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Simultaneous Solution of Smoothing, Filtering and Prediction Problems Using Integral Equations
Copyright @ IFAC 12th Triennial World Congress Sydney, Australia, 1993 '
SIMULTANEOUS SOLUTION OF SMOOTHING, FILTERING AND PREDICTION PROBLEMS USING INTEGRAL EQUATIONS D.G. Maksarov Depart~N of Engineering Scienu. University of Oxford. Parks Road. Oxford OX1 3P}. UK
Permanent address : Automatic CONrol Systems Department, Riga Technical University. 1 Kallcu St .• Riga. LV 1658. Latvia
Abstract. TIle go.'\l of this paper is to present an algorithm for simultaneous solution of stochastic smoothing, filtering and prediction problems. TIlis alglllithm was obtained when a stochastic process is desclibed by linear stochastic ho-Volterra integral equation. Thc Kalman filtering problem fOllnulation is used in the given paper. It has been shown that the gain of the optimal estimator and the covaliance of the estimation error are connected by a system of integral equations. TIle estimation process is described by a lincar integral equation of the second kind . An example of estimation in stochastic system is con sidered. Simulation results are presented.
Key Words. Smoothing; filtering; prediction; integral equation ; simulation.
applicatio n domains for many years (Verlan. et al.. 1986, 1988). Dynamics of a complicated stochastic system can be naturally presented by an integral equation driven by Ito stochastic integral. Integral equations represent an important class of mathematical models that can be successfully used in control theory. However, there has not been much development in the control theory of integral equations. As far as Volterra integral equations have been introduced as a more general mathematical model for description of a control system it is important to show that principal re sults in the control theory based on differenti al equations can be obtained from the mathematical model represented by Volterra integral equation. Moreover. integral equations allow to derive more general solutions as compared with those obtained on the basis of differential equations. Stochastic smoothing, filtering and prediction problems are considered in this paper.
1. INTRODUCTION
It is a question of primary importance which mathematical model to use when developing control algorithms. Classic al linear control theory has dealt with ordinary stationary differential equations. Later on nOllstationary ordinary differential equations were also considered. Many real world processes require taking into account a delay for an adequate modelling. Neutral type delay differential equations are used to model systems in biology. economy. engineering science. etc. This diversity of differential equations gives rise to certain difficulties. Every type uf differential equations requires unique algorithms for the design of control systems whose mathematical models arc described by these equations. In addition. real world processes can be presented by complicated integrodifferential equations with delays. These equations may not be successfully reduced to differential equations in some cases. However as industrial processes become more complicated and more involved it may become increasingly inadequate to model dynamics of a system by a set of linear differential equations. Real world processes arc ofll' n stochastic. Using Ito integral for noise representat ion leads to a strict mathematical description of stochastic dynamic systems. An adequate mathematical modd of a stnchasti c process can be obtained due to this approach .
The brief outline of the paper follows. The problem formulation is given in chapter 2. The main results are presented in chapter 3. In chapter 4 an example of estimation 111 a complicated stochastic system is co nsidered . Conclusions may he found in chapter 5.
2. PRORLEM FORMULATION
It can be inferred from what has been mentioned that in some cases effective solution of various control systems' design problems requires the implementation of more general mathematical models. In fact. one of the possible approaches is that some real world processes can more appropriately be modelled by integral equations or their mathematical models can be reduced to integral equations. Integral equations of Volterra type have been used for describing dynamics of complicated systems ill different
2 .1 Olltlilll' of thl' Probll'm.
