Robust Analysis of Handling Qualities in Aerospace Systems

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ROBUST ANALYSIS OF HANDLING QUALITIES IN AEROSPACE SYSTEMS A. Cavallo*, G. De Maria* and L. Verde** *Dipartilllfllto di IlIjormatica e Sistemistica , UIli! 'e nitiI degli Stlldi di .\'apuli, via C/alldio 21 , 80125 Napoli , Ital), ** Cl'lltru Italiall o Ricerche Aemspa ziali (C. I.R.A .), via iHaiorise. 8/0-/3 Capua (CE), Italy

Abstract. The modern Control Configurated Vehicle (C.C.V.) design technologies aims at taking simultaneously into account performances and maneuvrabllity requirements, and handling quality specifications. Therefore it is very important to provide the modern stability and control engineers with a tool which allows to determine the maximum allowable range of aircraft parameter variation according to handling quality requirements. By using decoupled linearized model of longitudinal and latero-directional dynamics, handling qualities are expressed in terms of pole location, with assigned tolerance, in the complex plane. With reference to a General Aviation aircraft Partenavia AP68 TPVIATOR, in this paper we present a procedure whichallows to compute an estimate of the domains in the planes altitude-speed (Flight Envelope) and weight-center of gravity location (Weight and Balance Envelope) inside which the specified handling qualities are respected. Kevwords . Aerospace engineering, Aerospace control, Dynamic stability, Robustness, System analysis.

INTRODUCTION The behaviour of an aircraft, considered as a rigid body, is essentially determined by its mass properties, propulsive forces, along with the aerodynamic forces and moments due to the interaction between the vehicle and the fluid medium which it moves through. The forces and moments acting on the aircraft depend on its speed, body geometry, angle of attack and on the air characteristics such as density, Reynolds number etc. Aircraft aerodynamic characteristics, the propulsive system and airframe characteristics substantially affect flight dynamics, aircraft performances and maneuverabi lity characterist ics. Moreover, airworthiness rules (MIL-F-8785B (ASG), 1969) require that the aircraft is safely cont rollable by the pilot without any exceptional piloting skill. The latter concerns with the dynamic aircraft behaviour and is referred in the literature as handling qualities. By using decoupled linearized model of longitudinal and laterodirectional dynamics, with reference to an equilibrium flight path and to fixed values of aerodynamic, propulsive, inertial and structur'\l parameters, the handling qualities are expressed in terms of pole location with an assigned tolerance in the complex plane . The modern Control Configured Vehicle (C. C. V. ) design technology aims at taking simultaneously into account performances and maneuverability requirements, and handling quality specifications. Therefore it is very important to provide the modern stability and control engineers with a tool which allows 1) to establish which parameters are important in terms of handling qualities and which are not; 2) to determine the maximum allowable range of parameter variation according to handling quality requirements, so that he can decide how to change the design of the aircraft in order to obtain better damping characteristics if this is so desired, or which type of stability augmentation to add. In order to solve the above problem, in the aero-

nautical literature many authors refer to the use of classical Routh-Hurwitz method for the stability analysis, or to root loci by gridding over one parameter for the stability analysis with respect to assigned regions of the complex plane (Roskam, 1982). In the very last years the automatic control literature has grown rich wi th new methodologies for the robust stability analysis, mainly due to the infl uence of the work of Ackermann (1980), Fam and Meditch (1978) and finally of the celebrated theorem of Kharitonov (1979). Starting from Kharitonov's theorem, many authors have proposed several procedures in order to solve the problem of robust stability analysis with respect to specified regions of the complex plane, in the case of structured perturbations, and, particularly when characteristic polynomial coefficients depend linearly on physical parameters (Tesi and Vicino, 1988; Barmish, 1989). However, such methodologies cannot be appl ied to aeronaut ical case due to highly nonlinear dependence on parameter uncertainties of characteristic polynomial coefficients. Recently the authors have proposed a computational tractable procedure in the case of nonlinearly dependent coefficient perturbations (Cavallo, Celentano and De Maria, 1989a). In this paper such a procedure is suitably modified in order to reduce the computational burden. The modified procedure allows to carry out the robust stability analysis with respect to assigned handling qualities, taking into account all the parameters which characterize the aircraft dynamic behaviour. With reference to a General Aviation aircraft Partenavia AP68 TP-VIATOR, such a procedure is applied in order to compute, in the case of l ongitudinal dynamics, an estimate of the stability domain in the planes h,V (Flight Envelope) and w, x (Weight and Balance Envelope) with respect to cg

specified handling qualities. Inside such domains the assigned handling qualities are respected.

