0 is the feedback gain, then the output error of the system is e(t + 1) = y m (t + 1) y(t + 1) = y m (t + 1) [g(t) + u c (t)] [ = y m (t + 1) g(t) + 1 ] ( g(t) + y m(t + 1) + pe(t)) = y m (t + 1) y m (t + 1) pe(t) = pe(t) (5) then it can be seen that if p < 1, the output of adjustable system can asymptotically track the reference model output y m (t + 1). Since g(t) and are unknown continuous functions, if they are substituted by fuzzy inference system g(t) and respectively, let the control law be then According to equation (4), u c (t) = 1 [ g(t) + y m (t + 1) + pe(t) y(t + 1) = g(t) + 1 [ ] g(t) + y m (t + 1) + pe(t) u c (t) = 1 [ g(t) + y m (t + 1)] + 1 pe(t) = u c(t) + p e(t) (8) where u c(t) = 1 [ g(t) + y m (t + 1)]. And the system error is e(t + 1) = [g(t) g(t)] + u c(t) u c(t) p e(t) (9) = g(t) g(t) + [ ]u c(t) p e(t) (10) Let g(t) = g(t) g(t), =, then e(t + 1) = g(t) + g(t)u c(t) p r(t) (11) where g(t) and are the identification errors of g(t) and. ] (6) (7)

5 Z. Li / Journal of Information & Computational Science 8: 9 (2011) Industrial Applications 3.1 Hydraulic AGC System Model Gauge control system is a typical double close loop control system; position control is the inner loop of gauge control system, and thus to establish system model of gauge control, the mathematical model of position control should be found first. The mathematical model of position control is composed of hydraulic cylinder, servo valve, position transducer and other segments [11]. As the main element, the hydraulic cylinder of AGC system dynamic mechanism model can be shown in Fig. 2. The thickness control system is composed of more than a combination of AGC control, foremost of which is electro-hydraulic position servo system. The servo system consists of oil pipelines, servo valves, back to the oil pipeline, hydraulic cylinders, sensors, control amplifier. Automatic gauge control system of the inner-loop is formed by the hydraulic pressure system based Automatic Position Control (APC) system. AGC feedback loop control system is measured by the gauge device of the thickness of the formed strip. AGC system uses the gauge outer loop and position inner loop mode or gauge outer loop and pressure inner loop mode. The difference between the two modes lies in the controlled variable, the gauge outer loop and position inner loop mode demands to calculate the relationship between δh and δs, then δs is added to the gap setting value as the required value of position inner loop to eliminate the thickness error δh. The gauge outer loop and pressure inner loop mode need to calculate the relationship of δh and δp, then δp is added to the pressure setting value as the required value of pressure loop to eliminate the thickness error. Gauge outer loop Gauge outer loop S SET +δs P SET +δp Position inner loop S Magnetic meter Pressure inner loop P Pressure transducer Servo valve Servo valve Fig. 2: Dynamic mechanism model of the hydraulic AGC system Based on the models of individual elements of the AGC system, the complete hydraulic AGC system block diagram can be shown in Fig. 3. The main parameters can be given as Table 1 [11]. The closed-loop hydraulic AGC system transfer function can be derived as G(s) = K vi K sv A P /K ce k ( ) ( s ω r + 1 s 2 ω ξ 0 s + 1 ω0 2 ) (12) Substitute the main parameters into the function: G(s) = s s s + 1 (13)

6 1688 Z. Li / Journal of Information & Computational Science 8: 9 (2011) U i K a S 2 2ζ ω s+1 v ω v K v ω 2 A/K a K s 2ζ 0 ω + s+1 r ( ) S 2 ωv 2 ω v +1 ( ) y K f /(T f +1) Fig. 3: Block diagram of the AGC system Amplifier gain K a Table 1: Control system parameters Hydraulic cylinder piston area A Electro-hydraulic servo valve intrinsic frequency ω v 1A/V 0.5m 2 50rad/s Electro-hydraulic valve gain K v Transducer gain K f Damping ratio ξ m 3 /As 400V/m 0.47 Load stiffness K Transducer intrinsic time constant T f coefficient of the spring coupled Ratio of height and damping in series ω r N/m 0.025s 0.105rad/s To illustrate the steady characteristics of the system, the Nyquist curves of the system can be plotted as in Fig.4. There are three poles as i, i, 19.19, which are all located in the left half plane and not surrounded with 1 point on the complex plane. From the Nyquist stability criterion, the system is determined stable. The dynamic response of the system can be shown in Fig. 5. Without controllers, the open-loop control system is stable, but the system response speed is too slow, the hydraulic AGC system is obviously does not meet the requirements of fast and must be corrected by adding controller to improve system performance, to function properly in production. Fig. 4: Nyquist curves of the AGC system Fig. 5: Open-loop step response curves of the AGC system

