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Identification and Adaptive Control of Wiener Type Nonlinear Processes
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IDENTIFICATION AND ADAPTIVE CONTROL OF WIENER TYPE NONLINEAR PROCESSES C. A. Pajunen /)'/}(II/II/{'1!1
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Abstract . This study is concerned with a problem of the on - I ine identi fi cation and adaptive control of Wiener type nonli nea r processes. It is assumed that the I inear dynamic part of the process can be approximated by a pulse transfer function model of a known order and stable inverse f 0 I l ow e d by a k no v, n pur e t i me - del ay. The s tat i c non I i ne a r i t y i sas sum e d to be monotonical function of its argument and it i s approximated by a piecewise - polynomial function model where the nurl ber of intervals, polynomial orders and the breakpoints should be chosen a priori. The output signal from the l inear part is not available for me asurement. The combined model is nonl i near in its paraCleters and it is characteri zed by a l arger number of parameters than the original Wiener mode l. Deterministic globally stable recursive parameter identi ficat ion algo rithms and the model reference adaptive control systems are developed for the Wiener process model. The asymptot i c properties in the presence of the input and output measurement noises and the influence of the error due to the approximation of a static non l i nearity by a piecewise polynomial funct i on of a finite order on the convergence properties of the proposed ident i fication schemes are analysed . Finally, the on - I ine identification and adaptive control of the pH - process in a continuous f l ow reactor were performed. The simulation and the experimental test resul ts are presented. Keywords. non l inear
Recursive ident i fication, adaptive control, systems, piecewise - polynomial approximation,
INTRODUCTION Most of the existing recurs i ve parameter estimat i o n and adaptive contro l a l gorithms are based on the I inear process models. Any rea l ist i c
process
may
be
l inear,
however,
on l y as a resu l t of some approximations. I n some cases when the non l inearity is severe in the operation region the non I i ne a r mo del s h 0 u I d be use din 0 r d er to obtain sufficient accurary. Thus the exist i ng theory shou l d be extended to nonl inear process mode l s. I n this work 14iener type nonl i near processes are cons i dered. The Wi ener model is composed of the single-input single - output (5150) li near dynamic system followed by a static non l inear e l ement. The Wi ener model is use fu l for represent i ng e . g. high blood pressure in man as a function of the c i rcu l ation load, or some phys i ological subsystems d i scussed by Otto and Wernstedt (1978) ,servo motors ( Desoer, Wang, 1979) or the pH p r ocess i n a continuous flow reactor present ed here as an example. The ide n tifica t ion prob l em of the process rep r esented by the Wiener mode l is d i fficult, not on I y because the process is non I i near, so the influence of the s t ochastic no i ses is mo r e severe and t he excitation signal must be chosen more carefu ll y t ha n in the I inear case, b ut espec i a ll y due to the fact t hat it is also nonl i near i n its parameters so t h e i dent i ficat i o n problem cannot be so l ved i n one s t ep even for t he
Wiener mode l pH - process.
deter mi nistic case. The methods so far used for i dentification of this k i nd of process are based ei ther on the iterative search algorith m, which does not assure global conver gence , so additional conver gence and stabi I i ty analysis is necessary (Dushkesas , Ka minskas , 1978), or on correlation techniques (8i I I ings, Fakhour i , 1977) so that when the input is a white Gaussian
noise
process,
the
identification
of the i mpu lse response funct ion of the linear dyna mics is decoupled from the characteristics of the nonlinear e l ement. Then the parameter estimates of the pu l se transfer funct i on mod el are obtained from this nonpara me tric impulse response mode l . In the present work the parametric identi fication ,·,ill be considered . The prob l em of the simultaneous ident i f ication and control of a process represent e d by the Wi en er mo del \., i t h ti me - va r y i n g parameters
has
not,
according
to
author ' s
knowledge , been considered at all i n the I ite r ature. So far al I adapt i ve contro l schemes for this kind of process with un known and time - varying both the parame t ers of the I inear part and the shape of the s t atic nonlinearity have been based on a I inear process model. WIENEP. MODEL Processes are studied which can be ade quately represented by the Wiener mode l The I inear dynam i c p r esented in F i g. 1. part i s represented by the pulse transfer
C ..-'1.. Paiullell function model
b u (t-d-l)
(1)
2: b u (t-s-d-l) s=1 s a
a
m
where A(q -1)
m
a
8(q-l)
where a
q-l is the backward shift operator, d represents the process time delay, {u(t) } and ( y(t) } are the process input and output measured signal sequences, respectively and x(t) is the internal unmeasurable signal. The orders 5 and the time delay d are assumed to be known. It is assumed that the static me moryless nonl inearity is one to one and can be approxi mated by a constant coefficients piecewise-polynomial funct ion wi th several terms:
): jir=ar' ji' for all r=O,I,oo.,a, and j = 1 ,2, ... , m and y. (t)
a,
m
n
Ya(t) =2: ~ : { 2: ', :. [x(t}]i ) j=I J i=OJI where x _ S x(t) < x. 1 j 1 J t)i = ( O otherwise
(2 )
j
O
(6)
+ 2: ~ . ( 2: 2: 0: .. y.(t-r) } j = 1 J i =0 r =0 J 1 r 1 = 1,
= 0,1, .. . , n
1
So obtained meters.
model
is
inear
in
its
para-
The nonl inearity approxi mation error € (y) can be transformed through the linear dynamic part to the input and co n sidered as an additive input noise ~ u(t). The measured input u(t) and output y(t) signals are possibly corrupted by additive measurement noises fl it) and ].l (t), respectively. IDENTIFICATION
and that its inverse can also be approximated by a piecewise-polyno m ial function with severa 1
te rms
m x
a
(t)
n
2: lJ.' . { 2: , .. [y(t}]i } j=1 J i=O JI
(4 )
The para meter identification algorithm developed for 1 inear systems using model reference approach is as follows (Landau, 1982) (8a) ~
( t) = ~ (t - 1) + F( t - 1 )
! ( t)[ 1+2 T (t) F ( t - 1 ) 2 (t-l)]-1 \.' 0(t)
where Yj_1 Sy (t)
T F- 1 (t) = Al (t}F- 1 (t-l) + \ 2(t} 2 (t) 2 (t) (8b) (5 )
In Eqs. (2) and (4) n is the polynomial order, m is the number of intervals and they are given a priori. In Eqs. (3)-(5) the functions Iji :(x) and tfi .(y) are denoted as t)i : and \jJ . fo-!- simplicity of representati6n. ThJ breakpoints xO,x 1 , . .. ,x m and YO'Yl"" 'Ym' for approximations (2) and (4) respectively, should also be chosen a priori. This means that some a priori knowledge of the shape of the static nonlinearity and the operating range is needed for correct choice of breakpoints, nu mber of intervals and the polynomial orders. It is not possible to identify the parameters of a linear dyna m ic part and a nonlinear static element independently. Since the steady state gain of the 1 inear part and the first coefficient of each polynomial approximating the static nonl inearity in each interval cannot be separated, the identification will always contain one arbitrary constant, but the model obtained will behave identically to the original system. It is possible to normalize the linear element so that its gain coefficient is set to one, or to set some parameter of the I inear model at a constant value. Here the parameter b is set to one, for O
convenience.
The combined process model which can be obtained by considering Eq. (1) in (2) is nonlinear
in
its
parameters .
Moreover,
it
contains an unknown internal signal sequence ( x(t) } , which should be estimated together with the unknown parameters in order to make this model applicable. The inverse of the combined process model is obtained by using the approximated signal xa(t) given by Eq. (4) instead of x(t} in a time-series model obtained from Eq. (1) with bO = 1 :
Here, this algorithm is applied for identification and/or tracking unknown and timevariable parameters in the combined model inverse given by Eq. (6) or in an original Wiener model given by Eqs. (1) and (2) or (4), each part identified separately, based on the measured input and output sequences. Then, the influence of the approximation error and the measure ment noises as well as the choice of excitation signal on the para meter convergence properties is considered. One-Step
Identification
Procedure
The identification procedure by which all parameters of the inverse of the combined process model given by Eq. (6) are identified simultaneously is here called the "onestep identification procedure" . It is well known (Landau, 1979), that for obtaining a globally stable adaptive syste m in a deterministic case, the system should be parametrized so that the adjustable system is linear in the para meters. That is why the series MRAS structure (input error method) is proposed in this case where the inverse of the combined process model is used as an adjustable system. The series identification scheme is shown in Fig.2. It is assumed that the static nonl ineari ty is one to one and that the linear part of the process is minimum phase (8(q-l) stable polynomial) in order to assure stabi I ity of this scheme. First, the noises are ignored and the deterministic identifier is delivered. series estimation model is given as
, b, m n u(t) = - L b (t}u(t-s)+ L t)i . { L s=1 s j=1 J i=O
The
\\'ic ll <:r T\vc \: oll li llca r Processes The d i sadvantage of the series mode l refe r e n ce adapt ive syste m (MRAS) is that the theory requires a positive rea l co n dition on the plant a l gorithm parameters to be satisfied, even in the no i se free case. In orde r to ful fi I this condition the a l gor i thm w i th extended esti mation mode l (Landau, 1978) i s used here. The extended parameter esti mates and regr e ssion vectors are defined as fo ll ows (lOa) [b (t), ... ,D - ( t) " " OO(t), .. • ,ex . _ (t) , . .. , l b J J Oa ."