This section contains aeronautical terms used thematical model of the zed with respect to the ters.

P T

a brief description of in this paper and the maaircraft fully parametrimost significant parame-

1)l

The model

Now we briefly outline the decoupled longitudinal equation of motion of an aircraft. Most aircraft dynamics text~ e. g. Etkin (1972), Roskam (1982), give more detailed versions of the derivation of this equation.

Notations A b

c C

00

wing aspect ratio (wing) span mean aerodynamic (geometric) chord drag coefficient at zero lift coefficient

c 1((

airplane lift curve slope - tail off

C

horizontal tail lift curve slope

100l

CL,C O

lift and drag coefficients (airplane)

C

airplane lift curve slope

LO: C Loe

variation of lift elevator angle

coefficient

with J.513_

coeffi c i ent at attack, zero elevator pitch rate

C

variation of pitch rate

C

airplane pitching moment coefficient

C

wing moment coefficient aerodynamic center

Lq

mac

1 i ft

lift

coefficient

C

variation of pitching moment with angle of attack

mo:

CmOe

variation of pitching coefficient with elevator angle

moment

C

variation of pitching coefficient with pitch rate

moment

IDQ

e g h i

l

Oswald's efficiency factor acceleration of gravity altitude horizontal tail zero lift attack

1

y l

q S S

dynamic pressure reference (wing) area horizontal tail surface

T

angle

of

engine thrust true speed

V

horizontal tail volume coefficient

x

location of aerodynamic center measured from the wing leading edge

H

ac

x

location of the center of gravity measured from the wing leading edge

r

flight path angle control elevator angle average downwash angle induced by the wing on the horizontal tail at zero angle of attack

C9

oe c o

C

0:

q S c C

(3)

y

(4)

2

V

T

(2)

where q=pV /2 is the dynamic pressure, and the air density p can be expressed as p=p(h). CL' Co' and C

are the

lift,

drag and pitching moment

aerodynamic coefficients. Such coefficients depend nonl inearly on 0:, q, V, the elevator deflection oe, the aircraft geometry and air characteristics. For low Mach number (M~O.4) and high Reynolds numbers, coefficients CL and C can be con-

pitching moment of inertia

mass (airplane) Mach number pitch rate

qS C + g sin(o:-fJ) + T coso: m 0 m

a= q

distance between airplane center of gravity and horizontal tail aerodynamic center

m M q

(1)

q I

wing zero-lift angle of attack I

equarigid plane writ-

qS C g Tsino: mV L + q + V cos(fJ-o:) -

-mv--

zero angle

pitching moment coefficient at angle of attack, zero elevator and zero pitch rate

mO

In the polar coordinate velocity form, the tions of the longitudinal motion of the aircraft, symmetrical wi th respect to X-Z and flying at small sideslip angles, can be ten as follows

the

C

1

Fig. 1. Partenavia AP68 TP-VIATOR

with

around

_

jz

zero angle of angle and zero

C

LO

pitch attitude angle air density effectiveness of the control surface dynamic pressure ratio at the horizontal tai 1

fJ

FLIGHT MECHANICS AND DEFINITIONS

sidered unchanged with V and 1 inearly dependent on 0:, q and oe (Roskam, 1982). With reference to a General Aviation aircraft Partenavia AP 68-VIATOR, wind tunnel tests guarantee that such hypothesis are broadly satisfied. By virtue of the above hypothesis the coefficients CL and C can be expressed as C L

C + C LO LO:

0:

+

qc CLq + C oe 2V LOe

(5)

