- Email: [email protected]
Dual Model Reference Adaptive Control on Paper Machine Headboxes
Copyright
© IFAC PRP 4 Automation, Ghent , Belgium 1980
DUAL MODEL REFERENCE ADAPTIVE CONTROL ON PAPER MACHINE HEADBOXES A. Nader* Imperial College of Science and Technology, Computing and Control Department, London, SW7, U,K ,
Abstract . When industrial processes operate under variable production conditions, linear models and linear controllers are generally not effici ent. When a paper machine operates at variable speed, the control of the headbox must be efficient for large operating point transitions. We present a dual model reference adaptive controller (DMRAC) that allows the control of linear time varying systems and the explicit on line identification of the process with delayed inputs. The application of the DMRAC to a pressurized headbox when large operating point t r ansitions are applied is presented. Finally a functional decomposition of the DMRAC algorithm for its implementation in a multi - Il,icroprocessor system is presented . Keywords. Adaptive Control, Adaptive Systems , Computer Control , Decoupling, Dual Control, Multi - processing , Multi - variable Control, Paper Industry, Process Control, Time Varying Systems . INTRODUCTION Most industrial processes operate under variable production conditions. Profit maxi mization is obtained through production and cost optimization. The operation of industrial processes on a wide spectrum of operating points obliges control designers to consider the non- linear nature of industrial processes.
These operating conditions require a good headbox control.
However, the state of the art in optimal multivariable non- linear control theory does not always allo.~ one to design suitable non- linear controllers (e . g . non- linearities in the states and inputs of the process). Linear controllers have excel:ent performance when they operate near an operating point of the process , but they behave poorly when the process operates far from the operating point the control system was designed for.
At present control of headboxes gene r ally implemented with two single PI controller s . The level and the total head at the slice are controlled . However, it is well known that there exist strong interactions between these two loops - furthermore it is necessary to control the consistency and the substance flowrates. We will present first a dual model reference adaptive controller (DMRAC) that allows the control of linear time varying systcns and the explicit on- line identification of the process with delayed inputs. A simple application example is presented. We then present the application of the DMRAC to a pressurized headbox when large operating point transitions are applied.
Between linear and non-linear control systems are the adaptive control systems . Linear Time Varying systems can be effi ciently controlled by adaptive controllers.
Finally a functional decomposition of the algorithm for its implementation in a multi micro- processor system is presented .
In the paper industry , in order to optimlze production, it is necessary to control the paper machine ' s speed. The paper machine operates at variable speeds according to the evaporation capabilities of the drying plant .
239
A. Nader
240 DUAL MODEL REFERENCE ADAPTIVE CONTROLLER
In this section we show a new multi - variable dual model reference adaptive controller that has been developed based on model reference adaptive systems MRAS. uMRAC take int o account linear time varying systems explicitly and can also treat the case of processes with delays. Model reference adaptive systems can be traced back to 1958 with Whitaker, Yarmon and Kezel . The introduction of stability theory concepts into the design procedure of adaptive systems was first developed by Parks in 1966 . Landau (Landau , ~_97 8 ) developed in 1973 a multi - variable continuous model reference adaptive controller. Several discrete monovariable approach es have been developed (Bethoux and Courtiol, 1973 ) DMRAC was developed from one of them. The basic idea behind the DMRAC is the applic ation of the adaptive control to an ad j ustable model parallel to the process . This model is adjusted through an on line identification mechanism according to the variation of parametres of the process . Most industrial processes are non- linear. When an operating point transition is performed , one can interpret it as a variation of parametres of the linearized model from the parametres of the ini tial operating point to the parametres of the final operating point . This variation of parametres is identified and the parallel model to the process is updated on line .
DMRAC have in addition to the MRAC an internal adjustable model parallel t o the process (fig . 1). The dual controller is formed by two subsystems ( fig. 2) structurally identical and identical t o the MRAC structure Notice the duality between the reference model and the process; the adjustable system and the adjustable model; and between the adaption law and identification law . DMRAC Design The design strategy is entirely based on Popov's hyperstability theory. We first enunciate a hyperstability theorem . Theorem (Pop::lv): Let ~+l vk
= ~+B~ = C~ + ~
be a linear system where (A, B) is completely controlable and (A,C) is completely observable. The linear system is feedback by the non-linear system w k
=f
(v,k,l,) l~k
satisfying the Popav in-equality:
v.. k_.L> k0 where
02
is a finite positive constant.
A necessary and sufficient condition to ensure that the linear system feedback by the non-l inear is asymptotically hyperstable lS that the transfer matrix
Reference Model
H(z ) lS
Adaptiou MKhauilm
=E
+ C (zI- A) - lB
strictly real positive.
Now we will present the design procedure of the DMRAC , considering first the control system and afterwards the identification system .
.------->\Rdef1!DC Modf1i 1 - - - - - - - - - ,
Set Point
FIGURE 2
FIGURE 1
MRAC
Dual Model Reference Adaptive Control
24 1
Y + = [BpKm + Bp (6Kp(k+l ) - Kp)] Y + APY k k l k Bp [ Ku +6 Ku(k +l ) ] zk + BpKme
" ' -.,
J
k
Linear
SYltem 1 _____ ---oJ
.. ~ I
'---~I
NOD Linear Sy.t e m
I
vir.
