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Iterative feedback tuning of controllers in cold rolling mills
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
14th World Congress ofIFAC
1-3b-08-5
Copyright (~ 1999 IFAC . 14th Triennial World Congress~ Beijing, P.R. Chlna
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD ROLLING MILLS· Hakan Hja1rnarsson'* Mario T. CalDeron **
* Dept. of Signals 7 Sensors and Systems;, The Royal Institute of Technology, 8-100 44 Stockholm, Sweden. hakan.hjalmarsson~s3wkth.se
** ABB Industrietechnik GmbH, D-68167 Mannheim, Germany. mario.cameron0deind.mail.abb.de Abstract: Preliminary results on the potential of Iterative Feedback Tuning (1FT) for tuning controllers in cold rolling mills are presented. The tuning of the strip thickness controller in a simulated one-stand cold rolling mill is considered. Preliminary simulation studies show that for the used model, 1FT handles deterministic disturbances in material hardness and roll gap friction very well. The performance of the algorithm for deterministic variations in entry thickness is some\vhat worse but still a significant increase in performance can be obtained. Copyright ©1999 IFAC Keywords: PI controller, non...linear models, non-stationary systems, metals
1. INTRODUCTION The development of drive and automation solutions for cold rolling mills is subject to a very high innovation pressure because of the increasing demands on both: the quality of cold rolled products and the operational performance of the mills. A permanent improvement in the control of these plants shall have to contribute to the fulfillment of these demands. Strip thickness and flatness play a fundamental role among the several variables that deterlnine the quality of cold roIled products. Indeed, strip thickness and flatness are the only that can be directly influenced by control. Tighter tolerances in these variables can, hov.rever, only be reached, if at the same time the strip tension is controlled4 One of the most important practical problems to deal with here is the adaptation of the controllers for all possible rolling conditionS4 These conditions are given by the combination of all possible thermal and mechanical states of the rolling mill, all classes of materials to be used and products to be rolled, as well as all possible rolling speeds. Another important practical problem to solve is the final controller tuning based on the real bellavior of the mill. In the last 30 years, however, most of the works published has only focused on the multivariable character of the plant. Many authors have worked on the design of optimal multivariable controllers for strip thickness and tension in multi-stand mills: Hcess (1969) was the first in designing a LQG-based controller~ Geddes and Postlethwaite (1993) where the first in designing an Hoc based controlIer4 The adaptation of all designs to the changes inter and intra rolling passes, and the final tuning based on the real
behavior of the plant, however, did not receive the necessary attention in any of these papers in particular, and in the control community in general. Some authors have been conscious of the adaptation problem~ but even in that case they dedicated only a short remark to a possible solution: gain schedule. This, however, simply understood as the result of several designs - one for each of all possible rolling conditions. Considering the huge number of rolling conditions, it is easy to see that this approach is not tractable, even for classical designs with relatively much fewer parameters to adapt than for the modern designs. This is one important reason why most of the designs published in the last 30 years for strip thickness, tension, and flatness contr·ol in cold rolling mills have not been implemented in practice. Valuable exceptions to this trend represent the works of Hoshino et al. (1988) and Estival and Huguel (1993). To get the adaptation problem tractable, a good controller design in rolling m.ills has to ensure that the controller parameters can be calculated analytically from the design model. T'his model is obtained by reducing the ideal plant model, \vhich of course, contains uncertainties, even though cold rolling mills have been extensively modeled. Hence, even if the controller design is model based, it is desirable to do a final tuning based on the real behavior of the rolling mill. Additionally, it can be shown that in this case with the implementation of a gain schedule to solve the adaptation problem one changes the plant dynamics at the same time. Hence, even under ideal conditions the controller performance obtained during design cannot be achieved when implemented, see e.g.
4670
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
(Moreno~ 1994). In this paper the parameters of the strip thickness controller of a simulated cold rolling mill win be tuned using Iterative Feedback Tuning (1FT).
1FT is a model free tuning method, which uses experimental data to iteratively search for the optimal controller settings. 1FT was originally developed for linear time-in"'ariant systems (Hjalmarsson et al., 1994) but has been shown to work for slowly varying periodic linear timevarying systems as well (Hjalmarsson, 1995)~ Recently, some interesting and encouraging results have been obtained for cert.ain non-linear systems. In (Hjalmarsson et al., 1998) and (Hjalmarsson et al., 1997) some real industrial applications from the chemical process industry are reported. In (Hjalmarsson, 1998a) a controller for a DC-servD with backlash ~"as successfully tuned using 1FT under laboratory conditions~ Here an initial analysis for the nonlinear case ,vas given which provides some intuition as to why 1FT works on npn-linear sY8tems~ In (Hjalmarsson, 1998b) the simulation results of a non-linear system is shown where 1FT outperforms system identification and model based controller design~ These encouraging results makes it interesting to apply 1FT for the final tuning of controllers in cold rolling mills. Other interesting papers related to 1FT where tuning of controllers for simulated non-linear systems is considered are (De Bruyne et al.) 1997) and (Sjoberg and Agarwal, 1996). Another interesting model-free tuning method is presented in (Spall and Cristion, 1998)~ The outline is as follo\vs. In Section 2 the simulation model and the experimental conditions are briefly described. Section 3 introduces 1FT for the IvlIMO case. Section 4 presents simulation results on the final tuning of the strip thickness controller in a single stand cold rolling mill. The paper finishes with the conclusions in Section 5.
