A method of optimising aircraft autostabilizer systems

l

THE COLLEGE OF AERONAUTICS

CRANFIELD

A METHOD OF OPTIMISING AIRCRAFT

AUTOSTABILIZER SYSTEMS

by

MAY,1957 T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D A Method of O p t i m i z i n g A i r c r a f t A u t o s t a b i l i z c r Sjratems b y -J . W o l k o v i t c h , M . S c . ( E n g . ) , Grad.R.Ac.S. SUMMi-jRY

A novel procedure for the optimization of aircraft autostabilizer systems is presented. The procedure is straightforward, and its application does not result in

demands for autostabilizer systems of prohibitive complexity. Many important non-linear effects may be included v/ith only

slight extra complication in the required calculations. The procedure is applicable, in the first place, to piloted aircraft,

- the essence of the procedure being the assumption that the purpose of the autostabilizer is to reduce the effort demanded

of the pilot in executing a given manoeuvre or attaining a given response. Although the presence of the pilot is explicitly taken into account in the calculations no form of pilot's transfer function need be specified.

It is shovm how the procedure may be modified to form an approximate procedvire for the optimization of auto stabilizers for pilotless aircraft having linear autostabilizer character-istics and linear aircraft dynamics. The results of some calculations presented herein support a suggestion that this approximate optimization procedure may also be frequently applied with success to pilotless aircraft having certain non-linearities, either in the autostabilizer system or in the aircraft dynamics.

Page List of Symbols

Introduction 1 The General Optimization Procedure for 4

Piloted Aircraft.

Example 1, Longitudinal Short-Period 6 Response to an Impulsive Pitching Moment.

Exam.ple 2. Longitudinal Short-Period 10 Response to a Sliarp-edged Gust.

Example 3» Longitudinal Short-Period 11 Response to a Step Deflection of Elevator.

Example 1+. Longitudinal Short-Period 15 Response to an Impulsive Pitching Moment

of an Aircraft having a Non-Linear

Variation of Pitching Moment Coefficient with Incidence.

Discussion of the General Optimization 18 Procedure.

Optimization of some Non-Linear Auto- 22 stabilization Systems (introduction)

Evaluation of the Coefficients of the 23 PolTOomial Approximation to the

Auto-stabilizer Chars.ct eristic.

Example 5. Longitudinal Short-Period 27 Response to an Imp\alsive Pitching Moment

with Flicker m Autostabilization. q

Evaluation of Some I n t e g r a l s Required f o r 32 t h e Optimization of Non-linear A u t o s t a b i l i z e r

Systems.

Example 6, Longitudinal Short-Period Response 34 t o an Impulsive P i t c h i n g Moment with Limited

A Brief Exposition of Cardinal Spectrum 3é iinalysis.

The Application of Cardinal Spectnm /malysis 43 to the Optimization Procedure.

Example 7. Lateral Response to a Sharp-Edged 44 Gust.

Optimization of Non-Linear Syntems by means 49 of Cardinal Spectrum Anal^rsis.

The Optdjiazation of Autostabilizer Systems 49 for Pilotless Aircraft.

A Suggested Procedure for the Optimisation 54 of AubostabiJizer Systems for Pilotless

Airoraf^t with Non-Lineai'ities.

Some Alternative Methods of Optimization. 55 The Method of Variation of Derivatives. 5^ Optinization of Frequency Response. 57

The Method of Standard Forms. 58

Other Methods 60 Conclusions. el Acknowledgments. 62 Notes on Chapters 63

References 65 Appendices: A.1. List of Derivatives. 67

EiSRIV/JB^F.S

The convention.'J. Eriti^^h sj^stem of n o n - d i m e n s i o n a l i z e d d e r i v a t i v e s •sf Ref. 1 i s eDpioyxid w r . h t h e f o i l e v r l n g m o d i f i c a t i o n s ,

( i ) ^ , i s v . T i t t e n sJXiicly a s ;j 1

( i i ) The deriva-'iive r?^,. i ^ /; timer, t h a t of R e f . 1 .

( i i i ) The ncn-di!r;e.a?iov:£i:'.;.'n;:I rai;e of V)l;;-h, -r~ i s w r i t t e n a s q . F o r I i a t e r a l Sta"; 11 it:y- t h e r-iii^plement.ary n o t a t i o n of M i t c h e l l ( R e f . 2 . ) i s eiEployed ( w i t h t r - . U l i n g a l t e r a t i o n s ) . I n t h i s s y s t e m , ^ c '

Ce __^A

e

2 J

V

/ = II ' "1 = _ > A'. ^ • o ' - n n i / C' , e t c . , = - n

V

OTIÏER SYIGOTS imwi CASE ( i ) s e e E q u a t i o n 2 . 4 8 ") I a n e g a t i v e c o n s t a n t ( i i ) s e e !h;cj_ia;\tion 3»23 a , a , a , a , see EquaMon o 0 ' 1 * 2 ' 3 '

a .a .a ,a. CoexVicicnts of the polynomial P(x)

1 ' 3 ' 5 ' 7 e , e , e e 0 1 2 3 m ee E.T-iairioii 6 . 6 . . w i t h a - . i t c s t a b i l i z a t l o n / b a s i c ra 'vV W h , h . h , k s e e Eq-oaciün 6 , 9 . 9 ' 1 * 2 * 3

k (i) °L/2

( i i ) X. _'/•'^l,l k' m vrith e.utcEtabjJ.isat?. on / b a s i c m

m ,m see Equation 2.ltJ+.

1 ' 3

n see Equation 3.28.

p see Equation 2.15. («• positive constant)

s T h e Laplace Transfonn Variable

t A meas'-ire of time (see Equation 5«ï4 a^d Chapter 4 )

w see Equation 3.27.

-(_

Iha

input signal to the autostabilizer.

Y,, T h e maximum value of X that need b e considered.

M

Xa

^^'1® value of X at which sat-oration occurs.

y A general conbrol surface deflection due to the

autostabi3.izer

y. The autostabilizer characteristic.

y The limiting value of y

a (±)

Angle of ^ c i d e n c e

(ii) see Equation 6.14.

/? & e \ A r

U^

1%

(r)

6 r? ^ 3 ^F

(for Time Vectors) = - v , Tiiiere v = sideslip angle.

see Equation 3.28

the Cardinal Spectrum Interval

a n arbitrarily short time

P(:^) -y^^

a n elevator deflection produced b y the pilot

a step elevator deflection of magnitude 6

r)

see A 77 (r )

an elevator deflection due to the autostabilizer

the amplitude of 77 for flicker autostabilization.

i A ,A 1 3

©

s

s

H,H'

I

h

I

1

I

2

I

00

^0

J *\,\> = ( i )

( i i )

cfener^JLLzed forms of a , a , a , a

'- 1 ' 3 ' 5 ' 7

a Cardinal spectrum

the d i f f e r e n t i a t i o n operator

defined from Equation 6,14.

n with a u t o s t a b i l i z a t i o n / bawic n

P P

an i n t e g r a l , as defined i n t h e t e x t

II II II II II II II

an i n t e g r a l of t h e 1 s t kind (see Cü'iapter 3.Section 4)

M II II II 2]:id " C " " » )

see Equation 3.30.

see Equation 3 , 3 1 .

= K - 1

H - R where H = -Urn + m z

w q w

K.K' = n with a u t o s t a b i l i z a t i o n / basic n

P defined by Equation 3.28

P = e -°-^34T ^^^ I" rp _77.05°] ) (from Equation 3.22)

P ,P , P ,P Coefficients of A

2 ' 4 ' 6 ' 8 1

P (x) The polynomial approximation to the autostabilizer characteristic Q see Equation 3.20 Q ,Q ,Q ,Q Coefficients of A 4 ^2'^4'^6'^8 — - — ^ -- "3 Q (s) a quadratic expression in s R = 2 ( ^ _w - z^) • E (r) a general response

R .R ,R ,R Coefficients of A^, 2 ' 4 ' 6 ' 8 5 S , S , S ,S C o e f f i c i e n t s of A-, 2 ' 4 ' 6 ' 8 I T ( i ) s- measure of t i m e ( f r e q u e n t l y 11,51 r ) ( i i ) a c o n v e n i e n t ( f i x e d ) time " ^ d e n o t e s p o l y m u l t i p l i c a t i o n A ( s ) a q u a r t i c e x p r e s s i o n i n s A 77 ( r ) a s t e p e l e v a t o r ; d e f l e c t i o n of ni:ignitude 6 77 SUFFICES

D a quantity associated with the desired response, (e.g. \ ^ , ^'^ ,)

S a control sui'face deflection due to the autostabilizer (e.g. ^3)

(This should not be confused with the Laplace transform of an elevator deflection 77 (r) , which is written 77(3) ) p a control surface deflection due to the pilot (e.g. 77 ) % (e,g,r7 ) t h e c o n t r o l s u r f a c e d e f l e c t i o n demanded of t h e ^D p i l o t t o a t t a i n t h e d e s i r e d r e s p o n s e . A few o t h e r t r a n s i e n t symbols a r e d e f i n e d a s n e c e s s a r y i n t h e t e x t , The n o t a t i o n of C a r d i n a l Spectram A n a l y s i s i s e x p l a i n e d i n C h a p t e r 4 . 1 F o o t n o t e s a r e n o t employed: s u p e r f i x e s i n t h e t e x t thiis r e f e r t o t h e s e c t i o n headed 'Notes on C h a p t e r s ' .

