Application of Active Disturbance Rejection Control to a Reel-to-Reel ...
Application of Active Disturbance Rejection Control to a Reel-to-Reel System Seen in Tire Industry Rafał Mado´nski? , Mateusz Przybyła?? , Marta Kordasz† , Przemysław Herman‡ , Chair of Control and Systems Engineering, Pozna´n University of Technology, ul. Piotrowo 3A, 61-138, Pozna´n, Poland, e-mail: ? [email protected], ?? [email protected], † [email protected], ‡ [email protected]
Abstract—In this paper, an Active Disturbance Rejection Control (ADRC) scheme is used to solve a sag control problem that has been noticed in the tire production process. Constant length of the rubber sag is crucial in order to achieve high quality of tire reinforcement, which has its direct impact on drivers’ safety. The ADRC method was compared in this research with a popular model-independent technique, i.e. Proportional-IntegralDerivative (PID) controller, widely used in engineering practice. Both of these techniques were implemented and experimentally verified on a model of a tire reinforcement production stage that can be described with a reel-to-reel system (R2R). The effect of various radius in the rubber winding phase makes the R2R a nonstationary dynamic plant. The ADRC approach resulted in better performance both energetically and for tracking a reference square trajectory, even with the presence of artificially added external disturbance. Additional perturbation imitated unpredictable and surprisingly often phenomena, which are results of incidents that happen in other stages of the manufacturing process.
strips is winded the bigger the radius is. Other problem is the changeable speed of the production line caused usually by the unpredictable incidents on prior and subsequent stages of the manufacturing process. The reasons mentioned above make the tire reinforcement process a nonstationary dynamic system with acting variable disturbances. Most common control methods for solving the sag control problem are based on a simple PID control schemes. The PID controller does not need a model of the system but as a result of constant tuning parameters can sometimes suffer from poor robustness due to changes in the plant during operation.
I. I NTRODUCTION A typical rubber tire is made of numerous components and its manufacturing process is complex and problematic. According to [3], tire structure can be divided into two main parts: a rubber carcass and a reinforcement (usually made of steel, see Figure 1). The latter is essential because it has direct impact on the tire performance. Reinforcement gives tire its strength, dent resistance, and flexibility. Features like adhesion, fatigue, and tensile make reinforcing process crucial in terms of end-users safety (for details see [2], [11]). These characteristics have to be chosen properly and stay consistent even in variable operating conditions (e.g. temperature, humidity, roadbed friction). Manufacturing defects can be substantially reduced by proper manufacturing practices. In reinforcement stage for example, rubber sheets are joined with layers of thin steel. Then, the winding reel (driven by a DC motor) collects the reinforced tire. Rubber sag (sometimes called festoon) between conveyor belt and the winding reel is the main concern for the operators. The lowest point of the sag1 needs to follow desired trajectory at each time sample of the production process. Sag control is also a challenging engineering problem because the radius of the reinforced rubber on a winding reel varies. It is a function of time, which means that the more 1 It
can be considered as a tip of a “parabolic” shape of the sag.
Figure 1.
A typical car tire construction (based on [1], [3]).
Therefore, this research is motivated by the sag control problem and tries to choose a control algorithm which will not need exact dynamics equations of the system and at the same time stay robust to internal (uncertainties of the plant dynamics) and external disturbances. The Active Disturbance Rejection Control (ADRC) is a control method which was originally proposed by Han in [4]2 . It is based on an idea, that in order to control a process user does not need to have an explicit mathematical model of the plant. By the use of a state observer, both internal and external disturbances are estimated and rejected in each sampling period. In this paper a novel methodology for precise sag control in a reeling system is presented. The aim of the work is to implement ADRC method on a reel-to-reel system and compare its performance with the PID strategy. Based on 2 Earlier papers regarding ADRC were published in Chinese. However, some references to them can be found in [4] or [6].
authors current knowledge, it is a first attempt to incorporate Han’s method on this type of system. II. R EEL - TO - REEL SYSTEM DESCRIPTION Due to the tire company privacy policy, authors were unable to perform planned experiments on real factory system and publish their results. However, for the purpose of this work a laboratory setup was built to imitate the behavior of the machinery that is responsible for gluing reinforcement to the tire. A. Mathematical model The real reinforcement phase seen in tire production can be described with a simplified reel-to-reel physical model (R2R, see Figure 2), where the first reel winds reinforced rubber (R1) and the second one can be used to produce additional disturbance to the system (R2). The R2R system was built in Chair of Control and Systems Engineering3 at the Pozna´n University of Technology [10].
