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Identification-Based Algorithm of Mine Ventilation Control
Copyright © IFAC Automation in Mining , Mineral and Metal Processing , Helsink i, Finland , ) 983
IDENTIFICATION-BASED ALGORITHM OF MINE VENTILATION CONTROL V. O. Bogdanov and D. V. Kneller Institut e of Control Sciences, Moscow, USSR
Abstract. The paper is concerned with a mine ventilation system based on the preliminary identification of the plant during its functioning. The sought control is determined by a gradient optimization method using the estimated Green's functions of a complex system.
Keywords. Identification. Green's function, Newton-Raphson method; coal mine ventilation; methane concentration.
INTHODUCTION
in large dimensionality of the problems solved.
One of the vital activities in mining research is development of methods and facilities for timely and adequate co al mine ventilation, i.e. removal of methane from the galleries. 11ethane is excreted by the ore during its mining and at concentrations exceeding the maximal acceptable presents a risk for miners. The ventilation is carried out by a powerful fan and a system of shutters i n the galleries.
At present there is quite a simple identification-based approach to solution of the control problem. According to it the fan and shutter Green's functions are identified with the subsequent derivation of the control vector. The authors sugp,est a method which does not require preliminary specification of the type of equations describing the process, but handles only known time re al izations of input and output signale obtained from the functioning plant.
The paper aims a t developing an al g orithm of the ve ntilation system control. Such control must reduce the material losses due to mine emergency downtime. Attempts at designing a mine ventilation control system were numerous. All of them were based on the physics of the gas excretion. This required involving a theory of nonlinear partial differential equations which resulted 709
A two-unit control system is proposed for mine ventilation. The input of the first unit of the system , is the L vector of control u(t) • The first L of its components determine the position of the air shutter controllers, whereas the -th one is a controlling parameter of the fan.
-1
L
v.
710
O. Bogdanov and D. V. Kneller
At the output of the first unit the sensors measure a vector of air consumption rate V (t) , coming as an input to the second unit. The 1vector of methane concentration C{t) is measured at the output of the second unit. Identification is used to a~sess the Green's functionsc1(') and <:;. C·) of the first and second units, respectively. The r.m.s. distance between identification results-based values of vectors 1)'Y.{t) ,c~ et) and the measured values of vectors 'VC t) and C{t) ,respectively, is taken to be a criterion of closeness of the Green's functions obtained to their true value~. Values of the functions and er (.) obtained are usecJ. to obtain an optimal control U. (t). The optimality criterion of the control vector u ( t) and restrictions to its components are given below.
11
G(.)
where 11= [v:, ('X/t) } "'/ V I1 ex/tJ] 7' is a vector of output variables, U. :: = [U1(~,t)I"" Ll.L(~,t)]-r is a vec.tor of input variables, ~ (.) = =[qmt>]l<,JJ 'JL is a matri~ of the If m,i, .../'.( sought weight functions. Similar relations can be written for the second unit. Attempts undertaken to solve the set of Fredholm I equations obtained due to its non-regularity result in computational difficulties and, as a rule, fail. To reduce the system to a number of independent equations we introduce a, notion of partial moment characteristics of random fields Ut and Uj " which are covariations of their va~ lues centered relative to the conditional expectations (Bogdanov and Rajbman, 1981) 0 O . \
Kuu./u_. . (~,1J'r,S )'='-N{[UJ~I't)- M(Ll JP)JU~tl) Y); • J tJJ AA" (T! 5)/ (,q ") K*L~ j fJE..2..,;YE~ ][U,/f,S)- rt( ~j : ' U..: l/ , Y ; 0
IDENTIFICATION The state of the system considered is characterized by a random vector field and described by differential equations in partial derivatives. An identification problem is stated, i.e. a problem of using the known values of input and output to develop a plant model, an output signal of which would in terms of a certain criterion, be close to an output of a real system. It is well known that with the r.m.s. criterion involved, application of correlation methods would lead to identification equations which are similar to Wiener's filter for a multi-dimensional distributed system (Rajbman, 1978). KUi1,(x)~lt)-- Si) 9iXJ~,S)j(",u/;)~, Z'-s ) d~c/s +...+
'.()~+ S: S.2:1' 9f/:x.J~' S)K.t~u/~ 17;r:-S)t;/~5 KVHU[x,p/L)=-r S1J11M{X,~,s)Ki{fLt!~' 'f,(-5)d~Js+..~ (1 ) (
0
+I 52Jf~HL ex, ~,S)KULU!f,-,~,t-sJd~s
0
0
/(tL 'j ;""E~~iYE:Ty]}
where
(2)
U( -=:. u.;-Mu.f.'