C

Chapter 1, INTROIUOTION

It Trill bc generally agreed that the subject of aircraft autostabilization has rapidly grown in importance in recent jrears, The reason for this is not hard to find; it is simply that recent great advances in aircraft performance have demanded aircraft oon=: figurations which often have irJierently poor stability and response characteristics. Examples of this are legion; to quote only three, the adoption of sweepback as a means of insreasing critical Mach number has led in some cases to an undesirably high value of •& at high G.s ; the high operational altitudes now common, result in poor danping of the lateral and longitudinal oscillations; and the

inertia distribution of many modem high performance aircraft is such that inertia! cross-coupling in roll is readily induced. The reader will doubtless be familiar with many ftirther examples of this trend of reduced stability with increased performance.

The problem that confronts us is, then, how we may Improve an aircraft's stability and response characteristic's without sacrifice of performance. Some improvement can be achieved by careful design of the basic airframe. For example, the above-mentioned excessive e ^ due to sweep maji be reduced by the adoption of anhedraH, and a large

fin may alleviate the undesired effects of inertial cross-coupling in' roll. However, the gains that can be attained in this manner are . limited by the restriction that the aircraft's performance must not be reduced, and in many cases autostabilization must be resorted to if satisfactory response and stability characteristics are to be attained.

If an autostabilizer system of unlimited weight, complication and expense were permitted the stability and response characteristics of a given aircraft could certainly be made quite satisfactory «nder all conditions. In practice, of course, all three of the above factors will be limited and it should be appreciated that any

discussion which fails to take into accoiont the possible effects of such limitations may be somewhat unrealistic. Thus, for example, a stuc3y of the effects of changing lateral derivatives may show that

satisfactory lateral stability characteristics may be obtained with a value of n several times that of the basic (i.e. non-autostabilized) aircraft. But it may be that the power available for the autostabilizer system is ina.deq\:iate to generate the control surface deflections required to attain this value of n at moderate and large angles of sideslip. Even if sufficient power is available, the aircraft designer may well

decide to limit the maximum amplitudes of the control surface deflections due to the autostabilizer so that in the event of a nan-away catastrophic divergence will not occur.

One important reason i/vhy comparatively little attention has been given to the more practical considerations of autostabilizer design such as the above-mentioned, is simply that the problem is non-linear; i.e. the mathematical formulation of such a problem reduces to a set of non-linear differential equations. ?/hereas non-linear dynamic systems of great complexity may be described by differential equations having fairly

straightforward methods of solution, the complications involved in solving even a simple non-linear equation may be considerable. For aircraft motion having several degrees of freedom it is frequently found that no analytic solution of the res\alting set of non-linear equations is known, Step-by-step nunerical solution is usually a tedious process and recourse has usually to be made to analogue computation. The procedure then adopted is to solve the equations of motion for different values of the adjustable autostabilizer parameters within the preselected limits. The adjustable parameters of the autostabilizer system are then fixed at those values which have been shovm to yield response characteristics acceptably close to the desired response characteristics.

The above procedure suffers the disadvantage of demanding analogue computer equipment - perhaps of considerable expense - and the procadure has a certain crudity in that the final (optimijm) values of the adjustable autostabilizer parameters cannot be attained directly but are arrived at by trial and error. Nevertheless, for the design of autostabilizers for pilotless aircraft having non-linear equations of motion with several degrees of freedom this procedure is probably the best practical technique

available. Analytic meéhods are only likely to prove superior in problems of very limited complexity,

For piloted aircraft even the recourse of analogue computation may fail. It is well knovm that the stability requirements for piloted and pilotless aircrafb differ. For example, spiral instability may be quite acceptable in a piloted aircraft, whereas in a pilotless aircraft it would be catastipphic. In a piloted aircraft, longitudinal or lateral oscillations having a very short period may cause confusion and

discomfort to the pilot whereas in a pilotless aircraft these characteristics may be quite unobjectionable - or even desirable, since such short periods are usually associated v/ith rapid rates of response. Considerations such as these show that we cannot simply assume that the optimum setting of the adjustable parameters of an autostabilizer calculated on the assijimption of pilotless flight will necessarily be suitable for piloted flight. The above-mentioned analogue computer equipment may (with further expense) be extended to form a flight simulator, but the representation of flight in such a device may be too limited to be satisfactory. Actual flight tests using the autostabilizer equipment can, of course, only be undertaken when the aircraft is complete and it is obviously desirable to have the design of the autostabilizer equipment finalized (at least within limits to allow for possible inaccuracies in the data used for computation) well before the completion of the aircraft. Might we then attempt an eir^lytical solution of the problem of optimizing the autostabilizer system of a piloted

aircraft by representing the pilot mathematically by a suitable transfer function in the equations of motion ? For a pinrely linear system (i.e. linear aircraft and autostabilizer characteristics) this would be possible, but for a non-linear system the equations of motion would be even more complicated than in the case of a pilotless aircraft and the ch^mces of an analytic solution being known even less. Apart from these consider-ations, however, at the present time no satisfactory transfer function to describe the pilot is available, though it has recently been shown that an expression for the transfer function of a pilot may be obtained

2 for certain very restricted types of manoeuvre such as pure yawing.

Por more complicated manoeuvres it seems tliat the human pilot is actually able to vary his transfer function to suit the conditions of flight and the demands ms.de on hirn. In viev;- of this, the com-plications of such an analjrtical solution starting from the equations

of motion, as suggested in this paragraph, become fonrddable.

From the above survey it might seem that an analytical solution of the problem of optimizing the autostabilizer system of a piloted airfract having non-linear characteristics (either in the aircraft dynamics, or in the autostabilizer system, or both) is, at present, hardly to be hoped for. In fact, this is not the case, and the pturpose of this thesis is to present a straightforward technique developed by tlie present wr'iter which yields eaact solutions for the optimiom values of the adjustable parameters of a speoii'ied auto-stabilizer system fo\- linear cases and approximate solutions of good accuracy for many noa-linear oases of impoz-tance. This technique

piloted and ^, , . . . . 13 a^rJ.icable to both/pilotless aircraft having linear characteristics (inclv.-üng the autostabilizer system) and to piloted aircraft having c e r t a i n n o n - l i n e a r c h a r a c t e r i s t i c s ( e i t h e r i n t h e a i r c r a f t dynamics, t h e a u t o s t a b i l i z e r , or b o t h ) .

CHAPTER 2 .

2 , 1 . THE OPTIMIZATION PROCEDURE FOR PILOTED AffiCR^^FT

The most g e n e r a l p r o c e d u r e c o n s i s t s of a number of s t e p s a s

d e t a i l e d below. ^n any p a r t i c u l a r example i t w d l l u s u a l l y b e p o s s i b l e t o t e l e s c o p e two o r more of t h e s t e p s i n t o one.

S t e p 1 .

The r e s p o n s e of t h e b a s i c a i r c r a f t ( t h e a i r c r a f t vri.th no

a u t o s t a b i l i z a t i o n ) t o a s p e c i f i e d i n p u t i s c a l c u l a t e d . Tliis r e s p o n s e i s c a l l e d t h e ' b a s i c r e s p o n s e ' .

Sfeep 2.

The desired response to the input is specified, and compared with the basic response. In general the basic response vdll differ

appreciably from the desired response and autostabilization will be required if the desired resoonse is to be attained or closely

approached. Step 3.

The type of autostabilization to be used is selected (i.e.

A m , A n , An , etc.) and any limitations or non-linearities specified. Step 4.

It is assumed that the desired response is attained exactly, through the combined actions of the autostabilizer and the pilot. Step 5.