transport. Sag length signal is used for the outer control loop designed in PC-based VisSim environment. For the sake of simplicity, instead of rubber belt, rigid line is used. A mathematical model of the R2R is presented below [10]: y(t) ˙
= y˙ R1 (t) + y˙ R2 (t) + d,
(1)
which can be further formulated as: y(t) ˙
=
r(t)η ω(t) + y˙ R2 (t) + d, 2
(2)
V where: y(t)[ ˙ s ] is the linear velocity of the mass (M) attached to the end of the sag, y˙ R1 (t) stands for the momentary, circumference velocity of the line winded by motor R1, y˙ R2 (t) is the momentary, circumference velocity of the line winded by motor R2 (external disturbance), r(t) denotes radius of the line winded up on the reel R1, 0 < η < 1 is the transmission ratio, ω(t) is the angular velocity of the motor R1, and d represents other and unknown disturbances acting on the system.
Figure 3.
The R2R system control block diagram.
B. User-system communication
Figure 2.
The R2R system that imitates the tire reinforcement process.
The main control task in the R2R setup is to wind reinforced rubber while stabilizing lenght of the sag. Others problems, like trajectory tracking, can be considered here as well. In the R2R system vertical position of the sag is controlled with a reel driven by a DC motor. First motor (R1) is working in an inner control loop (speed control) with a PID controller and a feedback signal from a rotary encoder. The reference angular speed of the DC motor is the input signal for the inner loop (see Figure 3). Current vertical position of the sag (L) is measured with a linear wire encoder placed on the upper base of R2R and attached to the load (M, it is also considered as the lowest point of the sag). The additional mass on the sag is a design parameter that can be interpreted as a weight of winded rubber 3 www.control.put.poznan.pl/en
Communication between R2R and VisSim is carried out with a PCI I/O card (Figure 4). Module works with VisSim and with RealTimePRO add-on it allows real-time operation. Second motor (R2) is the source of external disturbance added to the vertical position of the sag. Controlling second reel is similar to the first one. Different types of programmable perturbations can be chosen in VisSim and added to input signal of the inner loop (speed control). The R2R system main hardware parts are presented in Table 1.
Figure 4.
A simplified data flow scheme in the R2R system.
Table I T HE MAIN HARDWARE COMPONENTS OF THE R2R SYSTEM . Part DC motor (R1) DC motor (R2) DC motor controller Wire encoder Limit switches I/O card PC
Model Maxon 350370 Maxon 350336 Dunkelmotoren BGE 3515 Micro-Epsilon WPS-MK Crouzet 83133 DAS1602/12 Athlon 2600+, 512 MB RAM, WinXP
C. Real plant vs R2R model Some main differences between the real plant at the tire reinforcement stage and the considered R2R system can be pointed out. First difference is concerned with the type of used drive. In the real system, the winding reel is driven by a industrial servomechanism, not a DC motor like in R2R. The second distinction is in the type of measurement tool for the load position. The R2R laboratory setup uses wire encoder, however the real plant has a light barrier with light beams defining several regions of load height. The third difference lies in the control unit. A standard Programmable Logic Controller (PLC) is used in real process to control the drive. In the constructed system, the controller is implemented on a PC and both control and output signals are transmitted with a I/O card. With the above (and others) differences, the proposed solution may not be transferable to the industrial level application. At this point however, authors are unable to verify that. A practical demonstration of considered proposition seems to be crucial, since it could provide user with better insight. III. ACTIVE D ISTURBANCE R EJECTION C ONTROL The main idea of ADRC method is to combine the essence of widely used linear controllers with disturbance rejection feature. An Extended State Observer (ESO) is the heart of the ADRC method and by its characteristic structure, the observer defines the nonlinear system dynamics variations, system model uncertainties, and external disturbances as an augmented, virtual state variable to be actively estimated and canceled out in the control signal (details can be found in [6]). This procedure takes place in each sample period, thus results in reducing the process to a standard linear and time-invariant form. One can perceive this procedure as an indirect dynamic feedback linearization of the system. The ESO strategy results in the disturbance rejection and inherent robustness of the controller. Some practical examples of ADRC implementation can be found in [7], [8], [12]. In order to incorporate the ADRC approach, the R2R system from (2) can be first rewritten as: ( x˙ 1 (t) = r(t)η 2 ω(t) + y˙ R2 (t) + d = b(t)ω(t) + f (t, ·), y(t) = x1 (t). (3) where: y(t)[V ] = x1 (t)[V ] is a state variable representing actual position (height) of the load (M) attached to the end of the sag, b(t) = r(t)η is the unknown variable system 2 parameter, and f (·) = y˙ R2 (t) + d is the unknown general