C

C

0:

+

C

qc 2V + CmOe oe

(6)

mO

C

+

mo:

IDQ

as concern the drag coefficient Co' it can be expressed as

C

0

rate of change in downwash with angle of attack

70

C

00

+

2 C L nAe

(7)

The coefficients CLex ' CLq ' CLlle ' CmC(' Cmq' CmIle ' referred in literature as stability and control derivatives, for the total airplane can be expressed as

The model obtained by linearization can be written as

where

(8)

C

Lex

C Lq

2 1)t V c H lext

(9)

C Llle

S t 1)t S- c lext

and x ' u are derived as described above. o o Now, if we denote wi th n the vector of physical parameters, nominal speed, nominal flight path

(10)

angle and nominal altitude n:(nav

T

10 ho)T we can

o

rewrite Eqn. (17) as follows C

..ex

(x - x cq

ac

le

lex

i - 1) V (l-c le t H ex 1ext 1

Cmq

-2 1)t c lext VH c

(11)

t

x : A(n) x + B(n) u + b .

(19)

The dynamic behaviour of the aircraft and its handling qualities are determined by the roots of the characteristic polynomial of the dynamic matrix A

( 12)

p(s,n) : det(sI - A(n»

(13)

: (20)

and the coefficients C and Cme are given by LO

mO

mac

+c

i

1 a. w

(x

c9

-x

ac

) -1)

l

Vc

H I at

(i

t

-c )

n

As reported in MIL-F-8785B (1969), there are three flight handling qualities levels, which one uses to classify the dynamic behaviour of the aircraft. These levels are characterized by maximum and minimum allowable values of damping ratio < and natural frequency "'n of the modes associated wi th

(15)

0

where

the linearized model (19) period and < ,'" P

(16)

a

np

«,s '"ns

for the short

for the phugoid mode respecti-

vely) . Such limits for the Category B flight phase (cruise) with respect to the Cooper-Harper pilot Opinion Rating scale are

Denote with >r-=( ex v q 9) T the state vector, wi th u=(lle T)T the control vector and with n

(n)

sn-I + •• + a

I

(14)

C:C

(n)

sn + a

the vec-

level

tor of geometric, mass properties and aerodynamic parameters. Then the non-linear longitudinal equations of the motion can be linearized with respect to equilibrium state and control vectors Xo and u respectively. For given nominal altituo de ho, speed Vo ' flight path angle ' 0 : 9 -ex and 0 o for assigned values of the components of n, the

I

Sho~t per iO~ ns

s

1 2

0.30-2 . 0 0.20-2.0

3 . 1-7 2.5-8

level 3:< >0 .15, T P

2p

~55

sec.

and control vector o u can be derived from eqs. (1)-(4) with ~:O. o To validate the model, with reference to the General Aviation AP 68-TP VIATOR, in Fig.2 the so-

Now the problem of the robust stability analysis is to find the maximum allowable regions in the parameter space in order to confine the roots of the characteristic polynomial (20) inside domains in the complex plane established by the above limits as a function of the handling qualities levels.

l ut ions of the equat ion ~O for various fl ight conditions has been compared with flight tests.

ROBUST STABILITY TEST

a

equilibrium angle of attack ex

l 0r--~-----~----------,

Now we present a stabi I i ty test which allows to determine the allowable parameters uncertainities in order to preserve the specified handling qual! ties. Stability test of a convex combination of polynomials .

,

.

::~

Consider a linear time invariant dynamic system of order n, and suppose that a number p of system parameters are uncertain. Let n:(n ... n )T be the I

vector of uncertain parameters

. ,oL-_~--_--~_-_--:---,-:-:------:-:-:-------c 80 90 100 110 L::O no 140 150 160 170

p

belonging

to a

compact subset IT of the parameter space RP. Then a description of the system includes a whole family of characteristic polynomials of degree n

Fig. 2.

71

p(s.ll)=a (ll)+a (ll)S+ .. +sn=sn+vT(s)a(ll).llell. 1

2

and at least one polynomial of the family (27) has ~h roots lying inside the domain Vh'

(21)

h=1 •..• m. where where: n .