Problems of filtering. smoothing and prediction arise in many areas of engineering. economics. ecology, etc. The main aim of the solution of these problems is to estimate the most probable values of a stochastic process. The dilTerence among these problems is in the amount of information available for estimating values of a stochastic
413
process. This infonnation is available from the noisy observations. For the solution of these problems mathematical models in the fonn of stochastic dynamic systems can be formulated. A linear Volterra integral equation of the second kind driven by Ito stochastic integral is introduced as an adequate description of a stochastic process in this paper.
processes that are considered satisfy the following oonditions
When fonnulating a problem one may pose the question which model to consider - continuous or discrete time. A continuous time version of the problem is considered in the given paper. Time is generally perceived to be continuous. In practice a continuous time signal is sampled and the subsequent data processing is done in a discrete time mode. Continuous Volterra integral equation best fits the given formulation. Although it is originally considered to be continuous, a discrete version of that equation can be naturally obtained using a wide range of discretization methods. This approach utilizes advantages of both continuous and discrete time models. So the question what happens to the signal when the sampling time gets smaller, is clear in this case. The relationship between continuous and discrete time algorithms based on Volterra integral equations depends only on the method used for calculating the values of the integral, e.g. rectangle, trapezoidal or even based on Gaussian quadratures.
where the symbol E(·) denotes mathematical expectation and cov(·,e) denotes the covariance function of the two processes.
Et v(t)]- E[ wet») -E[x(to»)-O, cov[v(t),w('V»)-O, cov[ w(t),x(to») - cov[ v(t),x(to») -0,
(3)
cov[x(to),x(to)]-P0'
Covariance functions of the system and observation noises are described by the following expressions min(t.~
f V(t,~ )V('V,~ )d~,
R(tm-
'.
J W(I,~ )W('V,~)~
(4)
min(t~)
Q(t,'V)-
.
A problem formulation that is similar to the traditional Kalman filtering problem formulation is used for construction of a solution of smoothing, filtering and prediction problems. An estimation process is defined according the condition of nondisplacement that can be shown by the fonnula (5)
X(I)- E[x(t)/,(s).o
2 .2 Mathematical Formulation of the Problem. Throughout this section all random variables are assumed to be defined within the same fixed probability space. Further, they will be taken to have zero mean and finite variance, i.e. to belong to the Hilbert spaces, a class of linear spaces, but not being limited to finite dimensions. Dynamics of a stochastic system are described by a linear stochastic integral equation of Volterra type of the second kind x(t)+
j:(t,~ )x(~ )dt - jv(t,~ )dw(~ )+X(tO>,
The variance of the estimation error is described by the formula P(t,t)- cov[ (x(t)- x(t»,(x(t)- X(I» J.
(6)
The proposed estimator is used for constructing a nondisplacement estimation (5) of a stochastic process (1) by means of the observations (2) and according to the condition of the variance of the estimation error (6) to be mmunum.
3. OPTIMAL ESTIMATOR EQUATIONS
(1)
~
where x(t) is an output of a stochastic system; K(t,e) is a kernel of an integral operator representing dynamics of the system; W(t,e) is a kernel of integral operator representing the system's noise covariance; t is time; 10 is starting time. Observations given by a measurement scheme are described by the observation equation z(s)-b(s)x(s)+
f(S'~ )dv(~), t~s~,
Theorem .
For a system that is described by Volterra integral equation (1) and is being observed according to equation (2) taking into consideration conditions (3) and formulae (4) an optimal estimator is described by the following integral equations ,
P(t,'V)+ j<(t,~
(2)
'. where z(s) is an observation process; b(s) is a function representing dynamics of an observation process; V(s,') is an integral operator kernel representing covariance of an observation noise.
'0
,
~
'V
+ j(t,~) j(~
'0
t
-
'V
)P(~ ,'V)d~ + j(I,~ )K('V,~ )d~ +
,,, )K('V,,,)d,, d~ -
(7)
Po+Q(t,'V)-
t
'p(t,~) jR(~ ,,, )-b(e; )P(e; ,,, )b(TJ »D('V,,, )d~ dr], ~
System and observation noises wet) and vet) are random Wiener processes. The stochastic integral in equations (1) and (2) is considered in Ito sense. All functions in equations (1) and (2) are considered to be bounded on an infmite time interval. It is assumed that all conditions required for existence of Ito integral and an unique solution of a Volterra integral equation are valid. The stochastic
414
P(t,'V)b('V)+
j:(t,~ )P(~ ,'V)b('V)d~ ~
t
(8)
p(t,~ )(R(~ ,'V)-b(e; )P(~ ,'V)b('V»)d~ +Pob('V),
'0 where P(t,'t) the covariance of the estimation error,
D(t, 't) the integral operator kernel representing the gain of the estimator - are unknown functions.