disturbance, which is a sum of assumed internal and external disturbances. In the ADRC, an assumption is made that no analytical expression of f (t, ·) is required as long as the user can provide a close estimate of this element. In order to provide the estimate of general disturbance a state observer is designed. A. Extended State Observer State x2 (t) is an augmentation introduced to the state space model from (3): x˙ 1 (t) = b(t)ω(t) + f (t, ·), (4) x˙ 2 (t) = f˙(t, ·), y(t) = x1 (t). Two states (i.e. x1 (t) and x2 (t)) can be estimated with a linear version of ESO (with an assumption that f˙(t, ·) = 0): ( z˙1 (t) = z2 (t) − β1 eˆ(t) + b0 (t)ω(t), (5) z˙2 (t) = −β2 eˆ(t), where: b0 (t) denotes a variable, which is an approximation of b(t) from equation (3), z1 (t), z2 (t) are estimates of states x1 (t), x2 (t) respectively, eˆ(t) = z1 (t) − y(t) is an estimation error, and β1 , β2 are the observer positive gains. A parametrization method based on pole-placement approach was proposed in [5] to simplify the observer gains tuning procedure. For the considered system the observer gains can be expressed as functions of just one parameter i.e. ω0 : β1 = ω02 , β2 = 2ω0 ,
(6)
where: ω0 is the observer’s bandwidth. It is usually chosen as a subjective compromise between convergence speed of states estimation and the influence of noise and sampling time [5], [9]. B. The controller The type of controller used in ADRC should be related to a given task and technical specifications of the plant. Nevertheless, for the sake of simplicity the control signal in ADRC is presented below for a classic linear PD controller, thus the input signal for the system is as follows: ω(t) =
kp e(t) + kd e(t) ˙ − z2 (t) u0 (t) − z2 (t) = , b0 (t) b0 (t)
(7)
where: z2 (t) is the estimated general disturbance, u0 (t) is the output signal from the controller, e(t), e(t) ˙ are the tracking error and its derivative, kp , kd are the controller gains (proportional and derivative gains respectively). Assuming that the observer is well tuned (i.e. z2 (t) ≈ f (·) and b0 (t) ≈ b(t)), equation (3) with a new control signal (7) is transformed to: kp e(t) + kd e(t) ˙ − z2 (t) , (8) y(t) ˙ ≈ f (·) + b(t) b0 (t) which gives: y(t) ˙ ≈ kp e(t) + kd e(t). ˙
(9)
State space model in (3) is reduced to a simpler form: ( x˙ 1 (t) ≈ kp e(t) + kd e(t), ˙ y(t) ≈ x1 (t).
(10)
The ESO provides dynamical compensation of the general disturbance and therefore reduces the problem of controlling nonlinear, time-varying system to a simpler control of a linear one. Thus, classic linear control techniques can be incorporated, like the considered PD controller. Tuning procedure can be also based on [5]. The controller gains are usually chosen in the ADRC scheme as the result of pole-placement procedure: kp = ωn2 , kd = 2ξωn ,
(11)
where: ωn and ξ are the desired closed loop natural frequency and damping ratio. Figures 5 and 6 show PID and ADRC block diagrams for controlling the R2R system.
Figure 5.
A block diagram of PID controller for the R2R system.
Figure 6.
A block diagram of ADRC method for the R2R system.
4) Derivative gain kd is added if unacceptable overshoot after step 3) appears and is gradually increased until output signal reaches desired behavior. In many cases, thanks to the particular features of the ADRC concept, no integrating action is needed (see e.g. [6]). 5) Parameter b0 (t) is scaling the control signal so it effects the tracking error (ˆ e(t)) convergence rapidity. Choosing b0 (t) should be related to a given control task, defined by the system operator. The most important part of ADRC tuning is obtaining a minimum and fast converging estimation error, thus the ESO parameters are tuned first. Secondly, the PD regulator is tuned to achieve minimum settling time with no signal overshoot. These parameters are chosen empirically and left constant for the experiment. Parameters for the ADRC algorithm are chosen as follows: ω0 = 3rad/s, b0 (t) = 1, kp = 20, kd = 2, and ξ = 1 (selected to avoid oscillations in the output signal). Tuning values for the classic PID are: kp = 5 (proportional gain), ki = 0.025 (integral gain), and kd = 1 (derivative gain). No parameter retuning is done in PID or ADRC after the occurrence of added disturbance. A square reference signal is chosen with frequency: fq = 0.05Hz and sampling time: Ts = 0.01s. Time frame for the experiment is set for Texp = 120s. Additional disturbance (from motor R2) is switched on at t = 60s. Its signal profile is presented on Figure 7.
