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67ТЕХНИЧЕСКИЕ НАУКИ
Nguyen Van Bang, Dang Cong Vu, Vu Quang Luong
Air Defence - Air Force Academy, Ha Noi, Viet Nam
APPLICATION OF LEARNING FEED-FORWARD CONTROLLER BASED ON MODEL REFERENCE ADAPTIVE SYSTEM FOR MISSILE STABILIZATION
DOI: 10.31618/ESSA.2782-1994.202LL68.11 Abstract. The article presents the results of research, analysis and how to build learning feed-forward controller based on model reference adaptive system in the remote control loop for missile stabilization. The controller structure is simple, adaptive control law applying Lyapunov stability theory fast convergence and sustainable. The simulation results have shown the advantages of using algorithm, the missile is always stable when there is a parameter change due to the effects of flight conditions.
Keywords: Learning feed-forward controller, Model reference adaptive system, Missile, Stabilize.
1. INTRODUCTION
The nature of applying an adaptive control system according to a reference model is to design the controller so that the system to achieves the desired properties given by a mathematical model (reference model) [2]. When the properties of the real system different from the ideal properties of the reference model, the system is changed by adjusting the parameters of the controller or create additional subsignal [1].
The learning feed-forward controller is based on a model reference adaptive system capable of automatically adjusting the parameters of the controller
according to the tendency to bring erro (e) between the reference model and the process (missile) forward gradually go to zero. The advantages of this controller are fast adaptive speed, high stability and less sensitivity to noise [1, 2].
Missiles in the remote control loop is kinetic stage with change parameters, and therefore need to be stabilized. The change in the missile normal accelerations depends on the wing deflection angle in groove nod described by the transmission function [3, 4, 5, 6]:
VP) =
P-Tv
TpP2 + 2tpTpp+1
(1)
Inside; Kp - Transfer coefficient of the missile Vp - Missile velocity Tp - Time constant
- The attenuation coefficient fluctuates individually
Tv - Aerodynamic time constant Tv depends on the aerodynamic arrangement of the missile, the geometrical and aerodynamic characteristics of the missile's elements. They change
with flight conditions (altitude, velocity, change of
vv
attack angle ...) [5, 6]. In particular, the — coefficient
Tv
varies greatly, depending on the dynamic pressure and make amplification coefficient of the control system also change within a wide limit.
There have been a number of articles providing solutions to stabilize the missile by application of classical control theory [5]. However, only response within a certain range the dependence of the open control loop gain on the aerodynamic pressure [5]. Stabilizing the missile stage parameters requires complex equipment, multiple sensors, each parameter needs a separate stabilizer [4, 6]. Although stable solutions have been implemented, but in actual the missile stage parameters still change, so the quality of the control loop will decrease, the missile stage parameter is different from the calculated parameters [5, 6].
There have been a number of articles providing solutions to stabilize the missile by application of adaptive control theory [1, 2]. However, the solutions proposed today mainly use the feedback linearization method, complex stability algorithm, which requires many measuring set (or evaluation) of the missile's kinematic parameters [1, 4]. Therefore, the realization of the algorithm is very difficult.
Seeing that, to improve the accuracy of destroying the target, the missile needs to be stable during flight. Therefore, the article proposes how to stabilize the missile on the application of learning feed-forward controller based on model reference adaptive system. The controller has a simple structure, the law adapted fast convergence and sustainable. Algorithm is verified through simulation, has reliable results and is able to realize the algorithm in current technological and technical conditions.
2. DESIGN ADAPTIVE LEARNING FEEDFORWARD CONTROLLER ACCORDING TO THE REFERENCE MODEL
The structure depicted in figure 1 can be used as an model reference adaptive system [1, 2]. The process has a mathematical model of the second order, which is controlled with the help of the learning feed-forward controller. The parameters of this controller are am,bm,cm. The change parameter of the process
V
p
к
UB
ansa
(missile) is a, b, c. Normal acceleration requires of the missile stage input is ar (control signal).
Figure 1. Structure diagram ofprocess, reference model, learning feed-forward controller The reference model is described by:
Vm
ar p2 + 2$mü>m.p + ü>$n
(2)
The process is described by:
u T£p2 + 2l;p.Tp.p + 1 a.p2 + b.p + c
(3)
With a = —, b = ^^eIl c = —.
