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Self-Tuning Control of Stochastic Extremal Plants
CoplTight © I r,-\c Stochastic COlltrol \ 'ilnius , Lithualliall SSI{, L'SSR. lll ~ ti
SELF-TUNING CONTROL OF STOCHASTIC EXTREMAL PLANTS V. Kaminskas, t. Tallat-Kelpsa and K. Sidlauskas lll.ltitutl' <1'PhYsim/ 11 lid T I'(h 11 iw/ Prub/I' IIIS o/'ElIl'l'gl'ti('s, Lithuallilll/ Amdl'IN\' <1'S61'11(,I'.\ , 1\(1 U 11(1,1, CSSR
Abstract. Self-tuning control with recursive identification of extremal objects is considered. The objects can be represented by combinations of linear dynamic and extremal static elements, and their outputs are disturbed by a colour noise with a general fractional-rational spectral density. Minimum-variance controllers for Wiener-Hammerstein-, Wiener-, and Hammerstein-type objects are designed. The estimates of unknown parameters in the equations of the minimum-variance controllers are obtained in the identification process of the controlled object in a closed loop. The efficiency of self-tuning control algorithms is illustrated by statistical simulation. On the basis of the methods worked out adaptive systems for optimization of fuel combustion and steam condensation processes in 300 MW thermal power uni ts are developed. Keyword s . Stochastic extremal objects; minimum-variance control; selftuning control; control system synth~sis; recursive identification. PROBIEM STATEMENT
INTRODUCTION
Let us consider discrete-time single input -single output extremal dynamic objects wi th an observed output signal Ut defined by
Industrial processes often involve stochastic dynamic objects with extremal characteristics. The search methods (Rastri gin, 1974; Kazakevitch and Rodov, 1977) are most widely a pplied for the control law synthesis in designing control systems for such objec ts. However, the search stra tegies of extremum control require trial signals to be supplied to the object's input, this being not always possible. Moreover, the search methods are not optimal in providing extremal operation \Vi th the least errors arising due to uncontrolled disturbances.
Ut =~(Z-~f3)j(~ ; 8)+H (z-~h){t' ~
(1)
= W,(Z'1j c{)Xt '
(2)
where X - an observed input (control) t signal; an unobserved white noise sequence with a zero mean and a finite variance 6/ ;
It -
Often the control law synthesis for linear stochastic objects is based on the minimization of the variance of output signal deviations from the desired value (Astrom, 1970; Isermann, 1981). Such an approach deals with the desi gn of an optimal predictor of the controlled object's output and with the control strategy determination according to the equality condition between a corresponding number step-prediction value and a desired one. In selftuning control systems the unknovm parameters are replaced by their current estimates obtained in the identification process in a closed loop.
ne
'
na
1=0
j(
I-I
'
W(z-;.~) = L ~z-Ij( 1+ L;= alz-'), ,
fI d
(3)
nf
'
W (Z-~ 3)= C ~Z-/(1+ C oJL/ 2
/
I
=0
I
(4)
= I dl
(5 )
- fractional-rational transfer functions of the linear dynamic parts of the input (con trol ) channe 1 ( W, and Wz ) and the disturbance channel ( H ) with parameters
Recent works by Keviczky, Vajk and Hetthessy (1979 ) ; Kaminskas snd Tallat-Kelps a (1983); Kaminskas and Sidlauskas (1984, 1985) dea 1 with the applica tion of minimum-variance control strategies for the control of certain types of extremal dynamic objects. In this work problems of the design of self-tuning control algorithms for stochastic extremal objects of a more general class are considered. The objects consist of vario us interconnections of linear dynamic and extremal static elements and their outputs are corrupted by disturbances with a general fractional-rational spectral density.
d,T =
(a !' aZ] '" ] al1. , a
P of] p""p ) cC' j)11.6
(i,' 92" 'in;, do. d, ' " dad) f/= ( r; , r; rn. , P1 'Pz , .. , Pn,P )
ftT=
}
;
(6 )
7'"
-/
Z
-
rator;
an
,r'
t -step backward time-shift ope-
- an extremal characteristic of the static T element with the parameters 8 : (80 ,8t • 8z ); 37
38
\ ' . Kaminskas.
c:. Tall a t-K e lp sa and K. Sidlallskas
T - the transposi tion sign.
~
The equations (1)-(7 ) specify the most general class of stochastic extremal objects when the extrernal static element stands between two linear dynamic parts and the object's output is disturbed by a colour noise with a general fractional-rational spectral density (see Fig. la). Such objects are usually called stochastic extremal Wiener-Hammerstein-type objects. In particular cases, by removing the first linear dynamic part stochastic extremal Hammerstein-type objects are obtained, and by removing the second linear dynamic part we obtain stochastic extremal Wiener-type objects (see Fig. Ib and lc,respectively). Let us assume that the channels of control and disturbanoe signals are stable and minimum-phase, the polynomials in the numerator and the denominator of each of the transfer functions ()-( S) have no common roots, and the linear dynamic parts in the control channel have a unit gain: ~ ( Ijot) =- ~ (Ii;a) ~
Let it be required to -(7) from the initial terized by the val ues and uoto the state of
I.