IV. E XPERIMENT A. Study preparation For this particular work, methodology from (11) did not provide good results. That is why a different approach is used for tuning the PD controller. Since information about the parameters of R2R mathematical model is assumed to be unknown, both the observer and the controller are tuned empirically using simple tuning proceeding4 : 1) Parameters kp , kd , and ω0 (t) are positive and set close to zero. Parameter b0 (t) is set to one. 2) Parameter ω0 (t) in ESO is gradually increased until state variable z1 (t) estimates system output y(t) closely. 3) Proportional gain kp is gradually increased until output signal reaches desired behavior. 4 The tuning process is aimed to provide fastest nonoscillary error convergence for a square reference trajectory. Similar approach was successfully used in [8].
Figure 7.
Artificial, external disturbance generated by motor S2.
B. Experimental results Results of the performed experiments are shown in Figures 8-10. It can be noticed that before the additional disturbance occurs (t < 60s), both of the controllers perform similarly for trajectory tracking case. No overshoot or intolerable error in the steady state is seen in this part of the test (Figure 8). Temporary error seen for the first few seconds in the output signal (Figures 8 and 9) is a consequence of the convergence time of the observer signals. The ADRC method, thanks to the disturbance rejection feature, managed to oppose the acting perturbation much better than the PID controller (t < 60s). Also faster error convergence and no significant overshoot in the output signal
is seen for the ADRC (Figure 8). It is also confirmed in Figure 9. Control signal generated by the ADRC is slightly more shattered but less in amplitude than PID (Figure 10). The jagged character of the control signal has no direct impact on the visual smoothness of the R2R movement.
Figure 8.
Vertical position of the load M (sag length).
V. C ONCLUSIONS AND FUTURE WORK The Active Disturbance Rejection Control method was used in this research for a rubber sag control problem seen on tire production line. Applied technique could detect automatically the effect of the disturbances acting on the system, and then compensate for them in real-time. Thus, the ADRC did not rely on the mathematical expression of the considered system and it showed good robustness and adaptability features. It can be seen that the ADRC strategy reduces the control of a nonstationary and uncertain process to a much simpler problem. It was also noticed that even with the use of the Extended State Observer, the implementation and tuning difficulty did not increase significantly. The ADRC algorithm was verified to be an interesting alternative to widely-used PID controller and obtained results have potential importance to the tire industry. Future work will focus on comparing ADRC method with other minimum model approaches including (but not limited to) adaptive control and model-predictive control since these techniques can also generate good results without analytical model of the process. Such a study could provide evidence of potential advantages of ADRC algorithm over alternative control strategies. Authors also started to implemented several techniques on a Programmable Automation Controller (PAC) that can potentially replace the PLC at the tire reinforcement stage and allow to incorporate more advanced control algorithms. R EFERENCES
Figure 9.
Figure 10.
Vertical position error of the load M (sag length).
Control signals generated by each controller.
[1] A. Abe, T. Kamegawa, Y. Nakajima, Optimization of construction of tire reinforcement by genetic algorithm, Optimization and Engineering, vol. 5, pp. 77–92, 2004 [2] R. S. Bhakuni, G. W. Rye, S. J. Domchick, Adhesive and processing concepts for tire reinforcing materials, American Society for Testing and Materials International, pp. 122-128, 1979 [3] J. C. Dixon, Tires, suspension, and handling, Second Edition, Society of Automotive Engineers, chapters 2.1-2.3, 1996 [4] Z. Gao, Y. Huang, J. Han, An Alternative Paradigm for Control System Design, IEEE Conf. on Control and Decision, pp. 45784585, Orlando, 2001 [5] Z. Gao, Scaling and bandwidth-parametrization based controller tuning, IEEE Proc. of the American Control Conference, pp 4989-4996, Denver, 2003 [6] J. Han, From PID to Active Disturbance Rejection Control, IEEE Trans. on Industrial Electronics, vol. 56, no. 3, 2009 [7] Y. Hou, Z. Gao, F. Jiang, B. T. Boulter, Active Disturbance Rejection Control for Web Tension Regulation, IEEE Conf. on Decision and Control, 2001 [8] M. Kordasz, R. Mado´nski, M. Przybyła, P. Sauer, Active Disturbance Rejection Control for a Flexible-Joint Manipulator, Lecture Notes in Control and Information Sciences, vol., 422, pp. 247-256, 2012 [9] W. S. Levine, The Control Handbook, CRC Press., access: www.books.google.com, New York, 1996 [10] M. Michałek, ZB2 - Laboratory setup user’s manual (internal report, in Polish), Chair of Control and Systems Engineering, Pozna´n University of Technology, 2008 [11] The History of the tire industry: radial tire spawned industry revolution, International Tire Exhibition and Conference, http://www.itec-tireshow.com/history_of_tires.html, 2011 [12] Q. Zheng, Z. Gao, On Practical Applications of Active Disturbance Rejection Control, Proceedings of the 29th Chinese Control Conference, pp. 6095-6100, Beijing 2010
Abstract—In this paper, an Active Disturbance Rejection Control (ADRC) scheme is used to solve a sag control problem that has been noticed in the tire production process. Constant length of the rubber sag is crucial in order to achieve high quality of tire reinforcement, which has its direct impact on drivers’ safety. The ADRC method was compared in this research with a popular model-independent technique, i.e. Proportional-IntegralDerivative (PID) controller, widely used in engineering practice. Both of these techniques were implemented and experimentally verified on a model of a tire reinforcement production stage that can be described with a reel-to-reel system (R2R). The effect of various radius in the rubber winding phase makes the R2R a nonstationary dynamic plant. The ADRC approach resulted in better performance both energetically and for tracking a reference square trajectory, even with the presence of artificially added external disturbance. Additional perturbation imitated unpredictable and surprisingly often phenomena, which are results of incidents that happen in other stages of the manufacturing process.
