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Model predictive control helps to regulate slow processes-robust barrel temperature control
ISA TRANSACTIONS® ISA Transactions 41 共2002兲 501–509
Model predictive control helps to regulate slow processes—robust barrel temperature control Tien L. Chia, Ph.D. ControlSoft, Inc., 5387 Avion Park Drive, Highland Heights, Ohio 44143, USA
共Received 14 May 2001; accepted 15 December 2001兲
Abstract Slow temperature control is a challenging control problem. The problem becomes even more challenging when multiple zones are involved, such as in barrel temperature control for extruders. Often, strict closed-loop performance requirements 共such as fast startup with no overshoot and maintaining tight temperature control during production兲 are given for such applications. When characteristics of the system are examined, it becomes clear that a commonly used proportional plus integral plus derivative 共PID兲 controller cannot meet such performance specifications for this kind of system. The system either will overshoot or not maintain the temperature within the specified range during the production run. In order to achieve the required performance, a control strategy that utilizes techniques such as model predictive control, autotuning, and multiple parameter PID is formulated. This control strategy proves to be very effective in achieving the desired specifications, and is very robust. © 2002 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Model predictive control; Autotuning; Barrel temperature control; Injection molding machine; Extrusion
1. Introduction Temperature control is one of the common control loops encountered in industry. The dynamic responses of temperature loops cover a wide spectrum, from seconds to hours. Closed-loop control of slow temperature loops, such as barrel temperature control in injection molding and extrusion machines, can be difficult because of nonlinearity and slow dynamics. When, as is sometimes the case, a proportional plus integral 共PI兲 controller is used in such a loop, the control may develop a limit cycle. The limit cycle often goes undetected because of the slow dynamics and the long trend time, which means that a snapshot of the temperature record may look fine at any particular instant. This is a common problem in many slow temperature loops as well as in integrating processes. Such temperature loops behave more like an integrating
process than a self-regulating process. For such processes, a change of control output will cause a process variable to increase 共or decrease兲 continuously until it reaches a limit. One such example is the control of liquid level in a tank. Although proportional plus integral plus derivative 共PID兲 control has been used widely by industry for barrel temperature control, there have been many efforts to improve control performance by introducing more advanced control techniques. In Ref. 关1兴, efforts are devoted to developing an autotuning formula for barrel temperature control, while in Ref. 关2兴, autotuning is incorporated into a PID controller modified with a ‘‘cutback’’ feature. The cutback feature tends to reduce overshoot in the closed-loop response by establishing a temperature band about the set point such that the integral action of the PID control is activated within the band but is turned off when the temperature is
0019-0578/2002/$ - see front matter © 2002 ISA—The Instrumentation, Systems, and Automation Society.
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outside the band. In Ref. 关3兴, the use of an additional thermocouple sensor is proposed as a means of improving the control, with one sensor located inside the barrel and the other on the outside surface of the barrel. This is similar to cascade control of temperature in a continuous stirred tank reactor 关4兴 and provides the same benefits. However, this is done at the expense of additional instrumentation. Model predictive control 共MPC兲 关5,6兴 is a promising new technology that has been used in many process control applications to improve the control. The approach is based on the use of an internal model of the process to predict its response to input changes. MPC is particularly effective in dealing with processes characterized by large dead times, multivariable systems with dynamic interactions, and systems with inequality constraints 共such as saturation limits兲. In Ref. 关7兴, two different approaches of multivariable model predictive control are analyzed and compared with respect to constraint handling, tuning, and decoupling. An application of state space based control to the barrel temperature control problem is described in Ref. 关8兴. Barrel temperature control involves multiple interacting zones with common disturbances affecting adjacent zones, i.e., the heat loss from zone 2 to zone 3 is the same as the heat gain from zone 3 to zone 2. With the additional information, the constrained model-based adaptive control approach 关9,10兴 can be used for such applications to take advantage of available system information to improve control performance. However, due to the difficulties of field implementation, i.e., process knowledge required and high maintenance of an MPC based approach, this is not the ideal solution for barrel temperature control. Most users in industrial plants lack the time, special knowledge, and experience required for setting up and maintaining a complex control system. This paper proposes a control strategy which incorporates the essence of model predictive control as part of the strategy. The resulting solution inherits the benefits of MPC without its limitations. The resulting solution is robust, effective, and easy to use. Section 2 examines the control and stability of PI and PID control for an integrating-type process with interpretations relevant to barrel temperature control. Section 3 describes the barrel temperature
Fig. 1. Block diagram of control of an integrating process.
