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Aircraft Control Based on Localization Method
Copyrigth ~ I~ ."C Motjon Control for Intelligent AUtOr:-.l:','r. Perugja. !laly. October 27·29. 199~
AIRCRAFT CONTROL BASED ON LOCALIZATION METHOD M. BLACHUTA·, K. WOJCIECHOWSKI· and W. YURKEVICHu ·Silesian University of Technology. Control Theory Group. 16 Kalowicka Street. PL 44 101 Gliwice. Poland uNovosibirsk Institute of Elecuical Engineering, 20 K. Marks Avenue, 630092 Novosibirsk. Russia
Abst iQct l= :he paper syntbesis of the aircraft motion control is presented. The control t~~: ::; formulated as tracing problem for Euler angles. Syntbe~ is method is based on :::" Localization Principle . Estimates of time de:ivatives of otput variables are taker: ;;..0 c.:guments of the control law and high gain coefficients in feedback are applied . _-\:; a result output variables bave desired dynamic properties and do not depenci er:: :~-: model uncertainity. J{ e)1lJ.OTCi \' 2nlinear control , localization method, motion control, rigid body control, aircro.:-: 2,',.;tomatic control.
1
Introduction
Synthesis of an ai:;:d2.ne control system is still act ual area of both theoretical and application research. (\'ukobrato\'lc d al., 1988) Control system de::ifn is the most attractive when it achieves very accurate tracking and rejects a very broad class of disturbances (including parameter variation~) with minimal complexity. The simplest approach uses !inearized model. Then classical frequency (Crosley et al.,19i7; Kouvaritakis et a/.,19i9) , HOC or LQ methods (Mc Rouer et al., 19i3, Brousard et al., 1985; Grimble et 01.,1991) are applied . The above methods give satisfactory results if parameters of the airplane are known and they are restricted to small perturbation from the static work points. Model reference and adaptive controllers (Bukov, 19Si; Chandrasekhar et al., 198i) are less sensitive to nonlinearities and parameter ....ariations. Methods connected with inverse model (Petrov et 01.,1981) work well with nonlinear models and large perturbations but the exact model of the plant must be known. Variable structure sliding mode (Calise et 0/.,1983) algorithms require less information and seem to be very promising. The performance of presented controllers which are based on the Localization Method (Vostrikov 1988,1990; Vostrikov ft ai.,1991) is comparable with that of sliding mode algorithms. When com1 Supponed
by KB!'\.
gr&rl15
paring them with variable structure controllers, the main feature is that they use continuous control signal instead of producing a discontinuous one. Elimination of noniinear dynamic interactions is achieved by introducing large gains and feed backs not only from outputs but also from estimates of their time derivatives. The synthesis method presented in the paper does not require any linearization of the model and enables to obtain desired dynamic properties of the closed system. The method was tested for the case of longitudinal flight control (Yurkevich et 0/.,1991a,b) whereas in the present paper the spatial flight case is addressed. The paperl is organized as follows. In section 2 we give a model of an airplane. Then we introduce an idea of airplane control based on the Localization Method. In section 4 we discuss a more realistic control algorithm which instead of true output derivatives uses their estimates. In section 5 we give a simple design procedure which is followed by a design example in the next section. Finally, some remarks about computer simulation of the control system are given.
2
Model of aircraft motion
The following reference frames, i.e. right-banded orthogonal coordinate systems are introduced
3068391 and BK4&4
II - 19
BL ACH CTA M ..
YL~ hE \ · l C H
a) Inertial system (O,x,y , z) fixed to earth with the coordinate origin 0 located at ground surface with th · = axis of a vertically downwards sense b) Air( t-carried earth axis system (CM, Z 9' Yg, z ) with origin located at the airplane center of ~ass , with the axis parallel to those of the inertial system . c) Body-axis system (CM, x. , y" z.) with the origin located at the airplane center of mass ; x, axis is in the forward direction and z. axis is in th~ plane of symetry and of vertically downwards sense . d) Flight path axis system (CM,zo,Ya,zo) where Xa and velocity vector are colinear, with Zo in the plane of symmetry and the centre located at the airplane centre of mass.
v. .. W CJJC IEC HOV. S;';' l r. .
0'0(0', P)
SOlCP -SOlSP
[
+ cq'lc~
-s8 ] sq'lc8 cq'lc()
[n
-sq'l ] cq'lsOlc() cq'l/c8
~:: ]
= [
~ ] - 0'9 [ ::: ]
W
n
D;, [ :]
(10)
(11)
= diag(Sz,Sy,Sz)
We assume that aerodynamic force and moment coefficients depend on angles 0' and p and on control surfaces deflections Oh , ov, 0/ and can be presented with sufficient accuracy as follows :
(2)
• + c~O' • + c~~0'2 • + + r!! (3 + r!!~ p2 + C~Oh + + c~:30~ + cio" + c~~6: + + c::3°f = m? + mr 0' + mf f3 + + m~oh + miov + m~ol
=
Let 0' denotes an angle of attack and {3 denotes a sideslip angle. The relationship between 0'. (3. velocities u, v, w and wind velocities V wz , tiWI/' tiwz is given by equations (3) through (6) .
