European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanm¨ aki, T. Rossi, S. Korotov, E. O˜ nate, J. P´ eriaux, and D. Kn¨ orzer (eds.) Jyv¨ askyl¨ a, 24–28 July 2004
ON EFFICIENT IMPLEMENTATION OF THE KALMAN FILTER BANK FOR SCENARIO ANALYSIS Julia V. Tsyganova? and Alexey E. Kondratiev† ? Department of Mathematics and Mechanics, Ulyanovsk State University, 42 L. Tolstoy Str. 432970 Ulyanovsk, Russia e-mail:
[email protected] †
Department of Mathematics and Mechanics, Ulyanovsk State University, 42 L. Tolstoy Str. 432970 Ulyanovsk, Russia e-mail: mouse
[email protected]
Key words: Linear Stochastic Systems, Kalman Filter, System Identification, Adaptive Filtering, Fault Point Detection, Scenario Analysis. Abstract. The practical aspects of efficient implementation of the Kalman Filter Bank for scenario analysis are considered. Such implementation have to shorten running time, computational cost, and memory demands, required for computer model of the Kalman Filter Bank. The first requirement is met by concurrent computing. The second and third requirements are satisfied by substitution the Equivalent Kalman Filter Bank for the Standard Kalman Filter Bank. The paper contains theoretical and practical results, which can be used for scenario analysis and other problems (fault point detection, system identification, pattern recognition, and others). The work is partially supported by the Russian Ministry of Education (grants No. T0203.2-3427 and A03-3.16-409).
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Julia V. Tsyganova and Alexey E. Kondratiev
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INTRODUCTION
In many cases dynamical stochastic systems are considered as multi-mode systems. The switching from one mode to another is determined by some scenario. There exists the problem of efficient implementation of the detector of possible system modes switching. The basic requirement on such implementation is to shorten: • running time, • computational cost, and • memory demands. In the linear case the Standard Kalman Filter Bank (SKFB) can be considered as the model of linear stochastic multi-mode system [1], [2], [3]. However, the practical implementation of the Bank is a very costly algorithm in the case of systems with large dimensionality. The aim of this paper is to investigate the efficient implementation of SKFB computer model. It should satisfy the above mentioned requirements. The first requirement is met by concurrent computing. The practical computer experiments show the considerable running time shortening. The second and third requirements are satisfied by substitution the Equivalent Kalman Filter Bank (EKFB) [2] for SKFB. The equivalency of SKFB and EKFB is confirmed in this paper by MatLab and Borland Delphi numerical experiments. The mean residual errors here are less than 10−10 to 10−15 for the 5-dimensional system with 16 possible scenarios. The paper contains theoretical comparison of numbers of arithmetical operations needed for SKFB and EKFB. The obtained areas of efficiency are confirmed by the fact that the number of arithmetical operations for EKFB is 30% less than for SKFB in the case of systems with large dimensionality. Finally, parameters of linear transformation blocks in EKFB are sparse matrices. This fact allows us to apply sparse matrix packing algorithms for the shortening of memory demands. All obtained results confirm the efficiency of considered Kalman Filter Bank implementation for scenario analysis and other problems (for example, fault point detection and identification problems). 2
PROBLEM STATEMENT Let the discrete-time system be characterized by the following equations: x(ti ) = Φx(ti−1 ) + Ψu(ti−1 ) + Γw(ti−1 ),
(1)
z(ti ) = Hx(ti ) + v(ti ),
(2)
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Julia V. Tsyganova and Alexey E. Kondratiev
where x(ti ) is the n-dimensional state vector, z(ti ) is the m-dimensional system output, u(ti ) is the control input, and {w(t0 ), w(t1 ), . . .} and {v(t1 ), v(t2 ), . . .} are mutually independent zero-mean Gaussian sequences of independent vectors. Without loss of generality, their covariances Q and R are assumed to be reduced to identity matrices: Q = I and R = I. This can be easily done by normalizing input noise in (1) and measurements in (2). The sequences are considered independent of Gaussian initial x(t0 ) with mean x¯(t0 ) and P (t0 ). Let us assume that the system (1), (2) is time invariant, that allow to consider using a steady-state constant-gain Kalman filter model, which is denoted with the subscript k. xk (ti ) = Φk xk (ti−1 ) + Ψk u(ti−1 ) + Γk wk (ti−1 ), zk (ti ) = Hk xk (ti ) + vk (ti ),
(3) (4)
where xk is the Kalman filter model state vector, Φk is the Kalman filter model state transition matrix, Ψk is the Kalman filter model control input matrix, u is the system input vector, Γk is the Kalman filter model noise input matrix, wk is an additive white discrete-time dynamics noise input used in the Kalman filter model, with zero mean and identity covariance, zk is the Kalman filter model measurement vector, Hk is the Kalman filter model output matrix, vk is an additive white measurement noise input that is used in the Kalman filter model. It is assumed to be independent of wk , and zero mean with identity covariance. The Kalman filter algorithm uses this model to define time propagation and measurement update equations of the Kalman filter state estimates and the Kalman filter state estimates covariance matrix. Thus we have Time propagation xˆk (t− ˆk (t+ i ) = Φk x i−1 ) + Ψk u(ti−1 ), Vector Kk (ti ) Ck (ti ) νk (ti )
+ T T Pk (t− i ) = Φk Pk (ti−1 )Φk + Γk Γk .