equation of Volterra type has the form X(/)- jF(1; )+
The nondisplacement estimation process (5) satisfies the following linear integral equation, describing the structure of the optimal estimator
~(Tl ,I; )dTl )x(1; )d1; ~
t.
(11)
- :r(1; )dw(1; )+x(/o)' to
i(/)+
f(t,1; to
)X(I;
)dl; -
P(t,1;
)(z(1; )-b(1; )X(I; »dl;. (9)
The measurements (2) of the stochastic process (10) are available on the time interval from 10 to 'to The most probable values of the stocha~tic process (10) are to be estimated, using the given measurements. When t<'t smoothing and when t>'t prediction problems have to be solved.
to
Thus the evaluation of the best linear estimate of the state is described by a linear integral equation (9) driven by the observation process through the gain operator. In the case of smoothing and prediction problems this equation is to be solved as Fredholm integral equation. In the filtering problem case this equation may be solved as Volterra integral equation. The gain operator is obtained by solving a system of integral equations (7)-(8) which could be considered as an integral analog of the Ricatti equation. These equations do not depend on an observation process and yield the covariance of the estimation error. Formally, the situation is entirely analogous with a classical finitedimensional Kalman filter, described by differential equations. The estimator incorporates a full description of the state dynamics, process and observation models. In particular, this estimator can be applied to any problem solved by traditional state estimation methods based on differential equations, but the given solution of the formulated problem is more general.
Integral operators kernels and functions in the corresponding equations (1) and (2) are assumed to be the following
K(I,I; )- - (F(I; )+
F(I; )--151;, N(s,1; )--(20-4(s-1; »l(s-1; -h), G(I; )-gl'
(I for I~O where 1(1)- 0 for 1<0.
(12)
The figures given below illustrate simulation results based on the numerical solution of the equations (7)-(8) and calculation of the estimation process according to equation (9). While calculating the estimation process the optimal estimator uses only the noisy observations of the stochastic process that is being estimated and the gain of the estimator that has been computed prior to simulation. Initially, noise characteristics are assumed to be g,=O and g2=0 and the suggested algorithm is tested. Simulation results are shown in Fig.! and Fig.2. While Fig.! corresponds to the smoothing problem solution, Fig.2 corresponds to the transItive smoothing-prediction problem. It can be seen that the estimation process coincides with the original process. This result could be expected in the absence of noise. Now noise parameters are a~sumed to be the following g,=0.5 and g2=0.1. The simulation results are shown in Fig. 3-6. In Fig. 3,5 curve 1 represents stochastic process that is being estimated, curve 2 represents the noisy measurements of the stochastic process that are available for estimation, curve 3 represents estimation process. On Fig. 4,6 curve 1 represents the simulated estimation error e(t)=x(t)~(t) that corresponds to the previous figure (Fig. 3,5 respectively) and curve 2 represents limit mean square deviation of the estimation error, as it was computed apriori - ± 3\jP(t,t), where P(t,t) is the variance of the estimation error, taken form the equations (7)-(8). Fig. 3-4 refer to the smoothing problem solution, while Fig. 5-6 refer to the transitive smoothing-prediction problems solution.
The filtering problem most frequently arises in technical applications. Thus the filtering problem is regarded as an important special case of the estimation problem. Assuming t='t in equations (7)-(9) Kalman filtering algorithm can be easily derived. A solution to the filtering problem was presented in the paper (Maksarov, 1992), including a numerical example of filtering in a stochastic delay system. The interrelationship between Kalman and Wiener filtering algorithms in this case wa~ also considered in (Maksarov, 1991).