{ ==
i ,.. , L.
Using the system (1) a nd definition (2) we yield equations each one of . which gives a possibility of determining the sought weight functions.
Kv. u /u (X''f,t,fi)=St ~il77d.:x) F,t)S) ,.., f
-f
-OQ
J
X
xKue ~/Ldf.},s})(/~ ds, f'= f.. , L .tr(~j, ., 11.(3) The derived identification equations are also valid when input and output, noisy signals are measured. It is required only that the noise vector is uncorrelated with the input vector.
'rhus equations (3) were derived using the partial correlation functions. Each of the equations permit determining the sought Green's functions whereas application of coventional correlation methods requires solving
Identification-based Algorithm of Mine Ventilation Control
the system (1). Input and output signals can be treated as scalar uni-dimensional random fields. An input concentrated variable tit- which is a random process, presents an exception. By I1photographingl1 the process we obtain input and output time realizations. A plant specified in such a way can be treated as a multi-dimensional one with concentrated parameters for which the above postulations hold true. For the process considered the best r.m.s. estimates of the weight functions C(-) and C, (-) yielded by the correlation methods are: l' K7f",ut {u-t. (t);:; 50 ~",e ('C)K~udu_~ (t-t.) dI: , (4) KCi1fM
5: 'ii.n\J~)KVfT/.lIMIU.I)t(t-"t) d~
/1f..",(tj =
(5)
where KVml-{tfu,.e ,K.;y",/~"" KUeue/U_(,1
=/,'0 i 9",,( {f)KUei.le fu.e (ft.-p ),
Kcy",/V..m{n)=£
q.", ({ 0) KVM V..,. {v.. .... (n-f) ,
f:o(J
~ :: mii1 (n,P), E; =min.{if,P''ji ~~1, ..., I; C~ 1,. .}Li,,"=i, .. ,H.
The systems of linear algebraic equations obtained are solved by the Gauss method with the preliminary regularization (Tikhonov and Glasko, 1964) •
The estimates of the Green's functions are calculated in six points of the
711
axis 't (P~ P "= 6). The nwnber of these points is a significant parameter influencing the alg orithm promptness. To obtain the estimates of conditional expectations, variation ranges of signals Ut. and 1fM are divided uniformly. The quality of the model. obtained can be assessed by the r.m.s. criterion, or by the variance ratio R {C~ft)l:=;() M dt }IU t (t) ,t': 1.. .. L ~~) J ;z,[CJt)] , (-~ ...J.
i fC
H
In case of complete adequacy of the. model to a real process R.{c~{t)}:::.1. •. In a g eneral case, the measure of variance ratio lies within O~ ==-RiC;(t)}~j (Rajbman, 1978 ). The method discussed yields a best linear estimate of a nonlinear system being identified. The nonlinearity degree is also determined by expression (6)
3. CONTROL Components of the vector-function U(t) can be treated as constant on a segment (t K- 1 ,tK } , Kc;1,2 . ...
fu;<, U/,-"coflst, iE(il_f,t,..]}}
Li.e{t) =
(7)
This assumption does not contradict the physics of the processes considered. The model output c*{t) as the function of the vector u(t) is presen.,. ted by the following double convolution L M ~ ct(t)=I? (j~(qme1t {.(j!))) i=l,. .., I. (8) e;f m=i d' Denote the Green's functions obtained as a result of the identification within the tim~ interval (t(-l It",] as C;:;I::(.) and C; "'-(-) ,respectively, . whereas the methane concentration calculated thereby as C~ .... , i. -; 1, ... / I Assume that the identification yield~d the function values tq K(.) and G;'C(') in points tK,j =:::t~cilz. ,j=O,. ..,J. Hence, using the definition of a con-
v.