measurement update T −1 = Pk (t− i )Hk Ck (ti ) , − T = Hk Pk (ti )Hk + Im , = z(ti ) − Hk (ti )ˆ xk (t− i ),
(5) − Pk (t+ i ) = (In − Kk (ti ))Hk Pk (ti ), xˆk (t+ ˆk (t− i ) = x i ) + Kk νk (ti ),
where νk (ti ) is the Kalman filter residual vector. The following sequences Nk = {νk (t1 ), νk (t2 ), . . . , νk (ti )}
(6)
consists of mutually independent entries νk (tτ ) = zk (tτ ) − Hk xˆk (t− τ ), τ = 1, 2, . . . , t with zero mean and the Kalman filter-computed residual covariace matrix Ck , subject to the corresponding Kalman filter model. The steady state Kalman filter gain Kk and the steady state Kalman filter residual covariance matrix Ck can be precomputed using (5), 3
Julia V. Tsyganova and Alexey E. Kondratiev
and therefore do not need to be computed in real time. The steady state Kalman filter equations become xˆk (t− ˆk (t+ i ) = Φk x i−1 ) + Ψk u(ti−1 ), + − xˆk (ti ) = xˆk (ti ) + Kk νk (ti ),
(7) (8)
The set of Kalman filter models forms the Standard Kalman Filter Bank (SKFB). Further, let us follow [2] to apply expressions for the equivalent residual of a Kalman filter using the residual of another Kalman filter and the known differences between the two Kalman filter models. The subscripts j and k denotes the two different Kalman filter models. We consider model differences in the state transition matrix Φ, the input matrix Ψ, the noise input matrix Γ, and the output matrix H. 3
THE STANDARD KALMAN FILTER BANK AND ITS EFFICIENT IMPLEMENTATION: THE EQUIVALENT KALMAN FILTER BANK Let us use the following notation: ∆Ψkj ∆Hkj ∆Φkj ∆Γkj ∆Kkj
= = = = =
Ψk − Ψj → Ψj = Ψk − ∆Ψkj , Hk − Hj → Hj = Hk − ∆Hkj , Φk − Φj → Φj = Φk − ∆Φkj , Γk − Γj → Γj = Γk − ∆Γkj , Kk − Kj → Kj = Kk − ∆Kkj .
The difference in state estimate errors is defined in the following way: + + ∆²jk (t+ ˆj (t+ ˆk (t+ i ) = ²j (ti ) − ²k (ti ) = [xT (ti ) − x i )] − [xT (ti ) − x i )] = + + + + + xˆk (ti ) − xj (ti ) → xˆj (ti ) = xˆk (ti ) − ∆²jk (ti ).
Let us define the residual of filter number j: + νj (ti ) = z(ti ) − Hj Φj xˆk (t+ i−1 ) − Hj Ψj u(ti−1 ) + Hj Φj ∆²jk (ti−1 ).