II+-----J
(Am-BpKm )e
e + k l
BpKp - BpKin]Y
k
k
+ ~-Ap- Bp~p (k+l) +
+ [m - Bp (Ku+6)
taking the perfect model following conditions. Am - Ap = Bp (Kin- Kpo)
o Where Kpo and Ku are the unknown values of Kp and Ku that guarantee perfect model followin g Kpo = lim ~P-: \Kp( k+l
D
11._00
Kpo = lim [Ku+,6Ku(k+ID
kA
EQuation I can then be rewritten
with As(k +l)
[Ap+BP.L\Kp(k+l) - BpKp + Bp~
Bs(k+l)
Bp [Ku +!\Ku(k+l)=
and posing K = As ( k+l) Let us discuss the control system . The reference model is defined:
and the adjustable internal model is:
Bp~~
=1~&- ,,11 (v ,k,l )
+
"2
(v , k) + As (0)
Bs (k+l) = k> ~i (v , k,l ) + ~~ (v , k) + Bs (0) 1=0 one obtains 6Kp(k+l)
=l~t"l (v , k , l) + " 2(v , k ) +!:,.Kp(o)
6Ku(k+l )
=1~6~1 ( v,k,l ) + i 2 ( v- , k) + LKu(o)(~)
A particular solution for "1' "2' i l , We postulate the following control law:
that satisfies Popov
i~eQuality 1S:
Where Kp(k) ;;; Kp - ..:i Kp (k+l) Ku(k) ;;; Ku +/ Ku(k +l)
i 2 (v , k) = SPvk +1
So the output of the adjustable internal model is:
Equations 3 and L are defined as:
I.A .A. -R
i2
(3)
A. Nade r
242
~Kp(k+l) = ~~pl(k+l) +~ KpP(k+l)
The transfer function is:
~K/(k+l) =~K/(k)
H(z) = DBp+D[Am- BpKm][zl- Am+BpKm] - lBP
+ FV k +l [GyJ T
Applying the Kalman Szego Popov real pos i tive theorem one can show that:
i<:pP(k+l) = FPv + [Gy) T k l
~Ku(k+l) = ~Kul(k+l) +l::,KUP(k+l) 1 l::,KU (k +l) = ~Kul(k) + Sv k+l P /\Ku (k+l) = SPv + [Hz J T k l
where P i s the solution of the Lyapounov equation T
P(Am- BpKm) - P = Q'
Then equation 2 is:
(Am- BpKm)
e + =/,ACl- BpKm)e + Bp [Kp- KpO -6K pl (k)] Yk k k l
Fol l owing a similar procedure for the identification system , if the output of the fixed part of the int Ernal model is:
+ Bp[Ku
O
1 - Ku---6Ku (k) ] zk - BP[(F+FP)YkTGYk
Yk+l = Amyk + Bm~ + (S +SP)zk THzk ]
v + k l and the postulated control law
Defining
then ( 5)
where 1 1 0 Yk +l = As (k)Y k + Bs (k)Zk + Ke k 1 As (k) 1 Bs (k)
1 Ap + BP0J
e 2k = yprocess
k
- Yk
v 2k +l = [1 + D2 BJl2(k)] -1
v~k+l
Where
K = BpKm Equation 5 can be rewritten as:
and
e o+ = (Am- BpKm)e + Bp [ Kp-Kp 0- LKp 1 (k}j~ Y k k l k O + Bp[Ku - KU-6 Ku1 (k)] zk
Ku2 +I\Ku2 (k+l)
=~Kp~
Then -
v k +l = [ I + DBpN(k)j
-1
0
v k+l
o v k+1 c"-:~ be calculated wi :hout a~proxi matlon as the control law lS 2?plled to a known model . We decided that instead of . 0. . calculatlng v. 1 :et could be approxunated by v. as prop§~ed by Bethoux and Court iol for f~e monov~riable case in order to save computing time (Sethoux and Courtiol, 1973)
Applying t~e Popov hyperstability theorem we ~ave ~ow to ca~culate the matrix D figure 3 so that the linear syste:n has a strictly real positive transfer fur.ction .
FlGrRE J
L\Kp~(k+l)
(k+l)
= FP 2 v2 k +l
+!\K 1 (k+l) - p2 LG YkJ T
L\KP; (k+l) =I\KP; (k) + F2 v2 k +l [GYk] T P 6 Ku2 (k+l) = /\Ku 2 (k+l ) +/\ KU; (k+l) T 6" (k+l) = SP2 v k+l [H uk] ."uP 2 2
6 Ku 2
I
(k+l) =6KU; (k) + S2 V2 k+l [Hu,KJ
T
The D~ffiAC obtained is shown In figure 4 . As the control structure developed ~ e re iYlcludes an inter.1al model parallel to the process , the basic idea of the Schmidt corrector can ::.J so be applied . DR~-1AC with delays is shown in figure 5.
Dua l Mode l Refe r ence Adaptive Con t rol
A.
243
Internal Model Control System. 1.
Reference model output.
2.
Adjustable model output. ~ +l = Ac~ + Bc~
3.
Pr ocess output.
4.
New set point.
5.
D. C. component suppression. X +
FIGURE ..
k l
-
xo yo
Zk+l - zo YO~+l
= yom k + l - yo
6.
Computation of vO~+l
7.
Predictive computation of vO~+ 2
o
vO~ + 2 ~ vO~ + l
. . approxlmatlon
r-
Il -l o
vO~ +2 = LI+D BC< Ku2(k)) N(k+ll.;
vo~+2
T
N (k+l) = (F+FP) yom + G yom + k l k l T
+ (S +SP)zok+l Hzo k +l
8.
Computation of~Kp(k+ 2) .6 Kl(k +2)
~KpI(k+2)
FPvo~+2
[ Gyank+1J
=.6 Kp I(k+l) +
T
Fvo~;+2 [
GYaT'k+1JT 6Kp(k+2) =.6 Kp P (k+2) +.6 KpI(k +2)
9. FIGURE 5
Canputation of.6Ku(k+2) r ]T .6KuP (k+2) SP vO~ + 2 LHzo + k 1
I I 6 Ku (k+2) =6 Ku (k+l) + S
DMRAC Algorithm
vO~+2
[
Hzok+~T
Initial izations .6Ku(k+2) =.6KUP(k+2) +6 KuI (k+2)
Loop 10.
Computation of Ku(k+l) and Kp(k+l) Ku(k+l) = Ku +
Ku(k+2)
A. Nader
244
Kp(k +1 ) = 11.
Kp -
APPLICATIONS
Kp ( k+2)
Simple Example
Conputati on of uO + k 1 uO + = - Kp (k+1 )yon + + Ku(k+1)zo k 1 k 1 k+1 + Kr!lxo k +1
B. Identification System. 12 .
Here we present a basic example to show the behaviour and function of the DMRAC . The DMRAC is applied to a first order linear system , assumed not well known . This is equivalent to a step parametric disturbance. The process is:
Computation of v0 2 + k 1 v0 2k +1 = D2 (yok+1 - YO~ +l)
The fixed part of the adjustable model is: 1 3.