Jn
- ( t ) , h 1 ( t) , . .. ,h - ( t) ] J na b
0 ( t) , . . . , :"
( l ab) T
!e(t}
~
'
-
-
[ - u ( t -I ), .. . ,-u ( t - b) ,y ( t), . . . ,y(t - a) , ... ,
y n ( t) , ' .. ,y n ( t - a) ,\; ( t - I ) , .. . , , ( t - b) ] It/h
ere
"( t )
T ( t) : (t) - u(t - d - I ) °ej -e
Th e para met e r es ti mat ion al g orithm has the for m ( 7 ) bu t us ed for th e e xtended para mete r and
regre ss io n
ve c tors.
The probl em of selecting an i nput signal for syste m excitation is much more difficult in the cas e o f a nonl inear mode l than in the cas e of a I inear one and must be solved very carefully in order to obtain a consistent resul t. It should be persistently exci ting and cov e r th e inter i or of the whole operat ion r eg i on. I t was checked by simu l ation that if the input signal to the static part of the model is unifor mly d i str i buted be t ween the m in i mum and maxi mu m value
in
the current
interval the r e aso n able convergence speed for the whole interval is obtained . For this purpos e a special stochastic generator has been dev e loped as shown in Fig.3, where u'(t) is a rando m signal uniform l y d i str i buted between the minimum and maximu m va l ue of the p r ocess input in a current interva l . Depending o n the choice of parameters AI and \ 2 different type of adaptation gain i s obtain e d (Landau, Lozano, 1979) . In a case when th e constant para meters should be i de n ti f ied, then the time - decreasing adaptation gain should be used and the series i dentification sc heme shovJn in Fig . 2 is g l obal l y as ymp totically stable i n deter minis t ic environ ment if the persistency of excita t ion condition is fulfil l ed and the input signal covers the i nterior of the whole operation r e gion . When the track i ng of t i me - vary i ng parameters should be achieved, the best resul ts were obtained using the "constant-tracell adaptat i on gain .
The nu mber of unknown para mete r s is [ 2b + (n+ l )(a+ l )]m and increases by + l)m when the de g re e 0 f po I y no m i a I s n i sin c rea s ed by one, while i n the origina l Wiener mode l there are only + b + (n + l) m parameters. This means that there i s a certain amount of redundancy in the values of the parameters wh i ch necessitates determina t ion of the particu l ar set of va l ues (if needed) wh i ch
(a
cation p r ocedure". I n ST EP t he no nl inear pa r t of the Wie n er mode l i s i ne a r i zed nea r the opera ti ng po i nt and the pa r ameters of the I i near dynamic part are estimated , using t h e a l go r i t hm based on the pa r a ll e l MRAS with an extended est i mation mode l This scheme is g l oba ll y ( Landau, 1978). asymptotically stable in determin i stic environment
under
persistence
of
exc i tation
restr i ctions on the input signa l (Duga r d, Landau,1980) . I n STEP 11, the coeff i c ien t s in the piecew i se - polyno mial approximat i on of the nonl inear ele ment a r e estimated from the internal signal between the I inear and non li near parts of the Wiener model and the process outp u t, u s ing the least - squares method. This int e rnal signal can be obta in ed by pass i ng the i npu t signa I to the process through the f i I ter being the inverse of t h e I inear dyna mi c part of the process once t h e para meter esti mates of the I inear part of the mode l are available . Since the es t i mation model is static , no SPR cond i t i on is requ i red, and a g l obally asy mptotical l y stable identif i cation sche me is obtained. Because the static nonl inea r ity i s approx imated by a piecew i se - po l ynomia l funciton, the input signal to esti mation mode l shou l d cover the inter i or of the whole operat in g region. I t was found by si mu l at i on tha t if the exitation signa l i s uniformly distributed between the minimum and maximum value of the process input in e ach i n t erva l mul tip l ied by t h e steady - state gain of the linea r pa rt of the mode I then the good convergence speed for the whole interval i s obtained. Using this sche me it i s assumed, tha t t he parameters change slow l y, that they can be considered as constants from the ident i f ication po i nt of vie w (they do not change before the ident i fication algorithms of both steps converge). That is why t he pa r a meter identif i cation algorithms with a t i me - decreasing adaptation ga i n a r e used. I nf l uenc e of Stochastic Noises The adapt i ve schemes discussed above are determin i stic, so the influence of the ignored noises should be exa mined in orde r to check whether asymptot i c global conve rgence of these sche mes can be achieved . The in flu e n c e 0 f the f 0 I l ow in g no i se s i s considered: input and output measurement noises ~ (t) and ~ (t) r espective l y, which are assumed to be zero - mean, independent norma l ly d i str ibuted random variables, the nonlinearity approx i mation error, the noise due to mismatch of the li n ear part of the process and its es t imated mode l i n a case of the two - step ident i ficat i on procedure - Step 11.
a
gives
the mini mu m r ms erro r .
Two - Step
Identificat i on Procedure
Another poss i bi l ity is to sp l i t i dentifi cation p r ocedure into two steps. The i denti f ica ti on method by wh i ch t h e parameters of the I i near and nonl inear parts are est imated separate l y using the pa r a l le l MRAS structure is he r e ca l led "two - step iden ti fi -
Since the i dent i fication in open -l oop i s cons i dered, there is no coup l i ng between the i nput signa l and t he output noise or be t ween the i npu t signal a n d t he pa r ame t e r es t ima t es. It c a n be sh own ( Pajune n, 198 4 ), app l yi n g Th eorem 2 due t o Duga r d a n d Landau ( 198 0 ) t hat t he one - step i d e nt i f i ca ti on proce dur e desc r i bed above w i th ex te n ded es t i ma t i on model and a t i me - dec r easing adaptat i o n gain ( A (t)= I , " 2(t) = A )is strong l y 2 1
-l9ti
(; . :\ . l'ajullcll
co nsistent i f the i nput sig n al is persistent l y excit i ng a nd covers the i nterior of the who l e operat i on region. th e output measuremen t no i se and the non l inearity approxi mation error are neg l igib l e and the transfer function 1 /(B +H- I ) - \ 2/2 i s strictly positive real (SPR).
where T (q -I ) polynomial .
Simila rl y (Pajunen. 1984) the t ,"o - step id ent i f i cat i on procedure desc ribed above i s s tr ongly consistent i f the i nput measurement noise i s negligib l e and - for Step I . the error due to linearization of the stat ic nonl inearity at the working point i s neg l i gib l e. the transfer function 1 /( H+A -I ) - ) 2 /2 i s SPR and the input signal i s pe r sistent l y exciting. - for Step I I . the noise due to m is ma tch of the I inear dynamic pa rt of the process and its estimated model is neglig i ble. the mean value of the nonlinearity approximat i on e rr or is zero and the inp ut signal covers the i nter i or of the ,·,hole ope r at i on re g i on.
I ) Predictor for the I inear system g i ven as follows (Landau . 1982)
ADAPTIVE
an
asymptotically
Syste m for
= R(q and the
- I
- I
)B(q
,"here
degR = d.
degT = t
are the known. that for regu l ation
I ntroducing th e fo l l ow in g nonlinear transformation of th e mode l r efe rence ou t put signa l
m L
l,. . {
n L
j =1
)
i =O
expressed
as
the
dyna mics
T ( q- l)y(t+d +l )
- I
s} i r = 5 r 'r j
for a l l i=l, ... ,n; j= l, ... , m r = 0 , ... , 5
and y(t)
= T(q-I) y(t)
2) The control l aw y(t +d+I)= YM(t +d+l l follo", i ng
I
i s computed so that I t can be given in
m T [ L _ j~j~l(t)l J =I
( 18)
e re
~} = [ : j
j
j
I •. •. • Yj n • s I 0 • . ..• s j I 5 •...• s nO' ... •
Sjns·gj· .. · .gYd+b)
1
.!; ( 1
:: ~ (t) =
[y A I ( t +d + I ) •...• yAn ( t +d + I )
2;(t)
[ - YI (t) • ... ' - YI (t - 5) •...• - yn(t) •. . .•
t)
- Yn(t - s) . - u(t - I) • .. . • - u(t - d - ti)
: .. y M. ( t ) : )
1
(I
3)
1
and
1
defined O.
the
form
follows:
Th e objectives of cont r ol in a deterministic environment are: TraCking: The contro l signa l shou l d be designed so that the pla nt output i s equa l to the output of the reference mode l . wh i ch i s given b y Eq. ( 11) and r (t) is the bounded refer e nce signal. Re gulat ion: Th e contro l signa l shou ld be designed so that in regulat i on (r( t )=O) the in i t i al disturbance (y(O) ;la) is elim i nated wi t h
degS = s =
wi t h
\",h
where AM(q-l) and BM(q - l ) stable polynom i als . Note purposes r(t) o.
