PD-Type Iterative Learning Control for the Trajectory Tracking of a Pneumatic X-Y Table with Disturbances

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1 520 PD-Type Iterative Learning Control for the Trajectory Tracking of a Pneumatic X-Y Table with Disturbances Chih-Keng CHEN and James HWANG In this paper, a proportional-valve controlled pneumatic X-Y table system is built to perform position tracking control experiments. The pneumatic system is subjected to external loads and parameter changes during the control. ILC (Iterative Learning Control) controllers are implemented in the experiments to show their ability to reject disturbances. The P and PDtyped updating laws with delay parameters are used respectively for the repetitive trajectory tracking control of X-Y table. Pre-saved control signals for different types of disturbances are also used to compare control performances. Experimental results show that under the disturbances, the PD-typed ILC controller is superior to the P-typed one and can effectively control the system to track the given circular trajectory. Key Words: Proportional Valve, Pneumatic System, Iterative Learning Control (ILC), Two-Dimensional System, Disturbance 1. Introduction The valve-controlled pneumatic system is essentially a nonlinear system. Due to air compressibility, the pneumatic system is highly nonlinear and load affected. System performances are sensitive to the changes of external load and parameter variations. To control the pneumatic system, especially for trajectory tracking applications, advanced controllers involving complicated computational procedures are often required. Linearized models developed from complicated procedures are generated in order to apply the classical or modern controller design (1), (2). The modeling and control of a light-weight pneumatic robot system has been studied for the position tracking and end-effector force control (3). This approach depends fully on the nonlinear dynamic model for the controller design. Fuzzy and sliding surface control schemes are used to control the position of a propositional-valvecontrolled pneumatic rodless cylinder (4). The ILC (Iterative Learning Control) was first proposed by Arimoto et al. (5) in The PID-type learning algorithm was proposed to ensure the convergence of tracking error between the system output and a reference Received 2nd August, 2004 (No ) Department of Mechanical and Automatic Engineering, Da-Yeh University, 112 Shan-Jeau Rd., Chang-Hwa 515, Taiwan, R.O.C. ckchen@mail.dyu.edu.tw Department of Mechanical and Automatic Engineering, Da-Yeh University input. Theoretically, under the assumption of the same initial conditions, the tracking error should converge to zero as the number of iterations increases. Different schemes for the learning control are also provided by Amann et al. (6),Bien (7),Kurek (8), Moore et al. (9), (10) and Rosser (11) for comparison and improvement. The ILC system operates in two dimensions, one is for time and the other is for trial number. This complicated 2-D analysis has also been studied by Arimoto et al. (5),PadieuandSu (12), and Geng et al. (13) Some practical implementations of ILC controllers are applied to the position control of mechanical systems (14), especially for hydraulic and pneumatic systems (15), (16). In this research, a proportional-valve controlled pneumatic X-Y table system is built for trajectory tracking control experiments. The pneumatic cylinders system controlled by proportional valves is an essential nonlinear system. Thus, the constant-gain linear controller, like PID, can not track the position reference input accurately. Here, a PD-type ILC algorithm with time delay parameter is used to control the X-Y table to follow a desired circular trajectory under different types of disturbances. The PDtype ILC control scheme is discussed in section 2. The experimental apparatus used for this research is designed and shown in section 3. In section 4, the controllers are implemented in the platform to verify the system s tracking ability under different external disturbances. Experimental results using the P-type, PD-type and pre-saved ILC controllers are compared to study the convergence speed