71 2
O. Bogdanov and D. V. Kn e ller
volution and considering a constant nature of components of the vector u.. K. ~ we obtain 1'4 Cl(*" ,j.:' K-t (t&.~ )(f ,1 ( =fo ~~u.' [.;fJt•.!:~{(;rf ~l~: lz.)dtJd)( 9)
z-""
fC
tt
w~/ere V'::il~~~~"'1 ....,.e ,f=miH.ffJ~"'t':?!:-tl:-Jlt.j. IV
is a set of positive integers.
Denoting the expression in brackets 01<- 'I. as &te) i..~1.I""·II}t~i""IL ,we rewrite (9) as fOliws: C/{¥'" t/{-2 1:-1 i. =L.~ it tI~ } i.. :: j, ... , I . ~=11 c=( Since when obtaining tL , {=l, ... }L J U;~2.t<:1r-,l.l~J,-P-are believed to be already knowi,~introducing t: ,5" 17 K - 2 K- ' 2 . 1 (;(!.. =LL 'Oil" LI~ , L=1, ... , . 2=( e=( we 0 bt aln L
e
J( ~ /? K C K -f='Z~+~ifUft:
t;
10
Values can be approximately calculated using the trapezium formula (Krylov, 1976). The sought control vector must yield . the minimum of the following objective function (e(CJ" : U.;)-=cf.~(C~~) +j3j(U[), (11)
~{(4_C)~ "t {-t . StCdt)dt ~ ~ 0 i " t' l,<-I i. ~1 't7-i.-J-t:', Ct et) cft Le.
c:A ( fli'.'''/= L' ~ "L.
(12) where jOiS a given convex function, and C ia a given constant. Define 1. j ta:. 1. 1; -t t cJtJdt ~c. ~ =-,,, 1:.-1 ~-I i...
{
0
I
t It.
~t
Jt t
, -,
",
/1(- /
cJtJdtcc.
i=-1,. .• )[
and, considering (10) and (12) we revITite (11) as follows:
re{UI€J =J..IA.Ja::-c +1.. ti1 u(':y- +ft f ( uf).
(13)
i d.
~;;;1
~
I(
~
)
I
Denoting J:. t =2/.-C ,f(""cL.II. C ) 1.= {)''' ' we obtain r L ffL{K) = aLL. rJ;"K+Lti.~Cft.K./+jj{ut) . (14) <-= L
u.. trVk ...,
e:f...
Obviously, each of the two addends in (14) is a convex function of the ar/( gument u.. (the former is a sum of linear functions squared, the latter -by definition). A set of acceptable controls is determined by the following system of inequalities:
U ",Dt)(
t:: 1.
I
1 "-,
L-{
r..:C-IIe
(15 )
(16 )
!£ Ut.. - UirrO
-If * where u,...·"I Um,ik tU""", >! , U ....<\..\ i are given constants. The objective function (14) with restrictions (15), (16) is minimized as follows. We first minimize re ( U K) \'Ii thout re gard for the r es trictions u s ing the Hey/ton - Raphson meth od (Kar manov, 1980; Aoki, 1977). An al gorithm of obtai- . ning the vector U t:. looks like follows:
u!Io] == U:(;;. COftst , UK[n,] =: u1"n -f.]-H-{ r:,u(fUf [tL-iJ)
(17)
H-
()
=l
L
U mc'K ~ u..~~ £
1
H
Where is a reverse matri x to of second derivatives of the function (f.1."-j with respect to the compo nents of the vector L.(K. , 'If((u.') is a gradient of r ( UI( ) • The pro-:cess (17) is continued till the absolute value of the vector U. C [11-] and U. It. [tt-- iJ difference become s less than the given c:> 0 which characte'r rizes the desired accuracy of iteraUK) and tive process. The gradient matrix are calculated as follows:
r
Vcer
H
d
'lU (LtI()=L~ a'f{UK) d V'("'~)7r ( duI( ) ... , --. ~ J
r
ors~;')=2. ~ t' L- t H:=:[
' rAt (~~t t:;~~)~~l.!df("tJ
dzerU ....) l
ou.t.