=
and
fulfil
) - b ' O . - I . - I . - s + ... +S jis q Sji(q )=Sj iO+ Sjil q
u(t) = 1)0
x M( t)
R should
I n Wiener mode l case subst i tuting an unknown signa l x(t) in Eq. ( 14) by its approx i mat i on xa(t) and using Eqs. (4) and (7) the pred i ctor i s given as fo l l ows : ( 17) m n m n ,', -I L '! ' . L " ,, [ y" (t+d+ I) 1 i = L ~' . L S .. (q )y. (t) j=1 Ji= 1 JI 1 j=1 )i=1 )1
( 11 )
be
( 15 )
. -I +G " (q )u(t)+bou( t )
Th e mode l reference teChnique. ,"h i ch i s based on the solut i on to the I inear mode l following contro l pro bl em. cons i ders the generation of the po l ynomia l s G. S a nd the transfer funct i on H i ns uch a way t hat the c I 0 sed - I 00 p res p on s e 0 f y(t)tothe reference signal r(t) follo",s the response of the reference model which speci fies the design objectives given by the uSer :
can
( 14 )
)
the polynomials Sand f 0 I 10'" in g i d e n t i t y
Known
T he general control l e r st ruct ure for the Wi ener type nonl i near process i s presented in Fig.4. I n t hi s figure the nonlinear preinverses are th e inv erses of the stat i c nonl ineari ty of the process and are used for the I inearizat ion of the feedback system.
xM(t)
be
T (q -I );(t+d+l) = S(q - l)x(t)+G(q - l)u(t)
G(q
signal
can
max ( t-d -I .~ - I ) .
The purpose of th i s part of the work is to design an adaptive control system for tracking and regu l at i ng this class of p roc esses where parameters of the l i near pa rt and the s h ape of the nonl ineari ty are both u nkno wn o r unknown and ti me - vary i ng using a mo del reference approac h .
the
stable
I n o r de r t o desing the controller an impli c i t re fe r e n c e mo del s t rat e g y i sus e d . T he design procedure i s in two steps: p r edictor des i gn and contro l calcula tion.
CONTROL
Design of the Control Process Para meters
is
Adaptive
Sche me
The i mp l icit MRAC scheme is proposed and the parameters of the predictor are adjusted using an input error method. Since i t i s not possible to identify the paramete r s of a l inear dynamic part and nonlinear static e l ement in dependently. the paramete r b O is set at one.
by The
i nput
error
approach
is
as
fo l l ows:
\\'i cllcr T"l)c :\olllille, II' Proccsscs w hat i nput u(t-d-I) shou l d have been used so t hat the predictor output y(t} is equa l to the process output y(t}? Rearrang i ng Eq. ( 17 ) and substitut in g the unknown parameters vl i t h th e ir estimates . an adaptive pr e d i c tor can be vi r i t ten in the f 0 I low in g for m : ( 19)
~2(t)= [YI(t+d+I ) • . . . • y~(t +d+l ) ' - YI (t)., . .• - y(t - 5). ...• - yn(t). ··· '-Yn(t - 5) . - u(t - I). ...• - u(t - d-b)]T Unknown para me ter s can be es ti ma ted using thefollowing recursive parameter adaptation a l gorithm ( L andau. 1982 )
?- J. ( t + d + I ) = -~ J. ( t + d ) + F ( t ) -: ~ T (t) where F is (8b) and
the
v O(t)=T(q-l) Th en
the
F (t)
~
(2 I ) (t ) [ I+
(t) ] - I '. 0 (t +d + I )
adaptation
gain
y(t) _ ~ T(t - I ):
contro l
J
la w
-
g i ven
by
Eq.
(t - d -I )
i s given
as
follows
-I ~ li
titration functi on (o r its inverse) nearly exactly. The parameter estimates for the linear and non l inear pa rt s of the mode l obta i ned by the one - step ide n tification pro c e d u r e 'vI ere, how eve r, \-J 0 r 5 e t h ant h e parameter estimates obtained by the two step indentification method due to the output measurement no i se. the nonl inearity approx i ma tion error. the larger number of pa ra mete rs in the estimation mode l and the redundancy in the values of the parameters. Th en the computer contro l system (Pajunen. 1984) implemented at the Contro l En gineer in g Laboratory of Helsinki University of Techno l ogy . was used for the exper i mental Checking of the theoretical and simulation r esu l ts discussed above. Th e same chem i cal co mpo nents were used as for the si mulation studies, Th e re vi as both input ( reagent flow) and output (pH of eff l uent) me asure me nt noise . Since the process was very slow. only one interval "Ias cons idered in a mo del, The t ime decreasing adaptation ga in was used w ith 7 " 2 = 0 . 99 and F =dia g(10 )
o
Considering the two - step identification procedure the following parameter estimates of the I inear dynamic part were obtained after lOO m inutes of ident i f i cation: ,
,
b
(2 0)
A block d i agram of the mode l reference adaptive contro l (MRAC) scheme used to control Wiener type nonlinear processes with unknown stat i c nonlinearity is shown in Fig,S, I n the case when the reference s i gna l is constant or piecewise constant. ther e wi I I a l ways b e a steady - s t ate error unless the parameters converge to thei r true values . Th at i s why an i ntegral mode should be i ncluded in the contro ller (Pajunen . 1 984) . pH - PROCESS REPRESENTED LINEAR WIEN E R MODE L
BY
A NON-
I t is assumed that the p H process considered here can be described by the Wiener mode l where the I inear dynamic part represents the lin ea r m i x i ng dynam i cs of the concen t rat i on va r i a b I e san d the s tat i c non linea r i t Y represents a titration function vl hich is a highly non l inear momenta ry relat i on between the p H-varia b l e and the reagent concentration (Niemi . Jutila. 1977). Th e titration curve of the considered process is s ho wn i n Fi g. 10 - the curve I . Th e reagent concentration on the p ro cess input (or the reagent flo W) and the pH of the effluent are measu r ed . I t was found exper imenta II y that the I i near dynamic part can be represented by a fi rst order st a b l e pu l se transfer function mode l wi th ste ad y state ga i n equa I to one . Th e n o nlin ea r, conti n uous, static, memory l ess one to one titration function and its inve r se are app r oximated by a piecewise polynomial functions. I dentifica t ion S im u lation t est results ( Pajunen. 1 984) show that both a l gor ith ms work correc tl y. T he approximating curves obtained ma tc h th e
= 0 . 105 and b = 0.047. Thes e O l estimates cor r espond quite well with the phys i cal system parame t ers. The result i ng curve obta i ned in Step I I after lOO minutes of ident i fication and the th eore ticall y ca l cu l ated titration curve are shown in Fi g.6. by continuous and broken lines. r espect ively. T hen th e mo d e l i s frozen and con ne c t e d i n par a I I e I wit h the pro c e s s. The ramp input signa l is used for excitation of the mode l and the process. T he responses are shown in Fi g. 7. The parameters of the combined process mode l inve rse we re estimated next using the one - step identification procedure. After 50 m inutes of ident i fication the paramete r est i mates no l onger change much. The expe ri ment was continued for 200 mi n utes and the pa ra meters of the l i near and non I i ne a r par t s vi ere the n c a I cu i ate d fro m the parameter es ti mates of the comb i ned process model minimiz i ng the r ms error. Th e results were as follows : ~1= D.87. I = 0 . 045. The est i ma t e d 0 = 0 .082 and inverse of the titration curve and the theoreticall y calculated inver se of the titration funct i on are s hown in Fig.8 by co n t i nuous and broken l ines. respectively. Finall y. the combined process mode l inverse was frozen and connected in series with the process . The input to the process was a r amp funct i on. The p r ocess output was used as the i nput to the mode l and the ou tput of the mode l was compared with the input of the p ro cess. The resu I ts are s h ow n in Fig.9. The m i smatch is due to the influe n ce of the process noises and th e insu ff ic ien t est i mation time.
6
Adap ti ve
5
Control
Th e th eore t ically ca l culated t i tration functions for t he four p r ocess feeds cons id ered (Pajunen. 1984) are shown in Fig.IO. The inverse o f the t itration func t i on was approximated by a p i ecewise polynomial function. In all tests a
G. A. Pajulle ll t ime -va ry i ng " con s tant - t race " adaptat ion ma tri x was used . Th e initial values of the parameter estimates ~ (O) =0.
-
-
Responses of t he simula t ed process to the s inu so i d al and piecew i se - constant reference signal are s ho wn in Fi gs 1 1 a n d 1 2, respectively . Finally regulat i on result s are shown for s i mu l ated and real process in Figs 13 and 14, respectively. T he p r ocess feed and fl ow are marked in the figures. CONCLUSION S Thi s \'10rK considered the on - I ine i dentif icat ion and adaptive contro l of Wiener type nonl inear processes. I t was shown t hat the developed ident i ficat i on methods can be app l i ed succesfu l l y i f prio ri knoVlledge of the static non l ineari ty i s so good that th e number of i n t erva l s, breakpoints and the po l y no m i al orders can be chosen such that the nonlinearity approx i mat ion error i s negl igib le. Thi s condition may somet i mes be hard to sa ti sfy , especial l y when the shape of th e no nl inear it y changes a lot and further research is ne ede d in order to design an algorith m which automat i cal l y adjusts t he breakpo i nt s to ge th er with the unknown coefficients in the piecew i se polynom ial approximation function. I n a specia l case, l'ihen the nonlinearity is odd and symmetr i c with respect to the origin and i t can be approximated by a single polynomia l w i t h only odd terms, the two step identification me thod can be used succesfu l l y even i f the nonl i near i t y approximation error i s cons id e ra ble . Th e advantage of these methods in pH - process ident i f i cation is that the compo nen ts of the p rocess so lu tion do not need to be known. On the other han d i f th i s know l edge is ava i 1ab l e, i t is not possib l e to in corporate i t , which is a clear disadvantage.
