GB2448599A - Utilising a semi-martingale algorithm for predicting the motion of a target - Google Patents

Abstract

A method is disclosed for ascertaining the motion of at least one target object (A, B, B1, B2). The method comprises a mathematical method of estimation with the aid of a filter solution for estimating and predicting the position and/or orientation of the target object. The filter solution utilises a semi-martingale algorithm. The method is particularly suited for predicting the motion of a highly moveable target such as a ballistic missile B so that it may be intercepted by an interceptor missile C. The motion of the ballistic missile B can be predicted via a semi-martingale process such that abrupt changes in flight motion of the target can be reliably predicted.

Description

Ascertaining and predicting the motion of a target The invention

relates to a method with a system for ascertaining and predicting a motion of at least one target object, according to the precharacterising portion of Claim 1, and to a system according to Claim 10.

So-called tracking systems, which are able to register the motion of an object by means of a model assumption, are known. These systems utilise so-called filter equations. A tracking system serves for guiding an interceptor missile that is intended to hit a target object, for example a ballistic target missile. A target object -for example, a ballistic target missile -is distinguished in that it does not have a propulsion unit of its own. Such a missile is designated in German technical language as an unmanned, autonomous flying vehicle, so that ordinary powered aircraft are not covered by this concept.

The filter equations permit filter methods that, in principle, are mathematical methods of estimation. Known filter methods accordingly use model assumptions for estimating the state of the target missile.

A filter solution that has been preferred up until the present time is based on a so-called Kalman filter or extended Kalman filter.

In the event of abrupt changes of state or abrupt changes in the flight motion of the target missiles, these methods have a poor performance as a rule.

Filters of this type are described in, for example, the publication by G. Minkler, J. Minkler: Theory and Application of Kalman Filtering, Magellan Book Company, Palm Bay, 1993.

Other known filters, such as the so-called particle filter or the unscented filter, also do not solve this problem.

Unscented filters are known from, for example, the publication by B. Ristic, S. Arulampalam, N. Gordon: Beyond the Kalman Filter, Artech House Publishers, Boston, 2004.

Because these filter methods do not permit, or only inadequately permit, the modelling of abrupt changes of state or changes of manoeuvre, the filters react sluggishly or defectively in the event of such manoeuvres.

This is because the underlying model assumptions concerning the motion of the object cannot, for mathematical reasons, map random, discontinuous changes in the characteristics of the target objects, such as accelerations for example. In practice, non-liriearjtjes arise in addition, so that an estimation based on these methods is inaccurate.

In order to lessen this problem, it is known to provide an upstream application of integrity algorithms. In this case, use is made of a filter bank with several filters. If deviations of the predicted data, based on the model assumptions, from the measured data are too great, the measurements are discarded, or switching takes place to a different filter of the filter bank. By this means, although the problem can be alleviated in practice, the sluggishness still persists by reason of the behaviour of the target object, which is difficult to model.

Interception of highly manoeuvrab].e targets such as ballistic missiles, in particular by means of an interceptor missile, is likewise rendered distinctly difficult in these cases, resulting in a low hit-rate of the interceptor missile.

An object of the invention is to specify a method that permits an interception of highly manoeuvrable targets. * 3

The invention is defined in claim 1. The use, according to the invention, of stochastic process theory, which is based on semimartingale theory, creates filters that are fit for practical use and improved in comparison with the state of the art for ascertaining both states and transitions of manoeuvre. By this means, it is possible for the hit reliability of an interceptor missile to be distinctly improved.

The invention also allows discrimination; that is, targets of different kinds can also be recognised and differentiated. A detection of transitions of state, in particular a change in the configuration of the target -such as, for example, a change in aerodynamics, a change in behaviour as a result of component faults or such like -is also possible.

The resulting filter equations lead to optimal results, with a solution of equations of the semi-martingale algorithm in the continuous case or in the case of a non-discontinuous change in the target only requiring the solution of a stochastic differential equation. A solution of equations can therefore be found with relatively little computing capacity. Numerical inaccuracies are relatively slight.

In an advantageous further development of the method according to the invention, the invention provides that the motion is registered by a Markov process or semi-Markov process with the aid of stochastic differential equations.

The application of these processes based on general semi-martingale theory enables the creation of optimal filters.

These also permit discontinuous processes -that is to say, discrete state transitions -in the modelling. The only requirement is a model that forms a smooth semi-martingale, this being a very general requirement and one that is easy to satisfy in practice.

A semi-martingale representation in the compensator and martingale portions is possible in straightforward manner.