1'.:. {
tdr.,L r
dt
o~f u;
)
/
1,. "JL
dUe.
I
L
I
K
):.2I .(/f"+-J, S. (jY( ClLK) iE,'-!< 'I 't 'fL 'fj3 ~ UKT , (18) L L J;>1. =f('f~L I.t' -- /f'r0 ptL, o. i" L • K
The vector value U obtained as a result of the gradient minimization is checked to satisfy the inequaliti-:es (15), (16). If the value of a certain component of vector U'" turns out to be less than U or more than Uh1.P..~ (respectively, U1<~i~ and l\, fA. ntP.. )< for the L-th component) then it is replaced by £11>'\.(' 1'1. or U. ma..)( (respectively, tL~it and C,[ ;;'''' \i ). It it satisfies the inequality (15) /t£{ 'f\.
Identification-based Algorithm of Mine Ventilation Control
(respectively, (16», then it remains unchanged. Thus a new quasi-optimal . vector tilt is formed, which is a solution to the minimization problem with restrictions (15), (16).
REFEHENCES Aoki, M. (1977). Introduction to optimization techniques. M., Nauka Publishers, 344. Bogdanov V.O., and N.S. Hajbman (1981) Partial correlation functions and their application to identification of distributed plants. In: Mathematical statistics and its applications. Tomsk, Tomsk University Publ. House, 5 - 20 (in Russian). Karmanov, V.G. (1980). Mathematical, programming. :M., Nauka Publishers, 256 (in Russian). Krylov, V.I. et al. (1976). Computational technique. Vol. 1, 11., Nauka Publidhers, 304. (in Russian). Rajbman, N.S. (Ed.) (1978). Fundamentals of process control. M., Nauka Publishers, 440 (in Russian) • Tikhonov, A.N., and V.B. Glasko (1964) On the approximated solution of Fredhol1m integral equations of the 1st type. Zhurn. Vychisl. matem. i matem. fizikif Vol. 4, No. 3, 564 - 571 (in Russian).
713
IDENTIFICATION-BASED ALGORITHM OF MINE VENTILATION CONTROL V. O. Bogdanov and D. V. Kneller Institut e of Control Sciences, Moscow, USSR
Abstract. The paper is concerned with a mine ventilation system based on the preliminary identification of the plant during its functioning. The sought control is determined by a gradient optimization method using the estimated Green's functions of a complex system.
Keywords. Identification. Green's function, Newton-Raphson method; coal mine ventilation; methane concentration.
INTHODUCTION
in large dimensionality of the problems solved.
One of the vital activities in mining research is development of methods and facilities for timely and adequate co al mine ventilation, i.e. removal of methane from the galleries. 11ethane is excreted by the ore during its mining and at concentrations exceeding the maximal acceptable presents a risk for miners. The ventilation is carried out by a powerful fan and a system of shutters i n the galleries.
At present there is quite a simple identification-based approach to solution of the control problem. According to it the fan and shutter Green's functions are identified with the subsequent derivation of the control vector. The authors sugp,est a method which does not require preliminary specification of the type of equations describing the process, but handles only known time re al izations of input and output signale obtained from the functioning plant.
The paper aims a t developing an al g orithm of the ve ntilation system control. Such control must reduce the material losses due to mine emergency downtime. Attempts at designing a mine ventilation control system were numerous. All of them were based on the physics of the gas excretion. This required involving a theory of nonlinear partial differential equations which resulted 709
A two-unit control system is proposed for mine ventilation. The input of the first unit of the system , is the L vector of control u(t) • The first L of its components determine the position of the air shutter controllers, whereas the -th one is a controlling parameter of the fan.
-1
L
v.
