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Software for Tasks of Forming the Monitoring and Control Algorithms
SOFTWARE FOR TASKS OF FORMING THE MONITORING AND CONTROL ALGORITHMS V. V. Petrov, A. V. Zaporojets, V. M. Kostukov and I. N. Polyakov Moscow A ircraft Inst itut e, Moscow, U. S. S. R.
Abstract. On the base o~ the use of ~nformation theory methods the algorithms of ~orming the limiting probability characteristics of processes taking place in stochast1c systems are produced in this work. The method of extreme additional determination of particular probability characteristics of processes to the complete description with the use of guaranteed result princple is discussed. The information algorithms of stochastic optimisation of dynamical monitoring and control systems for limiting mathematical models of processes are developed. The formulated principle of developing systems makes it possible to estimate the optimum block diagram and information-controlling complex parameters and highly improve its accuracy indices. Keywords.Optimal control; computer applications;control theory; information theory. INTRODUCTION When des~gning the monitoring systems place when creating the modern dynathe problem of choice of the system mical monitoring and control systems parameters providing the extreme inis to obtain the highest accuracy of dices of their functioning takes synthesized systems.Modern theory place. gives the possibility of solving this Software for design tasks include the task for the class of systems wIth algorithms of forming the models desthe prescribed structure of enteraccribing the environment and the objtion of input processes and given ect behaviour and also the algorithms varied characteristics. of searching the optimum parameters The additional problem limiting the and block-diagrams of synthesized majotity of task of this class is systems. that only the tasks of input signal The existing apparatus methods of processing in open structures can analysis of statistic characterisbe examined. tics of processes give only particuAs a result the large volume of tasks lar random process description and for which the mentioned limitation basically don't allow to find their do not operate remains out of discomplete probability description. cussion. The solution of analysis and syntThe first part of the report contahesis tasks o~ control system characins the method of extreme additional teristics under the conditions of determination of particular probabithe initial date incompleteness can lity characteristics of stationary lead to non-complience of the guarandom processes, developed by the ranteed result principle, to baseauthors, when the arbitrary non-Galessness of solution according to opussian probability distribution tatimum rules and to incorrectness of kes place using the guaranteed resolution of analysis and synthesis sult principle. tasks. The comp1icasy of the useful random Because of the mentioned circumstanprocess processing made by the sysces the problem of determination of tem and the complicasy of compencomplete probability characteristics sating the random hindrance influence under the same conditions is when the particular characteristics determined by the rato of forecasare known is considered to be one of ting the random process, so that the most important problems in the the most unforecastable process is system theory. the most unfavourable one. One of the main tasks which takes As the forecasting of the process
41 3
V. V. Pe t r ov et
414
Vet) value in moment of time t is determined by the average quan~ity of mutial ~~formation between the reading~: 'Jlv1 /¥oJ:: M[-toflf{v'/voJ/P(VI)] where P(vtjVO), p(vl}.. conditional and unconditional probability distribution density of the process Vet) in the moment of time t , so the extreme additional dete~ination of the probability characteristios according to particular description of the random process Q~Dfp(v)] is effected as a solution of variation task for] the c~nditional extremum min~[v1/vo
for 5D[p(VIJ}:a
(1)
The solution of the above task in general is discussed in the report[4] and the pointed method for two most practically important oases is used below:1.The determination of the extreme estimation of two-measured law of probability distribution along the given interval of the possible value of random process [-Ao, LloJ and maximally possible value of the rate of the random process V(t)changes. 2.The determination of one-measured law of probability distribution of the process for the first and for the second cases is made on the basis of the extremalizing the degenarated information characteristic-entropy and comes to the variation task;
H{VO)lcv(t)]:[~~. AoJ "'~fl-X [J °/J(VOJt:q~VO)riVO (VO) -Ilo (2) P(VD)'" fi,o \~,~~~
m[l.)(
and has ~ solution:
The determination of the conditional distribution of the process V(t)values probability differs for the first and for the second cases respectively. With given internal and maximum rate change of the random process V(t)the aooessibility intervals~[v~~Jwhile the process changes from value VO in time rt are determined and +.hen mathematically the task of additional determination looks like follows: p(VI/VD rr) :fmUL ~ [VI/VD,1;']; L1 , - L1[VO,"i]} , P(II'/VO/I) P{VD) : H(VD)i.6 :~D}
[max P(Vo)
- interval of the effective process values [)]. The solution is made by the numerical methods with the usage of Lagrangian multipliers S. When solving the task of extremalization the theorem of convergence described by R.Blahut [7] was used. In general the conditional distribution is: vl)exp S(VI-VD)2j P(YI/VO,S) '"
~ p(V/)exp[S{VI -v
(3)
The accessibility intervals of mathematical model of the process are determined: .t1[VO. S] = f f/Xp[?,P(v'/vo,s)fu{lp(v'/VO,S)j
a~.