Vaz
] = T. [
with J = diag(Jz,JI/,Jz),S and L = diag(L z , LI/' Lz).
(1)
[
~
=
cd>s8s~ - sq'lc~
cq'l sq'ls8lc8 sq'l I c()
(9)
+
D. g (8 , q'l,~) = c8s~
COl
=
Let us first introduce some transformation matrices . If 'sin ' and 'cos ' are abbreviated to 6 and c then
stps()s~
(7)
The system of state equations is as follows :
Assuming that an airplane is a rigid body with six degrees of freedom , the state of aircraft motion is defined by twelve coordinates. The following state vector is adopted [u ,v, w,p , q , r, 8,q'l ,~, x,y,zV. Here U , V,ll' and p , q, r are projections of velocity and angular velocity onto the x,, Y. and z. axis of the body axis system respectively. (see Vukobratovic et al.,(1988) for the meaning of Euler angles, velocities and coordinate systems)
c8c1j; sqJs()c1j; - cq'lsv' [ cq'ls()cv' + sq'lsv'
= [C;~P -~psP -~O']
(3)
Vwz
for i
c~
= z, Ji, z.
(4)
3 (5)
(6)
Idea of a control law
We state the control task as a tracing problem of the Euler angles, which determine orientation of the airplane with respect to the inertial coordinate system:
We are now able to define the last transformation matrix 010
II - 20
lim 9(t)
__ 00
=60(t),
lim ,pet) = ,po(t) '
__ 00
AIRCRAFT CO '
lim .p(t)
'_00
,W L BASED 0 '. LOC A LlZA 11 0 '. ~ 1i: THO D
= .po(t).
It is also convenient to choose Kl in we form :
where Bo(t),
(15) where H = diag(h9 , h~,h",)
(16)
Substituting equations (14) and (15) into equation (12) and equation (12) into equation (14) we get:
(17)
and
(12)
where
fi C) = li(8 ,
(18)
i=8,4J,1/J Letting k tend to infinity we get asymptotic formulae
and
(19)
and In normal flight conditions we have detB(·) =F 0 and the values of functions 1liO I, i = B,,p,,p are bounded . Let us assume that the desired dynamics may be determined by a set of mutually independent differential equations:
B(2) ,p(2) ,p(2)
F,(e(l),B,Bo)
=
F~(
(13)
= F",(,p(l),,p,,po)
l~..~ [ :~ ] = 0/
B-
1 {
[
~:
F",
]- [
J:H ] }
1",(-)
(20) From equation (19) we see that when using control algorithm (18) with increasing values of the gain k the asymptotic dynamics of the controlled airplane is the same as the desired dynamics defined in (13). The control is realizable (Yurkevich, 1986) if
We now choose a control law in the form
[
lim {[6;:(t)F + [6:(t)]2
~; ] = KoK,
'-00
.{[~:~~(t:(~,B;~) ]_ [:~:~ ]} F",(,p(1),,p, ,po)
=
(14)
,p(2)
where Kl diag(k" le", k",) is a matrix of amplification factors and Ko is a matching matrix. The choice of matrix Ko will be discussed in section 5.
+ [6r(tW} < 00
(21)
This condition can be satisfied only if the set point is chosen from equilibrium conditions of stationary flight. In other situations we get a quasi-stationary trajectory in which Euler angles are stabilized but velocities U,v,W change in time. Such trajectories are called 'degenerate trajectories' (Ftantsuzova, 1986) and they may be stable or not. In the case of finite-time manoevers condition (21) plays no role.
II - 21
BLACHL'TA M .. YURKEYI-':H
4
w .. WOJCIECHOWS~l
Substituting equa~ion (23) into equation (12) we get
Cont"'ol algorithm
Time derivatives of Euler angles are practically not a vailable.They can be estimated using differential filters
+ 2del-'eO(1) + 0 1-'~~(2) + 2d~I-'~~(1) + ~ = 1-'~0(2)
l-'~tiP) + 2dv-I-'v-.,jP) +,j;
=
K
()(2)
tP(2)
[
1/;(2)
1= [/e(-) 1+ /~(-)
B(-)KoK 1 ·
/"'(-)
() 4;
(22)
(27)
.,p
A more realistic control algorithm can be obtained using estimates 0(1), ~(l), ,j;(l), 0(2), ~(2), ~(2) in (18) instead of exact values ()(1),4;(l),.,p(1),()(2),
Next, combining equations (26) and (27) we have
4P),.,p(2).
[:~I 1 :"
= KoKl · (28) (23)
Time constant 1-'0 = max(1-'8, 1-'4J) 1-'",) plays the role of small parameter so that properties of the closed-loop system can be analysed by distinguishing two subsystems: a slow one and a fast one. The slow subsystem refers to the plant variables whereas the fast subsystem refers to control signal and controller variables. From equation (22) it is clear that letting 1-'0 - 0 we get the slow subsystem with variables u,\',w,p,q,r,e, 4;,.,p whose equations are the same as in (8)-(11) and (17) i.e. as when using ideal time derivatives. To get the equations of the fast subsystem let us denote:
L(I-',p)
=
diag[le(l-'ep), 1~(I-'~p), lv-(I-''''p)] I-'
= (jl8,1-'~,1-'''']
=0 tPl = ~l, ()l
1
,
()2 tP2
= ff2 = ~2
where functions 'Pe(t), 'P~(t), 'Pop(t) depend on the 'frozen' values of U,V ,w ,p,q,r,(), 4;, t/J=const and solution to the equation (24) . To make properties of the control system with differential filters close to the ideal system we have to ensure asymptotic stability of the fast subsystem and make its transients fast enough. This can be achieved by a proper choice of matrices Ko, Kl and parameters I-'e, I-'~, I-'~ such that eigenvalues of system (29) have desired values for all possible arguments of matrix B(·).
