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• Model ∑ Inference Mechanism Adaptive Fuzzy Controller The reference model given in the FMRLC system characterizes the desirable design criteria, such as the stability, rise time, overshoot, and settling time. As shown in Figure 1, the input reference signal r(kT ) is applied to the reference model. The reference model output y m (kT ) is the desired value of our system to act. The desired performance of the controlled process is met if the learning mechanism forces y e (kT ) to be kept at a very small value for the entire time. Hence, if the performance is met, i.e. y e (kT ) ≈ 0, then no significant modifications are performed by the learning mechanism to the adaptive fuzzy controller [13]. The learning mechanism, which is the most important part of the controller, consists of 2 parts: a fuzzy inverse model and a knowledge-base modifier. Fuzzy inverse model with variable adaptation gain ( N p ) For adaptive control systems, a fuzzy inverse model was enhanced in the linguistic self-organizing control structure by investigating methods to reduce the problems using the inverse process model. Procyk and Mamdani’s [14] use of the inverse process model depended upon the use of an explicit mathematical model of the process, and eventually assumptions about the underlying physical process. In applying this approach, dependence on a mathematical model of the process often causes considerable difficulties. The aim of employing a fuzzy inverse model is to characterize how to change the plant inputs u(kT ) to force the plant output y(kT ) to be as close as possible to the reference model output y m (kT ) . It should be noted that, similar to the fuzzy controller, as shown in Figure 1, the fuzzy inverse model contains normalizing scaling factors, namely N ye , N yc , and N p , for each space of discourse of the inputs and output. Selection of the normalizing gains, N ye , N yc , and N p , can affect the overall performance of the system [13]. In this study, the input scaling factors of the fuzzy inverse model ( N ye and N yc ) are conventionally defined as constants. Moreover, the output scaling factor of the fuzzy inverse model ( N p ) is defined as a new variable adaptation gain depending on the error, y e (kT ) , as given by Eq. (5): N p (kT ) = a + b × |y e (kT ) | , (5) where a and b are roughly set constants. The system changes the amount of adaptation according to the error, i.e. a big error means a bigger adaptation, and a small error means a smaller adaptation. The variation curve of the variable adaptation gain ( N p ) is shown in Figure 2. The adaptive fuzzy control system with variable adaptation gain has become more dynamic. a + b Error, y e (kT ) Figure 2. Variation curve of the variable adaptation gain ( N p ) . In the adaptive fuzzy controller and fuzzy inverse model, the same types of triangular membership functions ( m 1 – m 11 ) are used for the 2 inputs and an output as shown in Figure 3. The rule base array for the fuzzy inverse model and the fuzzy controller is shown in Table 1 [12]. Y e and Y c denote the fuzzy sets associated with y e (kT ) and y c (kT ) , respectively, and P denotes the fuzzy sets quantifying the desired process input change p(kT ) that it is multiplied by the variable adaptation gain. -1 - 0.8 - 0.6 - 0.4 - 0.2 0 0.2 0.4 0.6 0.8 1 Input (a) Figure 3. Membership functions of a) the adaptive fuzzy controller ( e , c) and b) fuzzy inverse model ( y e , y c ) . Table Rule base for the fuzzy controller and fuzzy inverse model. P, U Y c , C m 1 m 2 m 3 m 4 m 5 m 6 m 7 m 8 m 9 m 10 m 11 Y e, E action u(kT − T ). This modification involves shifting these triangular membership functions by an amount specified with p(kT ) = [p 1 (kT ).....p n (kT )] T , thus: c n (kT ) = c n (kT − T ) + p n (kT ). (8) Modeling of a TVRS system Permanent magnet DC (PMDC) motors are commonly used as an actuator in control systems. They directly provide rotary motion or moment and, coupled with wheels or drums and cables, can provide transitional motion or force [17]. In