Dushkesas, M.I. , Kaminskas, W.A . (1978). Para me ter estimation i n Wiener -t ype mode l by nu mer i ca l optim izat ion methods. Liet . T SR Moks l u akad. Darbai S ser. (USSR), 3, 83 - 93 (in Russ~-Landau, 1. 0-:-(1978). Elimination of the real posit i vity condition in the design of para l l e l MRAS. IEEE Trans. Aut. Control, AC - 23, 1 0 15- 1020 . Landau, 1. 0. ( 1 979). Adapt i ve control the modern reference approach. Marcel Dekker, New York. Landau , 1.0. ( 19 82). Determ i nist i c and stoc hasti c adapt i ve po l es - zeros p l ace ment for m i nimum phase systems . Prepr. o f the Workshop on Adaptive Cont~ I S I S Flor ence, I taly, 207 - 244. L andau, 1 .0., Lozano, R. ( 1979) . Design and eva l uation of d i screte time exp l ic i t MRAC for tracking and regu l at i on. Report L AG 79.14, Dept. of Automatic Control, Grenobl e I nstit u te of Techno logy, France. Nie m i, A . ,Jut il a , P.K . ( 1977) . Process mo dels and contro l of acidity. Co mputer applic atio ns in the ana l ysis of chemical data a nd plants. CHEMDATA, 77,Flnland . Otto, P., Wer nstedt ,~978""J.Develop me nt and app l i cat i on of subopt i ma l test signa l sequences for i det i fy in g non I inear systems . Prep. of the 7t h IF AC Triennia l World Congre ss , Helsink i, Finl and, 1927 -1 933. Pajunen, G.A. ( 1984). Appl i cat i on of a mode l reference adaptive techn i que to the identification and control of Wiener t ype nonlinear p rocess. Acta Polytechnica Scand i navica, EE ser i es No 2..£, He l sinki, 1 30 pp.
!NONlI STATIC NE Aql T
LI NEAR DYNAM ICS
Y Y _____
;----<.~,
L . .. (Q)
Th en the control of a pH -pr ocess i n a continuous flow r eactor was co n sidered. F ig.l. The Wi ener mode l Simulation results proved that the adaptive controller was necessary for la rge changes in the pr ocess flo w and the s ha pe of t h e titration curve. Th e developed MRAC scheme with a nonlinear adaptive controller based on the p iece wise - cubic approxi mation of a static nonlinearity a nd a fast refe r ence mo del was best in tracking a s inusoid a l reference s i g nal . For the piece wise - constant or constant reference s i gnal the scheme w ith th e linear adaptive control ler, which was _u..;.,_lt_: -,.v I ---' based on the piecevlise - I inear or linear approxi mat ion of the stat ic nonlinearity, ~:tl wo r ked bet t er t ha n wit h the non I i ne a ran e . The s i mulation and experimental result s proved the effectiveness of th is adapti ve scheme w ith a linear adaptive cont r o l l er and a slow observer dynam i cs fo r t he compensation Fig.2. Series ident i ficat i on of la rge changes in th e process gain DO - fo l d)
MODEL
P it)
~-------~~~.-----------------"
and at the same process flow.
time
a
2 - fold
change
in
scheme.
the STO(HASTlC G,N ERATOR I (A'..C~LAT,::W
(
Si I I ings, S.A., Fakhouri S.Y. ( 1977 ). Identification of nonlinear sy s te ms using the Wiener mode l . Electron. Lett., 13,502-504. Desoer, C.A., Wang, Y.T.(1979) . Therobu st nonlinear servomechanism problem. Int.J. Cont rol , 29, 803 - 828. Dugard, L, Landau, 1 . 0. ( 1980). Recursive output e rror i den t ification algorithms. Th eory and eva l uation . Automatica,~, 443-462.
i
;g:S~~::~TTION ~'~------,
REFERENCES
F(y)
__ F_IU_I__._t l
~
i g.
3.
A s t ochastic generator excitation s ignal.
l '".,.>;
for
the
yltl
499
Wiener Type \!on linear Processes
CRO. ,O'
l
:mo le/ll !re ]
~ --
22.21 ~
_ / / /.h
/
/
/"
/
~ ~ ~ ':"
()(f-_ __ _ _ _ _--r-_ _ .o: PR'JC £SS _ '_
1. '. ~ I
.. "
~~:~ ~~~~~
:::: //
,. ,
1.5' 5.6 IL
+., ~
e-L .-L
_ __
1 ~
_ _- - , -_ _ _: - , : -_ _
60
6,
68
-:::-~ 7.2
pH
Fi g. 8 . The one - s te p identificat i o n procedure e x peri me ntal result s . The esti mated in verse of the t i trati o n function (contin uous line ) and the the o retically calculated inverse of the titrati on function (broken line) .
.
J
cR
10- !mole/ll tre J
2.8
~ ~I I J tl rf\
c.
2.6
f.
2.' 22
.orl) 20 18
Fig.4. Block diagram of the MRAC sch eme u sed to control Wiener type nonlinear proce sses .
_ _--'-_ _ 90 5'
16 ~-~L--~--~
o
, 8
36
--'-~
Time (1'T'1n
--'r--.··~ -~"':'~J'
em"'"
rl l , I _ _ __
rr it'fP" ( (I
i" I: '
,
"(l '. t IN~:' :' -- :"t;~ ". ~;.:":--
InOul
J
Fi g . 9. Validation test r esults: the process input (brok e n I ine ) and the output of the frozen mode l ( c ont inuous I ine ) c o nn e cted in series wi th the
p'o,m.
~ -: " -'-'-~~~ ~~~:~s~
Fig. 5. General controller structu r e for Wiener type nonlinear systems .
eR lmole/litre I
0.004 0.008 0.012 0.016
L
0[,
I so ·
Fig.IO. Theoretically calcu la ted titration functions for th e four process feeds cons i dered.
55L-~~-·~-~-1 5t. 1 63 1&21 9 6 " 2 1 1 2(.
~IO" ) lmOle/ il lre l
Fig. 6. The two - s t e p i den t i f i cat ion procedure - expe ri mental results. Es tima t ed s tati c nonlinearity (co ntinuou s I ine) and the theoretic ally calcu l ated titration curve (broken line)
m
:1
pH 72
i\
75'-
00
5.6
i
0 pH
32
I
o (a)
B,O
10.0 Tlme (mm l
Fig.7. The re spo nse of the process (continuous I ine) and the frozen model (broken I ine) to the ramp inpu t signa I.
V
V 60
I
8 00
~
A
. lJ LJ A\ n n
·00
V 110
I
.
I~
i
5 e~
C, .011
I !
I
72\-
I
4
6 I ;J~- _ /
I ! I
IV lBO
\I 240
IJ tl me/m ln
·006
0
60
120
180
240
c..
time/mm
(b)
Fig.ll. Tracking the sinusoida l reference signal (simulation r es ult s) using the adaptive system shown in Fig.4 with a nonI inear controller ( n= 3) and adaptive gai n reinitial i zati on at t=150 min. a) Reference model output YM(t} and process ootput pH(t). b) Reagent concentration Process feed I11 for time 0-150 min, p rocess fee d I for time 150-300 min.
:iOO
( ; . .-\ . 1"lllIll l' 11
tpH
C' I
:::: - - : ..E --
JO'------~--
o
.e
:6:
. .. :, . r e
~
r
(b I
( Q)
Fig.12 . Tracking a piecew i se - constant reference signal (si mulat ion results) "Iith a I i near contro l ler (n=1 ). a) Reference si g nal pHr{t) and proc es s output pH{t) . b) Reagent concentration. Process feed I I I for t=0 - 150 min and for t=150 - 300 min . Process flOlv I I/"'in for t=0 - 250 min and 0 . 5 I/m i n for t=250-300 min.
prl l ill , IT ~
III
!I!
G0 -
~
0 ..
2 0~
o_
o
!_L ~
') S ,,.'rrdfi
1 !/rn ln -L--l." • I
r llJ'IJ
I
80
( b)
(a)
F ig.13 . Regulation (simulation results) using the adapt i ve system i nc l uding i ntegral mode a) Process output pH{t) . b) Reagen t concentration CR(t). Process feed I II fot t=0 - 30 min, IV for t=30-60 min , I I I for t=60 - 90 mi n, I for t=90 - 180 min, I I for t= 180 - 270 mi n, I for t=270 - 315 min and I I I for t =3 15- 360 min. Pr ocess f low I I /min for t= I- 315 min and 0.5 I/m i n for t =3 15- 360 min.
ID
fLOW
0'> I!m,n 30
bO
90
120
1'>0
lBO
110
11.0
I'
=·-'Ii-m~ ~ ' - ..L_ .. ..-LlL _____ .L __ _ o
s:
I
i
270 t,me /mm
Fi g. 14. Regulat ion (experimenta l resul t s) us i ng t he ad a pti ve s c heme wi th an in t eg ral mode. The process fe eds a nd fl ow ar e ma rk ed i n t he f igure .
