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Multivariable Self-Tuning Control of a Paper Machine
Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993
MULTIVARIABLE SELF-TUNING CONTROL OF A PAPER MACHINE Li Qing-Quan*, Zhang Wei-Cun* and Chen Chang-Xiang** *Department of Automation. Tsinghua University. Beijing 100084. PRC **Wuhan Automatic Technology Research Institute. Wuhan 430010. PRC
Abstract: This paper presents an industrial application of multi variable combined self-tuning control techniques to a paper machine. The papermaking process is taken as a MIMO system with unequal time-lag, some time-varying parameters and unstahle inverses. In order to design a suitable control scheme for the process, a new control strategy, called multi variable combined self-tuner(MCST), with the U-D factorization recursive algorithm is pro· posed. The capability of the proposed scheme is demonstrated by its application to the basis weight loop and the moisture loop of the Long Wire Paper Machine(LWPM) at Juxian Paper Mill, Shandong, P.R.China. Keywords: Papermaking industry, Multivariable self-tuning control, U-D factorization algorithm.
INTRODUCTION
simultaneously, while the U-D factorization algorithm has good numerical stability.
The papermaking process is quite complicated. It can be viewed as a multi variable time-varying stochastic system. Basis weight(weight per unit area) and moisture control are important quality variables because they affect product quality and production economy. The specific aims are to keep the basis weight and moisture as close as possible to a chosen setpoint and the mean moisture content as high as possible; the deviations should have minimum statistical variance. Some results have been published on the modelling and adcontrol of paper machines. Beecher(1963), vanced Dahline(1970), Astrom(1967), Cegrell and Hedgqvist(1975) and Fjeld and Wilhelm(1981) implemented the modelling and the self-tuning minimum-variance control of paper machines respectively . These experiments on paper machines have shown that this type of stochastic identification and self-tuning control has many attractive properties. However these authors focused attention mostly on moisture control. In the course of an extensive research, it was found that a number of problems remained to be solved in case of applying these methods to basis weight control. The main problem is that the zeroes of the basis weight process may be unstable or poorly damped . Because of this, the self-tuning minimum-variance regulator(Cegrell and Hedqvist, 1975; Fjcld and Wilhelm, 1981) cannot be applied to the basis weight control loop. For the sake of settling the matter, Li Qing-Quan and Liu Hai-Yi(1988) implemented the modelling and the combined self-tuning control of paper machines. Long-time production run of several paper mills have shown that the combined self-tuner(CST) has good robustness and the satisfactory performances of servo-tracking and stochastic regulation. However CST is a single-input single-output controller and furtherance of the control performances was severely restricted. In order to raise the control performances still further and to meet the needs of production a new multivariable combined self-tuner(MCST) with the U-D factorization recursive algorithm was proposed. Extensive field experiments of the LWPM at Juxian paper Mill have shown that the MCST not only preserves the advantages of the single variable CST, but also overcomes the drawback that the performances of the basis weight loop and the moisture loop could not be satisfied
PROCESS FEATURES Determination of the papermaking process models based on physical considerations has been described by Beecher(1963) and Dahline(1970). However practical use of the resulting model was rather limited. The main reason is that the theoretical model is so simpe that it disagrees with dynamics of most paper machines. Li Qing-Quan and Liu Hai-Yi (1988) combined physics with information from experiments to obtain the dynamic model of the papermaking process.ln order to cover the shortage that the performance of the basis weight loop and the moisture loop can not be satisfied simultaneously, it is important that the papermaking process should be considered as a system with two input and two output. Preliminary analyses and experiments have shown that the above mentioned process has the following special features: (a) The papermaking process is a coupling two-input two-output system. It can be described two second-order differential equations with unequal time delay. (b )Because the thin stock flow and the machine speed directly influence the transport delay in the pipeline and the paper transport delay respectively, all the delay time of the system may be time-varying or slowly time-varying. (c)When the continuous time system with time-varying delay is sampled with a constant sampling interval, the discrete time system obtained often has zeroes outside the stability region. Design methods for the sampled systems which are based on cancellation of process zeroes can thus fail. Another consequence of the unstable zeroes is that such systems may be excited by unbounded input pulse sequencc giving a zero sampled output signal. (d)The system parameters vary with the thick stock concentration, the thin stock flow and the outlet velocity of the headbox. (e)There may be a constant or slowly-varying component in this system. In fact • there must still be a stochastic component. IDENTIFICATION The above stated process features are closely related to oper-
239
ating conditions. We have thus performed experiments to determine the model of the LWPM under various operating conditions. The design of experiments and some practical results obtained are as follows:
Table I. Estimates of the parameters 1
2
3
a" a"
-0.5609 -0.2050
-D.7222 -D.1509
-D.4983 -D.3572
a 21
-0.6167 -0.3835
-D.5265 -D.4846
-D.6736 -0.4784
0.1083 -0.03441
0.1184 -0.02991
-0.1103 -0.2223
0.1030 0.05718
0.09079 -0.05237
-0.1773 0.6545
b' 211
-0.2162 0.07728
-0.3657 -0,2462
-0.3463 -0.1935
b' 220 b' 21 !