this study, the TVRS system actuated with a PMDC motor that has the electric circuit of the armature and the loaded body diagram of the rotor are shown in Figure 4. The rotary servo plant and PMDC motor parameters are given in Table 2. J l I V b m m e k = , m m – – Figure 4. Equivalent circuit of the TVRS system. Table Rotary servo system and load parameters. Symbol Definition Value V Nominal voltage of the DC motor 6 V R Motor armature resistance 6 Ω L Motor armature inductance 0.18 mH k t Motor torque constant 0.00767 N m/A k b Motor back-EMF constant 0.00767 V/(rad/s) N g Total gear ratio (N 2 /N 1 ) 70 η g Gearbox efficiency 0.90 J m Motor inertia 6 × 10 −7 kg m 2 B m Motor viscous damping coefficient ∼ = 0 (negligible) J l in Initial load and gearbox moment of inertia 83 × 10 −7 kg m 2 J l sub Subsequent load and gearbox moment of inertia 83 × 10 −6 , kg m 2 B l in Initial load viscous damping coefficient 41 × 10 −6 N m/(rad/s) B l sub Subsequent load viscous damping coefficient 41 × 10 −5 N m/(rad/s) From Figure 4 we can write the following dynamic equations based on Newton’s law combined with Kirchhoff’s law [17,18]: V (t) = Ri(t) + L di(t) dt b (t), (9) τ m (t) = J m dt 2 + B m dt + τ l (t), (10) where i(t) is the armature current, e b (t) is the back emf voltage, τ m (t) is the motor torque, τ l (t) is the load torque, and θ m (t) is the angle of the armature. The motor torque, τ m (t) , is related to the armature current, i(t) , by a constant factor k t . The back emf, b (t) , is related to the rotational velocity of armature, ω m (t) or θ m (t) dt , by the following equations: τ m (t) = k t i(t), (11) e b (t) = k m ω m (t) = k m dt
• In the TVRS system, the rotational angle of the load, θ l (t) , transmitted by the gear box from the armature angle, θ m (t) , and variable load torque , τ l (t) , may be expressed as: θ l (t) = 1 N g τ l (t) = 1 N g η g dt 2 + B l dt ), (14) where N g is the total gear ratio and η g is the gearbox efficiency. Using Eqs. (9)–(14), the Simulink model of the TVRS system with variable load parameters ( J l , B l ) has been obtained, as shown in Figure 5. Teta V, ref Out Torque, Tm=kt*I Motor Angle (Teta) Gear Factor Derivative2 Current (I) Back Emf, Eb=km*Wm Figure 5. Simulink model of the TVRS system with a variable load ( J l , B l ) . Fuzzy model reference learning control of the TVRS system The MATLAB-Simulink model of the simulation and experimental system used in this study is shown in Figure 6, where the rule base adaptation and modification process are carried out in the MATLAB s-function and the first-order reference model is defined [19,20]. The variable adaptation gain ( N p ) is performed according to Eq. (5) and the TVRS system shown in Figure 5 is carried out as a subsystem in Figure 6. ym(kT) y(kT) Output.mat Nc 0.025 Derivative1, K=1 K (z-1) Ts z Controller Output Figure 6. Simulink model of the FMRLC for the simulation and experimental system. A photograph of the experimental system is given in Figure 7a. In the experimental system, a digital signal processing-based Quanser SRV-02 servo module has been used. As can be seen in Figure 7a, this module can be loaded with variable loads. A Q3 control PaQ-FW data acquisition board, which has its own built-in amplifier, is also utilized. Moreover, the MATLAB-Simulink based QuarC 2.0 software, which executes the FMRLC algorithm, has been used [21]. A block diagram of the experimental system is shown in Figure 7b. Figure 7. Photograph (a) and block diagram (b) of the experimental system. Simulation and experimental results The reference model for the simulation and experimental system was chosen to represent the somewhat realistic performance specifications and is expressed by Eq. (15) in the Laplace (s) domain. T ref (s) = 5 s + 5 (15) In both the fuzzy controller and fuzzy inverse models’ inputs and outputs, 11 fuzzy sets are defined with a triangular shape, with a base width of 0.4 membership functions, shown in Figure 3. Table 1 is used for the fuzzy inverse model as a rule base. For both the simulation and experimental