From comparing the accessibility intervals we have: S;, [VO, ~J.'l' vl-VO 2 ('d hence p(VI!VO,'7:)'" ex 0 't' [VI- VO]2] When the autocorre ation function is known the value of one.measured distribution P(VO) is calculated the conditional disctibution can be determined after solving the following variation task:
Th e
'll{Wro·>J/J[",/lfV{)'t'rhJ/'It,,.,% k . so~v~
,/.
P(vI/VD,S)=
0
PM e
[S
'~as VI-VD 2
e
~s:
:E.p, VI eXPfS[vl-voJ2 VI
•
(5") (6)
•
The autocorrelat~on funct~on of the mathematical model of the process: f, (S) ::: Kv(7:)
is confronted with the known correlation functionS= ~(~so and the needed law is
prVIIvO,'t) -
p(vl)e)(p[tL~)(vI _ VO)2]
- ?;p(VI)eXPl'f'('t')(V/-VO)2]
(7)
The developed approach was used by the authors for determining the statistic characteristics of the output coordinates of the objects, involved into the stabilisation circuit at the stage of the initial design. When principally the statistic characteristic of the measured object coordinates are absent. In the second part of the report the tasks of the stochastic optimization of the dynamic systems of the arbitrary structure which are influenced by the non-stationary Gaussian processes, and the stochastic processes characteristics (the density of probabilities, the correlation function) are fully known. Those are the limits of functioning of the above algorithms. The final criterion of the variational task is to minimize the dispersion of the error. The solution of these tasks suggests that the aharaoteristics of the synthesized systems should be optimized and the choice of the generalized structure of the input processes should be well-founded. The generalized structure systems and the way of optimizing them are the main objects of the investigation made in this article. For the system of this class the report considers:1.the solution of the task of the stochastic optimization of the non-stationary systems of generalized structure in accordance with the accuracy criterion;2.the solution of the stochastic optimization task of the non-stationary systems of the generalized structure in accordance with the information criterion;). the usage of the principle of developed systems for the stochastic optimization of the synthesized systems. The first question was raised because in the publications there were no
415
Monitoring and control algorithms
solution of the aocuracy optimization tasks for the generalized structure systems and in order to make it possible to compare the optimum indices for the systems of the mentioned class it is necessary to have the evident solution of the task. The solution of the information synthesis task for the generalized struoture systems is explained by the fact that the usage of the information criterion for the systems of the considered olass let clear up the inside possibilities of the accurate operation and point out the way of increasing the accuracy induces while developing the structure (the term "developing of the structure" means the introduction of the new connections and new variables being varied). On the basis of the examined two tasks the principle of the developed systems or~-criterion was formulated. It points to the possibility of receiving the higher accuracy indices under the prescribed initial data when using the average quadratical approach. In general non-stationary dynamical systems of generalized structure can be shown as the following system of equations: Xz(i t-{) =Az (t)X,z (t) f 8 2 (f)(j;(t)xdt) f h.z (t)] - vector difference equation of the optimized system; X, (HI)
=Adt)X,(t) f 8((t)n.dt)
- vector difference equation of the useful signal X3 (tH)= A.3 (t)'I3 (t) + 83 tt)nd{t) - vector difference equation of the input hindrance. rntJ:tri:;nIinn - vector discrete random process of the "white noise" type having the average value equal to zero and intensivity matrix: N(t) but N (t:> is input hindrance. and X~lt)'[~XJiXJ1 is the final output of the processing system. r r T T rl Por the random process X (t)"L/,;Xz.;X3 the difference equation for the dispersional matrix is as follows: 9J(ltf) =A (t)