The optimum adjustment of the variable parameters of the autostabilizer system is ass'umed to have been made when the effort demanded of the pilot is a minimum. ¥e use the tenn 'effort' in a broad sense to include mental strain as well as physical exertion. The mathematical representation of effort by means of an 'effort function' is discussed later.

Step 6.

With this criterion eqjations for the optr^num values of the adjustable parameters of the autostabilizer system are produced and solved.

Step 7.

The time history of the control surface deflections demanded of the pilot with the optimum autostabilization is calculated. If these appear difficult to attain it is necessary to proceed to Step 8, Step 8.

The response to the specified input with the optimum autostab-ilization but with no pilot action (other than such as may be included in the specified input) is calculated. This response is then compared with the desired response. If it is acceptably close to the desired response the optimum autostabilization may be regarded as satisfactory; if not, we conclude that the type of autostabilization chosen is

inherently incapable of producing a satisfactorily close approximation to the desired response even when adjusted to its optimum value, and some other type of autostabilization must therefore be selected.

Discussion of the above procedure is delayed lontil the end cf this chapter. Four examples now follow. In each we assume a pilot

/•CO ^

effort function of the form / I 77 (r) 1 ^ d r . where 77 ( T )

L I

PD

J V

is the elevator deflection that must be applied by the pilot to attain the desired response, andr is a measure of time. Thus we shall assume that the optimum autostabilization is that which

minimizes this integral. (The procedure is by no means restricted to effort functions of this type and the use of other types of effort function is described later).

2.2. EXj'-,MPLg 1. LONGITUDINAL SHORT-PERIOD RESPONSE TO AN IMPULSIVE PITCHING MOMENT

The standard non-dimensional equations of motion for short-period longitudinal response are

(D - 2 ) w (r) - ^ (r) = 0 (2,1) ^ D + m ^ J w (r) ^ | i D - ^ ) ^ (r) = m (r) (2.2) assuming 2(7") to be negligible with D = •^-dr , A / \ d9 and q (r) = -g^

lie assume that an impulsive pitching moment (due to, for example gun recoil) is applied such that / m(r) dr = 10 where e may be made as small as we please. °

Appljdng the Laplace transform to Equations 2.1 and 2.2 we ebtain (s - z) w (s) - ^ (s) = 0 (2,3) whence w(s) = ip 10 s^ + 2 R + R^ + J^ s (2.5) /

With negligible error, the following relations hold for e arbitrarily

small.

CO

e'^^dt = ^ ' (2.15)

e

CO

-pt

<''

e . sin« t dt =

"P't 2 ..X JO. -nZ . o ,.2

P % '<'' (2.16)

= Pl+^_M_'_

(2.17)

, ^ P (P' +

4 M ' )

Squaring Eqiiation 2.14 PJ^d integrating the resiilt between limits of

e and «> , making U3e of Equations 2.15, 1ö, 17, yields after some reduction

an expression for I of the form I = C.4^8 k' '••- 0.854 k' + terms not

involving k' (2,18)

For I^ stationary g f = 0 v^en k' , k'optimum = 9.53

We must now ascertain the nature of the control deflection demanded

of the pilot with the optimum value of k' (Step 7 ) .

77

(T)

is obtained from Equation 2,14, noting that because of the impulsive

^D

nature of

m^ir)

the solution for ^ (^) is, strictly, valid only for T > e .

Thus we obtain

Aim,

j7. n (T) = e"^'^^ (-0.8(?6 sin 11.51 T- 0.122 cos 11.51 T)

h ^ (2.19)

Equation 2.19 i s graphed i n F i g , 2 . I t w i l l be seen t h a t the required

IT, (T) can hardly be a t t a i n e d , i f only because t h e i n i t i a l ( r >e) amplitude

D

i s non-zero. However even i f the p i l o t holds the s t i c k quite fixed

( s t e p 8 ) , s u b s t i t u t i o n of k' = 9.55 i n Equation 2.15 e t . s e q . y i e l d s

w(r) = 1.225 e"^'^^^ sin 11.17^

k ' = 9.53.r7 = 0, 11.17 (2.20)

which, as may be seen from the graph (Fig. 2) is a close approximation

to the desired response.

We conclude that a satisfactory approximation to the desired response

is attained even if the pilot holds the stick quite fixed. The optimum

autostabilization may, therefore, be regarded as satisfactory.

2.3 EXAMPLE. 2. LcONGITUDIN/^iL SHORT-PERIOD RESPONSE TO A SHABP-EDCED COST

(it is important to avoid the appearance of divergent integrals in the expression for the effort function. This, and the next example show how this may be accomplished even with specified inputs of infinite duration,)

It is assiomed that at r = 0 the aircraft flies into a sharp-edged downgust of constant magnitude and infinite duration. The limited realism of this assumption should be appreciated - use of such a simplified representation leads to straightforward working, however,

and there is no essential difficulty in extending the technique illustrated by this example to deal with more complicated forms of gusts.

If the initial change in incidence due to gust is - ¥ , then application of the Laplace transform to Equations 2.1 and 2.2 sdelds, for the response in w(r) of the basic aircraft,

(s - z^) w(s) - q(s) = -Jff^ (2,21) - K s + \ ) ^(s) + (I3 s - m^) >^ (s) = m(s) + n^^ w^ ^2,22)

whence w(s) = I "*" ^B J

''o s' + 2 Rs + R ^ + J^ (2.23)

with the same notation as the previous example for m(s) = 0 (i.e. no applied moment by the pilot or autostabilizer)

Using the same numerical data as Exainple 1, and applying the inverse lB.place transform to Equation 2.23 we obtain

E ( I ) ^ e-1,706r ^_^^^ ^^^^^ ^^ ^^^^^ ^^^ 11.5ir) (2,24) w

o

Equation 2,24 describes the response of the basic aircraft, The calculation now proceeds in a similar manner to Example 1, Note how in formulating the initial conditions for Equations 2.21 and 2.22 we have chosen the origin of w such that w^^^= 0.

In this way the appearance of divergent integreils in the expression for

is avoided.

•

There is no further new^ point to be made by completing the example,

so we pass on to Example 3. in vrfiich there is rather more difficulty in

eliminating divergent integrals.

2.4. EXAIPIE 3. LONGITUDINAL SHORT-PERIOD RESPONSE TO A STEP

DEB^LECTION OF EIEVATOR

(This manoeuvre is of some importance as it may represent a

stressing case. As we shall show, the steady-state response must be

considered separate].y from the transient response due to the appearance

of divergent integrals in the expression for /

1%

(v) 1 ^ d7"

7fe write the equations of motion as °

( D - z ^ ) w (r) _

(1(T)

^ 0 (2.1)

- / m^ D + m^ \ w(r) ^iP, ^ " m. ^ ^ ( 0 = m^ . np(r) + m^ .A TJ (r) (2.25)

whereA 77 (r) represents the step deflection of the elevator,

A 77 = 0 forr < 0, A 77 = 6 77 for r > 0

and rj (r) is the additional 3levator deflection due to the pilot.

Appli.cation of the Laplace transform to Eq-Jiations 2.1, 2.25 yieH.ds for 7 7 = 0

P

(s ^ z^p w (ti) - q (s) = 0 (2.3)

-/'nj^s + n ^ \ V<s) + / j ^ s - m ^ ^ ^ ( s ) =

m

^ (2.26)

i

/

\ u n /

s

,-. w(s) = a^-'-r, •

s (3^ + 2 Rs + R^ +

f)

(2.27)

Applying the inverse Laplace transform to Equation 2.27 with the same

notation as Example 1. we obtain for the basic response in w(7-),

U s i n g t h e n u m e r i c a l d a t a of Example 1 . , we o b t a i n ,

' ' ' - V - \f

-_ -^ . . , — ,

KT) = 9 . 1 3 I 1 - e " ' ' ' ^ ^ ^ ' ' (cos 11.51 r+ 0.1482 s i n 11.51 r)\ ( 2 . 2 9 )

w _ V,, ^ - 1 . 7 0 6 ^ , ^ ^ ^. r-. _ . n j,,Or> „A^ AA !ZA ^ N / O 1tk\ (cos 11.51 r+ 0.1482 s i n 11.51 r)

where w^^ i s t h e s t e a d y - s t a t e r e s p o n s e i n W Now f o r t h e s t e a d y - s t a t e . E q u a t i o n s 2,1 and 2,25 become

- Z w A w - a = 0 m S ™77 • 77 - m*. w - m (r

where q is the steady-state response in q Thiis

w

m . & T 7 17 m K q w "1 - m q

-

nv

^ m^ 6 T7 u m - m z ^ w q w (2.38) (2.31) (2.32) (2.33)

(i.e. the steady state change of incidence is inversely proportional to the manoeuvre margin). Now re-writing Equation 2. 25 in terms of the desired response,

. „ .