160
_ 1
240
IDENTIFICATION AND ADAPTIVE CONTROL OF WIENER TYPE NONLINEAR PROCESSES C. A. Pajunen /)'/}(II/II/{'1!1
0/
Ild,illl.-i {'lIi",'/",!ll "/ I~dill"/' <~'T. F,/JOo. lil//,!IIr!
/:/('(/11(/// /:"1I,1;/I/{"'IIII,I':-.
Abstract . This study is concerned with a problem of the on - I ine identi fi cation and adaptive control of Wiener type nonli nea r processes. It is assumed that the I inear dynamic part of the process can be approximated by a pulse transfer function model of a known order and stable inverse f 0 I l ow e d by a k no v, n pur e t i me - del ay. The s tat i c non I i ne a r i t y i sas sum e d to be monotonical function of its argument and it i s approximated by a piecewise - polynomial function model where the nurl ber of intervals, polynomial orders and the breakpoints should be chosen a priori. The output signal from the l inear part is not available for me asurement. The combined model is nonl i near in its paraCleters and it is characteri zed by a l arger number of parameters than the original Wiener mode l. Deterministic globally stable recursive parameter identi ficat ion algo rithms and the model reference adaptive control systems are developed for the Wiener process model. The asymptot i c properties in the presence of the input and output measurement noises and the influence of the error due to the approximation of a static non l i nearity by a piecewise polynomial funct i on of a finite order on the convergence properties of the proposed ident i fication schemes are analysed . Finally, the on - I ine identification and adaptive control of the pH - process in a continuous f l ow reactor were performed. The simulation and the experimental test resul ts are presented. Keywords. non l inear
Recursive ident i fication, adaptive control, systems, piecewise - polynomial approximation,
INTRODUCTION Most of the existing recurs i ve parameter estimat i o n and adaptive contro l a l gorithms are based on the I inear process models. Any rea l ist i c
process
may
be
l inear,
however,
on l y as a resu l t of some approximations. I n some cases when the non l inearity is severe in the operation region the non I i ne a r mo del s h 0 u I d be use din 0 r d er to obtain sufficient accurary. Thus the exist i ng theory shou l d be extended to nonl inear process mode l s. I n this work 14iener type nonl i near processes are cons i dered. The Wi ener model is composed of the single-input single - output (5150) li near dynamic system followed by a static non l inear e l ement. The Wi ener model is use fu l for represent i ng e . g. high blood pressure in man as a function of the c i rcu l ation load, or some phys i ological subsystems d i scussed by Otto and Wernstedt (1978) ,servo motors ( Desoer, Wang, 1979) or the pH p r ocess i n a continuous flow reactor present ed here as an example. The ide n tifica t ion prob l em of the process rep r esented by the Wiener mode l is d i fficult, not on I y because the process is non I i near, so the influence of the s t ochastic no i ses is mo r e severe and t he excitation signal must be chosen more carefu ll y t ha n in the I inear case, b ut espec i a ll y due to the fact t hat it is also nonl i near i n its parameters so t h e i dent i ficat i o n problem cannot be so l ved i n one s t ep even for t he
Wiener mode l pH - process.
deter mi nistic case. The methods so far used for i dentification of this k i nd of process are based ei ther on the iterative search algorith m, which does not assure global conver gence , so additional conver gence and stabi I i ty analysis is necessary (Dushkesas , Ka minskas , 1978), or on correlation techniques (8i I I ings, Fakhour i , 1977) so that when the input is a white Gaussian
noise
process,
the
identification
of the i mpu lse response funct ion of the linear dyna mics is decoupled from the characteristics of the nonlinear e l ement. Then the parameter estimates of the pu l se transfer funct i on mod el are obtained from this nonpara me tric impulse response mode l . In the present work the parametric identi fication ,·,ill be considered . The prob l em of the simultaneous ident i f ication and control of a process represent e d by the Wi en er mo del \., i t h ti me - va r y i n g parameters
has
not,
according
to
author ' s
knowledge , been considered at all i n the I ite r ature. So far al I adapt i ve contro l schemes for this kind of process with un known and time - varying both the parame t ers of the I inear part and the shape of the s t atic nonlinearity have been based on a I inear process model. WIENEP. MODEL Processes are studied which can be ade quately represented by the Wiener mode l The I inear dynam i c p r esented in F i g. 1. part i s represented by the pulse transfer
C ..-'1.. Paiullell function model
b u (t-d-l)
(1)
2: b u (t-s-d-l) s=1 s a
a
m
where A(q -1)
m
a
8(q-l)
where a
q-l is the backward shift operator, d represents the process time delay, {u(t) } and ( y(t) } are the process input and output measured signal sequences, respectively and x(t) is the internal unmeasurable signal. The orders 5 and the time delay d are assumed to be known. It is assumed that the static me moryless nonl inearity is one to one and can be approxi mated by a constant coefficients piecewise-polynomial funct ion wi th several terms:
): jir=ar' ji' for all r=O,I,oo.,a, and j = 1 ,2, ... , m and y. (t)
a,
m
n
Ya(t) =2: ~ : { 2: ', :. [x(t}]i ) j=I J i=OJI where x _ S x(t) < x. 1 j 1 J t)i = ( O otherwise
(2 )
j
O
(6)
+ 2: ~ . ( 2: 2: 0: .. y.(t-r) } j = 1 J i =0 r =0 J 1 r 1 = 1,
= 0,1, .. . , n
1
So obtained meters.
model
is
inear
in
its
para-
The nonl inearity approxi mation error € (y) can be transformed through the linear dynamic part to the input and co n sidered as an additive input noise ~ u(t). The measured input u(t) and output y(t) signals are possibly corrupted by additive measurement noises fl it) and ].l (t), respectively. IDENTIFICATION
and that its inverse can also be approximated by a piecewise-polyno m ial function with severa 1
te rms
m x
a
(t)
n
2: lJ.' . { 2: , .. [y(t}]i } j=1 J i=O JI
(4 )
The para meter identification algorithm developed for 1 inear systems using model reference approach is as follows (Landau, 1982) (8a) ~
( t) = ~ (t - 1) + F( t - 1 )
! ( t)[ 1+2 T (t) F ( t - 1 ) 2 (t-l)]-1 \.' 0(t)
where Yj_1 Sy (t)
T F- 1 (t) = Al (t}F- 1 (t-l) + \ 2(t} 2 (t) 2 (t) (8b) (5 )
In Eqs. (2) and (4) n is the polynomial order, m is the number of intervals and they are given a priori. In Eqs. (3)-(5) the functions Iji :(x) and tfi .(y) are denoted as t)i : and \jJ . fo-!- simplicity of representati6n. ThJ breakpoints xO,x 1 , . .. ,x m and YO'Yl"" 'Ym' for approximations (2) and (4) respectively, should also be chosen a priori. This means that some a priori knowledge of the shape of the static nonlinearity and the operating range is needed for correct choice of breakpoints, nu mber of intervals and the polynomial orders. It is not possible to identify the parameters of a linear dyna m ic part and a nonlinear static element independently. Since the steady state gain of the 1 inear part and the first coefficient of each polynomial approximating the static nonl inearity in each interval cannot be separated, the identification will always contain one arbitrary constant, but the model obtained will behave identically to the original system. It is possible to normalize the linear element so that its gain coefficient is set to one, or to set some parameter of the I inear model at a constant value. Here the parameter b is set to one, for O
convenience.
The combined process model which can be obtained by considering Eq. (1) in (2) is nonlinear
in
its
parameters .
Moreover,
it
contains an unknown internal signal sequence ( x(t) } , which should be estimated together with the unknown parameters in order to make this model applicable. The inverse of the combined process model is obtained by using the approximated signal xa(t) given by Eq. (4) instead of x(t} in a time-series model obtained from Eq. (1) with bO = 1 :
Here, this algorithm is applied for identification and/or tracking unknown and timevariable parameters in the combined model inverse given by Eq. (6) or in an original Wiener model given by Eqs. (1) and (2) or (4), each part identified separately, based on the measured input and output sequences. Then, the influence of the approximation error and the measure ment noises as well as the choice of excitation signal on the para meter convergence properties is considered. One-Step
Identification
Procedure
The identification procedure by which all parameters of the inverse of the combined process model given by Eq. (6) are identified simultaneously is here called the "onestep identification procedure" . It is well known (Landau, 1979), that for obtaining a globally stable adaptive syste m in a deterministic case, the system should be parametrized so that the adjustable system is linear in the para meters. That is why the series MRAS structure (input error method) is proposed in this case where the inverse of the combined process model is used as an adjustable system. The series identification scheme is shown in Fig.2. It is assumed that the static nonl ineari ty is one to one and that the linear part of the process is minimum phase (8(q-l) stable polynomial) in order to assure stabi I ity of this scheme. First, the noises are ignored and the deterministic identifier is delivered. series estimation model is given as
, b, m n u(t) = - L b (t}u(t-s)+ L t)i . { L s=1 s j=1 J i=O
The
\\'ic ll <:r T\vc \: oll li llca r Processes The d i sadvantage of the series mode l refe r e n ce adapt ive syste m (MRAS) is that the theory requires a positive rea l co n dition on the plant a l gorithm parameters to be satisfied, even in the no i se free case. In orde r to ful fi I this condition the a l gor i thm w i th extended esti mation mode l (Landau, 1978) i s used here. The extended parameter esti mates and regr e ssion vectors are defined as fo ll ows (lOa) [b (t), ... ,D - ( t) " " OO(t), .. • ,ex . _ (t) , . .. , l b J J Oa ."