control problem—its characteristics and requirements. This is followed by a detailed discussion of the proposed control strategy. Section 4 discusses the results of such a control application in a real industrial environment. 2. Closed-loop control of an integrating process The block diagram of a typical process control loop is shown in Fig. 1, where G p( s ) denotes the transfer function of the process plus actuator, and G c( s ) denotes the controller. A reasonable approximation of a barrel zone temperature control process is an integrator plus dead time plus lag 共which may represent the actuator dynamics兲 with the transfer function given by Eq. 共1兲:
process:
G p共 s 兲 ⫽
K * e ⫺Ts , s 共 1⫹ s 兲
共1兲
where K is the process gain, T is the process dead time, and is the lag time constant. 2.1. Proportional plus integral controller Consider a proportional plus integral 共PI兲 controller applied to the above process:
PI controller:
G c共 s 兲 ⫽K p共 1⫹1/ is 兲 ,
共2兲
where K p is the proportional gain and i is the integral time constant of the controller. In the case of negligible dead time 关i.e., T in Eq. 共1兲 is zero兴, the open-loop transfer function for the system of Fig. 1 is
G OL共 s 兲 ⫽G c共 s 兲 G p共 s 兲 ⫽K OL*
共 1⫹ is 兲 , 共3兲 s 2 共 1⫹ s 兲
where K OL⫽KK p / i denotes the open-loop gain. This system has two poles at the origin—one from the process integrator, the other from the integral action of the controller; there is also a pole
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at ⫺1/, and a zero at ⫺1/ i , where may characterize the actuator time constant and i is the integral time constant. The root locus diagram of Fig. 2共a兲 shows that when i is set smaller than , the system is unstable irrespective of the controller gain. On the other hand, when i is set larger than , the system is stable for all values of controller gain. However, as shown in Fig. 2共b兲, the closed-loop response tends to be oscillatory for values of controller gain high enough to satisfy typical speed of response specifications. We note further that as the controller gain is increased, the response becomes more and more oscillatory. With PI control, there are only two degrees of freedom available 共i.e., choice of K c and i兲 for satisfying performance specifications. In addition, the range of values for K c consistent with allowable overshoot, results in a relatively slow response and limited disturbance rejection. If further realistic values of process dead times are considered, these limitations become increasingly critical. 2.2. Proportional plus integral plus derivative (PID) controller The addition of derivative action to the PI controller of Eq. 共2兲 results in the PID controller given by Eq. 共4兲:
PID controller:
G c共 s 兲 ⫽K p共 1⫹1/ is 兲 ⫻ 共 1⫹ Ds 兲 ,
共4兲
where D is the derivative time constant. This third tuning parameter provides an additional degree of freedom in satisfying control specifications. The open-loop transfer function of Fig. 1 with PID control becomes
G OL共 s 兲 ⫽G c共 s 兲 G p共 s 兲 ⫽K OL
共 1⫹ is 兲共 1⫹ Ds 兲 . s 2 共 1⫹ s 兲 共5兲
Thus PID control adds a second zero to the openloop transfer function which has the effect of pulling to the left the two branches of the root locus that originate at the origin. This is illustrated in Fig. 2共c兲 where, consistent with good design practice, the derivative time constant D is set to a value lower than the integral time constant i .
Fig. 2. 共a兲 Root locus for PI controlled system with i ⬍ . 共b兲 Root locus for PI controlled system with i⬎ . 共c兲 Root locus with PID control.
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Fig. 3. Injection molding machine diagram.
This has two desirable effects on the response of the closed-loop system as compared with PI control: 1. It tends to stabilize the system. 2. The response tends to be faster. Therefore derivative action is particularly desirable for an integrating-type process because it permits a higher controller gain with an acceptable degree of overshoot. The presence of dead time tends to destabilize the system resulting in more severe behavior in terms of overshoot and oscillation, as well as restricting further the range of proportional gain K p corresponding to a stable response. Temperature processes characterized by large thermal lags exhibit similar behavior to that of the integrating process given by Eq. 共1兲. This is because the slower the dynamics, the larger the time constant and hence the closer the process pole is to the origin.
3. Barrel temperature control 3.1. Performance specifications Barrel temperature control in an extrusion process or an injection molding machine 共Fig. 3兲 presents severe challenges to achieving satisfactory control. In injection molding, plastic in the form
of a granular solid is fed into the barrel, where it is melted so that it can be injected into the mold in a liquid state. The control of the temperature within the barrel is critical to the quality of the molded product. This is particularly true for some temperature-sensitive plastics which can tolerate only very small temperature deviations from the set point. Therefore it is important to have a very tight temperature control during production to ensure consistent product quality. Further, when the machine is started up from a cold state, it is desired that the temperature be brought to its set point 共usually several hundred degrees Fahrenheit兲 as rapidly as possible and without much overshoot. Since it may take from 15 minutes to a couple of hours, depending on the size of the barrel and the power rating of the barrel heater, to bring the barrel temperature to its set point, the startup period can have an important effect on productivity. The specification on overshoot is also often critical, as in the case of large, well-insulated barrels that take a long time to cool down. Summarizing, the performance specifications for barrel temperature control are: 1. Bring the barrel temperature up to set point as fast as the system allows without overshoot. 2. Maintain the barrel temperature within plus or minus one to two degrees Fahrenheit during the production run.
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3.2. Challenges in barrel temperature control Like many other slow processes, barrel temperature control can tolerate a very large proportional gain and, with proper choices of i and/or D , the system is always stable. However, the response may be oscillatory as indicated by the root locus diagrams of Figs. 2共b兲 and 共c兲. For processes with significant dead time, the allowable range of proportional gains is more limited, although the gain may still range over one to two orders of magnitude. However, since the dead time tends to make the period of closed-loop oscillation very large, poor temperature performance can often go undetected by the operator. Although stability itself is not a major concern, satisfying overall performance specifications for barrel temperature control is a challenge due to the nature of the system, as described in the next section. Several factors contribute to the complexities of barrel temperature control: 1. The process response is very slow, approaching that of an integrating type of system. 2. The dynamic response of the heating cycle is typically very different from that of the cooling cycle. 3. The barrel consists of several zones with significant interaction effects between neighboring zones. Further, the zones often exhibit different dynamics. 4. The process exhibits significant dead time ranging from a few seconds to several minutes. 5. For extrusion processes, the temperature control is further complicated by the shearing energy generated by the rotating shaft. This effect may be so great that a specified desired temperature profile may not even be attainable. 6. Rigorous requirements on the control system, including fast startup, no overshoot, and being able to maintain each zone within a couple of degrees of its respective set point during a production run. According to points 2 and 3 above, the control design problem is characteristically nonlinear and multivariable in nature. Due to the dynamic nature of the system and the analysis in Sec. 2, a conventional PI or PID controller will not work well for
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such an application. The problem is further complicated by the presence of significant dead time which, as already pointed out, leads to closed-loop instability 关4兴, and tends to limit the allowable range of controller gain. Hence simple PI or PID control, irrespective of the parameter settings, cannot generally satisfy all the stringent performance requirements, such as point 6 above. If an aggressive PID control is applied, the temperature tends to overshoot and be oscillatory. A less aggressive PID control may reduce the overshoot but at the expense of increasing the time to reach the set point, and in decreasing the ability of suppressing disturbance effects during the production run. The advanced control strategy developed in this paper uses a combination of control techniques to counter the adverse characteristics of the process. The control strategy incorporates autotuning, model prediction, and multiple PID’s applied to different barrel zones and different regimes of control action, as shown in Fig. 4. Autotuning is designed to generate optimal tuning parameters for such an integrating type of system; a form of model predictive control is incorporated which, while still predictive in nature, does not burden the user with the difficulties in setting up a conventional MPC. Multiple PID controllers are used to compensate for the system nonlinearities, and the controllers for each zone are coordinated to take into account the interaction effects of neighboring zones.