strips is winded the bigger the radius is. Other problem is the changeable speed of the production line caused usually by the unpredictable incidents on prior and subsequent stages of the manufacturing process. The reasons mentioned above make the tire reinforcement process a nonstationary dynamic system with acting variable disturbances. Most common control methods for solving the sag control problem are based on a simple PID control schemes. The PID controller does not need a model of the system but as a result of constant tuning parameters can sometimes suffer from poor robustness due to changes in the plant during operation.
I. I NTRODUCTION A typical rubber tire is made of numerous components and its manufacturing process is complex and problematic. According to [3], tire structure can be divided into two main parts: a rubber carcass and a reinforcement (usually made of steel, see Figure 1). The latter is essential because it has direct impact on the tire performance. Reinforcement gives tire its strength, dent resistance, and flexibility. Features like adhesion, fatigue, and tensile make reinforcing process crucial in terms of end-users safety (for details see [2], [11]). These characteristics have to be chosen properly and stay consistent even in variable operating conditions (e.g. temperature, humidity, roadbed friction). Manufacturing defects can be substantially reduced by proper manufacturing practices. In reinforcement stage for example, rubber sheets are joined with layers of thin steel. Then, the winding reel (driven by a DC motor) collects the reinforced tire. Rubber sag (sometimes called festoon) between conveyor belt and the winding reel is the main concern for the operators. The lowest point of the sag1 needs to follow desired trajectory at each time sample of the production process. Sag control is also a challenging engineering problem because the radius of the reinforced rubber on a winding reel varies. It is a function of time, which means that the more 1 It
can be considered as a tip of a “parabolic” shape of the sag.
Figure 1.
A typical car tire construction (based on [1], [3]).
Therefore, this research is motivated by the sag control problem and tries to choose a control algorithm which will not need exact dynamics equations of the system and at the same time stay robust to internal (uncertainties of the plant dynamics) and external disturbances. The Active Disturbance Rejection Control (ADRC) is a control method which was originally proposed by Han in [4]2 . It is based on an idea, that in order to control a process user does not need to have an explicit mathematical model of the plant. By the use of a state observer, both internal and external disturbances are estimated and rejected in each sampling period. In this paper a novel methodology for precise sag control in a reeling system is presented. The aim of the work is to implement ADRC method on a reel-to-reel system and compare its performance with the PID strategy. Based on 2 Earlier papers regarding ADRC were published in Chinese. However, some references to them can be found in [4] or [6].
authors current knowledge, it is a first attempt to incorporate Han’s method on this type of system. II. R EEL - TO - REEL SYSTEM DESCRIPTION Due to the tire company privacy policy, authors were unable to perform planned experiments on real factory system and publish their results. However, for the purpose of this work a laboratory setup was built to imitate the behavior of the machinery that is responsible for gluing reinforcement to the tire. A. Mathematical model The real reinforcement phase seen in tire production can be described with a simplified reel-to-reel physical model (R2R, see Figure 2), where the first reel winds reinforced rubber (R1) and the second one can be used to produce additional disturbance to the system (R2). The R2R system was built in Chair of Control and Systems Engineering3 at the Pozna´n University of Technology [10].
transport. Sag length signal is used for the outer control loop designed in PC-based VisSim environment. For the sake of simplicity, instead of rubber belt, rigid line is used. A mathematical model of the R2R is presented below [10]: y(t) ˙
= y˙ R1 (t) + y˙ R2 (t) + d,
(1)
which can be further formulated as: y(t) ˙
=
r(t)η ω(t) + y˙ R2 (t) + d, 2
(2)
V where: y(t)[ ˙ s ] is the linear velocity of the mass (M) attached to the end of the sag, y˙ R1 (t) stands for the momentary, circumference velocity of the line winded by motor R1, y˙ R2 (t) is the momentary, circumference velocity of the line winded by motor R2 (external disturbance), r(t) denotes radius of the line winded up on the reel R1, 0 < η < 1 is the transmission ratio, ω(t) is the angular velocity of the motor R1, and d represents other and unknown disturbances acting on the system.