Predictive computation of v02 + k 2
~+l = ac ~ + bc ~
v02°K+ 2 <- v0 2 k + 1.' a pproximation v02 + = k 2
[I +D2BcN2(k+l)0
-1
The reference model is: 0
v02 + k 2
T
N2( k+l) = (Fi+ FP2 )YO~+ 1 G YO~+l
where the process is :
+ (S 2+SP2) uo k +l Huo +l k 14.
Computation ,\ Kp2
P
ap bp
of~Kp2(k+2)
(k+2) = FP2v02 +2 k
LGyO~+ 1 J
T
The assumed model is: ac bc
~Kp2I(k+2 ) = ~Kp2 I(k+l) + F2V02 + [ k 2
JT
Gy om k + 1
0 . 8153 0.1846
The reference model is:
P
I
LKp2 ( k +2) =,6,Kp2 ( k+2) +L:::,.Kp2 (k+2)
15.
0.6049 0.3950
amr bmr
0 . 8187 0 . 1812
Computation ofL:::,.Ku2 (k+2) P l::,Ku2 ( k+2) = SP2v02 + [HUOk+lJ T k 2
~Ku2 I ( k+2 ) =L:::,.Ku2 I (k+l) + S2v02 k +2 [ ~T Huok+1J
The adapt ion and identification gains are : f
fp = s = sp = f'2 = fp 2 = s 2 = sp 2 = 100
g
h = g2 = h2 = 1
The corrector d
d 2 = 0.1034
and the gains: 16 .
Computation of Kp2( k+l) and Ku2(k+l ) Kp2( k+l )
Kp2 -6.Kp2( k+2 )
Ku2( k+l )
Ku2 + L:,Ku2( k+2 )
Computation of UOQk+l uom + = - Kp2 ( k+1 )yon + + Ku2 k 1 k l (k+l ) uO + k l
: 8.
Cor:trol signal U
Close loop
+
= km2
= O.
kp = - 0.01818 ku = 0.9818 kp2 = 0.0 ku 2 = 1.0 xo = yo = ymo
umo = uo
To achieve the fastest convergence of parametres we apply a pseudo random binary sequence of ~ax~um period of 63 to zk' The sampling per10d 1S 1 sec . The convergance of parametres is show!'. in o f1. . 6 . The values of kp 0 , ku 0 , kP2 and 8 ku found by the J~.ffiAC guarantee perfect 2 moiiel following:
k +l = uOk +l + uo uO~ + l
km
uo~
k 0 Po ku 0 kP2 0 kU 2
- 0 . 5411 0 . 4588 1 . 139 2.139
Dual Mode l Reference Adaptive Control ' , " CD (x' - y,) The crlterlOn L1 1
2
=
0 . 272 .
245
DMRAC Applied to a Pressurized Headbox
i =l This simple example shows that identification and perfect model following are achieved with the DMRAC .
We first present th e non- linear model of a pressurized headbox , fig . 7 . This model has been verified experirnentally (N ader , 1976 ) .
screen Qe
Qp
Ce
Cp Qd Cd
Fig . 7 - Head Box Circuits The state non linear model is: dh
X
dt
2
_ l (f
dX 2 dt
h P
+
lx 2
2 2
1 28
(Q
2
e
AX )2 1_ g 2 j
s
1 dP dt
= ;.. (H
-h)
m
RT=(Ql' - Qo ) + PAX 2 :
dCc _ Qe [ Cp( t-r )- Cd(t - r ) Qp(t - r) + Cd(t - r)-Cc~J' dt - Ah Qe(t-r ) Where :
Q ()
=K 0
P ( _a_ ) l /¥ \ ~ -------=----- \ /(P+P ) p.. Pa (¥ - ~/¥ P+P
a
V¥- l
a
a Ll- (p+p ) a
This model shows the multi variable non linear character of the headbox . The non linearities present in this model are o f a ge neral type. The stock flow Qp is a delayed input . The essential objectives of the control system are : decoupling the multi variable system and tracking for operating point transitions. We now present the applications of the D~ffiA C to the non linear model of the headbox. Two sL~ulation results will be presented ; the first in which the m,r uc has been dl':'signed using the linearized model of the initial operating point of the headbox and the second in which the D~ffiAC was designed using a wrong model of the process .
A. Nader
246
Q
_-------:-0
o
~
\
\
o
Dua l Mode l Refe r ence Adap t ive Co ntr o l The in i tial operating point is characterised by: ho
0 . 1176 m
Qeo
0 .0201 m3 /s
Po
26253 Pa
Kio
6 2 0.7454 x 10- m
Cco
4 . 2 gm/l
Qpo
. 00444 m3 /s
T
283
01
. 00679 m
A
2 0 . 34 m
Pk
31200 Pa
RLZ
0 . 40 m
Cp
1 5 gm/l
Pa
101325 Pa
Cd
1. 25 gm / l
Ko
6 2 0 . 3591 x 10- m
0
K
The discrete reference model
r-)
"---
0 . 95 12
Amr
~- ./
0 . 8187
C)
0.1324 x 10- 1
(
0.18126
/
=1
The sampling period
Cont r ol a l gor ithm ' s decomposition should take into account computing and control criteri a : load balanc i ng , interprocess communication , graceful degradation , control functions , etc .
sec .
The adapt ion and identification gains are : \1000
I r -,
G
1000
F
I
F
= FP = S
l
500
I F2
(
, /
500 (
. 08
6J SP
F2
J
= FP2 = S2 = SP2
Th r ee operating point step changes were performed for each simulati on : change of level , air pressure and headbox concentration set po i nts . ho
0 . 11 7 m to !"lo
0 .1 7 m
Po
26253 Pa to Po
27000 Pa
Cco
4 . 2 gm /l to Cco
= 5. 2
FUNCTIONAL DECOMPOSITION OF THE DMRAC ALGORITHM
The DMRAC structure is particularly well suited for its implementation in a multi - micro processor architecture . A functional decompos ition of the algorithm can increase the reli ability of the new controller .