(8)
drive the object (1) steady state characof the signals Xo extremal operation
Bo - B// ftez
u"
==
f(x*; 8) =
x*
=
tl!tJ e;h j (x, 8) = - ~ /282
}
Q/c) =
t=O,I, ... } (10 )
and the optimal current control signal value X;.~ at the discrete time t must be determined fro m the c ondition / ~ t1I7j milt Qt (xi {J ) (11 ) XI.,
i+{
..
where M - the sign of mathe matical expectation. Criterion (10) may be transformed into
Qt(Xf +()
==
where U-t..
J c Iz?') I
=
!1{ tlt.Jc lt7;}- u*(+ z [ 1- H-( £; hJ] Ut
O'j2)
(12 )
+
Wz (i;J} f [~ (iJ ri,) X'M
B 1) (1) - a one-step ahead prediction of the output signal at the discrete time t (Kaminskas, 1985); ~ (Xt >{, Xi' '' ' Ut' ili-( ''' ' ) the vector of inp ut and output observations used by the predic tor (1); c T ~ ( 0(" T, ~T, hT, eT) _ para me ters of the con trolled object (1)-(7). + H- (i;h)
j
J;T
The object's parameters being unknown it is impossible to calculate the optimal current control signal value as given in (11). If the parameters are replaced by their current estimates
x;{
(14)
(15)
t+co
'
where C - the admissible area for the pa"'* - the c urrent estimate of ra meters C; Ut u'" , obtained by substituting B t into (9). MINIMUM-VARIANCE CONTROLLERS Ta king into account equation (12) it is evident that condition (11) is equivalent to the following one:
aItj ~ Mlu.t.Jcl~ ) -u*}Z (17) t Then, simple transformations yield the eq uation of a mini mum-variance controller for stochastic extremal Wiener-Hammerstein -type objects (Kaminskas and Sidlauskas, 1985) : X
:{=
x:.(
t.,
=
~t{(£;rX,) (- ~ /2~
ft': max{ 0,
( 9)
2
[ue - u..e(c l~_{)] ,
a self-tuning system including control and identification algorithms can provide only the minimum value of the asymptotic variance . i ~¥}z (16) Q( xtd ) = f<-m M I u__ .( - Ut '
1t
=
u
=
± Vft-
)
[W/(i;I}(1t+~)-ui'] /eJ
z [H-« h)- fJ[Ui
and to ens ure the minimum variance of the errors arising due to uncontrolled dist urbances. Therefore the control performance criterion can be defined by
Qt(Xi'~)= t1!u_H -u...,,"}2,
1 t
Tt;
-
~ (.i ;f)f(lf ; g)]
.(18)
if k ~tJ u...J;.{ (c IzI;) , if kd-I, t-2, ...
{U'"
H
The block diagram of this controller shown in Fig. 2.
is
Seq uence ~t in (18 ) is a realization of an autoregressive-moving a verage process because it can be expressed as (19)
Therefore at the rando m discrete times k with «I< = 0 feedba ck is interrupted. Thus it is impossible to obtain the control error seq uence in the form of white noise wi th a variance This gives us the main principal difference as compared with the minim um-variance controllers for linear objects (Astrom, 1970 ; Isermann,
(J'/.
1 981) .
Substit uting W, {/IIC } -= 1 into eq uations (18 ) yields the e quat ion of a minimum-variance controller for stochastic e xtre mal Ha mrnerstein-ty pe obje c ts (Kaminskas and Ta 11a t-Ke Ips a, 198) :
x~{
=
-
~ /2B2 t Yf;
1t = z[fi(£~It)-f][Ut -
} ,(20)
~ (i ; J)J (Xijg)]
and in case ~ (z- ;! ) =- f , the e quation of a minimum-variance controller for stochastic extremal Wiener-type objects is obtained (Kami nskas and Sidlauskas, 1984) :
39
Self-luning Conlrol of SlOchaSlic Extremal Planls
SIMULATION RESULTS
= ~({Z-~(t.)(-~j2~
X;+1 Vt =
~t
tYf)
. (21)
max{O, ~t/Bz}
=z[f/7z-~h}-fJ[Urj(~j8)]
Figures 3 and 4 illustrate the processes of the minimum-variance control with the exactly known parameters of the object Cl) -(7). In Fig. 3a are given the results of the control of the determined extremal Wiener-Hammerstein-type object defined by 1 if:
t
12-171z(J+ZV- V2J 1- f3z- 1fI78£z t t
IDENTIFICATION IN A CLOSED LOOP
et
1982):
= Ct + ri+! JCt
Jet
=- *t+1
{,
- mt/~ { I, ?""mo.X_
t+! -
(J
(22)
(Ct )
Ct{-!
t+ 1
.