(22)

n

1

The proof of this theorem can be found Cavallo. Celentano and De Maria (1989b).

(23)

a(ll) := (a (ll) . . . a (ll»T.

hence Il is a hyperrectangle in the parameter space !RP. By denoting with a(ll) the image of Il in IIln. it is always possible to cover a(ll) by means of a convex hyperpolyhedron 1fc vi th a suitable number v

h

It is well known that the entire family of polynomials 'v'ae1f

Remark 2. The easiest way to cover the image a(ll) in the coefficient space is that of utilizing an n hyperrectangle whose vertices VI' i=1 •.. 2 are expressed by

(24)

c

can be generated by means of a convex combination of vertex polynomials of 1f

E;>. I p vi (s)

(25)

where vi

a+)T n - T a ) n

V = (a~ a n 2 2

- T a ) n

v

1=1

p

(a+ a + 1 2 (a+ a + 1 2

v

v p(s.a)

(s) = sn + vT(s)v

I

•

i=1 •..• v.

in

lytic form (e.g. by means of parametric equations) or in numerical form. Moreover the test (29) can be carried out by means of simple geometrical consjderatjrns on t~e complex numbers Pvl(S)PVl(S ). i=2 •..• V.'v'SEaV • without computing h their arguments.

(a(Il)~1f). c

I

(30)

Remark 1. Note that theorem 1 turns out to be a very easy D-stability test. In fact it is necessary only to perform the test (29) on the arguments of the vertex polynomials. With respect to this it suffices that aV is given either in ana-

In practice the components of vector II take on arbitrary values in prescribed intervals (ll~.ll:).

of vertices v

•

1 2

(31 )

where

(26)

a+=max a (ll)' a-=min a (ll). i 1 I llEIl I • I llEIl I = •..• n.

Now we introduce the Convex V-Stability Test. which stands for stability test with respect to assigned domains of the complex plane by using a convex covering of a(Il).

(32)

However the requirement of covering the image a(Jl) by means of a convex hyperpolyhedron generally whether involves burdensome computations or leads to very conservative results.

Theorem 1. Let the polynomial pes) be the convex combination of v assigned monic polynomials PVI(S) of degree n :

Nonconvex O-stabl1lty test. v

E \PVI (s)

pes)

• seC.

1=1

\>:0

v

E;>. I =1

Partition the set Il in the parameter space IRP by means of hyperrectangles Il • J=1 •... qP. Let J

(27)

1=1

where

(33) n

p (s)=s +a vi

In

s

n-1

+ .. +a

11

• a Jee.

(28)

l

be the succession of such partitions. such that

i=1 •.. • v.J=1 •.. • n. Consider

max O(Tf) I

compact connected domains V in the h complex plane. with V rlD =0. 'v'i~J. and denote l J with ao the boundaries of such domains. m~n

(~)~o 'v'~eav • i=1 •.. • v. h=1 •..• m. vi h Then the family of polynomials pes) has ~h roots inside the domain Oh' h=1 •..• m if and only if I
~eao

vi

(~»)-
vJ

(~») I
I

and

lim max o(Tf)=O. q_ I

Rn and the smallest hyperrectangle ~1~a(Il~). whn ose vertices v~J' J=1 •.. 2 are expressed by

h

Suppose that p

sup

> max o(Tf+l).

where o(ll) is the measure. according to a given metric. of the set Il. Consider the image a(rr7) in the coefficient space

q 11 q v 12

v (29)

(a q + a q+ 11 12 (a q + a q+ 11 12

aq+)T In q- T a ) In

(a q - a q11 12

a

h

e=arg(p vi (~». 'v'i.J=1 •..• v.i>J.'v'h=1 ...• m.

v

12

where ee[-ll.ll). denotes arg(ze- Je ); arg(z) denotes (arg(z)e[ -ll. ll»;

the

main

argument

where of

q

ZEe

72

n

q- T ) In

(34)

a q+ I

J

max a () n; neIf J

repeat this step with the next domain lIl+l ' else

a qI

min q aJ(n), nell

J

I

suspende current elaboration and go to step 4. Note that pq domains are tested, then the algorithm returns to the point where the previous elaboration had been suspended. If no more suspended elaborations exist go to the next step.