2. SIMULATION MODEL AND EXPERLVIENTAL CONDITIONS As already noted, in this study a single-stand cold rolling mill is considered. The ideal mill configuration is shown in Figure 1. The simulation model can be illustrated in block diagram form as in Figure 2. For simplicity the transversal effects across the lllill and strip are neglected and reduced nonlinear models for the material deformation in the roll gap, mill stand t strip tension and transport, coiling and llllcoiling process, as well as for the positioning and drive systems are considered. The roll gap variables are determined using the lllodel presented in (Cameron, 1996). To solve the multivariable algebraic loops during simulat.ion the direct approach presented in (Camerou, 1998) is used. The simulated mill describes then, the fundamental non-linear and non-stationary behavior of an ideal mill during the rolling of a coil, i.e. during the acceleration phases, constant speed phases and deceleration phases. Figure 3 shows the variaton of the rolling speed and coil radius at uncoiler and coiler during an ideal rolling pass: the rolling speed accelerates at the beginning, is then keep constant and finally decelerated. These speed variations and the resulting strip mass transfer from uncoiler to coiler give rise to the non-stationary behavior of the plant.
Min slanJ
T-r"
Fig.
00f
Fig. 1. Idealized configuration of a single stand cold rolling ruill: (HC) thickness control, (TC) tension control, (Se) speed control, (PC) position control.
2~ Block diagram of the simulation model for a single stand cold rolling mill.
Fig. 3. Ideal rolling pass. \lariation of rolling speed (top) and coil radius at uncoiler and uncoiler (bottom). For purposes of the study it will be assumed that during the 1FT apart from speed changes the other rolling conditions remain const.ant. T'his 4671
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R ...
Disturbance
Wavebnn
Strip thickness (welding point) Strip thickne~s (begin and cnd) Strip thickness and tension
_ _ _----'1--
00 -
0-
aAer prcl>Cli~ strip bacdness
Roll gap frichon
Table 1. Typical waveforms for some disturbances in cold rolling mills.
assumption will be released in Section 4. During tuning, measurable and non-measurable input disturbances will take place Table 1 shows that in practice many exogenous disturbances can be modeled as deterministic signals. For the sake of simplicity only disturbances in the thickness and hardness of the incoming material, as well as in the roll gap friction are considered. These disturbances are modeled as square waves. A
For the tuning experiments with 1FT the strip thickness and tension in the simulated singlestand cold rolling mill are controlled as in practice. To control these variables, very complex control configurations exist. One can show that most of tllese configurations are a subset of the general configuration presented in Figure 4. To achieve tighter tolerances not only output feedback control elements, but also presetting, decQupling and feedforward control elements are used. This control configuration work as follows: if the reference for the rolling speed rw remains constant and vlhether disturbances (v-m, v n ) nor noise affect the system, an ideal presetting control element should generate the initial value for the control signal u, that achieves the desired thickness tolerance. If changes in the reference for the rolling speed r w take place, an ideal disturbance control element should generate the additional control signal u to eliminate possible effects, at least asymptoticaly. The same function has the fcedfoTward control element for changes in the measurable disturbances V m ~ It is, however, clear that in practice all these control elements can be only designed based on the design model. ~foreover non-measurable disturbances Vn, as well as noise (n y and nd) affect both: the plant and the controller. The practical necessity of an output feedback is then clear~ The output feedback can \\Tock based alone 011 a timedelayed measurement of the exit strip thickness, or combined with a dead-time free estimation of
it. For the strip tension control (TC) only the feedforward and decoupling elements are active. For the strip thickness control (HC) all of them. The control signal of this controller is the reference for the position control (PC). For simplicity only the case with the dead-time free estimation of the exit thickness through mass flow is considered
14th World Congress of IFAC
here. This is the reason for the strip speed sensors shown in Fig. 1. To solve the ada.ptation problem each of the controller elements has in practice only few parameters to be set. In particular for the output feedback a tuning element in series with the base control element of I-type is used. The tuning element has the form
C(p) == Po
+ Pt
q-l
Here po and Pl are the parameters to be tuned and q-l is the discrete time backward shift operator q-ls(t) = s(t - 1). The resulting output feedback element is then of PI-type. The sampling time is 0.1 s. For the base control element base tuning based on a very simple design model, that gives reasonable performance for all rolling conditions is considered~ For the gain schedule the measurements of the rolling speed and the roll force are used.
I--~ y .... I
- - - - - - - - - - -.....
~-------------"
Fig. 4. General configuration of the strip thickness and tension controller in a cold rolling mill: (tb) output feedback, (ps) presetting, (ff) feed-forward, (de) decoupling.
3. ITERATI\lE FEEDBACK T'UNING In this section, the 1FT algorithm for tuning the parameters in a single input/single output (8I80) controller block is briefly reviewed. The origina.l derivation for 8180 systems can be found in (Hjalmarsson et al., 1994). The MIMO case is treated in full in (Hjalmarsson and Birkeland, 1998), see also (Hjalmarsson and Braslavsky, 1999) in these proceedings. "Ve consider an unknov.."n true system described
by
the discrere
t~t
)T: :(~
)Odel (1)
where t represents the discrete time instants, G is the (generalized) plant represented by a transfer function matrix, rt E Rnr represents external signals such as set-points, reference signals etc, 'Vt E R n v represents unmeasurable signals such as (process) disturbances and noise, Wt E Rn w represents the sensed outputs and Ut ERn", represents the control signals. Furthermore Yt E Rn y represents the variables that "Till be included in the control criterion (e.g. measured outputs 4672
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
14th World Congress of IFAC
o
0---..1
v
u:J Fig. 5. Feedback system vlhere each channel in the controller is independently parameterized. The generalized plant Gj is independent of the parameter vector p and control signals). These signals can be filtered (frequency weighted) versions of measured signals in the systemw 'Ye shall assllme, that {Vt} is a zero mean weakly stationary (see e.g. (Ljung, 1987» random process, but other disturbance assumptions can also be made. The system may be controlled by a multivariable controller and even though it is possible to tune all elements in this controller using 1FT, ""ill, for simplicity, consider the tuning of one single SISO controller block
we
Uj
=
Gj(p)Wj
parameterized by a parameter vector p ERn". Using this parametrization the system can be rearranged such that the relation between the controller block Cj and the output y is given by
Fig. 6. Block diagram of second experiment.
