Parametric identification of a model for the dynamics of helicopter

Paper N° 93 E • 07

PARAMETRIC IDENTIFICATION OF A MODEL FOR THE

DYNAMICS OF HELICOPTER MOTION USING OPTIMAL

CONTROL THEORY METHODS

L.N. Nikiforova

Kamov Helicopter Scientific & Tecnology Company, ·Russia

ABSTRACT

This paper i s Lhe result of a series o:f expei~iment.s lo identify parameters of a model for an object C a system) using methods based on the theory for achei vi ng optimal control of a system C or object.).

The advantages of solving dynamics, by the use of one strated. The dual problem peing and control optimization.

this dual-problem in flight and t-he same method are demon-that. of model id.ent-i.:f'ication An example i s given. Nam£:::-ly,model identificat-ion for coaxial helicopter mot-ion during a manoeuvre in a veJ' t.i cal pl ar>e, usi t'lg one of the optimal control theory methods.

I. I NTRODVCTI ON

Experience in t..he :field of' control systems design has clearly shown that the development of any control system is reduced to solving two inLerconnecled tasks: . task of creating an object model and of developing a control algorithm corres-ponding t.0 the given model and the given pUI· pose of the f'unclioning system. For a given system C or object.) paramet.r-ic identi:fication may be carried out at the same lime as system control optimization.

Though a lot of work has been done in the field of the control system Cor object) model identification [1 . . . 4J in this case helicopters [ 1 J by the progress of the identification t.heory and practice new tasks appear and get. solved. One o:f such problem is that o:f an integreled approach t.o the solving of the flight. dynamics t.wo-sided "task i . e . model iderYlificat..ion and control opt..imizat..ion.

The i nvest.i gat..i ons carr· i ed out proved the expediency of applying the same methods to idet'ltification tasks as to control optimization ones. Opt.imal co11trol theory approximation methods are considered to be promising as they may successfully be applied to both linear and non-linear models.One of such methods will be examined below namely the method of successive improve-ments in control [ 5,6 J based on V.F.Krotov theory valid in general conditions adequate for optimality.

The use of this improvement method development of the method for model parametric by means of a run of experiments. This melhod is linear and non-linear models. Non-lineality is helicopter dynamics models.

permitted the ident.ificat.ion applicable to

a fealuJ~e of

The second distinctive feature i s the use of a run of ex-periments permitting a decrease in the effect of random factors. The third distinctive feature is the continuity of both flight dynamics task parts C fig. 1 ) which in practice reduce the volume of calculat.ions when solving t.he t.ask as a whole.

,-2

n1odel parameters object mathematical

computed values

1 m.odel 2

1

model adequacy identification

conditions

3 task 4

'

t

flight optimal control control

testing

5 theOJ'Y method 6 optimization task 7,

I

"

t

J

1 ••. 6 identification task components

2;p 6, 7 optimization task corn.ponents

Fig.l Block diagl'am defining the steps of the

f l i g h t dynamics t.asks

solving-II PROBLEM !DEFINITION

I t i s assumed that a run of experiments was regarding the object in measur·ing cont.rol vector U Cl), part or the object state vector time variables:

yCU = {X CU, . . . X CU . . . X CD)

1 t m

XlCt)=FCt,Xl,Ul,A) t=l . . . n XCl)- N dimensional representation phase vector; UCt)- N order control vector;

YCt)- R R :$ N order measured parameters vector~

made and

( 1 ) ( 2 )

A S dimensional representation corrected pai~ameters vector;

M number of experirnents C number of modes at t e s t i n g ) .

The pr·oblem consists of det.erming vec"lol~ A so as to minimize t.he functional which consists of the squared diffe-rence integral values of the measured C XT ) and calculated C Xc ) phase veclor values coJ~respor~ding "lo solutions of (2) equa t..i on syst..em.:

t k M R I =

f

_E

_E

t J;::: i H D. C J

c

3)

Dj is the weight coet't'icient of the phase state j component.

Further, the functional minimum i s found in a similar way to the solvution of the control optimization tasks i.e. a conjugat-ed syslem C vect.oi~-funct.ion) is delermined by solving an auxiliary equat-ions syst...em

2X J 2 -2X. CE 'f'j F

t/

J J ;::: i N = t = 1. .M

. .. c

4)

The initial conditions lions system C4) for

for integrating the auxiliary egua-t.he ident..ificat...ion ~ask are zero:

lf

c

tk) = 0

J l.

