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13
1.РАДЮЕЛЕКТРОН1КА
УДК 681.515.62
THE SYNTHESIS OF A DIGITAL FUZZY CONTROLLER OF A CONTROL SYSTEM BY THE OBJECT "THE STEERING MACHINE + MISSILE"
V.I.Gostev, S.A.Magluy, V.O.Chmelev
Викладено синтез цифрового нечШкого регулятора систе-ми керування нестацюнарним об'ектом "кермова машина + ракета" з урахуванням ланки затзнювання, нелтшностей i жорсткого зворотного зв'язку в кермовш машит.
Изложен синтез цифрового нечеткого регулятора системы управления нестационарным объектом "рулевая машина + ракета" с учетом звена запаздывания, нелинейностей и жесткой обратной связи в рулевой машине.
The synthesis of a digital fuzzy controller of a control system of the non-stationary object " the steering machine + missile " is explained in view of a link of delay, non-linearities and direct feedback in the steering machine.
INTRODUCTION
In the papers [1-5] the synthesis of digital adaptive controllers and digital fuzzy controllers for control of the non-stationary object "the steering machine + missile" is explained and is shown by a method of mathematical simulation, that the digital fuzzy controllers allow to receive more high quality of automatic control systems, which is characterized by errors in the transient and steady-state operation modes of systems. The synthesis is fulfilled at an essential assumption that the steering machine is represented by a linear integrating link. The actual electrical, pneumatic, hydraulic steering machines have considerably more sophisticated mathematical description [6]. The synthesis of a digital fuzzy controller for control of the non-stationary object "the steering machine + missile" is below explained in view of a link of delay, non-linearities and direct feedback in the steering machine.
THE DESCRIPTION OF THE CONTROLLED
OBJECT AND SYNTHESIS OF A CONTROLLER
As an example we shall consider the pneumatic steering machine, which has practical application. The force intensity in pneumatic and hydraulic drives amounts 200-300 kg/cm2 (in electrical 4-6 kg/cm2), therefore these drives with the same overall dimensions and weights have higher speed, compared to electrical [6].
The schematic diagram of the pneumatic steering machine with direct negative feedback on an error angle of a rudder is represented in a fig. 1. The compressed air motor (pneumatic motor) has the movable cylinder 1 and nonmovable pistons 2. In the cylinder there are holes, through which air (or other
gas) from an ink-jet pipe 3 supplied. The ink-jet pipe is connected with the magnetoelectric device turning in a magnetic field of a permanent magnet when a control voltage m = m (t) is fed into a winding. The balance springs 4 hold the ink-jet pipe in a neutral position at absence of a signal m = m (t) (pressures p^ and P2 are being identical in cavities of the cylinder). At moving an ink-jet pipe in relation to holes of the cylinder the pressure drop p = p^ -P2 will be received. Thus the cylinder is transferred to the side opposite to rotational displacement of an ink-jet pipe, and through traction 5 it turns a rudder 6. For the reason that rotational displacement of a rudder and moving of the cylinder are coupled by transmission number n , the negative direct feedback on an error angle of a rudder is carried out. The turning moment in the magnetoelec-tric device is balanced by the moment of balance springs 4.
Pe
I I — m(l) —11
Figure 1
Motion of an ink-jet pipe can be described by the equation of an ideal link of delay
P(t) = k1m(t - T), (1)
where P(t) - is an angle of rotation of an ink-jet pipe from a neutral position, m(t) - is a controlling signal, ki - is a constant coefficient.
Motion of the cylinder can be described approximately as follows [6]:
y = k2[9(e) - Cjsing(y)]; e = P - k3y ; y = n5 , (2)
where k2 - is a constant coefficient, y(t) - is displacement of the cylinder, 5(t) - is an error angle of a rudder, k^ - is a coefficient of transformation of linear movement in angular, 9(e) - is a non-linearity of "limitation" type (see fig.2,a).
N
ç 0
< 1,
grating link: ko=k2 n = 100. A feedback factor: l = k3 n = 0,5.
The steering machine is switched on an input of the non-stationary controlled object - a non-wing missile with aerodynamic control. If to assume an output coordinate of a missile as an angle of attack a2 ( t) , and an input coordinate as
an angle of rudder rotation §(t) , we shall determine a transfer function of a missile as [2]
(3)
a2 ( s )
Gp(s) = W
Ka
T2 s2 + 2ç Ts + 1 '
where N - is a force of a dry friction of the cylinder against pistons, S - is a square of the piston, P0 - a pressure in an ink-jet pipe, ç - a coefficient which is taking into account losses of pressure.