4, EXAMPLE
Taking into account the result~ described 111 the paper (Kochetkov, et.al, 1978) a linear elastic aircraft model based on considering nonstationary flow may be described by the following stochastic integro-differential equation dx(/) _ F(I)x(I)+ fo(I,1; )x(1; )dl; + G(I) dW(I) , ~ ~ ~
~(s,1; )ds), W(/,I; )-G(I; ), ~
(10)
where F(t), G(t), N(t,-) are functions that are a~sumed to be bounded on the required time interval; wet) is a random Wiener process with Gaussian distribution. The right hand side of the equation (10) describes the generalized aerodynamic forces that depend on the history of the process. This equation may be reduced to differential equation in only a few special cases and the order of the corresponding differential equation is uncertain. But the kernel of the integral operator may have a shape when the given equation has no analog in the class of differential equations. In fact, the most convenient and natural way to solve the problem is to reduce equation (10) to the integral equation of Volterra type. The corresponding integral
5. CONCLUSIONS This paper has suggested using Volterra integral equations as a more general mathematical model in describing the dynamics of control systems. With the help of integral
415
second kind. The gain of the estimator and the covariance of the estimation error are connected by the corresponding system uf the integral equations that has been presented in the paper. The gain is used in computing an estimation pro cess .tfter the solution of an integral form of Ricatti equation. It can be seen that more general solutions can be derived with help of integral equations as compared with those of diffcrential equations. Numerical example of estimation in the nonstationary stochastic system with delay has been examined. Simulation results are shown. Using integral equations in control theory may lead to more general solutions of control engineering problems.
z.g
1.6 1.2 U
11.4 11 Fig. 1. The original process (curve 1). the meas urements (curve 2) and the estimation process (curve 3).
2.Q
1.6 1.2 1.8 1.4
Fig. 5 . TIle original stochastic process (curve 1), its measurements (curve 2) and the estimation process (curve 3).
• Fig. 2 . TIle original process (curve I), the mea surements (curve 2) and the estimation process (c urve 3).
11.6 11.4 11.2 11
11.9 11.8 9.7 &.6
-9,2
-9.4
11.5 9.4 9.3 &.2
-9.6
1.2
11.1 11
1.8
2.4
Fig. 6 . The simulated estimation error (curve I) and limit mean square deviation (curve 2).
Fig. 3. The original stochastic process (curve I), its measurement (curve 2) and the estimation process (c urve
3). (, . REFERENCES Ko c he tkov. Yu.A , V.K. Tomshin . (1978). Optimal control uf oeterminis ti c system oescribed by integro-differential equations. Alltoll/ation and Refllote Control. I, p.5-ll. Maksarov , D .G . (1991). Using Volterra Integral Equations in Filtering Theo ry, ProC£'edin;:s of the Latl'ian Probability Sefllinar . 1. p.116-126. Maksarov. D.G. (1992). Filtering in Stochastic Systems Described by Voltl' rra Integral Equations, Cybernetics (/l1d Srsteflls·92 . Proceeding.I' of the' 11th European MC'l'Iing Oil Cyhl'l'nC'tics and Systems Research . p.189-
H.4 &.2 &
-11.2 -&.4
Fig. 4 . The simulated estimation error (curve I) and limit mean square devi ation (curve 2) .