.,w

.

m. _ w D + m

.) V')*(i"-V)%(^)-n,^

77 (T") l'^ d r t o e x i s t ri (r= « ) must equal z e r o . 6 77 (2.34) For ' "D -* "D Thus y»'^ and 6_ must s a t i s f y E q u a t i o n 2 . 3 3

i . e . w^ D e o 00 ' ^-Q w

= 4.

Hence, the steady-state response must be adjusted to the desired value by autostabilization or other means before attempting to improve

the transient response.

Let us suppose this has been done, so that w in Equation 2,30 is equal to the desired steady-state incidence change w » , Thus (strictly) the manoeuvre margin has been fixed and any further auto-stabilization that may be introduced to improve the transient response

raust leave the value of the manoeuvre margin unch;Maged. Hence the derivatives contained in the expression for the maroeu-'^re margin are no longer at our disposal and any autostabilization that we may -wash to introduce must either consist of the variation of m. or the effective

w

introduction of derivatives such as m. , m- . etc. In practice, however, it is ijnlikely that the steady-state response will be specified exactly, as we have assumed here, - it is more likely tliat the static margin only will be specified, leaving derivatives other than m at our disposal for autostabilization piirposes. In these circumstances the desired response should be specified in such a manner that its steady state is zero (e.g. D ^^-Jj) should be specified, rather than w-,(r) as in this exanple.

Equation 2.29 is graphed in Fig.3. The basic response in w(r) is seen to be markedly oscilla.tory with a large initial overshoot. ¥e assume a desired resuonse of the form

m 6 = 9 . 1 5 1 - e " ^ ' ° ^ ( G O S 1 1 . 5 i r + 0.1504 s i n 11.51 T) whence Dw_^(r) " ^ r 7 ^ - 5 . 0 ' (112,0 s i n 1 1 . 5 i r + 30.8 COS 11.51 r) Prom E q u a t i o n s 2 . 1 , 2 . 3 4 , 2»35,

(2.34)

(2.35)

m _ . Ö 77 77 whence = 21.45 + e ' ^ ' ° ^ ( l 0 9 . 8 4 s i n 11.517" + 9.35 cos 11.517-) 1^77.077 = e - 5 , 0 T ( 1 , 2 1 8 . 2 cos 11.51 r _ 656,5 s i n 11.51 ^ ) (2.36) (2.37) We assume that the autostabilizer available is of such a type

3nal to Dw may be produoed, Thf we nay solve Equatitn 2,25 for that an elesrator deflection proportional to Dw may be produc^ed, Then puttine h = w with Piitostabilizer

m.

Thuï

}i . I . T\, ( O = e'"^-^'' ["(15.81 + 9.26 h) cos 11.51r +(-760.9 + 35.7 h)

"D

C

s m 11,51 r

+ a n e g l i g i b l e constant term due t o rounding-'ïff e r r o r s

P u t t i n g T = 11.51 r we obtain a f t e r some reduction

c o s ' T

(2.38)

]

2N - . 8 6 8 T 2

-^ . 1 . Hp (T)-i2 = (2.93.6 h + 85.75 h )e"

'^ "^ ^ -I + (-51,300 h + 1,136,0 h^) e - ^ ö ^ sin^ T

+ (-6,473 h + 312.0 h^)e"'®^®^ s i n 2 T

+ terms not involving h

I n t e g r a t i n g Equation 2,39 between 0 and " , making use of the

i n t e g r a l formulae of Eqra.tions 2,15, 2.16, 2.17 with e ^ 0 vre

obtain eventually

(2,39)

M . 1 . r)

( T )

3 n ^

= 710 h^ - 27,525 h

+ tenns not involving h

For t h i s i n t e g r a l t o be s t a t i o n a r y

h = h

cptimum

_7iox"2

27^525 = 19.6

(2.40)

(2.40

Substituting this value of h in Equation 2,58, the elevator

deflection demanded cf the pilot with the optimum autostabilization, 77 (r), is given by ^D -5.OT

198,71 cos 11.517" - 100,9 s i n 11,51^

(2,42)

£

1-

\

(^)

= e

ip 6 ^D ^ 77

Owing to the non-zero initial amplitude this deflection cannot be

attained. However, proceeding to Step 8, we find that the stick-fixed response with the optimum autostabilization is described by

w ( r | . 9,13 { ^ - e-^'^^''

m

cos 10,8r + 0.417 s i n 10,8r )1 - 43)

V T]

Equation 2,43 is graphed in Fig,4. It will be seen that it

approximates well to the desired response. The optimum autostabilization may, therefore, be regarded as satisfactory.

2^5^EXAIIPTJE 4 . SHORT-PERIOD LONGITUDINAL RESPONSE TO AT<f IMPULSFTE PITCHING MatJETCT OF AN AIRCR.'\FT HAVING A NON-LIIvTEAR VA.EIATION OF PITCHING Ma/Erg WITH INCIDP]NCE

( T h i s example i l l u s t r a t e s how t h e o p t i m i z a t i o n p r o c e d u r e may be e x t e n d e d t o d e a l w i t h n o n l i n e a r d e r i v a t i v e s and s e r v e s a s an i n t r o -d u c t i o n t o t h e t e c b j i i q u e u s e -d i n t h e s u c c e e -d i n g c h a p t e r s f o r t h e

o p t i m i z a t i o n of a u t p s t a b i l i z e r systems h a v i n g n o n - l i n e a r c h a r a c t e r i s t i c s , ) L e t t h e s t a t i c v a r i a t i o n of p i t c h i n g moment c o e f f i c i e n t w i t h

i n c i d e n c e be of t h e form C = A» + B « ' , where A -and B a r e c o n s t a n t s ,

m '

Then, for a conventional aircraft having the wing positioned near the

C.G., we may allow for this non-linearity in the equations cf motion simply

by replacing the term m . w(r) in Equation 2,2 by a cubic expression in

w(r). With this exception, we use the numerical data of Example 1,

The equations

ot

motion become

(D -

zj

w (r) - ^ (r) = 0 (2.1.)

-m^^ Dw(r) + m^ W ( T ) + m^. w^(7-) + (i^ D - m ) 'q (r) = m (r) (2,4!f)

~ IT n

For the same specified input as Example 1, and with the same desired

response in w

Wj^(r) ^ 0.1064 e~^*°^ sin 11.51T

Choosing Ai_ m = 132.3 and /£ m = -13,230.0

the static variation of C with « is of the foim shown in Fig. 5.

m

(This form is chosen in ordor to represent the characteristics of

'pitch-up'). The elevator deflection demanded of the pilot if the

desired response is to be attained exactly is given by

g r , • np^(0 = - ^ . r)w(r) H. m^.w^(r) + ^ . T

.^^{r)

V ^ D ^ r )

^D

Substituting the numerical data of Example 1, we obtain

to„ 77 (r) ^ g-0,43^T r (.,^23 sin T - 8.982 cos T)+k'(-0.2135 sin T 1 •i -PD J •3 •- +0.930 cos T) + 0.930 cos T ) - 15.98 e"^'^°^^ sin^T - /^ . m^(r) ïg ^ (2.46) where T = 11.51 r

As in Example 1, our criterion for the optimum k' is that

r

/ ^—77 . ''p^ ( T ) d T i s t o be a minimum, with e as

small as we p l e a s e .

Squari-ng Equation 2.4^ we obtain a f t e r some reduction

tiE^n ^^ (T)

l - ^

V . p-

D

2 ^ (e-ö-^34rg.^ y)2 (-0.8205 k " + 16.176 k ' )

+ e-°'^^^ (0.866 k'^ - 16.7 k' )

+ e-^'^^ s i n 2T (3.O6 k' - 0.1984 k " )

(2.47)

+ 2.96 k ' e"''*"^^^'^ s i n ' T cos T

+ terms not inv^l^^lng k'

With n e g l i g i b l e e r r o r , for very small e ,

e'-^ dt = — , / e~^ s i n wt dt =

P -'

ƒ- ' p^ + <^' (2.15,16)

^^ (e ^ . s i n t ) dt may be evaluated by means of a

t a b l e of Lap!].ace transforms or may be read from the gr-aphs (Pigs, I 8 & 19).