5
Now the equation (22) and their time derivatives can be symbolically written as
L(~,p) [ j ]
Changing the time scale to T = 1-'0 it and letting 1-'0 -. 0 from equations (8) -(11) we get U,v,w,p,q,r,(), 4;, t/J=const for finite values oft. As a result equations of the fast system take the form:
Controller design
We first choose parameters that determine fast dynamics of the controller. From equation (29) we see that it would be convenient to have the matrix B(·)K o diagonal. This would be possible by taking
(24)
L(~,p) [ :: J
=
(25)
L(~,p) [ ~]
=
(26)
As the values of 0' and f3 are not measured we cannot implement this formula in our design. However both 0' and f3 are small in normal flight conditions from which results that D. a (O',f3) is close to the unity matrix. Therefore a convenient choice is
n - 22
AIRCRAFT COSTIWL BASED O~ LuC A LlZA 11 0 " ~1 E THO D
Denote diagonal elements of matrix tJ- 2 B(·)Ko by k 16 , k~~, k",,,, . Assuming that the off diagonal elements are negligible the system (29) becomes diagonal _:; d is characterized by three characteristic polynomials : (32)
6
Design example
For an airplane whose parameters are given in tab.! through tab.2 parameters of the multivariable regulator have been calculated. Table 1.
for i = (), 4J, 1/; .
J" Jz
We have to choose di and I-'i, for i = (), 4J, 1/;, so that characteristic polynomials take the form
+ 2d Oil-'Oi). + 1
l-'~i).2
Lr
(33)
L"
L.
for i
= (), 4J, 1/;.
Denote 1-'0
=
max(/-l09, I-'Oql> 1-'0"')
Then good separation of the fast subsystem from the slow one occurs when 1-'0 ~ 0.lmin(T9 , T~ ,
m~
Ly Lz
m2 m2
p
kgJm,j mJs< NJkg%
9
Design assumptions are
T",)
l-loi(1 + ~pki)! doi ( l + ~pki)!.
=
Lz
le
Observe that if Ko and Kl are chosen according to formulae (30) and (15) respectively and u ~ tio then k99 ~ kt;>~ ~ kIP" ~ ~p . From (32) and (33) we get finally I-'i di
value 2000 2000 5000 10000 0.5 0.5 0.5 0.5 2.0 10.0 1.2 9.81 0.01
unit kg kg me.< kg m 2 kg m 2 m m m
symbol m Jr
Te
= TIP = T", = 5[5J
0e
=
OI~
=
01",
= 0.7
(34)
doe = do~ = do", = 0.2
for i
= (), 4J, 1/; .
The choice of values of k9, k~, kIP will be discussed later . Let the desired dynamics be given by a set of second order differential equations T - 2 [-2a:e Te()(1)
()P )
e 'T;2[-2a:~'T~4P)
+ 00 ] 4J + 4>0]
1~:lmar
= 1~~lml1r = 1~~lml1r = 0.0174[rad]
Table 2
- ()
1
d/'"
caI
Ca I
CIo2
d/
0.0 0.0 -8.6
-0.002 0.0 0.0
0.0 -0.005 0.0
0.0 0.0 0.0057
= (35) 1/;(2 ) = T,;2[-2a:",T",1/;(1) - 1/; + 1/;0]
X
z
-0.2 0.0 -0.15
Parameters 'T9,T~,'T1/; and 0I9,0I~,0I", have very well known meaning and must be chosen by the designer. Parameters k and h" h., h,p determine the steady-state error . Denote 0., 4>., 1/;. the steady-state values of Euler angles and ~; = 00 - 0., ~~ = 4>0 - 4>., ~~ = tPo - tP. the steady-state control errors. Assume that we require I~:I ~ 1~:lml1r for i = O,4>,tP. From equations (12) and (14) results
i
cf
cf:l
CV
CV'" I
c~'"
X
0.0 0.0 0.0001
-0 .002 0.0 0.0
0.0 -0.0025 0.0
-0 .002 0.0 0.0
-0.002 0.0 0.0
4J (2)
[tt, ]
= _k-l[B(·)KoKlTtl [
~~
~:n 1",(-)
y
Y
z
Using formulae from section 5 we have calculated following values of controller parameters p,
]
(36)
L .
~.
=
Lh .
~.
= p.
=P,p =0.5[8]
dB = 1.4, d. = 3.3, d", = 3.0
where
· (-2 T = d tag Te ,T~-2 ,'T,p-2) Combining equations (30) and (36) we see that the values of k and h" h~, h,p can be chosen from relations
I
h,
=0.8, h. = 4.4, h,p = 3.6, It = 100
Table 3.