2 521 of the tracking error. The inertia disturbance is also added to test the controller s performance. The results show the effectiveness of the proposed PD-type ILC controller with delay parameter under external disturbances. Nomenclature f(.), h(.) : model of a nonlinear system. e = [e x,e y ] T : tracking errors. K P : proportional learning gain matrix for the ILC scheme. K D : derivative learning gain matrix for the ILC scheme. d : time delay number. u = [u x,u y ] T : ILC controller output signals. u i = [u xi,u yi ] T : control signals of the inner PID control loop. y = [x,y] T : control system outputs. y d = [x d,y d ] T : reference inputs. 2. PD-Type ILC Controller The concept of ILC is to use error information from the current control trial to update the control signal of future trials such that the tracking error between output and the reference input is reduced. Many results have been developed for linear and nonlinear systems to guarantee the convergence of the tracking error (8). Consider the following nonlinear time-invariant discrete-time system: x(t +1) = f(x(t), u(t)) (1) y(t) = h(x(t)) where x R n is a state vector, u R m is an input vector, y R p is the output vector and f(x) andh(x) are vector functions with appropriate dimensions. The problem can be formulated as follows. Given the system (1) with initial condition x(0) = x 0, reference output y d (t) and tolerance ε>0, find the control sequence u(t) by the learning algorithm such that the system output follows the reference trajectory under the condition yd (t) y(t) <ε. The reference output y d (t) does not change for different trials, however, u(t) will be updated by the learning rule between two iterations. In 2-D (two-dimensional) representation, the ILC system dynamics can be discussed in two directions: time t and the number of iterations k. The first direction is for the system dynamics in time, i.e., time response. The second one is to reflect the dynamics of iterative learning. The 2-D diagram of the ILC control system is shown in Fig. 1. In the 2-D representation system, the tracking error at the k-th iteration can be defined as: e(t,k) = y d (t) y(t,k) (2) By the tracking error, the error change between two adjacent trials is thus defined as: e(t,k) = e(t,k) e(t,k 1). The PD-type learning rule can be written as Fig. 1 Two dimensional diagram of the ILC system u(t,k+1) = u(t,k)+ K P e(t+1,k)+ K D e(t+1,k) (3) = u(t,k)+ u(t,k +1) where u(t,k) andu(t,k + 1) denote the control input u at the k-th and (k + 1)-th learning, and u(t,k + 1) is the updating control signal at the (k + 1)-th iteration and K P, K D R m p are respectively the proportional and derivative parts of learning gain matrices. Equation (3) is called PDtype learning law since the revision in the control signal is a summation of proportional and difference parts of the tracking error. Furthermore, the difference gain K D = 0, Eq. (3) is thus reduced to a P-type ILC scheme only. Since the desired trajectory and system output are different at different trials, the error can be used to update the control inputs at the next trials until the tolerance requirement is satisfied. The learning gain may affect the system stability and the error convergence. The roles of the learning gains K P and K D in Eq. (3) are analogous to the proportional and derivative gains in the PD controller. A larger gain can increase the speed of error convergence but causes severe error oscillations within some range. In contrast, smaller learning gains will take more iteration to fulfill the error requirement. Adaptive learning gain has been proposed to make the convergence more stable and faster (13), (14). In this research, the learning gains are chosen as constants which are determined by a series of searching experiments. When the system delay is considered in the learning rule, one can use the future error e(t d,k) to update the signal at the (k + 1)-th trial, where d denotes the time delay parameter. The learning control rule (3) can be modified as follows: u(t,k+1) = K P e(t+1+d,k)+k D e(t+1+d,k) (4) In other words, Eq. (4) uses the error in the last trial with d samples ahead from the current trial to update the control signal at it. The real delayed time is dt = t s d, where t s is the system sampling time. In the experimental

3 522 study, it can be shown that the proper choice of the delay parameter d can improve the tracking performance. However, in this way, one can only find the control sequence u(t,k + 1), t = 0,1,..., N 1 d from the tracking error e(t,k)fort = d +1, d +2,..., N. To generate the last d control signals u(t,k+1), t = N d,..., N 1, the signals e(t,k), t = 1,2,..., d are used to extend the error sequence, that is, one can assume e(t+n,k) = e(t,k)fort = 1,2,..., d to compute the last d control signals. This can be done since the control system is operated to the periodic input signal. 