In expedient manner, the Markov process has an associated indicator process of the form: X = X0 + I Ek A.Xs,k ds + M, from 0 to t or dX = Ek A.Xt,k + M. X is renamed as Y. Therefore the formula reads X = X0 + I ?.,i ds + M, from 0 to t or dX = + M In this case the simplest case is considered, in which the motion of the target object is completely described by the diSCotjnuou process. This is the case, for example, if the target object assumes an acceleration that is discontinuous but constant between the discontjnujtjes, each spatial dimension is considered separately, and a so-called tracking station measures the velocity of the target object.

In the general case, the model consists of a discontinuous process that describes the discontinuous change, and of a continuous model that builds upon the discontinuous process.

This situation arises, for example, if an attacking missile suddenly changes its guidance method internally or the aerodynamics of the missile suddenly change, for example as a result of flaps being extended or as a result of a section of the missile being jettisoned.

This situation will now be described in more detail.

By way of target-object model, a smooth semi-martingale X will be considered, dXt=Atdt+M

I

Serni-martingales are very general stochastic processes. For the application under consideration, the following specialjsatjon is expedient: Let a process Y be given, in the simplest case a Markov process. A target-object model can now frequently be described as follows as a stochastic differential equation: dX = f(Y, X) dt + dM where X is the process -inter alia the multidimensional process -to be modelled, Y is the underlying discontinuous process, and M is a martingale. The solutions X of this differential equation are again semi-martingales; therefore this form of modelling is a specialisation of the above model.

In the simplest case the following model will be considered: dX = f(Y) dt dM If Y is a Markov process as described above, then it is expedient to model X as an indicator function or as a vector of indicator functions over all indices I: x=fly The indicator function is defined as follows: 11:= 1 if A holds, and hA:= 0 if A does not hold.

In this case, the model accordingly results as dX = X,i + In this connection: X = a state of the indicator process, = a transition rate, N = a martingale, Y such a Markov process.

A measurement satisfies, in particular, a stochastic differential equation of the form: dZ = F dt + at dW with the following meanings: F and a = an adapted stochastic process, Z = a measurement or a measured value and W = a Brownjan motion.

An exact and optimal (implicit) filter equation can be derived in this way. For preferred target objects and other special cases which involve discrete state transitions that are modelled by a Markov process or semi-Markov process, closed formulae arise. Depending on the modelling, this enables, for instance, the rapid recognition of a target type or the estimation of the probability of the existence of a state or manoeuvre. For example, a discontinuous change in the aerodynamics of the missile to be hit can be quickly established.

For the purpose of solving these equations, a numerical integration method is preferably employed. This method can be processed efficiently with computers. These computations can be solved with relatively slight numerical inaccuracies.

In preferred manner the state Y of the discrete process is assigned to a state of a state automaton in a guidance-strategy module of the target object and/or to an index of target-types of target objects. As a result, on the one hand diverse target objects -for example, diverse ballistic missiles -can be reliably distinguished from one another and, on the other hand, a change -for example, in the flight property of the target object, for example as a result of discontinuous changes of the flaps or as a result of parts of the object being jettisoned -can be established, and an exact prediction of its flight motion can be made. This decisively enhances the hit reliability of an interceptor missile.

Changes of manoeuvre of the target object can be easily recognised by means of an explicitly solvable stochastic differential equation. For example, an interceptor missile can be appropriately adjusted to this change of manoeuvre in a short time, even during its flight motion. In this connection, mathematical principles, such as those concerning the theory of Browniari motion and the theory of semi-martingales, can be utilised.

An important advantage of these measures consists in the fact that the formulae are easily transferable into the multidimensional domain. An inverse calculation and advance calculation or, to be more exact, a smoothing and a prediction are possible in a manner similar to conventional filters.

Accordingly, it is highly favourable that the stochastic differential equations describe, inter alia, the mathematical probability of the existence of a manoeuvre of the target object, with computation data relayed immediately to an interceptor missile. Because ballistic missiles are able to move at very high speeds, with known methods it is difficult to hit the target by means of an interceptor missile. Through the accurate prediction of the flight motion of the target on the basis of the method according to the invention and this advantageous further development, the target can be hit. This applies, in particular, to target missiles having discontinuous changes of manoeuvre.

An air-defence system or a tracking system, for example, can make use of the invention. This system includes a measuring station, in particular a radar measuring station. The latter can, using proven technology, provide data pertaining to the target. A processing-and-control unit for processing measured motion data or flight data is programmed in accordance with the mathematical model. The unit exhibits, in particular, one or more microprocessors.