(27)
0.1
The current estimates used instead of of the unknown parameters of the object (1)-(7) in the equations (18) of the minimum-variance controller, are obtained in the identification process in a closed loop by a recursive algorithm (Kaminskas,
Ct+1
I
o}
Vt
=
1- 0.8 Z-1 f 175 z-2 Xt
It was required to drive the object (27) from the initial steady state characterized by the values of the signals X = 0. and iJp :: J to the extremal sta te Xf:; ~ , U:t; '" • For this purpose two types of control sequences were applied: a step input X : x~, t t:; 1,2}... ,and an optimal control signal defined by (18). Figure 3b illustrates the control process for the stochastic extremal object of the same type which is defined by
where
12-0.fz- f
z
1
= - - - - - {3-f2V-V )+---f t 1-fJz-1f08z- Z t t 1-0.9[7 t
t1.
ro = -T (ltff= !1t -
Jtff).~f!0-!7t).tl-f~~/
+[Jf)..;f,r7t).i+1]ihf~;ff'
t
for
J
(23)
1=0
<: Ite
_
(" -
(It ). tf! T
1 fA IN !1t ). tf!
fl tff =[
(24)
I-.i t +f )i+IJ °t
U1
)..t+!
(C) =:
=tit +f ~ "ift
titfj(c/Jt}j
(25)
(Ct)j
(26)
rtf! -
a scalar factor providing that the trajectory of current estimates is within the admissible area C , as r::O:II .Jet Uis the distanc",e in the direc tion Jet from the poin t Ct to the boundary of this area j 0. , 1- zero and identity matrices correspondingly, their dimensions being ItcXlle 1Z,=lZa+IZCf'1Z
-I-lZd+ltr+Ilf'+llo+2
-
t~e
number of the &knolVn parame ters; ~ ciff (Ct ) - the value of the prediction error gradient (i.e. the vector of the first deriva ti ves with re spec t to pa ra me ters C ) at ,the point et ; a small positive constan t; 11' 11 - the sign of vec tor length.
er -
t
1-0.8z-1+0.52.-Z
Xt
The aim of the control is to drive the object (28) from the initial steady state to the extremal one providing the minimum variance of the sequence of the output signal Ut deviations from its extremum ",:If= It . As in the example above the diagrams of control and output signals in the cases of both step and optimal (18) inputs are given. In Fig. 4 diagrams of the autocovariance function estimates are given for the following sequences: for the white noise for the random disturbance signal H(Z-~Iz)lt in the output, and for the deviations of the output signal Ut from the extremal value uf . The estimates are obtained according to the results of a lOOO-step control process employed on the object (28) using algorithm (18) and eliminating the influence of transients at the initial stage. Figure 4 illustrates the conclusion that the minimum-variance controllers for stochastic extremal objects are not able to provide an uncorrelated control error sequence \Vi th a variance equal to tha t of the sequence In this case the variance of the control error sequence is an intermedia te value between the variances of the input and output signals in the disturbance channe 1.
It '
for t ~ Ite ;
t
V=
'(28)
0.1
It'
We must note that the results resembling those shown in Fig. 3 and 4 are obtained also in cases of the minimum-variance control of extremal Hammerstein-type and Wiener-type objects.
40
V. Ka minskas . ( :. Tallat-Kelpsa and K. Sidlauskas
Self-tuning control with recursive identification is illustrated by employing it on the stochastic extremal Wiener-Hammerstein -type object defined by IL=
{
16
f-tO.6z- f
V =
t
(2V- V2 )-r
t
t
0. 6
1-0.4 ['
1
1-0.8£'
I.) t
CONCLUSIONS
. (29)
t
t
The estimation of the parameters o"lu'#" '1,8 82 of the object (29) was performed by "means of the component version of the recursive identification algorithm (22)-(26) (Kaminskas, 1985). The estimates had zero initial values. The control sequence was obtained according to the minimum-variance control algorithm (18) in which the unknown parameters were substi~ " " t):!te~ by" th~ir c urrent estimates o." &O,jl '
iu,
do' If,
8"
c ursive identification. Figures 7 and 8 illustrate the self-t uning control of steam condensation and f uel combustion processes.
Bz •
In Fig. 5 the behaviour of the estimates of the parameters in the process of simultaneous control and identification is shown. Figure 6a gives the diagrams of the control and output signals at the initial stage when parameter estimation errors are large and at the stage when the parameter estimates are close to their true val ues. In Fig. 6b corresponding diagrams for the case of minimum-variance control with exactly known parameters of the object (29) are given.