3.3. Advanced control strategy for barrel temperature control In order to meet the strict performance requirements, different control techniques are used at different stages. In this section, these control strategies are described in some detail.
3.3.1. Startup and autotuning Normally, the system is started cold, and it takes some time to heat the barrel to set point temperature. In order to bring the temperature up to the set point as fast as possible, the control output is initially set to its maximum value. Autotuning is performed at this time if requested by the operator.
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Fig. 4. Control strategy.
During the tuning stage, model parameters such as gain, time constant, and dead time, as in Eq. 共6兲, are identified and saved. Either a reaction curve technique 关4兴 or least-square method can be used to identify the process model. Two sets of PID tuning parameters are determined, startup PID and run-time PID 关as in Eqs. 共7兲 and 共8兲 below兴, based on the identified model parameters. Methods for calculating these tuning parameters can be found in many articles and textbooks 关5,6兴. The design procedure for selection of the parameters for Eqs. 共7兲 and 共8兲 are explained in Sec. 3.3.3.
Model: Startup PID Run-time PID:
ˆ e ⫺Tˆ s /s 共 ˆ s⫹1 兲 , K
共6兲
K p1 共 1⫹1/ i1 s 兲共 1⫹ D1 s 兲 , 共7兲 K p2 共 1⫹1/ i2 s 兲共 1⫹ D2s 兲 . 共8兲
The tuning procedure is enhanced by recognizing that different barrel zones behave differently; for example, since the nozzle zone has a higher heat dissipation rate than the other zones it behaves more like a lag process than an integral process, permitting a faster integral action by the PID controller. 3.3.2. Model predictive control After tuning is completed, a preset point 共PSP兲, given by Eq. 共9兲, is calculated using an internal model of the process response. This PSP is a temperature below the actual set point that is determined to minimize the overshoot during the startup phase. When the temperature reaches the preset point, power to the heater is turned off for a prespecified time called the wait time 共WT兲, given
by Eq. 共11兲. If the system is perfectly described by Eq. 共6兲 then, given the parameter values for the model, the current control output, and the current temperature, the predicted temperature at the end of the wait time period can be calculated. However, since the model is never perfect, the DSP of Eq. 共10兲 is designed conservatively so that the temperature at the end of the WT is always below the set point,
PS P⫽S P⫺ ␣ DS P,
共9兲
ˆ Tˆ ⫹100 Kˆ 共 Tˆ ⫹ ˆ 兲 , DS P⫽100K
共10兲
WT⫽2Tˆ ⫹ ˆ ,
共11兲
where 1⬍ ␣ ⬍2, 1⬍  ⬍2, and SP is the user specified set point. The factor 100 reflects the fact that the controller output is 100% at the autotuning phase. Both ␣ and  should initially be set to 1, however, they can be set to be greater than 1; ␣ is selected to accommodate modeling errors, and  is adjusted to compensate for different zone behaviors as described in Sec. 3.3.5. The preset point is designed to ensure that, with the control off during the duration of the wait time, the zone temperature at the end of the wait time period is close to but does not exceed the user-specified set point. Even though power to the heater is off, the temperature will continue to rise during this time due to the thermal inertia characterizing the process behavior. If the wait time is selected too short, the temperature will overshoot; if it is too long, the temperature will take a longer time to reach the set point.
