APPLIED INDUSTRIAL CONTROL SOLUTIONS
ApICS LLC.
Accumulator Tower Tension Regulation
and
System Identification Techniques
B. T. BOULTER, H. W. FOX
© ApICS ® LLC 2000
ABSTRACT
Reports of tension and current major loop instabilities observed during the commissioning of a Hot Dip Coating Line (HDCL) at an integrated steel producing facility necessitated an investigation of the outer loop regulators in the entry and delivery tower areas. Experience had shown that instabilities were evident with certain products, certain tower operating conditions and certain regulator configurations. To gain a better understanding of the controlled system two methods of modeling the product and tower are implemented. The first, a more traditional method, employs a free-body diagram of the tower mechanics and the corresponding s-domain equations. The second, utilizing well established system identification techniques, uses empirical data obtained on site. Several models of the product and tower are obtained, and a final model used for analysis is derived.
Tension and current loop regulator designs, used during commissioning, were analyzed using the final model of the product and tower. From these analyses, conclusions explaining tension instabilities during commissioning are presented along with suggestions for improving the existing system's response.
INTRODUCTION
Figure 1 shows a simplified one line diagram of the entry tower area of the Hot Dip Coating Line at the facility. The diagram is drawn facing the machine from the operator's side of the line. The entry section is to the right of the entry tower and the process section is to the left. When the entry section speed matches that of the process section speed the tower carriage is down and the entry tower is full. During coil changes the entry tower empties, allowing the process section to continue uninterrupted.
Figure 1. Entry Tower Section
The entry tower tension zone is the tension zone between bridle #1 and bridle #2. Bridle #1 is the lead entry section and determines the speed at which new strip enters the entry tower tension zone. When the entry and process section speeds are equal, entry tower carriage position is maintained by a proportional only position loop closed around bridle #1. Two resolvers provide tower carriage position feedback to the tower position regulator. During an entry section coil change, the proportional only position regulator is disabled and the entry section follows a ramped speed reference to a stop.
The entry tower carriage motor is rated for stall torque operation and provides strip tension in the entry tower tension zone. Motor torque is transferred to tension via two sprockets and four chains connected to the tower carriage in a pull-up or pull-down arrangement (ref. Figure 1). A system of cable connected counter weights eliminates the requirement for pull-up operation under normal operation and reduces the required motor torque. Chain pull-down force is transmitted through four stacks of Schnorr disk springs to the tower carriage. The ends of the chains which are connected to the top side of the carriage, are not under any significant tension except for the tension provided by the jockey sprockets. The entry tower carriage drive has two modes of operation while the line is running, they are current and tension. Current mode operation provides strip tension by regulating armature current. Friction and inertia compensation help maintain constant strip tension during entry and/or process section acceleration or de-acceleration. A separate motor field controller maintains constant shunt field strength by regulating field current. The tension mode uses a tensiometer located on tower roll #12 as feedback to a tension regulator.
The rate at which strip exits the entry tower is ultimately determined by the temper mill, which is the lead process section. Seven tension zones employing 56 separate motors and static drives transport the strip from the entry tower to the temper mill. The strip is heated and coated in this region, changing the physical and metallurgical properties of the strip. In order to keep the complexity of the system model reasonable, only bridle #2 was included in the derivation of the analytical stall condition system model.
Chapter 1 describes an analytical model of the HDCL entry tower and bridle #2 at the facility. The entry tower is first represented with a free body diagram. Next, equations in the s-domain are written and solved for each mass position. Transfer functions describing the tower carriage motor torque to strip tension and the tower carriage motor torque to motor speed are derived. Bode plots were generated for both analytical transfer functions. A listing of all system parameters used in the analytical models can be found in Appendix A.
Chapter 2 describes empirically derived models of the HDCL entry tower and delivery tower at the facility. These models were obtained using stimulus/response ARX (Auto-Regressive) system identification techniques. The software used to generate the forcing function and to sample and save the data are described. Plots of the forcing function and sampled data are included along with the resulting system identification bode plots.
