SYNTHESIS OF THE MANEUVERING TARGET COORDINATE DETERMINATION ALGORITHM ON THE BASIS OF APPLICATION OF THE INTERACTIVE MULTI-MODEL ADAPTIVE FILTERING TECHNIQUE Текст научной статьи по специальности «Медицинские технологии»

ТЕХНИЧЕСКИЕ НАУКИ

Nguyen Van Bang, Hoang Van Ngoi, Nguyen Duy Tien

Air Defence - Air Force Academy, Ha Noi, Viet Nam Nguyen Ngoc Tuan

Military Technical Academy, Ha Noi, Viet Nam

SYNTHESIS OF THE MANEUVERING TARGET COORDINATE DETERMINATION ALGORITHM ON THE BASIS OF APPLICATION OF THE INTERACTIVE MULTI-MODEL ADAPTIVE

FILTERING TECHNIQUE

Abstract. The article presents the method of synthesizing the algorithm to determine the maneuvering target coordinates, building the appropriate state space model to apply the interactive multi-model filtering techniques, aimed at improving the evaluation accuracy of the target's motion parameters. Coordinate determination algorithm including accelerated evaluation filter and interactive multi-model adaptive filter band consisting of N filters; each filter in the filter band performs evaluation of the target's motion parameter with each acceleration value around the evaluated acceleration value. The target coordinate determination system is simulated through target fake creation, algorithm implementation and error evaluation.

Keywords: Target, Maneuverability, Interactive multi-model, Kalman filter, Adaptive filtration.

1. INTRODUCTION

There are many methods of building a kinetic model of the target, with a suitable model for a particular movement [3, 6]. In the general case it is not possible to choose a suitable model during the evaluation process as well as represent different types of maneuvering of the target. Therefore, using the Kalman filter algorithm with a specific model to synthesize the target coordinate determination system leads to an increase in the follow closely error when in fact the moving target does not match the selected model [5].

There are many adaptive filtering techniques that can be used in determining maneuvering target coordinates. Single-model adaptive filtration techniques often have a simple filter structure, however the error at the time of model transfer and the error reduction time is large [2]. In addition, defining thresholds to switch between models is often imposing. With the multi-model filtering technique, although more complex in structure, it avoids the selection of the maneuvering moment detection threshold as well as the smaller evaluation error compared to the single-model filtering technique [1, 3].

Kalman filters are widely used to evaluate a target's position, speed, and acceleration. However, when the maneuvering target the evaluation accuracy will decrease. To fix this, adaptive techniques are used. Adaptive techniques such as input evaluation, state space expansion [4, 5]... although the algorithm is simple, it is difficult to determine maneuvering moment detection threshold. Multi-model adaptive filtering techniques fix this disadvantage. Therefore, the article presents the application of interactive multi-model adaptive filtering technique to synthesize the maneuvering target coordinates determination algorithm. Multi-model adaptive filter is used to

improve the evaluation accuracy of the target's maneuvering parameters from the acceleration parameters provided by the Kalman filter.

2. INTERACTIVE MULTI-MODEL ALGORITHM Considering the linear step-jumping system represented by the model [3, 4]:

x(k) = F[M(k)]x(k - 1) + w[k - 1,M(k)]

(1)

(2)

z(k) = H[M(k)]x(k) + v[k,M(k)]

Inside, M (k) - model "at time k", it acts during the period of sampling cycle, ends at k. Hypothesis that, at time k, the system belongs to one of the N models:

r 1N

M(k) e \Mj] i; x - the state vector at time k; z -

measured value vector; F - state transition matrix; H -measurement matrix; v,w - The central Gaussian white noisy is not correlated

With the interactive multi-model evaluation algorithm, at time k, the state evaluation is calculated on the basis of the state combination of N filters, with each filter using the first condition mixed from the models. The algorithm takes the following form:

- Assign the first conditions:

+ Pij, i,j = 1,2,...N: Probability of model transfer, ie probability, at (k - 1) the model is in mode i and that k is in mode j. This probability is assumed to be invariant over time.

+ Hj(0): Model probability at the time of initialization.