For a better understanding of the invention, embodiments of it will now be described, by way of example, with reference to the accompanying drawings, in which: Fig. 1 is a schematic representation of an intercept system that is operated in accordance with the invention, Fig. 2 is a schematic representation of various flying objects, and Fig. 3 is a representation of probability curves with data calculated by the method in accordance with the invention in the case of two flying objects.

The following examples illustrate a method with a system, in particular an air-defence system, for ascertaining the motion of a target object, in particular a ballistic missile, by using a mathematical method of estimation with the aid of a filter method relating to the model assumption for estimating the motion of the missile. The system includes a tracking system or may be designated as such.

Instead of a ballistic missile, any other target object may also be considered.

In accordance with the invention, the filter method utilises a semi-martingale-based algorithm for estimating the motion of the missile. The invention utilises serni-martingale theory for solving various problems.

Certain processes that, in particular, are significant for a definition of a general stochastic integral are generally designated in stochastics as semi-martingales. Processes such as Brownian motion fall under the semi-martingales. A martingale is likewise a semi-martingaje in the sense of semi-martingale theory.

A probability space (Q, Z, F) with associated filtration (st) -that is to say, a filtered probability space (Q, Z, (3), P) -will be considered. A probability space is a concept from the mathematical subdomain of probability theory. It is basically a mathematical model for describing random events that are represented by the triple (Q, Z, F) . In this connection, ( =, ) form a measuring space. In general, C = denotes an outcome set, Z denotes the outcome sigma algebra, and P denotes a probability measure on.

Let the model be given in the form of a semi-martingale, in particular a multidimensional semi-martingale. For the purpose of simplification, the theory will be elucidated on the basis of a one-dimensional integral. A semi-martingale is a stochastic process: X = X0 + A + M where X is almost certainly finite and so-measurable. M is a local martingale. A is a process of finite variation.

This breakdown is, in general, not uniquely defined.

Here the following holds: = Xo + A + (M=(M) is a martingale) where Xc, is a omeasurable random variable, A=(At) is a adapted stochastic process with the following representation: At=fafromOtot The class of the semi-martingales is stable under many operations. Many discontinuous processes are semi-martingales, since they are of limited variation. *1 10

In probability theory a martingale is a stochastic process in which the conditional expectation of an observation at time t is equal to the observation at an earlier time s.

(Me), (3t) } t e R+ is a stochastic process with the set of non-negative real numbers R+.

M is called a martingale with respect to a filtration 3 if M is iritegrable for each t E T, is adapted to the filtering 3, and E (MJ3) = M holds for all s = t.

A Markov process (Markov chain) is a special stochastic process. The aim of the process is to specify probabilities for the occurrence of future events. The special feature of a Markov chain is the property that through knowledge of a limited previous history equally good prognoses are possible concerning the future development as in the case of knowledge of the entire previous history of the process.

For the purpose of describing Markov processes, the concepts of filtration and of conditional expectation are needed. An example of a continuous Markov process is Brownian motion.

Markov chains are eminently suitable for modelling random changes of state of a system.

Fig. 1. illustrates an air-defence system in respect of attacking flying objects, in particular missiles A or B. The system includes a radar measuring station R for measuring missile A or B (A is shown in Fig. 2). Moreover, a processing-and-control unit, which is not shown, for processing measured motion data or flight data pertaining to missile B is present. The control unit controls the flight direction, flight speed, etc. of an interceptor missile C for intercepting the target object, a ballistic missile B in this case. The interceptor missile C is launched from a launching ramp AR. It is preferably a ballistic interceptor missile and is equipped with means for implementing the method according to the invention. These means may be control means, computing means, for example with a microprocessor, guidance means and/or sensor means.

The target object may, however, suddenly change its flight direction, acceleration and/or flight speed -i.e. it may suddenly enter into a change of manoeuvre. Therefore a reliable prediction is needed in order to hit the target object. The system that is shown is a tracking system that is able to register a motion of an object by means of a model assumption according to the invention.

Important discontinuous processes are also distinguished by a smooth semi-martingale representation. Thus in the case of the discrete Markov process with transition rates ik the following holds: Yt = Y0 + S Ek 2Ly5,k ds + M, from 0 to t.

The associated indicator process X=(Xt), X = 11 Y has the following representation: Xt = Xo + S ?y ds + M, from 0 to t.