1 0.4877 x 10-
Bmr
Figures 11 , 12 and 13 represent set point changes of the level , pressure and concentrat ion in the headbox when the process is simulated by the non linear model and the D~ffiAC has been designed using a li near model that does not represent the headbox neither at the initial nor at the final operating point . Here again the decoupling is perfect , except for the leve l set point change in which the pressur e changes very little at the beginning of t he step. The tracking of the refe r ence model is perfe ct fo r the level change . For the pressur e the tra cking is not well achieved . The concent rat i on cannot track the reference model because of the effect of the deadtime .
The advent of cheap mi cro processors a l lows one to develop specialized, sophisticated contollers that can be localized in the plant site as wi t h analogtechnol ogy at pr esent . Distributed Comput er Control Systems (DCCS) allow one to distribut e functionally and/o r geogr a phically the computing power . DCCS are also r equi r ed when the performance and/or reliabi l ity of a single computer is inadequate to mee t the r e quirements of the control system .
1S:
0 . 9867
24 7
gn/l
Figures 8, 9 and 10 represent set point changes of the level , pressure and concent r ation in the headbox when the process is SiMUlated by the non linear model and the DMRAC has been designed using a linear model established around the intial operating point . The decoupling and tracking qualities of the DMRAC are excellent. Perfect model following and per fect decoupling is achieved .
Funct i onal decomposition of centralized control algorithms is natural since control blocks have a functional meaning . However , different levels of decomposition are possible : the control function level , the matheMatical funct i on l evel , the instruction level , the logical function level , etc. The increase of software processes in order to obtain maximum parallelism implies the multiplication of interprocess communication and synchroniz ation, then the increase of complexity of software processes management and decrease o f performance . The trade off of level of decoMposition) degree of complexity of software process manageMent and conservati on o f control meaning has to be considered. The major objectives of D~·~AC algorith:!l decomposition can be to minimize its execution time while conserving some control functions meaning . Taking into account these factors we will consider fo r the mffiAC deCOMposition each one of the algorithm steps (1 to 18) mentioned in t he algorithm section as indivisible sequential software processes except steps 5 and 18 . Step 5 is divided into three (5a , 5b , 5c) steps . 5a performs the DC component suppress -
A. Nade r
248
ion of the reference model and set point only (x , z). Simil arly 5b considers the parallel model DC suppression (y m) and 5c the process (y) DC suppression .
tasks ( i.e. 1,4 , 5a, 6 , 7 , 8 , 9 ,10 , 11 , 18a ) and t he other the identification tasks (i.e . 2 , 3 , 5b , 5c,12 , 13 , 14 , 15,16 ,17,1 8b) fig. 15. yom , KU 2
Step 18 is divided i n : 18 a computation o f t he process co~t r ol signal (u) and 18b computation o f the model cont r ol signal ( urn ) .
Uo
Process
We pres ent in figure 14 one possib le decompositi on of the D;,lF.AC algoritiur. for multi processing using a Petri Net description (Nader 1979) .
y
fig. 15. The processors communication needed is only the of vectors ~om , uo and,the matrix Ku2 . The 1nterest of th1S apprach 1S that each processor has a very well defined function: processor 2 identifies the process and communi cates the value of the internal model output ym and KU to allow processor 1 to compute 2 the control law . The minimum sampling period is then defined by the addition of the execution times of processes 2 , 5 , 6 ,7, 8 , 9 ,10 , 11, 18a . valu~s
The main disadvantage of this approach is that if a processor or communication link failure occurs, a recovery procedure would be very complex . The other approach is dynamic assignment in which tasks are executed in either one or the other processors. This approach offers higher reliability but does not maintain a defined function for each processor . CONCLUSION The dual model reference adaptive controller is an alternative approach for the cont r ol of non linear systems . The performance of t he controller applied to a pressurized paper machine headbox when it is submitted to large operating point transitions has been shown . The possibility of multi process ing the algorithm and one possible implementation in a dual processor has been presented . The conservation of the i de ntification and control functions for each processor enhances the technical application of the control l er.
FIGURE 14 Having deconposed t he control algorithm t h e problem of assig~ent of software processes to processors a rises . T~o strategies are possible : static or dyn ami c assignment. One int er es ti ng approach is to consider a two processor architecture an d a static assignment . One processor will execute the control
Dual Model Reference Adaptive Cont r ol
REFERENCES
249
APPENDIX LIST OF SYMBOLS
BETHOUX C. ~~D COURTIOL B. (1973) . A hyperstable discrete model reference adaptive control system. Proc. 3rd IFAC Symposium on sensitivity adantivity and optimality.
Qe(t)
Slice outflow. Suspension level ln the headbox.
Ischia , Italy , 1973. LANDAU I . D. (1978) . Adaptive Control . the model reference approach. DEKKER
Stock flow into the headbox.
S
s
Slice opening area . Slice outflow velocity .
New York , 1978 . p(t ) NADER A. (1978). ~lodelisation et commande adaptive multivariable des caisses de tete de machine papier . These Docteur Inge nieur . Institut Nat ional Polytechnique de Grenoble , 1978 .
Suspension density - Water density .
a
NADER A. (1979) . Petri Ne ts for realtime control algorithms decomposition . IFAC 1979 Workshop on distributed computer control systems . Tampa, Florida 1979.
Relative air pressure inside headbox .
pa
Air density .
A
Headbox free surface area .
m{t)
Air mass inside headbox .
Qi
Air mass inflow.
Qo
Air mass outflow .
H
Headbox height .
T
Absolute air temperature inside headbox .
m
Specific heats relation for air. Pa
Atmospheric pressure.
Mc
Pulp mass inside the headbox.
C
Input suspension concentration.
C
Output suspension concentration .
e s
C
c
Suspension concentration ins i de the headbox . Relative pressure before alr i nflow valve .
R
General gas constant for alr .
g
Gravity acceleration .
01
Slice opening .
Cp
Tmck stock concentration .
Cd
White water concentration .
