SYNTHESIS OF PARAMETER RECOGNITION ALGORITHM AND STATE EVALUATION FOR FLIGHT DEVICE Текст научной статьи по специальности «Медицинские технологии»

ТЕХНИЧЕСКИЕ НАУКИ

Nguyen Van Bang*, Nguyen Tat Tuan

Air Defence - Air Force Academy, Ha Noi, Viet Nam

Nguyen Ngoc Tuan Military Technical Academy, Ha Noi, Viet Nam

Dang Tien Trung Electric Power University, Ha Noi, Vietnam

SYNTHESIS OF PARAMETER RECOGNITION ALGORITHM AND STATE EVALUATION FOR FLIGHT DEVICE

Abstract. This paper presents the results build of parameter recognition algorithm and state evaluation for flight device, purpose to realize some control laws. The results were surveyed and simulated on Matlab-Simulink software, showing that the algorithm ensures accuracy in evaluating state variables and flight device parameters during flight device control.

Keywords: Flight device; Recognition; Parameter evaluation; State evaluation.

1. INTRODUCTION

When researching flight device remote control systems, three major issues need to be addressed [1]:

Flight device stabilization: Improved dynamic properties for flight device.

Guidance methods and flight device control laws: Guidance methods are intended to determine the desired trajectory of the flight device and control laws ensure that the flight device is flying in that desired trajectory.

Stabilize the remote control loop: Improved dynamics for control loop when taking into account the dynamics of all stages in the control loop.

In recent years, the research on flight device control mainly focused on synthesizing, perfecting the guidance methods [2, 3, 4, 5, 11, 12, 13] and synthesizing the stability algorithm for the self-guided missile class [14, 15]. Some documents and theses have focused on synthesizing control laws for remote controlled missiles [1] or UAV [6] that have not yet been published on model parameter recognition and state evaluation for flight device. The common point of the new guidance methods and control laws is that when the synthesis is completed, there exist components of the position and acceleration's speed of the flying device in the algorithm [1, 3, 5, 11, 12]. Therefore, realizing those algorithms in practice will face many challenges and difficulties.

As for the flight device control system (real system) that can not directly measure all the flight device state variables during flying, there are no published articles that have proposed a technical solution define of those state variables. In practice, the acceleration of the fly device (output signal) can be measured with the accelerometer, but some optimal control laws and adaptive control laws require must determine the derivative component of the acceleration [1, 3, 4, 5, 8, 11, 12, 13, 14, 15]. Therefore, this paper will propose the method of determining the state variables and parameters of the flying device on the basis of the extended Kalman filter [10].

2. SYNTHESIS OF FLIGHT DEVICE STATE EVALUATION FILTER

The Kalman filter mentioned problem go estimating the state of a control object is modeled in a discrete-time by a random linear equations of the form [7, 9, 10]:

x(k) = Fx(k - 1) + Bu(k - 1) + %x(k - 1)

(1)

Where;

x(k) - State vector, size (n x 1).

u - Control signal vector, size (r x 1).

k - Discrete time.

Qx(k) = Mtfx(k)?x(k)} - The intensity of the noise %x(k).

F - State transition matrix, it is applied to the previous state x(k - 1)

B - Input control matrix.

The observation equation has the form [7, 9, 10]:

z(k) = Hx(k) + $z(k)

(2)

z(k) - Observations vector, size (m x 1), with

m <n.

H - Matrix of observations, size (m x n), it transfers the measured state space into the observed state space.

Qz(k) = Mtfz(k)^(k)} - The intensity of the noise ^z(k).

In the case of process parameters exactly unknown and change (Kp,Tp,$p) [1, 11, 12, 13, 14, 15], can be used simultaneously algorithms evaluate the status and parameters using the extended Kalman filter. However, with this method, the size of the state space will be large, so the number of differentiation equations to be solved will be large [10]. Therefore, we will use two separate filters to evaluate separately the state and parameters of the flight device.

Parameters

Figure 1. State evaluation and parameter recognition The fligth device has the following transmission function [10, 11, 12, 13, 14, 15]:

V

yp(p) _ i u(p) t£p2+2Çptpp+i

ïp^pVp — Up

yP = —2tP">pyp — œ%yP + Kpœ%u

1

Where, Tv - time constant and wv =—;

P P Tp

u - the control input;

yp - the output of system.

Set: xlp = yp; X2P = yp.

Where, x1 and x2 represent the state variables;

We have, the equation of state of the fligth device:

L2p

— 2%pMpX2P ^p^ip +

From (5), the state space model of the fligth device in the discrete time domain has the form:

yv(k)-yv(k-1)=yp(k-l)

AT

)yPm-yp(k-1) _ AT

-c±.yp (k — 1) — c0.yp(k — 1) + b0.u(k — 1)

Where, c1 = 2%pœp, c0 = œp, b0 = Kpœp, AT - sampling cycle. Kp - amplification factor, - attenuation coefficient.