tests, a step-wave signal for 3 reference points ( + π 4 , 0, − π 4 rad.) is used. In the adaptive fuzzy controller, the normalizing controller gains for the error, change in error, and the controller output are obtained as N e = 0.025, N c = 0.025, and N u = 11, respectively. The adaptive fuzzy controller sampling period is defined as T = 0.1 s. In the fuzzy inverse model, the normalizing controller gains associated with y e (kT ) , y c (kT ) , and p(kT ) are obtained as N ye = 0.025, N yc = 0.025, and N p (kT ) = 0.7 + 0.5 × |y e (kT ) |, respectively, by trial and error. In both the simulation and experimental implementation, the servo system is initially loaded with the parameters J l = 83 × 10 −7 kg m 2 and B l = 41 × 10 −6 N m/(rad/s). After a while (at t = 125 s), the servo system is subsequently loaded with the parameters J l = 83 × 10 −6 kg m 2 and B l = 41 × 10 −5 N m/(rad/s). The responses of the simulation and experimental system, error signals, and control signals have been observed. In the control systems, performance indices are used to measure the system performance. In this study, the performance values of the simulation and experimental system have been measured using the integrate absolute error (IAE ) index given by Eq. (16). IAE = t ∫ |e(t)| dt (16) The IAE index is widely used in the literature because it is more selective in transient response and can be easily calculated. The IAE values of the simulation and experimental system was calculated for both the constant and variable values of the N p parameter. Simulation results The simulation process is carried out regarding Figure 6 based on a MATLAB-Simulink environment. The input signal ( r (kT )), reference model response ( y m (kT )), and servo position responses ( y (kT )) of the simulation are shown in Figure 8 and the zoom area is shown in Figure 9. Over time, the servo position response overlaps with the reference model response, as shown in Figure 8. At t = 125 s, the load is increased. At this time the value of the error increases but over time, this error value decreases again, as shown in Figure 10. The error, which is the difference between the reference model and the servo position response, gets smaller with time. Figure 11 shows the output or control signal of the adaptive fuzzy controller. The adaptive fuzzy controller produces a control signal, again at t = 125 s, and manages to control the system response on the reference model. The change in the knowledge base of the adaptive fuzzy controller before and after the simulation is shown in Table 3. Experimental results The experimental implementation of the proposed system is based on a Quanser SRV-02 servo module, as shown in Figure 7. The input signal ( r (kT )), the experimental response of the reference model ( y m (kT )), and the servo position response ( y (kT )) are shown in Figure 12. The servo position response overlaps the reference model output exactly, as shown in Figure 12. At t = 125 s, the load is increased. At this time, the value of the 50 100 150 200 -1 Time (s) 5 10 15 20 25 30 35 40 -0.1 Time (s) ym(kT) r(kT) Figure 8. Input signal, reference model, and servo position responses of the simulation system. 50 100 150 200 Time (s) 50 100 150 200 -0.1 Time (s) Figure 10. Error that is the difference between the reference model and the TVRS system output of the simulation system. Table Knowledge base of the adaptive fuzzy controller before and after the simulation. Mem. Initial centers Adapted (final) centers Func. C e C c C u C e C c C u c 1 –0 –0 –0.8 –000000 –000000 –0.800000 c 2 –0 –0.8 –0.6 –000000 –0.800000 –0.600000 c 3 –0.8 –0.6 –0.4 –0.800000 –0.600000 –0.400000 c 4 –0.6 –0.4 –0.2 –0.742232 –0.542232 –0.342232 c 5 –0.4 –0.2 0.0 –0.558066 –0.358066 –0.158066 c 6 –0.2 0.0 0.2 –0.199997 0.000003 0.200003 c 7 0.0 0.2 0.4 0.380312 0.580312 0.780312 c 8 0.2 0.4 0.6 0.358069 0.558069 0.758069 c 9 0.4 0.6 0.8 0.400000 0.600000 0.800000 c 10 0.6 0.8 0 0.600000 0.800000 000000 c 11 0.8 0 0 0.800000 000000 000000 error has been increased, but the servo position response overlaps the reference model output exactly in time. Figure 13 shows the error, which is the difference between the reference model and the TVRS system output. The adaptive fuzzy controller output, which is the control signal, is shown in Figure 14. The adaptive fuzzy controller produces a control signal again at t = 125 s, and manages to control the system response on the reference model. 50 100 150 200 -1 -0.8 -0.6 -0.4 -0.2 Time (s) 50 100 150 200 Time (s)