In Chapter 3 the difference between the analytical and empirical models is discussed and a final mechanical plant model useful in the design and analysis of the tower tension and current regulators is proposed.
In Chapter 4, the entry and delivery tower tension and current regulators used during commissioning of the HDCL at the facility are integrated with the final mechanical plant model obtained in chapter 3. Block diagrams depict the different regulator configurations. Open loop transfer functions are derived and bode plots are plotted using the tuning that existed on various dates during commissioning,
In Chapter 5 a summary of observations made during the analysis in Chapters 1 through 4 is presented. Conclusions are draw from these observations and finally some recomendations are made concerning improvement of regulator performance and system stability.
Appendices are included that contain mechanical and regulator parameters; derivations of equations describing tension variation during acceleration; equations of speed to tension transfer function for strip; analytical and empirical poles and zeroes; chart recordings of empirical data used in this report. Finally a Bibliography with usefull references is included.
OBSERVATIONS AND CONCLUSIONS
Observations
1) The analytical results and empirical data for the entry tower clearly show a resonance of 2-3 [rad/sec]. This resonance was also observed in the empirical data for the delivery tower, but with more damping.
2) The entry tower was observed to go unstable at the same frequency. This occurred (during the data collection) when the #1 bridle was stopped and the tower began emptying. This can be verified in Figure 10.
3) 2-3 [rad/sec] resonance was considerably more damped when the line was running. This can be verified in Figure 9 (stall) and Figure 12,13 and 14 (running).
4) The chains that coupled the motor sprocket to the carriage springs were noticed to sway at approximately 2-3 [rad/sec] when the instability was observed. Also when this occurred the resulting ARX system model was poor. This indicated that the resonance in the tension behaved as if it was the result of an external noise source (such as the chains swaying and pulling on the carriage). This led to a poor correlation between the input stimulus and the output response.
5) During line run conditions many resonances of varying frequency and damping were observed between 20-100 [rad/sec]. These dynamics are similar to those present in the analytical model that relate to bridle #2 dynamics. They may be attributed to the dynamics of process line tension zones and speed regulators adjacent to the entry tower.
6) Between 12/4/92 and 1/12/93, tension loop gain of #2 bridle was decreased by a factor of 3. Bridle #3 was also detuned. With this softened #2 and #3 bridle tension control we did not observe any significant change in system behavior with strip gauges below .048" that were run while we were taking data. It was noted in other reports, that were made when the #2 and #3 bridle tension control was stiffer, that the system was very sensitive to strip gauge over .050" (ref. memo 12/8/92 by D. J. Unite)
7) Observations from the Regulation Bode plots.
| |
Regulation
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w co
|
Phase Margin
|
Stability
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Type
|
|
A
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Entry Tower Ten. Maj. w. Spd. Int.
|
2.53
|
53.8
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good
|
2
|
|
B
|
Entry Tower Ten. Maj. w. Curr. Int.
|
0.25
|
64
|
fair
|
1
|
|
C
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Entry Tower Curr. Maj. w. Spd. Int.
|
0.73
|
34
|
poor
|
2
|
|
D
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Delivery Tower Ten w. Spd. Int.
|
3.0
|
64
|
good
|
2
|
|
E
|
Delivery Tower Curr. Maj. w. Spd. Int.
|
2.3
|
71
|
fair
|
2
|
|
F
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Entry Tower Curr. Maj. w/o Spd. Int.
|
32
|
130
|
excellent
|
1
|
A & D) This can be seen to be type 2 from the denominator of equation 4.5. Two s's can be factored out of DM , DI, and NS .
B) This can be seen to be type 1 from the denominator of equation 4.9. One s can be factored out of DM .
C & E) This can be seen to be type 2 from the denominator of equation 4.13. Two s's can be factored out of DM , DI, and NS .
Another interesting observation is that the open-loop transfer functions of the Current Major Loop with Speed Intermediate control configurations had the plant denominator polynomial in the numerator. This led to the 2-3 [rad/sec] pole pair appearing as an under-damped lead rather than a lag in the open-loop transfer function.