One "\1.I"ay of finding a (local) minim.um of J(p) is to use a descent search such as 8J . "+1' p~ = pt -l'iRi 1 Bp (pT.). (6) Here Ri is some appropriate positive definite matrix~ typica.lly a Gauss-Nelvton approximation of the Hessian of J, Vt-.-hile 'Yi is a positive real scalar that determines the step size~ To carry out the minimization it is necessary to have an expression for the gra.dient of this criterion with respect to the controller parameters. The novel contribution of the 1FT approach (Hjalmarsson et al., 1994) was to show that, in contrast to previous approaches an unbiased estimate of the gradient can be obtained entirely from input-output data collected on the actual closed loop system, by performing one special experirnent on that system. It can be shown that by the following procedure an unbiased estimate of the gradient can be obtained: i) Perform a closed loop experiment of length N "'''here the desired reference signal r is used Zj = 0 and with the parameter vector p in the controller set to the value for which the derivative is to be computed, see Figure 5. Collect measurements of y and Wj during this experiment. ii) Perform a second experiment where the reference r is set to zero, Zj = Wj and collect the outputs y, which we denote by y2, from this experiment, see Figure 6. iii) Filter each of the measured outputs from the second experiment, i.e. the elements of y2 t through the vector-valued transfer function consisting of 8Cj /8p. Collect all these signals j
(2) Uj == CjWj (3) where the generalized plant Gj is obtained from G and all controller blocks except C j .
\Vhenever signals are obtained from the closed loop system with the controller C J (p) operating, we ~"ill indicate this by llsing the p-argument; on the other hand, to ease the notation v.,..e will from now on omit the time argument of the signals~ Thus, y(p) will denote the signals in the control criterioll when the controller C j (p) is used~ In the 1FT design scheme the following quadratic control criterion is adopted:
J(p)
= 2~E
[t
[Yt(p)f WlYt(P)]
F.i -_
(4)
1\-"here E[~] denotes expectation "v.r.t. the weakly stationary disturbance v. The weight matrix lVtY, which for simplicity will be set to the identity, can be used to minimize the settling time at setpoint changes: see Hjalmarsson et al. (1998). This criterion include many commonly used control criteria. In particular it can, with proper definition of y, handle model reference following.
The optimal controller parameter p* is defined by p* = argminJ(p), (5) p
in a n y x n p dimensional matrix
Bp
~ [8ih] T -
-1 L--
N
t=l
Bp
~ and form
Yt (P)
(7)
By replacing 8Jlap in (6) an algorithTIl is obtained which under suitable choice of the step-size /1i converges to a stationary point of the criterion (4) if the signals stay bounded.
4.
TU~7ING
RESULTS
The control criterion ","'as taken as J(p) = E[y; (p)]
(8)
where E[-] denotes expectation w.r.t. the disturbances and Yt is the exit thickness_ The purpose 4673
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
is to evaluate the capability of 1FT to minimize this criterion for a number of €"-"Periment scenarios that are presented below.
4.1 Variations in roll gap friction In this example, a square wave with 48 s period time and amplitude 0.0018 was used to represent roll gap friction variations. During the gradient experiment these variations were set to zero. No other disturbances were applied. In Figure 7~ a contour plot of the criterion (8) is shown. The optimum is denoted by a + in Figure 7. In the figure the trajectories of 1FT for one initial point is shown. It can be seen that 1FT
It is also of interest to examine the behaviour of the control signal. In Figure 9 the control signal with the initial controller and the controller after 4 iterations are shown. The energy in the signals are quite similar as are their extremunl values. Thus, the tuning does not add any extra strain on the
mill.
converge to the optimum.
Fig. 9. Control signal. a) "rith initial controller. b) After 4 iterations. 4~2
Fig~
7. Contour plot of the criterion (8) Vi""hen roll gap friction varies as a. square-wave. + is the optimum point. x denotes the trajectory of 1FT \vhen the initial point is p
=
(1,0).
In Figure 8, the exit thickness variations during the normal experiment are shown with the initial controller p = (1,0) and after 4 iterations p = (2.289, -1.561).
Variations in entry thickness
In this example, the entry thickness varied as a square wave vlith 48 s period time and amplitude 0.04. During the gradient experiment these variations were set t.o zero. 1\0 ot.her disturbances were applied. In Figure 10, a contour plot of the criterion (8) is shov.rn together V1lith the trajectories of the controller parameters "\\rhen 1FT is used for the initial points (1,0) and (3~-1.5). When the algorithm Vlas started at (1,0), the step-size was kept at 1, except for the two last iterations ,,,"here the step-size was O~5 and O.2. respectively. For the other initial point~ the step size was kept at 1. The optimum is denoted by a + in Figure 10. t
-~t _'-0"..
--
---
~-
-
------"1~-
·-'-------'1s.{i.-
.''-
--. ~--:.o
. -- -_.
~50
Fig. 8. Exit thickness variations during normal experiment. Square-wave variations in roll gap friction. Top figure ,vith initial controller p = (1, 0), J = 8.26. Bottom figure after 4 iterations p :::=: (2.289~ -1.561) ~ J == 2.97. Tests \\Tith variations in roll position control or material hardness gave similar results, i.e. 1FT Vlas able to come very close to the optimum tuning.