0.2841 0.009165
0.7352 -D. 1344
0.4129 -0_02523
5.7182-EI2
-5 .2374-EI2
6.5449-E11
9.1646-E13
-1.3444-E11
-2.5230-EI2
case
I. Model
In accordence with the process features given above. a discrete time model corresponding to the wet basis weight loop and the moisture content loop can be given by A(q - ')y(k) = B/(q - 'lurk) + C(q - ')(k) (la) where A(q -')=diag{Ai(q - '). i= 1.2)
A
a" b'110
,
b'll!
Ai(q - ')= I + Laijq - j j- ,
b'120
b / 121
B/(q - ')= rq - d'I B/ij(q - ')]. I ~dii ~dij
,
B
B/ij(q - ')= Lb/i,q - I. b' iP ~O.i.j= 1.2
b'210
'_ 0
(Ib)
C(q - ')=diag{Ci(q-').
i=I .2}
C i (q - ')=I+c il q-' .i=I.2
--
y(k) = ry , (k).y 2 (k)] T c II
u(k) = [u, (k).u 2 (k)] T
---
C
c 21
«k) = re, (k).e 2 (k)] T and «k) denotes a vector of 2 uncorrelated sequences of ran dom variables with zero mean and covariance E ( e(k) eT (k)) = R. d,/ is the integral, time delay of the ijth clement of polynomial matrixB (q - '), q -, is the unit delay operator. i.e. q - i y(k) = y(k - i) For the sake of convenience all the variables are transformed into the normalizing signals.
The above results show that the data from experiments can be well modelled by the second-order models. There are very good agreements between the identification results and the the-
2. Samplinl( period
oretical models.
The choice of the sampling period is connected with many factors. such as process dynamics and characteristics of disturbances. the purpose and required accuracy of the model to be established. control computer performances and others. In practice. its choice is mainly based on some rule of thumb. The sampling period has been 30 seconds after all things considered. In this way the process time delay are about d l1 =2. d 12 =2. d 21 =2. d 22 = I.
MULTIVARIABLE
SELF-TUNING
It can be seen from Table 2 that the poles and the zeroes of the process may be unstable or poorly damped. Because of this. the self-tuning minimum-variance regulator (Cegrell and Hedqvist. 1975; Fjield and Wilhelm. 1981) cannot be applied to the LWPM . The results of feasibility study have show that the multivariable combined self-tuner (Li Qing-Quan and Zhang Wei-Cun. 1991) can be applied to the paper machine. Consider a multi-input-multi-output linear discrete-time randomly disturbed sy~tem described by the model A(q -')y(k) = B/(q - 'lurk) + C(q -')C(k) (2a)
3. Fillerinl(
All kinds of the noise are involved in the measured values due to the effect of various factors. We have found that there are the high frequency noises in the signals filtered through the pre-Iow-pass filters . The filter effects are thus supplemented with the high pass filters. 4. Experiments The determination of the system order is identical with single-variable CST( Li Qing-Quan and Liu Hai-Yi. 1988) in method. which the orders of the system are determined by an F-test. The experiments indicate that the second-order models are appropriate. The results obtained when the parameters are estimated under various operating conditions are given in Table 1. In Table 2 we also summmarize the obtained poles and zeroes of the second-order models. Table 2.
COMBINED CONTROL
where A(q -') = dial({A ,(q -'), i = 1.2 •. ..•p)
'.
+ L a'l q - I
A ,( q - ') = I
J- ,
B'(q -') = rq B
I
,iq
-1
- d" B' ,/q - ')]. '~ ~,
.-.
) = L. b ,~q
- A:
'l
(2b)
•
C(q -') = dial({ C ,(q - '),
Characteristic parameters of the second-order models
I ~ dl/ ~ d
i = 1.2 •. .. •p)
C,(q -')= 1 + IC,lq - 1 J- ,
I 1 - -f--- -0.8130.-0.2521 Poles -1.0001.-0.3834 1 -- -0.3177 case
2
3
0.9008.-0.1692
0.8967.-D.3984
1.0075.-0.4810
B(q -')= [q -d,.. B'(q - ')] 'I
1.I061.-D.4325 I---
0.2526
Let
-2.0154
f--- - - .--
Zeroes
_.