The remaining i n t e g r a l i s eval\aated by i n t e g r a t i ' i n by p a r t s , thus:

e- .sin'' t . c o s t . dt =

e , s m t

^at . 4 ,

, A a t . A, J.

ifa, e . s i n t dt

4

-a / (e'~ s i n t ) dt

4 (2.48)

(2.49)

e

which l a s t i n t e g r a l i s evaliiated as described above.

I n t e g r a t i n g Equation 2,47, making M^Q cf t h e above i n t e g r a l s ,

v.-e obtain a f t e r reduction

r wn„ 77^ (T) -i = 0.5255 k'^ - 10.785 k'

e + terms nest involving k' (2.50) whence k' , . ^^ = 10.785 =10.25

optamom 21^0^25

Substituting this value of k' in Equation 2.43 vre find that the elevator deflection demanded of the pilot if the desired response is to be attained exactly is given by,

HB^ ^ (T) = e~°*^^^(0.558 cos T - 0.955 sin T)

^ ^ -15.98 e-^-^°2T^in' ^ (2.5l) This is graphed in Pig.5. The required V (T) can hardly be attained,

mainly because the initial (T >e] jmiplitude is non-zero. It is necessary, therefore, to proceed to step 8 of the optimization procedure.

Step 8 presents more difficulty than hitherto due to the non-linearity of the equations of motion. The procedure adopted is as followH.

Equations 2.1, 2,44 are combined to give T^ / m. m , / \ TN , m k z Er + (-_w -_2,k - z j D + ^ i m + _2, w

^ ^B ^ ^

w(r)+/im w'(r) _ Mnu, (T)

(2.52)

where

TCIAT)

i s the (non-dJmensionalized)

applied impulsive moment

Since f m j r ) dr = 10"^, f

^J^(T)

d r = 1.225 (2.53)

i I

"B

S u b s t i t u t i n g the appropriate nijmerical data i n Equation 2.52 we obtain

r if + iO,4tfD + 150.61 w(r) - 1 3 230 w' (r) = 1,225 X

•- -• (2.54) where _L denotes a unit impiolse

Equation 2.54 is solved by Tustin's regression equation technique (see Ref. 8) - an extension of Cardinal Fp'=5ctrvim Analysis - using a step interval of —^ of an airsecond and replacing the 'init impulse

1 by a triangular pulse of equal strength (i.e. 'area') and. of base -rx

of finite-difference techniques such as that employed, the initial value is in error as (to a lesser extent) is the second value.

Allowance has been made for this in drawing the curve describing the solution of Equation 2.54. These errors could be reduced (but never elininated) by taking a smaller step-interval.

Ceiirparison of the desired response (Fig.l) with that graphed in Pig, 6, shows good agreement between the tv/o. We conclude therefore that the optimism autostabilization is satisfactory.

Discussion of the Optimization Procedure Step 1.

The piorpose of this step is merely to confirm the necessity for autostabilization. For non-linear cases the calculation required for this step may be considerable and if it is reasonably certain that the basic response is unsatisfactory this step may be omitted. Thus in Example 4 since the non-linearity is mild for small w it is

reasonable to suppose that the basic response of the aircraft will be somewhat similar to the basic response of Example 1., and Step 1 may saf aly be emitted.

Step 2.

There should be little difficulty in specifying the form of the desired response - though there may be considerable difficulty in assessing the merit of amy particular form of response chosen, For example, the author chose the form of the desired response in Example 1.

TCp(r) = 0.1064 e"^*°'''sin 11.51 r

aimply because it is smooth and highly damped and therefore likely to be pleasing to the pilot. Many other similar fonus of response would have been equally acceptable and it is not easy to formulate a numerical criterion for the relative merits of the possible response forms,

It is true that (military) aircraft specifications frequently demand that certain response and stability criteiia should be met

-f or exanrrple, a minimum value o-f the logarithirdc decrement e-f the longitudinal en.d lateral oscillations is commonly specified.

Criteria of this kind are chosen on the basis of pilots experience and preferences^ (see for example Ref, 5) tiu-t although these criteria define the oo-ondaries between acceptable and unacceptable response characteristics they provide little guidance on the relative merits of various acceptable responses. Optimxm forms of response are coramcnly specified for servomechanisms (see Ref. 10) usxially forms which minimize a certain function of output error (E) such as

i ^ ^^

- but OTving to the large number of freedoms possessed by an aircraft and th-© wide range of flight conditions under v/hich it may operate it hardly seems practicable to extend this concept of optimum response to aircraft flight. Certainly any attempt to do so would be beyond the scope of this present report.

Whilst this difficulty of assessing the merit of a given response should not be overlooked, we believe that it is of a philosophical rather than of practical importance. For any given aircraft one will always be able to suggest a suitable form for the desired response, even though one may be unable to define the optimum response.

Step 3.

It is obviously desirable that the type of autostabilization chosen should be capable of attaining the desired response without making excessive demands on the pilot. Otherwise, effort will have been wasted in fruitless calculation. For linear systems the time vector method of presentation provides an excellent means of predicting the probable effects of various types of autostabilization. The vector polygons for the short period longitudinal oscillation of the aircraft of Examples 1 to 3 are given in Fig.31. From inspection of these polygons one can deduce the tj/j^e of autostabilization most likely to achieve the high damping associated with the desired response. Although for the short period longitudinal oscillation one could deduce as much from the coefficients of the auxiliajry equation

/m m, \ iim „ ^2 - ( q + w + z ) \ - ' ^ w + m z = o

^ -"^ ^

for the more complicated lateral oscillation this is hardly possible and the use of time vector presentation is very desirable if an intelligeni; approach is to be made to the problem of choosing the type of autostabilization most likely to achieve the desired response. Step 4.

For general (i.e. non-optimum) autostabilization this ass-umption must be regarded as a mathematical artifice rather than an assertion of what is physically feasible. A check on the validity of this assumption vdth the optimum adjustment of the autostabilization is provided by Step 7.

Step 5.

A good discussion of the effects of pilot effort on aircraft response is contained in Ref. 4 v/hich see. This supports our view that the purpose of the autostabilizer is to relieve the strain on the pilot so that more of his attention may be devoted to tasks such as navigation, weapon aiming, etc, and so that he may have greater

reserves available for emergencies. Unless this view is accepted it is hardly possible to optimize the autostabilizer system of a piloted

aircraft as such, and one is reduced to improving the response character-istics of the (sam.e) aircraft in the (supposed) absence of a pilot, As shown in the Introduction this procedure may be somewhat unrealistic,

(An optimization procedure for pilotless aircraft is developed later in this report.)

The choice of effort function must be made on onpirical grounds, as there is insufricient data at present available on the psychological and physical strain experienced by a pilot in attempting a given task. In Example 1 to 6 an 'integrated displacement-squared' effort function

ƒ

CO

[77 (r) 1^ d r was employed. Although the

0 L % J

pilot actions demanded to attain the desired response wex'e in each example physically unattainable, with the optimum autostabilization the

aniplitude of 77 iJi each example was so small that the stick-fixed response was very close to the desired response. This state of affairs is more likel^T- to be achieved by the use of an effort function which is a function of control deflection rather than by the use of a more refined effort function dependent on time deriva-tives of the pilot's control deflections. For this reason the writer prefers to use simple 'displacement' effort functions rather than more refined types. Use of a more refined type of effort fionction ¥»'ould result in a m.ore physically feasible 77 but the

^D

results of the pilot failing to achiex'-e this would, in general, be less satisfactory.

However, many other types of effort functions may be employed. Duddy (in Ref. 4) suggests that the best control system (autostabilizer system in the present context) would be that which demands the simplest transfer function of the pilot. This criterion is much less easy to use as a basis for optimizing a given type of autostabilizer system since there is great difficulty in formulating a quantitative criterion'for the complexity of transfer functions. Step 6.

The minimization of the effort functions of Examples 1 to 6 has been performed by the standard procedure of the differential calculus. This will generally be possible(as we later demonstrate) even if the desired response is specified in a non-analytic manner, provided the effort function is of an analytic form, e.g.

/ ^r^ (^^) '^^ rather than, say / r? ^ (r)

J 0 L P]3 J •' 0 P D

a non-analytic effort function recourse may have to iterative methods of minimization,

d r . With be made to

Step 7.

Although the main purpose of this step is to examine the feasibility of the demanded pilot action, it also checks that the stationary value of the effort function obtained in Step 6 is a minimum and net a maximum.