7 Z. Li / Journal of Information & Computational Science 8: 9 (2011) Adaptive Fuzzy Hydraulic AGC System Control For the FMRAS, the controller can be PID controller, with corresponding adjustable parameters k p, k i and k d. Let the system input is r, the reference model output y m should be rises and approached to r. When the object structure changes or the parameters deviate, the function of adaptive fuzzy mechanism is to amend k p, k i and k d continuously to make y follow y m, i.e. t 0, make e = y m y 0. In this research, the unit of rolling range of roll gap error e is ( 200µm, +200µm), the derivative of roll gap error ec is ( 15µm, +15µm). Define the domain of two fuzzy sets as e, ec = { 3, 2, 1, 0, 1, 2, 3}, and its subset is e, ec={nb, NM, NS, O, PS, PM, PB}, which elements representing negative large, negative middle, negative small, zero, positive small, positive middle, positive large. The domain of K p, K i, K d are set by { 0.3, 0.2, 0.1, 0, 0.1, 0.2, 0.3}; { 0.06, 0.04, 0.02, 0, 0.02, 0.04, 0.06}; { 3, 2, 1, 0, 1, 2, 3}. The subset is chosen as {NB, NM, NS, O, PS, PM, PB}. Membership functions according to the basic principles must be followed to the roll gap error e, for example, the membership function is shown in Fig. 6. In the middle of the input and output membership functions, strong resolution trigonometric functions are used; in the minimum and maximum margins, relatively slow changing large left-right down small decreasing function (i.e. Z-type membership function) and large left-right down big increasing function (i.e. S-type membership function) are adopted. The output of Kd, Ki, Kp surfaces on the domain can be shown in Fig. 7. Fig. 6: Membership function of the roll gap deviation e The basic principle of selecting fuzzy control rules are: when the roll gap position deviation getting large or larger, the control variable should be mainly chosen to eliminate the amount of the error as soon as possible; when the error is small, the selection of control input should be taken to avoid overshoot with the stability of the system taken as the main point. The definite rules from empirical datum are shown as Table 2 to Results and Analysis According to the above established controller and fuzzy controller, create the two controllers based on the hydraulic AGC system simulation module in MATLAB, the input for the unit step signal and the signal involved in testing the performance of the controller. After repeatedly adjusting the parameters of controller, it can be derived that K p = 0.2, K i = 0, K d = 0.001, when the controller can make the system get better control performance. Three parameters in the PID controller

8 1690 Z. Li / Journal of Information & Computational Science 8: 9 (2011) Fig. 7: The output of Kd, Ki, Kp surfaces on the domain Table 2: Fuzzy rules of K P K P NB NM NS ZO PS PM PB ec NB PB PB PM PM PS ZO ZO NM PB PB PM PS PS ZO NS NS PM PM PM PS ZO NS ZS ZO PM PM PS ZO NS NM NM PS PS PS ZO NS NS NM NM PM PS ZO NS NM NM NM NB PB ZO ZO NM NM NM NB NB e are chosen to regulate the process, although they can make the system stable, but the process is too cumbersome and difficult to get the right parameters, and fuzzy adaptive controller can change according to the output data automatically to adjust the three parameters ensuring the stability control system. The PID controller based unit step response curve and fuzzy adaptive PID controller step response curves are shown in Fig. 8 and Fig. 9. From Figs. 8 and 9, the comparison can be seen under the control of fuzzy PID controller settling time AGC system is significantly less than the PID controller, fuzzy PID controller has

9 Z. Li / Journal of Information & Computational Science 8: 9 (2011) Table 3: Fuzzy rules of K I K I NB NM NS ZO PS PM PB ec NB NB NB NM PM PS ZO ZO NM NB NB NM PS PS ZO NS NS NB NM NS PS ZO NS NS ZO NM NM NS ZO PS PM PM PS NM NS ZO PS PS PM PB PM ZO ZO PS PS PM PB PB PB ZO ZO PS PM PM PB PB e Table 4: Fuzzy rules of K D K D NB NM NS ZO PS PM PB ec NB PS NS PB NB PS NM PS NM PS NS PB NM PS NS ZO NS ZO NS NM NM ZO NS ZO ZO ZO NS NS NS NS NS ZO PS ZO ZO ZO ZO NS ZO ZO PM PB NS PS PS NM PS PB PB PB PM PM PS NM PS PB e Fig. 8: AGC step response curves with ordinary PID controller Fig. 9: AGC step response curves with adaptive fuzzy controller been in the 0.5s before the system stable, while PID controller in the 0.5s before the system is stable, and the fuzzy PID controller overshoot than the PID controller is much smaller overshoot. Hydraulic AGC system is highly uncertain, so it really should be used by the control algorithm with strong robustness. To illustrate the adaptive performance, the controlled object model can be change and then compare ordinary PID and fuzzy PID controller s results. The hydraulic AGC

10 1692 Z. Li / Journal of Information & Computational Science 8: 9 (2011) system closed-loop transfer function is changed to G(s) = 8/( s s s + 1). The step response curves of AGC system can be observed under two controllers, as shown in Figs. 10 and Fig. 10: AGC step response curves with ordinary PID controller Fig. 11: AGC step response curves with adaptive fuzzy controller For the step response, it can be observed from figure 10 and 11, when the controlled object model changes, the response curves under the control of fuzzy PID controller correspond to no significant changes, overshoot and settling time are still very small; the curves under the control of PID controller volatile with the rapid increase in overshoot. It can be seen that the ordinary PID controller is weak to be adapt to external parameters, when the parameters of the controlled object changes it can not make the necessary adjustments; the adaptive fuzzy PID controller is with strong ability of resisting the disturbances with better robust performance, to make the hydraulic AGC control system achieve satisfactory results. 4 Conclusions In accordance with the features of AGC system for cold rolling mill, a model reference adaptive controller design based on fuzzy inference system tuning to achieve the hydraulic servo control system for rolling mill gauge is developed. The fuzzy adaptive controller based on the fuzzy-tuning mechanism according to the size of the input signal, direction and trends and other characteristics can make decisions by fuzzy inference on the ratio of differential and integral adjustment parameters online to make the controlled object with desired dynamic and static features. Simulation results show that, FMRAS in the severe model mismatch case, still can get better control performance, with enhanced satisfied self-adaptability and the resistance ability to internal and external disturbances than the conventional control system significantly. The design is relatively simple without requirement of the object model, which is apt for engineering applications. It is expected that the presented work can give a reference for further development of advanced schemes of adaptive fuzzy control and for industrial applications. References [1] I. D. Landu, Adaptive Control-model Reference Approach, Marcel Dicker, New York, 1979

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