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D) Control in Telephone Cable Insulation Process
1f-02 6
Copyright © 1996IFAC 131h Triennial World Congn::ss. San FranciS(."O, USA
CAPACITANCElDIAMETER (CID) CONTROL IN TELEPHONE CABLE INSULATION PROCFSS
Jukka Pulkkinen*, Heikki N. Koivo" , and Jean·Charles Zaramella' 'Teol/isuuden Voima Ltd.. FlN·27160 Olkiluoto. Finland "Helsinki University a/Technology. Control Engineering Laboratory. FIN 02150 Helsillki. Finland
INokia-Maillefer Ltd.• Telecommunications Business Area, CH-1024 EcublensLausanne. Switzerland
Abstract. Foam insulation is used in the production of plastic insulated telephone cable, because it both saves material costs and improves the electric characteristics of the telephone cable. Using foam makes control of the process quite hard. because it causes a severe nonlinearity and generates strong interactiuns between the outputs. the capacitance (C) and the diameter (D) of the cable. In this paper a new robust control scheme is presented which significantly improves the existing control systems in use. A model is developed to compensate the nonlinearity. Multivariable transfer function model at several operating points is used to model both the process interactions and the uncertainty in the modeL Experiments show that the designed control system achieves robust perfonnance. which guarantees reliable operation of the system under varying operating conditions. Keywords. Robust control, multi variable control systems. telephone cable. extrusion
1. INTRODUCTION
varying operating conditions. In this paper a new control scheme is developed which is based on robust control theory (Doyl•• 1.C., Wall . I.E. and G. Stein 1982; Morari. M. and E. Zaviriou. 1989). A multi variable transfer function model with uncertainty is developed for the process. The severe nonlinearity in actuation is measured and a model is constructed. Based on the modeling effort an efficient new tuning method using Edmund's (1979) algortihm is presented.
Telephone cable is produced in telephone cable foam line. where the copper wire is insulated with foam material. International standards detennine the electric and mechanical properties of telephone cables. Foam insulation is used in the production of plastic insulated telephone cable, because it both saves material costs and improves the electric characteristics of the telephone cable. Using foam makes control of the process quite difficult, because it causes a severe nonlinearily and ge nerales strong interactions between the outputs. the capacitance (C) and the diameter (D) of the cable.
The paper first discusses the process and its modehng. Control structure and control tuning are presented next. Compensation of the nonlinearity is then discussed and the paper concludes with ex~rimental results and conclusions.
The existing constant parameter conventional controllers are unable to meet tbe strict production specifications under
725
motorized telescope is introduced. With this the foaming time can be controlled. The capacitance of the telephone wire is controlled with the step motor moving the telescope.
2. PROCESS MODELING
The use of polyethene foam in telephone cable production saves material costs and improves the electric characteristics of the telephone cable, but makes the control of the process
A strong interaction exists between the diameter and the capacitance control loops. That is, if a change is made in the screw speed of the extruder, it effects both the capacitance and the diameter of the telephone wire. Foaming occurs very quickly after the die and cooling takes place slowly after it, which causes a significant nonlinearity in the capacitance control loop.
difficult. Raw material savings can be quite significant, because percentage of foam can be 60% and in such a case
insulation material is 60% air and 40% polyethene. The following variables are controlled in telephone cable foam line: diameter, capacitance and tension of the cable, temperature, screw speed of the polyethene extruder and pressure of foam substance. In this paper only CID control is discussed.
4. DYNAMIC MODEL OF FOAM LINE 3. OPERATION OF FOAM LINE
A nominal model P(s) ofthe process is used in the design and tuning ofthe controller. The uncertainty of the model is .1(s) .
Foam plastic insulation is a continuous process, in which the wire is unwound from the input reel, coated with insulation plastic, cooled and finally wound on the output reel. Aschema of the process is shown in Fig. 1.
