Designing an Adaptive System to Improve Piloting Characteristics of Transport Space-craft

DESIGNING AN ADAPTIVE SYSTEM TO IMPROVE PILOTING CHARACTERISTICS OF TRANSPORT SPACE-CRAFT

A.D. Aleksandrov, Moscow, USSR

K.'r. Tsaturyan, Moscow, USSR SUMMARY The necessity of creat,ing an adaptive syst em for the improvement of piloting characteristics of a transport space aircraft is substantiated. On the base of stability condition, a specific synthesis of laws for correcting coefficients in damping, stability and controllability circuits is carried out. The laws derived assume different versions of practical realization. The technique of choosing the preferable variant, according to the criteria combination, taking into account, parallel with dynamic indices, weight, realiability and so on is discussed. In recent years great attention has paid to the idea of creating space aircraft of the recoverable type /1,2,3/. The design of transport space aircraft, developed at the present time, to specify the use of winged craft or craft with lifting fuselage, able to land on the required airfield, according to a conv entional airplane configuration. Methods of automatic landing were pres ented for such craftes /4/. Besides automatic control, such system must provide the possibility of a piloted r eentry from the orbit and landing. It should be taken into account, chat the expansion of the speed range and altitude even of modern supersonic airplanes results in an essential change of their design and deterioration of piloting charact e ristics. It is quite natural to suppose, that the cont rol a transport space aircraft will be impossible without int r oducing special au tomatic devices, that make easi er the piloting process. Div., ...'se on-board automatic devices from an intricate equipment complex, providing the solut ion of navigation and control problems . 'r he p r oc e s s of desi c;ning such complex is made e asier by sub dividinc the e 'luipmcnt into simpler complexes such as: navi gation and trajec t or:r control system and the s;ystem to improve piloting characteristics. The last system must provide manual and automa tic control with high quality. Th e sy st em i mproving pilot ing characteristics, forms a single whol e with t he craft and constitutes, together with t he latt e r "an cr'aft wit h required characteristics". As the craft charac teri 3ti cs change in a wide range , depending on th e environment a l conditions, there arises the probl em of making an adaptive system for improving piloting characteristics. Such sy s tem

580

A. Aleksandrov, K. Tsaturyan gives the possibility to test the new craft under conditions, when accurate craft and control surfaces characteristics are not known. If, during tests, there reveals the necessity to improve the craft or control surfaces, this does not influence the process of designing the navigation and the trajectory control system, which from the very begining of the design process is fitted to one "craft with required characteristics". Figure 1 shows a blok-diagram of system for improving the craft's characteristics, which includes contours for improving damping, stability and controllability. The coefficients in these contours are corrected from a selfadaptive system. If we consider the control stick deviation as the input, and 'the vertical acceleration as the output, the simplified mathematical model of short-period longitudinal motion of the craft contains three parameters:

81 = 2~Q;

Bo=Kp; where

B2,=Q,

"-

Kg> - static gain coefficient; ~ - relative damping coefficient; Q - natural oscillation frequency of short-

period motion. Using experimental estimates according to the Cooper scale, it is possible to choose in the region of parameters eo, 8i and El? ,all the points of which are admissible from the point of view of pilots. In this region must lie the parameters of a real craft provided by a system for improving pilQting characteristics. Parameters 8 0 , 8i and ~2 of the craft itself, for the most part of possible regimes, are outside this region, that results in the necessity of adjusting gain coefficients (see figure 1). One of the best known methods of selfadjustment of aircraft control systems is the method of a reference model /5, 6/. Let us use the method, proposed in work /7/, and carry out a specific synthesis of the coefficient correction laws. Disregarding the dynamics of sensors and of actuator, we shall, have a design diagram (see figure 2). The synthesis is carried out supposing, that the system satisfies the quasistationary conditions, introduced in work /8/. It is also supposed, that the behaviour of the reference model describes the short-Qeriod motion of the aircraft with constant parameters Bo* , Bi* and 82.*' The values of these parameters are chosen inside the above mentioned region, acceptable from the point of view of pilots. Let us introduce vectors:

[ rL~]

~-~~

[Cf] ; X=

0 ; 'J*

f,tL*]

[L'.tL 1 r I'V* -

t:t; "~= :"ftt*- :~. n..'j-]

581

&l

'"

Control stick

W

Gain

L-.-

Controllability coefficient

~

Vertical acceleratio_n Actuator

rL--

r---

Vehicle

.

:0-

> I-' Cl>

I>'i"' Cl>

Damping coefficient

r--

~ < .

Speed gyroscope

o

~

Stability coefficient

.......--

Accelerometer

>-3 Cl>

~ >:!

