CONTROL OF OSCILLATIONS IN MANUFACTURING NETWORKS

PHYSCON 2009, Catania, Italy, September, 1 September, 4 2009 CONTROL OF OSCILLATIONS IN MANUFACTURING NETWORKS Alexander Y. Pogromsky Department of Mechanical Engineering Eindhoven University of Technology Eindhoven, The Netherlands a.pogromsky@tue.nl Boris Andrievsky Institute for Problems of Mechanical Engineering of Russian Academy of Sciences Saint Petersburg, Russia bandri@yandex.ru Jacobus E. Rooda Department of Mechanical Engineering Eindhoven University of Technology Eindhoven, The Netherlands j.e.rooda@tue.nl Abstract The paper is devoted to supression of the manufacturing network oscillation, induced by the combined influence of control saturation, input signal fluctuation and presence of the integral component in the control law. A feedback-feedforward observer-based control strategy is proposed, significantly reducing magnitude of oscillation. Key words Oscillations Control; Manufacturing Systems 1 Introduction The production control of manufacturing systems, i.e. how to control the production rates of machines such that the system tracks a certain customer demand while keeping a low inventory level, has been a field of interest for several decades. Simple discrete-event manufacturing systems can be controlled by policies such as PUSH, CONWIP or Kanban (see e.g. (W. J. Hopp and M. L. Spearman, 2000)). However, as manufacturing systems become more complex, these policies become less effective. A more structured approach for the control of manufacturing systems was proposed in the 1980 s, i.e. supervisory control theory in (Ramadge and Wonham, 1987), which is also based on a discreteevent description of the manufacturing system. A disadvantage of this approach, however, is that if it comes to the control of large manufacturing systems (or networks of such systems), supervisory control is not very suitable due to the high level of detail they deal with, which causes the corresponding control problem to grow intractably large. Therefore, there is need for a simple, straightforward control strategy for manufacturing systems, that does not rely on predictions of the future demand. In (W. A. P. Van den Bremer et al., 2008; R. A. Van den Berg, 2008) such a strategy is derived by using feedback control of continuous systems. A simple PI controller is used to set the production rate in the ODE model of a manufacturing system such that the production meets a certain demand. The combination of control action saturation and the integrator in the PI controller leads to a phenomenon called integrator windup. When the actuator saturates, the effective control signal cannot exceed some value, which affects the system behavior and therefore again the control signal. As a result of this, the closed-loop performance of the system can deteriorate, and in some situations the system can even become unstable. By adding a socalled anti-windup controller to the system, this loss of performance can be counteracted by turning off the integrator in the controller when the machine saturates. In this paper and a different solution to this problem is provided, based on employing the observer in the framework of feedforward-feedback control strategy of (Andrievsky et al., 2009; Kommer et al., 2009). The paper is organized as follows. The problem statement and the continuous-time representation of a line of manufacturing machines are given in Section 2. Decentralized control strategy for a manufacturing line is proposed in Section 3. The numerical example is given in Section 4, where the simulation results in continuous domain and discrete-event representation are given. Concluding remarks and the future work intentions are presented in Section 5.

2 Problem statement Consider a line of N manufacturing machines M 1, M 2,..., M N, which are separated by buffers B j 1,j, j = 1,..., N with infinite capacity, see Fig. 1. The first machine M 1 is supplied by raw material, the Nth machine M N produces finished product. Each machine M j takes out a raw product from the corresponding input buffer B j 1,j and puts a processed product to the output buffer B j,j+1. In what follows suppose that there is always sufficient raw material to feed the first machine, i.e. that the buffer B 0,1 is never exhausted. Figure 1. Schematics of a line of N manufacturing machines. M j machines, B i,j buffers, i, j = 1,..., N. Summarizing, we obtain the following manufacturing line model ẏ 1 (t)=u 1 (t), ẏ 2 (t)=u 2 (t) sgn(w 2 (t)),...... ẏ N (t)=u N (t) sgn(w N (t)), (3) where sgn(z) = ( 1, if z > 0 0, otherwise ). The control aim is tracking the non-decreasing reference production variable y d (t). Since the finished product of the manufacturing line is an output of the Nth machine (see Fig. 1), the system accuracy is expressed in terms of the reference error e(t) e N (t) = y d (t) y N (t). Let us represent the demand y d (t) as a sum of a linear on time t function and a casual term as follows Following (van den Berg et al., 2006; R. A. Van den Berg, 2008; Andrievsky et al., 2009; Kommer et al., 2009), at the stage of the control law design a continuous approximation of the discrete-event manufacturing machine is used. Assume that a manufacturing machine produces items continuously in time t R with a certain production rate u j (t) R, where j = 1,... N is a number of the machine. The total amount of items produced by jth machine is described by a continuous variable y j (t) R. Interaction between the machines is described by the buffer content variables w j (t) = max(y j 1 y j, 0), j = 2,..., N. The case of w j (t) = 0 means absence of the row material in the input buffer of jth machine and, therefore, the machine M j work is suspended. The above reasons lead to the following continuous model of the manufacturing machine: ẏ j (t) = { u j (t), if w j (t) > 0, 0, otherwise, (1) where t R stands for continuous time argument; j = 1,..., N is a machine number. The production rates u j are bounded by u max due to machine capacity limitation. In the sequel we assume, without loss of generality, that all the machine capacities in the line have the same upper bound u max. Since the production rates u j can not also be negative, the following bounds are valid for u j (t): 0 u j (t) u max, j = 1,..., N, t 0. (2) Inequalities (2) lead to a saturation effect in the system. This effect restricts the production rate, and complicates design of the controller and the system performance analysis. y d (t) = y d,0 + v d t + ϕ(t), (4) where y d,0 denotes the bias in the production demand, v d is a constant, representing the average desired production rate, ϕ(t) is a bounded function, describing fluctuation of the production demand from the linear trend y d,0 + v d t. This fluctuation may be caused by market seasonal variations, for example. Suppose that ϕ(t) has a zero mean in a some sense because its averaged value may be referred to y d,0. 3 Control strategy 3.1 Wind-up effect for the case of PI-control and input saturation Since the demand (4) has a part v d t that is linear, it can be argued by means of the final value theorem from linear control theory that for ϕ(t) 0 a controller with integral action should be used to track the error e(t) = y d (t) y(t) to zero. The simplest controller with integral action is a PI controller for which the controller output at time t is given by: u(t) = k P e(t) + k I t 0 e(τ)dτ, (5) with k P and k I the controller parameters. Using the Routh-Hurwitz stability criterion, it can be concluded that the closed loop system is stable iff k P and k I 0 are both positive. A specific choice of these parameters has to be made based on performance criteria, for instance certain demands for the sensitivity and complementary sensitivity functions. To demonstrate the windup effect let us consider the following numerical example. Let the single manufacturing machine be modeled by (1), the control signal is bounded by u max = 1, the