Figure 3.
The R2R system control block diagram.
B. User-system communication
Figure 2.
The R2R system that imitates the tire reinforcement process.
The main control task in the R2R setup is to wind reinforced rubber while stabilizing lenght of the sag. Others problems, like trajectory tracking, can be considered here as well. In the R2R system vertical position of the sag is controlled with a reel driven by a DC motor. First motor (R1) is working in an inner control loop (speed control) with a PID controller and a feedback signal from a rotary encoder. The reference angular speed of the DC motor is the input signal for the inner loop (see Figure 3). Current vertical position of the sag (L) is measured with a linear wire encoder placed on the upper base of R2R and attached to the load (M, it is also considered as the lowest point of the sag). The additional mass on the sag is a design parameter that can be interpreted as a weight of winded rubber 3 www.control.put.poznan.pl/en
Communication between R2R and VisSim is carried out with a PCI I/O card (Figure 4). Module works with VisSim and with RealTimePRO add-on it allows real-time operation. Second motor (R2) is the source of external disturbance added to the vertical position of the sag. Controlling second reel is similar to the first one. Different types of programmable perturbations can be chosen in VisSim and added to input signal of the inner loop (speed control). The R2R system main hardware parts are presented in Table 1.
Figure 4.
A simplified data flow scheme in the R2R system.
Table I T HE MAIN HARDWARE COMPONENTS OF THE R2R SYSTEM . Part DC motor (R1) DC motor (R2) DC motor controller Wire encoder Limit switches I/O card PC
Model Maxon 350370 Maxon 350336 Dunkelmotoren BGE 3515 Micro-Epsilon WPS-MK Crouzet 83133 DAS1602/12 Athlon 2600+, 512 MB RAM, WinXP
C. Real plant vs R2R model Some main differences between the real plant at the tire reinforcement stage and the considered R2R system can be pointed out. First difference is concerned with the type of used drive. In the real system, the winding reel is driven by a industrial servomechanism, not a DC motor like in R2R. The second distinction is in the type of measurement tool for the load position. The R2R laboratory setup uses wire encoder, however the real plant has a light barrier with light beams defining several regions of load height. The third difference lies in the control unit. A standard Programmable Logic Controller (PLC) is used in real process to control the drive. In the constructed system, the controller is implemented on a PC and both control and output signals are transmitted with a I/O card. With the above (and others) differences, the proposed solution may not be transferable to the industrial level application. At this point however, authors are unable to verify that. A practical demonstration of considered proposition seems to be crucial, since it could provide user with better insight. III. ACTIVE D ISTURBANCE R EJECTION C ONTROL The main idea of ADRC method is to combine the essence of widely used linear controllers with disturbance rejection feature. An Extended State Observer (ESO) is the heart of the ADRC method and by its characteristic structure, the observer defines the nonlinear system dynamics variations, system model uncertainties, and external disturbances as an augmented, virtual state variable to be actively estimated and canceled out in the control signal (details can be found in [6]). This procedure takes place in each sample period, thus results in reducing the process to a standard linear and time-invariant form. One can perceive this procedure as an indirect dynamic feedback linearization of the system. The ESO strategy results in the disturbance rejection and inherent robustness of the controller. Some practical examples of ADRC implementation can be found in [7], [8], [12]. In order to incorporate the ADRC approach, the R2R system from (2) can be first rewritten as: ( x˙ 1 (t) = r(t)η 2 ω(t) + y˙ R2 (t) + d = b(t)ω(t) + f (t, ·), y(t) = x1 (t). (3) where: y(t)[V ] = x1 (t)[V ] is a state variable representing actual position (height) of the load (M) attached to the end of the sag, b(t) = r(t)η is the unknown variable system 2 parameter, and f (·) = y˙ R2 (t) + d is the unknown general