For the changes of manoeuvre to be modelled and later to be estimated, these indicator processes prove to be particularly useful. For instance, the state Y of the discrete process may denote the state of a state automaton in the guidance algorithm/strategy of the target, or the index of one of various target configurations/types. In the multidimensional case, yet further indices or continuous model parameters may form the state vector.

The measurement in the simplest case will now be described by a stochastic differential equation, which in general is multidimensional, of the form dZ = F dt + at dW Here F=(E') and a=(at) is a suitable -adapted or (t2) -adapted stochastic process, so that the stochastic differential equation has a strong solution. For at, in addition at > 0 is assumed. In the multidimensional case the positive definiteness holds. For F, arbitrary functional dependencies of X are allowed.

The estimation of X that is optimal with respect to the estimation variance in the case of existing measurements (Z =) is given by the conditional expectation E(XtItz).

Also possible are other variants which will be considered later.

From semi-martingale theory it follows -as elucidated in greater detail in G. Kallianpur: Stochastic Filtering Theory, Springer-Verlag, New York, 1980 -that the conditional expectation satisfies the following filter equation, with only the one-dimensional formula being specified here: dE(XtJtz) = (E(atIZtz) -a'(E(X Fttz) - (E(XtIZtz) E(FtJtz) ) a1 E(FtI3t2) dt + at' (E (X F I tz) -E (Xt I tz) E (F I tZ) ) at1 dZ.

The specified filter formula is in general not sufficient for ascertaining the estimation: in general, the variables E(atItZ), E(Ftftz), E(X Ftftz) are not known. Here, in particular, approximative methods or the (mostly infinitely) recursive ascertainment of the conditional expectations concerning the above filter equation are applied. These come into operation in respect of at, F, X, F. For the most important case of the indicator processes X for the changes of manoeuvre that have been described, the variables are simplified to give E (at I 3tz) = E ( Yt,i 1l I 2) = Ek ki E (llvt=k I Ztz) E (X F I Ztz) = E (llYti F I 3tz) = E (llyt-i F I 3tz, X) C I F z -. ,fl z xiLt,t -i IIYti rt t and are consequently easy to ascertain. Estimations may be carried out simultaneously for all states i. If these results are inserted into the filter equation, in the case of the indicator processes an explicitly solvable stochastic differential equation arises.

With the aid of the procedure that has been described, changes of manoeuvre, for example, can be recognised, and the interceptor missile can be controlled appropriately.

Further embodiments are possible. For instance, in one variant a stochastic functional dependence between a model noise and measuring noise is taken into account.

If M and W are not independent of one another, the filter equation is extended by one term which includes the derivative of the quadratic covariation coefficient of the two processes. Solvability nonetheless obtains.

A multidimensional measurement and/or a multidimensional model is/are also possible. As already explained, the formulae are transferable into the multidimensional domain.

As already indicated, discontinuous processes X other than Markov processes are also possible. If, for example, X is a semi-martingale process, then the use of indicator functions of the form presents itself. In this connection, t denotes the time that has elapsed since the last discontinuity in the process X. If an estimating time is not equal to a measurement time, an inverse calculation or advance calculation, or, to be more exact, a smoothing or prediction are possible. The derivation of the filter equations for this case is undertaken in analogous manner.

In the case of a discrete sampling, the discrete sampling case can be realised by replacing the (temporally) continuous semi-martingales with (temporally) discrete semi-martingales by prediction with respect to the next measurement times and/or by discretisation of the stochastic integral.

If the filter is operated within the context of an air-defence system, the filter makes additional information available concerning the attacking missile by ascertaining the type of manoeuvre or other internal variables.

Consequently the air-defence system is capable of communicating such additional information to the interceptor missile in addition to the trajectory data for the intercept course. The interceptor missile is thereby put in a position to optimise its guidance law with respect to the circumstances of the attacker.

Fig. 2 shows the missile B, which in this example is approaching a target ballistically. The missile B is continuously surveyed by the radar station R. It is known that the type of missile originates from a set of two possible types A and B. One type is a conventional ballistic missile; the other missile exhibits folding wings.

The latter type is able to change its configuration by unfolding the wings at a random instant (configurations B1 and B2).

By way of model, use is made of, for example, a simple Markov process having three states. The associated transition diagram is represented in Fig. 2.

Fig. 3 illustrates characteristics or traces of a filter that has been realised in accordance with the method according to the invention and that specifies a probability of a missile type and flight characteristics. *1 15

The top trace FK represents a simulated trajectory of a missile of type B. First of all, this corresponds to the values 2 (for B1) and 3 (for B2), which are present on the y-axis.