Jn
- ( t ) , h 1 ( t) , . .. ,h - ( t) ] J na b
0 ( t) , . . . , :"
( l ab) T
!e(t}
~
'
-
-
[ - u ( t -I ), .. . ,-u ( t - b) ,y ( t), . . . ,y(t - a) , ... ,
y n ( t) , ' .. ,y n ( t - a) ,\; ( t - I ) , .. . , , ( t - b) ] It/h
ere
"( t )
T ( t) : (t) - u(t - d - I ) °ej -e
Th e para met e r es ti mat ion al g orithm has the for m ( 7 ) bu t us ed for th e e xtended para mete r and
regre ss io n
ve c tors.
The probl em of selecting an i nput signal for syste m excitation is much more difficult in the cas e o f a nonl inear mode l than in the cas e of a I inear one and must be solved very carefully in order to obtain a consistent resul t. It should be persistently exci ting and cov e r th e inter i or of the whole operat ion r eg i on. I t was checked by simu l ation that if the input signal to the static part of the model is unifor mly d i str i buted be t ween the m in i mum and maxi mu m value
in
the current
interval the r e aso n able convergence speed for the whole interval is obtained . For this purpos e a special stochastic generator has been dev e loped as shown in Fig.3, where u'(t) is a rando m signal uniform l y d i str i buted between the minimum and maximu m va l ue of the p r ocess input in a current interva l . Depending o n the choice of parameters AI and \ 2 different type of adaptation gain i s obtain e d (Landau, Lozano, 1979) . In a case when th e constant para meters should be i de n ti f ied, then the time - decreasing adaptation gain should be used and the series i dentification sc heme shovJn in Fig . 2 is g l obal l y as ymp totically stable i n deter minis t ic environ ment if the persistency of excita t ion condition is fulfil l ed and the input signal covers the i nterior of the whole operation r e gion . When the track i ng of t i me - vary i ng parameters should be achieved, the best resul ts were obtained using the "constant-tracell adaptat i on gain .
The nu mber of unknown para mete r s is [ 2b + (n+ l )(a+ l )]m and increases by + l)m when the de g re e 0 f po I y no m i a I s n i sin c rea s ed by one, while i n the origina l Wiener mode l there are only + b + (n + l) m parameters. This means that there i s a certain amount of redundancy in the values of the parameters wh i ch necessitates determina t ion of the particu l ar set of va l ues (if needed) wh i ch
(a
cation p r ocedure". I n ST EP t he no nl inear pa r t of the Wie n er mode l i s i ne a r i zed nea r the opera ti ng po i nt and the pa r ameters of the I i near dynamic part are estimated , using t h e a l go r i t hm based on the pa r a ll e l MRAS with an extended est i mation mode l This scheme is g l oba ll y ( Landau, 1978). asymptotically stable in determin i stic environment
under
persistence
of
exc i tation
restr i ctions on the input signa l (Duga r d, Landau,1980) . I n STEP 11, the coeff i c ien t s in the piecew i se - polyno mial approximat i on of the nonl inear ele ment a r e estimated from the internal signal between the I inear and non li near parts of the Wiener model and the process outp u t, u s ing the least - squares method. This int e rnal signal can be obta in ed by pass i ng the i npu t signa I to the process through the f i I ter being the inverse of t h e I inear dyna mi c part of the process once t h e para meter esti mates of the I inear part of the mode l are available . Since the es t i mation model is static , no SPR cond i t i on is requ i red, and a g l obally asy mptotical l y stable identif i cation sche me is obtained. Because the static nonl inea r ity i s approx imated by a piecew i se - po l ynomia l funciton, the input signal to esti mation mode l shou l d cover the inter i or of the whole operat in g region. I t was found by si mu l at i on tha t if the exitation signa l i s uniformly distributed between the minimum and maximum value of the process input in e ach i n t erva l mul tip l ied by t h e steady - state gain of the linea r pa rt of the mode I then the good convergence speed for the whole interval i s obtained. Using this sche me it i s assumed, tha t t he parameters change slow l y, that they can be considered as constants from the ident i f ication po i nt of vie w (they do not change before the ident i fication algorithms of both steps converge). That is why t he pa r a meter identif i cation algorithms with a t i me - decreasing adaptation ga i n a r e used. I nf l uenc e of Stochastic Noises The adapt i ve schemes discussed above are determin i stic, so the influence of the ignored noises should be exa mined in orde r to check whether asymptot i c global conve rgence of these sche mes can be achieved . The in flu e n c e 0 f the f 0 I l ow in g no i se s i s considered: input and output measurement noises ~ (t) and ~ (t) r espective l y, which are assumed to be zero - mean, independent norma l ly d i str ibuted random variables, the nonlinearity approx i mation error, the noise due to mismatch of the li n ear part of the process and its es t imated mode l i n a case of the two - step ident i ficat i on procedure - Step 11.
a
gives
the mini mu m r ms erro r .
Two - Step
Identificat i on Procedure
Another poss i bi l ity is to sp l i t i dentifi cation p r ocedure into two steps. The i denti f ica ti on method by wh i ch t h e parameters of the I i near and nonl inear parts are est imated separate l y using the pa r a l le l MRAS structure is he r e ca l led "two - step iden ti fi -
Since the i dent i fication in open -l oop i s cons i dered, there is no coup l i ng between the i nput signa l and t he output noise or be t ween the i npu t signal a n d t he pa r ame t e r es t ima t es. It c a n be sh own ( Pajune n, 198 4 ), app l yi n g Th eorem 2 due t o Duga r d a n d Landau ( 198 0 ) t hat t he one - step i d e nt i f i ca ti on proce dur e desc r i bed above w i th ex te n ded es t i ma t i on model and a t i me - dec r easing adaptat i o n gain ( A (t)= I , " 2(t) = A )is strong l y 2 1
-l9ti
(; . :\ . l'ajullcll
co nsistent i f the i nput sig n al is persistent l y excit i ng a nd covers the i nterior of the who l e operat i on region. th e output measuremen t no i se and the non l inearity approxi mation error are neg l igib l e and the transfer function 1 /(B +H- I ) - \ 2/2 i s strictly positive real (SPR).
where T (q -I ) polynomial .
Simila rl y (Pajunen. 1984) the t ,"o - step id ent i f i cat i on procedure desc ribed above i s s tr ongly consistent i f the i nput measurement noise i s negligib l e and - for Step I . the error due to linearization of the stat ic nonl inearity at the working point i s neg l i gib l e. the transfer function 1 /( H+A -I ) - ) 2 /2 i s SPR and the input signal i s pe r sistent l y exciting. - for Step I I . the noise due to m is ma tch of the I inear dynamic pa rt of the process and its estimated model is neglig i ble. the mean value of the nonlinearity approximat i on e rr or is zero and the inp ut signal covers the i nter i or of the ,·,hole ope r at i on re g i on.
I ) Predictor for the I inear system g i ven as follows (Landau . 1982)
ADAPTIVE
an
asymptotically
Syste m for
= R(q and the
- I
- I
)B(q
,"here
degR = d.
degT = t
are the known. that for regu l ation
I ntroducing th e fo l l ow in g nonlinear transformation of th e mode l r efe rence ou t put signa l
m L
l,. . {
n L
j =1
)
i =O
expressed
as
the
dyna mics
T ( q- l)y(t+d +l )
- I
s} i r = 5 r 'r j
for a l l i=l, ... ,n; j= l, ... , m r = 0 , ... , 5
and y(t)
= T(q-I) y(t)
2) The control l aw y(t +d+I)= YM(t +d+l l follo", i ng
I
i s computed so that I t can be given in
m T [ L _ j~j~l(t)l J =I
( 18)
e re
~} = [ : j
j
j
I •. •. • Yj n • s I 0 • . ..• s j I 5 •...• s nO' ... •
Sjns·gj· .. · .gYd+b)
1
.!; ( 1
:: ~ (t) =
[y A I ( t +d + I ) •...• yAn ( t +d + I )
2;(t)
[ - YI (t) • ... ' - YI (t - 5) •...• - yn(t) •. . .•
t)
- Yn(t - s) . - u(t - I) • .. . • - u(t - d - ti)
: .. y M. ( t ) : )
1
(I
3)
1
and
1
defined O.
the
form
follows:
Th e objectives of cont r ol in a deterministic environment are: TraCking: The contro l signa l shou l d be designed so that the pla nt output i s equa l to the output of the reference mode l . wh i ch i s given b y Eq. ( 11) and r (t) is the bounded refer e nce signal. Re gulat ion: Th e contro l signa l shou ld be designed so that in regulat i on (r( t )=O) the in i t i al disturbance (y(O) ;la) is elim i nated wi t h
degS = s =
wi t h
\",h
where AM(q-l) and BM(q - l ) stable polynom i als . Note purposes r(t) o.