I

(35)

i=1, .. ,qP; j=1, .. ,n; q=1, 2, ..

q

P

Step 3. If in step 4 no radius had been stored (see below) increase r and go to step 1, else exit with the minimum value of r among those stored. This value is the robustness index according to the given metric.

Then 1~1~1 guarantees a covering, generally nonconvex, of a(ll), i.e. q

a( !I) =

P

q

U a(If) 1=1

P

UJtIcl

s;

I

(36)

1=1

Step 4. Part i t ion the current domain III by subdomains

Denote wi th (37)

I

SIMULATION RESULTS As remarked in the introduction, it is important for the modern stability and control engineers to assure handling quality requirements in the total flight envelope and for all airplane configurations, i.e . weight and cg location . In the first stage of the project, the tool presented in the previous section can be used in order to determine the maximum allowable regions around a nominal point in the parameter space so that handling quality requirements are respected. If very high performance requirements (high manoeuvrability, low fuel consumption, high efficiency) are required, it is not possible to satisfy handling quality requirements, then it is important to establish which kind of stability augmentation system to add . Many authors solve this problem by deSigning several controllers in a finite number of points of the flight envelope and for fixed values of cg locations . Then by simulation tests they determine rectangular regions around the nominal points, with respect to each controller has been designed, where the assigned handling qualities are respected . Then a switch among the controllers occurs, depending on flight conditions. By using the procedure illustrated in the previous section it is possible to give a robustness index with respect to flight conditions (h,V,w,cg location), in terms of an estimate of the regions inside flight envelope (in h-V plane) and weight and balance envelope (in w-x plane) where the

q

X cl

a(lI)

(38)

.

Now we apply the above theory to the case we are studying. The problem of robust analysis can ~ expressed as follows: given the nominal vector n , find the largest neighbourhood of nO, according to a given metric, so that the handling qualities are guaranteed. By using a weighted 1", norm, such a neighbourhood is a hyperrectangle ll(r)={neIR P: n~-rwI "n l "n~+rwI' wl;,O, i=1, .. , p},

(39)

where r is a suitable positive number. Theorem 1 allows to perform a D-stability test in a very short time on the convex covering of the image a(ll) in the coefficient space (convex D-stability test). Since in the problem we are dealing with the link between coefficients and parameters is highly non-linear, the convex D-stability test would give too conservat i ve resul ts, hence a non-convex D-stability test is required. A crucial point for efficiency of the procedure is the rule according to which the partition of the domain II in parameter space is carried out. We now present a recursive algorithm which solves this problem by parti tioning only those domains lI~ whose covering XCI is a too conservative ap-

C9

specified handling qualities are guaranteed . The following Figs.3,4 represent such regions for levels 1 and 2 handling qualities requirements in the plane w-x

proximation of the real image a(~).

C9

They have been obtained starting algorithm 1 from a given nominal point in which handling quality requirements were assured . Moreover, the points stored by the algori thm at step 4 have been depicted too. An approximation of the real region in parameter space can be obtained by starting the algorithm from different points as shown in Figs. 5,6 for the h,V plane. The points stored at step 4 of the algori thm can be considered as boundary points of the exact domain in parameter space.

Algori thm 1. Step 1. Inizialization: inizialize r, partition the domain lI(r) by pq domains III and consider the succession of covering

~

q

of a(lI(r)).