demand y d is given by (4) and has the following parameters: y d,0 = 0, v d = 0.75. Fluctuation signal ϕ(t) in (4) has a harmonic form, ϕ(t) = ϕ 0 sin(ωt), where ϕ 0 = 2.5, ω = 0.2 s 1. Let us apply the pole placement technique to find the PI-controller (5) parameters for the case of non-saturated (linear) system. It may be easily checked that choice of the gains k I = 9 S 1, k P = 5 of PI-controller (5) ensures the Butterworth distribution of the closed-loop system eigenvalues s 1,2 as s 1,2 = 2.5 ± 1.66i, s 1,2 = 3 s 1. In the absence of the control signal saturation, the close-loop system has a trancient time about one second and, in the steady-state mode, the error signal magnitude e(t) max 0.011. The system behavior is dramatically changed due to saturation in control, as it is evident from Fig. 2, where the simulation results for the considered saturated system are depicted. The tracking error e(t) in this case 3.2 Observer-based feedback controller for a single machine To start, let us recall the observer-based feedback control strategy of (Andrievsky et al., 2009; Kommer et al., 2009) for a single machine. Assuming that y(t), v d may be measured and used to form the control signal u(t), the following feedforward-feedback control law may be used u(t) = sat [0,umax] ( kp e(t) + v d ), (6) where e(t) = y d (t) y(t) denotes the tracking error, k p is the controller parameter (a proportional gain), sat( ) denotes the saturation function sat [a,b] (z) = min ( b, max(a, z) ). It may be easily seen that, in the absence of control saturation, the control strategy (6) leads to asymptotically vanishing error e(t) for linear on t demand y d (t). Assuming that only the error signal e(t) can be measured and used to form the control action, in (Andrievsky et al., 2009) was proposed to replace v d by its estimate ˆr(t), provided by the observer, which employs only available signals e(t) = y d (t) y(t) and u(t). Luenberger s design method (D. G. Luenberger, 1971) leads to the following reduced-order observer Figure 2. Windup effect in the manufacturing process; u max = 1, y d (t)=0.75t+2.5 sin(0.2t), k I =9 s 1, k P =5. { σ(t) = λσ(t) λ 2 e(t) + λu(t) ˆr(t) = σ(t) + λe(t), (7) has a form of irregular oscillations of the magnitude about 10. The windup-caused oscillations may be reduced by means of the anti-windup control, see e.g. (Hippe, 2006) for details. The PI-controller with an antiwindup control strategy for a single manufacturing machine is proposed and thoughtfully studied in (van den Berg et al., 2006; R. A. Van den Berg, 2008). This controller ensures asymptotically vanishing tracking error e(t) for constant ϕ(t) and, also, independence of the asymptotic system behavior of the initial conditions if fluctuations and disturbances take effect on the system (the so called convergence property. 1 Supression of system oscillation may be also ensured by means of the observer-based control strategy proposed in (Andrievsky et al., 2009; Kommer et al., 2009). Controller of (Andrievsky et al., 2009; Kommer et al., 2009) implements a proportional (P-) control law with an estimator of the average desired production rate. Absence of the integral component makes possible to avoid the anti-windup compensator of (van den Berg et al., 2006; R. A. Van den Berg, 2008) in the controller. 1 Recall that this property means that the system, being excited by a bounded input, have a unique bounded globally asymptotically stable steady-state solution, see (Pavlov et al., 2004; Pavlov et al., 2005) for details. where λ > 0 is the observer parameter (observer gain), setting the transient time for the estimation procedure. Finally, the control action u(t) takes the form u(t) = sat [0,umax] ( kp e(t) + ˆr(t) ), (8) where e(t) = y d (t) y(t), ˆr(t) is governed by (7). Equations (7), (8) describe the first-order feedback controller. The control signal u(t) is calculated based on the error e(t) measurement only. The gains k P > 0 and λ > 0 are the controller parameters. Let us continue the above given example. Choice the control gain k P = 5 and λ = 25 s 1 in (7), (8) lead to the error signal shown in Fig 3 (solid line), where for the sake of comparability the error signal of the PIcontrolled system (1), (5) is also depicted (dash-dotted line). The simulation results demonstrate that the oscillations magnitude for the observer-based controller (7), (8) is about five times less than that for the PI-controller (1), (5). 3.3 Control strategy for a line of machines The control strategy (7), (8) was extended to control of a line of the manufacturing mashines in (Pogromsky et al., 2009). The direct usage of (7), (8) for each machine is rather unpractical, because in this case the buffer