- Calculate the mixing probability, that is, the probability at the time (k - 1) appears the ith model and the time k appears the jth model:

1

ßilj(k-l) = -Pij^i(k-1), i,j = 1,...,N; cj = Yiri=iPijßi(k - 1), j = 1,...,N

c i

- Calculate the first condition for the ] filter:

+ Initial state: x0'(k - 1) = Zri=1 xl(k - 1)/iiU(k - 1);j = 1,...,N

(4)

+ Correlation of initial evaluation error:

P0 j(k -1) = Zri=i^i\j(k - 1){Pi(k -1) + [xl(k -1)- x°j(k - 1)][xl(k -1)- x°i(k - 1)]7}

] = 1.....N (5)

- Perform filtering algorithm for jth model with - Calculate the rational function for the jth filter:

initial conditions have been mixed.

Aj(k) = N [z(k);zj[klk - 1;x0j(k - Ilk - 1)],Sj[k,P0j (k - Ilk - 1)]]

It mean Aj(k) = N[ej(k);0;Sj(k)], inside, N[ ] - Gaussian distribution density function, ej -tracking error of the jth filter, Sj - correlation of the ej tracking error.

- Update jth model probabilities:

Vj(k) = ~Aj(k)Cj; c = Ylrj=iAj(k)c^

(6)

(7)

- Combination of state evaluation and error correlation matrix:

X(k) = Tlrj=iXj(k)ßj(k); P(k) = Tr=1ßj(k){p}(k) + [&(k) - x(k)][x'(k) - x(k)]T}

(8)

0 1 0 0

xa(t) = 0 0 1 xa(t) + 0

0 0 -a. 1

3. STATE SPACE MODEL OF THE TARGET

To evaluate position, speed and acceleration, the Singer model is often used [1, 2, 5]:

wa(t) (9)

Inside, xa = [(p <p <p]T - location coordinates, speed and acceleration of the target; wa(t) - the central Gaussian white noise with intensity a2; a -maneuvering frequency.

The time discrete model of model (9) has the form:

Xa(k + 1) = FaXa(k) + Wa(k) (10)

Inside,

Fn =

1 T ßi 0 1 ß2 L0 0 ß3\

with

pt = (aT-1 + e-aT)/a2; p2 = (1- e-aT)/a; = e aT; T - sampling interval. The process noise correlation matrix is determined by:

Qx

00 00

0 0 ol

;oi = o2(1 - e )

Seeing that if maneuvering frequency a and fixed maneuvering intensity a2 are selected, the follow closely accuracy of the coordinate components will decrease as the target performs different maneuvers.

Assuming known target acceleration, then the target model can be described in terms [3]:

xv(k + 1) = Fvxv(k) + G(u(k) + wv(k)) (11)

Inside, xv(k) = [(p(k) <p(k)]T; u(k) = <p(k);

Gwv(k) - The process noise correlation matrix is determined by:

00

Qx ~ [Q T2\ aa, with a2 intensity of noise

Wv(k)

Seeing that, when using model (10) to perform Kalman filtering algorithm, we will get an evaluation the target's state, including the acceleration component Pp(k). The evaluation accuracy is highly dependent on the actual movement of the target.

The main idea here is that, the target acceleration used in model (11), u(k) is determined to be in a range around the value of the acceleration Pp(k) rated by using model (10). That is, choose the specified acceleration range eN-1 and form a set of values for the acceleration

{<p(k) <p(k) ±£i ... <p(k) ± £(N-i)/2}, in which each acceleration value is separated by a distance 1 = e/N — 1. With the assumption, the observation channel can only measure the position component, we receive a state space model for the combination of the maneuver target tracking system as follows:

- The model uses the evaluation of coordinate parameters:

xlv(k + 1) = Fvxv(k) + G(U(k) + wv(k)),

i = 1,2,...N (12)

z(k) = Hvxv(k) + v(k);Hv = [1 0]

- The process equation using acceleration evaluation:

xa(k + 1)=Faxa(k)+Wa(k) (13)

z(k) = Haxa(k) + v(k);Ha = [1 0 0]

F„

= [0 T;G =

T2/2

T

Inside, z(k) - observations received at time k; Hv, 4.1. Kalman filter algorithm for acceleration

Ha - matrix of observations; v(k) - the observed evaluation filter

channel noise, assumed to be the central Gaussian Applying Kalman filter algorithm [6, 7] for

white noise with the correlation matrix Qz = - acceleration evaluation filter according to the model

channel noise intensity observed. (13).