The relationship between the transfer function
pes) describing the real behavior of the process and the nominal model
pes) used in the design is
pes) = (l + .1(s» Pcs)
(1)
in which the frequency dependent uncertainty is determined with the biggest singular value /1(.1(im» (Maciejowski, 1989). The nominal model
pes) can be identified and the
uncertainty of the model is .1(s) estimated by performing appropriate process experiments (Ljung, 1987). The nominal model has the structure Fig. 1. Schema of the telephone cable foam line.
Pcs) =
The cable diameter is measured after cooling. Although the primary objective is to control the diameter of the cable, in practice the computed surface area of the insulation is used. The reason is that the surface area has a more linear relationship with the control variable. The plastic granules are melted in the plastic extruder. The screw in the extruder barrel transports the molten mass to the die. Il is then fed through the die on top of the copper wire to fonn an insulation layer. The screw speed is used to control the diameter of the telephone cable. The screw speed has its own control loop, for which the
r~II(S) ~12(S) 1
IPlieS) P (s)
(2)
21
where PlI(S) and P,l(S) are the transfer functions from the extruder screw speed to the computed insulation surface area and the capacitance of the telephone cable and
Pl2 (S)
and
P22(S) from the telescope position control to the computed insulation surface area and the capacitance of the telephone cable. From physical reasons it is easy to see that the transfer
CID controller computes the setpoinl.
functions
Pds)
and
Pzls) are of the form of an integrator
and a delay, because the telescope acts as an integrator and there is a delay in the diameter measurement. Foaming occurs so quickly that it has practically no dynamics compared with the the rest of the process. Based on measurement data first
After the extruder the foaming line has a cooling process, in which the hot insulation is cooled with water. The foaming of polyethene occurs very quickly after the extruder and foaming ends in cooling of insulation. Before actual cooling a
726
order models with delay are sufficient to describe the transfer functions PII(S) and P,,(s).
5. TUNING OF CID CONTROLLER
Several experiments are made at different operating points by changing the diameter and the capacitance of the telephone cable, and the line speed. Based on these the nominal model (2) and the uncertainty for the process were detennined. The gains in the model elements changed significantly as a function of diameter. Therefore the model gains are represented as functions of the diameter d, First order polynomial was sufficient to describe the dependence in all cases.
Proportional-Integral (PI) controller is used in the diameter control loop. Proportional (P) controller is sufficient in the capacitance control loop, since the loop already contains an integrator. This implies that the structure of the CID controller is
1 K,,(\ +-T )
The elements in the nominal model (2) are
;;;;;
=
C(s)
Kll(d) e- 65.' 5.0s+ 1 ; K,,(d) ;;;;; 1253d -224, K,,(d)
e- 50 .,
6.3 s+ J
=
K 12(d) e-'"
s
(4)
K,,(d) = 6.87 d - 7.76 ,
The control parameters are computed with Edmund's (1979) algorithm, in which the control parameters are optimized in frequency domain in such a way that the closed loop transfer function is as close as possible to the desired one. The desired transfer function is chosen as
(5)
(6)
; K22(d) = 7.68 d - 11.1
T/s) = diag
the diameter. It was decided to represent it at three different values of the diameter d 0.6,0.9, and 1.1 mm. These are correspondingly
.1(s) = diag {0.07(1
+ !Os), 0.45(1 + 3s)}
,
+ 4S)}
.
{u
1+ 1 ' t s \
where the time constant 't acts as a tuning parameter. The advantage of the tuning procedure is the possibility to use the multivariable process model to tune the scalar controllers in order to decrease the loop interactions, The tuning parameter 't is chosen so that the robust performance objective of the system is achieved, This implies that the closed loop system is stable and realizes the desired performance objective in spite of the uncertainty of the nominal model. The performance objective is defined by the weight transfer function
(8)
(9)
_ 1
W,(s) - 'tS' .,
The uncertainty is required to test robust performance, In such a case the uncertainty is decomposed into the form .1(s)
= Wl 3 W, '
(12)
I}'
(7)
+ 20s), 0.13(1 + !Os)} ,
.1(s) = diag {O.27( J + 8s), O.44( I
(11 )
o
where Kd is the control gain of the diameter, 1'; is the integration time, and Kc is the control gain of the capacitance.