Conclusions
Very often, the first concern, when modeling the mechanics of a process line tension regulator, is the resonance that will be present due to the spring-like nature of the strip material. In Chapter 1 the analytical model contained a strip resonant frequency equal to approximately 50 [rad/sec] with the tower nearly full and a strip cross sectional area of 0.84 [in2]. Although this resonant frequency could be as low as 24 [rad/sec] with the lightest strip, it was clear that dynamics other than the strip resonance were lower in frequency and would therefore be more dominant.
The analysis shows clearly that the resonance at 2-3 [rad/sec] is a problem. The natural frequency of the tower chains had also been observed to be approximately the same frequency. Accumulator tower chains have been shown to have a natural frequency of approximately 1.8-2.2 [rad/sec] ([6] pp.4) at other sites. The inability of the regulation schemes to control this resonance, which occurs close to the crossover of the tension loops, may be explained by the chains swaying 180 degrees out of phase with the response of the regulator to the tension transients caused by the swaying of the chains. Under this assumption, the energy supplied by the regulator to suppress the resonance would be fed into the swaying of the chains. Had this non-linear aspect of the system been included in the analytical model it may have been more readily visible during the analysis.
An important phenomena that is somewhat hard to explain, is the change in the low frequency under-damped resonance to that of an over-damped response when changing from the stalled condition of Figure 9 to the running condition of Figure's 12, 13 and 14 (this can be seen in the various empirical model pole positions in Appendix E). From the derivation shown in Appendix D, we see that the effect of bridle #1 introducing strip into the tower tension zone changes the transfer function of carriage speed to strip tension from a pure integration to a first order lag (ref. D.5). However, even with bridle #1 at full speed the transfer function around the frequency band of interest is largely unaffected. We believe that by releasing the brake on bridle #1 and the brakes on sections downstream of bridle #2, a "softening" of the system is introduced, that both reduces the low frequency resonance and provides damping. This is analogous to allowing movement of the ground plane shown in the top of Figure 2. The concept of a soft lead section providing damping is not a new one (ref. [5] p.2) and could explain a number of observations made during commissioning. Stopping bridle #1, as is the case when the entry tower is raising, will stiffen the system. Heavier strip and/or more responsive adjacent section regulation will also stiffen the system. De-tuning adjacent tension zones will "soften" the system and, as observed, reduce the sensitivity of the system to variations in strip gauges. Any stiffening of the system, as observed empirically, will tend to peak the 2-3 [rad/sec]resonance.
If the mechanical system and all adjacent sections are relatively stiff, the only spring that must be considered is the strip. With a material such as steel, this spring constant is extremely high and the gain in the transfer function from torque to speed is negligible at and below the tension crossover frequency. This would allow a simplification where-in this block may be eliminated entirely. This is equivalent to opening the speed loop. Further, if the line speed is small compared to the length of material in the tension zone, the system model in a stall condition is a good approximation of the line in a run condition (see Appendix D). This is the approach taken during the original design of the tower tension regulator. However, since the tower has a low frequency resonance in the 2-3 [rad/sec] area, this simplification cannot be made.
Analysis of the Tension Major Loop with Speed Intermediate control showed that the plant poles are canceled out of the open loop transfer function. This is true under the assumption that the speed and tension plant transfer function denominators are similar to each other (see equation 4.5). This implies that the speed intermediate loop provides very good damping of the plant resonances in the open loop transfer function.
The Tension Major Loop with Current Intermediate control was found to be far more sensitive to changes in the damping of the plants resonances. It is a type 1 system, the other tension/current major loop control configurations were found to be type 2.
The Current Major Loop with Speed Intermediate control Bode plot did not look at all as expected. The 2-3 [rad/sec] plant resonance, as observed above, places an unexpected 2nd order under-damped lead at a critical frequency of the open loop transfer function. In this configuration a lowering of the proportional gain or the lead frequency of the current major loop regulator will induce instability rather than suppress it. This would make the regulator hard to tune if the on-site engineer assumed that lowering the gain and the Wldc values would reduce instability. (see Figures 34 and 37). This may explain why, during commissioning, the Current Major Loop was uncharacteristically found to be marginally stable.