when
A{t)=
[. It, (0
0 0 J; Bft):[B,ft) D DJ 0 Bzm 0
BAt)C(t) Az (t) 0
o
0
A3 (t)
0
0 F3ft)
Thus the task of synthesizing of the non-stationary generalized structure systems upon the accuracy criterion can be formulated in the followin~ way: when the matrixes A1 (t).B (tJ, BJa~Nff~re known it is necessary 1to find such a sequence of matrixes A2 (t) and B?(t) that the error dispersion between the vector X4 (t) i-component and vector X1 (t) i component would be minimum. In this case the initial functional can be expressed like follows:
44 14 V(t+1) Dii(t+1) - 2Dii(t+1) or T T roT V(tt1) ZiPZi+2Zihi+Bii - 2lZigi+hi] when _ (9) r:zT _
LJ--
I;
ni
[BiT -AIT]_ 2:,z J
0 1 , e ~ 8.z J1.jz - A., 2)J.z )
iT
-= [ A3
i'i_ l:
T
iT
:
iT
] _
[A;i:~~3A;/B;~3 8~J~ pJC1)"C;Nzz :c~z1
9. -[A, ~'3A3
N,JB.J], .I. ji = [A;T?J H A; + 8;TN,J 8;J; tB,
L!Z).u C
:
i>zd
The expression (9) reaches the minimum if [Bz :Azl=UE:"O)(Af~ftcrtBflV,tiA,~/I.)-
- (A /J)~/CT + B3 N32 ; A3 2)32) JG-
(10)
when E is the unity matrix, G=p- 1 If X (t)=0. and the processes. affecting 3the processing system, are not correlated thus the algorithm (10) comes to the results equal to those of Kalman algorithm. The task of information syntesis of the non-stationary dynamic systems of the generalized structure is formulated in the following way: to find such a sequence of matrix A2 (t) and B2 (t), so that the average quatity of information concerning the i - component of the vector X1 (t), included into the i-component of the vector X4 (t) should be maximum. SInce L _ L [ [
J[X, (tV)(~{t)]::-z &9 1-1)rJt)~it(t)
the optimazing functional oan be written in the following way: (11) [Zrtt)9i (t)+.{il(t)]Z
V{t+f) [l[(tW(t)Zdt)+Z lj(t)h;fl)fpii(l)] =.
The solution of that task is [BAt); {[M/: 0] [A, '.l)"eT+B,A'u: A, 'llI2] -
\]=
~~)
-[A/J:JJ,C fI33~i!:A~ ~HJ16
~'= [H(7W- A}~HA[-B4A1HBrlii
where
7
19("W-A(~f~Al-8,Nt38_D-i
H~[A!>'1:Jj/l' + ~~1. i AS~JzJ 9= [A, ~/' C7+ B,Na; A, 9Jdi
N, = f j('j l
L
Examening the expressions t10} and (12) it is easy to conclude that the system optimal upon the average quadratical criterion is not optimal upon the accuracy criterion.It means that on the one hand it is no US8 of minimizing the dispersion error for the maximization of the information indices, on the other hand the possibility of obtaining the highest accuracy indices is not fully used and there are structures with the same initial data which can lead to the less errors than during the realization of the algorithm (10). On the base of the obtained results the principle of the developed systems [6] was formulated: in order to decrease the error dispersion and to increase the average quantity of information it is necessary to
416
V. V. Petrov et aL .
put into the system the new alternating variables till the number -~ will be equal to 1. In this case the error dispersion obtains the limit minimum value and varying transfer functions will be optimum in accordance with the dispersional and informational methods. The algorithms developed in this article make it possible to create the adequate mathematical models of processes, flowing in the dynamic systems and to determine the parameters and the structure of the information-controlling complexes of moving objects, highly increasing their accuracy indices. The develo-
ped algorithms can be used when designing the wide class of monitoring and control systems. REFERENCES
1.Sveshnikov A.A. The applied methods of the random functions theory. Moscow, Nauka, 1968. 2.Iosimura Soeda.Practically realized filter for the systems with the unknown parameters.In "Dynamical systems and the control", 197), N 4. ).Bovitsky N.V.The principles of the information theory of the measurement devices.Moscow,Energy, 1968.
