Set-valued linear dynamical system state estimation with anomalous measurement errors

Abstract

The article deals with set-valued dynamical system state estimation problem when there is no statistical information on initial state, disturbances and noises but sets of their possible values are available. In practice some measurements can be performed with anomalous errors (failures, fallings out). That is at some time instants the measurement errors were realized outside of the given set of possible values. The article considers the level of measurement error falling out that can be reliably recognized using set-valued state estimation.

Keywords

• linear dynamical systems
• set-valued estimation
• anomalous measurement errors

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( 1 ) ( 2 ) ( 3 )

Introduction

Nowadays set-valued dynamical system state vector estimation under condition of statistical uncertainty is being developed in control and identi¯cation theory. A lot of publications are devoted to this area [2, 4, 8, 10, 11, 12, 13, 14, 18]. The main subject of set-valued estimation is a feasible set, that is a set of all possible dynamical system states at a time instant.

The processes in the control system are described with equations: xk+1 = Axk + Buk + ¡wk; yk+1 = Gxk+1 + Hvk+1; k = 0; 1; : : : N: where xk 2 Rnx , uk 2 Rnu , wk 2 Rnw , yk 2 Rny ,vk 2 Rnv denote state, control, disturbance, measurement, noise vectors at time instant k correspondingly; A, B, ¡, G, H are known matrices. The system ( 1 ) is supposed to be controllable and observable.

Initial state x0, disturbances wk and noises vk are unknown but they can take some value from given convex sets: x0 2 X0; wk 2 W;

vk 2 V:

Xk+1=k = AX¹k + ¡W: Copyright °c by the paper's authors. Copying permitted for private and academic purposes.

X¹k+1 = Xk+1=k \ X[yk+1]: ( 4 ) ( 5 ) Operations in ( 3 )-( 5 ) are set operations, that is Minkowski sum, set intersection, linear set transformation. According to equations ( 3 )-( 5 ) the shape and size of feasible sets depend not only on the sets X¹0, W and V but also on the values of disturbance wk and measurement error vk. The feasible set is calculated as intersection of reachable set Xk+1=k and measurement consistent set X[yk+1], the location of these sets depends on the values of realized disturbance wk and measurement error vk. The smaller feasible set is constructed the more accurate set-valued estimate we have. Therefore if a feasible set is a point it means that the exact value of the dynamical system state is calculated.

Feasible set shape and structure can be complex that is it can contain a lot of vertices and facets. But when the system state vector dimension increases troubles in performing set operations in real-time occur. Then feasible set outer approximations with some canonical forms, like ellipsoids [3, 8, 9, 10], parallelotopes [7] and zonotopes [18] can be applied.

In practice some measurements can be performed with anomalous errors (failures, fallings out) [1, 5, 11, 15]. Anomalous measurements can appear because of abrupt violation of data measurement equipment performance condition. Anomalous measurements can lower e±ciency of classical data processing algorithms [5, 15]. That is why it is important to timely recognize anomalous measurements and exclude them from the following processing. There are heuristic approaches to anomalous measurement errors ¯ltration like application of appropriate threshold criteria for selection and elimination of failures, least absolute deviation method and others. However these methods are e®ective only when the anomalous measurements signi¯cantly di®er from other measurements. Besides these methods do not allow to get borders of parameter estimation errors. 2

Anomalous measurements

Let us consider that at a time k the measurement error vk was outside the given set of possible values vk 2= V . Two cases from feasible set construction algorithm ( 3 )-( 5 ) depending on the realized value of the measurement error vk are possible: ² the feasible set is empty; ² the feasible set is not empty.

If the feasible set X¹k is empty at a time k (¯g.1), it means that at this time k or earlier the falling of vk out of the set V has happened, i.e. vk 2= V . In this case estimation algorithm can be restarted with new initial data, for example the borders of set V can be extended. Another way is to exclude this measurement from data processing and do not perform estimation at this time k. If the feasible set X¹k is not empty (¯g.2) the constructed set may not contain the real system state xk 2= X¹k. But in this case it is impossible to recognize the failure. Therefore the guaranteed feature of anomalous measurements at set-valued estimation is the empty feasible set. For comparison when Kalman ¯lter is used for system state estimation there are no guaranteed features of anomalous measurements because the system state estimation is probabilistic [6, 16, 17].

Let us consider what level of measurement error vk falling out can be reliably recognized using set-valued state estimation that is at what measurement error vk falling out the feasible set is empty. The feasible set X¹k is empty if the corresponding reachable set Xk=k¡1 and measurement consistent set X[yk] do not intersect (¯g.3). These sets certainly do not intersect if their projections on coordinate axes do not intersect . Let us consider projections of the sets Xk=k¡1 and X[yk] on coordinate axes, on which the measurements are performed.

