APPENDIX A
APPENDIX A
Proposed S D M Presentation Integrated Aeroservoelastic Tailoring of Lifting Surfaces Thomas A. Zeiler* Kentron International, Inc.
Hampton Technical Center N. Armistead Ave.
Hampton, Virginia 23666 Phone 804-838-1010 and
- Terrence A. Weisshaar*
School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana 47907 Phone 317-494-5975 Abstract of Paper Proposed for the 27th AIAA Structures, Structural Dynamics and Materials Conferencet May 1986 San Antonio, Texas *Members, AIAA Address all correspondence to the second author.
tProposed for the Structural Dynamics session.
The design of an aerospace structure involves a complicated sequence of operations requiring multiple, interdisciplinary interac- tions. The overall design process has a single objective, superior per- formance subject to a multitude of constraints. Unfortunately, perfor- mance has a multiplicity of definitions, depending upon the specific discipline involved within the design process. Worse yet, sometimes these measures of performance are at odds with one another. Future com- petitive aerospace structural designs Will increase the need for creative interaction among the various disciplines and also require accounting for these interactions early in the desfgn process. This paper will discuss the integration of two of these areas, the optimal design'process for structures and active control of such a structure.
While the results presented are limited in scope, they nonetheless illustrate benefits of integrating the aeroservoelastic design process.
This integrated design process is referred to as integrated aeroservoe- lastic tailoring.
The objective of this study was to determine how to maximize the stable flight envelope of an idealized, actively controlled aeroelastic system shown in Figure 1. This 4-degree-of-freedom system consists of a 3-degree-of-freedom, typical-section airfoil mounted on a rigid support with a stabilizing tail surface; the model is free to pitch about a pivot. This model is Intended to simulate a flexible wing with an important body freedom. This model has the potential for simulating high frequency classical flutter behavior, low frequency body-freedom flutter and classical divergence.
An analytical formulation of the equations of motion of this model was developed, including unsteady aerodynamic loads in an s-plane or time domain form. The result was a state-space representation of the equations of motion. The structural design variable was taken to be the shear center position with respect to the airfoil midchord, denoted as a in Figure 1. This parameter is nondimensional with respect to the e airfoil semi-chord dimension, b, and is taken to be positive if the shear center lies aft of the airfofl midchord. As a result, the limits to a are -1 < ae < 1.
e Optimal steady-state linear quadratic regulator theory (SSLQR) was used to synthesize full-state fee6back control laws to stabilize the model at different airspeeds (represented in nondimensional form as
-
) and different values of a . Figure 2 shows the "open-loop", 'DES e control-off, stability boundaries for the model dimensions chosen, but taken to be a design parameter capable of being chosen arbi- with a e trarily. In Figure 2, the parameter bCT/b represents the ratio of tail surface area to wing surface area. Note that full body pitch restraint or "clamping" the fuselage reduces the flutter and divergence boundaries to those of the 3-degree-of-freedom airfoil alone. The use of SSLQR theory to synthesize control laws with the shear center at various posi- tions uses a measure of state and control activity at a fixed design
airspeed, i ? as a cost function, J, to be minimized. Values of this
DES '
function J for this idealization are plotted versus a and design e
-
While the absolute value of J has no phy- airspeed, i n Figure 3.
'DES 9 sical significance, the relatively large valws of J near a = -0.4 and e
-
= 6 indicate that the active control is experiencing difficulty 'DES stabilizing the system fn this region.
These regions of relatively high cost correspond to configurations for which the system experiences near-uncontrollability of unstable aodes. This is indicated in Figure 4 by the close proximity of zeros of the loop transfer matrix to some system poles (eigenvalues) near the j w axis.
A similar contour plot for control cost was constructed for the airfoil model with rigid body pitch freedom suppressed. This contour indjdates that high cost regions are also plot, shown in Figure 5, present, particularly where open-loop divergence is to be stabilized.
It would appear that a design procedure that can select low control c o s t regions at a fixed design airspeed would be sufficient to the integrated opti?al design task. Such is not the case, as indicated in Figure 6 .