The uncenainty ofthe nominal model Li(s) also depends on
.1(s) = diag {0.28(1
o
(3)
K 12(d) = 3852 d - 1844,
P12 (s)
is
=
(13)
This implies that the time constant of the closed loop transfer function must be smaller or equal to 't,. in spite of the uncertainty in the nominal model.
(10)
To check the nominal performance the test system
where W j and W1 are transfer functions with minimum phase and 1.1(s) L ,;; 1 (DoyJe, Wall, and Stein, 1980).
727
Q(s) =
lQII(S) QI2(S)j
the telescope position and capacitance is measured when the telescope position changes between 0 to 100%. A second order polynomial is fitted to the data
(14)
Q21(s) Q22(S)
Fig. 2 shows both the measured and computed values of the capacitance. The fit is sufficiently good for practical purposes. The second order polynomial is given by
is defined. Here (dropping the argument s and using notation of Doyle, Wall, and Stein (1980)
c =c(P) =0.053 p2
(15)
- 10.61 P + 2078,
(22)
where C = capacitance, P= position of the telescope and e(P) is the characteristic curve.
(16)
c,--___-----, "'"
(17)
1900 1800
Q" =
-W,PCII +PCr' Wt·
(18)
"00 1000 1500 0
Robust performance is achieved, if the structural singularity
IQ L < I (Doyle, Wall, and Stein, 1980; Morari andZaviriou, Because the diameter has significant effect on model gains and the magnitude of uncertainty, the controller is tuned at three different valuesofthediameter,d=0.6, 0.9 and 1.1 mm. Based on these the diameter dependent control gains are detennined. The integration time remains constant, because its variation with different diameter values is small. The time constant 't".1 dctcnnining the performance objective, is chosen to be 320 s,
The gain can now be computed by taking the derivative of c( P) which is
K(P) = cCP) = 0.011 P - 10.61 .
parameters of the diameter controller are
+0.00030,
T, = 6.63,
(19) (20)
kiP)
K, = 0.0027 d + 0.0016.
(23)
When the dependence of the process gain of the telescope position is known, the static nonlinearity caused by this can be compensated by changing the gain according to the operating point or the position of the the telescope. In previous section the controller was tuned when telescope position was 30%, so the change in gain is realized by multiplying the output of the controller with
which is sufficient for the telephone cable foam line. The
= -0.()()()18d
P
Fig. 2. The characteristic curve of a telescope, measured points C*), fitted curve C-).
1989).
K"
100
(21)
c(30)
- 7.31
(24)
= c'(P) = 0.11 P - 10.61
This implies that the product k(P)K(P) is constant. 6. COMPENSATION OF NONLINEARITIES
independent of the operating point. The process nonlinearily has been compensated and the value for control gain determined during tuning is used at telescope position 30%.
The position of the telescope creates a static nonlinearity in the process. Foaming occurs quickly after the extruder. This implies a large gain from the telescope position to capacitance, when the telescope is near thc cxtruder and a small gain when it is further from the extruder.
7. EXPERIMENTAL RESULTS In normal production the tolerance for variation in diameter is
To compensate for this non linearity the dependence between
± 20 mm and in capacitance ± 2 pF/m. Fig.
728
3 displays the
the desired perfonnance specification during all operating conditions. REFERENCES Doylc, I.e., Wall , I.E. and G. Stein (1982). Perfonnance and robustness analysis for structured uncertainty. IEEE Conference all Decision and Control. Orlando PL. Edmunds, J.M. (1979). Control system design and analysis using closed-loop Nyquist and Bode array•. Int. J. Control. 30, 773-802. Ljung, L. (1987). Sysrem Identification: Theory for User. Prentice Hall. New Jersey. Maciejowski, J. (1989). Multivariable Feedback Design. Addison-Wesley Publ. Co., Reading, Mass. Morari, M. and E. Zaviriou (1989). Robust Process Control. Prentice Hall. New Jersey.