Fig. 10. Contour plot of the criterion (8). The entry thickness varies as a square-wave. + denotes the optilllUIIl point~ * denotes trajectory of 1FT vlhen the initial point is p == (3, -1.5). x denotes trajectory of 1FT vlhen the initial point is p == (1, O)~ As can be seen from Figure 10, the 1FT algorithm approaches the optimum but is then pushed away in a certain direction direction. "that happens in the last iterations is that the low frequency gain is decreased. This indicates that the algorithm "believes" that there is an unnecessary integrator 4674
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
in the system which should be cancelled. Other experimental conditions may alleviate this problem. In any case it should be noted that it was possible to reduce the cost function from 5.34 to 1.72 in three iterations. The corresponding exit thickness variations are shown in Figure. I!.
14th World Congress ofIFAC
tuning procedure able to t.une controllers during a single rolling pass. In this case the assumption of constant rolling conditions can be released. 6. REFERENCES eamer-on, M.T. (1996). A cold rolling model for control applications. In: Proc. 15th lASTED Conf. on Modelling} Identification and Control. Insbrouk, Austria. pp. 187-190. Cameron, M.T. (1998). Digital simulation of flat rolling mills: linear and noninear approaches. In: Proc. 9th
IFAC Symposium on Mining, Minerals and Metals Processing. Cologne, Germany. pp. 391~396. De Bruyne, F ... B.D.O. AndersOll, M. Gevers and N. Linard --"--...] - - -.. ~.~
Fig. 11. Exit thickness variations during normal experiment. Square-wave variations in entry thickness. 'Top figure with initial controller p == (1,0), J = 5.34.. Bottom figure after 4 iterations p (1.75, -1.02), J == 1.72.
=
In Figure 12 the control signal with the initial controller and the controller after 3 iterations are shown. The energy in the signals are quite signals as are their extremum values. Thus, also here the tuning does not add any extra strain on the mill.
~;~
I
:;[E:~ I,
t/I
Fig.. 12. Control signal. a) \Vith initial controller. b) After 3 iterations. 5. CLOSING
(1997). Iterative controller optimization for nQnlinear
------"'".&(]~--~------::!
REM~t\RKS
For deterministic disturbances in material hardness and roll gap friction alOl0st optimal performance was obtained. The perforrnance obtained for variations in entry thickness was not optimal but still a significant increase in performance compared to the initial controller "\\ras obtained. Considering the control signals, the tuning does not add any extra strain on the mill. These results seem promising considering that the tuning was based 011 no a priori knowledge of the process. Further examinations are still needed to achieve a
systems. In: Proc. 36th IEEE Conf. on Decision and Control. San Diego, California. Estival, J .L. and A. Huguel (1993). Predictive thklcness control in an aluminum cold rolHng mill. RGE 5, 3239. In French. Geddes~ E.J .~. and 1. Postleth\vaite (1993). Robust control of a tandem cold rolling mill using H oo optimization. In: Proc. 1 st. Int. Con!. on Modelling of Metal Rolling Processes. pp. 389-404. Heess, G. (1969). Optimal control 0 a multi-stand cold rolling mill. Regelungstecknik 17 pp. 245-292. In German. Hjalmarsson, H. (1995). Model~free tuning of controllers: Experience with time-varying linear systems. In: Proaedings of European Contro~ Conference. Rome, Italy. pp. 2869-2874. Hjalmarsson, 11. (1998a). Control of nonlinear systems using Iterative Feedback Tuning. In: Proc. 1998 American Control Conference. Philadelphia. pp. 2083-2087. Hjalmarsson, H. (1998b). Iterative Feedback Tuning. In: IFAG Workshop an Adaptive Control and Signal Processing. Strathc1yde~ Glasgow, UK. pp. 109-116. HjaImarsson , H. and J.H. Brasla.vsky (1999). Tuning of controllers and generalized hold functions in sampleddata systems using Itera.tive Feedback Tuning. In: 14th. IFAC World Congress. Beijing, P.R. China. Accepted for publication. HjalmarssQn~ H. and T. Birkeland (1998). Iterative feedback tuning of linear time-invariant fylIMO systerns. In: GDC-9B. Tarnpa. Hjalmarsson) H., M. Gevers and O. Lequin (1997). Iterative feedback tuning: theory and applications in chemical process control. Journal A 38(1), 16-25. Hjalmarsson, ll.r ~. Gevers and O. Lequin (1998). Iterative feedback tuning: theory and applications. IEEE Control 8ystems Magazine 18(4), 26-41. IIjalmarsson) H., S. Gunnarsson and M. Gevers (1994). A convergent iterative restricted complexity control design scheme. In: Proc. 33rd IEEE eDC. Orlando, Florida. pp. 1735-1740. Hoshino, 1., Y. MaekaVl.'a, T. Fujimoto, H. Kimura and H. Kimura (1988). Ohserver-ba,.c.ed multivariabJe control of the aluminum cold tandem mill. Automatica 24 1 741-754. Ljung, L. (1987). System Identification: Theory for the User. Prentice-HaIl. Engle",,~ood Cliffs, NJ. Moreno, J .A. (1994). Gain-schedule-based control for nonlinear systems. PhD thesis. University of the Federal Army, Halllbllrg. In GerITlaTI, Sj5berg~ J. and ~1. AgarwaI(1996). Model-free repetithre control design for nonlinear systems. In: Proc. 35th
Conference on Decz"sion and (,'ontrot. KQbe, Japan. Spall, J .C. and J .A. Cristion (1998). Model-free control of nolinear stochastic systems \vith discretetime measurements. IEEE Trans. Automatic Control 43, 1198-1210.