196. Veri an. A .F., V.S . Sizikov (1986). Intellral equations: II/etll(lds . alxoritllll/s. prox rams. Naukova Dumka, Kiev . (in Russian) Veri an, A.F., S .S . Mosknluk (1988) . Mathematical II/odl'/lillg nf omtim(o/l.\' dynamic systems. Naukova Dumb, Kiev . (in Russian)
equations it is possible to de scribe a number of important classes of dynamic systems s uch as nonstatiunary continuous stochastic delay systems, systems described by integro-differential equation, etc. This paper has dealt with the simultaneous solution of smoothing, filtering and prediction problems when a stochastic process is desc ribed by the stochastic Ito- Volterra integral equation of the second kind. It has been shown that an estimation process is described by the deterministic integral equation of thc
416
SIMULTANEOUS SOLUTION OF SMOOTHING, FILTERING AND PREDICTION PROBLEMS USING INTEGRAL EQUATIONS D.G. Maksarov Depart~N of Engineering Scienu. University of Oxford. Parks Road. Oxford OX1 3P}. UK
Permanent address : Automatic CONrol Systems Department, Riga Technical University. 1 Kallcu St .• Riga. LV 1658. Latvia
Abstract. TIle go.'\l of this paper is to present an algorithm for simultaneous solution of stochastic smoothing, filtering and prediction problems. TIlis alglllithm was obtained when a stochastic process is desclibed by linear stochastic ho-Volterra integral equation. Thc Kalman filtering problem fOllnulation is used in the given paper. It has been shown that the gain of the optimal estimator and the covaliance of the estimation error are connected by a system of integral equations. TIle estimation process is described by a lincar integral equation of the second kind . An example of estimation in stochastic system is con sidered. Simulation results are presented.
Key Words. Smoothing; filtering; prediction; integral equation ; simulation.
applicatio n domains for many years (Verlan. et al.. 1986, 1988). Dynamics of a complicated stochastic system can be naturally presented by an integral equation driven by Ito stochastic integral. Integral equations represent an important class of mathematical models that can be successfully used in control theory. However, there has not been much development in the control theory of integral equations. As far as Volterra integral equations have been introduced as a more general mathematical model for description of a control system it is important to show that principal re sults in the control theory based on differenti al equations can be obtained from the mathematical model represented by Volterra integral equation. Moreover. integral equations allow to derive more general solutions as compared with those obtained on the basis of differential equations. Stochastic smoothing, filtering and prediction problems are considered in this paper.
1. INTRODUCTION
It is a question of primary importance which mathematical model to use when developing control algorithms. Classic al linear control theory has dealt with ordinary stationary differential equations. Later on nOllstationary ordinary differential equations were also considered. Many real world processes require taking into account a delay for an adequate modelling. Neutral type delay differential equations are used to model systems in biology. economy. engineering science. etc. This diversity of differential equations gives rise to certain difficulties. Every type uf differential equations requires unique algorithms for the design of control systems whose mathematical models arc described by these equations. In addition. real world processes can be presented by complicated integrodifferential equations with delays. These equations may not be successfully reduced to differential equations in some cases. However as industrial processes become more complicated and more involved it may become increasingly inadequate to model dynamics of a system by a set of linear differential equations. Real world processes arc ofll' n stochastic. Using Ito integral for noise representat ion leads to a strict mathematical description of stochastic dynamic systems. An adequate mathematical modd of a stnchasti c process can be obtained due to this approach .
The brief outline of the paper follows. The problem formulation is given in chapter 2. The main results are presented in chapter 3. In chapter 4 an example of estimation 111 a complicated stochastic system is co nsidered . Conclusions may he found in chapter 5.
2. PRORLEM FORMULATION
It can be inferred from what has been mentioned that in some cases effective solution of various control systems' design problems requires the implementation of more general mathematical models. In fact. one of the possible approaches is that some real world processes can more appropriately be modelled by integral equations or their mathematical models can be reduced to integral equations. Integral equations of Volterra type have been used for describing dynamics of complicated systems ill different
2 .1 Olltlilll' of thl' Probll'm.