(yP(k) = yp(k-1)+AT.yp(k-1) \yp(k) = (1-ATci).yp(k - 1)-AT.c0.yp(k

yp(k) 1 AT \yP(k — 1)

jp(k). —c0AT 1 — c1AT\ iyP(k — 1)J

1) + AT.b0.u(k — 1) u(k — 1)

0

+ b0AT_

Process equations: x(k) = F. x(k - 1) + B. u(k - 1) (9)

In there: x(k) —

yP(k) yP(k)

; F —

1

AT

—c0AT 1 — c1AT

; B —

0

b0AT

(3)

(4)

(5)

(6)

(7)

(8)

On the flying device using measuring set of the acceleration (accelerometer), then the observation equation takes the form:

zx(k) = Hx(k) + fz(k) (10) No loss of generality, can be considered: H = [l 0]

With the process equation (9) and the observational equation (10), the Kalman filter equation takes the form:

x- — Fx(k — 1) + Bu(k — 1), x(0) — x0 x(k) — x-(k) + Kx(k)[zx(k) — Hx-(k)], x(0) — x0 Kx(k) — D-(k)HT[HD-(k)HT + Qz]-1 Dx(k)—[E — Kx(k)H]D-(k) D-(k) — D(k — 1) + Qx(k — 1)

(11) (12)

(13)

(14)

(15)

Xip — X2p

For the flight device, because the parameters F11 = 1, F12 = AT, B1 = 0 are constants, the parameter vectors need to be evaluated:

n = [^1 = [F21 F22 B2]t = [-c0AT 1 - c^T b0AT]

Details of the expression (11) - (15), receiving:

(x-(k) = oc1(k - 1) + ATx2(k - 1) [x-(k) = n1x1(k - 1) + n2x2(k - 1) +n3u(k - 1)

e(k) = zx(k) -x-(k) (x1(k)=x-(k)+KXi(k)e(k) \x2(k) = x-(k) + KX2(k)e(k)

fD-ii(k) = DXn(k - 1) + QXn

\D-i2(k) = D^k) = DXi2(k - 1) 'D~(k) = D^(k - 1) + Q^

DXll(k) = (1 - KXl(k))D-11(k) DXl2(k) = DX2i(k) = (1 - KXi(k))D-i2(k) [DX22(k) = -KX2(k)D-i2(k) + D^k)

\KX2(k) =

DXl1(k) + Qz

Px12(k)

(16)

(17)

(18)

(19)

(20)

(21)

DXll(k) + Qz

The n1, n2, D3 parameters are unknown, so it was replaced by the evaluation of it H1, h2, h3 via a

parametric evaluation filter.

zß) ^ e(k)

K

+

AT

+

Delay

Q,

-xgx

u(k)

Q3

+

+(8)

Q}2 Delay

K

+

Figure 2. Schematic structure of missile state evaluation filter

3. SYNTHESIS OF FILTERS EVALUATION OF FLIGHT DEVICE PARAMETERS

Can be rewritten (1) to:

xp(k) = <Pxp(k-1) + ^Xp(k-1)

In there,

xv = ixT uT]T - Extended state vector, size m = n + r.

(22)

1

2

<P = [F B] - The state transition matrix expands, size (m x m). Kxv = K 0T - Gaussian white noise vector.

QXp (ft) = M {$Xp (k)&v (ft)} - The intensity of the noise $Xp (ft). 0 - Vector whose elements are zero, size r.

The parameter vector are unknown: fi(ft) = [(P1 <P2 . . &m]T <Pi - Matrix ith row of the matrix <P.

Due the state is evaluated by the private filter, so in the filter the evaluation of the state parameter of the process is known. Therefore observation equation of the form:

zn (ft) = xp (ft) + (ft) (23)

Combining formulas (22) and (23) received:

zn(k) —

Xp(k — 1) 0

0

0

xp(k — 1) 0

0 0 0

xp(k — 1).

<Pi

™m

+ SXp(k)

Rewrite (24) in the following format:

zn(k) — Mp(k)H(k) + $x(k)

Mp(k) —

xp(k — 10 0

0

0

xp(k — 1) 0 0

0 0 0

xp(k — 1)_

(24)

(25)

(26)

Normally, the coefficients in the matrix <P (vector can be considered constant or almost constant. Such

fi) are functions of variation with time slower than the processes can be modeled by a random process with

xp states. Therefore, during the observation period, <P almost zero speed, i.e:

fi(ft)=fi(ft-1) + ^n(ft-1) (27)

In there; (n - Gaussian white noise on centre with With process equation (27) and observational

Qn intensity. equation (25), applying Kalman filter algorithm to

Qn is selected depends on the fi parameter change evaluate fi(ft), receive: speed.

Û(k) — Ù(k — 1) + Kn(k)[xp(k) — Mp(k)Ù(k — 1)], /2(0) — ß0 Ka(k) — Dß(k)Mp(k) [Mp(k)D-(k)M^(k) + QXp(k)]-1

Da(k) — [E — K(k)Mp(k)]D-(k) D-(k) — Dn(k — 1) + Qn(k — 1), Dn(0) — Dng

(28)

(29)

(30)

(31)

In there;

Dn - The posterior error correlation matrix of Ü. D- - The a priori error correlation matrix of Ü. aa, Dn0 - First condition. In the case the x components of xp cannot be measured, then in equations (26), (28) - (31), x is

replaced by x, Qx is replaced by Dx(ft). In which, x is generated from the own filter, Dx(ft) is the correlation matrix of x.