Correclion of the unknown parameters vector A the contr·ol correclion when solving lhe cont-rol task ar1d is done by iterations:

is similar "Lo op"Limizalion

Aj.= _{Aji - 1} + oAJ ( 5 ) L

j= 1. . . . S~ i - number of iteration~

A correction may be dol"le both at each step

c

par amet.eJ~ A varies il"l lime):

aH 1 M oA.C l ) ~ _K.

E

( 6 ) J J ₁₌₁

a

A _j al"ld integrally: lk aH 1 M oA. ~ _K.

E

s

dl _{( 7 )} J J ClA 1 = 1 j l H

2.1 Peculiarities of the problem solution

The problem of vector A def'init..ion is solved when one of the following conditions i s ach.e>ived:

(a) - convergence of the calculated funcliol"lal value with

a pr-edetel~mined degree of accur-acy:

I - I

.

'-1

.£; ::= L " ~ .C

1 I i 1m

(b) convergence of the componel"ll of vector A to the nearest approximation:

,; aj = A - A j j \.. L-1 the parameters of

Cc) _{integral convergence of the} paramete~ of vector A to

the norm: c = a A. - A , l L -1 A. L-1

:s

.£ a m 2.2 Weight coefficients A. L

The necessity of specifying weight coefficients Dj (3) in the parametric idel"ltificalion problem is dictated by the physical meaning of the problem to be solved, since the helicopter motion parameters chal"lge within different limits. Thus, the change of flight speed i s measured in the order of one to tens m/s~ t..he change of rotor angular speed is measu-red in the order of 0.1 to 0.5 rad/s, the change of a maneu-vering helicopter altitude - of up to a radian etc.

Since the total functional consists of several compo-nents C3) i t is l"leccessary to speci:fy the weight o:f each component according to the engeneering evalua~ion of ~his or

~hal parameter, which is imporlant in lhe mode examined. So, for example, i f for praclical reasons. the following devia-tion values are taken as equal lo: 1 m/s for flight speed; 0.05 rad for pilch angle; 0.1 rad/s for main rotor angular speed, lhen corresponding values of weight coefficienls must be selected lo conform lo the following ratios: D1: Dz: 03 ~

1: 500 : 100 Since the sensitivity of the problem change in one or olher of lhe componenls of lh~ vector staLe,

depe11ds on lhe values of weight coefficients, i t is quite evident l~1al engeneering experience will be necessar_ to evaluate correctly vector D.

2.3 The met-hod As t-he componenls

dimensional~ t.wo t.ypes applied:

applicat-ion algorit-hm

of the vector determined are not. one-of improvement algorithms may be a) The method of successive eli mi na t.i on of each of the componen"ls or the vect.or, t.hal i s the one component. of vecloi~

A i s deLe!" mined, A1 for example, C "Lhe rest. Aj a.r e no"l

changed i.e. in formulae (6) and C7) lhe corresponding coeffi-cienU> are laken equal lo zero: Kj ~ 0 , j t 1 ) and lhe calcu1alion i s done by convergence in £I or in. .£:C<. • Then t.h.e

obtained value is recorded and the next parameter is deter-mined etc., so at each step the correction is calculated only for one parameler. Afler lhe delerminalion of lhe vecloJ· A last component the calculation returns to the f i r s t component. A wi lh new Aj C j#l) values and so i l goes on up lo lh,; loi..al convergence in veclor A. In fig. 2 lhe change of lhe func-tional and the corrected parameters are presented according to t..he it..erations in the process of minimiza-Lion, upon lhe

given diagram;

b) The melhod of parallel delerminalion of componenls with a simultaneous cor·recli on of t..he corrected paramet.e1~

vect.o1~ componen:ls. As .. t-he c::omput-i ng pract-ice demonst..rat-es t

lhis me~hod is reliable enough and works fasler lhan lhe first. one. They ar-e compared in fig. 2.

I

\ \

\

~

consislenl eliminat-ion mel hod

\

_"'l

- - -

parallel melhod

\"'

..._

--

1--A~

I.2

/

'

I. I

/,.--r--·

-

-

-

_r-

-I.O

v

A2

I. I

I.O

k--

JZ[~__.__I

N I I I l

Ill

A;,

I.O

0.9

i\

'

-Fig.2 It-eralional paramet...ers

~

I

_'

_;

I

_I

_I

-

...,__

_{/ i t.e1- a t i or1}

change of funclional I and correcling A , A . A

1 2. 3

5

III IDENTIFICATION OF A COAXIAL HELICOPTER MOTION MODEL IN A MANOEUVRE IN A VERTICAL PLANE

The analysis of the flight test results of a helicopter performing manoeuvres in t.he vertical plane C sleep climbs, dives ) proves that though the integral parameters like overload are rather well represented by the model, but the flight. 'lest result.s demonstrate some difference as regards the flight speed, pitch angle and main rotor speed values.