Step 2. For each current domain III and the associated

convex

covering

XCI

and recursively repeat

returns to the point of the step 2 where the previous elaboration had been suspended testing the next domain II 1+1

the parameter space. Clearly by increasing q, one obtains a covering which better fits a(ll). Taking into account that the real vectorial function a(n) is a continuous one and the set II is a compact one, it is also possible to prove that

n

j=1, .. ,pq,

algorithm stores the current value of the radius r, 1. e. the distance between the center of the current domain II and the nominal point nO , then

the succession of covering of a(ll) in the coefficient space associated to the succession 1'q in

q=1 1=1

lIlJ'

step 2 for each of these domains. The partition procedure halts when a minimum dimension of the domain III has been reached. As this occurs, the

perform

the

D-stability test, which is satisfied if condition (29) of theorem 1 holds. If the test is satisfied

7 :~

CONCLUSIONS

.. 10 1

"

In this paper, after a description of aeronautical terms and the mathematical model of the longi tudinal dynamics of an aircraft fully parametrized with respect to the most significant parameters, a robust stability test with respect to assigned domains of the complex plane has been presented. Such procedure allows the modern stabi li ty and control engineers il to establish which parameters are important in terms of handling qualities and which are not; ii) to determine the maximun allowable range of aircraft parameter variations according to handling quality requirements. The proposed procedure has been used in this paper in order to compute, in the case of longitudinal dynamics, an estimate of the domains in the planes h, Y (Flight Envelope) and w, x (Weight and Balance Envelope) inside

\\CI hl.ulO.J I>JIJro..'C' c",elope

3

1

(.

2

I

1

I 10

~O

15

].5 .. 10 '

25

30

35

40

45

50

Lcvc:12 ' ,abll,,) domlln

C9

WCI htllldlul;lf1('Ccnvcl

~

which the specified hand 1 ing qual it ies are respected, wi th reference to a General Aviat ion aircraft Partenavia AP68 TP-YIATOR. Since the proposed procedure is computat ionally tractable, work in progress is concerned with the development of an aircraft control system which enlarges the handling qualities domains in aircraft parameter space.

'

21

f. ~

REFERENCES

11

I

0

10

11

20

21

30

31

40

.,

Ackermann J. (1980). Parameter space design of robust control systems. IEEE Trans. Automat. Control, 25, 1058-1072. Barmish B.R. (1989). A generalization of Kharitonov's four polynomials concept for robust stability problems with linearly dependent coefficient perturbations. IEEE Trans. Autom. Contr., 34, 157-165. Cavallo A., Celentano G. and De Maria G. (1989a). Robust stability analysis of uncertain linear time invariant dynamical systems. Accepted for presentation at 28th CDC. Cavallo A., Celentano G. and De Maria G. (1989b). Robust stabi 1 i ty analysis of polynomials with linearly dependent coefficients perturbations. Submitted to IEEE Trans . Autom. Contr. Etkin B. (1972). Dynamics of atmospheric flight. J. Wiley & sons. Fam A.T. and Meditch J.S. (1978). A canonical parameter space for 1 inear system design. IEEE Trans. Automat. Control, 23, 454-458. Kharitonov V. L. (1979). Asymptotic stabi I ity of an equilibrium position of a family of systems of linear differential equations. Differential Equations, li, 1483-1485. HIL-F-8785B (ASG) military specification . (1969J. Flying qualities of piloted airplanes. Roskam J. (1982). Airplane flight dynamics and automatic flight controls. Parts I,ll. Roskam aviation and engineering corporation, Ottawa, Kansas . Tesi A. and Vicino A. (1988). Robustness analysis of uncertain dynamical systems with structured perturbations. Proc. of the 27th CDC, 519-525.

10

Ccnter of gravny IOClllon 1% of c l

Figs. 3,4. Stabi 1 ity domains in w-x

plane with C9

h =1S25 m and Y =60 mls. e e Ltvcl I

"lOO

~ l lbl l llY

dom:llns

7000 6000

Fh hi Cn\'c!olC

1000 • 000 3000 2000 1000 0 0

20

120 Speed Im/s l

Lt vcl 2 ~ I Jb,hl) "Om;1In5 M1(J(1

7~)(l ..

(0001,1

rill:htC'n\clt.

,ono .aCO')

1000 2000 1000 0 0

20

120 Sp«d I nllsl

Figs . 5,6.

Stability domains in h-Y plane with w=28743 N and the cent er of gravity located at 25.5X of the mean chord c.

74