Methods for self-tuning control of stochastic extremal objects are considered. The objects consist of vario us interconnections of linear dyna mic parts and an extremal static element, and their outputs are disturbed by a colour noise with a general fractional-rational spectral density. Minimum-variance controllers are designed for extremal Wiener-Ha mmerstein-, Wiener-, and Hammerstein-type objects. It has been determined that the controllers are not able to provide (unlike the minimum-variance controllers for stochastic linear objects) an uncorrelated control error sequence \Vi th a variance equal to that of a white noise s equence at the input of the dist urbance channel. Closedloop parametric identification algorithms have been designed to estima te the unkno\'ffi parameters in the eq uations of the controllers. The efficiency of self-tuning control algorithms with rec ursive identif ication is illustrated by statistical simUlation. The methods worked out have allowed to design adaptive optimization systems for fuel combustion and steam condensation processes in thermal power units. REFERENCES
APPLICATION Designed self-tuning control algorithms were applied for the optimization of fuel combustion and steam condensation processes in 300 MW power units of a thermal power plant. The main characteristics of subsystems of the power units are mostly extremal. During operation process these characteristics change due to ageing of the equipment, contamination of working surfaces and under the influence of other uncontrolled disturbances. Therefore it is possible to provide opti mal operation of separate subsystems and the power unit as a whole only by applying self-tuning control. The aim of the control may be the minimization of the given to a user active power deviations from its highest pos s ible value at a fixed expenditure of fuel. The process of steam condensation is controlled by changing the circ ulation pumps loading that determines the absol ute pressure in the condenser, the latter being considered as an observed input signal. Control of fue l combustion process is carried out by changing the ventilator wings angle Which influences on the oxygen concen tra tion in smo ke gases, the la t ter being considered as an input signal. The stochastic extremal Hammerstein-type model has been used for the description of above-mentioned processes. Preliminary identification of the object in an open loop has been carried out by applying pseudorandom input seq uences. Estimates of the parameters obtained in this way have been used as the initial values in the algorithms of self-tuning control with re-
As trom, K.:J. (1970). Introd uc tion to Stochastic Control Theory. Academic Press, New York. I sermann, R. (1981 ) . Digital Control Systems. Springer , Berlin. Kaminskas, V. (1982 ) . Dynamic System I dentification via Discrete-Time Observations. Part 1 - Statistical I.lethod Foundations. Es tlmation In Llnear Systems. Mokslas, Vllnius (In Russian) . Kaminskas, V. (1985 ) . Dynamic Sys tem I dentification via Di screte-Tlme Observations. Part 2 - Estima tion In Non Hnear Systems. Mokslas, Vllnius (in RUSSlan) . Kaminskas, v., and C. Tallat-Kelpsa (1 983 ). Con trol of extremal Hammerstein-type sys tems. In A. Nem ura (Ed.), Sta tistical Problems of Contro l, Issue 60. Vllni us. pp. 27-51 (ln Russ ian ) . Kaminskas, V., and K. Si dl auskas (1 984) . Control of extremal Wiene r -type systems. In C. Pa ula uskas (Ed.), Statistical Problems of Control, Issue 63. Vl1nius. pp. 146-156 (In Russ ian ) . Kaminskas, V., and K. Sidla usl
41
Sel f-tullill g Control of Stoc hastic Lxtrcmal Plants
y,
!It
':f'
a
c
Fig. 1. Stochastic extremal objects of Wiener-Hammerstein (a), Hammerstein (b ) , and Wiener (c) t ypes .
ICONTROLLER-
--------,
I
I
I .-------,
I
OBJECT
r PREDICTOR -
-
-
-
-l
-
I
Fig . 2 . The m ~n~ mum -varia nce controller for a stochastic extremal Wiener-Hammerstein-type object.
Xt
.. ..
....'
: XXX X XX :~~l~~~~.·········
'
O ~-~-~-~_L-_L
o
. ..
x xxxx , x •••••• x xxx xxxxxxx x
10
20
R{/J
e•
O' ~-~-L--~·~~~L
0. o x -
••••
10.
20.
1
0.2 x -
0-2
•
•• 2
•••
1
· -2 Ut
5
0.1 XX
x ' ••• ¥ ••• t ••• I ••• t'~~¥~". xxx
0' c......._
o
X X
u·· ;;
•• 3
••• •••
••••• ••••••
, 4
•• ••
•••• •••••••
......_ t 4 10 20. I
3 0.
10.
6
20.
t
Fig. 3. The res ul t s of the minimum-variance control applied to the object ( 27 ) (a) and to the obj e c t (28) (b): 1 - the step input being used, 2 - the optimal control being used.
Fig. 4. The autocovariance function estimates for the object (28): 1 - for the white noise, 2 - for the disturbances at the output, J - for the control error.
42
V. Kamin skas ,
C.
Tallat-Kelpsa a nd K. Sidlauskas
"2
fi p: 2
/
_
,
_::::
e ~8' ,~:: -. ,-
,; ;
2
-2o
H
I
t
I
t
- .------:::;:~.~=:::::;;;::~====~===== a; a, n
-
\
- ;
,
I
500
150
/000
Fig. 5 . The c urrent para meter esti mates for the object (2 9) in the process of self-tuning control with recursive identification.
Fig. 6. The dia grams of the control and output signals of the object ( 29 ) in the case of self-tuning control with recursive identification (a) and in the case of optimal control with known parameters (b).
Fig. 7. The adaptive optimization of the steam condensation process (V - the absolute pressure in the condenser, PN - the active power given to a user).