-
-0.5027
0.5768
3.6919
-0.3574
-D.6732
-0.5685
-D.1828
-0.0611
- - - ---
-0.0326
240
(3a)
where BD (q - ')
H , (q - ') must ensure the outputy;(k) track s the command
+ B u/(q
B(q -') = [BJq - ')] = B D(q - ')
- ')
Y M(k) in the steady state and cancels the noise /ilter dynamics
= diaK( B uCq - ').i = 1.2 •.. .• p}
which would otherwise contribute to the servo response pole set. This leads to L;(q - ')=Am;(q - ') (10)
"" B,,(q - ')= L,b'l/ q - ) 1- ,
(3b)
BUL(q - ')=[BUL(q - ')]I}=O. B UL(q - ') = [B Ul.(q - ')]" B,lq -') = q -,
"I'.-,b
_,
i=j
HJq
= [B,lq - ')]. i,£ j
,,.q
and d,jo is the maximum time delay that may occur in the ijth
s
B ' (q -' ). When d ') < d 'jo' it
element of polynomial matrix
is always possible that the corresponding coefficients of (3b) are assigned to be zero to equate B with B'. In the course of the analysis. for the reasons given above. let (4) B(q - ') = B' (q - ') there is no less of generality. Thus the system to be considered can also be described by
state. (12) Making use of (9) to (\ 2). equations (7) and (8) may be rewritten as A ~,(I)C ,(q - ')
(5)
A(q -')y(k)= B(q -')u(k)+ C(q - ')C(k)
By means of (2b) and (3b) every output component of (5) may be described as A ,(q -')y ,(k) = B ,,(q - ')u , (k)
±
+
u,(k)=
, B;;CI)F,(q
B ,,(q - ')u ,(k)
•
1- 1.10'1
+C,(q - 'lC,(k).
where y , (k).u ,(k) and (,(k) a re y(k). u(k) and (k) respectively .
the
ith
element
B ,, (I)Am/q
,
1
) "
K(k)+ ,
,
- I
L, - - _-, [F ;(q 1-,.I.,C,(q )
- 1
)(B J q
)
B}1)B ; (q -' ) )]u/k) ' B,,('I)
(14)
A,(q - '),B,,(q - '),BJq - '). and C,(q - ') in the above equa-
• L,K,/q - ')u/k)
tions are replaced by their estimate provided that the parameters of the process are unknown. The recursive parameter estimator is a key part of self-tuning control and can present numerical difficulties, especially when a microprocessor with relatively short word-lengths is used for self-tunning control. The coefficients of the LWPM model are thus estimated by the u-o factorization recursive algorithm ofThornton and Bierman(l978).
(7)
'-'.1"
where y M(k) is the ith clement of the setpoint y , (k). F ,(q - ') • G ,(q -'). H,(q - ') and
_,
_, y(k)+ F(q
Y ,(k) =
H,(q - ') _, _ , Y,/k) F,(q )L,(q ) G,(q - ') - - - - Y(k)F , (q - ' ) '
(13)
A~;(I)B,,(q - ')
of
On the principle of the combined self-tuning control (Li Qing-Quan. 1986). the system (6) should be controlled with the combined self-tuning control law u,(k)
)Am;(q
B(I)
- I_LB::(I)ulk)
(6)
i=I .2 ..... p
(I la)
Where A .(I)=[A(q-')] _, , rn, rn, • (lIb) { B(I)=[B(q - ')] q - , , 11 11 It is clear that the second term of the right-hand side of (8) is the coupling error component of the output. In order· to equate the coupling error component to zero in the steady
i,£ j
- t.
Am;(I) _, ) = B;;(I) C;(q )
are
L/q - ')
polynomials in the unit delay operator q
- \
controller
.K ,,(q
- 1
) is the
impulse transfer operator for the compensating action . Combining (6) and (7) give the following subexpression of the closed-loop system
SIMULATION AND REAL-TIME CONTROL
Y,(k) - 1
_ -
_, [A,(q
- 1
B/q )H,(q ) _, _, _, )F, (q ) + B ,, (q )G,(q )]L , (q
.;. F,(q - ')(B Jq - ') - B ,,(q - ')KuCq
+ ...
_,
;:: A ,(q
+
_, A,(q
,
)F,(q
) + B ,,(q
C,(q - ')F , (q -')
,
)F,(q
where F,(q -') is monic.