Qp
Thick stock flow .
Qd
White water flow .
© IFAC PRP 4 Automation, Ghent , Belgium 1980
DUAL MODEL REFERENCE ADAPTIVE CONTROL ON PAPER MACHINE HEADBOXES A. Nader* Imperial College of Science and Technology, Computing and Control Department, London, SW7, U,K ,
Abstract . When industrial processes operate under variable production conditions, linear models and linear controllers are generally not effici ent. When a paper machine operates at variable speed, the control of the headbox must be efficient for large operating point transitions. We present a dual model reference adaptive controller (DMRAC) that allows the control of linear time varying systems and the explicit on line identification of the process with delayed inputs. The application of the DMRAC to a pressurized headbox when large operating point t r ansitions are applied is presented. Finally a functional decomposition of the DMRAC algorithm for its implementation in a multi - Il,icroprocessor system is presented . Keywords. Adaptive Control, Adaptive Systems , Computer Control , Decoupling, Dual Control, Multi - processing , Multi - variable Control, Paper Industry, Process Control, Time Varying Systems . INTRODUCTION Most industrial processes operate under variable production conditions. Profit maxi mization is obtained through production and cost optimization. The operation of industrial processes on a wide spectrum of operating points obliges control designers to consider the non- linear nature of industrial processes.
These operating conditions require a good headbox control.
However, the state of the art in optimal multivariable non- linear control theory does not always allo.~ one to design suitable non- linear controllers (e . g . non- linearities in the states and inputs of the process). Linear controllers have excel:ent performance when they operate near an operating point of the process , but they behave poorly when the process operates far from the operating point the control system was designed for.
At present control of headboxes gene r ally implemented with two single PI controller s . The level and the total head at the slice are controlled . However, it is well known that there exist strong interactions between these two loops - furthermore it is necessary to control the consistency and the substance flowrates. We will present first a dual model reference adaptive controller (DMRAC) that allows the control of linear time varying systcns and the explicit on- line identification of the process with delayed inputs. A simple application example is presented. We then present the application of the DMRAC to a pressurized headbox when large operating point transitions are applied.
Between linear and non-linear control systems are the adaptive control systems . Linear Time Varying systems can be effi ciently controlled by adaptive controllers.
Finally a functional decomposition of the algorithm for its implementation in a multi micro- processor system is presented .
In the paper industry , in order to optimlze production, it is necessary to control the paper machine ' s speed. The paper machine operates at variable speeds according to the evaporation capabilities of the drying plant .
239
A. Nader
240 DUAL MODEL REFERENCE ADAPTIVE CONTROLLER
In this section we show a new multi - variable dual model reference adaptive controller that has been developed based on model reference adaptive systems MRAS. uMRAC take int o account linear time varying systems explicitly and can also treat the case of processes with delays. Model reference adaptive systems can be traced back to 1958 with Whitaker, Yarmon and Kezel . The introduction of stability theory concepts into the design procedure of adaptive systems was first developed by Parks in 1966 . Landau (Landau , ~_97 8 ) developed in 1973 a multi - variable continuous model reference adaptive controller. Several discrete monovariable approach es have been developed (Bethoux and Courtiol, 1973 ) DMRAC was developed from one of them. The basic idea behind the DMRAC is the applic ation of the adaptive control to an ad j ustable model parallel to the process . This model is adjusted through an on line identification mechanism according to the variation of parametres of the process . Most industrial processes are non- linear. When an operating point transition is performed , one can interpret it as a variation of parametres of the linearized model from the parametres of the ini tial operating point to the parametres of the final operating point . This variation of parametres is identified and the parallel model to the process is updated on line .
DMRAC have in addition to the MRAC an internal adjustable model parallel t o the process (fig . 1). The dual controller is formed by two subsystems ( fig. 2) structurally identical and identical t o the MRAC structure Notice the duality between the reference model and the process; the adjustable system and the adjustable model; and between the adaption law and identification law . DMRAC Design The design strategy is entirely based on Popov's hyperstability theory. We first enunciate a hyperstability theorem . Theorem (Pop::lv): Let ~+l vk
= ~+B~ = C~ + ~
be a linear system where (A, B) is completely controlable and (A,C) is completely observable. The linear system is feedback by the non-linear system w k
=f
(v,k,l,) l~k
satisfying the Popav in-equality:
v.. k_.L> k0 where
02
is a finite positive constant.
A necessary and sufficient condition to ensure that the linear system feedback by the non-l inear is asymptotically hyperstable lS that the transfer matrix
Reference Model
H(z ) lS
Adaptiou MKhauilm
=E
+ C (zI- A) - lB
strictly real positive.
Now we will present the design procedure of the DMRAC , considering first the control system and afterwards the identification system .
.------->\Rdef1!DC Modf1i 1 - - - - - - - - - ,
Set Point
FIGURE 2
FIGURE 1
MRAC
Dual Model Reference Adaptive Control
24 1
Y + = [BpKm + Bp (6Kp(k+l ) - Kp)] Y + APY k k l k Bp [ Ku +6 Ku(k +l ) ] zk + BpKme
" ' -.,
J
k
Linear
SYltem 1 _____ ---oJ
.. ~ I
'---~I
NOD Linear Sy.t e m
I
vir.