=
and
fulfil
) - b ' O . - I . - I . - s + ... +S jis q Sji(q )=Sj iO+ Sjil q
u(t) = 1)0
x M( t)
R should
I n Wiener mode l case subst i tuting an unknown signa l x(t) in Eq. ( 14) by its approx i mat i on xa(t) and using Eqs. (4) and (7) the pred i ctor i s given as fo l l ows : ( 17) m n m n ,', -I L '! ' . L " ,, [ y" (t+d+ I) 1 i = L ~' . L S .. (q )y. (t) j=1 Ji= 1 JI 1 j=1 )i=1 )1
( 11 )
be
( 15 )
. -I +G " (q )u(t)+bou( t )
Th e mode l reference teChnique. ,"h i ch i s based on the solut i on to the I inear mode l following contro l pro bl em. cons i ders the generation of the po l ynomia l s G. S a nd the transfer funct i on H i ns uch a way t hat the c I 0 sed - I 00 p res p on s e 0 f y(t)tothe reference signal r(t) follo",s the response of the reference model which speci fies the design objectives given by the uSer :
can
( 14 )
)
the polynomials Sand f 0 I 10'" in g i d e n t i t y
Known
T he general control l e r st ruct ure for the Wi ener type nonl i near process i s presented in Fig.4. I n t hi s figure the nonlinear preinverses are th e inv erses of the stat i c nonl ineari ty of the process and are used for the I inearizat ion of the feedback system.
xM(t)
be
T (q -I );(t+d+l) = S(q - l)x(t)+G(q - l)u(t)
G(q
signal
can
max ( t-d -I .~ - I ) .
The purpose of th i s part of the work is to design an adaptive control system for tracking and regu l at i ng this class of p roc esses where parameters of the l i near pa rt and the s h ape of the nonl ineari ty are both u nkno wn o r unknown and ti me - vary i ng using a mo del reference approac h .
the
stable
I n o r de r t o desing the controller an impli c i t re fe r e n c e mo del s t rat e g y i sus e d . T he design procedure i s in two steps: p r edictor des i gn and contro l calcula tion.
CONTROL
Design of the Control Process Para meters
is
Adaptive
Sche me
The i mp l icit MRAC scheme is proposed and the parameters of the predictor are adjusted using an input error method. Since i t i s not possible to identify the paramete r s of a l inear dynamic part and nonlinear static e l ement in dependently. the paramete r b O is set at one.
by The
i nput
error
approach
is
as
fo l l ows:
\\'i cllcr T"l)c :\olllille, II' Proccsscs w hat i nput u(t-d-I) shou l d have been used so t hat the predictor output y(t} is equa l to the process output y(t}? Rearrang i ng Eq. ( 17 ) and substitut in g the unknown parameters vl i t h th e ir estimates . an adaptive pr e d i c tor can be vi r i t ten in the f 0 I low in g for m : ( 19)
~2(t)= [YI(t+d+I ) • . . . • y~(t +d+l ) ' - YI (t)., . .• - y(t - 5). ...• - yn(t). ··· '-Yn(t - 5) . - u(t - I). ...• - u(t - d-b)]T Unknown para me ter s can be es ti ma ted using thefollowing recursive parameter adaptation a l gorithm ( L andau. 1982 )
?- J. ( t + d + I ) = -~ J. ( t + d ) + F ( t ) -: ~ T (t) where F is (8b) and
the
v O(t)=T(q-l) Th en
the
F (t)
~
(2 I ) (t ) [ I+
(t) ] - I '. 0 (t +d + I )
adaptation
gain
y(t) _ ~ T(t - I ):
contro l
J
la w
-
g i ven
by
Eq.
(t - d -I )
i s given
as
follows
-I ~ li
titration functi on (o r its inverse) nearly exactly. The parameter estimates for the linear and non l inear pa rt s of the mode l obta i ned by the one - step ide n tification pro c e d u r e 'vI ere, how eve r, \-J 0 r 5 e t h ant h e parameter estimates obtained by the two step indentification method due to the output measurement no i se. the nonl inearity approx i ma tion error. the larger number of pa ra mete rs in the estimation mode l and the redundancy in the values of the parameters. Th en the computer contro l system (Pajunen. 1984) implemented at the Contro l En gineer in g Laboratory of Helsinki University of Techno l ogy . was used for the exper i mental Checking of the theoretical and simulation r esu l ts discussed above. Th e same chem i cal co mpo nents were used as for the si mulation studies, Th e re vi as both input ( reagent flow) and output (pH of eff l uent) me asure me nt noise . Since the process was very slow. only one interval "Ias cons idered in a mo del, The t ime decreasing adaptation ga in was used w ith 7 " 2 = 0 . 99 and F =dia g(10 )
o
Considering the two - step identification procedure the following parameter estimates of the I inear dynamic part were obtained after lOO m inutes of ident i f i cation: ,
,
b
(2 0)
A block d i agram of the mode l reference adaptive contro l (MRAC) scheme used to control Wiener type nonlinear processes with unknown stat i c nonlinearity is shown in Fig,S, I n the case when the reference s i gna l is constant or piecewise constant. ther e wi I I a l ways b e a steady - s t ate error unless the parameters converge to thei r true values . Th at i s why an i ntegral mode should be i ncluded in the contro ller (Pajunen . 1 984) . pH - PROCESS REPRESENTED LINEAR WIEN E R MODE L
BY
A NON-
I t is assumed that the p H process considered here can be described by the Wiener mode l where the I inear dynamic part represents the lin ea r m i x i ng dynam i cs of the concen t rat i on va r i a b I e san d the s tat i c non linea r i t Y represents a titration function vl hich is a highly non l inear momenta ry relat i on between the p H-varia b l e and the reagent concentration (Niemi . Jutila. 1977). Th e titration curve of the considered process is s ho wn i n Fi g. 10 - the curve I . Th e reagent concentration on the p ro cess input (or the reagent flo W) and the pH of the effluent are measu r ed . I t was found exper imenta II y that the I i near dynamic part can be represented by a fi rst order st a b l e pu l se transfer function mode l wi th ste ad y state ga i n equa I to one . Th e n o nlin ea r, conti n uous, static, memory l ess one to one titration function and its inve r se are app r oximated by a piecewise polynomial functions. I dentifica t ion S im u lation t est results ( Pajunen. 1 984) show that both a l gor ith ms work correc tl y. T he approximating curves obtained ma tc h th e
= 0 . 105 and b = 0.047. Thes e O l estimates cor r espond quite well with the phys i cal system parame t ers. The result i ng curve obta i ned in Step I I after lOO minutes of ident i fication and the th eore ticall y ca l cu l ated titration curve are shown in Fi g.6. by continuous and broken lines. r espect ively. T hen th e mo d e l i s frozen and con ne c t e d i n par a I I e I wit h the pro c e s s. The ramp input signa l is used for excitation of the mode l and the process. T he responses are shown in Fi g. 7. The parameters of the combined process mode l inve rse we re estimated next using the one - step identification procedure. After 50 m inutes of ident i fication the paramete r est i mates no l onger change much. The expe ri ment was continued for 200 mi n utes and the pa ra meters of the l i near and non I i ne a r par t s vi ere the n c a I cu i ate d fro m the parameter es ti mates of the comb i ned process model minimiz i ng the r ms error. Th e results were as follows : ~1= D.87. I = 0 . 045. The est i ma t e d 0 = 0 .082 and inverse of the titration curve and the theoreticall y calculated inver se of the titration funct i on are s hown in Fig.8 by co n t i nuous and broken l ines. respectively. Finall y. the combined process mode l inverse was frozen and connected in series with the process . The input to the process was a r amp funct i on. The p r ocess output was used as the i nput to the mode l and the ou tput of the mode l was compared with the input of the p ro cess. The resu I ts are s h ow n in Fig.9. The m i smatch is due to the influe n ce of the process noises and th e insu ff ic ien t est i mation time.
6
Adap ti ve
5
Control
Th e th eore t ically ca l culated t i tration functions for t he four p r ocess feeds cons id ered (Pajunen. 1984) are shown in Fig.IO. The inverse o f the t itration func t i on was approximated by a p i ecewise polynomial function. In all tests a
G. A. Pajulle ll t ime -va ry i ng " con s tant - t race " adaptat ion ma tri x was used . Th e initial values of the parameter estimates ~ (O) =0.
-
-
Responses of t he simula t ed process to the s inu so i d al and piecew i se - constant reference signal are s ho wn in Fi gs 1 1 a n d 1 2, respectively . Finally regulat i on result s are shown for s i mu l ated and real process in Figs 13 and 14, respectively. T he p r ocess feed and fl ow are marked in the figures. CONCLUSION S Thi s \'10rK considered the on - I ine i dentif icat ion and adaptive contro l of Wiener type nonl inear processes. I t was shown t hat the developed ident i ficat i on methods can be app l i ed succesfu l l y i f prio ri knoVlledge of the static non l ineari ty i s so good that th e number of i n t erva l s, breakpoints and the po l y no m i al orders can be chosen such that the nonlinearity approx i mat ion error i s negl igib le. Thi s condition may somet i mes be hard to sa ti sfy , especial l y when the shape of th e no nl inear it y changes a lot and further research is ne ede d in order to design an algorith m which automat i cal l y adjusts t he breakpo i nt s to ge th er with the unknown coefficients in the piecew i se polynom ial approximation function. I n a specia l case, l'ihen the nonlinearity is odd and symmetr i c with respect to the origin and i t can be approximated by a single polynomia l w i t h only odd terms, the two step identification me thod can be used succesfu l l y even i f the nonl i near i t y approximation error i s cons id e ra ble . Th e advantage of these methods in pH - process ident i f i cation is that the compo nen ts of the p rocess so lu tion do not need to be known. On the other han d i f th i s know l edge is ava i 1ab l e, i t is not possib l e to in corporate i t , which is a clear disadvantage.