11;(·)1. ~lor t. =(}.J. ./. (37) -> 2p-1T.~• 1~:lm= ,¥I, ¥I II - 23
1
m~
X
0.0 0.057 0.0
y
z
•
m'! • 0.0 0.0 -0.011
m~
•
0.0 -0.01 0.0
mV • -0.004 0.0 0.0008
m~ I
-0.04 0.0 -0.00002
BLACHt .:TA M .. )"'"RKEV 1CH W ..
Gri; .. ole,M., and S.Carr (1991) . Compa rison of LQG, HOO and classical designs for the pitch control of an ASTO L aircraf t , Report fCU/SS 1 , Indust rial Contro l Unit, Strath clyde Univer sity, Glasgo w .
Sim ulati on resu lts
7
Because of noniin ear nature of the airpian e and some approx imatio ns and simpiif ication s in reasoning made in previou s section s a numeric~l verification of results is necessary. Moreover, slmula tion of dynam ical system gives plots of variabl es which cannot be calcula ted analyti cally. Therefore a compu ter simula tion progra m has been constructe d which offers possibi lity of various experiments . We have perform ed a large numbe r of them changi ng stepwis e setpoin ts, wind velocity and the mass of the airplan e. In all cases we have found that the control system works well according to the assume d proper ties. BecaU.'le of the lack of space simula tion results will be presen ted at the Conference . Some examp les are displayed in Biachu ta et al.,(1992) .
8
WOJC1ECHO\l..:>~ k.
Kouva ritakis , B .,W.M urray and A .Mac Farlan e (1979) . Charac teristic frequen cy-gain design study of an autom atic flight control system , Int.J. Control, vo1.29,No 2,pp. 325-358. Petrov ,B. and P.Krut ko (1981). Synthe sis of flight control algorit hms based on solutio ns of inverse dynam ics tasks, Tekhni cheska ya Kibern etika, No 2-3. Mc Ruer, D. , I.Ashk enas and D .Graha m (1973) . A ircraft Dynam ics and Autom atic Contro l, Prince ton Univer sity Press , New Jersey. Medve dov, V. and A .Maksi mov (1975) . Analyt ical design of aircraf t control system , V IFAC Sympo sium on Autom atic Contro l in Space, Genov a
Refe renc es
Blachu ta,M., Z.Dud a , A.Pola nski and K.Wojciechowski (1992) . Visual feedback for rigid body motion control , in preprin ts of Worksh op on Motion Control for Intellig ent A utomatio n , Perugi a, Italy.
Vostrik ov , A . (1988) Autom atic Contro l Theory . Localization Princip le, NIEE Press, Novosi birsk . (in Russia n)
BukoY , V. (1987) . Adapti ve Predictive Flight C ontrol S ystems , Nauka, Moskow .
Vostrik ov, A. and V.Yurkevich (1991) . Design of control system s by the localiz ation metho d (survey ), Proceedings of the Intern ationa l Workshop, Novosi birsk.
Brousa rd ,R ., R .Down ing and W .Bryan t (1985). Design and flight testing of a digital optima l control genera l aviatio n autopil ot, Autom atIca , voJ.21 , No l.
Vostrik ov, A. (1990) . Synthe sis of Nonlin ear System s by the Localization Method, NIEE Press, Novosi birsk . (in Russia n)
Calise,A. and F.Kram er (1983) . A variabl e structure approa ch to robust control of VTOL aircraf t, AIAA Guidance and Contro l Conference , Gatlin burg . Chand rasekh ar, J . and M.Rao (1987) . A new model reference adaptiv e aircraf t controller . JO-th World IFAC Congress, vo1.6, pp.128 -143, Munic h . Crosley, T., N .Munro and K.Beth orn (1977). Design of aircraf t autosta bilizat ion system using Inverse Nyquis t Array method , IFAC Multiva riable Technological System s Sympo sium, Freder iction Frantsu zowa, G. (1986). On degene rated motion s in the proble m of multiv ariable system s stabilizati on, in Autom atic Contro l for Plants with Varying Characteristics, NIEE Press, Novosi birsk . (in Russia n) Gerash chenko , F. and S.Gera shchen ko (1975). A Method of Motion Separa tion and Optimiza tion of Non-Li near System s, Nauka , Moscow . (in Russia n'
Vukob ratovic , M. and R.Stoji c (1988) . Modern Aircra ft Flight Control, Lectur e Notes in Contro l and Inform ation Scienc es, 109, Spring er-Verlag, Berlin Yurkev ich, V. (1986) . Realiza bility condit ions of desired motion s and synthe sis of system s with the velocit y vector in feedba ck, Ph.D thesis, NOV08ibirsk Institu te of Electri cal Engine ering. (in Russia n) Yurkevich, V. (1988) . On the contro l stabili ty of dynam ic system s, in Autom atic Contro l of Plants with Varyin g Charac teristic s, NIEE Press, Novosi birsk. (in Russia n) Yurkevich, V., M.Bla chuta and K.Woj ciecho wski (1991) , Stabili zation system for the aircraf t longitU dinal motion , Archiw um Autom atyh i Roboty h, vol.XX XVI, No 3-4, pp.517 -535. (in Polish)
II - 24
AIRCRAFT CONTROL BASED ON LOCALIZATION METHOD M. BLACHUTA·, K. WOJCIECHOWSKI· and W. YURKEVICHu ·Silesian University of Technology. Control Theory Group. 16 Kalowicka Street. PL 44 101 Gliwice. Poland uNovosibirsk Institute of Elecuical Engineering, 20 K. Marks Avenue, 630092 Novosibirsk. Russia
Abst iQct l= :he paper syntbesis of the aircraft motion control is presented. The control t~~: ::; formulated as tracing problem for Euler angles. Syntbe~ is method is based on :::" Localization Principle . Estimates of time de:ivatives of otput variables are taker: ;;..0 c.:guments of the control law and high gain coefficients in feedback are applied . _-\:; a result output variables bave desired dynamic properties and do not depenci er:: :~-: model uncertainity. J{ e)1lJ.OTCi \' 2nlinear control , localization method, motion control, rigid body control, aircro.:-: 2,',.;tomatic control.