710
O. Bogdanov and D. V. Kneller
At the output of the first unit the sensors measure a vector of air consumption rate V (t) , coming as an input to the second unit. The 1vector of methane concentration C{t) is measured at the output of the second unit. Identification is used to a~sess the Green's functionsc1(') and <:;. C·) of the first and second units, respectively. The r.m.s. distance between identification results-based values of vectors 1)'Y.{t) ,c~ et) and the measured values of vectors 'VC t) and C{t) ,respectively, is taken to be a criterion of closeness of the Green's functions obtained to their true value~. Values of the functions and er (.) obtained are usecJ. to obtain an optimal control U. (t). The optimality criterion of the control vector u ( t) and restrictions to its components are given below.
11
G(.)
where 11= [v:, ('X/t) } "'/ V I1 ex/tJ] 7' is a vector of output variables, U. :: = [U1(~,t)I"" Ll.L(~,t)]-r is a vec.tor of input variables, ~ (.) = =[qmt>]l<,JJ 'JL is a matri~ of the If m,i, .../'.( sought weight functions. Similar relations can be written for the second unit. Attempts undertaken to solve the set of Fredholm I equations obtained due to its non-regularity result in computational difficulties and, as a rule, fail. To reduce the system to a number of independent equations we introduce a, notion of partial moment characteristics of random fields Ut and Uj " which are covariations of their va~ lues centered relative to the conditional expectations (Bogdanov and Rajbman, 1981) 0 O . \
Kuu./u_. . (~,1J'r,S )'='-N{[UJ~I't)- M(Ll JP)JU~tl) Y); • J tJJ AA" (T! 5)/ (,q ") K*L~ j fJE..2..,;YE~ ][U,/f,S)- rt( ~j : ' U..: l/ , Y ; 0
IDENTIFICATION The state of the system considered is characterized by a random vector field and described by differential equations in partial derivatives. An identification problem is stated, i.e. a problem of using the known values of input and output to develop a plant model, an output signal of which would in terms of a certain criterion, be close to an output of a real system. It is well known that with the r.m.s. criterion involved, application of correlation methods would lead to identification equations which are similar to Wiener's filter for a multi-dimensional distributed system (Rajbman, 1978). KUi1,(x)~lt)-- Si) 9iXJ~,S)j(",u/;)~, Z'-s ) d~c/s +...+
'.()~+ S: S.2:1' 9f/:x.J~' S)K.t~u/~ 17;r:-S)t;/~5 KVHU[x,p/L)=-r S1J11M{X,~,s)Ki{fLt!~' 'f,(-5)d~Js+..~ (1 ) (
0
+I 52Jf~HL ex, ~,S)KULU!f,-,~,t-sJd~s
0
0
/(tL 'j ;""E~~iYE:Ty]}
where
(2)
U( -=:. u.;-Mu.f.'
{ ==
i ,.. , L.
Using the system (1) a nd definition (2) we yield equations each one of . which gives a possibility of determining the sought weight functions.
Kv. u /u (X''f,t,fi)=St ~il77d.:x) F,t)S) ,.., f
-f
-OQ
J
X
xKue ~/Ldf.},s})(/~ ds, f'= f.. , L .tr(~j, ., 11.(3) The derived identification equations are also valid when input and output, noisy signals are measured. It is required only that the noise vector is uncorrelated with the input vector.
'rhus equations (3) were derived using the partial correlation functions. Each of the equations permit determining the sought Green's functions whereas application of coventional correlation methods requires solving
Identification-based Algorithm of Mine Ventilation Control
the system (1). Input and output signals can be treated as scalar uni-dimensional random fields. An input concentrated variable tit- which is a random process, presents an exception. By I1photographingl1 the process we obtain input and output time realizations. A plant specified in such a way can be treated as a multi-dimensional one with concentrated parameters for which the above postulations hold true. For the process considered the best r.m.s. estimates of the weight functions C(-) and C, (-) yielded by the correlation methods are: l' K7f",ut {u-t. (t);:; 50 ~",e ('C)K~udu_~ (t-t.) dI: , (4) KCi1fM
5: 'ii.n\J~)KVfT/.lIMIU.I)t(t-"t) d~
/1f..",(tj =
(5)
where KVml-{tfu,.e ,K.;y",/~"" KUeue/U_(,1
=/,'0 i 9",,( {f)KUei.le fu.e (ft.-p ),
Kcy",/V..m{n)=£
q.", ({ 0) KVM V..,. {v.. .... (n-f) ,
f:o(J
~ :: mii1 (n,P), E; =min.{if,P''ji ~~1, ..., I; C~ 1,. .}Li,,"=i, .. ,H.