Abstract. On the base o~ the use of ~nformation theory methods the algorithms of ~orming the limiting probability characteristics of processes taking place in stochast1c systems are produced in this work. The method of extreme additional determination of particular probability characteristics of processes to the complete description with the use of guaranteed result princple is discussed. The information algorithms of stochastic optimisation of dynamical monitoring and control systems for limiting mathematical models of processes are developed. The formulated principle of developing systems makes it possible to estimate the optimum block diagram and information-controlling complex parameters and highly improve its accuracy indices. Keywords.Optimal control; computer applications;control theory; information theory. INTRODUCTION When des~gning the monitoring systems place when creating the modern dynathe problem of choice of the system mical monitoring and control systems parameters providing the extreme inis to obtain the highest accuracy of dices of their functioning takes synthesized systems.Modern theory place. gives the possibility of solving this Software for design tasks include the task for the class of systems wIth algorithms of forming the models desthe prescribed structure of enteraccribing the environment and the objtion of input processes and given ect behaviour and also the algorithms varied characteristics. of searching the optimum parameters The additional problem limiting the and block-diagrams of synthesized majotity of task of this class is systems. that only the tasks of input signal The existing apparatus methods of processing in open structures can analysis of statistic characterisbe examined. tics of processes give only particuAs a result the large volume of tasks lar random process description and for which the mentioned limitation basically don't allow to find their do not operate remains out of discomplete probability description. cussion. The solution of analysis and syntThe first part of the report contahesis tasks o~ control system characins the method of extreme additional teristics under the conditions of determination of particular probabithe initial date incompleteness can lity characteristics of stationary lead to non-complience of the guarandom processes, developed by the ranteed result principle, to baseauthors, when the arbitrary non-Galessness of solution according to opussian probability distribution tatimum rules and to incorrectness of kes place using the guaranteed resolution of analysis and synthesis sult principle. tasks. The comp1icasy of the useful random Because of the mentioned circumstanprocess processing made by the sysces the problem of determination of tem and the complicasy of compencomplete probability characteristics sating the random hindrance influence under the same conditions is when the particular characteristics determined by the rato of forecasare known is considered to be one of ting the random process, so that the most important problems in the the most unforecastable process is system theory. the most unfavourable one. One of the main tasks which takes As the forecasting of the process
41 3
V. V. Pe t r ov et
414
Vet) value in moment of time t is determined by the average quan~ity of mutial ~~formation between the reading~: 'Jlv1 /¥oJ:: M[-toflf{v'/voJ/P(VI)] where P(vtjVO), p(vl}.. conditional and unconditional probability distribution density of the process Vet) in the moment of time t , so the extreme additional dete~ination of the probability characteristios according to particular description of the random process Q~Dfp(v)] is effected as a solution of variation task for] the c~nditional extremum min~[v1/vo
for 5D[p(VIJ}:a
(1)
The solution of the above task in general is discussed in the report[4] and the pointed method for two most practically important oases is used below:1.The determination of the extreme estimation of two-measured law of probability distribution along the given interval of the possible value of random process [-Ao, LloJ and maximally possible value of the rate of the random process V(t)changes. 2.The determination of one-measured law of probability distribution of the process for the first and for the second cases is made on the basis of the extremalizing the degenarated information characteristic-entropy and comes to the variation task;
H{VO)lcv(t)]:[~~. AoJ "'~fl-X [J °/J(VOJt:q~VO)riVO (VO) -Ilo (2) P(VD)'" fi,o \~,~~~
m[l.)(
and has ~ solution:
The determination of the conditional distribution of the process V(t)values probability differs for the first and for the second cases respectively. With given internal and maximum rate change of the random process V(t)the aooessibility intervals~[v~~Jwhile the process changes from value VO in time rt are determined and +.hen mathematically the task of additional determination looks like follows: p(VI/VD rr) :fmUL ~ [VI/VD,1;']; L1 , - L1[VO,"i]} , P(II'/VO/I) P{VD) : H(VD)i.6 :~D}
[max P(Vo)
- interval of the effective process values [)]. The solution is made by the numerical methods with the usage of Lagrangian multipliers S. When solving the task of extremalization the theorem of convergence described by R.Blahut [7] was used. In general the conditional distribution is: vl)exp S(VI-VD)2j P(YI/VO,S) '"
~ p(V/)exp[S{VI -v
(3)
The accessibility intervals of mathematical model of the process are determined: .t1[VO. S] = f f/Xp[?,P(v'/vo,s)fu{lp(v'/VO,S)j
a~.