)+B,,(q
,
, )G , (q
-'»
R = [0.01 0. ] 0. 0.01 The initial conditions for the simulation were as follows: Parameters matrix elements are all zeroes with exception of b 110 = 1. b 220 = 1. Gain matrix is prO) = 10001
, u lk)
)G ;(q
,
I. Simulation Take case 3. in table I. for example. The simulation results are shown in Fig.1 ( the first 30 steps were cut out ) • where. the covariance matrix of noises is:
, Y,,(k) )
)
(k) )'
(8)
The simulation results illustrate that the behavior of the algorithm is excellent.
It is clear that the third term of
the right-hand side is the noise component of the output. The condition that minimize its variance is given by (Astrom and Wittenmark. 1984) A unique solution is guaranteed if A,. B" are coprime and the polynominal degrees satisfy the conditions {degF; =degB ;; - I (9b) degG; = degA; - I If the servo closed-loop poles are determined by the zeroes of
2. Real-time con/rol
The theoretical model and the identification results have shown that two loops of the LWPM can be described by (I). The desired servo closed-loop characteristic polynomi al were chosen as: Am(q
_ ,)=[A~,(q ~:)l=[1 +06q ~ : J A ~2(q)
the chosen polynomial A ~,(q -' ) i.e. the ith expected charac-
I
+ 0.8q
In order to investigate the control efficiency of the MCST. a comparison was made between the performance of
teristic equation of the closed-loop subsystem, the choice of
241
conventional control and that of the multivariable combined self-tuning control. In actual operation we can now consistently achieve standard deviation of 1.32 g / m 1 wet basis weight, while it was greater or equal to 2.41 g / m 1 before the control computer was installed. The economy a year is calculated roughly at 540 thousand yuan. The interactive on-line self-tuning control program has been implemented on the computer AST 286 / 140. The program is programmed in C language. The task which realizes the above described control strategy is only a little part of the microcomputer control system. Several other tasks arc needed in order to ensure the real-time connection among the process, the computer and the operator.
,,1
3
o
·3
b
30
60
90
120 '150 1.80
lno
240 270 SIlO
30
60
90
120 1~0 180 210
40 270 300
,,2
1 CONCLUSIONS The identification and multivariable self-tuning control of a paper machine has been presented . The aim of our work was
- 1.
to improve the produet quality and production economy and to give some useful innovations to the control design of paper machines. The design of experiments to determine the dynamo ie model was preceded by the analysis of the most essen tial relationship of the process and the identification was performed without interrupting the normal production. Based on the identified model an adaptive control of the basis weight loop and the moisture loop was realized. It is found that the procedure of identification and the MCST are quite practical and useful for the control of the paper machines with unstable in · verses. The results clearly demonstrate the adavantage of the new MCST over conventional controller and the sclf-tuning minimum-variancc controller.
o Fig.1
ACKNOWLEDGEMENT: This work was supported by the National Natural Science Fund of China. REFERENCES Astrom, KJ (1967), Computer control of a paper machine-an application of linear stochactic control theory, IBM J . Res. Develop.• Vol. 11 , No.4. pp. 389-405. Astrom, K.J . and B. Wittenmark (1984). Computer ControlSystem-Theory and Design, Prentice-Hall. led Englewood Cliffs, N .J. Beecher, A.E . (1963), Dynamic Modelling Technique in the Paper Industry, TAPPI, Vol.48 , No .2, pp . 117-120. Cegrell, T.and T . Hedqvist (1975), Successful Adaptive Con trol of Paper Machines. Automatica. Vol. 11, No.I, pp .53-59. Dahline, E.B . (1970), Interaction Control of Paper Machines, Control Enlfineerinlf. Vol.l7, No. 1. pp . 76-81. Fjeld, M. and R.G.wilhclm (1981), Self-Tuning Regulators-the Software way, Control Enlfineerinlf. Vo1.28 . No .II , pp. 99-102. Li Qing-Quan (1986), Combined Self-Tuner, Acta Automatica Sinica. Vol. 12, No .2. pp . 138-145. Li Qing -Quan and Liu Hai-Yi (1988) , Identification and Self-Tuning Control of a Paper Machine, Presented at the 8th IFAC / IFORS Symposium on Identification and System Parameter Estimation, Beijing, China. Li Qing-Quan and Zhang Wei-Cun (1991), Design of Multivariable Combined Self- Tuner for Unknown Varying Time Delays, Presented at 1991 Annual Meetings on Control Theory and Application, Weihai, Shandong, China. Thornton, C.L. and G.J . Bierman (1978), Filtering and error analysis via the UDU convarivance factorisation, IEEE Trans. Aut. Control. AC-23 , No . 5. pp.901-907.