4675
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
1-3b-08-5
Copyright (~ 1999 IFAC . 14th Triennial World Congress~ Beijing, P.R. Chlna
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD ROLLING MILLS· Hakan Hja1rnarsson'* Mario T. CalDeron **
* Dept. of Signals 7 Sensors and Systems;, The Royal Institute of Technology, 8-100 44 Stockholm, Sweden. hakan.hjalmarsson~s3wkth.se
** ABB Industrietechnik GmbH, D-68167 Mannheim, Germany. mario.cameron0deind.mail.abb.de Abstract: Preliminary results on the potential of Iterative Feedback Tuning (1FT) for tuning controllers in cold rolling mills are presented. The tuning of the strip thickness controller in a simulated one-stand cold rolling mill is considered. Preliminary simulation studies show that for the used model, 1FT handles deterministic disturbances in material hardness and roll gap friction very well. The performance of the algorithm for deterministic variations in entry thickness is some\vhat worse but still a significant increase in performance can be obtained. Copyright ©1999 IFAC Keywords: PI controller, non...linear models, non-stationary systems, metals
1. INTRODUCTION The development of drive and automation solutions for cold rolling mills is subject to a very high innovation pressure because of the increasing demands on both: the quality of cold rolled products and the operational performance of the mills. A permanent improvement in the control of these plants shall have to contribute to the fulfillment of these demands. Strip thickness and flatness play a fundamental role among the several variables that deterlnine the quality of cold roIled products. Indeed, strip thickness and flatness are the only that can be directly influenced by control. Tighter tolerances in these variables can, hov.rever, only be reached, if at the same time the strip tension is controlled4 One of the most important practical problems to deal with here is the adaptation of the controllers for all possible rolling conditionS4 These conditions are given by the combination of all possible thermal and mechanical states of the rolling mill, all classes of materials to be used and products to be rolled, as well as all possible rolling speeds. Another important practical problem to solve is the final controller tuning based on the real bellavior of the mill. In the last 30 years, however, most of the works published has only focused on the multivariable character of the plant. Many authors have worked on the design of optimal multivariable controllers for strip thickness and tension in multi-stand mills: Hcess (1969) was the first in designing a LQG-based controller~ Geddes and Postlethwaite (1993) where the first in designing an Hoc based controlIer4 The adaptation of all designs to the changes inter and intra rolling passes, and the final tuning based on the real
behavior of the plant, however, did not receive the necessary attention in any of these papers in particular, and in the control community in general. Some authors have been conscious of the adaptation problem~ but even in that case they dedicated only a short remark to a possible solution: gain schedule. This, however, simply understood as the result of several designs - one for each of all possible rolling conditions. Considering the huge number of rolling conditions, it is easy to see that this approach is not tractable, even for classical designs with relatively much fewer parameters to adapt than for the modern designs. This is one important reason why most of the designs published in the last 30 years for strip thickness, tension, and flatness contr·ol in cold rolling mills have not been implemented in practice. Valuable exceptions to this trend represent the works of Hoshino et al. (1988) and Estival and Huguel (1993). To get the adaptation problem tractable, a good controller design in rolling m.ills has to ensure that the controller parameters can be calculated analytically from the design model. T'his model is obtained by reducing the ideal plant model, \vhich of course, contains uncertainties, even though cold rolling mills have been extensively modeled. Hence, even if the controller design is model based, it is desirable to do a final tuning based on the real behavior of the rolling mill. Additionally, it can be shown that in this case with the implementation of a gain schedule to solve the adaptation problem one changes the plant dynamics at the same time. Hence, even under ideal conditions the controller performance obtained during design cannot be achieved when implemented, see e.g.
4670
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
(Moreno~ 1994). In this paper the parameters of the strip thickness controller of a simulated cold rolling mill win be tuned using Iterative Feedback Tuning (1FT).
1FT is a model free tuning method, which uses experimental data to iteratively search for the optimal controller settings. 1FT was originally developed for linear time-in"'ariant systems (Hjalmarsson et al., 1994) but has been shown to work for slowly varying periodic linear timevarying systems as well (Hjalmarsson, 1995)~ Recently, some interesting and encouraging results have been obtained for cert.ain non-linear systems. In (Hjalmarsson et al., 1998) and (Hjalmarsson et al., 1997) some real industrial applications from the chemical process industry are reported. In (Hjalmarsson, 1998a) a controller for a DC-servD with backlash ~"as successfully tuned using 1FT under laboratory conditions~ Here an initial analysis for the nonlinear case ,vas given which provides some intuition as to why 1FT works on npn-linear sY8tems~ In (Hjalmarsson, 1998b) the simulation results of a non-linear system is shown where 1FT outperforms system identification and model based controller design~ These encouraging results makes it interesting to apply 1FT for the final tuning of controllers in cold rolling mills. Other interesting papers related to 1FT where tuning of controllers for simulated non-linear systems is considered are (De Bruyne et al.) 1997) and (Sjoberg and Agarwal, 1996). Another interesting model-free tuning method is presented in (Spall and Cristion, 1998)~ The outline is as follo\vs. In Section 2 the simulation model and the experimental conditions are briefly described. Section 3 introduces 1FT for the IvlIMO case. Section 4 presents simulation results on the final tuning of the strip thickness controller in a single stand cold rolling mill. The paper finishes with the conclusions in Section 5.