3. Experimental Apparatus The experimental apparatus, as shown in Fig. 2, is a proportional-valve-controlled pneumatic X-Y table. It consists of two pneumatic cylinders with the position feedback sensors at each individual axis. When high pressure air is pumped in, the proportional valves controlled by a computer controller can change the air flow rates, and thus change the velocities of the cylinders. The input voltage to the proportional valve ranges from 0 to 10 volt, and 5 V is for the valve s neutral position. The position sensors (accurate potentiometers) can return the table s two-axis position signals to the controller and complete the feedback control loops. The range of cylinder movement of each axis is 90 mm. Diameters of the cylinders used in the platform are chosen not too large (16 mm), and thus they can sensitively reflect the position changes due to external disturbances. The controller hardware is implemented by a notebook PC with a PCMCIA AD/DA interface card. It can measure signals from the two position sensors and send the control signals to both proportional valves. There are two ways of generating the disturbances to the system in the platform. In Fig. 2, part number 10 is a spring with the spring constant of 2 kgf/cm. It can be easily hooked (or unhooked) on the X-Y table to provide an external load in both X and Y directions. The load varies due to the change of table position. Furthermore, since the trajectory of the table is repetitive, the spring can generate a periodic load to the table. The second disturbance is generated by a mass block (part number 9) in Fig. 2. During the motion of the X-Y table, the mass can be dropped and attached to the table to change the inertial properties of the system. For different types of control schemes, the tracking error convergences can be compared when the disturbances are introduced to the system. The block diagram of this control system is shown in Fig. 3. In the inner loop, there is a PID controller for each axis. The reference signals for each PID control loop are denoted as u x and u y. The PID controller outputs denoted by u xi and u yi are sent to proportional valves to control the velocities of the two cylinders. The outer loops are the ILC control system with two tracking errors e x and e y as inputs to the ILC controller. The PD-type ILC scheme as in Eq. (4) is used to update the new learnt signals u x and Fig. 2 Fig. 3 Experimental configuration of the pneumatic X-Y table system Block diagram for ILC control system with disturbance u y at each iteration. The convergence of the ILC controller is evaluated by computing the MSE (Mean Square Error) from the tracking errors for each learning number. The formula of MSE is: ( N 1/2 MSE = (e 2 x(t)+ey(t))) 2 (5) t=1 where N is the number of time steps at one learning trial. The smaller and faster MSE convergence indicates the better ILC control performance. In the following experiments, the MSEs are plotted to show the effects of different control schemes under different external loads. 4. ILC Control Experiments In this section, the ILC controller is implemented based on Eq. (4) for the trajectory tracking control of the X-Y table in Fig. 2. The ILC controller and data acquisition are implemented by coding inside the GUI real-time control tool LabVIEW. In the following experiments, the sampling time is set to 0.01 sec and all the controller parameters are chosen as constants for all cases as shown in Table 1. At the first trial (k = 1), only the PID controller is used for the trajectory tracking control. The reference signals are sent as commands to the inner PID control loops, i.e. u x (t, 1) = x d (t) andu y (t, 1) = y d (t) as shown in Fig. 3.

4 523 Table 1 Parameters for ILC controller Fig. 5 MSE vs. learning number for ILC control with spring load Fig. 4 MSE vs. learning number for ILC control without spring load The circular reference signals are defined as: x d (t) = x 0 +Rcos(2π ft) and y d (t) = y 0 +Rsin(2π ft) (6) where (x 0,y 0 ) is the center of the circle, R is its radius and f is the frequency of the signal. The reference command used in the experiments here is parameterized as (x 0,y 0 ) = (50,50) (mm) and R = 30 (mm) with the frequency 0.5 Hz for most cases unless re-specified. In these experiments, three kinds of control schemes, P-typed, PDtyped and pre-saved ILC controller are used for the trajectory following control. The pre-saved ILC controller uses pre-stored control signals, u x and u y, which have been learned from the converged ILC control procedure. Their tracking performances are compared under different disturbances. Figure 4 shows the experimental results of the three control schemes for the pneumatic system without adding any disturbances. As discussed in section 3, the plot of MSE vs. the learning number can indicate the convergence speed and accuracy for ILC control. One can observe that the pre-saved case can rapidly converge in one or two iterations since its initial control signal is already well trained. The pre-saved control still needs some more training (by PD ILC) to reduce the error further. The PD-typed ILC controller can converge the error faster than the P-typed Fig. 6 Tracks of the X-Y table by ILC control with spring load at the 1st and 