Let us consider the outermost case when the real system state is on the border of reachable set Xk=k¡1 projection (¯g.3). If the measurement error was realized on the border vmaxi of the set V on the axis x(i) the measurement yk would get to the point yk0. Let us suppose that the falling out of measurement error happened and the measurement is in the point yk.

Then the value of measurement error falling out is equal to the distanse between points yk0 and yk: yk0 = xk + vmaxi ; yk = xk + vk: ±vx(i) = jvk ¡ vmaxi j = jyk0 ¡ ykj: Then the minimum falling out ±vx(i) when the projections' intersection is empty is equal to ±vx(i) = jvk ¡ vmaxi j = (d(Xk=k¡1)x(i) ¡ r(X[yk0])x(i)) + r(X[yk])x(i) = d(Xk=k¡1)x(i); ( 8 ) where d(²)x(i) denotes the projection of set diameter to the axis x(i), r(²)x(i) denotes the projection of set radius to the axis x(i).

Let us estimate the diameter of the set Xk=k¡1. The set Xk=k¡1 is constructed as Minkowski sum of sets AX¹k¡1 and ¡W . The set-valued estimate X¹k¡1 is not worse than the set V on measured coordinates. Then d(Xk=k¡1)x(i) = d(AX¹k¡1)x(i) + d(¡W )x(i) · d(AV )x(i) + d(¡W )x(i);

Therefore if the measurement error is outside of the given set V and the following condition is ful¯lled for any of the coordinates x(i)

±vx(i) ¸ d(AV )x(i) + d(¡W )x(i); then the measurement error falling out will be reliably recognized. ( 9 ) ( 10 ) Let us consider the model ( 1 ) with the following matrices:

A = ¡1 0 1 01:45 ¢ 10¡41

B2:28 ¢ 10¡2 10 CCA vk · B@1:45 ¢ 10¡4CC :

A Let us compute from ( 10 ) the measurement error falling out level that can be recognized for this model.

For the coordinate x( 1 ) the falling out level is: d(AV )x( 1 ) = 0:0024, d(¡W )x( 1 ) = 0:00036, ±vx( 1 ) ¸ 0:0024 + 0:00036 = 0:00276.

For the coordinate x( 2 ) the falling out level is: d(AV )x( 2 ) = 0:039170, d(¡W )x( 2 ) = 0:013917, ±vx( 2 ) ¸ 0:039170 + 0:013917 = 0:053087.

Notice that the measurement error set is much smaller that the disturbance set at the ¯rst coordinate. That is why the measurement error falling out can be recognized if the measurement error vk value is 19 times greater than the largest possible value of the set V at the ¯rst coordinate. For the second coordinate if the falling out is 2.3 times greater than the largest possible value of the set V at the second coordinate the anomalous measurement error can be recognized. However this estimate is upper-bound. In practice there are realizations when the smaller falling outs can be recognized. Let us consider a disturbance realization (¯g. 4). We considered some measurement errors realizations with di®erent falling outs at the steps k = 10 and k = 20. At the ¯rst realization the value of vk( 2 ) is equal to 1:5vmax2 = 0:0342 at steps k = 10 and k = 20. At the second realization the values of vk( 1 ) is equal to 3vmax1 = 4:35 ¢ 10¡4 at steps k = 10 and k = 20. The ¯gures 5-7 show the measurement error realization with falling outs which were recognized although the falling out value was smaller than the estimate ±vx( 2 ). For these realizations the feasible set is empty at the time k = 10. For the realizations from ¯g.8, 11 the measurement error falling outs were not recognized. The feasible sets at the times k = 10 and k = 20 are not empty but they do not contain the real system state xk (¯g.8, 11). ) −2 0 x 10−3 v,k1 2 4 0 −20 ) −2 The set-valued dynamical system state estimation with anomalous measurements was considered when the measurement error is realized outside the set of possible values. The reliable feature of anomalous measurement errors is empty feasible set. The anomalous measurement is reliably recognized if the falling out value ±ºk at a time k is greater than the sum of projections of the diameters of the sets AV and ¡W on any of coordinates. However in practice there are realizations when smaller falling outs can be recognized. When the anomalous measurement is recognized it can be excluded from the data processing or the estimation procedure can be restarted with new initial data.

Acknowledgements References

The work was supported by Act 211 Government of the Russian Federation, contract 02.A03.21.0011.

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