The expression in square brackets in the equations (2) represents non-linearity of "limitation+dead zone" type (see Fig.2,b).
where Kg is a conversion ratio of a missile, T - is a time constant, q - is damping factor.
While researching a control system we shall assume, that the dependences of missile parameters on time of flight are determined as following [2]:
T(t) = 0,9849 - 0,1188t + 0,006312 - 0,0001213 ; q(t) = 0,2970 - 0,0535t + 0,0043t2 - 0,0001113; Kgf( t) =16,5475 - 4,4469t + 0,4843t2 - 0,0231513+0,0004t4.
The closed-loop automatic control system of the common object "the steering machine+missile" with a digital fuzzy controller FC is represented in a fig. 4, where the angle of attack ^ t) is input action, and a 2( t) is output of the system.
Figure 2
The equations of the steering machine can finally be presented as
P(t) = k1 m(t- t) , 5 = — 91 (e), e = P - k3n5 . (4)
Block diagram of the pneumatic steering machine which is appropriate to the equations (4) is represented in a fig.3, where 91(e) - non-linearity H1, k0 = k^^n , l = k3n , and the non-linearity H2 characterizes limitation of deviation of a rudder.
Figure 3
While simulating we shall set the following parameters of the steering machine. For an ink-jet pipe: 100; t = 0,01s. For non-linearity H1 (see Fig.2,b): ^ = 0,1; C2 = L = 5. For non-linearity H2 (see Fig.2,a, in which e it is necessary to replace by 5): C2 = L = 5. A transmission factor of an inte-
Figure 4
The digital fuzzy controller FC is switched on between an analog-to-digital converter ADC and digital/analog converter DAC (zero-order hold with a transfer function H(s) ). The error of the system 9(t) = a 1 (t) - a2(t) is
quantized with a sampling interval h .
We research accuracy of improvement by the automatic control system with a digital fuzzy controller of the law of input action change, which is preset: a) by a single step-function and b) by a polynomial [2]
a1(t) = 1 - 1,3316x10 3 + 0,16532691 -- 0,478500812 + 0,103792813 - 8,8016x10-314 + + 3,404 x10-415 - 5,093x10-6t6.
While simulating the system we shall describe the dynamics of separate links, using approximating on a trapezoid rule.
For an oscillatory link with variable parameters:
Ci =
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ISSN 1607-3274 "PaaioeëeKTpoHiKa. iH^opMaTHKa. YnpaBëiHHfl" № 2, 2001
x1v x1v- 1 + 2 (x2v + x2v - 1) ;
X2v 4 + 2bvh0 + avh2 X2v-1 4 + 2b^o + «vA2X1V-1 2hr
4 - 2bv- 1h0 - avh0
- Xo
2(av + av- 1)h0
+-
'0
4 + 2 bvho + avh0°
(avuv + av- 1uv- 1)•
For an integrating link: xv = xv- 1 + — (uv + uv- 1) .
In the written formulas uv - is an input variable, and Xjv - is an output variable of a link; x—v - is intermediate variable. Step of simulation Hq = 0,05H . A sampling interval (interval, with which the data act on an input of fuzzy controller) H = 0,01 s.
For a simplicity of solution of the task of synthesis of an fuzzy controller we shall suppose, that number of terms, through which the linguistic variables (input parameters and output parameter of an fuzzy controller) an error of the system 9 , a speed of change (first derivative) of an error 9 , an
acceleration (second derivative) of an error 9 , a controlling action on the object m are evaluated, is minimum, i.e. is
equal Let's map ranges [9min' 9max] , [9min 9max] ,
[9min 9max] and [ mmin> mmax ] of changes of input and output parameters on uniform universal set Ut = [0, -1] = = [ 0, 1 ], where L{ = 2 - the number appropriate to quantity
of terms by each linguistic variable xt, i = 1, n , n = 4 . The conversion of a fixed value of parameter xi* e [xHi, xsi] into an appropriate element u* e [0, 1 ] is defined by the formulas [7-9]
U1* = (9* - 9min)/(9max - 9min) i u2* = ((3* - 9min)/(9max - 9min) ; u3* = (9* - ëmin)/(9max - 9min) ; u4* = (m* - mmin)/(mmax - mmin) •
u) = e-cu ; 2(u) = e-c( 1 - u), u e [0, 1 ]
(5)
(6)
(7)
(8)
On universal set U = [ 0, 1 ] we shall set two fuzzy subsets, which have the membership functions of exponential form (1 and 2) - see fig. 5.