17th trials one. This result shows the effects of the learning process by adding the D part (K D e) to the ILC. When the spring load is hooked on the table, the same experiments are carried out again as shown in Fig. 5. When at the initial position, (x(0), y(0)) = (80, 50), the spring elongation is 6 cm. The convergence speeds for P and PD ILC schemes become slower than the previous case under the influence of external load. They need more iteration to converge the error. In Fig. 5, the minimum MSE happens at the 17th trial and the trajectory of the X-Y table is shown in Fig. 6. Comparing the results by PID only (trial 1) and by ILC (trial 17) in the figure, one can see that the ILC controller can significantly improve tracking properties. The tracking error between the reference input and the system output is defined as e(t) = (e 2 x(t) + e 2 y(t)) 1/2 and plotted along a circular angle as shown in Fig. 7 in a polar plot format. Figure 8 shows the time responses of the ILC control at the 17th iteration in Fig. 6. The first plot in Fig. 8 is the tracking errors in the two axes which are within the range

5 524 Fig. 9 MSE vs. learning number by P-type ILC control with the spring load change disturbance Fig. 7 Polar plot of the tracking error e(t) = (e 2 x(t)+e 2 y(t)) 1/2 at the 17th trial Fig. 10 MSE vs. learning number by PD-type and pre-saved ILC control with the spring load change disturbance Fig. 8 Time responses of the ILC control system at the 17th trial of ±0.7 mm. The second plot in Fig. 8 is the input voltage signals to the proportional valves inside the inner PID control loop, i.e., the signals u xi and u yi. As shown in the figure, the two signals are shifted above from their neutral position of 5 Volt to resist the external load produced by the hooked spring. The third plot in Fig. 8 shows the learnt signals by ILC control scheme at the 17th iteration. One can see that the shapes of the learnt commands, u x and u y, to the inner loops are far from the original sinusoidal signals as defined in Eq. (6). Since the pneumatic system is very sensitive to load change, the effect of the disturbance can be obviously introduced and easily studied. In Fig. 9, the P-typed ILC scheme is applied to the same circular trajectory tracking control of the pneumatic system (with spring hooked) un- til the MSE falls to a minimum level. At the 40th trial, the spring is suddenly released from the platform to generate a disturbance change, and thus increases the value of MSE tremendously. After about 15 iterations, the MSE converges to a small range, and then the spring is re-hooked (at the 60th trial) to the table to produce another load change for the system. This causes the error to rise again and to decrease after about 20 iterations. The same experiments are carried out for the PD-typed and pre-saved ILC schemes as shown in Fig. 10. One can observe that the iteration number to converge the MSE after the disturbance is applied, is apparently reduced by the PD-typed control law when compared with the P-typed result. The disturbances from changing the inertial parameters are also applied to the system. In Fig. 11, the system without the spring load is controlled at a minimum MSE level by three ILC schemes. At the 40th iteration, a 1.92 kg mass block is dropped and attached to the platform to increase its inertial parameter, and again at the 54th iteration, another block with the same mass is added to the system. Figure 11 shows the MSE changes along

6 525 Fig. 11 Results of adding mass blocks for 0.5 Hz reference input without spring load Fig. 13 Results of adding mass blocks for 1 Hz reference input without spring load ILC control, the learning procedure can be switched off to prevent performance degradation, e.g. numerical divergence. Once the MSE jumps over the threshold again due to parameter change or disturbance, the learning controller can be reactivated to reduce the error. In addition, when the disturbances that could be encountered during the cyclical operations are known in advance, the corresponding trained signals for each potential disturbance can be recorded and stored for on-line use to make the system react rapidly to disturbances. 