1
Introduction
Synthesis of an ai:;:d2.ne control system is still act ual area of both theoretical and application research. (\'ukobrato\'lc d al., 1988) Control system de::ifn is the most attractive when it achieves very accurate tracking and rejects a very broad class of disturbances (including parameter variation~) with minimal complexity. The simplest approach uses !inearized model. Then classical frequency (Crosley et al.,19i7; Kouvaritakis et a/.,19i9) , HOC or LQ methods (Mc Rouer et al., 19i3, Brousard et al., 1985; Grimble et 01.,1991) are applied . The above methods give satisfactory results if parameters of the airplane are known and they are restricted to small perturbation from the static work points. Model reference and adaptive controllers (Bukov, 19Si; Chandrasekhar et al., 198i) are less sensitive to nonlinearities and parameter ....ariations. Methods connected with inverse model (Petrov et 01.,1981) work well with nonlinear models and large perturbations but the exact model of the plant must be known. Variable structure sliding mode (Calise et 0/.,1983) algorithms require less information and seem to be very promising. The performance of presented controllers which are based on the Localization Method (Vostrikov 1988,1990; Vostrikov ft ai.,1991) is comparable with that of sliding mode algorithms. When com1 Supponed
by KB!'\.
gr&rl15
paring them with variable structure controllers, the main feature is that they use continuous control signal instead of producing a discontinuous one. Elimination of noniinear dynamic interactions is achieved by introducing large gains and feed backs not only from outputs but also from estimates of their time derivatives. The synthesis method presented in the paper does not require any linearization of the model and enables to obtain desired dynamic properties of the closed system. The method was tested for the case of longitudinal flight control (Yurkevich et 0/.,1991a,b) whereas in the present paper the spatial flight case is addressed. The paperl is organized as follows. In section 2 we give a model of an airplane. Then we introduce an idea of airplane control based on the Localization Method. In section 4 we discuss a more realistic control algorithm which instead of true output derivatives uses their estimates. In section 5 we give a simple design procedure which is followed by a design example in the next section. Finally, some remarks about computer simulation of the control system are given.
2
Model of aircraft motion
The following reference frames, i.e. right-banded orthogonal coordinate systems are introduced
3068391 and BK4&4
II - 19
BL ACH CTA M ..
YL~ hE \ · l C H
a) Inertial system (O,x,y , z) fixed to earth with the coordinate origin 0 located at ground surface with th · = axis of a vertically downwards sense b) Air( t-carried earth axis system (CM, Z 9' Yg, z ) with origin located at the airplane center of ~ass , with the axis parallel to those of the inertial system . c) Body-axis system (CM, x. , y" z.) with the origin located at the airplane center of mass ; x, axis is in the forward direction and z. axis is in th~ plane of symetry and of vertically downwards sense . d) Flight path axis system (CM,zo,Ya,zo) where Xa and velocity vector are colinear, with Zo in the plane of symmetry and the centre located at the airplane centre of mass.
v. .. W CJJC IEC HOV. S;';' l r. .
0'0(0', P)
SOlCP -SOlSP
[
+ cq'lc~
-s8 ] sq'lc8 cq'lc()
[n
-sq'l ] cq'lsOlc() cq'l/c8
~:: ]
= [
~ ] - 0'9 [ ::: ]
W
n
D;, [ :]
(10)
(11)
= diag(Sz,Sy,Sz)
We assume that aerodynamic force and moment coefficients depend on angles 0' and p and on control surfaces deflections Oh , ov, 0/ and can be presented with sufficient accuracy as follows :
(2)
• + c~O' • + c~~0'2 • + + r!! (3 + r!!~ p2 + C~Oh + + c~:30~ + cio" + c~~6: + + c::3°f = m? + mr 0' + mf f3 + + m~oh + miov + m~ol
=
Let 0' denotes an angle of attack and {3 denotes a sideslip angle. The relationship between 0'. (3. velocities u, v, w and wind velocities V wz , tiWI/' tiwz is given by equations (3) through (6) .
Vaz
] = T. [
with J = diag(Jz,JI/,Jz),S and L = diag(L z , LI/' Lz).