242
MULTIVARIABLE SELF-TUNING CONTROL OF A PAPER MACHINE Li Qing-Quan*, Zhang Wei-Cun* and Chen Chang-Xiang** *Department of Automation. Tsinghua University. Beijing 100084. PRC **Wuhan Automatic Technology Research Institute. Wuhan 430010. PRC
Abstract: This paper presents an industrial application of multi variable combined self-tuning control techniques to a paper machine. The papermaking process is taken as a MIMO system with unequal time-lag, some time-varying parameters and unstahle inverses. In order to design a suitable control scheme for the process, a new control strategy, called multi variable combined self-tuner(MCST), with the U-D factorization recursive algorithm is pro· posed. The capability of the proposed scheme is demonstrated by its application to the basis weight loop and the moisture loop of the Long Wire Paper Machine(LWPM) at Juxian Paper Mill, Shandong, P.R.China. Keywords: Papermaking industry, Multivariable self-tuning control, U-D factorization algorithm.
INTRODUCTION
simultaneously, while the U-D factorization algorithm has good numerical stability.
The papermaking process is quite complicated. It can be viewed as a multi variable time-varying stochastic system. Basis weight(weight per unit area) and moisture control are important quality variables because they affect product quality and production economy. The specific aims are to keep the basis weight and moisture as close as possible to a chosen setpoint and the mean moisture content as high as possible; the deviations should have minimum statistical variance. Some results have been published on the modelling and adcontrol of paper machines. Beecher(1963), vanced Dahline(1970), Astrom(1967), Cegrell and Hedgqvist(1975) and Fjeld and Wilhelm(1981) implemented the modelling and the self-tuning minimum-variance control of paper machines respectively . These experiments on paper machines have shown that this type of stochastic identification and self-tuning control has many attractive properties. However these authors focused attention mostly on moisture control. In the course of an extensive research, it was found that a number of problems remained to be solved in case of applying these methods to basis weight control. The main problem is that the zeroes of the basis weight process may be unstable or poorly damped . Because of this, the self-tuning minimum-variance regulator(Cegrell and Hedqvist, 1975; Fjcld and Wilhelm, 1981) cannot be applied to the basis weight control loop. For the sake of settling the matter, Li Qing-Quan and Liu Hai-Yi(1988) implemented the modelling and the combined self-tuning control of paper machines. Long-time production run of several paper mills have shown that the combined self-tuner(CST) has good robustness and the satisfactory performances of servo-tracking and stochastic regulation. However CST is a single-input single-output controller and furtherance of the control performances was severely restricted. In order to raise the control performances still further and to meet the needs of production a new multivariable combined self-tuner(MCST) with the U-D factorization recursive algorithm was proposed. Extensive field experiments of the LWPM at Juxian paper Mill have shown that the MCST not only preserves the advantages of the single variable CST, but also overcomes the drawback that the performances of the basis weight loop and the moisture loop could not be satisfied
PROCESS FEATURES Determination of the papermaking process models based on physical considerations has been described by Beecher(1963) and Dahline(1970). However practical use of the resulting model was rather limited. The main reason is that the theoretical model is so simpe that it disagrees with dynamics of most paper machines. Li Qing-Quan and Liu Hai-Yi (1988) combined physics with information from experiments to obtain the dynamic model of the papermaking process.ln order to cover the shortage that the performance of the basis weight loop and the moisture loop can not be satisfied simultaneously, it is important that the papermaking process should be considered as a system with two input and two output. Preliminary analyses and experiments have shown that the above mentioned process has the following special features: (a) The papermaking process is a coupling two-input two-output system. It can be described two second-order differential equations with unequal time delay. (b )Because the thin stock flow and the machine speed directly influence the transport delay in the pipeline and the paper transport delay respectively, all the delay time of the system may be time-varying or slowly time-varying. (c)When the continuous time system with time-varying delay is sampled with a constant sampling interval, the discrete time system obtained often has zeroes outside the stability region. Design methods for the sampled systems which are based on cancellation of process zeroes can thus fail. Another consequence of the unstable zeroes is that such systems may be excited by unbounded input pulse sequencc giving a zero sampled output signal. (d)The system parameters vary with the thick stock concentration, the thin stock flow and the outlet velocity of the headbox. (e)There may be a constant or slowly-varying component in this system. In fact • there must still be a stochastic component. IDENTIFICATION The above stated process features are closely related to oper-
239
ating conditions. We have thus performed experiments to determine the model of the LWPM under various operating conditions. The design of experiments and some practical results obtained are as follows:
Table I. Estimates of the parameters 1
2
3
a" a"
-0.5609 -0.2050
-D.7222 -D.1509
-D.4983 -D.3572
a 21
-0.6167 -0.3835
-D.5265 -D.4846
-D.6736 -0.4784
0.1083 -0.03441
0.1184 -0.02991
-0.1103 -0.2223
0.1030 0.05718
0.09079 -0.05237
-0.1773 0.6545
b' 211
-0.2162 0.07728
-0.3657 -0,2462
-0.3463 -0.1935
b' 220 b' 21 !