disturbance, which is a sum of assumed internal and external disturbances. In the ADRC, an assumption is made that no analytical expression of f (t, ·) is required as long as the user can provide a close estimate of this element. In order to provide the estimate of general disturbance a state observer is designed. A. Extended State Observer State x2 (t) is an augmentation introduced to the state space model from (3): x˙ 1 (t) = b(t)ω(t) + f (t, ·), (4) x˙ 2 (t) = f˙(t, ·), y(t) = x1 (t). Two states (i.e. x1 (t) and x2 (t)) can be estimated with a linear version of ESO (with an assumption that f˙(t, ·) = 0): ( z˙1 (t) = z2 (t) − β1 eˆ(t) + b0 (t)ω(t), (5) z˙2 (t) = −β2 eˆ(t), where: b0 (t) denotes a variable, which is an approximation of b(t) from equation (3), z1 (t), z2 (t) are estimates of states x1 (t), x2 (t) respectively, eˆ(t) = z1 (t) − y(t) is an estimation error, and β1 , β2 are the observer positive gains. A parametrization method based on pole-placement approach was proposed in [5] to simplify the observer gains tuning procedure. For the considered system the observer gains can be expressed as functions of just one parameter i.e. ω0 : β1 = ω02 , β2 = 2ω0 ,
(6)
where: ω0 is the observer’s bandwidth. It is usually chosen as a subjective compromise between convergence speed of states estimation and the influence of noise and sampling time [5], [9]. B. The controller The type of controller used in ADRC should be related to a given task and technical specifications of the plant. Nevertheless, for the sake of simplicity the control signal in ADRC is presented below for a classic linear PD controller, thus the input signal for the system is as follows: ω(t) =
kp e(t) + kd e(t) ˙ − z2 (t) u0 (t) − z2 (t) = , b0 (t) b0 (t)
(7)
where: z2 (t) is the estimated general disturbance, u0 (t) is the output signal from the controller, e(t), e(t) ˙ are the tracking error and its derivative, kp , kd are the controller gains (proportional and derivative gains respectively). Assuming that the observer is well tuned (i.e. z2 (t) ≈ f (·) and b0 (t) ≈ b(t)), equation (3) with a new control signal (7) is transformed to: kp e(t) + kd e(t) ˙ − z2 (t) , (8) y(t) ˙ ≈ f (·) + b(t) b0 (t) which gives: y(t) ˙ ≈ kp e(t) + kd e(t). ˙
(9)
State space model in (3) is reduced to a simpler form: ( x˙ 1 (t) ≈ kp e(t) + kd e(t), ˙ y(t) ≈ x1 (t).
(10)
The ESO provides dynamical compensation of the general disturbance and therefore reduces the problem of controlling nonlinear, time-varying system to a simpler control of a linear one. Thus, classic linear control techniques can be incorporated, like the considered PD controller. Tuning procedure can be also based on [5]. The controller gains are usually chosen in the ADRC scheme as the result of pole-placement procedure: kp = ωn2 , kd = 2ξωn ,
(11)
where: ωn and ξ are the desired closed loop natural frequency and damping ratio. Figures 5 and 6 show PID and ADRC block diagrams for controlling the R2R system.
Figure 5.
A block diagram of PID controller for the R2R system.
Figure 6.
A block diagram of ADRC method for the R2R system.
4) Derivative gain kd is added if unacceptable overshoot after step 3) appears and is gradually increased until output signal reaches desired behavior. In many cases, thanks to the particular features of the ADRC concept, no integrating action is needed (see e.g. [6]). 5) Parameter b0 (t) is scaling the control signal so it effects the tracking error (ˆ e(t)) convergence rapidity. Choosing b0 (t) should be related to a given control task, defined by the system operator. The most important part of ADRC tuning is obtaining a minimum and fast converging estimation error, thus the ESO parameters are tuned first. Secondly, the PD regulator is tuned to achieve minimum settling time with no signal overshoot. These parameters are chosen empirically and left constant for the experiment. Parameters for the ADRC algorithm are chosen as follows: ω0 = 3rad/s, b0 (t) = 1, kp = 20, kd = 2, and ξ = 1 (selected to avoid oscillations in the output signal). Tuning values for the classic PID are: kp = 5 (proportional gain), ki = 0.025 (integral gain), and kd = 1 (derivative gain). No parameter retuning is done in PID or ADRC after the occurrence of added disturbance. A square reference signal is chosen with frequency: fq = 0.05Hz and sampling time: Ts = 0.01s. Time frame for the experiment is set for Texp = 120s. Additional disturbance (from motor R2) is switched on at t = 60s. Its signal profile is presented on Figure 7.