II+-----J
(Am-BpKm )e
e + k l
BpKp - BpKin]Y
k
k
+ ~-Ap- Bp~p (k+l) +
+ [m - Bp (Ku+6)
taking the perfect model following conditions. Am - Ap = Bp (Kin- Kpo)
o Where Kpo and Ku are the unknown values of Kp and Ku that guarantee perfect model followin g Kpo = lim ~P-: \Kp( k+l
D
11._00
Kpo = lim [Ku+,6Ku(k+ID
kA
EQuation I can then be rewritten
with As(k +l)
[Ap+BP.L\Kp(k+l) - BpKp + Bp~
Bs(k+l)
Bp [Ku +!\Ku(k+l)=
and posing K = As ( k+l) Let us discuss the control system . The reference model is defined:
and the adjustable internal model is:
Bp~~
=1~&- ,,11 (v ,k,l )
+
"2
(v , k) + As (0)
Bs (k+l) = k> ~i (v , k,l ) + ~~ (v , k) + Bs (0) 1=0 one obtains 6Kp(k+l)
=l~t"l (v , k , l) + " 2(v , k ) +!:,.Kp(o)
6Ku(k+l )
=1~6~1 ( v,k,l ) + i 2 ( v- , k) + LKu(o)(~)
A particular solution for "1' "2' i l , We postulate the following control law:
that satisfies Popov
i~eQuality 1S:
Where Kp(k) ;;; Kp - ..:i Kp (k+l) Ku(k) ;;; Ku +/ Ku(k +l)
i 2 (v , k) = SPvk +1
So the output of the adjustable internal model is:
Equations 3 and L are defined as:
I.A .A. -R
i2
(3)
A. Nade r
242
~Kp(k+l) = ~~pl(k+l) +~ KpP(k+l)
The transfer function is:
~K/(k+l) =~K/(k)
H(z) = DBp+D[Am- BpKm][zl- Am+BpKm] - lBP
+ FV k +l [GyJ T
Applying the Kalman Szego Popov real pos i tive theorem one can show that:
i<:pP(k+l) = FPv + [Gy) T k l
~Ku(k+l) = ~Kul(k+l) +l::,KUP(k+l) 1 l::,KU (k +l) = ~Kul(k) + Sv k+l P /\Ku (k+l) = SPv + [Hz J T k l
where P i s the solution of the Lyapounov equation T
P(Am- BpKm) - P = Q'
Then equation 2 is:
(Am- BpKm)
e + =/,ACl- BpKm)e + Bp [Kp- KpO -6K pl (k)] Yk k k l
Fol l owing a similar procedure for the identification system , if the output of the fixed part of the int Ernal model is:
+ Bp[Ku
O
1 - Ku---6Ku (k) ] zk - BP[(F+FP)YkTGYk
Yk+l = Amyk + Bm~ + (S +SP)zk THzk ]
v + k l and the postulated control law
Defining
then ( 5)
where 1 1 0 Yk +l = As (k)Y k + Bs (k)Zk + Ke k 1 As (k) 1 Bs (k)
1 Ap + BP0J
e 2k = yprocess
k
- Yk
v 2k +l = [1 + D2 BJl2(k)] -1
v~k+l
Where
K = BpKm Equation 5 can be rewritten as:
and
e o+ = (Am- BpKm)e + Bp [ Kp-Kp 0- LKp 1 (k}j~ Y k k l k O + Bp[Ku - KU-6 Ku1 (k)] zk
Ku2 +I\Ku2 (k+l)
=~Kp~
Then -
v k +l = [ I + DBpN(k)j
-1
0
v k+l
o v k+1 c"-:~ be calculated wi :hout a~proxi matlon as the control law lS 2?plled to a known model . We decided that instead of . 0. . calculatlng v. 1 :et could be approxunated by v. as prop§~ed by Bethoux and Court iol for f~e monov~riable case in order to save computing time (Sethoux and Courtiol, 1973)
Applying t~e Popov hyperstability theorem we ~ave ~ow to ca~culate the matrix D figure 3 so that the linear syste:n has a strictly real positive transfer fur.ction .
FlGrRE J
L\Kp~(k+l)
(k+l)
= FP 2 v2 k +l
+!\K 1 (k+l) - p2 LG YkJ T
L\KP; (k+l) =I\KP; (k) + F2 v2 k +l [GYk] T P 6 Ku2 (k+l) = /\Ku 2 (k+l ) +/\ KU; (k+l) T 6" (k+l) = SP2 v k+l [H uk] ."uP 2 2
6 Ku 2
I
(k+l) =6KU; (k) + S2 V2 k+l [Hu,KJ
T
The D~ffiAC obtained is shown In figure 4 . As the control structure developed ~ e re iYlcludes an inter.1al model parallel to the process , the basic idea of the Schmidt corrector can ::.J so be applied . DR~-1AC with delays is shown in figure 5.
Dua l Mode l Refe r ence Adaptive Con t rol
A.
243
Internal Model Control System. 1.
Reference model output.
2.
Adjustable model output. ~ +l = Ac~ + Bc~
3.
Pr ocess output.
4.
New set point.
5.
D. C. component suppression. X +
FIGURE ..
k l
-
xo yo
Zk+l - zo YO~+l
= yom k + l - yo
6.
Computation of vO~+l
7.
Predictive computation of vO~+ 2
o
vO~ + 2 ~ vO~ + l
. . approxlmatlon
r-
Il -l o
vO~ +2 = LI+D BC< Ku2(k)) N(k+ll.;
vo~+2
T
N (k+l) = (F+FP) yom + G yom + k l k l T
+ (S +SP)zok+l Hzo k +l
8.
Computation of~Kp(k+ 2) .6 Kl(k +2)
~KpI(k+2)
FPvo~+2
[ Gyank+1J
=.6 Kp I(k+l) +
T
Fvo~;+2 [
GYaT'k+1JT 6Kp(k+2) =.6 Kp P (k+2) +.6 KpI(k +2)
9. FIGURE 5
Canputation of.6Ku(k+2) r ]T .6KuP (k+2) SP vO~ + 2 LHzo + k 1
I I 6 Ku (k+2) =6 Ku (k+l) + S
DMRAC Algorithm
vO~+2
[
Hzok+~T
Initial izations .6Ku(k+2) =.6KUP(k+2) +6 KuI (k+2)
Loop 10.
Computation of Ku(k+l) and Kp(k+l) Ku(k+l) = Ku +
Ku(k+2)
A. Nader
244
Kp(k +1 ) = 11.
Kp -
APPLICATIONS
Kp ( k+2)
Simple Example
Conputati on of uO + k 1 uO + = - Kp (k+1 )yon + + Ku(k+1)zo k 1 k 1 k+1 + Kr!lxo k +1
B. Identification System. 12 .
Here we present a basic example to show the behaviour and function of the DMRAC . The DMRAC is applied to a first order linear system , assumed not well known . This is equivalent to a step parametric disturbance. The process is:
Computation of v0 2 + k 1 v0 2k +1 = D2 (yok+1 - YO~ +l)
The fixed part of the adjustable model is: 1 3.