Fig. 8. The adaptive optimization of the fuel combustion process (° 2 - the concentration of oxygen in smoke gases, PN - the active power given to a user ) .
SELF-TUNING CONTROL OF STOCHASTIC EXTREMAL PLANTS V. Kaminskas, t. Tallat-Kelpsa and K. Sidlauskas lll.ltitutl' <1'PhYsim/ 11 lid T I'(h 11 iw/ Prub/I' IIIS o/'ElIl'l'gl'ti('s, Lithuallilll/ Amdl'IN\' <1'S61'11(,I'.\ , 1\(1 U 11(1,1, CSSR
Abstract. Self-tuning control with recursive identification of extremal objects is considered. The objects can be represented by combinations of linear dynamic and extremal static elements, and their outputs are disturbed by a colour noise with a general fractional-rational spectral density. Minimum-variance controllers for Wiener-Hammerstein-, Wiener-, and Hammerstein-type objects are designed. The estimates of unknown parameters in the equations of the minimum-variance controllers are obtained in the identification process of the controlled object in a closed loop. The efficiency of self-tuning control algorithms is illustrated by statistical simulation. On the basis of the methods worked out adaptive systems for optimization of fuel combustion and steam condensation processes in 300 MW thermal power uni ts are developed. Keyword s . Stochastic extremal objects; minimum-variance control; selftuning control; control system synth~sis; recursive identification. PROBIEM STATEMENT
INTRODUCTION
Let us consider discrete-time single input -single output extremal dynamic objects wi th an observed output signal Ut defined by
Industrial processes often involve stochastic dynamic objects with extremal characteristics. The search methods (Rastri gin, 1974; Kazakevitch and Rodov, 1977) are most widely a pplied for the control law synthesis in designing control systems for such objec ts. However, the search stra tegies of extremum control require trial signals to be supplied to the object's input, this being not always possible. Moreover, the search methods are not optimal in providing extremal operation \Vi th the least errors arising due to uncontrolled disturbances.
Ut =~(Z-~f3)j(~ ; 8)+H (z-~h){t' ~
(1)
= W,(Z'1j c{)Xt '
(2)
where X - an observed input (control) t signal; an unobserved white noise sequence with a zero mean and a finite variance 6/ ;
It -
Often the control law synthesis for linear stochastic objects is based on the minimization of the variance of output signal deviations from the desired value (Astrom, 1970; Isermann, 1981). Such an approach deals with the desi gn of an optimal predictor of the controlled object's output and with the control strategy determination according to the equality condition between a corresponding number step-prediction value and a desired one. In selftuning control systems the unknovm parameters are replaced by their current estimates obtained in the identification process in a closed loop.
ne
'
na
1=0
j(
I-I
'
W(z-;.~) = L ~z-Ij( 1+ L;= alz-'), ,
fI d
(3)
nf
'
W (Z-~ 3)= C ~Z-/(1+ C oJL/ 2
/
I
=0
I
(4)
= I dl
(5 )
- fractional-rational transfer functions of the linear dynamic parts of the input (con trol ) channe 1 ( W, and Wz ) and the disturbance channel ( H ) with parameters
Recent works by Keviczky, Vajk and Hetthessy (1979 ) ; Kaminskas snd Tallat-Kelps a (1983); Kaminskas and Sidlauskas (1984, 1985) dea 1 with the applica tion of minimum-variance control strategies for the control of certain types of extremal dynamic objects. In this work problems of the design of self-tuning control algorithms for stochastic extremal objects of a more general class are considered. The objects consist of vario us interconnections of linear dynamic and extremal static elements and their outputs are corrupted by disturbances with a general fractional-rational spectral density.
d,T =
(a !' aZ] '" ] al1. , a
P of] p""p ) cC' j)11.6
(i,' 92" 'in;, do. d, ' " dad) f/= ( r; , r; rn. , P1 'Pz , .. , Pn,P )
ftT=
}
;
(6 )
7'"
-/
Z
-
rator;
an
,r'
t -step backward time-shift ope-
- an extremal characteristic of the static T element with the parameters 8 : (80 ,8t • 8z ); 37
38
\ ' . Kaminskas.
c:. Tall a t-K e lp sa and K. Sidlallskas
T - the transposi tion sign.
~
The equations (1)-(7 ) specify the most general class of stochastic extremal objects when the extrernal static element stands between two linear dynamic parts and the object's output is disturbed by a colour noise with a general fractional-rational spectral density (see Fig. la). Such objects are usually called stochastic extremal Wiener-Hammerstein-type objects. In particular cases, by removing the first linear dynamic part stochastic extremal Hammerstein-type objects are obtained, and by removing the second linear dynamic part we obtain stochastic extremal Wiener-type objects (see Fig. Ib and lc,respectively). Let us assume that the channels of control and disturbanoe signals are stable and minimum-phase, the polynomials in the numerator and the denominator of each of the transfer functions ()-( S) have no common roots, and the linear dynamic parts in the control channel have a unit gain: ~ ( Ijot) =- ~ (Ii;a) ~
Let it be required to -(7) from the initial terized by the val ues and uoto the state of
I.