0.2841 0.009165
0.7352 -D. 1344
0.4129 -0_02523
5.7182-EI2
-5 .2374-EI2
6.5449-E11
9.1646-E13
-1.3444-E11
-2.5230-EI2
case
I. Model
In accordence with the process features given above. a discrete time model corresponding to the wet basis weight loop and the moisture content loop can be given by A(q - ')y(k) = B/(q - 'lurk) + C(q - ')(k) (la) where A(q -')=diag{Ai(q - '). i= 1.2)
A
a" b'110
,
b'll!
Ai(q - ')= I + Laijq - j j- ,
b'120
b / 121
B/(q - ')= rq - d'I B/ij(q - ')]. I ~dii ~dij
,
B
B/ij(q - ')= Lb/i,q - I. b' iP ~O.i.j= 1.2
b'210
'_ 0
(Ib)
C(q - ')=diag{Ci(q-').
i=I .2}
C i (q - ')=I+c il q-' .i=I.2
--
y(k) = ry , (k).y 2 (k)] T c II
u(k) = [u, (k).u 2 (k)] T
---
C
c 21
«k) = re, (k).e 2 (k)] T and «k) denotes a vector of 2 uncorrelated sequences of ran dom variables with zero mean and covariance E ( e(k) eT (k)) = R. d,/ is the integral, time delay of the ijth clement of polynomial matrixB (q - '), q -, is the unit delay operator. i.e. q - i y(k) = y(k - i) For the sake of convenience all the variables are transformed into the normalizing signals.
The above results show that the data from experiments can be well modelled by the second-order models. There are very good agreements between the identification results and the the-
2. Samplinl( period
oretical models.
The choice of the sampling period is connected with many factors. such as process dynamics and characteristics of disturbances. the purpose and required accuracy of the model to be established. control computer performances and others. In practice. its choice is mainly based on some rule of thumb. The sampling period has been 30 seconds after all things considered. In this way the process time delay are about d l1 =2. d 12 =2. d 21 =2. d 22 = I.
MULTIVARIABLE
SELF-TUNING
It can be seen from Table 2 that the poles and the zeroes of the process may be unstable or poorly damped. Because of this. the self-tuning minimum-variance regulator (Cegrell and Hedqvist. 1975; Fjield and Wilhelm. 1981) cannot be applied to the LWPM . The results of feasibility study have show that the multivariable combined self-tuner (Li Qing-Quan and Zhang Wei-Cun. 1991) can be applied to the paper machine. Consider a multi-input-multi-output linear discrete-time randomly disturbed sy~tem described by the model A(q -')y(k) = B/(q - 'lurk) + C(q -')C(k) (2a)
3. Fillerinl(
All kinds of the noise are involved in the measured values due to the effect of various factors. We have found that there are the high frequency noises in the signals filtered through the pre-Iow-pass filters . The filter effects are thus supplemented with the high pass filters. 4. Experiments The determination of the system order is identical with single-variable CST( Li Qing-Quan and Liu Hai-Yi. 1988) in method. which the orders of the system are determined by an F-test. The experiments indicate that the second-order models are appropriate. The results obtained when the parameters are estimated under various operating conditions are given in Table 1. In Table 2 we also summmarize the obtained poles and zeroes of the second-order models. Table 2.
COMBINED CONTROL
where A(q -') = dial({A ,(q -'), i = 1.2 •. ..•p)
'.
+ L a'l q - I
A ,( q - ') = I
J- ,
B'(q -') = rq B
I
,iq
-1
- d" B' ,/q - ')]. '~ ~,
.-.
) = L. b ,~q
- A:
'l
(2b)
•
C(q -') = dial({ C ,(q - '),
Characteristic parameters of the second-order models
I ~ dl/ ~ d
i = 1.2 •. .. •p)
C,(q -')= 1 + IC,lq - 1 J- ,
I 1 - -f--- -0.8130.-0.2521 Poles -1.0001.-0.3834 1 -- -0.3177 case
2
3
0.9008.-0.1692
0.8967.-D.3984
1.0075.-0.4810
B(q -')= [q -d,.. B'(q - ')] 'I
1.I061.-D.4325 I---
0.2526
Let
-2.0154
f--- - - .--
Zeroes
_.