The problem demands deteJ'mination of the vector of con-stant coerficienls, correcting the calculat.ed values of the coaxial helicopter motion equations coefficients. for t.he vertical manoeuvres.

3.1 Motion equations

A

coaxial helicopter motion model developed by the Kamov Helicoplers Scienlific & TechtJ.ology Company, i s used as a mot.ion model. Due to the aerodynamic symmet.ry of t.he coaxial helicopter, the side motion may be disregarded, which considerably simplifies the system.

The equat.ions are written in the righl side coordinate system related to the main rotor shaft.

dV Accepted Vx, Vy w ,')-103 w z X, y Mz Oz designations:

- flight speed components in respect main rotor shaft related axes~

main rotor angular- speed; - p.it.ch angle;

main rotor selling angle;

to the

angular speed of rol.ation around OZ axis; components of forces acting upon the heli-copter and referred to the heliheli-copter mass; moment in respect lo OZ axis relerl~ed 'lo

the helicopter inertia longitudinal moment; main rotor shaft torque referred lo lhe main rotor inertia polar moment;

engin"e momenl on lh.e main rot.or shaft.. re-ferr·ed Lo main rolor· inerlia polar mometYl; main rotor resultant force deviation angle

at longitudinal control;

main rotor collective pitch angle.

X = XC

v )

XC

v )

+

err:--

X X V w -q [ si nC &-'!' ) - si nC & -

so )

+ y 2. - 3 0 3 0 -6z -~ +

X

l>oz +

X

tip dV y

=

err:--

CYCV )-YCV

X X

0

~ w +hw

-o... -p o 2

) +V w -gcosC

&-so )

+Y ~ l>6z+Y Apl C )

-y z 3 6.)0

V w -gcosC &-9 )

X Z 3

dw

z -vy -wz -Oz -If-">

AM C V ) -AM C V ) +~l · /~Vv+H Aw +M b6z+M b9

Z X Z X · z Z Z

of d& dt dw dt-= w z = L>M -K ):\B [ M C V ) K X w +!J.w L>M CV) + Mvyt>V K X I< y 0 +

M

CV ))( 0 ) 2 + M CV ) K X K X W 0 0 Engine the main dL.M

operation equation considering the operation rotor speed governer:

1

K):\B

= ~T""-ns- C -L.MK - K Lw )

,._, ,ll.B W

The model implies some non-1 i near dependences, i . e . - dependence on t-he pitch angle of the force of gr· a vi 1.-y

components project..ed on to lhe axes;

- dependence of lhe main rot.or· lhrus"L and torque upon t.he main rotor speed C main rotor angular speed);

- deper1dence of t.he forces and moments upon the flight speed. I t i s necessary to calculate the values of ll1e

coefficient..s A1 A2 A3 A4, A5

Mvy

₌

_{A C} MVy)

Mwz

₌

_{A C} Mwz) _M

-oz

₌

_A

_c

MOz)

z 1 z T z 2 z T z 3 z T Mvy

₌

v

MVy) K ! ( T

-oz

( MOz) M

₌

A ! ( ₅ K T so as t-est-s.

to reduce t-he difference bet-ween t-he model and

flight-3.2 Specification of basic standard dependences

The following mot-ion paramet-ers obt-ained from flight test-ing are accept-ed as the most valid C ref. fig. 3)

f 1 i ght. speed C Vx) ~ pi t.ch angle C & ) , main r ot.or r· evol uti OI":!S

C n ) . Except- for· these standard dependences, t-he change of control corresponding t.o e?Jc:h of t.he modes C Xa ) is also t-aken from the flight- 1.-esL results. In Lhe example presenLed, those modes are selected for identificat-ion, where only t.he longiLudinal cont-rol changes C Lhe value of t-he collective pit.ch angle is not changed in t.he process of t.he mode reali-zat.ion) i.e. the cont.rol vector is one-dimensional.

The number of modes ac:cept.ed as basic st.andard according to t-he flighL 1.-esL resulLs, is equal 1.-o 5.

So, in the equa"lion syst.em describing helicopter molion in t.he vert.ical plane, orders of cont.rol vect.ors and correc-Led coefficient-s et-c. following are defined:

Nu = 1 cont.rol vector order;

N = 7 init.ial equations system order;

S

=

5 correct.ed paramet.ers vect.or order;

R

=

3 changed parameters vector order;

M = 5 number of experiments.