2. SIMULATION MODEL AND EXPERLVIENTAL CONDITIONS As already noted, in this study a single-stand cold rolling mill is considered. The ideal mill configuration is shown in Figure 1. The simulation model can be illustrated in block diagram form as in Figure 2. For simplicity the transversal effects across the lllill and strip are neglected and reduced nonlinear models for the material deformation in the roll gap, mill stand t strip tension and transport, coiling and llllcoiling process, as well as for the positioning and drive systems are considered. The roll gap variables are determined using the lllodel presented in (Cameron, 1996). To solve the multivariable algebraic loops during simulat.ion the direct approach presented in (Camerou, 1998) is used. The simulated mill describes then, the fundamental non-linear and non-stationary behavior of an ideal mill during the rolling of a coil, i.e. during the acceleration phases, constant speed phases and deceleration phases. Figure 3 shows the variaton of the rolling speed and coil radius at uncoiler and coiler during an ideal rolling pass: the rolling speed accelerates at the beginning, is then keep constant and finally decelerated. These speed variations and the resulting strip mass transfer from uncoiler to coiler give rise to the non-stationary behavior of the plant.
Min slanJ
T-r"
Fig.
00f
Fig. 1. Idealized configuration of a single stand cold rolling ruill: (HC) thickness control, (TC) tension control, (Se) speed control, (PC) position control.
2~ Block diagram of the simulation model for a single stand cold rolling mill.
Fig. 3. Ideal rolling pass. \lariation of rolling speed (top) and coil radius at uncoiler and uncoiler (bottom). For purposes of the study it will be assumed that during the 1FT apart from speed changes the other rolling conditions remain const.ant. T'his 4671
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ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R ...
Disturbance
Wavebnn
Strip thickness (welding point) Strip thickne~s (begin and cnd) Strip thickness and tension
_ _ _----'1--
00 -
0-
aAer prcl>Cli~ strip bacdness
Roll gap frichon
Table 1. Typical waveforms for some disturbances in cold rolling mills.
assumption will be released in Section 4. During tuning, measurable and non-measurable input disturbances will take place Table 1 shows that in practice many exogenous disturbances can be modeled as deterministic signals. For the sake of simplicity only disturbances in the thickness and hardness of the incoming material, as well as in the roll gap friction are considered. These disturbances are modeled as square waves. A
For the tuning experiments with 1FT the strip thickness and tension in the simulated singlestand cold rolling mill are controlled as in practice. To control these variables, very complex control configurations exist. One can show that most of tllese configurations are a subset of the general configuration presented in Figure 4. To achieve tighter tolerances not only output feedback control elements, but also presetting, decQupling and feedforward control elements are used. This control configuration work as follows: if the reference for the rolling speed rw remains constant and vlhether disturbances (v-m, v n ) nor noise affect the system, an ideal presetting control element should generate the initial value for the control signal u, that achieves the desired thickness tolerance. If changes in the reference for the rolling speed r w take place, an ideal disturbance control element should generate the additional control signal u to eliminate possible effects, at least asymptoticaly. The same function has the fcedfoTward control element for changes in the measurable disturbances V m ~ It is, however, clear that in practice all these control elements can be only designed based on the design model. ~foreover non-measurable disturbances Vn, as well as noise (n y and nd) affect both: the plant and the controller. The practical necessity of an output feedback is then clear~ The output feedback can \\Tock based alone 011 a timedelayed measurement of the exit strip thickness, or combined with a dead-time free estimation of
it. For the strip tension control (TC) only the feedforward and decoupling elements are active. For the strip thickness control (HC) all of them. The control signal of this controller is the reference for the position control (PC). For simplicity only the case with the dead-time free estimation of the exit thickness through mass flow is considered
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here. This is the reason for the strip speed sensors shown in Fig. 1. To solve the ada.ptation problem each of the controller elements has in practice only few parameters to be set. In particular for the output feedback a tuning element in series with the base control element of I-type is used. The tuning element has the form
C(p) == Po
+ Pt
q-l
Here po and Pl are the parameters to be tuned and q-l is the discrete time backward shift operator q-ls(t) = s(t - 1). The resulting output feedback element is then of PI-type. The sampling time is 0.1 s. For the base control element base tuning based on a very simple design model, that gives reasonable performance for all rolling conditions is considered~ For the gain schedule the measurements of the rolling speed and the roll force are used.
I--~ y .... I
- - - - - - - - - - -.....
~-------------"
Fig. 4. General configuration of the strip thickness and tension controller in a cold rolling mill: (tb) output feedback, (ps) presetting, (ff) feed-forward, (de) decoupling.
3. ITERATI\lE FEEDBACK T'UNING In this section, the 1FT algorithm for tuning the parameters in a single input/single output (8I80) controller block is briefly reviewed. The origina.l derivation for 8180 systems can be found in (Hjalmarsson et al., 1994). The MIMO case is treated in full in (Hjalmarsson and Birkeland, 1998), see also (Hjalmarsson and Braslavsky, 1999) in these proceedings. "Ve consider an unknov.."n true system described
by
the discrere
t~t
)T: :(~
)Odel (1)
where t represents the discrete time instants, G is the (generalized) plant represented by a transfer function matrix, rt E Rnr represents external signals such as set-points, reference signals etc, 'Vt E R n v represents unmeasurable signals such as (process) disturbances and noise, Wt E Rn w represents the sensed outputs and Ut ERn", represents the control signals. Furthermore Yt E Rn y represents the variables that "Till be included in the control criterion (e.g. measured outputs 4672
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ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
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o
0---..1
v
u:J Fig. 5. Feedback system vlhere each channel in the controller is independently parameterized. The generalized plant Gj is independent of the parameter vector p and control signals). These signals can be filtered (frequency weighted) versions of measured signals in the systemw 'Ye shall assllme, that {Vt} is a zero mean weakly stationary (see e.g. (Ljung, 1987» random process, but other disturbance assumptions can also be made. The system may be controlled by a multivariable controller and even though it is possible to tune all elements in this controller using 1FT, ""ill, for simplicity, consider the tuning of one single SISO controller block
we
Uj
=
Gj(p)Wj
parameterized by a parameter vector p ERn". Using this parametrization the system can be rearranged such that the relation between the controller block Cj and the output y is given by
Fig. 6. Block diagram of second experiment.