Tien L. Chia / ISA Transactions 41 (2002) 501–509
3.3.3. Two stage robust control strategy At the end of the wait time period, automatic control using the PID controller takes over. During the autotuning stage, two sets of PID parameters are generated as in Eqs. 共7兲 and 共8兲. The startup PID of Eq. 共7兲 is a proportional control dominated action designed to bring the temperature up fast; this is used when the temperature is far from the set point at the end of the wait time period. If the model identified in Sec. 3.3.1 is perfect, then WT and PSP can be determined so that, at the end of the WT, the temperature will reach the set point and there is no need for the startup PID phase. However, since a perfect model is highly unlikely, there is a need for a startup PID phase. The purpose of this phase is to account for modeling errors and the intentionally inflated DSP in Sec. 3.3.2. The addition of the startup phase increases the robustness of the control strategy. If the model is accurate; this phase will be bypassed; if the temperature is far from the SP at the end of the WT, the startup PID will bring the temperature close to the set point without overshoot. Once the temperature is close to the set point, the system will switch to a more aggressive set of controller settings 关runtime PID as in Eq. 共8兲兴 designed to meet the stringent temperature control requirements during the production run. This run-time PID has sufficient derivative action to counter the integrating type of response of the barrel temperature as per the analysis in Sec. 2.2. This controller is too aggressive to use in the startup phase; hence the change from startup PID to run-time PID parameters happens only after the temperature has reached the neighborhood of its set point value 共typically 5–10°兲. This enables the control system to bring the temperature up fast, while still minimizing overshoot and providing adequate run-time control. The system may switch between the two controllers depending on how much the temperature deviates from the set point 共as caused by disturbances or set point changes兲. 3.3.4. Speed reconciliation Another complexity of this control problem is due to the differences in physical size of the zones. The response speed of the zone’s temperature varies dramatically with the size of the zone. Thus, because the nozzle zone is usually much smaller than the other zones, it may reach its set point within several minutes, compared to the one to
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two hours required in some very large middle zones. Since the production cannot start until all zones reach their respective set points, the faster temperature rise in the smaller zones results in a waste of energy and causes disturbances to adjacent zones. Thus it is desirable to reconcile the speeds of the different zones so that they each reach their respective set points in a comparable time frame. This is done by applying different control outputs for the different zones to ensure that they all reach their set points at about the same time. In other words, for smaller and faster zones, less than the maximum control outputs are used in the startup stage, as discussed in Sec. 3.3.1. 3.3.5. Fine tuning Further enhancement of performance can be achieved by fine tuning the system. Zone dynamics may be affected by their physical location; for example, some zones have only one adjacent zone, while others have two. In general, a zone having only one adjacent zone 共e.g., one side is exposed to the ambient兲 will have faster heat dissipation than a zone with both ends thermally connected to other zones. Therefore these two types of zones should be treated differently when tuning the control parameters. Also, how closely a zone behaves like an integrating system depends very much on the mass or thermal capacity of the zone. Minor adjustments according to the above variations enable the user to fine tune the system to his or her preference. 4. Field test results and conclusions The control strategy that employs a combination of several of the control techniques described above is proven to be very effective for this type of process. Unlike most advanced model-based control solutions which require the user to input extensive information concerning the process dynamics, this system requires no such inputs and is very easy to use. The control strategy can be embedded within any digitally implemented barrel temperature control system. The only information required of the user in setting up the system are the typical PID setup parameters, e.g., PID type, sampling time, etc. The above solution has been applied in many installations. Some of the field test data are pre-
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Fig. 5. Autotuning with heat/cool cycle and first-time startup.
sented here. Fig. 5 shows the startup behavior of a heat/cool extrusion system on the first startup 共i.e., prior to any tuning兲. Autotuning is done during this startup stage with the results stored in memory. Fig. 6 shows a typical startup of a fourzone injection molding machine, after autotuning. As shown in Figs. 5 and 6, the system reaches the set point as fast as the system allows 共in about 20 min for a set point of 350 °F兲 without any overshoot, and the temperature is maintained within a one to two degree band about the set point during the production run. The above control scheme provides a good solution for a heating-only type of barrel temperature control. If both heating and
cooling are involved, the control problem is more complicated due to the very different dynamics between the heating and cooling cycles. The above solution is very effective and also very robust. The robustness comes from the fact that PID-based controls are used and, unlike most model predictive control schemes, the control system does not require any a priori model information to set up. Performance can be further enhanced by inserting a user-adjustable aggressiveness factor for the model prediction scheme which enables the user to adjust the control to accommodate variations in
Fig. 6. Cold startup for a four-zone injection molding machine.
Tien L. Chia / ISA Transactions 41 (2002) 501–509
the system that were not captured in the autotuning and modeling stages. Acknowledgment The author would like to thank Dr. Irving Lefkowitz for his valuable inputs and comments on this paper.
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关8兴 Bulgrin, T. C. and Richards, T. H., Van Dorn Demag Corp., The Application of Advanced Control Theory to Enhance Molding Machine Performance. Injection Molding Expo, Akron, Ohio, 1994. 关9兴 Chia, T., Chow, P., and Chizeck, H., Recursive parameter identification of constrained systems. IEEE Trans. Biomed. Eng. 38, 429– 442 共1991兲. 关10兴 Timmons, W., Chizeck, H., Casas, F., Chankong, V., and Katona, P., Parameter-constrained adaptive control. Ind. Eng. Chem. Res. 36, 4894 – 4905 共1997兲.
References 关1兴 Searler, J. G., Delaney, B. P., Delaney, K. O., Hogan, K. M., and Schmidt, P. B., Injection Molding Machine Temperature Control System. United States Patent 1995. Patent Number 5,397,515. 关2兴 Installation and Operation Manual for Models 808 and 847 Digital Controllers-Eurotherm Corporation, 1987. 关3兴 User Manual for Model 1771-TDC Module, Rockwell Automation and David Standard, 1995. 关4兴 Stephanopoulos, G., Chemical Process Control. Prentice-Hall, Englewood Cliffs, NJ, 1984. 关5兴 Morari, M. and Zafiriou, E., Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ, 1989. 关6兴 Brosilow, C. B. and Joseph, B., Techniques of ModelBased Control. Prentice-Hall, Upper Saddle River, NJ, 2002. 关7兴 Chia, T. L. and Lefkowitz, I., Multivariable Control Technologies. ControlSoft Publication #PA401.
Tien-Li Chia received a B.S. degree from Tan-Kung University, Taiwan, in 1977 and M.S. and Ph.D. degrees in systems and control engineering from Case Western Reserve University, Cleveland, Ohio, in 1982 and 1985, respectively. Dr. Chia is the president of ControlSoft, Inc., and adjunct faculty at EE&CE Department, Cleveland State University, Cleveland, Ohio. He is experienced in the design and development of advanced control based solutions, including adaptive control, expert-based solutions, and multivariable control, for various industrial and manufacturing processes. At ControlSoft, Dr. Chia has designed and directed the development of many control software solutions that are now widely used in industry.