From comparing the accessibility intervals we have: S;, [VO, ~J.'l' vl-VO 2 ('d hence p(VI!VO,'7:)'" ex 0 't' [VI- VO]2] When the autocorre ation function is known the value of one.measured distribution P(VO) is calculated the conditional disctibution can be determined after solving the following variation task:
Th e
'll{Wro·>J/J[",/lfV{)'t'rhJ/'It,,.,% k . so~v~
,/.
P(vI/VD,S)=
0
PM e
[S
'~as VI-VD 2
e
~s:
:E.p, VI eXPfS[vl-voJ2 VI
•
(5") (6)
•
The autocorrelat~on funct~on of the mathematical model of the process: f, (S) ::: Kv(7:)
is confronted with the known correlation functionS= ~(~so and the needed law is
prVIIvO,'t) -
p(vl)e)(p[tL~)(vI _ VO)2]
- ?;p(VI)eXPl'f'('t')(V/-VO)2]
(7)
The developed approach was used by the authors for determining the statistic characteristics of the output coordinates of the objects, involved into the stabilisation circuit at the stage of the initial design. When principally the statistic characteristic of the measured object coordinates are absent. In the second part of the report the tasks of the stochastic optimization of the dynamic systems of the arbitrary structure which are influenced by the non-stationary Gaussian processes, and the stochastic processes characteristics (the density of probabilities, the correlation function) are fully known. Those are the limits of functioning of the above algorithms. The final criterion of the variational task is to minimize the dispersion of the error. The solution of these tasks suggests that the aharaoteristics of the synthesized systems should be optimized and the choice of the generalized structure of the input processes should be well-founded. The generalized structure systems and the way of optimizing them are the main objects of the investigation made in this article. For the system of this class the report considers:1.the solution of the task of the stochastic optimization of the non-stationary systems of generalized structure in accordance with the accuracy criterion;2.the solution of the stochastic optimization task of the non-stationary systems of the generalized structure in accordance with the information criterion;). the usage of the principle of developed systems for the stochastic optimization of the synthesized systems. The first question was raised because in the publications there were no
415
Monitoring and control algorithms
solution of the aocuracy optimization tasks for the generalized structure systems and in order to make it possible to compare the optimum indices for the systems of the mentioned class it is necessary to have the evident solution of the task. The solution of the information synthesis task for the generalized struoture systems is explained by the fact that the usage of the information criterion for the systems of the considered olass let clear up the inside possibilities of the accurate operation and point out the way of increasing the accuracy induces while developing the structure (the term "developing of the structure" means the introduction of the new connections and new variables being varied). On the basis of the examined two tasks the principle of the developed systems or~-criterion was formulated. It points to the possibility of receiving the higher accuracy indices under the prescribed initial data when using the average quadratical approach. In general non-stationary dynamical systems of generalized structure can be shown as the following system of equations: Xz(i t-{) =Az (t)X,z (t) f 8 2 (f)(j;(t)xdt) f h.z (t)] - vector difference equation of the optimized system; X, (HI)
=Adt)X,(t) f 8((t)n.dt)
- vector difference equation of the useful signal X3 (tH)= A.3 (t)'I3 (t) + 83 tt)nd{t) - vector difference equation of the input hindrance. rntJ:tri:;nIinn - vector discrete random process of the "white noise" type having the average value equal to zero and intensivity matrix: N(t) but N (t:> is input hindrance. and X~lt)'[~XJiXJ1 is the final output of the processing system. r r T T rl Por the random process X (t)"L/,;Xz.;X3 the difference equation for the dispersional matrix is as follows: 9J(ltf) =A (t)