The analytical expressions exponential membership functions look like:
(9)
At arrival on an fuzzy controller of values of input variables 9* , 9* and 9* with a sampling interval H the calculation of values u1*, u2* and u3* under the formulas (5)-(7)
and of membership functions |j/(u) , j = 1, 2 , is carried out.
We shall generate a linguistic rule of control (a working rule) of fuzzy controller as [7]:
If ( 9* = j) and (9*=a2) and ( 9* = «3), then
(m* =«4), (10)
Figure 5
where and o3 - is linguistic estimates of an error,
of a speed of change (of first derivative) of an error and of an acceleration (of second derivative) of an error considered as of the fuzzy sets, which are defined on universal set,
j = 1, 2 ; o4 - is linguistic estimates of a controlling action on the object, which are selected from term-set by a variable m . The linguistic estimates are selected from term-set of linguistic variables 9* , (9* , 9* and m* :
dj e {negative( 1 ),positive(2)} .
In other words, all signals (above-named linguistic variables) in the system of automatic control are characterized as negative ( j = 1 ) or positive ( j = 2 ).
Let |j ( Xi) ) is the membership function of the parameter Xi* e [xHp xsi] to an fuzzy term a,i, i = 1, 3 ; j = 1, 2 . Then |mj(9, 9, 9) - the membership function (depending from three variable (x1 = 9 ; x2 = 9 ; x3 = 9 )) of a vector of parameters to solution (to selected controlling action on the
object) mj , j = 1, 2, is defined from the system of the fuzzy logical equations:
|mj(x^ x2, x3) = |j(x1 ) a |j(x2) A |j(x3). (11)
Thus, |m1(x1, x2, x3) is the membership function of a controlling action to fuzzy set "negative", and |m2(x1, x2, x3) is the membership function of a controlling action to fuzzy set "positive". Resulting membership function for a controlling action according to working rule of fuzzy controller is written as
|m(x^x2,x3) = |m1(x1,x2,x3)v|™2(x1,x2,x3). (12)
+
In expressions (11) and (12) a - is logical and, v - logical or.
According to linguistic rules of control formalized by the system of the fuzzy logical equations (11), the membership
function of controlling action u4) to fuzzy set "negative" is bounded above by value:
A = min[^1(u1*),^1 (u2*), |1 (u3*)], (13)
and the membership function of controlling action | 2( u4) to fuzzy set "positive" is bounded above by value:
B = min[*2(u1*), *2(u2*), |*2(u3*)]
(14)
The resulting membership function for controlling action on the basis of expression (12) is defined as
*( u4 ) = I1 ( u4 )v| 2( u4 ), (15)
i.e. is acquired by forming of a maxima (line AB in a fig. 5): *(u4) = max[|1 (u4), |2(u4)]. (16)
To define the concrete value of controlling action m* the "resulting figure", limited by the resulting membership function, is formed.
Calculation of an abscissa of a barycentre sc=S(u , |*C) of a part of the square, surrounded by the resulting membership function |*(u) in limits of change of a variable u from u = U1 to u = U2, is made, using a numerical integration on a method of trapezoids (with step of digitization u0), under the formula
M - 1
ut = u„ =
1
M - 1
*0 + y * + *M 2 y ^ 2
i = 1
(17)
m* = mmin + (mmax - mmin)u4*
(19)
with a transfer function H(s) = (1 - e-hs)/s ), and then as continuous controlling action m (t) on the controlled object.
As first and second derivative from an error at simulation is used accordingly first and second difference, namely:
99(k) = [9(k) - 9(k- 1)]/h ; 99 (k) = [9 (k) - 9 (k -1)]/h =
= ([9( k) - 2 9( k - 1) + 9( k - 2)]/h2).
For obtaining satisfactory transients we fulfil an adjustment of a digital fuzzy controller by a variation of the parameter C in exponential membership functions and by a variation of ranges of change of input parameters and output
parameter [9min, 99max] , [9min, 99max] , [ mm
RESULTS OF SIMULATION OF THE SYSTEM
where ( U2 - U1)/M= u0 is step of digitization, M - is number of steps on an interval U2 - U1 , i = 1, 2, 3, ..., M - 1 .
While definiting the resulting membership function it is necessary to define abscissas of cross points of membership functions of fuzzy subsets (of terms "negative" and "positive") with horizontal straight lines.
For membership functions of exponential kind abscissa of a cross point are defined as
u* = -Cln*1(u*) and u* = 1 +Cln*2(u*). (18)
The obtained value u4* on the basis of the formula (8) will be converted to value of controlling action on the controlled object
D.0 0.6
a) 0.4 0.2 0.0 -0.2
f
50 40 30 20 10
b) 0 -10 -20 -30 -40.