Model predictive control helps to regulate slow processes—robust barrel temperature control Tien L. Chia, Ph.D. ControlSoft, Inc., 5387 Avion Park Drive, Highland Heights, Ohio 44143, USA
共Received 14 May 2001; accepted 15 December 2001兲
Abstract Slow temperature control is a challenging control problem. The problem becomes even more challenging when multiple zones are involved, such as in barrel temperature control for extruders. Often, strict closed-loop performance requirements 共such as fast startup with no overshoot and maintaining tight temperature control during production兲 are given for such applications. When characteristics of the system are examined, it becomes clear that a commonly used proportional plus integral plus derivative 共PID兲 controller cannot meet such performance specifications for this kind of system. The system either will overshoot or not maintain the temperature within the specified range during the production run. In order to achieve the required performance, a control strategy that utilizes techniques such as model predictive control, autotuning, and multiple parameter PID is formulated. This control strategy proves to be very effective in achieving the desired specifications, and is very robust. © 2002 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Model predictive control; Autotuning; Barrel temperature control; Injection molding machine; Extrusion
1. Introduction Temperature control is one of the common control loops encountered in industry. The dynamic responses of temperature loops cover a wide spectrum, from seconds to hours. Closed-loop control of slow temperature loops, such as barrel temperature control in injection molding and extrusion machines, can be difficult because of nonlinearity and slow dynamics. When, as is sometimes the case, a proportional plus integral 共PI兲 controller is used in such a loop, the control may develop a limit cycle. The limit cycle often goes undetected because of the slow dynamics and the long trend time, which means that a snapshot of the temperature record may look fine at any particular instant. This is a common problem in many slow temperature loops as well as in integrating processes. Such temperature loops behave more like an integrating
process than a self-regulating process. For such processes, a change of control output will cause a process variable to increase 共or decrease兲 continuously until it reaches a limit. One such example is the control of liquid level in a tank. Although proportional plus integral plus derivative 共PID兲 control has been used widely by industry for barrel temperature control, there have been many efforts to improve control performance by introducing more advanced control techniques. In Ref. 关1兴, efforts are devoted to developing an autotuning formula for barrel temperature control, while in Ref. 关2兴, autotuning is incorporated into a PID controller modified with a ‘‘cutback’’ feature. The cutback feature tends to reduce overshoot in the closed-loop response by establishing a temperature band about the set point such that the integral action of the PID control is activated within the band but is turned off when the temperature is
0019-0578/2002/$ - see front matter © 2002 ISA—The Instrumentation, Systems, and Automation Society.
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Tien L. Chia / ISA Transactions 41 (2002) 501–509
outside the band. In Ref. 关3兴, the use of an additional thermocouple sensor is proposed as a means of improving the control, with one sensor located inside the barrel and the other on the outside surface of the barrel. This is similar to cascade control of temperature in a continuous stirred tank reactor 关4兴 and provides the same benefits. However, this is done at the expense of additional instrumentation. Model predictive control 共MPC兲 关5,6兴 is a promising new technology that has been used in many process control applications to improve the control. The approach is based on the use of an internal model of the process to predict its response to input changes. MPC is particularly effective in dealing with processes characterized by large dead times, multivariable systems with dynamic interactions, and systems with inequality constraints 共such as saturation limits兲. In Ref. 关7兴, two different approaches of multivariable model predictive control are analyzed and compared with respect to constraint handling, tuning, and decoupling. An application of state space based control to the barrel temperature control problem is described in Ref. 关8兴. Barrel temperature control involves multiple interacting zones with common disturbances affecting adjacent zones, i.e., the heat loss from zone 2 to zone 3 is the same as the heat gain from zone 3 to zone 2. With the additional information, the constrained model-based adaptive control approach 关9,10兴 can be used for such applications to take advantage of available system information to improve control performance. However, due to the difficulties of field implementation, i.e., process knowledge required and high maintenance of an MPC based approach, this is not the ideal solution for barrel temperature control. Most users in industrial plants lack the time, special knowledge, and experience required for setting up and maintaining a complex control system. This paper proposes a control strategy which incorporates the essence of model predictive control as part of the strategy. The resulting solution inherits the benefits of MPC without its limitations. The resulting solution is robust, effective, and easy to use. Section 2 examines the control and stability of PI and PID control for an integrating-type process with interpretations relevant to barrel temperature control. Section 3 describes the barrel temperature
Fig. 1. Block diagram of control of an integrating process.