Based on the flight test- resulLs Cfig.3) t-he basic standard dependences are defined for calculation - change in rotor resultant force deviat.ion angle, change in. flight speed~

change in pitch angle and rotor change in angular speed. The dependeJJ.ces obtai ned fo1~ five cal cul a 'led modes are p1~

e-sent.ed at fig.4, where time is equal to zero at the beginnir1g of the longitudinal cont.rol change in each mode.

.,

_....

(Q

.

w

0 C"l >-o,:; t>l '0 ;, (])(!)

..,

(])

,..,

0 0

..,

,..,

~-

(+ :J

:r

(Q (]) Vl

:r

(+ (]) (!) "" (!) .... 'tl n 0 (1 '0 >-'(+ >-· (])

"

..,

6-!3 - 0 " (+ ;

...

:; 0 0 -!I! ... c'tl

<

w

..,

.,

!I!

w

3 Iii (+ ro

..,

Vl >-· :l (+

:r

(]) 'C

-.

0 0 (]) Vl Vl yaw roll control, control1 mm mm

"""

"""

0 0 0 0 C.Jl H 0 :::!:

~

_!P

:::!: 3 _ro 3 l1l Ul _Ul col. pitch angle, H 0 0 I long. control, I c:; 0 mm 0 [j) main rotor revol .. 1 % H OJ 0 0 0

1/

\

( +

--·

3 l1l tn C.Jl I H 0 altitude, m 0

\

\

"""

0

1\

flight speed, km/h N 0 0 0

'

yaw roll angle, angle, de g. de g. C.Jl N 0 0 0 0

~~\ffi~ ~-~

[j) [j) . Ul Ul

pitch angu- pitch lar speed, angle,

0 deg./s N 0 de g. I

"""

"""

0 0 0 ( +

3.

-11> [!; '-l

ci) (j) '0

~D

6

(j)

c

~

.2

4

0 u ~ (j) Of;

c

<tl

.c

u

2

0

0

-IO

'0 <I)

~6-20

0. <I)

-30

<I)

2

-,..-d

<tl l(j ~ ...

0

j -Of;(j)

§

~

_-2

<tl

'".c

.8

u

-4

0 '"'D

c

~

·~ 0. <tl liJ E

I

II

I

2

3

4

5

time, s

I

2

3

4

5

time, s

~

_~

I

~

_~

-...

r----:--::::

:::-:;.

t---

~ _p

2

3

4

5

time, s

5

time, s - - 0 - - modes

t r

o

-Fig. 4 Change of the longitudinal cont•-ol Coz), pitch angle ( 3 ) ,

speed C\lx) and rotor angular speed Cw) in t..he px~ocess

9

The minimized funcli onal i s wril"len in form of: l k !5 3 I =

f

_E

_E

D ( _x1

-

_x1 )2 _{dt =} t,I + AI + L',I j i 2 9 1=1 j = 1 _{J'l' Jc} l 0 t t k k !5 !5 AI =

D,f

r;c

t,V -t,V ) 2dt M =

Z\f

_EC

b& -b& ) 2dt i 1=1 x1 x1 2 1=1 1 1 l T c t T c 0 0 l k l 0 where D.

J weight coefficient in the functional component:

D1 - for the f l i g h t speed, Dz - for the pitch angle, for the rotor angular speed.

The weight coefficients D1 Dz and 03 values J'ela-tionship for t.he condit.ion o:f t.he quanlit.at.ive value compa-rabilit-y in t.he f'unctional components of' deviations from preliminarary determined i . e . flight speed bVx in 1 m/s, pitch angle .6& in 0. 05 rad. and rotor angular speed D.L0 in 0. 1 rad/s D1=1, D2= 500, 03= 100.

To i l l u s t r a t e the above, the relationship of the

func-tional components bi1 Mz bi3 at selected weight

coefficient values in the process of the functional minimization when solving the problem sta.rting with different i n i -lial values A = 0.5 and A

=

2, is presented at. f'ig. 5.

0 0 r---· _r·~J·

-I3

11

I~

_./;

\

'

_-~

\

1\

~

f:::-

:--;

' /

ld

'

...

'\:'

~

. /

v

·'./ .. A,doHn>Hon

f

an~-- . . .

-tr'

·---1--~

0-·-

-::::--

--~~

\

1\

A ₁ definition

r\.

range I

"'

\

,

v

I'..

"'

/ . /

-

_'

"'-...

_..

_{,_..}

_.-.:::.

~

I

!

...

,.