Kv
1
Kr,
Represents process as a state variable:
x =
-±1 01 rXi "1 0
= c b Iv1] + 1
A. a a. .a.
u = Ax + Bu
(4)
With A =
, B =
By means of a learning feed-forward controller, the state variable filter output signals can be used to create an inverse model of the process. We need to define the operating principle based on the errors between the output of the reference model (ym) and the output of the process (x), and adjust the parameters am, bm and cm so that they converge according to the parameters of process a, b and c respectively.
This shows that we can use the approach by classic Lyapunov stability theory to find the law of adaptation for the learning feed-forward controller parameters. - Step 1: Determine the differential equation for e Represent the reference model as a state variable:
Therefore, the design problems posed are: Find (stabilize) the adjustment law for the adjustment parameters am, bm and cm so that the error between the reference model (e) and the process progresses to 0, and adjust the parameters am, bm and cm so that they converge according to the parameters of process a, b and c respectively. Steps to design the controller adapted with Lyapunov stability theory as follows:
jml = Jm2 ; jm2 = £
ym =
ïnK hem-G oMB
(5)
(6)
2 m
K
1
x
V
V
0
1
b
c
a
a.
.a.
£
Represents a process as a state variable:
X1 = X2
X2 = —äX1 — aX2 + 1 (cmJm1 + ^mJm2) + 1 am. £
(7)
(8)
Xi 0 1 rX1l 0 0
X = = C b + m bm
Ï2. X2 -
a a a a
0 1 0 0 0
= C b X + Cm il Jm + am
a a a a a
M +
\Jm2] +
With error e is determined by the following formula:
e = ym-x e = ym-x
Replace (6) and (9) in to (11) we have:
(9)
(10) (11)
0 0 01 0
e = Cm il Jm - c b X + 1 -am
a a a a a
0 0 01 01 01 0
Cm bm Jm - c b Jm + c b Jm - c b X + 1-am
a a a a a a a a a
0
b b;
La a a
Jm +
With A, =
00
E — Em. b bm
a a a a
<Jm-x) + 0
1 —am
a
A =
e = AiJm + A2e + Bi£ (12) 0
B =
\ — zm
a
- Step 2: Select the Lyapunov V(e) function [1,2]
V(e) = eTNe + aTaa + bTpb
(13)
(14)
Inside, N - Symmetric matrix is determined a and p - The diagonal matrix contains the
arbitrarily positive. elements that determine the adaptation process speed.
a and b - The vectors contain nonzero elements of - Step 3: Determine the conditions for function
matrices A1 and B1. V(e) to determine negative
V(e) = (A1ym + A2e + B1e)T. N.e + eT. N. (A1ym + A2e + B1e) + 2a. a. aT + 2b. p. bT
= (A2e)T. N.e + eT. N. A2. e + 2eT. N. A1. ym + 2d. a. aT + 2eT. N.B.e + 2. b. p. bT (15)
From [1, 2]: ATN + NA = -Q either (A2e)T.N.e + eT.N.A2.e is always negative. Thus, eT(ATN + NA)e = -eTQe the stability of the system will be ensured if part
According to Malkin's theorem, Q is a positive 2eT. N. A1. ym + 2a. a. aT + 2eT. N. B. e + 2.b. p. bT deterministic matrix. This means that the value for part has zero value, that is:
eT. N. A1.ym + a. a.aT = 0 eT.N.B.e + b.p.bT = 0
a11 0
0 a22.
(16) (17)
With a = [a2i ^22], e = [ei e2], a=[
N = \nii ni2] A = N = \n2i n22],Al =
00
a21 a22
^m = LJ
After calculating, we get the following results:
From formula (9) we have:
a21 = ~ (el. n21 + e2n22)Jm1
a11 1
a22 = - — (ei.n2i + e2n22)jm2 a22
_ C Cm . _ 1 .
a2i — ^ a2i — cm
21 a a 21 a m
(18)
(19)
a
0
0
1
0
C C
m
1
To complete the parameter update process, cm is defined by the following expression:
crn = ~lKei.n2i + e2n22)ymi\ dt + cm(0)
an
From formula (13) we have:
(21)
b bn a„ =----
^ änn = -~b„
(22)
To complete the parameter update process bm, an are defined by the following expression:
K =-^J[(ei.n2! + e2U22)ym2\ dt + bm(0)
a22 ' a
am = -;rI[(ei.n2i + e2^22)£] dt + am(0)
P22
(23)
(24)
Where a22 and fi22 are called the speed of the adaptation process, n21 and n22 are the elements of the matrix N.