One "\1.I"ay of finding a (local) minim.um of J(p) is to use a descent search such as 8J . "+1' p~ = pt -l'iRi 1 Bp (pT.). (6) Here Ri is some appropriate positive definite matrix~ typica.lly a Gauss-Nelvton approximation of the Hessian of J, Vt-.-hile 'Yi is a positive real scalar that determines the step size~ To carry out the minimization it is necessary to have an expression for the gra.dient of this criterion with respect to the controller parameters. The novel contribution of the 1FT approach (Hjalmarsson et al., 1994) was to show that, in contrast to previous approaches an unbiased estimate of the gradient can be obtained entirely from input-output data collected on the actual closed loop system, by performing one special experirnent on that system. It can be shown that by the following procedure an unbiased estimate of the gradient can be obtained: i) Perform a closed loop experiment of length N "'''here the desired reference signal r is used Zj = 0 and with the parameter vector p in the controller set to the value for which the derivative is to be computed, see Figure 5. Collect measurements of y and Wj during this experiment. ii) Perform a second experiment where the reference r is set to zero, Zj = Wj and collect the outputs y, which we denote by y2, from this experiment, see Figure 6. iii) Filter each of the measured outputs from the second experiment, i.e. the elements of y2 t through the vector-valued transfer function consisting of 8Cj /8p. Collect all these signals j
(2) Uj == CjWj (3) where the generalized plant Gj is obtained from G and all controller blocks except C j .
\Vhenever signals are obtained from the closed loop system with the controller C J (p) operating, we ~"ill indicate this by llsing the p-argument; on the other hand, to ease the notation v.,..e will from now on omit the time argument of the signals~ Thus, y(p) will denote the signals in the control criterioll when the controller C j (p) is used~ In the 1FT design scheme the following quadratic control criterion is adopted:
J(p)
= 2~E
[t
[Yt(p)f WlYt(P)]
F.i -_
(4)
1\-"here E[~] denotes expectation "v.r.t. the weakly stationary disturbance v. The weight matrix lVtY, which for simplicity will be set to the identity, can be used to minimize the settling time at setpoint changes: see Hjalmarsson et al. (1998). This criterion include many commonly used control criteria. In particular it can, with proper definition of y, handle model reference following.
The optimal controller parameter p* is defined by p* = argminJ(p), (5) p
in a n y x n p dimensional matrix
Bp
~ [8ih] T -
-1 L--
N
t=l
Bp
~ and form
Yt (P)
(7)
By replacing 8Jlap in (6) an algorithTIl is obtained which under suitable choice of the step-size /1i converges to a stationary point of the criterion (4) if the signals stay bounded.
4.
TU~7ING
RESULTS
The control criterion ","'as taken as J(p) = E[y; (p)]
(8)
where E[-] denotes expectation w.r.t. the disturbances and Yt is the exit thickness_ The purpose 4673
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ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
is to evaluate the capability of 1FT to minimize this criterion for a number of €"-"Periment scenarios that are presented below.
4.1 Variations in roll gap friction In this example, a square wave with 48 s period time and amplitude 0.0018 was used to represent roll gap friction variations. During the gradient experiment these variations were set to zero. No other disturbances were applied. In Figure 7~ a contour plot of the criterion (8) is shown. The optimum is denoted by a + in Figure 7. In the figure the trajectories of 1FT for one initial point is shown. It can be seen that 1FT
It is also of interest to examine the behaviour of the control signal. In Figure 9 the control signal with the initial controller and the controller after 4 iterations are shown. The energy in the signals are quite similar as are their extremunl values. Thus, the tuning does not add any extra strain on the
mill.
converge to the optimum.
Fig. 9. Control signal. a) "rith initial controller. b) After 4 iterations. 4~2
Fig~
7. Contour plot of the criterion (8) Vi""hen roll gap friction varies as a. square-wave. + is the optimum point. x denotes the trajectory of 1FT \vhen the initial point is p
=
(1,0).
In Figure 8, the exit thickness variations during the normal experiment are shown with the initial controller p = (1,0) and after 4 iterations p = (2.289, -1.561).
Variations in entry thickness
In this example, the entry thickness varied as a square wave vlith 48 s period time and amplitude 0.04. During the gradient experiment these variations were set t.o zero. 1\0 ot.her disturbances were applied. In Figure 10, a contour plot of the criterion (8) is shov.rn together V1lith the trajectories of the controller parameters "\\rhen 1FT is used for the initial points (1,0) and (3~-1.5). When the algorithm Vlas started at (1,0), the step-size was kept at 1, except for the two last iterations ,,,"here the step-size was O~5 and O.2. respectively. For the other initial point~ the step size was kept at 1. The optimum is denoted by a + in Figure 10. t
-~t _'-0"..
--
---
~-
-
------"1~-
·-'-------'1s.{i.-
.''-
--. ~--:.o
. -- -_.
~50
Fig. 8. Exit thickness variations during normal experiment. Square-wave variations in roll gap friction. Top figure ,vith initial controller p = (1, 0), J = 8.26. Bottom figure after 4 iterations p :::=: (2.289~ -1.561) ~ J == 2.97. Tests \\Tith variations in roll position control or material hardness gave similar results, i.e. 1FT Vlas able to come very close to the optimum tuning.