In Examples 1 to 6 inspection of 77 (r) reveals the ^D

amplitude to be so small that one might woll surmise that the effect of the pilot taking no action whatsoever would be to cause only a slight divergence from the desired response. However it is always desirable to prove this by proceeding to Step 8, particularly so for non-linesir systems, where a pilot input of small amplitude may produce an unexpectedly large change in the aircraft response,

S t e p 8,

I t i s worthy of remark t h a t , f o r n o n - l i n e a r s y s t e m s , Step 8 ( t o g e t h e r w i t h Step 1) w i l l p r o b a b l y be t h e most t e d i o u s p a r t of t h e c a l c u l a t i o n .

More g e n e r a l coirnnents on t h e p r o c e d u r e a s a whole and comparisons v d t h p u b l i s h e d work a r e given t o w a r d s t h e end of t h i s r e p o r t .

CHAPTER 3'.

3 . 1 . OFJMIZATION OF SOIvS NON-LINE/iR AUTOSTABILIZER SYgTE^tS IMTRODUCTION

In general, the amplitude of the control surface deflection generated by the autostabilizer system will be limited. The limits may be chosen deliverately so as to avoid catastrophic divergence

in the event of an autostabili zer run-away, or may arise through limitations of available jack effort, or through installation difficulties. Provided the control surface deflection required to attain the desired response does not exceed the limiting value the methods cf the previous chapter may be applied, and the limits and the non-linearities arising therefrom need not be taken into account.

In this chapter v/e show how the general optimization procedure for piloted aircraft may be used to obtain the optimum

values of the adjustable parameters of such a 'limited' autostabilizer system for the more general case when these non-linearities cannot be excluded from the analysis. The procedure is applicable both to

'limited' or 'saturable' autostabilizer systems of the type described above, and to 'flicker^' or 'flip-flop' autostabilizer systems in which the magnitude of the control surface deflection is constant, its sign,

at any instant, being the same as that of BCTÏX? r*^^'*"?!'?'=^'^^ •^'^^'sp«riae parameter. In the latter case, we assume that the system parameter to be optimized is the magnitude of the control surface deflection.

We adopt the term 'Autostabilizer Sharacteristic' to denote the graph of the autostabilizer output (i.e. control surface deflection) against the input signal to the autostabilizer. It is first necessary to derive a family of continuous characteristic curves which approximate to the discontinuous ch^aracteristic of the actual autostabilizer. The seventh-power poljmomial approximation presented in the following section has been found to give solutions of good accuracy wdtliout introducing too great complication into the calciilaticn required for the optimizat1<ui px-ooedure.

^.2 To determine the Coefficients of the Polynomial Approx.-^-mation to the Autostabilizer" Characteristic.

\ \" i j f ^ v^. -"M • • — - > '

y^

•t ~ \~ ^ ^ y = P(x) 1 ^ ' - f — y A ( ^ , FIG. 3.1. In Fig,3.1.

y = the control surface deflection due to the autostabilizer

X = the input signal to the autostabilizer (Thus for, say, m autostabilization, y would be elevator deflection,

y, is the actual autostabilizer characteristic,

il

y., is the limiting value of y,

Xy. is the maximum value of the input signal that need be considered in any particijlar example, (Thus for m autostabilization, X|^ would be the maximum value of

%(r)

)

X is the 'saturation' x ,

s '

If the ma.::imum amplitude of control surface deflection is limited to prevent catastrophic divergence ensuing from autostabilizer fail-ure and run-avray, then y^ is fixed and the only parameter of y. at our disposal is x . For 'flicker' or 'flip-flop' autostabili f-at ion we

s

assimie that the parameter at our disposal is y,,.

In this section we seek to obtain expressions for the coefficients of the polynomial apj)roximations to y. . We select a poljnondal approxi^nation of the forra

y = a X +

X + a x^ + a x^ = P(x)

3 5 7

(3.1)

Oior c r i t e r i o n for the choice of a , a , a , a , i s t h a t

p cx> 1 3 5 7

P(x)

-^A

P u t t i n g P(x) - y

A = e

dx = a minimum

it will be seen that

(3.2)

'M

evaluated in two steps, 0 to x , and x to s s

discontinuous nature of y .

s dx must be

because of the

For 0 < X < X

yi M

(a ) x + a X + a x ^ + a x^

(3.3)

whence

f e ^ dx = 1 (a - ?-j5 x^% 2a3(a - V ) x= + f ! i (a - ^ )+ ^3

•n

a (a -

"^M)

+ a a x ' +

7 1 — 3 5 s s 2 11 3

+ a^ I

11 -• 11 2a a 13 ^2 _ i s + 5 7 X + a X

13 ^ -L ^

^^ 15

(3.4)

F o r X < X < s ^

(3.5)

= [-p(x)j« + 4 - % P W

1 "' '^ = ^M ^ - yM^ V - 2 % % 4 - 3 %. % 4 ' 4 ^M % ^

whence X . s a 2 3 2a a 3 5

^i

2a a + a ^ •1 g

7

V * | [ ^ ^ *%%] V

+ 3 7 2a a + a'' , , 2a a ^ 13 a

9

2 15 11

13 15

- y^/ x^ + y , a x / + -1 V, a_ x / + 4" y,, a , x * + f v , a_ X_Ö 'M s ^ -'M 1 2 "'M 3 s ^ 3 -'M s s ^ 4 -'M - a X - 2a a x^ - r 2a 1 s 1 g s 3 5 r 2a a + a T L 1 5 3 i

7

2_ £ _ s -a a + a a -1 X?

-E

2a a + a' 3 7 5_

'ii

• i s 2a a 1 3 a^ x ^ ^ 5 7 s - _I_ s 15

13

— r a a + a a n 7 L 1 7 3 s j

(3.0

1 = / e ^ d x = - " ^ m r 2 a x ' + 2 a x ^ X3 L 3 5 Summing E q u a t i o n s 3 . 4 and 3 . 6 we o b t a i n \

" [ - | a x ; - . 2 a x ; - . 2 a x j + 2 a x / j + ^ ( ^ ^ 3 )

r 2 l 4 l 6 l 8 l

j [ ^ ^1 - ^ 2 ^ ^ - ^ 3 %

"M

^ 4 ^ ,

^ J

1 6 1 8^ + - r a x + T a X T 3 a s 4 7 s

_ . . ^ ^i _ j ^ + 9 ^ % *^^5J^

- y^.. 2a a 3 1 3 5 ^ ï •*• 5 ^-l

7

2a_ a_ + a_ "-1 2a a 13 a^ xi^.* r «ia a. + a -1 . 2a a + _ 2 _ J L 11 5 7 L- 11 - 1 ^ * 13 "^ "^ 15

(3.7)

A necessary condition for Equation 3.2. to be satisfied is that

ai

aa

91

3a

31

3a

31

3a

= 0

(3.8/

D i f f e r e n t i a t i n g E q u a t i o n 3 . 7 we o b t a i n g 2a 2a 5 2a 2a

aa = " % ^ "^ % ^•s'' "^ ~3^ V "^ ~5~ "^ *~T "Si ^1~ ^

1 3 1 1 <» 1 4 _ ÏÏÜ = - 2 % " ^ + 2 ^ v l ' ' s + 5 2a

* 7 ^ 4 +

9 2a 2a 7 11 2a am . ^ 2a ?a 0 1 J. ^ i ^ L Y^ L ' L ^ ^ Z a a = - 3 •% ^vl + 3 •% ^ s * 7 ^ •*• 9 ^ * 11 ^ "^ 13

- 3 *% ^

(3.9)

^ 5 "Ms

(3.10)

13 2 6 ^ - 7 "^M s 2a 8 1 9 2a 2a 2a 3 . 1 1 5 __ 1 3 7

ai 1 e 1

^a = " 4 ^M ^ + 4 ^M "'s "^ r "^^ "*• T T ^vl + 13 "^l

F a r stutioii'^x-y I , t h e f o l l o w i n g m a t r i x e q u a t i o n r e s u l t s . 1 "I 1 1 1

3 5 7 V i

1 5 1 7 1 / 1 9 1 9

J...

.1 1 11 1

".3

' ' , ^

% 4

^ ^ 1 1 -•• 1 j a X,,

9 ÏV 'O

TJJ

L ' ^

^M ' " l k^ 2-Z 1 ^ 4 " 20

1 - ] £ .