Figure 6 shows the closed-loop stability boundaries of the actively controlled 3-degree-of-freedom airfoil model, as functions of shear
center position, a . Open-loop stability boundaries are superimposed on
e this figure. Figure 6 was constructed by choosing a large number of a e values and then constructing a control law with 5 held fixed at 6 . 0 .
DES
Thus, while tDES held fixed at 6.0, the control law changes with a in
e Figure 6 . The high control cost region near a = -0.4 also inclgdes e instabilities below the design speed. Thus, while the active control has extended the upper part of the flight envelope, in this region a new instability associated with off-design airspeeds has appeared. For this the cost function from SSLQR theory is inadequate as a sole per- reason, formance index to be used in the integrated design process. To remedy this, a combined design procedure based upon multi-level linear decompo- sition [ l ] of the aeroservoelastic system into structural and control subsystems was formulated. For this procedure, the overall design objective was maximization of the stable airspeed envelope with a struc- tural parameter (a ) and control parameters from SSLQR theory as design e variables.
With multi-level, linear decomposition, the subsystem designs are themselves in some measure optimal.
Optimal sensitivity derivatives are computed with respect to specified system parameters to aid in choosing a new design that is both optimal on the subsystem level, yet satisfies the global objectives at the system level. In this case, analytical expressions for the changes in the eigenvalues of the closed-loop system (subject to the constraint that the system is optimally controlled) were constructed using a method proposed by Gilbert [ 2 ] . No structural cost was associated with changes in the shear center position, representing a limiting case such as might be present in a laminated wing structure.
This assumption does not, however, limit the future applications of the procedure. To assess system stability, a stability index F is defined - - sj such that F - - l n ( C e sj p I= 1 where p is a weighting function (in this case, p = l ) , N is equal to the number of potentially critical eigenvalues, u is the real part of the ith eigenvalue and is the airspeed at which F is computed j sj
(nj # vDES). If F < 0 then the system is stable. If F > 0 then the
sj sj system is to be stabilized by finding the proper combination of a and e control parameters that will minimize F sj' The design procedure begins with the choice of initial values of ae Next, a control law is synthesized at and other system parameters.
-
. An airspeed vk is chosen for which the closed-loop system is
'DES
unstable (F The derivatives of Fsk with respect to ae and TDES
> 0 ) .
sk are computed, subject to the constraint that the active control law is optimal. In addition, derivatives of other stability indices at lower airspeeds,
(tj < zk), with respect to these variables are also com-
j' puted. An optimization procedure based upon a simplex algorithm uses this sensitivity information to choose changes in a
and FDES to minim-
e izc FskB without allowing other F values to become positive y j (unstable).
If the value of Fsk is found to be negative on a certain design
cycle (the system is thug stable at vk), the airspeed ck becomes a sub-
critical airspeed. A new, post-critical airspeed is then chosen as fk
and the procedure continues. The Frocedure terminates when is either k equal to the desired maximum stable airspeed or when no further stabili- zation' is possibie. Figure 7 illustrates this procedure.
For this example, the nondimensional airspeeds E at which stabil- j ity was required were chosen (arbitrarily) to be integers, thus 3 = j j in Figure 7 .
Initially, the system is unstable at Ek = 7 . 0 with a - con-
- trol design airspeed of vDES = 6 . 0 .
The first design iteration reduces the measure of instability F by instructing the "structures group" to sk shift the shear center aft towards the mid-chord and asking the "con- trols group" to reduce ;he value of its design airspeed. At design cycle 4 the actively controlled system is stable at = 7 . 0 so it is now required that the closed-loop system attempt stabilization at = 8 . 0 .
k This task is achieved at design cycle 7 . At this point, the requfrement
is changed to attempt closed-loop stability at tk = 9 . 0 . Figure 7(b)
-
indicates that this objective cannot be met; however, Fsk, nt Uk = 9 is minimized. The root locus plot (using airspeed as the gain) of the final, actively controlled system is shown in Figure 8 .
Figure 8 shows that the final design Is a compromise between flutter in two different modes. The paper discusses the reasons for arriving at this result.