IV. E XPERIMENT A. Study preparation For this particular work, methodology from (11) did not provide good results. That is why a different approach is used for tuning the PD controller. Since information about the parameters of R2R mathematical model is assumed to be unknown, both the observer and the controller are tuned empirically using simple tuning proceeding4 : 1) Parameters kp , kd , and ω0 (t) are positive and set close to zero. Parameter b0 (t) is set to one. 2) Parameter ω0 (t) in ESO is gradually increased until state variable z1 (t) estimates system output y(t) closely. 3) Proportional gain kp is gradually increased until output signal reaches desired behavior. 4 The tuning process is aimed to provide fastest nonoscillary error convergence for a square reference trajectory. Similar approach was successfully used in [8].
Figure 7.
Artificial, external disturbance generated by motor S2.
B. Experimental results Results of the performed experiments are shown in Figures 8-10. It can be noticed that before the additional disturbance occurs (t < 60s), both of the controllers perform similarly for trajectory tracking case. No overshoot or intolerable error in the steady state is seen in this part of the test (Figure 8). Temporary error seen for the first few seconds in the output signal (Figures 8 and 9) is a consequence of the convergence time of the observer signals. The ADRC method, thanks to the disturbance rejection feature, managed to oppose the acting perturbation much better than the PID controller (t < 60s). Also faster error convergence and no significant overshoot in the output signal
is seen for the ADRC (Figure 8). It is also confirmed in Figure 9. Control signal generated by the ADRC is slightly more shattered but less in amplitude than PID (Figure 10). The jagged character of the control signal has no direct impact on the visual smoothness of the R2R movement.
Figure 8.
Vertical position of the load M (sag length).
V. C ONCLUSIONS AND FUTURE WORK The Active Disturbance Rejection Control method was used in this research for a rubber sag control problem seen on tire production line. Applied technique could detect automatically the effect of the disturbances acting on the system, and then compensate for them in real-time. Thus, the ADRC did not rely on the mathematical expression of the considered system and it showed good robustness and adaptability features. It can be seen that the ADRC strategy reduces the control of a nonstationary and uncertain process to a much simpler problem. It was also noticed that even with the use of the Extended State Observer, the implementation and tuning difficulty did not increase significantly. The ADRC algorithm was verified to be an interesting alternative to widely-used PID controller and obtained results have potential importance to the tire industry. Future work will focus on comparing ADRC method with other minimum model approaches including (but not limited to) adaptive control and model-predictive control since these techniques can also generate good results without analytical model of the process. Such a study could provide evidence of potential advantages of ADRC algorithm over alternative control strategies. Authors also started to implemented several techniques on a Programmable Automation Controller (PAC) that can potentially replace the PLC at the tire reinforcement stage and allow to incorporate more advanced control algorithms. R EFERENCES
Figure 9.
Figure 10.
Vertical position error of the load M (sag length).
Control signals generated by each controller.
[1] A. Abe, T. Kamegawa, Y. Nakajima, Optimization of construction of tire reinforcement by genetic algorithm, Optimization and Engineering, vol. 5, pp. 77–92, 2004 [2] R. S. Bhakuni, G. W. Rye, S. J. Domchick, Adhesive and processing concepts for tire reinforcing materials, American Society for Testing and Materials International, pp. 122-128, 1979 [3] J. C. Dixon, Tires, suspension, and handling, Second Edition, Society of Automotive Engineers, chapters 2.1-2.3, 1996 [4] Z. Gao, Y. Huang, J. Han, An Alternative Paradigm for Control System Design, IEEE Conf. on Control and Decision, pp. 45784585, Orlando, 2001 [5] Z. Gao, Scaling and bandwidth-parametrization based controller tuning, IEEE Proc. of the American Control Conference, pp 4989-4996, Denver, 2003 [6] J. Han, From PID to Active Disturbance Rejection Control, IEEE Trans. on Industrial Electronics, vol. 56, no. 3, 2009 [7] Y. Hou, Z. Gao, F. Jiang, B. T. Boulter, Active Disturbance Rejection Control for Web Tension Regulation, IEEE Conf. on Decision and Control, 2001 [8] M. Kordasz, R. Mado´nski, M. Przybyła, P. Sauer, Active Disturbance Rejection Control for a Flexible-Joint Manipulator, Lecture Notes in Control and Information Sciences, vol., 422, pp. 247-256, 2012 [9] W. S. Levine, The Control Handbook, CRC Press., access: www.books.google.com, New York, 1996 [10] M. Michałek, ZB2 - Laboratory setup user’s manual (internal report, in Polish), Chair of Control and Systems Engineering, Pozna´n University of Technology, 2008 [11] The History of the tire industry: radial tire spawned industry revolution, International Tire Exhibition and Conference, http://www.itec-tireshow.com/history_of_tires.html, 2011 [12] Q. Zheng, Z. Gao, On Practical Applications of Active Disturbance Rejection Control, Proceedings of the 29th Chinese Control Conference, pp. 6095-6100, Beijing 2010