(8)
drive the object (1) steady state characof the signals Xo extremal operation
Bo - B// ftez
u"
==
f(x*; 8) =
x*
=
tl!tJ e;h j (x, 8) = - ~ /282
}
Q/c) =
t=O,I, ... } (10 )
and the optimal current control signal value X;.~ at the discrete time t must be determined fro m the c ondition / ~ t1I7j milt Qt (xi {J ) (11 ) XI.,
i+{
..
where M - the sign of mathe matical expectation. Criterion (10) may be transformed into
Qt(Xf +()
==
where U-t..
J c Iz?') I
=
!1{ tlt.Jc lt7;}- u*(+ z [ 1- H-( £; hJ] Ut
O'j2)
(12 )
+
Wz (i;J} f [~ (iJ ri,) X'M
B 1) (1) - a one-step ahead prediction of the output signal at the discrete time t (Kaminskas, 1985); ~ (Xt >{, Xi' '' ' Ut' ili-( ''' ' ) the vector of inp ut and output observations used by the predic tor (1); c T ~ ( 0(" T, ~T, hT, eT) _ para me ters of the con trolled object (1)-(7). + H- (i;h)
j
J;T
The object's parameters being unknown it is impossible to calculate the optimal current control signal value as given in (11). If the parameters are replaced by their current estimates
x;{
(14)
(15)
t+co
'
where C - the admissible area for the pa"'* - the c urrent estimate of ra meters C; Ut u'" , obtained by substituting B t into (9). MINIMUM-VARIANCE CONTROLLERS Ta king into account equation (12) it is evident that condition (11) is equivalent to the following one:
aItj ~ Mlu.t.Jcl~ ) -u*}Z (17) t Then, simple transformations yield the eq uation of a mini mum-variance controller for stochastic extremal Wiener-Hammerstein -type objects (Kaminskas and Sidlauskas, 1985) : X
:{=
x:.(
t.,
=
~t{(£;rX,) (- ~ /2~
ft': max{ 0,
( 9)
2
[ue - u..e(c l~_{)] ,
a self-tuning system including control and identification algorithms can provide only the minimum value of the asymptotic variance . i ~¥}z (16) Q( xtd ) = f<-m M I u__ .( - Ut '
1t
=
u
=
± Vft-
)
[W/(i;I}(1t+~)-ui'] /eJ
z [H-« h)- fJ[Ui
and to ens ure the minimum variance of the errors arising due to uncontrolled dist urbances. Therefore the control performance criterion can be defined by
Qt(Xi'~)= t1!u_H -u...,,"}2,
1 t
Tt;
-
~ (.i ;f)f(lf ; g)]
.(18)
if k ~tJ u...J;.{ (c IzI;) , if kd-I, t-2, ...
{U'"
H
The block diagram of this controller shown in Fig. 2.
is
Seq uence ~t in (18 ) is a realization of an autoregressive-moving a verage process because it can be expressed as (19)
Therefore at the rando m discrete times k with «I< = 0 feedba ck is interrupted. Thus it is impossible to obtain the control error seq uence in the form of white noise wi th a variance This gives us the main principal difference as compared with the minim um-variance controllers for linear objects (Astrom, 1970 ; Isermann,
(J'/.
1 981) .
Substit uting W, {/IIC } -= 1 into eq uations (18 ) yields the e quat ion of a minimum-variance controller for stochastic e xtre mal Ha mrnerstein-ty pe obje c ts (Kaminskas and Ta 11a t-Ke Ips a, 198) :
x~{
=
-
~ /2B2 t Yf;
1t = z[fi(£~It)-f][Ut -
} ,(20)
~ (i ; J)J (Xijg)]
and in case ~ (z- ;! ) =- f , the e quation of a minimum-variance controller for stochastic extremal Wiener-type objects is obtained (Kami nskas and Sidlauskas, 1984) :
39
Self-luning Conlrol of SlOchaSlic Extremal Planls
SIMULATION RESULTS
= ~({Z-~(t.)(-~j2~
X;+1 Vt =
~t
tYf)
. (21)
max{O, ~t/Bz}
=z[f/7z-~h}-fJ[Urj(~j8)]
Figures 3 and 4 illustrate the processes of the minimum-variance control with the exactly known parameters of the object Cl) -(7). In Fig. 3a are given the results of the control of the determined extremal Wiener-Hammerstein-type object defined by 1 if:
t
12-171z(J+ZV- V2J 1- f3z- 1fI78£z t t
IDENTIFICATION IN A CLOSED LOOP
et
1982):
= Ct + ri+! JCt
Jet
=- *t+1
{,
- mt/~ { I, ?""mo.X_
t+! -
(J
(22)
(Ct )
Ct{-!
t+ 1
.