when
A{t)=
[. It, (0
0 0 J; Bft):[B,ft) D DJ 0 Bzm 0
BAt)C(t) Az (t) 0
o
0
A3 (t)
0
0 F3ft)
Thus the task of synthesizing of the non-stationary generalized structure systems upon the accuracy criterion can be formulated in the followin~ way: when the matrixes A1 (t).B (tJ, BJa~Nff~re known it is necessary 1to find such a sequence of matrixes A2 (t) and B?(t) that the error dispersion between the vector X4 (t) i-component and vector X1 (t) i component would be minimum. In this case the initial functional can be expressed like follows:
44 14 V(t+1) Dii(t+1) - 2Dii(t+1) or T T roT V(tt1) ZiPZi+2Zihi+Bii - 2lZigi+hi] when _ (9) r:zT _
LJ--
I;
ni
[BiT -AIT]_ 2:,z J
0 1 , e ~ 8.z J1.jz - A., 2)J.z )
iT
-= [ A3
i'i_ l:
T
iT
:
iT
] _
[A;i:~~3A;/B;~3 8~J~ pJC1)"C;Nzz :c~z1
9. -[A, ~'3A3
N,JB.J], .I. ji = [A;T?J H A; + 8;TN,J 8;J; tB,
L!Z).u C
:
i>zd
The expression (9) reaches the minimum if [Bz :Azl=UE:"O)(Af~ftcrtBflV,tiA,~/I.)-
- (A /J)~/CT + B3 N32 ; A3 2)32) JG-
(10)
when E is the unity matrix, G=p- 1 If X (t)=0. and the processes. affecting 3the processing system, are not correlated thus the algorithm (10) comes to the results equal to those of Kalman algorithm. The task of information syntesis of the non-stationary dynamic systems of the generalized structure is formulated in the following way: to find such a sequence of matrix A2 (t) and B2 (t), so that the average quatity of information concerning the i - component of the vector X1 (t), included into the i-component of the vector X4 (t) should be maximum. SInce L _ L [ [
J[X, (tV)(~{t)]::-z &9 1-1)rJt)~it(t)
the optimazing functional oan be written in the following way: (11) [Zrtt)9i (t)+.{il(t)]Z
V{t+f) [l[(tW(t)Zdt)+Z lj(t)h;fl)fpii(l)] =.
The solution of that task is [BAt); {[M/: 0] [A, '.l)"eT+B,A'u: A, 'llI2] -
\]=
~~)
-[A/J:JJ,C fI33~i!:A~ ~HJ16
~'= [H(7W- A}~HA[-B4A1HBrlii
where
7
19("W-A(~f~Al-8,Nt38_D-i
H~[A!>'1:Jj/l' + ~~1. i AS~JzJ 9= [A, ~/' C7+ B,Na; A, 9Jdi
N, = f j('j l
L
Examening the expressions t10} and (12) it is easy to conclude that the system optimal upon the average quadratical criterion is not optimal upon the accuracy criterion.It means that on the one hand it is no US8 of minimizing the dispersion error for the maximization of the information indices, on the other hand the possibility of obtaining the highest accuracy indices is not fully used and there are structures with the same initial data which can lead to the less errors than during the realization of the algorithm (10). On the base of the obtained results the principle of the developed systems [6] was formulated: in order to decrease the error dispersion and to increase the average quantity of information it is necessary to
416
V. V. Petrov et aL .
put into the system the new alternating variables till the number -~ will be equal to 1. In this case the error dispersion obtains the limit minimum value and varying transfer functions will be optimum in accordance with the dispersional and informational methods. The algorithms developed in this article make it possible to create the adequate mathematical models of processes, flowing in the dynamic systems and to determine the parameters and the structure of the information-controlling complexes of moving objects, highly increasing their accuracy indices. The develo-
ped algorithms can be used when designing the wide class of monitoring and control systems. REFERENCES
1.Sveshnikov A.A. The applied methods of the random functions theory. Moscow, Nauka, 1968. 2.Iosimura Soeda.Practically realized filter for the systems with the unknown parameters.In "Dynamical systems and the control", 197), N 4. ).Bovitsky N.V.The principles of the information theory of the measurement devices.Moscow,Energy, 1968.