~ ~

Figure 1: Block-diagram of the system

w

Kv

tult


~Laws

of coefficient corrections Figure 2: Design diagram

._:-io

A. Aleksandrov, K. Tsaturyan In the vector matrix form the equations of the system and those of the model may be written in the following form:

~ =An.~+Bn,:t;

(1)

d~*_

(2)

cif; -A*~*+B*:t,

where y - system output vector; y. - model output vector. Relations for matrices, entering equation (1), have the following form:

A.J !+KKW~KKV

b

K"

T2>

o

o

o Matrices

and

A~

B~

An.-A+r ~ Bn,-B + A.

are given as the matrix sum:

Matrices A and B characterise an aircraft wi thout a control system, and matrices r and b. include adjusted coefficients

r =[_ KKw+0KkvKh, TZ

17

~Kw Ti

] ;

~=[ KK~K~ rZ Taking for the and for matrix assuming, that it is possible

matrix elements A* - An, the symbol exi,;' , el ements B. - Bh, the symbol g,~~ arrd the matrix coefficients are quasistationary, to have matrix equations:

t~~ J=-tJ~}];

0) 583

A. Aleksandrov, K. Tsaturyan

(4)

elements of matrix r ; elements of matrix l!". • Subtrating equation (1) from the equation (2), we shall have the equation of vector error in the following form: t;,~

where

OL~

-

~lJ' =A*~~+[A*-AI'\,] lJ+ [B*-B~J:r..

(5)

Considering elements (XL~ and f>i~ as additional state variables along with 0 l!".~ and 0 dD.tv/ dt , we shall construct a positive definite quadratic form from these variables: 2

2

V=t,f P 6~+.E [':=1

+= (~i,VX,tk +1L}J>t~), t=1 0

(6)

()

and 1i,~ -arbitrary positive constant ofOrelative dimensional values. The index T in equation (6) means transposition. The matrix P is defined from the matrix equation where

(7)

The matrix

Q

~' th[~ifOllOWli~

o

where ~i ~ and 1 i 2. sion values i - i se~ In our case

~2 -

,

~i (8~* -t- 82o*) +~2. 8~* 2.8 1* 82*

p=

584

form:

units of re la ti ve dimen-

~2, = 1sec .

A. Aleksandrov, K. Tsaturyan :Function V is reduced to zero at ~y.. = 0 , i.e. in the case, when the system parameters c~rrespond to the model parameters. Then~ the time derivative of \T is found. Using expressions(5) and (7), the derivative may be given in the following form:

where

~i - the j-th element of the vector y; :t,} - the j-th element of the vector x; p~ - the i-th column of the matrix P.

If elements cq~ (t) and r"L~ (t) are chosen according to laws a J 6

dJxL~

~t t.~T PL

cit=djh'~}

cit=then dv / dt is function of variables

!,r .

t In this case the Liapunov is provided.

6l1i

J.!L}

~:r

t.f Pt

(10)

11,} a negative semi-defini te Mrvj (Lt , OGL~ and Q

system stability according to

For our case, when equations (3) and (4) are taken into account the number of equations is three. Solving equations (9~ and (10), as to adjusted coefficients, we shall, finally, have

585

A. Aleksandrov, K. Tsaturyan

(12)

1

dKtt, _ _ (T:? )r~ ~n+ ~i+~28~ . ~tv]J_l_ dl1,~_ dt - '2Bz,*" KKV l i 2B i *8Z* dt hfLz2. dt -

~2i ~tv~ }

(13)

Thus, with assumptions made, we have succeeded in carrying through the synthesis procedure and in receiving the adjustment laws desired. Constant values }-i.2! and ~2i entering into equations (11), (12) and (13), have dimensions of sec 4 ,the constant ~zz - dimension of sec2 • The selfadjustment speed may be varied by the choice of these constants. But this speed is proportional to the expression given in round brackets, which depends on the real craft characteristics and must change as to time, in accordance with the programmed trajectory. This concerns the value ~ in equation (13), as well. The laws (11), (12) and (13) are complicated from the point of view of practical realization. Therefore, quite natural is the wish to make them simpler, without greatly breaking the stability conditions. For example" assumi~ th~t there is no information about speedsdAtt"dt and diL~ / dt , simplified laws are received and it is possible to have a constant coefficient in the damping circuit. Different simplifying assumptions result in different version of practical realization, some of which are not greatly inferior, than the principal one, from the point of view of stability. This happens because of the fact, that on the base of Liapunov~method it is possible to have only sufficient conditions of stability. However, the synthesis, carried out, is rather useful from the engineering point of view, as it gives the possibility to estimate solutions, which provided by the modern theory of model reference adaptive system. Thus, in the process of designing an adaptive system for the improvement of the craft piloting characteristics, there arises a number of versions, providing static and dynamic characteristics and acceptable from the point of view of pilots. Then it is necessary to choose the preferable variant of the system, taking into account contradictory criteria. Along with dynamic and static factors, 586

A. Aleksandrov,

x:.

Tsaturyan

these criteria take into account such factors as weight, reliability, power -consumption, cost of the system and so on. Then, the possible procedure of solving similar problems, taking into account work /9, 10/ is given. Let us consider the table of positive numbers ~L}, formed on the base of computed values of the j-th criterion for the i-th variant of the system Table Variants Criteria W;

...

Vi

. ..

Wit

·

'\Ch}

Vi-

~l

·

'lD't}

Vt'n

'LD-mi

·

W'm.,i-

.