Problems of filtering. smoothing and prediction arise in many areas of engineering. economics. ecology, etc. The main aim of the solution of these problems is to estimate the most probable values of a stochastic process. The dilTerence among these problems is in the amount of information available for estimating values of a stochastic
413
process. This infonnation is available from the noisy observations. For the solution of these problems mathematical models in the fonn of stochastic dynamic systems can be formulated. A linear Volterra integral equation of the second kind driven by Ito stochastic integral is introduced as an adequate description of a stochastic process in this paper.
processes that are considered satisfy the following oonditions
When fonnulating a problem one may pose the question which model to consider - continuous or discrete time. A continuous time version of the problem is considered in the given paper. Time is generally perceived to be continuous. In practice a continuous time signal is sampled and the subsequent data processing is done in a discrete time mode. Continuous Volterra integral equation best fits the given formulation. Although it is originally considered to be continuous, a discrete version of that equation can be naturally obtained using a wide range of discretization methods. This approach utilizes advantages of both continuous and discrete time models. So the question what happens to the signal when the sampling time gets smaller, is clear in this case. The relationship between continuous and discrete time algorithms based on Volterra integral equations depends only on the method used for calculating the values of the integral, e.g. rectangle, trapezoidal or even based on Gaussian quadratures.
where the symbol E(·) denotes mathematical expectation and cov(·,e) denotes the covariance function of the two processes.
Et v(t)]- E[ wet») -E[x(to»)-O, cov[v(t),w('V»)-O, cov[ w(t),x(to») - cov[ v(t),x(to») -0,
(3)
cov[x(to),x(to)]-P0'
Covariance functions of the system and observation noises are described by the following expressions min(t.~
f V(t,~ )V('V,~ )d~,
R(tm-
'.
J W(I,~ )W('V,~)~
(4)
min(t~)
Q(t,'V)-
.
A problem formulation that is similar to the traditional Kalman filtering problem formulation is used for construction of a solution of smoothing, filtering and prediction problems. An estimation process is defined according the condition of nondisplacement that can be shown by the fonnula (5)
X(I)- E[x(t)/,(s).o
2 .2 Mathematical Formulation of the Problem. Throughout this section all random variables are assumed to be defined within the same fixed probability space. Further, they will be taken to have zero mean and finite variance, i.e. to belong to the Hilbert spaces, a class of linear spaces, but not being limited to finite dimensions. Dynamics of a stochastic system are described by a linear stochastic integral equation of Volterra type of the second kind x(t)+
j:(t,~ )x(~ )dt - jv(t,~ )dw(~ )+X(tO>,
The variance of the estimation error is described by the formula P(t,t)- cov[ (x(t)- x(t»,(x(t)- X(I» J.
(6)
The proposed estimator is used for constructing a nondisplacement estimation (5) of a stochastic process (1) by means of the observations (2) and according to the condition of the variance of the estimation error (6) to be mmunum.
3. OPTIMAL ESTIMATOR EQUATIONS
(1)
~
where x(t) is an output of a stochastic system; K(t,e) is a kernel of an integral operator representing dynamics of the system; W(t,e) is a kernel of integral operator representing the system's noise covariance; t is time; 10 is starting time. Observations given by a measurement scheme are described by the observation equation z(s)-b(s)x(s)+
f(S'~ )dv(~), t~s~,
Theorem .
For a system that is described by Volterra integral equation (1) and is being observed according to equation (2) taking into consideration conditions (3) and formulae (4) an optimal estimator is described by the following integral equations ,
P(t,'V)+ j<(t,~
(2)
'. where z(s) is an observation process; b(s) is a function representing dynamics of an observation process; V(s,') is an integral operator kernel representing covariance of an observation noise.
'0
,
~
'V
+ j(t,~) j(~
'0
t
-
'V
)P(~ ,'V)d~ + j(I,~ )K('V,~ )d~ +
,,, )K('V,,,)d,, d~ -
(7)
Po+Q(t,'V)-
t
'p(t,~) jR(~ ,,, )-b(e; )P(e; ,,, )b(TJ »D('V,,, )d~ dr], ~
System and observation noises wet) and vet) are random Wiener processes. The stochastic integral in equations (1) and (2) is considered in Ito sense. All functions in equations (1) and (2) are considered to be bounded on an infmite time interval. It is assumed that all conditions required for existence of Ito integral and an unique solution of a Volterra integral equation are valid. The stochastic
414
P(t,'V)b('V)+
j:(t,~ )P(~ ,'V)b('V)d~ ~
t
(8)
p(t,~ )(R(~ ,'V)-b(e; )P(~ ,'V)b('V»)d~ +Pob('V),
'0 where P(t,'t) the covariance of the estimation error,
D(t, 't) the integral operator kernel representing the gain of the estimator - are unknown functions.
equation of Volterra type has the form X(/)- jF(1; )+
The nondisplacement estimation process (5) satisfies the following linear integral equation, describing the structure of the optimal estimator
~(Tl ,I; )dTl )x(1; )d1; ~
t.