729
Copyright © 1996IFAC 131h Triennial World Congn::ss. San FranciS(."O, USA
CAPACITANCElDIAMETER (CID) CONTROL IN TELEPHONE CABLE INSULATION PROCFSS
Jukka Pulkkinen*, Heikki N. Koivo" , and Jean·Charles Zaramella' 'Teol/isuuden Voima Ltd.. FlN·27160 Olkiluoto. Finland "Helsinki University a/Technology. Control Engineering Laboratory. FIN 02150 Helsillki. Finland
INokia-Maillefer Ltd.• Telecommunications Business Area, CH-1024 EcublensLausanne. Switzerland
Abstract. Foam insulation is used in the production of plastic insulated telephone cable, because it both saves material costs and improves the electric characteristics of the telephone cable. Using foam makes control of the process quite hard. because it causes a severe nonlinearity and generates strong interactiuns between the outputs. the capacitance (C) and the diameter (D) of the cable. In this paper a new robust control scheme is presented which significantly improves the existing control systems in use. A model is developed to compensate the nonlinearity. Multivariable transfer function model at several operating points is used to model both the process interactions and the uncertainty in the modeL Experiments show that the designed control system achieves robust perfonnance. which guarantees reliable operation of the system under varying operating conditions. Keywords. Robust control, multi variable control systems. telephone cable. extrusion
1. INTRODUCTION
varying operating conditions. In this paper a new control scheme is developed which is based on robust control theory (Doyl•• 1.C., Wall . I.E. and G. Stein 1982; Morari. M. and E. Zaviriou. 1989). A multi variable transfer function model with uncertainty is developed for the process. The severe nonlinearity in actuation is measured and a model is constructed. Based on the modeling effort an efficient new tuning method using Edmund's (1979) algortihm is presented.
Telephone cable is produced in telephone cable foam line. where the copper wire is insulated with foam material. International standards detennine the electric and mechanical properties of telephone cables. Foam insulation is used in the production of plastic insulated telephone cable, because it both saves material costs and improves the electric characteristics of the telephone cable. Using foam makes control of the process quite difficult, because it causes a severe nonlinearily and ge nerales strong interactions between the outputs. the capacitance (C) and the diameter (D) of the cable.
The paper first discusses the process and its modehng. Control structure and control tuning are presented next. Compensation of the nonlinearity is then discussed and the paper concludes with ex~rimental results and conclusions.
The existing constant parameter conventional controllers are unable to meet tbe strict production specifications under
725
motorized telescope is introduced. With this the foaming time can be controlled. The capacitance of the telephone wire is controlled with the step motor moving the telescope.
2. PROCESS MODELING
The use of polyethene foam in telephone cable production saves material costs and improves the electric characteristics of the telephone cable, but makes the control of the process
A strong interaction exists between the diameter and the capacitance control loops. That is, if a change is made in the screw speed of the extruder, it effects both the capacitance and the diameter of the telephone wire. Foaming occurs very quickly after the die and cooling takes place slowly after it, which causes a significant nonlinearity in the capacitance control loop.
difficult. Raw material savings can be quite significant, because percentage of foam can be 60% and in such a case
insulation material is 60% air and 40% polyethene. The following variables are controlled in telephone cable foam line: diameter, capacitance and tension of the cable, temperature, screw speed of the polyethene extruder and pressure of foam substance. In this paper only CID control is discussed.
4. DYNAMIC MODEL OF FOAM LINE 3. OPERATION OF FOAM LINE
A nominal model P(s) ofthe process is used in the design and tuning ofthe controller. The uncertainty of the model is .1(s) .
Foam plastic insulation is a continuous process, in which the wire is unwound from the input reel, coated with insulation plastic, cooled and finally wound on the output reel. Aschema of the process is shown in Fig. 1.