~

... ·

· ·

Wh, '\.6th,

.

'\JJttt. 'LD-m.n.

The criteria of the table may be subdivided into two categories: criteria of the efficiency type; criteria of the expenses type'. In order to compare criteria of different dimensions, the next normalization procedure is carried out. The minimum element min 'U}-ti and the maximum element max ~~ are chosen in ea~h column. If it is desirable to g1've the maximum value to the criterion W· (the criterion of the efficiency type), then, accordi~ to equation:

and, if it is desirable to give the minimum value (the criteria of the expences type), then according to equation: • ,~.• - mLI1" tl};. WLJ} L LJ'

the matrix of normalized losses id developed. Preference is given to the variant, for which the sum of normalized losses, accordirgto the criteria combination tt.

PL=E QL~

}=1

0

(14)

has the minimum value. When solving such tasks, different weights are attributed to different criteria on the base of recent experience or on the base of the method expert 587

A. Aleksandr6v, K. Tsaturyan estimates

/10/. But the process of criteria weighting can be regard-

ed from another point of view, connected with the objectiv e difficulty of reaching an optimum value according to a given particular criterion, taking into account the whole variants combination. If for some criterion the majority of variants have values, close to an optimum or far from some unallowable level, then it is not difficult to reach this criterion and it must be attached smaller weights. When choosing the preferable variant, it is recommended to carry out the criteria w~ighting on the base of matrices of normalized losses. By summing up the elements of matrix A= [a.L~J according to columns , it is possible to receiveOdata, reflecting the possibility degree of reaching the value may,: 'llYr or min UTi,~ according to the corresponding j-thtcriterion foraa great number of variants considered. In this case the number of variants must be strictly more then two. The normalization of the indicated sums is carried out according to the formula: m. I: Cl'· t,

~J'

.A.j.= om'"

l;: ~ ClL}

t~el

Values.A.} are as weight coefficients. Taking into account these weight coefficients the formula (14) will have following form: n,

PL=~ Yv}C1LjWhen summing up normallzed losses according to the matrix columns, the criterion, having the bi ggest sum, receives the highest wei ght; when the process of summing up is carried out according to line s , one prefers variant, where the criterion, which it is most difficult to achieve as to the whol e variant combination, has the lowest value. If the criterion, which it is most difficult to achive according to the variant combination bel ongs to the efficiency type, then the weighting process 0f the penalty matrix shifts to the side of the vari ant "progress ive in the technical meaning of the word" • If the c r it erion, which it is difficult to ac ~i ve , belongs to the expences type, then the weighting process shifts to the side of the variant "progressive in the economic meaning of the word". The existance of such duality in the process of technical-economical design analysis is known from literature. The presented procedure of comparing alternative variants, according to the combination of contradicting criteria, is very simple. It does not make necessary to enlist the service of experts in order to estimate the 588

A. Aleksandrov, K. Tsaturyan significance of the criteria. Weight coefficients received for a number of specific tasks by means of the method described, coincide in practice with weights, received with the help of expert estimates /11/. However, the criteria values can differ for different flight regimes of a space vehicle. If a common system has to be designed for all regimes, it is necessary to have a set of initial tables for different regimes, for example, for the regions of injection, reentry and landing. In this case, the normalization is carried out according -to the same formulas, the corresponding maximum and minimum values being assumed for all the criteria not from each table taken separately, but according to the whole combination of versions and regimes. The criteria weights are also defined taking into account all the variants for all regimes. A column of weight sums of normalized losses for each regime is received on the base of the procedure described. With the known possibility of appearing regimes refferred above, the furth er choice of the preferable version is carried out, using these possibilities as weight coefficients. However, even in this case, the use of the matrix weighting as to regimes gives the pos s ibility to estimate the relative difficulty of reaching an optimum value for different regimes for the whole criteria combination. This technique gives the possibility to design an efficient adaptive system for the improvement of piloting characteristics of a transport space craft. Bibliographical References

/1/ !!JyHevJ30BaHHH.

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/4/

/7/ Porter

B., 'i'atnall M.L. Performance characteri s tics of multivariable model refer ence adaptive system synthesizeq by Liapunov's direct method . Internat J. Control, 10, (1909),W- 3. 589

A. Aleksandrov. K. Tsaturyan

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1972.

/9/ !DDC P.,

P8~$8 X.~rp8 H pemeHHfi. UocxBa, ~3~. HHOCTp. nHTepaTyp8, I96I. /10/ qep~KeH Y., AxoW P•• ApHO$ n. BBe~eHHe B HCCne~OBaHHe onepa~HH. MOCKB8, HaYKa, I968. /11/ AneXC8H~pOB A.~., TIeTpoBa M.B., UaTYPfiH R.T. Ronz~e­ CTBeHHoe cpSBHeHHe anDTepHaTHBH8X B8pHaHTOB HJlH cTpaTerH~ B 38~8qax ynpaBneHHH. Tpy~ Y BcecolO3Horo COBe~8HHft no npo6neuau ynpaBneHHH. MOCKB8, Q8CTD I, Hayx8,

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590