20
15 10 5
c) 0 -5 -10 -15
\ L
\ /7 'A v )
v.-
0.0
0.2
0.4
t, c
^ 1 v
\ AIM
\ IU II11 1 1 \
nUffillt iUffl irifr—ii Irti lift n
lini№_UlnnL f m T 1 T V
I
\
0 0.2 0.4 t, c
-NW AAAA/
I 1 V V <
0.0
0.2
0.4 t. c
The controlling actions m* as the code m ( k) with a sampling interval h act at first on the DAC (zero-order hold
Figure 6
In a fig. 6 the results of research of improvement by the automatic-control system (see fig. 4) of an input single step action (of an angle of attack a1 (t)) are represented. Time of observation is 0,6 s. Graphs in figures are presented in the following order: a - is an input action a1 (t) of the system and current error 9(t) = a^t) - a2(t) in the system; b - is a signal on an output of a link of delay (see Fig.3) and restricted signal on an output of non-linearity H1 in the
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ISSN 1607-3274 "Pa^ioeëeKTpoHiKa. Ii^opMaTHKa. YnpaBëiHHfl" № 2, 2001
steering machine; c - is a signal on an input of non-linearity H2 (after the integrator) in the steering machine; d - is a signal of control §( t) on an output of the steering machine (error angle of a rudder).
In a fig. 7 the results of research of improvement by the automatic-control system of the law of change of input action given by the polynom (of the law of change of an angle of attack ^ (t)) is represented. Time of observation is 15 s. Graphs in figures are presented in the following order: a - is an input action a^ t) of the system; b - is a current error 9(t) = a 1(t) - a2(t) in the system; c - is a signal on an output of a link of delay in the steering machine (see Fig.3); d - is a signal of control §( t) on an output of the steering machine (error angle of a rudder).
1.2 1.0 0.8 0.6 0.4 0.2 0.0
/\ Ï- (t)
0.0
0.2
0.4
\ г- (t\
b)
Figure 8
1 о
CONCLUSION
Figure 7
In a fig. 8 the response of the system a2 ( t ) is presented: a) at single step action and b) at polynomial action.
At tuning of a digital fuzzy controller close to optimum (both for single step action, and for polynomial action on an input of the system) are the following parameters of a controller. Coefficient in exponential membership functions is c = 20. Ranges of change of input parameters and output parameter are
[Ömin' 9max] = [-1,5, 1,5] , [9mta, 9max] = [-0,6, 0,6] ,
[Ömin,ëmax] = [-1,5, 1,5] and [m^^, «max] = [-1,0, 1,0] .
The digital fuzzy controller provides fast response of the system and small enough current error in steady-state operation mode. It is necessary to mention, that synthesis of a linear digital controller for the investigated non-stationary object "the steering machine + the missile", in which the steering machine has delay, non-linearities and direct feedback (see fig. 3), is rather complicated task. Therefore, application of a digital fuzzy controller is expedient, more so as the algorithm of its operation is simple enough for controlling objects of any complexity.
REFERENCES
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[3] Гостев В.И., Баранов A.A., Чуприн А.Е., Худолий Д.А. Синтез цифрового нечеткого регулятора системы управления нестационарным объектом // Прац м1жнародноУ конфе-ренцм з автоматичного управлшня "Автоматика-2000": Льв1в, 11-15 вересня 2000 р.-Т.2-Льв1в: Державний HДi шформацшноУ ¡нфраструктури, 2000.-С.63-67.
[4] Гостев В.И., Чуприн А.Е., Лесовой И.П. Синтез цифрового регулятора системы управления нестационарным объектом на базе нечеткой логики // Мехашка та машинобудуван-ня.-2000.-№1.-С.128-133.
[5] Гостев В.И., Стеклов В.К. Системы автоматического управления с цифровыми регуляторами: Справочник.- К.: "Радю-аматор", 1998.-704 с.
[6] Системы управления и динамика полета ракет/Пугачев В.С., Казаков И.Е., Гладков Д.И., Евланов Л.Г., Мишаков А.Ф., Седов В.Д. / Под ред. Пугачева В.С. - М.: Издание ВВИА им. проф. Н.Е. Жуковского, 1965.- 616 с.
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[8] Архангельский В.И., Богаенко И.Н., Грабовский Г.Г., Рюм-шин Н.А. Досви розвитку i застосування систем фуцш-управлшня //Автоматизашя виробничих процеав.-1997.-№2(5).-С.1-10.
[9] Архангельский В.И., Богаенко И.Н., Грабовский Г.Г., Рюмшин Н.А. Системы фуцци-управления.-К.: Техника, 1997.- 208 с.