The relationship between the transfer function
pes) describing the real behavior of the process and the nominal model
pes) used in the design is
pes) = (l + .1(s» Pcs)
(1)
in which the frequency dependent uncertainty is determined with the biggest singular value /1(.1(im» (Maciejowski, 1989). The nominal model
pes) can be identified and the
uncertainty of the model is .1(s) estimated by performing appropriate process experiments (Ljung, 1987). The nominal model has the structure Fig. 1. Schema of the telephone cable foam line.
Pcs) =
The cable diameter is measured after cooling. Although the primary objective is to control the diameter of the cable, in practice the computed surface area of the insulation is used. The reason is that the surface area has a more linear relationship with the control variable. The plastic granules are melted in the plastic extruder. The screw in the extruder barrel transports the molten mass to the die. Il is then fed through the die on top of the copper wire to fonn an insulation layer. The screw speed is used to control the diameter of the telephone cable. The screw speed has its own control loop, for which the
r~II(S) ~12(S) 1
IPlieS) P (s)
(2)
21
where PlI(S) and P,l(S) are the transfer functions from the extruder screw speed to the computed insulation surface area and the capacitance of the telephone cable and
Pl2 (S)
and
P22(S) from the telescope position control to the computed insulation surface area and the capacitance of the telephone cable. From physical reasons it is easy to see that the transfer
CID controller computes the setpoinl.
functions
Pds)
and
Pzls) are of the form of an integrator
and a delay, because the telescope acts as an integrator and there is a delay in the diameter measurement. Foaming occurs so quickly that it has practically no dynamics compared with the the rest of the process. Based on measurement data first
After the extruder the foaming line has a cooling process, in which the hot insulation is cooled with water. The foaming of polyethene occurs very quickly after the extruder and foaming ends in cooling of insulation. Before actual cooling a
726
order models with delay are sufficient to describe the transfer functions PII(S) and P,,(s).
5. TUNING OF CID CONTROLLER
Several experiments are made at different operating points by changing the diameter and the capacitance of the telephone cable, and the line speed. Based on these the nominal model (2) and the uncertainty for the process were detennined. The gains in the model elements changed significantly as a function of diameter. Therefore the model gains are represented as functions of the diameter d, First order polynomial was sufficient to describe the dependence in all cases.
Proportional-Integral (PI) controller is used in the diameter control loop. Proportional (P) controller is sufficient in the capacitance control loop, since the loop already contains an integrator. This implies that the structure of the CID controller is
1 K,,(\ +-T )
The elements in the nominal model (2) are
;;;;;
=
C(s)
Kll(d) e- 65.' 5.0s+ 1 ; K,,(d) ;;;;; 1253d -224, K,,(d)
e- 50 .,
6.3 s+ J
=
K 12(d) e-'"
s
(4)
K,,(d) = 6.87 d - 7.76 ,
The control parameters are computed with Edmund's (1979) algorithm, in which the control parameters are optimized in frequency domain in such a way that the closed loop transfer function is as close as possible to the desired one. The desired transfer function is chosen as
(5)
(6)
; K22(d) = 7.68 d - 11.1
T/s) = diag
the diameter. It was decided to represent it at three different values of the diameter d 0.6,0.9, and 1.1 mm. These are correspondingly
.1(s) = diag {0.07(1
+ !Os), 0.45(1 + 3s)}
,
+ 4S)}
.
{u
1+ 1 ' t s \
where the time constant 't acts as a tuning parameter. The advantage of the tuning procedure is the possibility to use the multivariable process model to tune the scalar controllers in order to decrease the loop interactions, The tuning parameter 't is chosen so that the robust performance objective of the system is achieved, This implies that the closed loop system is stable and realizes the desired performance objective in spite of the uncertainty of the nominal model. The performance objective is defined by the weight transfer function
(8)
(9)
_ 1
W,(s) - 'tS' .,
The uncertainty is required to test robust performance, In such a case the uncertainty is decomposed into the form .1(s)
= Wl 3 W, '
(12)
I}'
(7)
+ 20s), 0.13(1 + !Os)} ,
.1(s) = diag {O.27( J + 8s), O.44( I
(11 )
o
where Kd is the control gain of the diameter, 1'; is the integration time, and Kc is the control gain of the capacitance.