This equation can be rewritten in the following way: xk (t+ νj (ti ) = νk (ti ) + Hj Φj ∆²jk (t+ i−1 ) i−1 ) + [Hj ∆Φkj + ∆Hkj Φk ]ˆ +[Hj ∆Ψkj + ∆Hkj Ψk ]u(ti−1 ). Let us write a recursive expression for the difference in state estimates between filter k + and filter j (∆²jk (t+ i )). It allows to compute ∆²jk (ti ) at each time sample and then the residual for the filter j. Thus, we have + xk (t+ ∆²jk (t+ i−1 ) i ) = (I − Kj Hj )Φj ∆²jk (ti−1 ) + [(I − Kk Hk )Φk − (I − Kj Hj )Φj ]ˆ +[(I − Kk Hk )Ψk − (I − Kj Hj )Ψj ]u(ti−1 ) + ∆Kkj z(ti ).
Now consider four cases: 4
Julia V. Tsyganova and Alexey E. Kondratiev
• differences in the input matrix Ψ, • differences in the output matrix H, • differences in the state transition matrix Φ, • differences in the noise input matrix Γ. Case 1. Ψk 6= Ψj , Hk = Hj , Φk = Φj , Γk = Γj , Kk = Kj . + ∆²jk (t+ i ) = (I − Kj Hj )Φj ∆²jk (ti−1 ) + (I − Kj Hj )∆Ψkj u(ti−1 ), νj (ti ) = νk (ti ) + Hj Φj ∆²jk (t+ i−1 ) + Hj ∆Ψkj u(ti−1 ).
Case 2. Ψk = Ψj , Hk 6= Hj , Φk = Φj , Γk = Γj , Kk 6= Kj . + ∆²jk (t+ xk (t− i ) = (I − Kj Hj )Φj ∆²jk (ti−1 ) + ∆Kkj z(ti ) + (Kj Hj − Kk Hk )ˆ i ), + − νj (ti ) = νk (ti ) + Hj Φj ∆²jk (ti−1 ) + ∆Hkj xˆk (ti ).
Case 3. Ψk = Ψj , Hk = Hj , Φk 6= Φj , Γk = Γj , Kk 6= Kj . + ∆²jk (t+ ˆk (t+ i ) = (I − Kj Hj )[Φj ∆²jk (ti−1 ) + Φk x i−1 )] + ∆Kkj νk (ti ), + νj (ti ) = νk (ti ) + Hj Φj ∆²jk (ti−1 ) + Hj ∆Φkj xˆk (t+ i−1 ).
Case 4. Ψk 6= Ψj , Hk = Hj , Φk = Φj , Γk 6= Γj , Kk 6= Kj . + ∆²jk (t+ i ) = (I − Kj Hj )Φj ∆²jk (ti−1 ) + ∆Kkj z(ti ), νj (ti ) = νk (ti ) + Hj Φj ∆²jk (t+ i−1 ).
So, the Equivalent Kalman Filter Bank (EKFB) consists of the source Kalman filter k and the set of linear transforms for producing the equivalent of the residual from another Kalman filters j (j = 1, . . . , M ). 4
COMPARISON OF THE OPERATION COUNT. AREAS OF EFFICIENT IMPLEMENTATION OF EKFB
Let us compare the operation count (number of multiplications and additions) that are needed to implement the Standard Kalman Filter Bank with the count needed to implement the Equivalent Kalman Filter Bank. Consider the following values: ns — dimension of state vector x, ni — dimension of control input u, nm — dimension of measurement vector z. Let us denote the number of multiplications as < ∗ > and the number of additions as < + >. SKFB. < ∗ > = ns (ns + ni + 2nm ), < + > = ns (ns + ni + 2nm − 1). 5
Julia V. Tsyganova and Alexey E. Kondratiev
EKFB, ∆Ψ 6= 0. Let nΨ — the number of nonzero columns of matrix ∆Ψ (1 ≤ nΨ ≤ ni ). Then < ∗ > = nΨ (ns + nm ) + ns (ns + nm ), < + > = nΨ (ns + nm ) + ns (ns + nm − 2).
(9)
Let us define the area of efficient implementation of the EKFB through examining the set of inequalities which describes the gain in the operation count. Choosing from (9) the stronger inequality, we have: nΨ ≤ ns (ni + nm )/(ns + nm ).
(10)
Let’s notice, that this inequality is meaningful always, since ns and nm are positive natural numbers. Taking into account the possible values of nΨ , we can draw a plot of the desired area (see figure 1). In this case efficiency is particularly obvious for systems with (ni = nH (2ns − nK − 1) + nK (ns + nK − 1) + ns (ns + nm ). There are two possible alternatives in this case: 1) If nH ≥ ns − nK , then nH ≤ (ns (ni + nm ) − nK (ns + nK ))/(2ns − nK );
(11)
2) If nH < ns − nK , then nH ≤ (ns (ni + nm − 1) − nK (ns + nK − 1))/(2ns − nK − 1).