The systems of linear algebraic equations obtained are solved by the Gauss method with the preliminary regularization (Tikhonov and Glasko, 1964) •
The estimates of the Green's functions are calculated in six points of the
711
axis 't (P~ P "= 6). The nwnber of these points is a significant parameter influencing the alg orithm promptness. To obtain the estimates of conditional expectations, variation ranges of signals Ut. and 1fM are divided uniformly. The quality of the model. obtained can be assessed by the r.m.s. criterion, or by the variance ratio R {C~ft)l:=;() M dt }IU t (t) ,t': 1.. .. L ~~) J ;z,[CJt)] , (-~ ...J.
i fC
H
In case of complete adequacy of the. model to a real process R.{c~{t)}:::.1. •. In a g eneral case, the measure of variance ratio lies within O~ ==-RiC;(t)}~j (Rajbman, 1978 ). The method discussed yields a best linear estimate of a nonlinear system being identified. The nonlinearity degree is also determined by expression (6)
3. CONTROL Components of the vector-function U(t) can be treated as constant on a segment (t K- 1 ,tK } , Kc;1,2 . ...
fu;<, U/,-"coflst, iE(il_f,t,..]}}
Li.e{t) =
(7)
This assumption does not contradict the physics of the processes considered. The model output c*{t) as the function of the vector u(t) is presen.,. ted by the following double convolution L M ~ ct(t)=I? (j~(qme1t {.(j!))) i=l,. .., I. (8) e;f m=i d' Denote the Green's functions obtained as a result of the identification within the tim~ interval (t(-l It",] as C;:;I::(.) and C; "'-(-) ,respectively, . whereas the methane concentration calculated thereby as C~ .... , i. -; 1, ... / I Assume that the identification yield~d the function values tq K(.) and G;'C(') in points tK,j =:::t~cilz. ,j=O,. ..,J. Hence, using the definition of a con-
v.
71 2
O. Bogdanov and D. V. Kn e ller
volution and considering a constant nature of components of the vector u.. K. ~ we obtain 1'4 Cl(*" ,j.:' K-t (t&.~ )(f ,1 ( =fo ~~u.' [.;fJt•.!:~{(;rf ~l~: lz.)dtJd)( 9)
z-""
fC
tt
w~/ere V'::il~~~~"'1 ....,.e ,f=miH.ffJ~"'t':?!:-tl:-Jlt.j. IV
is a set of positive integers.
Denoting the expression in brackets 01<- 'I. as &te) i..~1.I""·II}t~i""IL ,we rewrite (9) as fOliws: C/{¥'" t/{-2 1:-1 i. =L.~ it tI~ } i.. :: j, ... , I . ~=11 c=( Since when obtaining tL , {=l, ... }L J U;~2.t<:1r-,l.l~J,-P-are believed to be already knowi,~introducing t: ,5" 17 K - 2 K- ' 2 . 1 (;(!.. =LL 'Oil" LI~ , L=1, ... , . 2=( e=( we 0 bt aln L
e
J( ~ /? K C K -f='Z~+~ifUft:
t;
10
Values can be approximately calculated using the trapezium formula (Krylov, 1976). The sought control vector must yield . the minimum of the following objective function (e(CJ" : U.;)-=cf.~(C~~) +j3j(U[), (11)
~{(4_C)~ "t {-t . StCdt)dt ~ ~ 0 i " t' l,<-I i. ~1 't7-i.-J-t:', Ct et) cft Le.
c:A ( fli'.'''/= L' ~ "L.
(12) where jOiS a given convex function, and C ia a given constant. Define 1. j ta:. 1. 1; -t t cJtJdt ~c. ~ =-,,, 1:.-1 ~-I i...
{
0
I
t It.
~t
Jt t
, -,
",
/1(- /
cJtJdtcc.
i=-1,. .• )[
and, considering (10) and (12) we revITite (11) as follows:
re{UI€J =J..IA.Ja::-c +1.. ti1 u(':y- +ft f ( uf).