-
-0.5027
0.5768
3.6919
-0.3574
-D.6732
-0.5685
-D.1828
-0.0611
- - - ---
-0.0326
240
(3a)
where BD (q - ')
H , (q - ') must ensure the outputy;(k) track s the command
+ B u/(q
B(q -') = [BJq - ')] = B D(q - ')
- ')
Y M(k) in the steady state and cancels the noise /ilter dynamics
= diaK( B uCq - ').i = 1.2 •.. .• p}
which would otherwise contribute to the servo response pole set. This leads to L;(q - ')=Am;(q - ') (10)
"" B,,(q - ')= L,b'l/ q - ) 1- ,
(3b)
BUL(q - ')=[BUL(q - ')]I}=O. B UL(q - ') = [B Ul.(q - ')]" B,lq -') = q -,
"I'.-,b
_,
i=j
HJq
= [B,lq - ')]. i,£ j
,,.q
and d,jo is the maximum time delay that may occur in the ijth
s
B ' (q -' ). When d ') < d 'jo' it
element of polynomial matrix
is always possible that the corresponding coefficients of (3b) are assigned to be zero to equate B with B'. In the course of the analysis. for the reasons given above. let (4) B(q - ') = B' (q - ') there is no less of generality. Thus the system to be considered can also be described by
state. (12) Making use of (9) to (\ 2). equations (7) and (8) may be rewritten as A ~,(I)C ,(q - ')
(5)
A(q -')y(k)= B(q -')u(k)+ C(q - ')C(k)
By means of (2b) and (3b) every output component of (5) may be described as A ,(q -')y ,(k) = B ,,(q - ')u , (k)
±
+
u,(k)=
, B;;CI)F,(q
B ,,(q - ')u ,(k)
•
1- 1.10'1
+C,(q - 'lC,(k).
where y , (k).u ,(k) and (,(k) a re y(k). u(k) and (k) respectively .
the
ith
element
B ,, (I)Am/q
,
1
) "
K(k)+ ,
,
- I
L, - - _-, [F ;(q 1-,.I.,C,(q )
- 1
)(B J q
)
B}1)B ; (q -' ) )]u/k) ' B,,('I)
(14)
A,(q - '),B,,(q - '),BJq - '). and C,(q - ') in the above equa-
• L,K,/q - ')u/k)
tions are replaced by their estimate provided that the parameters of the process are unknown. The recursive parameter estimator is a key part of self-tuning control and can present numerical difficulties, especially when a microprocessor with relatively short word-lengths is used for self-tunning control. The coefficients of the LWPM model are thus estimated by the u-o factorization recursive algorithm ofThornton and Bierman(l978).
(7)
'-'.1"
where y M(k) is the ith clement of the setpoint y , (k). F ,(q - ') • G ,(q -'). H,(q - ') and
_,
_, y(k)+ F(q
Y ,(k) =
H,(q - ') _, _ , Y,/k) F,(q )L,(q ) G,(q - ') - - - - Y(k)F , (q - ' ) '
(13)
A~;(I)B,,(q - ')
of
On the principle of the combined self-tuning control (Li Qing-Quan. 1986). the system (6) should be controlled with the combined self-tuning control law u,(k)
)Am;(q
B(I)
- I_LB::(I)ulk)
(6)
i=I .2 ..... p
(I la)
Where A .(I)=[A(q-')] _, , rn, rn, • (lIb) { B(I)=[B(q - ')] q - , , 11 11 It is clear that the second term of the right-hand side of (8) is the coupling error component of the output. In order· to equate the coupling error component to zero in the steady
i,£ j
- t.
Am;(I) _, ) = B;;(I) C;(q )
are
L/q - ')
polynomials in the unit delay operator q
- \
controller
.K ,,(q
- 1
) is the
impulse transfer operator for the compensating action . Combining (6) and (7) give the following subexpression of the closed-loop system
SIMULATION AND REAL-TIME CONTROL
Y,(k) - 1
_ -
_, [A,(q
- 1
B/q )H,(q ) _, _, _, )F, (q ) + B ,, (q )G,(q )]L , (q
.;. F,(q - ')(B Jq - ') - B ,,(q - ')KuCq
+ ...
_,
;:: A ,(q
+
_, A,(q
,
)F,(q
) + B ,,(q
C,(q - ')F , (q -')
,
)F,(q
where F,(q -') is monic.