(27)
0.1
The current estimates used instead of of the unknown parameters of the object (1)-(7) in the equations (18) of the minimum-variance controller, are obtained in the identification process in a closed loop by a recursive algorithm (Kaminskas,
Ct+1
I
o}
Vt
=
1- 0.8 Z-1 f 175 z-2 Xt
It was required to drive the object (27) from the initial steady state characterized by the values of the signals X = 0. and iJp :: J to the extremal sta te Xf:; ~ , U:t; '" • For this purpose two types of control sequences were applied: a step input X : x~, t t:; 1,2}... ,and an optimal control signal defined by (18). Figure 3b illustrates the control process for the stochastic extremal object of the same type which is defined by
where
12-0.fz- f
z
1
= - - - - - {3-f2V-V )+---f t 1-fJz-1f08z- Z t t 1-0.9[7 t
t1.
ro = -T (ltff= !1t -
Jtff).~f!0-!7t).tl-f~~/
+[Jf)..;f,r7t).i+1]ihf~;ff'
t
for
J
(23)
1=0
<: Ite
_
(" -
(It ). tf! T
1 fA IN !1t ). tf!
fl tff =[
(24)
I-.i t +f )i+IJ °t
U1
)..t+!
(C) =:
=tit +f ~ "ift
titfj(c/Jt}j
(25)
(Ct)j
(26)
rtf! -
a scalar factor providing that the trajectory of current estimates is within the admissible area C , as r::O:II .Jet Uis the distanc",e in the direc tion Jet from the poin t Ct to the boundary of this area j 0. , 1- zero and identity matrices correspondingly, their dimensions being ItcXlle 1Z,=lZa+IZCf'1Z
-I-lZd+ltr+Ilf'+llo+2
-
t~e
number of the &knolVn parame ters; ~ ciff (Ct ) - the value of the prediction error gradient (i.e. the vector of the first deriva ti ves with re spec t to pa ra me ters C ) at ,the point et ; a small positive constan t; 11' 11 - the sign of vec tor length.
er -
t
1-0.8z-1+0.52.-Z
Xt
The aim of the control is to drive the object (28) from the initial steady state to the extremal one providing the minimum variance of the sequence of the output signal Ut deviations from its extremum ",:If= It . As in the example above the diagrams of control and output signals in the cases of both step and optimal (18) inputs are given. In Fig. 4 diagrams of the autocovariance function estimates are given for the following sequences: for the white noise for the random disturbance signal H(Z-~Iz)lt in the output, and for the deviations of the output signal Ut from the extremal value uf . The estimates are obtained according to the results of a lOOO-step control process employed on the object (28) using algorithm (18) and eliminating the influence of transients at the initial stage. Figure 4 illustrates the conclusion that the minimum-variance controllers for stochastic extremal objects are not able to provide an uncorrelated control error sequence \Vi th a variance equal to tha t of the sequence In this case the variance of the control error sequence is an intermedia te value between the variances of the input and output signals in the disturbance channe 1.
It '
for t ~ Ite ;
t
V=
'(28)
0.1
It'
We must note that the results resembling those shown in Fig. 3 and 4 are obtained also in cases of the minimum-variance control of extremal Hammerstein-type and Wiener-type objects.
40
V. Ka minskas . ( :. Tallat-Kelpsa and K. Sidlauskas
Self-tuning control with recursive identification is illustrated by employing it on the stochastic extremal Wiener-Hammerstein -type object defined by IL=
{
16
f-tO.6z- f
V =
t
(2V- V2 )-r
t
t
0. 6
1-0.4 ['
1
1-0.8£'
I.) t
CONCLUSIONS
. (29)
t
t
The estimation of the parameters o"lu'#" '1,8 82 of the object (29) was performed by "means of the component version of the recursive identification algorithm (22)-(26) (Kaminskas, 1985). The estimates had zero initial values. The control sequence was obtained according to the minimum-variance control algorithm (18) in which the unknown parameters were substi~ " " t):!te~ by" th~ir c urrent estimates o." &O,jl '
iu,
do' If,
8"
c ursive identification. Figures 7 and 8 illustrate the self-t uning control of steam condensation and f uel combustion processes.
Bz •
In Fig. 5 the behaviour of the estimates of the parameters in the process of simultaneous control and identification is shown. Figure 6a gives the diagrams of the control and output signals at the initial stage when parameter estimation errors are large and at the stage when the parameter estimates are close to their true val ues. In Fig. 6b corresponding diagrams for the case of minimum-variance control with exactly known parameters of the object (29) are given.