The second trace, P for missile type A, represents a result of the filter according to the invention in the case of simulated measurements by the radar station R. In this case the conditional expectation E(XtIZtz) specified above yields precisely the probability of the existence of one of the three possible missile configurations; by way of process X, use is made of the indicator process. As can be seen, missile A has the probability virtually zero.

The third trace, P5 for missile type B1, represents the probability of the missile B with the characteristic B1 (no wings).

The fourth trace, P for missile type B2], represents the probability of the missile B with the characteristic B2 (unfolded wings).

The discrimination between missiles A and B1 at the start of the simulation is undertaken very quickly and can barely be discerned in the characteristics that are represented.

After about 1.1 $ the missile unfolds the wings, the air resistance which changes as a result leads to a change of motion, which is detected immediately.

As can be seen from the third characteristic, the probability of the existence of type B1 falls, specifically to precisely zero. In analogous manner the probability of the existence of type B2 rises to virtually 100%.

Because the method according to the invention is able to map random and discontinuous changes in the characteristics of the target objects -such as accelerations, for example -the accuracy of estimation of the tracking system is high.

In addition, non-linearjtjes may also arise, something which frequently occurs in practice, so that an estimation based on these methods is extremely accurate.

Accordingly, because the filter method based on semi-martingale theory permits the modelling of abrupt changes of manoeuvre of missile C, in the event of such manoeuvres the filter that is employed does not react sluggishly and also does not react defectively.

The filter according to the invention is provided with hardware, preferably a microprocessor, and with a software program for implementing the method according to one of the preceding claims.

The invention is not restricted to this example; accordingly, other variants of semi-martingale filter equations may also be employed. The registering of other target objects is also possible. The launching ramp may also be a different launching apparatus. An interception! termination is not necessarily coupled with the tracking.

In practice, it makes perfect sense to track an object without interception.

LIST OF REFERENCE SYOLS

A first missile B second missile B1 first missile (without wing) B2 first missile (with wing) C interceptor missile AR launching ramp R radar station

Claims (14)

1. A method for ascertaining the motion of at least one target object (A, B, B, B2) by using a mathematical method of estimation with the aid of a filter method relating to a model assumption for estimating the motion and/or orientation of the target object, characterised in that the filter method utilises a semi-martingale algorithm -i.e. an algorithm based on model equations that can be represented by smooth semi-martingales -for estimating the motion.

2. A method according to Claim 1, in which the motion and/or orientation is/are ascertained by modelling with the assistance of stochastic differential equations.

3. A method according to Claim 2, in which the motion and/or orientation is/are ascertained by modelling with the assistance of a discontinuous process, for example a Markov process or semi-Markov process.

4. A method according to any preceding claim, in which the motion and/or orientation is/are ascertained by a numerical method.

5. A method according to any preceding claim, in which a measurement is represented by a stochastic differential equation of the form dZ = F dt + at dW, where F and a represent an adapted stochastic process, z represents a measurement or a measured value, and W represents a Brownian motion.

6. A method according to Claim 3, in which use is made of indicator processes X of the form x = or X = additional condition of a discontinuous process Y that may be considered for modelling, of the form: Xt = Xo + J Ay ds + M, from 0 to t where X is an indicator process, ?. represents a transition rate, M represents a martingale, and Y represents such a àrkov process.

7. A method according to any of Claims 3 to 6, in which the state Y of the discontinuous process is assigned to a state in a state automaton in a computer of the target object (B1, B2) and/or to an index of types (A, B) or configuration of target objects.

8. A method according to any of Claims 3 to 7, in which the conditional expectation of the drift process F of the model is ascertained by using the indicator process, E (F I 3tz) = E E (llYti F I tZ) where is the filtration generated by the measurement z.

9. A method according to any preceding claim, in which differential equations, in particular the stochastic differential equations, describe the probability of the existence of a certain manoeuvre of the target object (A, B, B1, B2), in particular of a flying target object.

10. A method according to any preceding claim, in which the target object is a missile (B) which is followed by an interceptor missile (C) controlled by the method.

11. A tracking system with means for operating the method according to any preceding claim.

12. A system according to claim 11, including a measuring station, in particular a radar measuring station (R), a processing-and-control unit for processing measured motion data, in particular flight data, and with at least one interceptor missile (C) controlled by the processing-and-control unit for intercepting each target object (A, B, B1, B2)

13. An interceptor missile (C) with means for implementing the method according to any of claims 1 to 10.

14. A filter with a program for implementing the method according to any of claims 1 to 10.