(1)
[
~
=
cd>s8s~ - sq'lc~
cq'l sq'ls8lc8 sq'l I c()
(9)
+
D. g (8 , q'l,~) = c8s~
COl
=
Let us first introduce some transformation matrices . If 'sin ' and 'cos ' are abbreviated to 6 and c then
stps()s~
(7)
The system of state equations is as follows :
Assuming that an airplane is a rigid body with six degrees of freedom , the state of aircraft motion is defined by twelve coordinates. The following state vector is adopted [u ,v, w,p , q , r, 8,q'l ,~, x,y,zV. Here U , V,ll' and p , q, r are projections of velocity and angular velocity onto the x,, Y. and z. axis of the body axis system respectively. (see Vukobratovic et al.,(1988) for the meaning of Euler angles, velocities and coordinate systems)
c8c1j; sqJs()c1j; - cq'lsv' [ cq'ls()cv' + sq'lsv'
= [C;~P -~psP -~O']
(3)
Vwz
for i
c~
= z, Ji, z.
(4)
3 (5)
(6)
Idea of a control law
We state the control task as a tracing problem of the Euler angles, which determine orientation of the airplane with respect to the inertial coordinate system:
We are now able to define the last transformation matrix 010
II - 20
lim 9(t)
__ 00
=60(t),
lim ,pet) = ,po(t) '
__ 00
AIRCRAFT CO '
lim .p(t)
'_00
,W L BASED 0 '. LOC A LlZA 11 0 '. ~ 1i: THO D
= .po(t).
It is also convenient to choose Kl in we form :
where Bo(t),
(15) where H = diag(h9 , h~,h",)
(16)
Substituting equations (14) and (15) into equation (12) and equation (12) into equation (14) we get:
(17)
and
(12)
where
fi C) = li(8 ,
(18)
i=8,4J,1/J Letting k tend to infinity we get asymptotic formulae
and
(19)
and In normal flight conditions we have detB(·) =F 0 and the values of functions 1liO I, i = B,,p,,p are bounded . Let us assume that the desired dynamics may be determined by a set of mutually independent differential equations:
B(2) ,p(2) ,p(2)
F,(e(l),B,Bo)
=
F~(
(13)
= F",(,p(l),,p,,po)
l~..~ [ :~ ] = 0/
B-
1 {
[
~:
F",
]- [
J:H ] }
1",(-)
(20) From equation (19) we see that when using control algorithm (18) with increasing values of the gain k the asymptotic dynamics of the controlled airplane is the same as the desired dynamics defined in (13). The control is realizable (Yurkevich, 1986) if
We now choose a control law in the form
[
lim {[6;:(t)F + [6:(t)]2
~; ] = KoK,
'-00
.{[~:~~(t:(~,B;~) ]_ [:~:~ ]} F",(,p(1),,p, ,po)
=
(14)
,p(2)
where Kl diag(k" le", k",) is a matrix of amplification factors and Ko is a matching matrix. The choice of matrix Ko will be discussed in section 5.
+ [6r(tW} < 00
(21)
This condition can be satisfied only if the set point is chosen from equilibrium conditions of stationary flight. In other situations we get a quasi-stationary trajectory in which Euler angles are stabilized but velocities U,v,W change in time. Such trajectories are called 'degenerate trajectories' (Ftantsuzova, 1986) and they may be stable or not. In the case of finite-time manoevers condition (21) plays no role.
II - 21
BLACHL'TA M .. YURKEYI-':H
4
w .. WOJCIECHOWS~l
Substituting equa~ion (23) into equation (12) we get
Cont"'ol algorithm
Time derivatives of Euler angles are practically not a vailable.They can be estimated using differential filters
+ 2del-'eO(1) + 0 1-'~~(2) + 2d~I-'~~(1) + ~ = 1-'~0(2)
l-'~tiP) + 2dv-I-'v-.,jP) +,j;
=
K
()(2)
tP(2)
[
1/;(2)
1= [/e(-) 1+ /~(-)
B(-)KoK 1 ·
/"'(-)
() 4;
(22)
(27)
.,p
A more realistic control algorithm can be obtained using estimates 0(1), ~(l), ,j;(l), 0(2), ~(2), ~(2) in (18) instead of exact values ()(1),4;(l),.,p(1),()(2),
Next, combining equations (26) and (27) we have
4P),.,p(2).
[:~I 1 :"
= KoKl · (28) (23)
Time constant 1-'0 = max(1-'8, 1-'4J) 1-'",) plays the role of small parameter so that properties of the closed-loop system can be analysed by distinguishing two subsystems: a slow one and a fast one. The slow subsystem refers to the plant variables whereas the fast subsystem refers to control signal and controller variables. From equation (22) it is clear that letting 1-'0 - 0 we get the slow subsystem with variables u,\',w,p,q,r,e, 4;,.,p whose equations are the same as in (8)-(11) and (17) i.e. as when using ideal time derivatives. To get the equations of the fast subsystem let us denote:
L(I-',p)
=
diag[le(l-'ep), 1~(I-'~p), lv-(I-''''p)] I-'
= (jl8,1-'~,1-'''']
=0 tPl = ~l, ()l
1
,
()2 tP2
= ff2 = ~2
where functions 'Pe(t), 'P~(t), 'Pop(t) depend on the 'frozen' values of U,V ,w ,p,q,r,(), 4;, t/J=const and solution to the equation (24) . To make properties of the control system with differential filters close to the ideal system we have to ensure asymptotic stability of the fast subsystem and make its transients fast enough. This can be achieved by a proper choice of matrices Ko, Kl and parameters I-'e, I-'~, I-'~ such that eigenvalues of system (29) have desired values for all possible arguments of matrix B(·).