control problem—its characteristics and requirements. This is followed by a detailed discussion of the proposed control strategy. Section 4 discusses the results of such a control application in a real industrial environment. 2. Closed-loop control of an integrating process The block diagram of a typical process control loop is shown in Fig. 1, where G p( s ) denotes the transfer function of the process plus actuator, and G c( s ) denotes the controller. A reasonable approximation of a barrel zone temperature control process is an integrator plus dead time plus lag 共which may represent the actuator dynamics兲 with the transfer function given by Eq. 共1兲:
process:
G p共 s 兲 ⫽
K * e ⫺Ts , s 共 1⫹ s 兲
共1兲
where K is the process gain, T is the process dead time, and is the lag time constant. 2.1. Proportional plus integral controller Consider a proportional plus integral 共PI兲 controller applied to the above process:
PI controller:
G c共 s 兲 ⫽K p共 1⫹1/ is 兲 ,
共2兲
where K p is the proportional gain and i is the integral time constant of the controller. In the case of negligible dead time 关i.e., T in Eq. 共1兲 is zero兴, the open-loop transfer function for the system of Fig. 1 is
G OL共 s 兲 ⫽G c共 s 兲 G p共 s 兲 ⫽K OL*
共 1⫹ is 兲 , 共3兲 s 2 共 1⫹ s 兲
where K OL⫽KK p / i denotes the open-loop gain. This system has two poles at the origin—one from the process integrator, the other from the integral action of the controller; there is also a pole
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at ⫺1/, and a zero at ⫺1/ i , where may characterize the actuator time constant and i is the integral time constant. The root locus diagram of Fig. 2共a兲 shows that when i is set smaller than , the system is unstable irrespective of the controller gain. On the other hand, when i is set larger than , the system is stable for all values of controller gain. However, as shown in Fig. 2共b兲, the closed-loop response tends to be oscillatory for values of controller gain high enough to satisfy typical speed of response specifications. We note further that as the controller gain is increased, the response becomes more and more oscillatory. With PI control, there are only two degrees of freedom available 共i.e., choice of K c and i兲 for satisfying performance specifications. In addition, the range of values for K c consistent with allowable overshoot, results in a relatively slow response and limited disturbance rejection. If further realistic values of process dead times are considered, these limitations become increasingly critical. 2.2. Proportional plus integral plus derivative (PID) controller The addition of derivative action to the PI controller of Eq. 共2兲 results in the PID controller given by Eq. 共4兲:
PID controller:
G c共 s 兲 ⫽K p共 1⫹1/ is 兲 ⫻ 共 1⫹ Ds 兲 ,
共4兲
where D is the derivative time constant. This third tuning parameter provides an additional degree of freedom in satisfying control specifications. The open-loop transfer function of Fig. 1 with PID control becomes
G OL共 s 兲 ⫽G c共 s 兲 G p共 s 兲 ⫽K OL
共 1⫹ is 兲共 1⫹ Ds 兲 . s 2 共 1⫹ s 兲 共5兲
Thus PID control adds a second zero to the openloop transfer function which has the effect of pulling to the left the two branches of the root locus that originate at the origin. This is illustrated in Fig. 2共c兲 where, consistent with good design practice, the derivative time constant D is set to a value lower than the integral time constant i .
Fig. 2. 共a兲 Root locus for PI controlled system with i ⬍ . 共b兲 Root locus for PI controlled system with i⬎ . 共c兲 Root locus with PID control.
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Fig. 3. Injection molding machine diagram.
This has two desirable effects on the response of the closed-loop system as compared with PI control: 1. It tends to stabilize the system. 2. The response tends to be faster. Therefore derivative action is particularly desirable for an integrating-type process because it permits a higher controller gain with an acceptable degree of overshoot. The presence of dead time tends to destabilize the system resulting in more severe behavior in terms of overshoot and oscillation, as well as restricting further the range of proportional gain K p corresponding to a stable response. Temperature processes characterized by large thermal lags exhibit similar behavior to that of the integrating process given by Eq. 共1兲. This is because the slower the dynamics, the larger the time constant and hence the closer the process pole is to the origin.
3. Barrel temperature control 3.1. Performance specifications Barrel temperature control in an extrusion process or an injection molding machine 共Fig. 3兲 presents severe challenges to achieving satisfactory control. In injection molding, plastic in the form
of a granular solid is fed into the barrel, where it is melted so that it can be injected into the mold in a liquid state. The control of the temperature within the barrel is critical to the quality of the molded product. This is particularly true for some temperature-sensitive plastics which can tolerate only very small temperature deviations from the set point. Therefore it is important to have a very tight temperature control during production to ensure consistent product quality. Further, when the machine is started up from a cold state, it is desired that the temperature be brought to its set point 共usually several hundred degrees Fahrenheit兲 as rapidly as possible and without much overshoot. Since it may take from 15 minutes to a couple of hours, depending on the size of the barrel and the power rating of the barrel heater, to bring the barrel temperature to its set point, the startup period can have an important effect on productivity. The specification on overshoot is also often critical, as in the case of large, well-insulated barrels that take a long time to cool down. Summarizing, the performance specifications for barrel temperature control are: 1. Bring the barrel temperature up to set point as fast as the system allows without overshoot. 2. Maintain the barrel temperature within plus or minus one to two degrees Fahrenheit during the production run.
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3.2. Challenges in barrel temperature control Like many other slow processes, barrel temperature control can tolerate a very large proportional gain and, with proper choices of i and/or D , the system is always stable. However, the response may be oscillatory as indicated by the root locus diagrams of Figs. 2共b兲 and 共c兲. For processes with significant dead time, the allowable range of proportional gains is more limited, although the gain may still range over one to two orders of magnitude. However, since the dead time tends to make the period of closed-loop oscillation very large, poor temperature performance can often go undetected by the operator. Although stability itself is not a major concern, satisfying overall performance specifications for barrel temperature control is a challenge due to the nature of the system, as described in the next section. Several factors contribute to the complexities of barrel temperature control: 1. The process response is very slow, approaching that of an integrating type of system. 2. The dynamic response of the heating cycle is typically very different from that of the cooling cycle. 3. The barrel consists of several zones with significant interaction effects between neighboring zones. Further, the zones often exhibit different dynamics. 4. The process exhibits significant dead time ranging from a few seconds to several minutes. 5. For extrusion processes, the temperature control is further complicated by the shearing energy generated by the rotating shaft. This effect may be so great that a specified desired temperature profile may not even be attainable. 6. Rigorous requirements on the control system, including fast startup, no overshoot, and being able to maintain each zone within a couple of degrees of its respective set point during a production run. According to points 2 and 3 above, the control design problem is characteristically nonlinear and multivariable in nature. Due to the dynamic nature of the system and the analysis in Sec. 2, a conventional PI or PID controller will not work well for
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such an application. The problem is further complicated by the presence of significant dead time which, as already pointed out, leads to closed-loop instability 关4兴, and tends to limit the allowable range of controller gain. Hence simple PI or PID control, irrespective of the parameter settings, cannot generally satisfy all the stringent performance requirements, such as point 6 above. If an aggressive PID control is applied, the temperature tends to overshoot and be oscillatory. A less aggressive PID control may reduce the overshoot but at the expense of increasing the time to reach the set point, and in decreasing the ability of suppressing disturbance effects during the production run. The advanced control strategy developed in this paper uses a combination of control techniques to counter the adverse characteristics of the process. The control strategy incorporates autotuning, model prediction, and multiple PID’s applied to different barrel zones and different regimes of control action, as shown in Fig. 4. Autotuning is designed to generate optimal tuning parameters for such an integrating type of system; a form of model predictive control is incorporated which, while still predictive in nature, does not burden the user with the difficulties in setting up a conventional MPC. Multiple PID controllers are used to compensate for the system nonlinearities, and the controllers for each zone are coordinated to take into account the interaction effects of neighboring zones.