4. TRACKING ALGORITHM TO - Assign the first value:

MANEUVERING TARGETS

xa(0) = [$0 Vo Vol' P(0,0) = E[(x(0) - E[x(0)]) X (x(0) - £[x(0)]f]

- Calculate the state predictions at time k:

{p-(k) = cp(k -1) + Tcp(k -1)+ p1cp(k - 1)

¿p-(k) = $(k-1)+p2$(k-1) (14)

$-(k)=p3$(k-1)

- Calculate the a priori error correlation matrix:

P-i(k) = Pn(k - 1) + 2TP!2(k - 1) + 2p!P13(k - 1) + T2?22(k - 1) + 2Tp1P23(k - 1) + p^P33(k - 1) P-2(V = Pi2(k - 1) + T?22(k -1) + (fii + Tp2)P2s(k - 1) + №13^ - 1) + Pip2Pss(k - 1)

P-2&) = P-i(k)

P-3 (k) = P-i (k) = №13 (k-1) + TP3P23 (k-1) + P1P3P33 (k-1) (15)

P-2(k) = P22(k - 1) + 2p2P23(k - 1) + P2P33(k - 1) P-3 (k) = P32 (k) = faP23(k -1)+ №3P33(k - 1) P33(k) = P2P33(k -1) + cri

- Calculate the Kalman amplification matrix:

Ki(k)=pSBi; K2(k)=K3(k)=^ (16)

x'(k-1)

xN (k-1)

Mi(k-1) Mn (k-1) x(k-1)

Figure 1. General block diagram of maneuvering target tracking system

- Calculate the state evaluation at time k:

cp(k) = cp-(k) + K1(k)[z(k) - cp-(k)]

$(k) = r(k)+K2(k)[Z(k) - rm

$(k) = $-(k) + K3(k)[z(k) - cp-(k)]

- Calculate the posterior error correlation matrix:

P11(k) = [1- K1(k)]P!-1(k);P12(k) = P21(k) = [1- K1(k)]Pl-2(k) P±s(k) = Ps1(k) = [1 - K1(k)]P-3(k); P22(k) = P-2(k) - K2(k)P-2(k) P2s(k) = Ps2(k) = P-3(k) - K2(k)P-3(k);P33(k) = P-3(k) - K3(k)P-3(k)

(17)

(18)

4.2. Building a filter to evaluate maneuvering target coordinates on the basis of interactive multi- ^ (0). model algorithm

Assign the first condition: , i,j = 1,2,... N,

- Calculate the mixing probability: ^¿^(k-l), Applying interactive muhi-m°del algorithm for itj = \t,,,tN according t0 the (3).

- Calculate the first condition for the jth Kalman filter:

model (12), which uses Kalman filter algorithm for each filter, received the following algorithm:

¿p°j(k-1) = ^cpi(k-1)ßiU(k-1)

î=I

V0j(k -1) = YtUy^k - 1)Hj(k - 1)

r

Pii (k - 1) = YIi=ißi\j(k - 1)[P1i(k -1) + (Pp\k -1)- p0j(k - 1))2] (19)

r

P?!(k - 1) = ^ßiU(k - 1)[Pi2(k - 1) + (Ppi(k -1)- p0j(k - 1))(Ppi(k -1)- Pp0j(k - 1))]

=i

Pi02(k -1) = P°i(k - 1)

r

P20l(k -1) = ^lU(k - 1) [Pii(k -1) + (Pp^k -1)- Pp0j(k - 1))2]

=i

- Perform filtering algorithm for jth model with mixed first condition:

pp-j(k) = pp0j(k - 1) + T(p0j(k - 1) + T2uj(k - 1)

pp-j(k) = pp0j(k -1) + Tuj(k - 1) (20)

- Calculate the a priori error correlation matrix:

Pi-ij(k) = Piio(k -1) + 2TPi20(k -1) + T2P220(k - 1)