•^ 42

1 kl

L 8 - 72 J

X, •. X ( 3 . 1 1 ) ( 3 . 1 2 ) 15 ^ „ 8

^ - 9 "^ s

(3.13)

(3.14)

T h i s e q u a t i o n :;ay be gen^^y^-lLtzQÜ. f o r a l l Xj^ and y „ b y p u t t i n g = Ai , "••—•— = A , e t c . The s o l u t i o n s f o r A , A , e t c . may be w r i t t e n i n t h e form A = P + P k^ + P k^ V P k^ + P k 1 0 2 4 6 8 A = Q + Q k'= + 3 0 2 A = R + 5 • A = S + 7 • + Q k" + R k + S k^ 8 (3.15)

- 2 7 - Kanaalstraat 10 - DELFT

The c o e f f i c i e n t s P , P , S , S are given in Table I both

» ' 2 ' e ' s o

as fractions and as numbers correct to 4 decimal places. It vidll

be observed that the determinants required for the solution of Equation

3.13 by Cramer's rule are ill-conditioned -hence the elements of these

determinants are left as fractions throughout the solution of Equation

3.13, only the final solutions for the coefficients P^, P„, S^, S^

being converted to decimal form.

3 5 7 /•

Graphs of y = A x + A x + A x + A x are plotted in Figs. 7-16

for k = 0, .1, .2, .3, .4, .5, .6, .7, .8, .9 It will be seen that

except for the lowest ks the polynomial approximation to the exact

autostabilizer characteristic is very close and we believe that the

accuracy of this approximation is sufficient for most practical

calcul-ations. We later give an ex-Miple for the k = 0 case (flicker

auto-stabilization), for v^ich the approximation is least accurate, the resiolts

of v*iich support this view.

3.3 Example 5. Longitiodinal Short-Period Renponse to an Impulsive

Pitching Moment Vvdth Flicker m Autostabilization

(This example illustrates how a flicker autostabilizer system

may be optimized using the same type of effort function used in the

previous examples.)

We shall employ the niimerical data of Example 1 and the same

magnitude of the applied impulsive pitching moment as in Example 1.

The non-dimensionalized equations of motion may be v;ritten as

( D . z^) w(r)

^\{T)

= 0 (2.1)

J^±

D ^. m

\

(r) A D - fla ) -^ (0 = "^(0 * m .

r,^{r)

. 77^(r)

where ni^('^) is the impulsive moment

T7 is the elevator deflection due to the autostabilizer

T? is the elevator deflection due to the pilot

From Exan^jle 1 the basic response in w(r) i s known t o be

. W(T) = 0.1064 e'"^*'^'^^''sin 11,51 T (2.8)

We assijme a s i m i l a r desired response to Example 1, i . e .

Wp(r) = 0.1064 e " ^ ' ° ^ s i n 1 1 . 5 l r (2.9)

The associated responses i n Dv/ ( r ) , Dq-,(r) & q-rCT") axe as given

i n Equations 2.10, 2t,11, 2.12. In p a r t i c u l a r y/e have 'in^ir) - e

(-0.284 s i n 1 1 . 5 l r + 1.225 cos 11.51 r ) . l

By t a l d n e the maximtmi value of the desired resoonse i n q, 8L ,

'^ ° -^ ^ a) max '

as 1.4 (the exact value i s 1.225 but t h i ö i s net c r i t i c a l ) the

c o e f f i c i e n t s of t h e polj.'nomial approximation t o the a u t o s t a b i l i z e r

c h a r a c t e r i s t i c are given by

A A A^ / \

'^s -~ ^P [ 5 . 3 8 0 3 %_ - 19.7388 / % _ V + 3 0 . 7 9 2 5 / ^ " ^ - 15.7104 / ^ \ ^

'' 1,4 v - V V'"-^/ v-^y

(3.17)

where n _, = the amplitude of the e l e v a t o r

deflo'jtion generatc^d by the

a u t o s t a b i l i z e r

V i s t h e system parameter at our disposal for optimization purposes.

For r > e ( e i s the duration of the impulsive moment) s u b s t i t u t i o n

of t h e numerical data i n Equation 3.16 y i e l d s , f o r m = - 0.205, and

for T = 11.51^

^ ^1 . T7 (T) ^ Q,3ö1 e'^"**-^^ (-0 532 s i n T + 1.225 cos T )

^ + 132.3.^0.1064 e " ° - ^ ^ ^ s i n T

+ e~^'^^^' (-12.68 s i n T - 9.385 cos T )

+ 0.760 e~°*^^^^ (-0.284 sin T + 1.225 cos T)

+ 25l.0r?p

[ 5 . 3 8 3 3 ( . Ï L )

- 19.7388r^D_j

+ 30.79251 %L - 15.7104

1.4^

5 (3.18)

= e

~°*^^^(1.0165 s i n T - 8.052 cos T )

+ 251.0 ri [ 5 . 3 8 3 5 Q - 19.7388 Q^* + 30,7925 Q= - 15.7104 Q^]

^ •' (3.19)-'

where Q = e''^'^^^'^ (0„875 oos T - 0.2039 sin T) (3.20)

whence,

- r?^ ( T ) = _ 0,003587 e " ° - ^ ^ ^ sin T + 0,0366625 P

PD • ^ 3

- üj, [5.3833 X (0,898 P ) - 19.7388 x (0.898 P )

+ 30.7925 x(0.898 Ff - 15.7104 x.(0.898 P f]

(3.21)

^'^ \ = ^ 8 - e-^-^^^^' S i . ( T - 7 7 . 0 ^ ) - ^^^^^^

Squaring Equation 3^21 we obtain

|2

[%<')]'

= vJ (23.5695 P ^' 138.2968 p S 378.6333 P '

•p 1 1 1

- 586.6059 P ° + 536.0218 p ^'^ - 266.7702 p ^ ^

+ 54.9029 F ^

+ T?p (-0.158493 i^ + 0.470if28 P * - 0.592057 P "

+ 0.22.36978 P ° + 0.16375 P e ~ ° ' ^ ^ ^ s i n T

- 0.04843 P ' e~°*'^^^sin T + 0.060978 P^ e " * ^ ^ ^ s i n T

- 0.0250995 P ^ e " ° ' ^ ^ ^ s i n T) / , ^^s

+ terms not involving rj

To obtain the effort function

/• 00 ^ ji'

j f^ (T)"1^ "TiT it is necessary

, ^ f» -D an jm „„J „^ 4.u^ .P

to evaluate integrals of the form ƒ P dT, and of the fprm

r

.„ 2n-1 -O.4321T . „ , „ ° ^ , ^ .,^ P, . e sin T d T . In order to avoid too great a digression at this point, description of the evaluation of these integrals is held over to the next section of this chapter. Suffice to state that general formulae and graphs axe given therein for the evaluation^of integrals of the forms

/ P ^ dT and ƒ P^"'' e^^ sin T d T where

aT ''° / > J''

P = e sin (T +/?), a and/? being constants.

Using the results of the next section with P = P we obtain,

r

i« 00 « 0 1

P ^ d T :5 1.097, / P / d T a 0.908, / P / d T = 0.852,

I f

d T = 0.807, fp/" d T = 0.774, fp/'' d T = 0.757

I p g-C.434T ^^^ T d T = 0.2461, j P " .e"°-^^'+'^ sin T d T = O.366,

p^5 ^^-0.434T sin T d T = O.41O5, / P ^ . e ~ ° ' ^ ^ ^ sin T d T = 0,422

• o

Integrating Equation 3.23, malcing use of the above integrals, vre obtain

after some reduction, -^

J ° [ r 7 ^ ^ ( r ) ] ^ d T . 2.3522 77/ -0^54959 r7p (3.24)

o

+ terms not i^ivclving 77„ whence

0,54959 X 57.296 ^ + 0 . 6 7 °

^'F optimum " 2 x 2.3522 • (3.25) (The positive sign indicates that the sign of r/ at any instant is

the same as that of q .)

Since the iiotcl elevator deflection, 7? + V , must be

'

PD

^

of a smooth nature in orf.or to obtain the desired response exactly, and since V (r) is discontinuous, it follows tha.t rj (r) must also be

s ' ' Pp' '

discontinuous. The pilot will certainly be unable to provide such a discontinuous V (r), so we may proceed at once to Step 8 of the

PD

optimization procedure, without actually calciolating the time history of V , wi.th the optimum adjustment of V .

?D F The stick-fixed response for 77 = 0. 67° and with the exact

autostabilizer charactexistic (not the polynomial approximation) has been calculated by piecewise application of standard linear response theory, the 'pieces' being the intervals between successive zeroes of q(7'). The resulting time histories of w(r), and q(r) are plotted in Pig. 17. It v/ill be seen thtt the response in w(r)

approximates well to the desired response and the optimum autostabilization

It will be observed that the solution for q 'ends' at the second zero of q. (T) , Thir is a familiar phenomenon in discontinuous autouiatic control systems may be explained as follows".