3.3. Advanced control strategy for barrel temperature control In order to meet the strict performance requirements, different control techniques are used at different stages. In this section, these control strategies are described in some detail.
3.3.1. Startup and autotuning Normally, the system is started cold, and it takes some time to heat the barrel to set point temperature. In order to bring the temperature up to the set point as fast as possible, the control output is initially set to its maximum value. Autotuning is performed at this time if requested by the operator.
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Fig. 4. Control strategy.
During the tuning stage, model parameters such as gain, time constant, and dead time, as in Eq. 共6兲, are identified and saved. Either a reaction curve technique 关4兴 or least-square method can be used to identify the process model. Two sets of PID tuning parameters are determined, startup PID and run-time PID 关as in Eqs. 共7兲 and 共8兲 below兴, based on the identified model parameters. Methods for calculating these tuning parameters can be found in many articles and textbooks 关5,6兴. The design procedure for selection of the parameters for Eqs. 共7兲 and 共8兲 are explained in Sec. 3.3.3.
Model: Startup PID Run-time PID:
ˆ e ⫺Tˆ s /s 共 ˆ s⫹1 兲 , K
共6兲
K p1 共 1⫹1/ i1 s 兲共 1⫹ D1 s 兲 , 共7兲 K p2 共 1⫹1/ i2 s 兲共 1⫹ D2s 兲 . 共8兲
The tuning procedure is enhanced by recognizing that different barrel zones behave differently; for example, since the nozzle zone has a higher heat dissipation rate than the other zones it behaves more like a lag process than an integral process, permitting a faster integral action by the PID controller. 3.3.2. Model predictive control After tuning is completed, a preset point 共PSP兲, given by Eq. 共9兲, is calculated using an internal model of the process response. This PSP is a temperature below the actual set point that is determined to minimize the overshoot during the startup phase. When the temperature reaches the preset point, power to the heater is turned off for a prespecified time called the wait time 共WT兲, given
by Eq. 共11兲. If the system is perfectly described by Eq. 共6兲 then, given the parameter values for the model, the current control output, and the current temperature, the predicted temperature at the end of the wait time period can be calculated. However, since the model is never perfect, the DSP of Eq. 共10兲 is designed conservatively so that the temperature at the end of the WT is always below the set point,
PS P⫽S P⫺ ␣ DS P,
共9兲
ˆ Tˆ ⫹100 Kˆ 共 Tˆ ⫹ ˆ 兲 , DS P⫽100K
共10兲
WT⫽2Tˆ ⫹ ˆ ,
共11兲
where 1⬍ ␣ ⬍2, 1⬍  ⬍2, and SP is the user specified set point. The factor 100 reflects the fact that the controller output is 100% at the autotuning phase. Both ␣ and  should initially be set to 1, however, they can be set to be greater than 1; ␣ is selected to accommodate modeling errors, and  is adjusted to compensate for different zone behaviors as described in Sec. 3.3.5. The preset point is designed to ensure that, with the control off during the duration of the wait time, the zone temperature at the end of the wait time period is close to but does not exceed the user-specified set point. Even though power to the heater is off, the temperature will continue to rise during this time due to the thermal inertia characterizing the process behavior. If the wait time is selected too short, the temperature will overshoot; if it is too long, the temperature will take a longer time to reach the set point.