(13)
i d.
~;;;1
~
I(
~
)
I
Denoting J:. t =2/.-C ,f(""cL.II. C ) 1.= {)''' ' we obtain r L ffL{K) = aLL. rJ;"K+Lti.~Cft.K./+jj{ut) . (14) <-= L
u.. trVk ...,
e:f...
Obviously, each of the two addends in (14) is a convex function of the ar/( gument u.. (the former is a sum of linear functions squared, the latter -by definition). A set of acceptable controls is determined by the following system of inequalities:
U ",Dt)(
t:: 1.
I
1 "-,
L-{
r..:C-IIe
(15 )
(16 )
!£ Ut.. - UirrO
-If * where u,...·"I Um,ik tU""", >! , U ....<\..\ i are given constants. The objective function (14) with restrictions (15), (16) is minimized as follows. We first minimize re ( U K) \'Ii thout re gard for the r es trictions u s ing the Hey/ton - Raphson meth od (Kar manov, 1980; Aoki, 1977). An al gorithm of obtai- . ning the vector U t:. looks like follows:
u!Io] == U:(;;. COftst , UK[n,] =: u1"n -f.]-H-{ r:,u(fUf [tL-iJ)
(17)
H-
()
=l
L
U mc'K ~ u..~~ £
1
H
Where is a reverse matri x to of second derivatives of the function (f.1."-j with respect to the compo nents of the vector L.(K. , 'If((u.') is a gradient of r ( UI( ) • The pro-:cess (17) is continued till the absolute value of the vector U. C [11-] and U. It. [tt-- iJ difference become s less than the given c:> 0 which characte'r rizes the desired accuracy of iteraUK) and tive process. The gradient matrix are calculated as follows:
r
Vcer
H
d
'lU (LtI()=L~ a'f{UK) d V'("'~)7r ( duI( ) ... , --. ~ J
r
ors~;')=2. ~ t' L- t H:=:[
' rAt (~~t t:;~~)~~l.!df("tJ
dzerU ....) l
ou.t.
1'.:. {
tdr.,L r
dt
o~f u;
)
/
1,. "JL
dUe.
I
L
I
K
):.2I .(/f"+-J, S. (jY( ClLK) iE,'-!< 'I 't 'fL 'fj3 ~ UKT , (18) L L J;>1. =f('f~L I.t' -- /f'r0 ptL, o. i" L • K
The vector value U obtained as a result of the gradient minimization is checked to satisfy the inequaliti-:es (15), (16). If the value of a certain component of vector U'" turns out to be less than U or more than Uh1.P..~ (respectively, U1<~i~ and l\, fA. ntP.. )< for the L-th component) then it is replaced by £11>'\.(' 1'1. or U. ma..)( (respectively, tL~it and C,[ ;;'''' \i ). It it satisfies the inequality (15) /t£{ 'f\.
Identification-based Algorithm of Mine Ventilation Control
(respectively, (16», then it remains unchanged. Thus a new quasi-optimal . vector tilt is formed, which is a solution to the minimization problem with restrictions (15), (16).
REFEHENCES Aoki, M. (1977). Introduction to optimization techniques. M., Nauka Publishers, 344. Bogdanov V.O., and N.S. Hajbman (1981) Partial correlation functions and their application to identification of distributed plants. In: Mathematical statistics and its applications. Tomsk, Tomsk University Publ. House, 5 - 20 (in Russian). Karmanov, V.G. (1980). Mathematical, programming. :M., Nauka Publishers, 256 (in Russian). Krylov, V.I. et al. (1976). Computational technique. Vol. 1, 11., Nauka Publidhers, 304. (in Russian). Rajbman, N.S. (Ed.) (1978). Fundamentals of process control. M., Nauka Publishers, 440 (in Russian) • Tikhonov, A.N., and V.B. Glasko (1964) On the approximated solution of Fredhol1m integral equations of the 1st type. Zhurn. Vychisl. matem. i matem. fizikif Vol. 4, No. 3, 564 - 571 (in Russian).
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