(11)
- :r(1; )dw(1; )+x(/o)' to
i(/)+
f(t,1; to
)X(I;
)dl; -
P(t,1;
)(z(1; )-b(1; )X(I; »dl;. (9)
The measurements (2) of the stochastic process (10) are available on the time interval from 10 to 'to The most probable values of the stocha~tic process (10) are to be estimated, using the given measurements. When t<'t smoothing and when t>'t prediction problems have to be solved.
to
Thus the evaluation of the best linear estimate of the state is described by a linear integral equation (9) driven by the observation process through the gain operator. In the case of smoothing and prediction problems this equation is to be solved as Fredholm integral equation. In the filtering problem case this equation may be solved as Volterra integral equation. The gain operator is obtained by solving a system of integral equations (7)-(8) which could be considered as an integral analog of the Ricatti equation. These equations do not depend on an observation process and yield the covariance of the estimation error. Formally, the situation is entirely analogous with a classical finitedimensional Kalman filter, described by differential equations. The estimator incorporates a full description of the state dynamics, process and observation models. In particular, this estimator can be applied to any problem solved by traditional state estimation methods based on differential equations, but the given solution of the formulated problem is more general.
Integral operators kernels and functions in the corresponding equations (1) and (2) are assumed to be the following
K(I,I; )- - (F(I; )+
F(I; )--151;, N(s,1; )--(20-4(s-1; »l(s-1; -h), G(I; )-gl'
(I for I~O where 1(1)- 0 for 1<0.
(12)
The figures given below illustrate simulation results based on the numerical solution of the equations (7)-(8) and calculation of the estimation process according to equation (9). While calculating the estimation process the optimal estimator uses only the noisy observations of the stochastic process that is being estimated and the gain of the estimator that has been computed prior to simulation. Initially, noise characteristics are assumed to be g,=O and g2=0 and the suggested algorithm is tested. Simulation results are shown in Fig.! and Fig.2. While Fig.! corresponds to the smoothing problem solution, Fig.2 corresponds to the transItive smoothing-prediction problem. It can be seen that the estimation process coincides with the original process. This result could be expected in the absence of noise. Now noise parameters are a~sumed to be the following g,=0.5 and g2=0.1. The simulation results are shown in Fig. 3-6. In Fig. 3,5 curve 1 represents stochastic process that is being estimated, curve 2 represents the noisy measurements of the stochastic process that are available for estimation, curve 3 represents estimation process. On Fig. 4,6 curve 1 represents the simulated estimation error e(t)=x(t)~(t) that corresponds to the previous figure (Fig. 3,5 respectively) and curve 2 represents limit mean square deviation of the estimation error, as it was computed apriori - ± 3\jP(t,t), where P(t,t) is the variance of the estimation error, taken form the equations (7)-(8). Fig. 3-4 refer to the smoothing problem solution, while Fig. 5-6 refer to the transitive smoothing-prediction problems solution.
The filtering problem most frequently arises in technical applications. Thus the filtering problem is regarded as an important special case of the estimation problem. Assuming t='t in equations (7)-(9) Kalman filtering algorithm can be easily derived. A solution to the filtering problem was presented in the paper (Maksarov, 1992), including a numerical example of filtering in a stochastic delay system. The interrelationship between Kalman and Wiener filtering algorithms in this case wa~ also considered in (Maksarov, 1991).