The uncenainty ofthe nominal model Li(s) also depends on
.1(s) = diag {0.28(1
o
(3)
K 12(d) = 3852 d - 1844,
P12 (s)
is
=
(13)
This implies that the time constant of the closed loop transfer function must be smaller or equal to 't,. in spite of the uncertainty in the nominal model.
(10)
To check the nominal performance the test system
where W j and W1 are transfer functions with minimum phase and 1.1(s) L ,;; 1 (DoyJe, Wall, and Stein, 1980).
727
Q(s) =
lQII(S) QI2(S)j
the telescope position and capacitance is measured when the telescope position changes between 0 to 100%. A second order polynomial is fitted to the data
(14)
Q21(s) Q22(S)
Fig. 2 shows both the measured and computed values of the capacitance. The fit is sufficiently good for practical purposes. The second order polynomial is given by
is defined. Here (dropping the argument s and using notation of Doyle, Wall, and Stein (1980)
c =c(P) =0.053 p2
(15)
- 10.61 P + 2078,
(22)
where C = capacitance, P= position of the telescope and e(P) is the characteristic curve.
(16)
c,--___-----, "'"
(17)
1900 1800
Q" =
-W,PCII +PCr' Wt·
(18)
"00 1000 1500 0
Robust performance is achieved, if the structural singularity
IQ L < I (Doyle, Wall, and Stein, 1980; Morari andZaviriou, Because the diameter has significant effect on model gains and the magnitude of uncertainty, the controller is tuned at three different valuesofthediameter,d=0.6, 0.9 and 1.1 mm. Based on these the diameter dependent control gains are detennined. The integration time remains constant, because its variation with different diameter values is small. The time constant 't".1 dctcnnining the performance objective, is chosen to be 320 s,
The gain can now be computed by taking the derivative of c( P) which is
K(P) = cCP) = 0.011 P - 10.61 .
parameters of the diameter controller are
+0.00030,
T, = 6.63,
(19) (20)
kiP)
K, = 0.0027 d + 0.0016.
(23)
When the dependence of the process gain of the telescope position is known, the static nonlinearity caused by this can be compensated by changing the gain according to the operating point or the position of the the telescope. In previous section the controller was tuned when telescope position was 30%, so the change in gain is realized by multiplying the output of the controller with
which is sufficient for the telephone cable foam line. The
= -0.()()()18d
P
Fig. 2. The characteristic curve of a telescope, measured points C*), fitted curve C-).
1989).
K"
100
(21)
c(30)
- 7.31
(24)
= c'(P) = 0.11 P - 10.61
This implies that the product k(P)K(P) is constant. 6. COMPENSATION OF NONLINEARITIES
independent of the operating point. The process nonlinearily has been compensated and the value for control gain determined during tuning is used at telescope position 30%.
The position of the telescope creates a static nonlinearity in the process. Foaming occurs quickly after the extruder. This implies a large gain from the telescope position to capacitance, when the telescope is near thc cxtruder and a small gain when it is further from the extruder.
7. EXPERIMENTAL RESULTS In normal production the tolerance for variation in diameter is
To compensate for this non linearity the dependence between
± 20 mm and in capacitance ± 2 pF/m. Fig.
728
3 displays the
the desired perfonnance specification during all operating conditions. REFERENCES Doylc, I.e., Wall , I.E. and G. Stein (1982). Perfonnance and robustness analysis for structured uncertainty. IEEE Conference all Decision and Control. Orlando PL. Edmunds, J.M. (1979). Control system design and analysis using closed-loop Nyquist and Bode array•. Int. J. Control. 30, 773-802. Ljung, L. (1987). Sysrem Identification: Theory for User. Prentice Hall. New Jersey. Maciejowski, J. (1989). Multivariable Feedback Design. Addison-Wesley Publ. Co., Reading, Mass. Morari, M. and E. Zaviriou (1989). Robust Process Control. Prentice Hall. New Jersey.
729