Predictive computation of v02 + k 2
~+l = ac ~ + bc ~
v02°K+ 2 <- v0 2 k + 1.' a pproximation v02 + = k 2
[I +D2BcN2(k+l)0
-1
The reference model is: 0
v02 + k 2
T
N2( k+l) = (Fi+ FP2 )YO~+ 1 G YO~+l
where the process is :
+ (S 2+SP2) uo k +l Huo +l k 14.
Computation ,\ Kp2
P
ap bp
of~Kp2(k+2)
(k+2) = FP2v02 +2 k
LGyO~+ 1 J
T
The assumed model is: ac bc
~Kp2I(k+2 ) = ~Kp2 I(k+l) + F2V02 + [ k 2
JT
Gy om k + 1
0 . 8153 0.1846
The reference model is:
P
I
LKp2 ( k +2) =,6,Kp2 ( k+2) +L:::,.Kp2 (k+2)
15.
0.6049 0.3950
amr bmr
0 . 8187 0 . 1812
Computation ofL:::,.Ku2 (k+2) P l::,Ku2 ( k+2) = SP2v02 + [HUOk+lJ T k 2
~Ku2 I ( k+2 ) =L:::,.Ku2 I (k+l) + S2v02 k +2 [ ~T Huok+1J
The adapt ion and identification gains are : f
fp = s = sp = f'2 = fp 2 = s 2 = sp 2 = 100
g
h = g2 = h2 = 1
The corrector d
d 2 = 0.1034
and the gains: 16 .
Computation of Kp2( k+l) and Ku2(k+l ) Kp2( k+l )
Kp2 -6.Kp2( k+2 )
Ku2( k+l )
Ku2 + L:,Ku2( k+2 )
Computation of UOQk+l uom + = - Kp2 ( k+1 )yon + + Ku2 k 1 k l (k+l ) uO + k l
: 8.
Cor:trol signal U
Close loop
+
= km2
= O.
kp = - 0.01818 ku = 0.9818 kp2 = 0.0 ku 2 = 1.0 xo = yo = ymo
umo = uo
To achieve the fastest convergence of parametres we apply a pseudo random binary sequence of ~ax~um period of 63 to zk' The sampling per10d 1S 1 sec . The convergance of parametres is show!'. in o f1. . 6 . The values of kp 0 , ku 0 , kP2 and 8 ku found by the J~.ffiAC guarantee perfect 2 moiiel following:
k +l = uOk +l + uo uO~ + l
km
uo~
k 0 Po ku 0 kP2 0 kU 2
- 0 . 5411 0 . 4588 1 . 139 2.139
Dual Mode l Reference Adaptive Control ' , " CD (x' - y,) The crlterlOn L1 1
2
=
0 . 272 .
245
DMRAC Applied to a Pressurized Headbox
i =l This simple example shows that identification and perfect model following are achieved with the DMRAC .
We first present th e non- linear model of a pressurized headbox , fig . 7 . This model has been verified experirnentally (N ader , 1976 ) .
screen Qe
Qp
Ce
Cp Qd Cd
Fig . 7 - Head Box Circuits The state non linear model is: dh
X
dt
2
_ l (f
dX 2 dt
h P
+
lx 2
2 2
1 28
(Q
2
e
AX )2 1_ g 2 j
s
1 dP dt
= ;.. (H
-h)
m
RT=(Ql' - Qo ) + PAX 2 :
dCc _ Qe [ Cp( t-r )- Cd(t - r ) Qp(t - r) + Cd(t - r)-Cc~J' dt - Ah Qe(t-r ) Where :
Q ()
=K 0
P ( _a_ ) l /¥ \ ~ -------=----- \ /(P+P ) p.. Pa (¥ - ~/¥ P+P
a
V¥- l
a
a Ll- (p+p ) a
This model shows the multi variable non linear character of the headbox . The non linearities present in this model are o f a ge neral type. The stock flow Qp is a delayed input . The essential objectives of the control system are : decoupling the multi variable system and tracking for operating point transitions. We now present the applications of the D~ffiA C to the non linear model of the headbox. Two sL~ulation results will be presented ; the first in which the m,r uc has been dl':'signed using the linearized model of the initial operating point of the headbox and the second in which the D~ffiAC was designed using a wrong model of the process .
A. Nader
246
Q
_-------:-0
o
~
\
\
o
Dua l Mode l Refe r ence Adap t ive Co ntr o l The in i tial operating point is characterised by: ho
0 . 1176 m
Qeo
0 .0201 m3 /s
Po
26253 Pa
Kio
6 2 0.7454 x 10- m
Cco
4 . 2 gm/l
Qpo
. 00444 m3 /s
T
283
01
. 00679 m
A
2 0 . 34 m
Pk
31200 Pa
RLZ
0 . 40 m
Cp
1 5 gm/l
Pa
101325 Pa
Cd
1. 25 gm / l
Ko
6 2 0 . 3591 x 10- m
0
K
The discrete reference model
r-)
"---
0 . 95 12
Amr
~- ./
0 . 8187
C)
0.1324 x 10- 1
(
0.18126
/
=1
The sampling period
Cont r ol a l gor ithm ' s decomposition should take into account computing and control criteri a : load balanc i ng , interprocess communication , graceful degradation , control functions , etc .
sec .
The adapt ion and identification gains are : \1000
I r -,
G
1000
F
I
F
= FP = S
l
500
I F2
(
, /
500 (
. 08
6J SP
F2
J
= FP2 = S2 = SP2
Th r ee operating point step changes were performed for each simulati on : change of level , air pressure and headbox concentration set po i nts . ho
0 . 11 7 m to !"lo
0 .1 7 m
Po
26253 Pa to Po
27000 Pa
Cco
4 . 2 gm /l to Cco
= 5. 2
FUNCTIONAL DECOMPOSITION OF THE DMRAC ALGORITHM
The DMRAC structure is particularly well suited for its implementation in a multi - micro processor architecture . A functional decompos ition of the algorithm can increase the reli ability of the new controller .