Methods for self-tuning control of stochastic extremal objects are considered. The objects consist of vario us interconnections of linear dyna mic parts and an extremal static element, and their outputs are disturbed by a colour noise with a general fractional-rational spectral density. Minimum-variance controllers are designed for extremal Wiener-Ha mmerstein-, Wiener-, and Hammerstein-type objects. It has been determined that the controllers are not able to provide (unlike the minimum-variance controllers for stochastic linear objects) an uncorrelated control error sequence \Vi th a variance equal to that of a white noise s equence at the input of the dist urbance channel. Closedloop parametric identification algorithms have been designed to estima te the unkno\'ffi parameters in the eq uations of the controllers. The efficiency of self-tuning control algorithms with rec ursive identif ication is illustrated by statistical simUlation. The methods worked out have allowed to design adaptive optimization systems for fuel combustion and steam condensation processes in thermal power units. REFERENCES
APPLICATION Designed self-tuning control algorithms were applied for the optimization of fuel combustion and steam condensation processes in 300 MW power units of a thermal power plant. The main characteristics of subsystems of the power units are mostly extremal. During operation process these characteristics change due to ageing of the equipment, contamination of working surfaces and under the influence of other uncontrolled disturbances. Therefore it is possible to provide opti mal operation of separate subsystems and the power unit as a whole only by applying self-tuning control. The aim of the control may be the minimization of the given to a user active power deviations from its highest pos s ible value at a fixed expenditure of fuel. The process of steam condensation is controlled by changing the circ ulation pumps loading that determines the absol ute pressure in the condenser, the latter being considered as an observed input signal. Control of fue l combustion process is carried out by changing the ventilator wings angle Which influences on the oxygen concen tra tion in smo ke gases, the la t ter being considered as an input signal. The stochastic extremal Hammerstein-type model has been used for the description of above-mentioned processes. Preliminary identification of the object in an open loop has been carried out by applying pseudorandom input seq uences. Estimates of the parameters obtained in this way have been used as the initial values in the algorithms of self-tuning control with re-
As trom, K.:J. (1970). Introd uc tion to Stochastic Control Theory. Academic Press, New York. I sermann, R. (1981 ) . Digital Control Systems. Springer , Berlin. Kaminskas, V. (1982 ) . Dynamic System I dentification via Discrete-Time Observations. Part 1 - Statistical I.lethod Foundations. Es tlmation In Llnear Systems. Mokslas, Vllnius (In Russian) . Kaminskas, V. (1985 ) . Dynamic Sys tem I dentification via Di screte-Tlme Observations. Part 2 - Estima tion In Non Hnear Systems. Mokslas, Vllnius (in RUSSlan) . Kaminskas, v., and C. Tallat-Kelpsa (1 983 ). Con trol of extremal Hammerstein-type sys tems. In A. Nem ura (Ed.), Sta tistical Problems of Contro l, Issue 60. Vllni us. pp. 27-51 (ln Russ ian ) . Kaminskas, V., and K. Si dl auskas (1 984) . Control of extremal Wiene r -type systems. In C. Pa ula uskas (Ed.), Statistical Problems of Control, Issue 63. Vl1nius. pp. 146-156 (In Russ ian ) . Kaminskas, V., and K. Sidla usl
41
Sel f-tullill g Control of Stoc hastic Lxtrcmal Plants
y,
!It
':f'
a
c
Fig. 1. Stochastic extremal objects of Wiener-Hammerstein (a), Hammerstein (b ) , and Wiener (c) t ypes .
ICONTROLLER-
--------,
I
I
I .-------,
I
OBJECT
r PREDICTOR -
-
-
-
-l
-
I
Fig . 2 . The m ~n~ mum -varia nce controller for a stochastic extremal Wiener-Hammerstein-type object.
Xt
.. ..
....'
: XXX X XX :~~l~~~~.·········
'
O ~-~-~-~_L-_L
o
. ..
x xxxx , x •••••• x xxx xxxxxxx x
10
20
R{/J
e•
O' ~-~-L--~·~~~L
0. o x -
••••
10.
20.
1
0.2 x -
0-2
•
•• 2
•••
1
· -2 Ut
5
0.1 XX
x ' ••• ¥ ••• t ••• I ••• t'~~¥~". xxx
0' c......._
o
X X
u·· ;;
•• 3
••• •••
••••• ••••••
, 4
•• ••
•••• •••••••
......_ t 4 10 20. I
3 0.
10.
6
20.
t
Fig. 3. The res ul t s of the minimum-variance control applied to the object ( 27 ) (a) and to the obj e c t (28) (b): 1 - the step input being used, 2 - the optimal control being used.
Fig. 4. The autocovariance function estimates for the object (28): 1 - for the white noise, 2 - for the disturbances at the output, J - for the control error.
42
V. Kamin skas ,
C.
Tallat-Kelpsa a nd K. Sidlauskas
"2
fi p: 2
/
_
,
_::::
e ~8' ,~:: -. ,-
,; ;
2
-2o
H
I
t
I
t
- .------:::;:~.~=:::::;;;::~====~===== a; a, n
-
\
- ;
,
I
500
150
/000
Fig. 5 . The c urrent para meter esti mates for the object (2 9) in the process of self-tuning control with recursive identification.
Fig. 6. The dia grams of the control and output signals of the object ( 29 ) in the case of self-tuning control with recursive identification (a) and in the case of optimal control with known parameters (b).
Fig. 7. The adaptive optimization of the steam condensation process (V - the absolute pressure in the condenser, PN - the active power given to a user).
Fig. 8. The adaptive optimization of the fuel combustion process (° 2 - the concentration of oxygen in smoke gases, PN - the active power given to a user ) .