-...__

_{- ,,1--.}

r--::

r---1'-l

-

r-

T

J

O.b

1.4

l1

0.6

I

1.4

Fig.5 Functional and i t s components change dependences at various initial values of corrected param.(.>ters

C - - Ao;::.O. 5, - - - Ao;::.2)

I~

3.3 Identification results

The results ol the calculat-ions on. t.he cor-reeled para-meters veclor determination are presenled al fig.6 in the form of comparing the helicopter motion parameters calculated change at one of the modes and the flight lest results. At the same fig.6 the change of the motion parameters defined upon t.he initial C noncorrecled ) helicopler motion model are marked for the purpose of comparison.

To illustrate the importantce of performing the run of expiriments to solve the identification task, the scatter of corrected coefficients values at identification is shown for one mode only and the result for all the five modes i s given. The iden:Lificat.ion deviation at. each of t.he modes separat.ely is frorn 10 to 20 % for various coefficienls.

0

I

3

4

5

time, s

-IO

cJ ill (j)----(j) E

-20

~ ill (j) OJ)

_-30

+'

c

~<0

-~

.c

_-40

c::; u

1---+-+--+-~-~~t~~:-rF-N;tj

-.. .._ -., _:.·

~

.... r .

---

1---... ·-

·--+---t--- ·--+---t--- 1 ·--+---t--- ·--+---t--- ·--+---t---... .

0.4

0

I

4

5

time, s cJ <0 (j) 0 01)

-;:: c

(j) <0

>

.c

0 u

2

0

- 2

I

I

2

3

___

t-~~--~~- -~---

----. -

---

-·-Fig .. 6 Comparison of motion parameters obtained through the model Cinilial and identified) and flight test results However i l should be kept in mind t.hat the inilial modes were selected carefully enough and did not show any explicile vio-lations of lhe mar1euver oerformance conditions (approximately equal maxi1nal lor1giludi~cil. control level and i t s change rate, constant collective pitch elc.) which explains

derably s1nall scatter of resul~s.

for a

consi-i nil. idednl.

t-able 1 ----·--correc"Led paramf?ters modes Ai A2 A~ A4

"'

1. I dent-if i cat-i or; for each mode separately

mode 1 1.205 0.978 0.868 1. 107 mode 2 1. 555 0.887 0.909 0. 991 mode 3 1. 017 1. 043 0.957 1. 016 mode 4 1. 249 1.192 0.996 1.358 mode 5 1. 340 0.906 0.858 1. 219 2.IdenLification foJ- 1. 216 1. 00 0.900 1.155 a r-un or f'i ve modes

3. Maxi mal deviation values at- identification !or·

each mode separately Cas 16 11.3 10.6 14.6 a ~ of the identification

result for the run)

IV CONCLUSION

The use of one and lhe same mathematical apparatus for the solution of bot-h general flight dynamics tasks i.e. :for "lhe object motion model identification and the conlrol optimization considerably reduces the volume of computations in solving problems because of lhe possibility of a consequ-ential program application for solving the both parts of the problem the model description in particular.

From the practical point of view an important feature of the proposed approach is also Lhe possibility of identifying a model nol only by means of one experiment. bul by a r-un o:f experiment.s which increases t.he degree of t.he model adequacy due t.o t.he averaging influence on random par-amet-ers.

REFERENCES

1 rpoyn )l CGroup D) MeTO-'lbl HAeHTH<jiHKaUI•lJ.l CHCTeH. M. MHp. 1979r., 302 c.Ctransla~ed from english to russian) 2 EepecTOB n.M. ,nonnascKHA E.K. ,MHpOillHHYeHKO n.~.

\.{aCTOTHble HeTO,llbl H,lleHTH<jiHKaUHH JieTaTeJibHbiX annapaTOS. M. , f1amHHOCTpoeHHe, 1985r. , 164.c.

3 3Axxo<ji<ji n.CEykhoff P) 0CH08hl H,lleHTH<jiHKBUHH CMCTeH

ynpasneHH>l. M. ,MHp,l975r. ,683c. (translated from english lo russian)

4 Himmelblau D Process analisis by sta~istical methods.1970 5 KpoToB B. \D.' EyKpees B. 3 .• rypHaH B. l-1. HoBble HeTO,llbl

sapHa-UMOHHOro HC~HC~eHt1~ B ~HHaMHKe no~eTa.M. ,MaWHHOCTp02HHe

1969r.

6 KpoToB B.~. ,rypMaH B.M.MeTO~bl H 3ana~H onTHMa~bHoro ynpas~eHM~.M. ,HayKa,1973r. ,384c.