Fig. 10. Contour plot of the criterion (8). The entry thickness varies as a square-wave. + denotes the optilllUIIl point~ * denotes trajectory of 1FT vlhen the initial point is p == (3, -1.5). x denotes trajectory of 1FT vlhen the initial point is p == (1, O)~ As can be seen from Figure 10, the 1FT algorithm approaches the optimum but is then pushed away in a certain direction direction. "that happens in the last iterations is that the low frequency gain is decreased. This indicates that the algorithm "believes" that there is an unnecessary integrator 4674
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ISBN: 0 08 043248 4
ITERATIVE FEEDBACK TUNING OF CONTROLLERS IN COLD R...
in the system which should be cancelled. Other experimental conditions may alleviate this problem. In any case it should be noted that it was possible to reduce the cost function from 5.34 to 1.72 in three iterations. The corresponding exit thickness variations are shown in Figure. I!.
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tuning procedure able to t.une controllers during a single rolling pass. In this case the assumption of constant rolling conditions can be released. 6. REFERENCES eamer-on, M.T. (1996). A cold rolling model for control applications. In: Proc. 15th lASTED Conf. on Modelling} Identification and Control. Insbrouk, Austria. pp. 187-190. Cameron, M.T. (1998). Digital simulation of flat rolling mills: linear and noninear approaches. In: Proc. 9th
IFAC Symposium on Mining, Minerals and Metals Processing. Cologne, Germany. pp. 391~396. De Bruyne, F ... B.D.O. AndersOll, M. Gevers and N. Linard --"--...] - - -.. ~.~
Fig. 11. Exit thickness variations during normal experiment. Square-wave variations in entry thickness. 'Top figure with initial controller p == (1,0), J = 5.34.. Bottom figure after 4 iterations p (1.75, -1.02), J == 1.72.
=
In Figure 12 the control signal with the initial controller and the controller after 3 iterations are shown. The energy in the signals are quite signals as are their extremum values. Thus, also here the tuning does not add any extra strain on the mill.
~;~
I
:;[E:~ I,
t/I
Fig.. 12. Control signal. a) \Vith initial controller. b) After 3 iterations. 5. CLOSING
(1997). Iterative controller optimization for nQnlinear
------"'".&(]~--~------::!
REM~t\RKS
For deterministic disturbances in material hardness and roll gap friction alOl0st optimal performance was obtained. The perforrnance obtained for variations in entry thickness was not optimal but still a significant increase in performance compared to the initial controller "\\ras obtained. Considering the control signals, the tuning does not add any extra strain on the mill. These results seem promising considering that the tuning was based 011 no a priori knowledge of the process. Further examinations are still needed to achieve a
systems. In: Proc. 36th IEEE Conf. on Decision and Control. San Diego, California. Estival, J .L. and A. Huguel (1993). Predictive thklcness control in an aluminum cold rolHng mill. RGE 5, 3239. In French. Geddes~ E.J .~. and 1. Postleth\vaite (1993). Robust control of a tandem cold rolling mill using H oo optimization. In: Proc. 1 st. Int. Con!. on Modelling of Metal Rolling Processes. pp. 389-404. Heess, G. (1969). Optimal control 0 a multi-stand cold rolling mill. Regelungstecknik 17 pp. 245-292. In German. Hjalmarsson, H. (1995). Model~free tuning of controllers: Experience with time-varying linear systems. In: Proaedings of European Contro~ Conference. Rome, Italy. pp. 2869-2874. Hjalmarsson, 11. (1998a). Control of nonlinear systems using Iterative Feedback Tuning. In: Proc. 1998 American Control Conference. Philadelphia. pp. 2083-2087. Hjalmarsson, H. (1998b). Iterative Feedback Tuning. In: IFAG Workshop an Adaptive Control and Signal Processing. Strathc1yde~ Glasgow, UK. pp. 109-116. HjaImarsson , H. and J.H. Brasla.vsky (1999). Tuning of controllers and generalized hold functions in sampleddata systems using Itera.tive Feedback Tuning. In: 14th. IFAC World Congress. Beijing, P.R. China. Accepted for publication. HjalmarssQn~ H. and T. Birkeland (1998). Iterative feedback tuning of linear time-invariant fylIMO systerns. In: GDC-9B. Tarnpa. Hjalmarsson) H., M. Gevers and O. Lequin (1997). Iterative feedback tuning: theory and applications in chemical process control. Journal A 38(1), 16-25. Hjalmarsson, ll.r ~. Gevers and O. Lequin (1998). Iterative feedback tuning: theory and applications. IEEE Control 8ystems Magazine 18(4), 26-41. IIjalmarsson) H., S. Gunnarsson and M. Gevers (1994). A convergent iterative restricted complexity control design scheme. In: Proc. 33rd IEEE eDC. Orlando, Florida. pp. 1735-1740. Hoshino, 1., Y. MaekaVl.'a, T. Fujimoto, H. Kimura and H. Kimura (1988). Ohserver-ba,.c.ed multivariabJe control of the aluminum cold tandem mill. Automatica 24 1 741-754. Ljung, L. (1987). System Identification: Theory for the User. Prentice-HaIl. Engle",,~ood Cliffs, NJ. Moreno, J .A. (1994). Gain-schedule-based control for nonlinear systems. PhD thesis. University of the Federal Army, Halllbllrg. In GerITlaTI, Sj5berg~ J. and ~1. AgarwaI(1996). Model-free repetithre control design for nonlinear systems. In: Proc. 35th
Conference on Decz"sion and (,'ontrot. KQbe, Japan. Spall, J .C. and J .A. Cristion (1998). Model-free control of nolinear stochastic systems \vith discretetime measurements. IEEE Trans. Automatic Control 43, 1198-1210.
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