• Figure 12. Input signal, reference model, and servo position responses of the experimental system. Figure 13. Error that is the difference between the reference model and the TVRS system output of the experimental system. 50 100 150 200 -0.1 -0.08 -0.06 -0.04 -0.02 Time (s) Figure 14. Experimental output of the adaptive fuzzy controller. The change in the knowledge base of the fuzzy controller before and after the experimental application is shown in Table 4. The performance measurements obtained by calculating IAE indices for the simulation and experimental system in the case of both constant gain ( N p = 0.7) and the newly defined variable gain ( N p = 0.7+0.5 ×|y e |) are given in Table 5. It is seen from Table 5 that the performance index (IAE ) has a lower value for the simulation and experimental system, in which the gain depends on the error ( y e (kT )). These results demonstrate that the proposed variable gain method provides a better performance compared to the traditional constant gain approach. Table Knowledge base of the adaptive fuzzy controller before and after the experimental application. Mem. Initial centers Adapted (final) centers func. C e C c C u C e C c C u c 1 –0 –0 –0.8 –000000 –000000 –0.800000 c 2 –0 –0.8 –0.6 –000000 –0.800000 –0.600000 c 3 –0.8 –0.6 –0.4 –0.800000 –0.600000 –0.400000 c 4 –0.6 –0.4 –0.2 –0.705990 –0.505990 –0.305990 c 5 –0.4 –0.2 0.0 –0.517573 –0.317573 –0.117573 c 6 –0.2 0.0 0.2 –0.200002 –0.000002 0.199998 c 7 0.0 0.2 0.4 0.252435 0.452435 0.652435 c 8 0.2 0.4 0.6 0.317571 0.517571 0.717571 c 9 0.4 0.6 0.8 0.400000 0.600000 0.800000 c 10 0.6 0.8 0 0.600000 0.800000 000000 c 11 0.8 0 0 0.800000 000000 000000 Table Performance measurements of the simulation and experimental system. Perf. index Simulation system Experimental system N p = 0.7 N p = 0.7 + 0.5 ×|y e | N p = 0.7 N p = 0.7 + 0.5 ×|y e | IAE 4527 6697 1567 0367 Conclusions
• In this paper, the simulation and experimental implementation of a fuzzy model reference learning control technique with a variable adaptation gain ( N p ) is studied for the position control of the TVRS system. The FMRLC structure provides an automatic method to synthesize the knowledge base. The learning and adaptation mechanism in the FMRLC continuously updates the knowledge base in the adaptive fuzzy controller. Hence, when unpredictable changes occur within the plant, the FMRLC can make on-line adjustments in the fuzzy controller to follow the reference model. Thus, the TVRS system response overlaps the reference model output. Moreover, the error, which is the difference between the reference model and system output, gets smaller with time. The reference model is overlapped more quickly by the adaptive fuzzy control system with the new definition of the variable adaptation gain ( N p ) . As can be seen from the simulation and experimental results and the performance measurements, the proposed system has become more robust against time-varying loads. As a result, the proposed FMRLC of the TVRS system has been successfully realized in the simulation and practical application system. Acknowledgment This study was supported by the Sel¸ cuk University Scientific Research Projects (BAP) Support Fund under contract number 09101057. References K.M. Passino, J. Layne, “Fuzzy model reference learning control for cargo ship steering”, IEEE Control Systems, Vol. 13, pp. 23–34, 1993. ˙I. Y¨ uksel, Otomatik Kontrol, Sistem Dinami˘ gi ve Denetim Sistemleri, Bursa, Vipa¸s, 2001. K. Narendra, A. Annaswamy, Stable Adaptive Systems, New Jersey, Prentice Hall, 1999.
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