Due to the presence of (unavoidable) time lags in the autostabilizer circuit 'switching' of the elevator does not occur until a short time A T after q. ( T ) changes sign. Hence the graph of 4 ( T ) against T at the second zero cf q is of the form drawn below.

Switclung occurs at the point B, when the q ( T ) graph oonmeiiices to follow the path B'-JDE. But switching in the opposite sense occurs at point D, whereupon q (T; commenoes to follow the path DFHG, which it does as far as the next smtchiag point H. The conditions at H axe similar to those at B and so the cyclic variation of q is repeated ad infiüitum. This phenometon is known as 'chatter:?_ng' and, for this example, is of theoret^lcal rather than practical interest

since an exact flicker characteristic is not practically attainable, and the pi'esenoe of unavoidable imperfections such as dead-bands

(see below) in the autostabilizer characteristic generally obviates chattering. Cc!i,'«rcl Surfa'3y D'2fl'='-ction ^ I I t Input Signal —m^é • Dead-band

The time history of W(T ) vrf.th chattering in q ( T ) , andT?(T), may

be found from Equation 2.1, a:'suming that the chattering amplitude

of q is negligible in coraparisoh with w(r), and Dw (r ).

With this assumption Equation 2.1 becomes

(D - z j w (r) = 0 (3.26)

This has the solution

(3.27)

/ \ +Z .T

w (r) = w . e w

^

'

c

where w is the amplitude of w at the

commencement of chatter

and T is measured from the commencement of

chatter

3.4. Evpluaticn of Some Integrals Required for the Optimization

of Non?lineex Autostabilizer Systems

As demonstrated in Example 5 we require to evaluate aiib«gj.-aln

of two kinds, (i) / P^" d T,

and (ii) / P^""'' e^^ sin T d T, where P = e^^ sin ( T + ^ )

It will be found that integrals of these kinds are frequently

req\iired v/hen optimizing non-linear systems in v/hich the non-linearity

is expressible as a finite powei' series in some response parameter and

it is convenient to evaluate these integrals once and for all for a

range of

^ , P ,

& n rather than separately for each example.

Evaluation of Integrals of the First Kind

Denoting these by I , we have

« '^ 1 00

I = ] i^" d T = 1^ e ^ ^ . sin^ (T +/?) d T (3.28)

The substitution t = T + /9 yields

I = e - 2 " ^ (I - I. ) (3.29)

\ ^ CO p '

where I = / (e^* sin t ) ^ " d t (3.30)

oo

I may be obtained from a table of Laplace transforms (e.g. Ref.lS) as

CO

00

I =

[

e^^* sin^^t d t = ^MJ.

^ (-2na)(2m^ + 2'')(2na'' + 4*^) (2na'+ 2n ') (3.32) I^ is plotted for n = 1,2,3,4,5,6,7 and -0,5 < a < -0.05 in Figs. 18 & 19. I has been evaluated by graphical integration for the same n and a as I , and for 0 < /S < 2'ir , Carpets cf the variation of ja with these parameters are given in Fig,s 20 - 26

Evaluation of Integrals of the Second Kind Denoting these by I , we have

I =

2

00 " ~ 2 '

= I P^^-^ e^* sin (T +/5) dT = f e ^ " ^ ^ s i n ^ " ^ (T + /?). sinT, d T

(3.33)

0 0

The s u b s t i t u t i o n t = T + ^ jrields

I = e ' ^ " ^ L e^""^^ s i n 2^-^ t , s i n ( t ^ ) dt (3.34)

2 ito P

-2x\afi r 2nat r . 2n . ^ . ^ . 2n-1. x 1 .4. /'T Tc^

= e e s i n t cosp - s m p , s i n t , cost J dt {3.35)

= e - 2 ^ cos ^ ( I „ - I^ )

-2na/5 . ^ / 2nat . 2n-1 . . •,. / , ,/:%

- e s m p ƒ e , s m t , c o s t , d t \3.3o)

h

The l a s t i n t e g r a l i s evalioated by i n t e g r a t i o n by p a r t s v/hich y i e l d s

with a < o ( i , e , a s t a b l e d e s i r e d response)

0 0 ' ^

2nat . 2n-1 ^ ^ ,. -1 . 2n ^ • 2 n ^ / 2nat . 2ia . ,,

e s m t cost dt = - ^ s m p , e - a / e s i n t dt

Z?" "" /? (3.37)

Thios s u b s t i t u t i n g from Equation 3.37 in Equation 3.36 we obtain

I^ = e-2na/? , ^ 3 ^ ( i ^ - I^) + ^ sin^^^^ /S + a s i n ^. e ' ^ ^ ( l ^ - I^)

(3.38)

, - . I = 1 ^ sin^^*''^ + e - 2 " ^ (cos /? + a s i n ^ ) ( l ^ - I ^ ) (3.39)

This equation enables us to express the integrals of the second kind in terms of integrals of the first kind, as below

I = -5- sin^'^V + (cos ^ + a sin^) I (3.40)

It will sometimes be found that for large a, n, and for /9 close to 2n- , I - I^ is given as a small difference between two approximately eq\ial numbers, and in such circumstances difficulty will occur in

evaluating I and I accurately. Rather than attempting great accuracy

1 2

ia the evaliiation of I and I^ , the best procediore is then to replace /? by a negative angle - 2w + yS = /? and to evaluate

1z - \ (e sin t) d t graphically or numerically.

•- 2n+/5

3.5. Example 6. Longitudinal Short-period Response to an Impulsive Moment wi-th Limited "^g Autostabilization.

(This example illustrates how a 'limited' or saturable autostabilizer system may be optimized by the use of the seventh-power polynomial approximation developed eaxlier in this chapter. An effort function similar to that of the previous exaniples is employed.)

With the numerical data, and desired response, of Example 1, and assuming a limiting elevator deflection of i 1.05°, the calculation proceeds similarly to Example 5, except that Equation 3,17 is replaced

'q SAT. being the (3.42)

A

saturation' value of q

and the coefficients P ,... S are as listed in Table I

o 8

The problem i s t o find the optimum k . P u t t i n g k^ - h the

equation corresponding to Equation 3,21 of Example 5 i s

- ^ p ^ ( T ) - ! ^ = ^ (-0.003387 e - ° ' « ^ ^ s i n T ., 0,040827 Z)

+ [P^Z-^V' ^ V ' -^^o^'l

+ h r P Z + Q Z ' +R Z^ + S z H

L '2 Z 2 2 J

+ h ^ r P Z + Q Z ' H . R Z ^ + S Z ^ l

L 4 4 4 4 i

+ h ' r P Z + Q Z ' + R Z ^ + S Z ^

|_ 6 6 6 6 .

+ h^ [P Z + ^ Z ' + R^Z= + S^Z' ] {1,1^)

where Z = 0.898 P , P being defined by Equation 3.22.

To evaluate the effort function,/ TJ (T) I*^ d T

we write Equation 3.42 as

^ -1.05 - ^ + ^

where a = - — ^ (-0.003387 e"^*^^'*^ sin T +0.040827 Z)

and b = the remainder of the r . h . s , of Equation 3»42

(3.43)

0 0

(3.44)

f[77 (T) x ^ ^ 2 ^ 1 ^ d T = r a * d T + f 2 a b d T + | b ' ' d T

0 jto 0 0 O

In Equation 3.44 j a^ d T need not be evaltiated since i t does

• 0

not involve h , and therefore will not appear in the eqviation for the optimum h, rr- T f^ ( T ) I'' d T = 0.

The evaluation of / 2 a b dT is straightforward since the integrals required are of the forms T P^ ^'^ dT, / P^"^ ^-0,4341 gjj.^ rp ^jj^ and these may be read from the list on Page 29 or more generally, for other problems, evaluated by means of the carpets of FigS. 20 to 26 Using these integrals / b* d T is easily calculated once b is known. However, b is comprised of no less than twenty terms and it

2

would be very tedious to have to evaluate b anew for each problem. b ^ has therefore been evaluated once and for all, for a general Z, the result being tabulated in Table II.

Making use of this table, we eventually obtain

2

[ ^ ' o l ^ ^vJ'^^ ] dT = -50.50289h + 328.96l6eh^ + 2,512.19225 h'

" +2,293.68167 h^ - 2,828,8175 h'

+2,098.11575 h* - 861.2662 h^

• 150,35009 h^

(3.45)

(Note that it is desirable to leave rounding-off until late in the

calculation, as far as possible. This is the reason for the appearance of such a large nuniber of significant figures in the coefficients of the