P-j(k) = Pi20(k -1) + TP220(k - 1) (21)

Pi-2 ( k) = P2-i ( k)

P-2(k) = PL(k -1) + T2oa2

- Calculate the Kalman amplification matrix:

Kj(k) = P}'(k) ; Kj(k) = Pl2(k) (22)

- Calculate the state evaluation at time k:

$i(k) = pp-j(k) + K((k)[z(k) - pp-j(k)]

¿pj (k) = p-j (k) + Kj (k) [z(k) - pp-j (k)] (23)

- Calculate the posterior error correlation matrix:

Ph(k) = [1-Ki(k)]P-i(k) Ph(k) = PÎi(k) = [1- Ki(k)]P-2j(k) (24)

PL(k) = P22 (k) - Ki(k)P-2j(k)

- Calculate the rational function for the jth Kalman filter:

ej (k) = z(k) - pp-j (k - 1) (25) Sj(k) = Hv[FvPj(k - 1)FT + QX]HVT + Qz Sj (k) = Pj(k - 1) + 2TPj (k - 1) + T2Pj (k - 1) + ol (26)

- Update jth model probabilities:

Aj (k) =^= exp (--— e2 (k)) (27)

\2nSj(k) 2S](k)

Hj (k) = i Aj (k)Cj; c = Fj=i Aj (k)Cj (28)

- Combination of state evaluation and evaluation error correlation matrix:

H(k) = Zrj=iVj(k)fij(k) <P(k) = Yj=i<Pj (k)ßj(k)

(29)

Pii(k) = ^ti](k){p1i(k) + №(k) - ¿p(k)]

=i

Pi2(k) = P2i(k) = Zrj=iPj(k){Pi2(k) + [<pj(k) - №)]W(k) - 9(k)]}

(30)

I

Pii(k) = ^ßj(k) {PÎi(k) + [$J(k) - $(k)]2}

=i

5. SIMULATION RESULTS AND ANALYSIS

Parameter of the interactive multi-model filter:

To simulate, the target is fake created in the <7% = 0,03(m2/s4)

horizontal plane and the follow-up is done in the x coordinate.

+ Number of filters; N = 7 + The acceleration range is around the evaluation

Creating fake targets with the following value; £ = 40(m/s2)

parameters:

- Target speed: Vm = -400(m/s2);

- Normal acceleration: y'm = - ajjm + um aj = 0,5(1/s);

+ Probability of the model state transition:

PiJ

M {0,

0,9when i = j 0,1/(N -1) = 0,0167wheni ± j 1,...,N

1

um(m/s2) = <

0whent < 5s 100when5s < t < 10s 0when10s <t<15s -100 wh n15 < < 20 0when20s < t < 25s

+ Model probability at the time of initialization: pj = 1/N = 0,1429, j = 1,...,N * Sampling frequency: 16 H z The evaluation error of the position, speed of the acceleration filter and the interactive multi-model filter

- Channel noise intensity observed: a? = 10(m2) is shown from figure 2 to figure 7. * Parameter of the acceleration evaluation filter: a = 0,5(1/s); a2 = 0,1(m2/s4)

Evaluate the position of the acceleration Alter

Time [s]

r

2

*

v.

Figure 2. Evaluate the position oof the acceleration filter

-100

-150

Evaluate the speed of the acceleration filter

-450

5 10 15 20

Time [s]

Figure 3. Evaluate the speed of the acceleration filter

25

Evaluate the location of the IMM filter

-200

-250

3_ -300

Time [s]

Figure 4. Evaluate the location of the IMMfilter Evaluate the speed of the IMM filter

5 10 15 2

Time [s]

Figure 5. Evaluate the speed of the IMMfilter

IJ«

Location evaluation error

Time [s]

Figure 6. Location evaluation error

Figure 7. Speed evaluation error

From the simulation results, it is found that the tracking system using interactive multi-model algorithm significantly improves both position error and speed when the target is maneuvering.

6. CONCLUSIONS

The target tracking system is synthesized on the basis of the acceleration evaluation filter and the interactive multi-model filter with small tracking errors in both position and speed in case of maneuvering targets.

References

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