Figure 3. Tracking error e(t) for the system with the observerbased controller (7), (8) (solid line) and PI-controller (1), (5) (dash-dotted line); u max = 1, y d (t)=0.75t+2.5 sin(0.2t). the system parameters (Andrievsky et al., 2009; Kommer et al., 2009): u p,max = 1.0, k p = 5, λ = 25. Choose k w = 30, w d = 1. Consider again the system behavior for the case of y d (t) = 0.75t+2.5 sin(0.2t). The simulation results are plotted in Fig. 4, where the tracking errors e j (t), j = 1,..., N for the line of N = 4 manufacturing machines with the PI-controller (1), (5) and the observer-based controller (7), (8) are depicted. contents are not taken into account, which may lead to exhaustion of some buffers or, alternatively, to stacking in buffers an extra amount of material. Besides, from implementation reasons, it is desirable to organize interactions between the neighboring machines only and avoid transferring the reference signal to each machine. Due to these reasons, the following modification of the control strategy (7), (8), intended to control of a manufacturing line has been proposed in (Pogromsky et al., 2009). The desirable constant level of the buffer contents( w d > 0 was introduced and the penalty term k w wd w j+1 (t) ), where k w > 0 is a certain gain (designed parameter) to jth control action u j (t) was added. The following demand signal for jth machine ensuring equality y j 1 (t) = y j (t) + w d in the steady-state nominal regime was used. This leads to the following control strategy for the line of manufacturing machines. Take the control law for Nth machine in the form (7), (8), namely let the control signal u N (t) be calculated as ( ) u N = sat [0,umax] kp ε N + ˆr N, σ N (t)= λσ N (t) λ 2 e(t)+λu N (t), ˆr N (t)=σ N (t)+λe(t), (9) where ε N (t) e(t) = y d (t) y N (t) is the reference error. Take the control law for machine M j, j =1,..., N 1 in the following form: u j = sat [0,umax](k p ε j +ˆr j + k w (w d w j+1 ) ), ε j (t) = w d + ε j+1 (t) w j+1 (t) σ j (t)= λσ j (t) λ 2 ε j (t)+λu j (t), ˆr N 1 (t)=σ N 1 (t)+λε N 1 (t), (10) where w j+1 (t) = y j (t) y j+1 (t); w d is the buffer contents demand. Formulas (9), (10) recursively specify the distributed controller for a line of N 2 manufacturing machines. 4 Numderical example. Consider the manufacturing line from N = 4 machines. Let us take the following numerical values of Figure 4. Tracking errors e j (t) for the line of N = 4 manufacturing machines with the PI-controller (1), (5) and the observerbased controller (7), (8); u max = 1, y d (t) = 0.75t + 2.5 sin(0.2t), w d = 1. e 1 dotted line, e 2 dashed line, e 3 dash-dotted line, e 4 solid line, 5 Conclusions Supression of the manufacturing network oscillation, induced by the combined influence of control saturation, input signal fluctuation and presence of the integral component in the control law is considered. Efficiency of a feedback-feedforward observer-based control strategy in reducing oscillation magnitude for a line of the manufacturing machines is demonstrated. Future work is aimed to studying the discrete-event implementation of the proposed control (the first attempt is presented in (Pogromsky et al., 2009)) and in generalisation of the proposed control strategy to manufacturing networks of more general topology. The problems of coping with imprecisions, missing data and delays in the system will be also considered. Acknowledgments The work was done when the second author was with the Eindhoven University of Technology. Partly supported by C4C project, by De Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), ref. # B 69-113 and by the Russian Foundation for Basic Research (RFBR), Proj. # 09-08-00803.

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