)+B,,(q
,
, )G , (q
-'»
R = [0.01 0. ] 0. 0.01 The initial conditions for the simulation were as follows: Parameters matrix elements are all zeroes with exception of b 110 = 1. b 220 = 1. Gain matrix is prO) = 10001
, u lk)
)G ;(q
,
I. Simulation Take case 3. in table I. for example. The simulation results are shown in Fig.1 ( the first 30 steps were cut out ) • where. the covariance matrix of noises is:
, Y,,(k) )
)
(k) )'
(8)
The simulation results illustrate that the behavior of the algorithm is excellent.
It is clear that the third term of
the right-hand side is the noise component of the output. The condition that minimize its variance is given by (Astrom and Wittenmark. 1984) A unique solution is guaranteed if A,. B" are coprime and the polynominal degrees satisfy the conditions {degF; =degB ;; - I (9b) degG; = degA; - I If the servo closed-loop poles are determined by the zeroes of
2. Real-time con/rol
The theoretical model and the identification results have shown that two loops of the LWPM can be described by (I). The desired servo closed-loop characteristic polynomi al were chosen as: Am(q
_ ,)=[A~,(q ~:)l=[1 +06q ~ : J A ~2(q)
the chosen polynomial A ~,(q -' ) i.e. the ith expected charac-
I
+ 0.8q
In order to investigate the control efficiency of the MCST. a comparison was made between the performance of
teristic equation of the closed-loop subsystem, the choice of
241
conventional control and that of the multivariable combined self-tuning control. In actual operation we can now consistently achieve standard deviation of 1.32 g / m 1 wet basis weight, while it was greater or equal to 2.41 g / m 1 before the control computer was installed. The economy a year is calculated roughly at 540 thousand yuan. The interactive on-line self-tuning control program has been implemented on the computer AST 286 / 140. The program is programmed in C language. The task which realizes the above described control strategy is only a little part of the microcomputer control system. Several other tasks arc needed in order to ensure the real-time connection among the process, the computer and the operator.
,,1
3
o
·3
b
30
60
90
120 '150 1.80
lno
240 270 SIlO
30
60
90
120 1~0 180 210
40 270 300
,,2
1 CONCLUSIONS The identification and multivariable self-tuning control of a paper machine has been presented . The aim of our work was
- 1.
to improve the produet quality and production economy and to give some useful innovations to the control design of paper machines. The design of experiments to determine the dynamo ie model was preceded by the analysis of the most essen tial relationship of the process and the identification was performed without interrupting the normal production. Based on the identified model an adaptive control of the basis weight loop and the moisture loop was realized. It is found that the procedure of identification and the MCST are quite practical and useful for the control of the paper machines with unstable in · verses. The results clearly demonstrate the adavantage of the new MCST over conventional controller and the sclf-tuning minimum-variancc controller.
o Fig.1
ACKNOWLEDGEMENT: This work was supported by the National Natural Science Fund of China. REFERENCES Astrom, KJ (1967), Computer control of a paper machine-an application of linear stochactic control theory, IBM J . Res. Develop.• Vol. 11 , No.4. pp. 389-405. Astrom, K.J . and B. Wittenmark (1984). Computer ControlSystem-Theory and Design, Prentice-Hall. led Englewood Cliffs, N .J. Beecher, A.E . (1963), Dynamic Modelling Technique in the Paper Industry, TAPPI, Vol.48 , No .2, pp . 117-120. Cegrell, T.and T . Hedqvist (1975), Successful Adaptive Con trol of Paper Machines. Automatica. Vol. 11, No.I, pp .53-59. Dahline, E.B . (1970), Interaction Control of Paper Machines, Control Enlfineerinlf. Vol.l7, No. 1. pp . 76-81. Fjeld, M. and R.G.wilhclm (1981), Self-Tuning Regulators-the Software way, Control Enlfineerinlf. Vo1.28 . No .II , pp. 99-102. Li Qing-Quan (1986), Combined Self-Tuner, Acta Automatica Sinica. Vol. 12, No .2. pp . 138-145. Li Qing -Quan and Liu Hai-Yi (1988) , Identification and Self-Tuning Control of a Paper Machine, Presented at the 8th IFAC / IFORS Symposium on Identification and System Parameter Estimation, Beijing, China. Li Qing-Quan and Zhang Wei-Cun (1991), Design of Multivariable Combined Self- Tuner for Unknown Varying Time Delays, Presented at 1991 Annual Meetings on Control Theory and Application, Weihai, Shandong, China. Thornton, C.L. and G.J . Bierman (1978), Filtering and error analysis via the UDU convarivance factorisation, IEEE Trans. Aut. Control. AC-23 , No . 5. pp.901-907.
242