5. Conclusion Fig. 12 Results of adding mass blocks for 0.5Ḣz reference input with spring load the number of iterations after the inertial disturbances are inserted. It can be observed that the convergence speed of PD-typed ILC is still faster than the P-typed one. With the spring hooked, the same experiments are performed again and the results are shown in Fig. 12. The mass blocks are added at the 39th and 53th iterations, sequentially. Concurrently with the spring load, the MSE is increased a little higher and attenuated after some iteration trials. The increase in error due to the inertial parameter change becomes more apparent if the moving speed of the system is increased. By increasing the frequency of the reference command in Eq. (6) from 0.5 Hz to 1 Hz, the results of the same mass-adding experiments are shown in Fig. 13. The figure shows an obvious increase in the MSE due to the raise of the inertial force. Similarly, in this case, the PDtyped ILC still shows its superiority over the P-typed one. In practical applications, a MSE threshold can be preset. When the MSE is lower than this value during In this study, a proportional-valve-controlled pneumatic X-Y table is built for control experiments. There are two different kinds of disturbances studied here, spring load and the attached mass block. The ILC control scheme is implemented for the trajectory tracking control of the X- Y table under the influence of these disturbances. The experimental results show that only the PID controller cannot follow the reference trajectory efficiently. With the ILC controller, one uses the tracking error information of the previous control trial to update the control signals for the current trial such that the tracking errors are reduced, and the system can follow the repetitive circular reference track after the learning procedure. The P-typed, PD-typed and pre-saved ILC schemes are implemented for the control of the pneumatic system. The experimental results show that the number of iterations needed to converge the error after the disturbance is applied is apparently reduced by the PD-typed control law compared to the P-typed. The PD-typed ILC shows better error-convergence performance when load disturbances are encountered. Also, the pre-saved ILC control demonstrates the use of the pre-trained control signals when subjected to different kinds of disturbances. In this way, the number of iteration trial for MSE convergence

7 526 can be effectively reduced. The iterative learning control has the advantage of minimal complexity, in both formulation and computation. At the same time, it can maintain the repeated trajectory tracking even when disturbances are introduced to the system. This can make it attractive to industry due to its ability to learn on-line in a relatively simple way compared to other model based methods. Acknowledgement The work was supported by the National Science Council in Taiwan, Republic of China, under the project number NSC E References ( 1 ) McCloy, D. and Martin, H.R., Control of Fluid Power, 2nd ed., (1980), Ellis Horwood. ( 2 ) Watton, J., Fluid Power System, Modeling, Simulation, Analog & Micro-Computer Computer Control, (1987), Prentice-Hall. ( 3 ) Bobrow, J.E. and McDonell, B.W., Modeling, Identification, and Control of a Pneumatically Actuated, Force Controllable Robot, IEEE Trans. on Robotics and Automation, Vol.14, No.5 (1998), pp ( 4 ) Renn, J.C., Position Control of a Pneumatic Servo Cylinder Using Fuzzy-Sliding Surface Controller, Int. J. of Fluid Power, Vol.3, No.3 (2002), pp ( 5 ) Arimoto, S.S., Kawamura, S. and Miyazaki, F., Bettering Operation of Robots by Learning, J. of Robotic System, Vol.1, No.2 (1984), pp ( 6 ) Amann, N., Owen, D.H. and Roger, E., Iterative Learning Control Using Optimal Feedback and Feedforward Actions, Int. J. of Control, Vol.65, No.2 (1996), pp ( 7 ) Bien, Z., Xu, J.X., Ed., Iterative Learning Control, Analysis, Design, Integration and Applications, (1998), Kluwer Academic Publishers. ( 8 ) Kurek, J.E. and Zaremba, M.B., Iterative Learning Control Synthesis Based on 2-D System Theory, IEEE Trans. on Automatic Control, Vol.38, No.1 (1993), pp ( 9 ) Moore, K.L., Iterative Learning Control for Deterministic System, Advances in Industrial Control Series, (1992), Springer, London. (10) Moore, K.L. and Xu, J.X., Special Issue on Iterative Learning Control, Int. J. Control, Vol.73, No.10 (2000), pp (11) Roesser, R., A Discrete State-Space Model for Linear Image Processing, IEEE. Trans. on Automatic Control, Vol.20, No.1 (1975), pp (12) Padieu, F. and Su, R., An H-Infinity Approach to Learning Control Systems, Int. J. of Adaptive Control and Signal Processing, Vol.4 (1990), pp (13) Geng, Z., Lee, D.J., Carroll, R.L. and Haynes, L.H., Learning Control System Design Based on 2-D Theory An Application to Parallel Link Manipulator, IEEE. J. of Robotics and Automation, Vol.6, No.2 (1991), pp (14) Barton, A.D., Lewin, P.L. and Brown, D.J., Practical Implementation of a Real-Time Iterative Learning Position Controller, Int. J. Control, Vol.73, No.10 (2000), pp (15) Chen, C.K. and Hwang, J., Iterative Learning Control for Position Tracking of a Pneumatic Actuated X- Y Table, Control Engineering Practice, Vol.13 (2005), pp (16) Chen, C.K. and Zeng, W.C., The Iterative Learning Control for the Position Tracking of the Hydraulic Cylinder, JSME Int. J., Ser. C, Vol.46, No.2 (2003), pp