5
Now the equation (22) and their time derivatives can be symbolically written as
L(~,p) [ j ]
Changing the time scale to T = 1-'0 it and letting 1-'0 -. 0 from equations (8) -(11) we get U,v,w,p,q,r,(), 4;, t/J=const for finite values oft. As a result equations of the fast system take the form:
Controller design
We first choose parameters that determine fast dynamics of the controller. From equation (29) we see that it would be convenient to have the matrix B(·)K o diagonal. This would be possible by taking
(24)
L(~,p) [ :: J
=
(25)
L(~,p) [ ~]
=
(26)
As the values of 0' and f3 are not measured we cannot implement this formula in our design. However both 0' and f3 are small in normal flight conditions from which results that D. a (O',f3) is close to the unity matrix. Therefore a convenient choice is
n - 22
AIRCRAFT COSTIWL BASED O~ LuC A LlZA 11 0 " ~1 E THO D
Denote diagonal elements of matrix tJ- 2 B(·)Ko by k 16 , k~~, k",,,, . Assuming that the off diagonal elements are negligible the system (29) becomes diagonal _:; d is characterized by three characteristic polynomials : (32)
6
Design example
For an airplane whose parameters are given in tab.! through tab.2 parameters of the multivariable regulator have been calculated. Table 1.
for i = (), 4J, 1/; .
J" Jz
We have to choose di and I-'i, for i = (), 4J, 1/;, so that characteristic polynomials take the form
+ 2d Oil-'Oi). + 1
l-'~i).2
Lr
(33)
L"
L.
for i
= (), 4J, 1/;.
Denote 1-'0
=
max(/-l09, I-'Oql> 1-'0"')
Then good separation of the fast subsystem from the slow one occurs when 1-'0 ~ 0.lmin(T9 , T~ ,
m~
Ly Lz
m2 m2
p
kgJm,j mJs< NJkg%
9
Design assumptions are
T",)
l-loi(1 + ~pki)! doi ( l + ~pki)!.
=
Lz
le
Observe that if Ko and Kl are chosen according to formulae (30) and (15) respectively and u ~ tio then k99 ~ kt;>~ ~ kIP" ~ ~p . From (32) and (33) we get finally I-'i di
value 2000 2000 5000 10000 0.5 0.5 0.5 0.5 2.0 10.0 1.2 9.81 0.01
unit kg kg me.< kg m 2 kg m 2 m m m
symbol m Jr
Te
= TIP = T", = 5[5J
0e
=
OI~
=
01",
= 0.7
(34)
doe = do~ = do", = 0.2
for i
= (), 4J, 1/; .
The choice of values of k9, k~, kIP will be discussed later . Let the desired dynamics be given by a set of second order differential equations T - 2 [-2a:e Te()(1)
()P )
e 'T;2[-2a:~'T~4P)
+ 00 ] 4J + 4>0]
1~:lmar
= 1~~lml1r = 1~~lml1r = 0.0174[rad]
Table 2
- ()
1
d/'"
caI
Ca I
CIo2
d/
0.0 0.0 -8.6
-0.002 0.0 0.0
0.0 -0.005 0.0
0.0 0.0 0.0057
= (35) 1/;(2 ) = T,;2[-2a:",T",1/;(1) - 1/; + 1/;0]
X
z
-0.2 0.0 -0.15
Parameters 'T9,T~,'T1/; and 0I9,0I~,0I", have very well known meaning and must be chosen by the designer. Parameters k and h" h., h,p determine the steady-state error . Denote 0., 4>., 1/;. the steady-state values of Euler angles and ~; = 00 - 0., ~~ = 4>0 - 4>., ~~ = tPo - tP. the steady-state control errors. Assume that we require I~:I ~ 1~:lml1r for i = O,4>,tP. From equations (12) and (14) results
i
cf
cf:l
CV
CV'" I
c~'"
X
0.0 0.0 0.0001
-0 .002 0.0 0.0
0.0 -0.0025 0.0
-0 .002 0.0 0.0
-0.002 0.0 0.0
4J (2)
[tt, ]
= _k-l[B(·)KoKlTtl [
~~
~:n 1",(-)
y
Y
z
Using formulae from section 5 we have calculated following values of controller parameters p,
]
(36)
L .
~.
=
Lh .
~.
= p.
=P,p =0.5[8]
dB = 1.4, d. = 3.3, d", = 3.0
where
· (-2 T = d tag Te ,T~-2 ,'T,p-2) Combining equations (30) and (36) we see that the values of k and h" h~, h,p can be chosen from relations
I
h,
=0.8, h. = 4.4, h,p = 3.6, It = 100
Table 3.