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3.3.3. Two stage robust control strategy At the end of the wait time period, automatic control using the PID controller takes over. During the autotuning stage, two sets of PID parameters are generated as in Eqs. 共7兲 and 共8兲. The startup PID of Eq. 共7兲 is a proportional control dominated action designed to bring the temperature up fast; this is used when the temperature is far from the set point at the end of the wait time period. If the model identified in Sec. 3.3.1 is perfect, then WT and PSP can be determined so that, at the end of the WT, the temperature will reach the set point and there is no need for the startup PID phase. However, since a perfect model is highly unlikely, there is a need for a startup PID phase. The purpose of this phase is to account for modeling errors and the intentionally inflated DSP in Sec. 3.3.2. The addition of the startup phase increases the robustness of the control strategy. If the model is accurate; this phase will be bypassed; if the temperature is far from the SP at the end of the WT, the startup PID will bring the temperature close to the set point without overshoot. Once the temperature is close to the set point, the system will switch to a more aggressive set of controller settings 关runtime PID as in Eq. 共8兲兴 designed to meet the stringent temperature control requirements during the production run. This run-time PID has sufficient derivative action to counter the integrating type of response of the barrel temperature as per the analysis in Sec. 2.2. This controller is too aggressive to use in the startup phase; hence the change from startup PID to run-time PID parameters happens only after the temperature has reached the neighborhood of its set point value 共typically 5–10°兲. This enables the control system to bring the temperature up fast, while still minimizing overshoot and providing adequate run-time control. The system may switch between the two controllers depending on how much the temperature deviates from the set point 共as caused by disturbances or set point changes兲. 3.3.4. Speed reconciliation Another complexity of this control problem is due to the differences in physical size of the zones. The response speed of the zone’s temperature varies dramatically with the size of the zone. Thus, because the nozzle zone is usually much smaller than the other zones, it may reach its set point within several minutes, compared to the one to
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two hours required in some very large middle zones. Since the production cannot start until all zones reach their respective set points, the faster temperature rise in the smaller zones results in a waste of energy and causes disturbances to adjacent zones. Thus it is desirable to reconcile the speeds of the different zones so that they each reach their respective set points in a comparable time frame. This is done by applying different control outputs for the different zones to ensure that they all reach their set points at about the same time. In other words, for smaller and faster zones, less than the maximum control outputs are used in the startup stage, as discussed in Sec. 3.3.1. 3.3.5. Fine tuning Further enhancement of performance can be achieved by fine tuning the system. Zone dynamics may be affected by their physical location; for example, some zones have only one adjacent zone, while others have two. In general, a zone having only one adjacent zone 共e.g., one side is exposed to the ambient兲 will have faster heat dissipation than a zone with both ends thermally connected to other zones. Therefore these two types of zones should be treated differently when tuning the control parameters. Also, how closely a zone behaves like an integrating system depends very much on the mass or thermal capacity of the zone. Minor adjustments according to the above variations enable the user to fine tune the system to his or her preference. 4. Field test results and conclusions The control strategy that employs a combination of several of the control techniques described above is proven to be very effective for this type of process. Unlike most advanced model-based control solutions which require the user to input extensive information concerning the process dynamics, this system requires no such inputs and is very easy to use. The control strategy can be embedded within any digitally implemented barrel temperature control system. The only information required of the user in setting up the system are the typical PID setup parameters, e.g., PID type, sampling time, etc. The above solution has been applied in many installations. Some of the field test data are pre-
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Fig. 5. Autotuning with heat/cool cycle and first-time startup.
sented here. Fig. 5 shows the startup behavior of a heat/cool extrusion system on the first startup 共i.e., prior to any tuning兲. Autotuning is done during this startup stage with the results stored in memory. Fig. 6 shows a typical startup of a fourzone injection molding machine, after autotuning. As shown in Figs. 5 and 6, the system reaches the set point as fast as the system allows 共in about 20 min for a set point of 350 °F兲 without any overshoot, and the temperature is maintained within a one to two degree band about the set point during the production run. The above control scheme provides a good solution for a heating-only type of barrel temperature control. If both heating and
cooling are involved, the control problem is more complicated due to the very different dynamics between the heating and cooling cycles. The above solution is very effective and also very robust. The robustness comes from the fact that PID-based controls are used and, unlike most model predictive control schemes, the control system does not require any a priori model information to set up. Performance can be further enhanced by inserting a user-adjustable aggressiveness factor for the model prediction scheme which enables the user to adjust the control to accommodate variations in
Fig. 6. Cold startup for a four-zone injection molding machine.
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the system that were not captured in the autotuning and modeling stages. Acknowledgment The author would like to thank Dr. Irving Lefkowitz for his valuable inputs and comments on this paper.
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关8兴 Bulgrin, T. C. and Richards, T. H., Van Dorn Demag Corp., The Application of Advanced Control Theory to Enhance Molding Machine Performance. Injection Molding Expo, Akron, Ohio, 1994. 关9兴 Chia, T., Chow, P., and Chizeck, H., Recursive parameter identification of constrained systems. IEEE Trans. Biomed. Eng. 38, 429– 442 共1991兲. 关10兴 Timmons, W., Chizeck, H., Casas, F., Chankong, V., and Katona, P., Parameter-constrained adaptive control. Ind. Eng. Chem. Res. 36, 4894 – 4905 共1997兲.
References 关1兴 Searler, J. G., Delaney, B. P., Delaney, K. O., Hogan, K. M., and Schmidt, P. B., Injection Molding Machine Temperature Control System. United States Patent 1995. Patent Number 5,397,515. 关2兴 Installation and Operation Manual for Models 808 and 847 Digital Controllers-Eurotherm Corporation, 1987. 关3兴 User Manual for Model 1771-TDC Module, Rockwell Automation and David Standard, 1995. 关4兴 Stephanopoulos, G., Chemical Process Control. Prentice-Hall, Englewood Cliffs, NJ, 1984. 关5兴 Morari, M. and Zafiriou, E., Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ, 1989. 关6兴 Brosilow, C. B. and Joseph, B., Techniques of ModelBased Control. Prentice-Hall, Upper Saddle River, NJ, 2002. 关7兴 Chia, T. L. and Lefkowitz, I., Multivariable Control Technologies. ControlSoft Publication #PA401.
Tien-Li Chia received a B.S. degree from Tan-Kung University, Taiwan, in 1977 and M.S. and Ph.D. degrees in systems and control engineering from Case Western Reserve University, Cleveland, Ohio, in 1982 and 1985, respectively. Dr. Chia is the president of ControlSoft, Inc., and adjunct faculty at EE&CE Department, Cleveland State University, Cleveland, Ohio. He is experienced in the design and development of advanced control based solutions, including adaptive control, expert-based solutions, and multivariable control, for various industrial and manufacturing processes. At ControlSoft, Dr. Chia has designed and directed the development of many control software solutions that are now widely used in industry.