4, EXAMPLE
Taking into account the result~ described 111 the paper (Kochetkov, et.al, 1978) a linear elastic aircraft model based on considering nonstationary flow may be described by the following stochastic integro-differential equation dx(/) _ F(I)x(I)+ fo(I,1; )x(1; )dl; + G(I) dW(I) , ~ ~ ~
~(s,1; )ds), W(/,I; )-G(I; ), ~
(10)
where F(t), G(t), N(t,-) are functions that are a~sumed to be bounded on the required time interval; wet) is a random Wiener process with Gaussian distribution. The right hand side of the equation (10) describes the generalized aerodynamic forces that depend on the history of the process. This equation may be reduced to differential equation in only a few special cases and the order of the corresponding differential equation is uncertain. But the kernel of the integral operator may have a shape when the given equation has no analog in the class of differential equations. In fact, the most convenient and natural way to solve the problem is to reduce equation (10) to the integral equation of Volterra type. The corresponding integral
5. CONCLUSIONS This paper has suggested using Volterra integral equations as a more general mathematical model in describing the dynamics of control systems. With the help of integral
415
second kind. The gain of the estimator and the covariance of the estimation error are connected by the corresponding system uf the integral equations that has been presented in the paper. The gain is used in computing an estimation pro cess .tfter the solution of an integral form of Ricatti equation. It can be seen that more general solutions can be derived with help of integral equations as compared with those of diffcrential equations. Numerical example of estimation in the nonstationary stochastic system with delay has been examined. Simulation results are shown. Using integral equations in control theory may lead to more general solutions of control engineering problems.
z.g
1.6 1.2 U
11.4 11 Fig. 1. The original process (curve 1). the meas urements (curve 2) and the estimation process (curve 3).
2.Q
1.6 1.2 1.8 1.4
Fig. 5 . TIle original stochastic process (curve 1), its measurements (curve 2) and the estimation process (curve 3).
• Fig. 2 . TIle original process (curve I), the mea surements (curve 2) and the estimation process (c urve 3).
11.6 11.4 11.2 11
11.9 11.8 9.7 &.6
-9,2
-9.4
11.5 9.4 9.3 &.2
-9.6
1.2
11.1 11
1.8
2.4
Fig. 6 . The simulated estimation error (curve I) and limit mean square deviation (curve 2).
Fig. 3. The original stochastic process (curve I), its measurement (curve 2) and the estimation process (c urve
3). (, . REFERENCES Ko c he tkov. Yu.A , V.K. Tomshin . (1978). Optimal control uf oeterminis ti c system oescribed by integro-differential equations. Alltoll/ation and Refllote Control. I, p.5-ll. Maksarov , D .G . (1991). Using Volterra Integral Equations in Filtering Theo ry, ProC£'edin;:s of the Latl'ian Probability Sefllinar . 1. p.116-126. Maksarov. D.G. (1992). Filtering in Stochastic Systems Described by Voltl' rra Integral Equations, Cybernetics (/l1d Srsteflls·92 . Proceeding.I' of the' 11th European MC'l'Iing Oil Cyhl'l'nC'tics and Systems Research . p.189-
H.4 &.2 &
-11.2 -&.4
Fig. 4 . The simulated estimation error (curve I) and limit mean square devi ation (curve 2) .
196. Veri an. A .F., V.S . Sizikov (1986). Intellral equations: II/etll(lds . alxoritllll/s. prox rams. Naukova Dumka, Kiev . (in Russian) Veri an, A.F., S .S . Mosknluk (1988) . Mathematical II/odl'/lillg nf omtim(o/l.\' dynamic systems. Naukova Dumb, Kiev . (in Russian)
equations it is possible to de scribe a number of important classes of dynamic systems s uch as nonstatiunary continuous stochastic delay systems, systems described by integro-differential equation, etc. This paper has dealt with the simultaneous solution of smoothing, filtering and prediction problems when a stochastic process is desc ribed by the stochastic Ito- Volterra integral equation of the second kind. It has been shown that an estimation process is described by the deterministic integral equation of thc
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