(12)
Let us consider these alternatives in order: 1) The inequality (11) is meaningful always, because nk ≤ ns , that is 2ns − nK 6= 0. Taking into account restrictions on nH , it is possible to draw a plot of desired area (see figure 2). In this case efficiency grows at increase ni . 2) The denominator of fraction in the right part of an inequality (12) can be equal zero in case nK = 2ns − 1, i.e. nK − ns = ns − 1, but since nK − ns ≤ 0, and ns − 1 ≤ 0. It is possible only at ns = 1. But then ns = 1, nK = 1 and 1 ≤ 0, since nH ≤ ns − nK . We come to contradiction. Hence, a denominator of fraction is not equal to zero at any allowable values of parameters, therefore the inequality (12) is meaningful always. Now we can draw a plot of desired area subject to restrictions on values nH (see figures 3). In this case efficiency quickly decreases at increase of system dimensions. EKFB, ∆Φ 6= 0 (→ ∆K 6= 0). < ∗ > = ns (2ns + 3nm ), < + > = ns (2ns + 3nm − 1) − 2nm . In this case the property of matrix ∆Φ to be sparse does not give advantages since it has usually one nonzero line and one nonzero column. When ∆Φ is multiplied with another matrix, the product is no longer a sparse matrix. The condition of efficiency in this case looks as follows: ni ≥ ns + nm . It is equivalent to the previous case without taking a sparse structure into account. The plot of the area of efficiency is shown on figure 4. 7
Julia V. Tsyganova and Alexey E. Kondratiev
n =(n (n +n )−n (n +n ))/(2n −n ) H
s
i
m
K
s
K
s
K
nH=nm nH=ns−nK 1500
nΨ
1000
500
0 1000
−500 0
800 200
600 400
400
600 Area of efficiency
200
800 1000
0
ni
n
s
Figure 2: Area of efficiency in Case 2.1
The computational savings are obvious for systems with (ni >> ns ) & (ni >> nm ). EKFB, ∆Γ 6= 0 (→ ∆K 6= 0). < ∗ > = ns (ns + 2nm ), < + > = ns (ns + 2nm − 1) − nm . Here the sparse structure of matrix ∆Γ does not give advantages too, however computational savings are evident based on inequalities ni ≥ 0 and ni ns + nm ≥ 0. All obtained results allow at comparison EKFB with SKFB for the systems of large dimensionality to speak about considerable computational savings during EKFB implementation. 5
NUMERICAL EXAMPLE
As a practical numerical example consider the following system taken from the inertial navigation [4]:
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Julia V. Tsyganova and Alexey E. Kondratiev
1200
nH=n+m−nK
NH=(ns(nm+ni−1)−nk(ns+nk−1))/(2ns−nk−1) 1000 800 600
n =n
nΨ
H
m
400 200 0 −200 1000 800 600 400 200
Area of efficiency
0
600
800
1000
400
200
0
ni
n
s
Figure 3: Area of efficiency in Case 2.2 xt+1 =
0.75 0.09 0.0 0.0 0.0 "
zt =
1−e 0
−1.74 0.91 0.0 0.0 0.0 0 1−g
−0.3 −0.0005 0.95 0.0 0.0 0 0
0.0 0.0 0.0 0.55 0.0 0 1−h
−0.15 −0.008 0.0 0.0 0.905 1−f 0
xt +
0.0 0.0 24.64 0.0 0.0
0.0 0.0 0.0 0.835 0.0
0.0 0.0 0.0 0.0 1.83
wt
#
xt + vt ,
e, f, g, h = {0, 1}