1 0.4877 x 10-
Bmr
Figures 11 , 12 and 13 represent set point changes of the level , pressure and concentrat ion in the headbox when the process is simulated by the non linear model and the D~ffiAC has been designed using a li near model that does not represent the headbox neither at the initial nor at the final operating point . Here again the decoupling is perfect , except for the leve l set point change in which the pressur e changes very little at the beginning of t he step. The tracking of the refe r ence model is perfe ct fo r the level change . For the pressur e the tra cking is not well achieved . The concent rat i on cannot track the reference model because of the effect of the deadtime .
The advent of cheap mi cro processors a l lows one to develop specialized, sophisticated contollers that can be localized in the plant site as wi t h analogtechnol ogy at pr esent . Distributed Comput er Control Systems (DCCS) allow one to distribut e functionally and/o r geogr a phically the computing power . DCCS are also r equi r ed when the performance and/or reliabi l ity of a single computer is inadequate to mee t the r e quirements of the control system .
1S:
0 . 9867
24 7
gn/l
Figures 8, 9 and 10 represent set point changes of the level , pressure and concent r ation in the headbox when the process is SiMUlated by the non linear model and the DMRAC has been designed using a linear model established around the intial operating point . The decoupling and tracking qualities of the DMRAC are excellent. Perfect model following and per fect decoupling is achieved .
Funct i onal decomposition of centralized control algorithms is natural since control blocks have a functional meaning . However , different levels of decomposition are possible : the control function level , the matheMatical funct i on l evel , the instruction level , the logical function level , etc. The increase of software processes in order to obtain maximum parallelism implies the multiplication of interprocess communication and synchroniz ation, then the increase of complexity of software processes management and decrease o f performance . The trade off of level of decoMposition) degree of complexity of software process manageMent and conservati on o f control meaning has to be considered. The major objectives of D~·~AC algorith:!l decomposition can be to minimize its execution time while conserving some control functions meaning . Taking into account these factors we will consider fo r the mffiAC deCOMposition each one of the algorithm steps (1 to 18) mentioned in t he algorithm section as indivisible sequential software processes except steps 5 and 18 . Step 5 is divided into three (5a , 5b , 5c) steps . 5a performs the DC component suppress -
A. Nade r
248
ion of the reference model and set point only (x , z). Simil arly 5b considers the parallel model DC suppression (y m) and 5c the process (y) DC suppression .
tasks ( i.e. 1,4 , 5a, 6 , 7 , 8 , 9 ,10 , 11 , 18a ) and t he other the identification tasks (i.e . 2 , 3 , 5b , 5c,12 , 13 , 14 , 15,16 ,17,1 8b) fig. 15. yom , KU 2
Step 18 is divided i n : 18 a computation o f t he process co~t r ol signal (u) and 18b computation o f the model cont r ol signal ( urn ) .
Uo
Process
We pres ent in figure 14 one possib le decompositi on of the D;,lF.AC algoritiur. for multi processing using a Petri Net description (Nader 1979) .
y
fig. 15. The processors communication needed is only the of vectors ~om , uo and,the matrix Ku2 . The 1nterest of th1S apprach 1S that each processor has a very well defined function: processor 2 identifies the process and communi cates the value of the internal model output ym and KU to allow processor 1 to compute 2 the control law . The minimum sampling period is then defined by the addition of the execution times of processes 2 , 5 , 6 ,7, 8 , 9 ,10 , 11, 18a . valu~s
The main disadvantage of this approach is that if a processor or communication link failure occurs, a recovery procedure would be very complex . The other approach is dynamic assignment in which tasks are executed in either one or the other processors. This approach offers higher reliability but does not maintain a defined function for each processor . CONCLUSION The dual model reference adaptive controller is an alternative approach for the cont r ol of non linear systems . The performance of t he controller applied to a pressurized paper machine headbox when it is submitted to large operating point transitions has been shown . The possibility of multi process ing the algorithm and one possible implementation in a dual processor has been presented . The conservation of the i de ntification and control functions for each processor enhances the technical application of the control l er.
FIGURE 14 Having deconposed t he control algorithm t h e problem of assig~ent of software processes to processors a rises . T~o strategies are possible : static or dyn ami c assignment. One int er es ti ng approach is to consider a two processor architecture an d a static assignment . One processor will execute the control
Dual Model Reference Adaptive Cont r ol
REFERENCES
249
APPENDIX LIST OF SYMBOLS
BETHOUX C. ~~D COURTIOL B. (1973) . A hyperstable discrete model reference adaptive control system. Proc. 3rd IFAC Symposium on sensitivity adantivity and optimality.
Qe(t)
Slice outflow. Suspension level ln the headbox.
Ischia , Italy , 1973. LANDAU I . D. (1978) . Adaptive Control . the model reference approach. DEKKER
Stock flow into the headbox.
S
s
Slice opening area . Slice outflow velocity .
New York , 1978 . p(t ) NADER A. (1978). ~lodelisation et commande adaptive multivariable des caisses de tete de machine papier . These Docteur Inge nieur . Institut Nat ional Polytechnique de Grenoble , 1978 .
Suspension density - Water density .
a
NADER A. (1979) . Petri Ne ts for realtime control algorithms decomposition . IFAC 1979 Workshop on distributed computer control systems . Tampa, Florida 1979.
Relative air pressure inside headbox .
pa
Air density .
A
Headbox free surface area .
m{t)
Air mass inside headbox .
Qi
Air mass inflow.
Qo
Air mass outflow .
H
Headbox height .
T
Absolute air temperature inside headbox .
m
Specific heats relation for air. Pa
Atmospheric pressure.
Mc
Pulp mass inside the headbox.
C
Input suspension concentration.
C
Output suspension concentration .
e s
C
c
Suspension concentration ins i de the headbox . Relative pressure before alr i nflow valve .
R
General gas constant for alr .
g
Gravity acceleration .
01
Slice opening .
Cp
Tmck stock concentration .
Cd
White water concentration .
Qp
Thick stock flow .
Qd
White water flow .