Specifically, for flight device (the missile stage), due the F11 = 1, F12 = AT, B1 = 0 parameters are constants, the parameter vectors to be evaluated are:

n — [Oi n2 ^^ — [F21 F22 B2]T — [—c0AT 1 — CiAT boAT]1

Then, the observation equation takes the form:

zn(ft) = *2(ft) = Mp(ft)

+ tX2(k)

(32)

Mp(k) — [Sc1(k — 1) x2(k — 1) u(k — 1)]

Ûi(k) —Ûi(k—1) + Kni(k)en(k); Ûi(0) — Oio

Ô2(k) — Ô2(k — 1) + K02(k)en(k); ^(0) — O0

Ô3(k) — Ü3(k—1) + Kq3(k)en(k); ^(0) — 030

en(k) — X2(k) — 0iXi(k — 1) — n^Mk — 1) — ^u(k — 1)

0

0

0

1

2

3

_ Dä11(k)Sci(k- -1)+Dni2(k)x2(k- -1)+D-13(k)u(k- -1)

_ Dn12(k)jc1(k- MS -1)+D-22(k)ic2(k- -1)+D-23(k)u(k- -1)

_ ün13(k)X1(k- MS -1)+D-23(k)iC2(k- -1)+D-33(k)u(k- -1)

Km (k) =

Ka3(k) =

(33)

MS

With the denominator MS is determined by:

MS = Dn11(k)x12(k- 1) + Dn22(k)x22(k- 1) + D^33(k)u2(k - 1) + 2Dn12(k)x1(k - 1)x2(k - 1) +2D~ni3(k)x1(k- 1)u(k- 1) + 2D~n23(k)xi(k-1)u(k-1) + D^k)

The components in the Dn and D- matrices are for state evaluation and parameter recognition the flight determined according to (30) and (31), paying attention device is shown in Figure 3. to the symmetry of the matrix to reduce the computational mass. The structure diagram of the filter

Zx(k)

e(k)

AT

+

-XgK-

Delay

Q,

-Kg*

u(k) Q3 +(

+(8>

Delay

K„

xg)->

Delay

Xg)—►

Delay

x1(k)

q2 Delay

x2(k)

Qj

[2,

02,

Figure 3. The filter for state evaluation and parameter recognition offlight device

4. SIMULATION RESULTS AND ANALYSIS The simulation, filter survey, state evaluation and parameter recognition of flight device are considered in

the components of the remote control loop. Simulation blocks diagram is shown in Figure 4.

Figure 4. Block diagram of the flight device remote control loop

2

+

+

- Command setting system using the command creating method with the 3-point guidance method.

- Target parameter:

- Target velocity; Vmt = 350m/s

- The target's flight altitude; H = 6ftm

- The target's horizontal distance; Dmtx = 30ftm

- The flight device with transfer function; Kp (p) = —f-, motion with velocity

TpP +2$pTpP+1

Vp = 720 m/s.

Case 1: No white noise.

The trajectory of the flight device - the target

Guidance error

Horizontal distance [m]

Figure 5. The trajectory oof the flight device - the target

Flight time [s]

Figure 6. Straight deviation

Amplification factor

Time constant

Flight time [s]

Figure 9. The change oof parameter Kp

Flight time [s]

Figure 10. The change oof parameter Tp

Attenuation coefficient fluctuates

Figure 11. The change oof parameter L

Comment: The errors of the filters are small, the time to establish is small. Convergence algorithm rapid and sustainable. Small guidance error. The algorithm always accurately evaluates the state variables of the

flying device (acceleration, derivative of the acceleration) and recognizes the unknown parameters (Kp, Tp, ^p).

Case 2: There is white noise

The trajectory of the flight device - the target

ouuu -Target -Flight device

5000

~ 4000

©

S 3000 n £

Ol 2000

1000

0.5 1 1.5 2 2.5 Horizontal distance [m] x 104

Figure 12. The trajectory of the flight device - the target

Guidance error

Flight time [s]

Figure 13. Straight deviation

H* ipHi of dmi|»«f HwfW Met KaknUM

miMHmmiimimimmtmtm 10

Figure 14. Acceleration of flight device

Amplification factor

Flight time [s]

Figure 16. The change of parameter Kp

Figure 15. Speed changes the acceleration of flight

device

Time constant

10 15 20

Flight time [s]

Figure 17. The change of parameter Tp

Attenuation coefficient fluctuates

Flight time [s]

Figure 18. The change of parameter

Comment: When the target is complex, the algorithm still ensures well the determination of state variables and evaluation of the flight device parameters. Small filter error and guidance error.

5. CONCLUSION

The paper presented the results of synthesizing the flying device model recognition algorithm. Essentially, a synthesis algorithm of simultaneous evaluation of state and flight device parameter evaluation using the extended Kalman filter. The results are simulated by the Matlab-Simulink software, showing that the synthetic filter is capable of accurately estimating state variables and flight device parameters, this is an important basis for the command setting system to actualize control laws.

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