{wt } and {vt } are zero-mean white Gaussian sequences with covariances Qt = I3 , and Rt = I2 (In is the n-dimensional identity matrix). These equations describe the damped Shuler loop driven by the exponentially correlated 3-dimensional noise wt [4]. The mode of the stochastic system, or system scenario, is determined by matrix H. Thus, 24 = 16 types of system scenario are possible. Table 1 shows experimental data, which confirm the equivalence of residuals obtained from the Standard Kalman Filter Bank and the Equivalent Kalman Filter Bank. This experiment was conducted for the chosen example and 1000 quantums of time. The source filter corresponds to scenario S0 . The mean residual errors here are less than 10−14 for 9
Julia V. Tsyganova and Alexey E. Kondratiev
2000
ni=ns+nm
1500
ni
1000
500
0 1000 800 600 400
0 200
Area of efficiency
400
200
600 0
800 1000
nm
ns
Figure 4: Area of efficiency in Case 3
the 5-dimensional system with 16 possible scenarios. 6
SCENARIO ANALYSIS
Suppose that there is a control interval [t0 , tN ] on which the system behavior is subject to some scenario S0 . It means that system parameters are Φ0 , Ψ0 , Γ0 , and H0 . Further, consider the time moment ts ∈ [t0 , tN ] at which the system behavior can be changed, i. e. the nominal scenario S0 is changed to one of possible scenarios Sk , k = 1 . . . , M . Each scenario Sk is considered as the set of the system parameters {Φk , Ψk , Γk , and Hk }. Our goal is to choose the right scenario of system behavior after the moment ts . One of the possible solving method is the Wald sequential probability ratio test [5], which requires the computation of likelihood ratio function λp (ti ) at each moment ti . Thus, we have λp (ti ) = λp (ti−1 ) + µp (ti ), 2µp (ti ) = ln det C0 (ti ) − ln det Cp (ti ) + (s0 (ti ) − sp (ti )), 10
Julia V. Tsyganova and Alexey E. Kondratiev
Type of Scenario S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 256 ensemble average
Difference of Residuals ν1 0 4.64 · 10−14 4.79 · 10−14 4.79 · 10−14 4.31 · 10−14 4.37 · 10−14 4.32 · 10−14 4.32 · 10−14 1.45 · 10−14 1.37 · 10−14 1.33 · 10−14 1.33 · 10−14 3.12 · 10−15 3.12 · 10−15 3.12 · 10−15 3.12 · 10−15 4.59 · 10−14
Difference of Residuals ν2 0 1.91 · 10−14 6.02 · 10−15 5.12 · 10−16 2.70 · 10−14 2.65 · 10−14 6.02 · 10−15 5.12 · 10−16 2.18 · 10−14 2.39 · 10−14 6.02 · 10−15 5.12 · 10−16 2.21 · 10−14 2.42 · 10−14 6.02 · 10−15 5.12 · 10−16 1.78 · 10−14
Table 1: Average Difference between SKFB and EKFB residuals.
s0 (ti ) = ν0T (ti )C0−1 (ti )ν0 (ti ), sp (ti ) = νpT (ti )Cp−1 (ti )νp (ti ), where ν0 (ti ) is a system residual for scenario S0 , νp (ti ) is a system residual for scenario Sp (1 ≤ p ≤ M ), and matrices C0 (ti ) and Cp (ti ) have to be precomputed up to steady state values C0 and Cp . The value of λp (ti ) is then tested against two threshold levels A and B (where A > B). If a’priori probabilities of different types of scenario are equal, then initial value of λp (t0 ) = 0. Decision rule for choosing desired scenario is as follows [6]:
1. 2.