The Current Major Loop without Speed Intermediate control does not have any plant dynamics in the open loop transfer function. This would explain why it was found to be more stable and easier to tune than the previous configuration. One important drawback to the use of this configuration is the loss of speed limiting during a strip break. A comparison of carriage speed to entry and process strip speed provides strip break detection, but is subject to nuisance trips when the line is stopped and the tower tension is turned on.
Analysis of the empirical data from the entry and delivery towers indicated that in stall the transfer functions were similar. Further analysis indicated that the tension major with speed intermediate loop control configuration was a stable choice on both towers. The only significant difference found between the two towers from the point of view of the control loops was the current major loop with speed intermediate loop tuning. By using the instrument variable system identification algorithm it was found that the torque to speed and torque to tension plant poles during line run were very similar for both towers (this was not as apparent with the ARX method).
RECOMMENDATIONS
A) The entry tower tension regulator should be returned to the tension control configuration and tuning as of 9-21-92.
REASONS
1) The Bode plot for this configuration was seen to be the most stable.
2) The speed intermediate loop was shown to provide cancellation of the plant poles in the open loop transfer function . The 2-3 [rad/sec] plant resonance was effectively canceled. This is equivalent to saying that it provides damping of the plant resonances.
3) Speed limiting is provided during a strip break or inadvertent tail out.
B) All towers should be configured without a cascaded speed intermediate loop in the current mode.
REASONS
1) A speed intermediate loop is very useful when it provides speed regulation that reduces disturbances introduced into its governing major loop by process line accelerations. It achieves this by quickly reacting to speed reference changes. However, as seen in the speed intermediate loop analysis (ref. Figure 31, where the open loop response of the intermediate speed loop is shown. This would be common to both major loop configurations.) the speed loop gain is low enough that it does not provide any significant speed control with a taut strip.
2) The pole zero cancellation provided by the speed intermediate loop in the tension mode is not a benefit in the current mode since the plant poles do not appear in the open current loop transfer function.
3) A case could be made for a speed intermediate loop based on the speed limiting it provides during a strip break. However, the strip break detection, already implemented for the entry tower has been shown to serve as protection against a strip break.
C) The ramped speed reference for the entry, process and delivery sections should be adequately softened with an s-curve block, as installed.
REASONS
1) While the frequency content of a pure ramp is less than that of a step, it is still rich enough in harmonic content to excite the chains connecting the carriage to the motor sprocket. By ramping the speed with a soft s-curve the harmonic content is greatly reduced as is the inclination of the change in speed to excite the chains. This is especially true when the entry tower begins emptying.
2) With softening of the reference speed ramp, the error between the actual entry, process, and delivery section speeds and their references is reduced. As a result of this, feedforward signals to the accumulator towers' speed intermediate loops would more accurately reflect the actual speed. This results in reducing the output required of the tension regulator. Certainly any reduction in tension regulator output corresponds to a reduction in tension regulator input, or in other words tension error.
D) Employ a fixture that will inhibit the lateral motion of the chains connecting the carriage to the motor sprockets on all accumulator towers.
REASONS
1) As noted in Chapter 2 pp. 32, and the observations and conclusions above, the natural resonance of the mass-spring pair consisting of the carriage connection spring and the motor/gearbox (2-3 [rad/sec]) was the same as the frequency the chains were observed to sway at when the entry tower broke into a severe tension oscillation. Inhibiting the lateral motion of the chains would prevent excitation of this uncontrollable phenomena. A more thorough mechanical analysis of the chain configuration may shed more light on this phenomena.
E) Confirm that the Schnorr disk springs between the chains and the carriage do not become active during any normal operating conditions.
REASONS
1) The 2-3 [rad/sec] resonant frequency will increase in proportion to the square root of the disk spring constant. Increasing the resonant frequency from approximately 2-3 [rad/sec] up to 4-6 [rad/sec] will move the resonance away from the tension loop crossover.
2) This higher resonant frequency may reduce the tendency of the chains to sway (needs further analysis).
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