- Step 4: Determine N from ATN + NA = -Q with Q =
Qu Ri2 R21 Q22
W-21 n22] + №21 ^J
1
Qll Rl2 %2l R22
(25)
-(n.2l+n22)
b
Z.n2l
C
~.n22
Ull
nl2
c
~.n22
a
+ n
b
-.nl2
t c a 2
Rll Rl2 R2l Q22
(26)
The results are as follows:
nl2 = n2l = 2 Qll
l i-a2 1 a \
n-ll = n.22 =-(~i.qll+-q22)
(27)
(28)
From formula (21), (23) and (24), the design of the reference adaptive system with Lyapunov's stability learning feed-forward controller based on model theory in figure 1 is redrawn in figure 2 as follows:
a a
a
c-
0
0
b
c
1
a
U-
a-
n
l
2l
Figure 2. The learning feed-forward controller based on model reference
adaptive system
3. SIMULATION RESULTS AND ANALYSIS
The algorithm survey is performed within the
remote missile control loop [5, 6]. The simulation organization diagram is in the form of figure 3, with the parameters are selected follows:
J, Creating fake Target coordinate determination system Command setting ar The missile has adaptive system
targets system
Missile coordinate determination system
Figure 3. Structure diagram of control loop using learning feed-forward controller based on model reference
adaptive system
- Target has speed Vt = 450(m/s), flying in, the horizontal distance D = 29(km), altitude H = 6(km), maneuver start time at moment ^ = 10(s), maneuver finish at moment t2 = 15(s), maneuver 30(m/s2).
Missile
velocity Vp = 900(m/s).
a = 0,011; b = 0,15; c = 1.
- Controller parameters: wm = 10; ¡¡m = 0,7.
Q = \4] thi P21 * 0,0385; P22 * 0,078.
- The command setting system creates command according to the 3-point guidance method.
2000 -
1.5
Horizontal distance [m]
Figure 4. Missile trajectory - target
Guidance error
Time [s]
Figure 5. Error at meeting point
ив
НЩШ
The position error
Time [s]
Figure 6. Error of position
Speed error
Figure 7. Speed error
Comment: The missile is always stable during flight. The controller is adaptable to changes in kinetic parameters, the error at the meeting point (straight deviation) is small. Error between the reference model and missile is small, the law of adaptation is fast convergence and sustainable.
4. CONCLUSION The controller structure is simple, adaptive control law according to the reference model applied Lyapunov stability theory fast convergence and sustainable. Simulation results show that, when using an adaptive mechanism, it is more stable than when not in use. This is the basis of improving the accuracy destroy targets, meet in the actual conditions when the missile's flight conditions change.
However, the adaptive laws (21), (23) and (24) only apply to the process and the reference model has the second order transfer function. Thus, when the process has a higher order transfer function, we must approximate that transfer function to degree order 2. The learning feed-forward controller only compensates and corrects for the process with the second order transfer function, which is a limitation of this method.
REFERENCES
1. Kadriye Tiryaki Kutluay, Adaptive control of guided missiles, A Thesis Submitted to The Graduate School of Natural and Applied Sciences of Middle East Technical University, 2011.
2. Pankaj K., Kumar J.S. and Nema R.K., Comparative Analysis of MIT Rule and Lyapunov Rule in Model Reference Adaptive Control Scheme, Innovative Systems Design and Engineering, Vol.2, 2011, pp. 154-162.
3. Nguyen Van Bang, Nguyen Tat Tuan, Synthesis of parameter recognition algorithm and state evaluation for flight device, East European Scientific Journal, Vol 2, No.66, 2021, pр. 46-54.
4. Nguyen Van Bang, Synthesis of remote control law when taking into dynamics and nonlinear of the missile stage, Intelligent Systems and Networks, Springer, No.22, April 2021, DOI:10.1007/978-981-16-2094-2_22.
5. Канащенкова. А.И и Меркулова. В.И. Авиационные системы радиоуправления (том 2), Радиотехника, Москва, 2013.
6. Анатолий Борисович Скорик, Системы управления современными зенитными ракетами. Математическое моделирование контура наведения, Сучасци информащйя технологи сфере безпеки та оборони (том 1), 2013.