Dushkesas, M.I. , Kaminskas, W.A . (1978). Para me ter estimation i n Wiener -t ype mode l by nu mer i ca l optim izat ion methods. Liet . T SR Moks l u akad. Darbai S ser. (USSR), 3, 83 - 93 (in Russ~-Landau, 1. 0-:-(1978). Elimination of the real posit i vity condition in the design of para l l e l MRAS. IEEE Trans. Aut. Control, AC - 23, 1 0 15- 1020 . Landau, 1. 0. ( 1 979). Adapt i ve control the modern reference approach. Marcel Dekker, New York. Landau , 1.0. ( 19 82). Determ i nist i c and stoc hasti c adapt i ve po l es - zeros p l ace ment for m i nimum phase systems . Prepr. o f the Workshop on Adaptive Cont~ I S I S Flor ence, I taly, 207 - 244. L andau, 1 .0., Lozano, R. ( 1979) . Design and eva l uation of d i screte time exp l ic i t MRAC for tracking and regu l at i on. Report L AG 79.14, Dept. of Automatic Control, Grenobl e I nstit u te of Techno logy, France. Nie m i, A . ,Jut il a , P.K . ( 1977) . Process mo dels and contro l of acidity. Co mputer applic atio ns in the ana l ysis of chemical data a nd plants. CHEMDATA, 77,Flnland . Otto, P., Wer nstedt ,~978""J.Develop me nt and app l i cat i on of subopt i ma l test signa l sequences for i det i fy in g non I inear systems . Prep. of the 7t h IF AC Triennia l World Congre ss , Helsink i, Finl and, 1927 -1 933. Pajunen, G.A. ( 1984). Appl i cat i on of a mode l reference adaptive techn i que to the identification and control of Wiener t ype nonlinear p rocess. Acta Polytechnica Scand i navica, EE ser i es No 2..£, He l sinki, 1 30 pp.
!NONlI STATIC NE Aql T
LI NEAR DYNAM ICS
Y Y _____
;----<.~,
L . .. (Q)
Th en the control of a pH -pr ocess i n a continuous flow r eactor was co n sidered. F ig.l. The Wi ener mode l Simulation results proved that the adaptive controller was necessary for la rge changes in the pr ocess flo w and the s ha pe of t h e titration curve. Th e developed MRAC scheme with a nonlinear adaptive controller based on the p iece wise - cubic approxi mation of a static nonlinearity a nd a fast refe r ence mo del was best in tracking a s inusoid a l reference s i g nal . For the piece wise - constant or constant reference s i gnal the scheme w ith th e linear adaptive control ler, which was _u..;.,_lt_: -,.v I ---' based on the piecevlise - I inear or linear approxi mat ion of the stat ic nonlinearity, ~:tl wo r ked bet t er t ha n wit h the non I i ne a ran e . The s i mulation and experimental result s proved the effectiveness of th is adapti ve scheme w ith a linear adaptive cont r o l l er and a slow observer dynam i cs fo r t he compensation Fig.2. Series ident i ficat i on of la rge changes in th e process gain DO - fo l d)
MODEL
P it)
~-------~~~.-----------------"
and at the same process flow.
time
a
2 - fold
change
in
scheme.
the STO(HASTlC G,N ERATOR I (A'..C~LAT,::W
(
Si I I ings, S.A., Fakhouri S.Y. ( 1977 ). Identification of nonlinear sy s te ms using the Wiener mode l . Electron. Lett., 13,502-504. Desoer, C.A., Wang, Y.T.(1979) . Therobu st nonlinear servomechanism problem. Int.J. Cont rol , 29, 803 - 828. Dugard, L, Landau, 1 . 0. ( 1980). Recursive output e rror i den t ification algorithms. Th eory and eva l uation . Automatica,~, 443-462.
i
;g:S~~::~TTION ~'~------,
REFERENCES
F(y)
__ F_IU_I__._t l
~
i g.
3.
A s t ochastic generator excitation s ignal.
l '".,.>;
for
the
yltl
499
Wiener Type \!on linear Processes
CRO. ,O'
l
:mo le/ll !re ]
~ --
22.21 ~
_ / / /.h
/
/
/"
/
~ ~ ~ ':"
()(f-_ __ _ _ _ _--r-_ _ .o: PR'JC £SS _ '_
1. '. ~ I
.. "
~~:~ ~~~~~
:::: //
,. ,
1.5' 5.6 IL
+., ~
e-L .-L
_ __
1 ~
_ _- - , -_ _ _: - , : -_ _
60
6,
68
-:::-~ 7.2
pH
Fi g. 8 . The one - s te p identificat i o n procedure e x peri me ntal result s . The esti mated in verse of the t i trati o n function (contin uous line ) and the the o retically calculated inverse of the titrati on function (broken line) .
.
J
cR
10- !mole/ll tre J
2.8
~ ~I I J tl rf\
c.
2.6
f.
2.' 22
.orl) 20 18
Fig.4. Block diagram of the MRAC sch eme u sed to control Wiener type nonlinear proce sses .
_ _--'-_ _ 90 5'
16 ~-~L--~--~
o
, 8
36
--'-~
Time (1'T'1n
--'r--.··~ -~"':'~J'
em"'"
rl l , I _ _ __
rr it'fP" ( (I
i" I: '
,
"(l '. t IN~:' :' -- :"t;~ ". ~;.:":--
InOul
J
Fi g . 9. Validation test r esults: the process input (brok e n I ine ) and the output of the frozen mode l ( c ont inuous I ine ) c o nn e cted in series wi th the
p'o,m.
~ -: " -'-'-~~~ ~~~:~s~
Fig. 5. General controller structu r e for Wiener type nonlinear systems .
eR lmole/litre I
0.004 0.008 0.012 0.016
L
0[,
I so ·
Fig.IO. Theoretically calcu la ted titration functions for th e four process feeds cons i dered.
55L-~~-·~-~-1 5t. 1 63 1&21 9 6 " 2 1 1 2(.
~IO" ) lmOle/ il lre l
Fig. 6. The two - s t e p i den t i f i cat ion procedure - expe ri mental results. Es tima t ed s tati c nonlinearity (co ntinuou s I ine) and the theoretic ally calcu l ated titration curve (broken line)
m
:1
pH 72
i\
75'-
00
5.6
i
0 pH
32
I
o (a)
B,O
10.0 Tlme (mm l
Fig.7. The re spo nse of the process (continuous I ine) and the frozen model (broken I ine) to the ramp inpu t signa I.
V
V 60
I
8 00
~
A
. lJ LJ A\ n n
·00
V 110
I
.
I~
i
5 e~
C, .011
I !
I
72\-
I
4
6 I ;J~- _ /
I ! I
IV lBO
\I 240
IJ tl me/m ln
·006
0
60
120
180
240
c..
time/mm
(b)
Fig.ll. Tracking the sinusoida l reference signal (simulation r es ult s) using the adaptive system shown in Fig.4 with a nonI inear controller ( n= 3) and adaptive gai n reinitial i zati on at t=150 min. a) Reference model output YM(t} and process ootput pH(t). b) Reagent concentration Process feed I11 for time 0-150 min, p rocess fee d I for time 150-300 min.
:iOO
( ; . .-\ . 1"lllIll l' 11
tpH
C' I
:::: - - : ..E --
JO'------~--
o
.e
:6:
. .. :, . r e
~
r
(b I
( Q)
Fig.12 . Tracking a piecew i se - constant reference signal (si mulat ion results) "Iith a I i near contro l ler (n=1 ). a) Reference si g nal pHr{t) and proc es s output pH{t) . b) Reagent concentration. Process feed I I I for t=0 - 150 min and for t=150 - 300 min . Process flOlv I I/"'in for t=0 - 250 min and 0 . 5 I/m i n for t=250-300 min.
prl l ill , IT ~
III
!I!
G0 -
~
0 ..
2 0~
o_
o
!_L ~
') S ,,.'rrdfi
1 !/rn ln -L--l." • I
r llJ'IJ
I
80
( b)
(a)
F ig.13 . Regulation (simulation results) using the adapt i ve system i nc l uding i ntegral mode a) Process output pH{t) . b) Reagen t concentration CR(t). Process feed I II fot t=0 - 30 min, IV for t=30-60 min , I I I for t=60 - 90 mi n, I for t=90 - 180 min, I I for t= 180 - 270 mi n, I for t=270 - 315 min and I I I for t =3 15- 360 min. Pr ocess f low I I /min for t= I- 315 min and 0.5 I/m i n for t =3 15- 360 min.
ID
fLOW
0'> I!m,n 30
bO
90
120
1'>0
lBO
110
11.0
I'
=·-'Ii-m~ ~ ' - ..L_ .. ..-LlL _____ .L __ _ o
s:
I
i
270 t,me /mm
Fi g. 14. Regulat ion (experimenta l resul t s) us i ng t he ad a pti ve s c heme wi th an in t eg ral mode. The process fe eds a nd fl ow ar e ma rk ed i n t he f igure .
160
_ 1
240