If ∀p λp (ti ) ≤ B, then system scenario S0 is chosen. If ∃! q ∀p 6= q (λq (ti ) ≥ A)&(λp (ti ) < A), then system scenario Sq is chosen. 3. If ∀p B < λp (ti ) ≤ A, the process is repeated for ti+1 . 4. If ∃p, q (p 6= q): (λp (ti ) ≥ A)&(λq (ti ) ≥ A), then the process is terminated with the choice of scenario Sl , for which the value of likelihood function is maximum, that is λl (ti ) = max{λp (ti ), λq (ti )}. 11
(13)
Julia V. Tsyganova and Alexey E. Kondratiev
The probability of the right choice of scenario varies within the limits of 60%-95%, that directly depends on parameters of a concrete system. 7
CONCURRENT COMPUTING
To satisfy the first requirement mentioned in Section 1, the concurrent computing was implemented for SKFB and EKFB computer models. A shortening of running time within the framework of the given experiment to estimate difficultly valid its transiency, however on 10000 passes of corresponding testing intervals the following parameters of efficiency of EKFB have been obtained in comparison with SKFB (The testing was carried out on CPU Intel Pentium II Original, 64 Mb RAM, OS Microsoft Windows 98 OSR2, the source code was compiled in Borland Delphi 5 Enterprize Edition Version 5.0 Build 5.62.): • EKFB (concurrent programming did not applied) ≈ 11%; • EKFB (concurrent programming was applied, thread priority is tpNormal) ≈ 68.8%; • EKFB (concurrent programming was applied, thread priority is tpHigher) ≈ 66.2%; • EKFB (concurrent programming was applied, thread priority is tpHighest) ≈ 65.03%; The given proposal is probably effective for problems with large, growing in time volumes of complex calculations. 8
IMPLEMENTATION OF SPARSE MATRICES
Distinctions between models in the bank of filters are often represented by sparse matrices, therefore it is reasonable to apply packing algorithms for their storage and processing. The next way of packing has been applied for realization of experiments (an initial code in Borland Delphi 5): TFloat = Extended; TMatrixBaseClass = class end; TInfoRec = record i, j: Integer; Data: TFloat; end; TPackedMatrixArray = array of TInfoRec; TPackedMatrix = class(TMatrixBaseClass) private Fi: Integer; Fj: Integer; 12
Julia V. Tsyganova and Alexey E. Kondratiev
FMatrix: TPackedMatrixArray; function GetElement(i, j: Integer): TFloat; procedure SetElement(i, j: Integer; Value: TFloat); public property I: Integer read Fi; property J: Integer read Fj; property Element[i, j: Integer]: TFloat read GetElement write SetElement; default; constructor Create(i, j: Integer); virtual; destructor Free; end; The packed matrix is submitted as an array (TPackedMatrixArray) of open length which elements are information records (TInfoRec). Elements of a matrix are numbered in the lines from the first column to the last. Such implementation gives serious economy of operative memory for a data storage and gives an opportunity rather easy in sense of amount of operations of search of a necessary element of the packed matrix. Here it is necessary to mention accuracy of storage of matrix elements. Insufficient accuracy can lead to the situation in which at definition of distinctions between system models required sparse matrices will turn out zero, that certainly will lead to occurrence of incorrect results. 9
CONCLUSIONS
The purpose of the given work was theoretical research, development of idea [2], [3] and computer implementation of efficient bank of Kalman filters, experimental confirmation of its equivalence to the Standard Kalman Filter Bank, and comparison of operation count. The possibility of using the Equivalent Kalman Filter Bank in scenario analysis with application of concurrent programming and Win32 threads technology was also considered. There was obtained the following theoretical results: the idea of EKFB was developed, theoretical number of arithmetical operations for EKFB in comparison this required number of operations for SKFB was obtained in the form of inequalities and graphically as the areas of efficiency. The practical results are the following: the efficient (equivalent) bank of Kalman filters with application of concurrent computing and packing algorithms of sparse matrices was implemented, its equivalence to SKFB was experimentally proved. Thus, considered implementation of the Kalman Filter Bank is really efficient, because it meets the three basic requirements: shortening of running time, computational cost, and memory demands. This implementation can be used for scenario analysis, fault point detection, system identification, pattern recognition, and other problems.
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REFERENCES [1] Newbold, P., M. and Ho Yu-Chi. Detection of Changes in the Characteristics of a Gauss-Markov Process. IEEE Trans. on Aerosp. and Electron. Systems, Vol. AES4(5), 707–718, 1968. [2] Hanlon, P., D., Maybeck, P., S. Interrelationship of Single-Filter and Multiple-Model Adaptive Algorithms. IEEE on Aerospace and Electronic Systems Vol. 34(3), 934– 946, 1998. [3] Hanlon, P., D., Maybeck, P., S. Equivalent Kalman Filter Bank Structure for Multiple Model Adaptive Estimation (MMAE) and Generalized Likelihood Ratio (GLR) Failure Detection. Proc. of the 36th Conference on Decision & Control: San Diego California USA, 5, 4312–4317, 1997. [4] Stratton, A. Combination of Inertial Navigation and Radio Systems. In Borisov, N., I. (ed.): Problems of Inertial Navigation, Mashinostroienie, 1961. [In Russian] [5] Wald, A. Sequential Analysis. New York, Wiley, 1947. [6] Semoushin, I., V. The Quickest in the Mean Manoeuvre Detection with the Guaranteed Probability Error (Methods). Shipbuilding: Computing Techniques, 26, 3–7, 1990 [In Russian]
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