11;(·)1. ~lor t. =(}.J. ./. (37) -> 2p-1T.~• 1~:lm= ,¥I, ¥I II - 23
1
m~
X
0.0 0.057 0.0
y
z
•
m'! • 0.0 0.0 -0.011
m~
•
0.0 -0.01 0.0
mV • -0.004 0.0 0.0008
m~ I
-0.04 0.0 -0.00002
BLACHt .:TA M .. )"'"RKEV 1CH W ..
Gri; .. ole,M., and S.Carr (1991) . Compa rison of LQG, HOO and classical designs for the pitch control of an ASTO L aircraf t , Report fCU/SS 1 , Indust rial Contro l Unit, Strath clyde Univer sity, Glasgo w .
Sim ulati on resu lts
7
Because of noniin ear nature of the airpian e and some approx imatio ns and simpiif ication s in reasoning made in previou s section s a numeric~l verification of results is necessary. Moreover, slmula tion of dynam ical system gives plots of variabl es which cannot be calcula ted analyti cally. Therefore a compu ter simula tion progra m has been constructe d which offers possibi lity of various experiments . We have perform ed a large numbe r of them changi ng stepwis e setpoin ts, wind velocity and the mass of the airplan e. In all cases we have found that the control system works well according to the assume d proper ties. BecaU.'le of the lack of space simula tion results will be presen ted at the Conference . Some examp les are displayed in Biachu ta et al.,(1992) .
8
WOJC1ECHO\l..:>~ k.
Kouva ritakis , B .,W.M urray and A .Mac Farlan e (1979) . Charac teristic frequen cy-gain design study of an autom atic flight control system , Int.J. Control, vo1.29,No 2,pp. 325-358. Petrov ,B. and P.Krut ko (1981). Synthe sis of flight control algorit hms based on solutio ns of inverse dynam ics tasks, Tekhni cheska ya Kibern etika, No 2-3. Mc Ruer, D. , I.Ashk enas and D .Graha m (1973) . A ircraft Dynam ics and Autom atic Contro l, Prince ton Univer sity Press , New Jersey. Medve dov, V. and A .Maksi mov (1975) . Analyt ical design of aircraf t control system , V IFAC Sympo sium on Autom atic Contro l in Space, Genov a
Refe renc es
Blachu ta,M., Z.Dud a , A.Pola nski and K.Wojciechowski (1992) . Visual feedback for rigid body motion control , in preprin ts of Worksh op on Motion Control for Intellig ent A utomatio n , Perugi a, Italy.
Vostrik ov , A . (1988) Autom atic Contro l Theory . Localization Princip le, NIEE Press, Novosi birsk . (in Russia n)
BukoY , V. (1987) . Adapti ve Predictive Flight C ontrol S ystems , Nauka, Moskow .
Vostrik ov, A. and V.Yurkevich (1991) . Design of control system s by the localiz ation metho d (survey ), Proceedings of the Intern ationa l Workshop, Novosi birsk.
Brousa rd ,R ., R .Down ing and W .Bryan t (1985). Design and flight testing of a digital optima l control genera l aviatio n autopil ot, Autom atIca , voJ.21 , No l.
Vostrik ov, A. (1990) . Synthe sis of Nonlin ear System s by the Localization Method, NIEE Press, Novosi birsk . (in Russia n)
Calise,A. and F.Kram er (1983) . A variabl e structure approa ch to robust control of VTOL aircraf t, AIAA Guidance and Contro l Conference , Gatlin burg . Chand rasekh ar, J . and M.Rao (1987) . A new model reference adaptiv e aircraf t controller . JO-th World IFAC Congress, vo1.6, pp.128 -143, Munic h . Crosley, T., N .Munro and K.Beth orn (1977). Design of aircraf t autosta bilizat ion system using Inverse Nyquis t Array method , IFAC Multiva riable Technological System s Sympo sium, Freder iction Frantsu zowa, G. (1986). On degene rated motion s in the proble m of multiv ariable system s stabilizati on, in Autom atic Contro l for Plants with Varying Characteristics, NIEE Press, Novosi birsk . (in Russia n) Gerash chenko , F. and S.Gera shchen ko (1975). A Method of Motion Separa tion and Optimiza tion of Non-Li near System s, Nauka , Moscow . (in Russia n'
Vukob ratovic , M. and R.Stoji c (1988) . Modern Aircra ft Flight Control, Lectur e Notes in Contro l and Inform ation Scienc es, 109, Spring er-Verlag, Berlin Yurkev ich, V. (1986) . Realiza bility condit ions of desired motion s and synthe sis of system s with the velocit y vector in feedba ck, Ph.D thesis, NOV08ibirsk Institu te of Electri cal Engine ering. (in Russia n) Yurkevich, V. (1988) . On the contro l stabili ty of dynam ic system s, in Autom atic Contro l of Plants with Varyin g Charac teristic s, NIEE Press, Novosi birsk. (in Russia n) Yurkevich, V., M.Bla chuta and K.Woj ciecho wski (1991) , Stabili zation system for the aircraf t longitU dinal motion , Archiw um Autom atyh i Roboty h, vol.XX XVI, No 3-4, pp.517 -535. (in Polish)
II - 24