It is interesting to note that the optimal actively controlled structural configuration does not correspond to the structural configuretion that one would find If only passive tailoring were used to increase stability.
The SSLQR theory requires the user to choose weighting matrices In the cost function J . These elements are f o m d to have a significant effect, in some cases, upon the appearance of sub-critical stability regions. As a result, a second example was choseu to illustrate the use of a state weighting element Q , , (the weighting on rigid body pitch), as a dnsigr parameter. In addition, the position of the airfoil with respect to the pivot, given as the dimension b % in Figure 1 was also treated as a design parameter, together with EDES and a .
e Figure 9 shows the design cycle history for the 4-degree-of-f reedom The initial objec- wing configuration, which includes rigid-body pitch.
= 4 . 0 and 5 . 0 using Q , and CDEs as tive was to stabilize the system at
design variables. Note that,the closed-loop system is stable at - U = 6 . 0 . By design cycle number 7, the procedure was experiencing dif- ficulty meeting its objectives. At this point, the position of the wing, with respect to the pivot, by, was allowed to change, together
with ae, for the next two iterations. and x were held
After cycle 9, a e fixed and optimization continued using and Q , as design variables.
DES The effects of the use of b ; and a as parameters can be seen in e Figure 1 0 . Figure 10 plots the partial derivatives of the stability indices, with respect to Q , , as functions of design cycle number. This figure indicates that changes in by and a increase the lnagnitudes ox' e these derivatives.
As a result, Q , becomes more effective as a design parameter.
Figure 11 shows the root-locus plots, with 5 as a gain, for the
initial closed-loop design and the final closed-loop design.
The final
design is Been to be a compromtse between flutter at F = 7.04 and
flutter in a hump mode at around 5 - 5.0. If one were to try to further
increase the 5 = 7.04 flutter speed, the stability constraint at = 5.0
would be violated.
The paper will describe additional features of this integrated design technique. Included will be additional data indicating why several of the features observed in the previous figures occur as they do, I n addition, a discussion will be included as to how this procedure may be expanded to include control synthesis by techniques other than SSLQR theory .
References 1. Sobieszczanski-Sobieski, J., James, B., and Dovi, A., "Siructural Optimization by Fiultilevel Decomposition," AIAA/kSME/ASCE/AHS 24th Structures, Structural Dynamics and Materials Conference Proceedings, A I A A paper no. 83-0832-CP, F l a y 2-4, 1963.
Gilbert, M.G., "Optimal Linear Coatrol Law Design Using Optimum 2.
Parameter Sensitivity Analysis,'' Proposal for Doctoral Dissertation F.esearch, Purdue University, School of Aeronautics and Astronautics, 1983.
f
-7
2 +
G I - . %
c
i u a 0) v!
d A A I I "3 I r-l f I I n I a I m I .
c '10
/
\ /
JJ Open-Laop
.
= 5 E' I I I I - - - I . - -. 0 -0.6 -0 - 5 -0.4 - 0 . 3 -0.2 -0.1 0 a e
Figure 6 - Open and closed-loop stability boundaries as a
function of a for the 3-dof aeroelastic
model; controf laws are synthesized at u
= 6.0 DES 1.
A U d .
rl P m 01 I U I- A u L .
W Q M d f4 z
Figure 8 - Velocity root l o c i , design c y c l e # l l for the
3-dof a e r o e l a s t i c model.
. - 'del x Q0 'des F i n a l Values
ii 5
0 5 10 15 20 .
No. of Cycles a. Design Parameters No. of Cycles b . S t a b i l i t y I n d i c e s
Figure 9 - Design c y c l e h i s t o r i e s f o r the 4-dof a e r o e l a s t i c
model example.
20 -
redesign
-
1 1 5 #6 #7 US # 9 # l o aFs j
-
(xlO'4) j =5 -5- Figure 10- The value of aF /ae, as a function of d & g n cycle number for the 4-dof aeroelastic model.
" 9
d l I
- -
d
'I I
i:
APPENDIX B
APPENDIX B The attached document summarizes the development of a two-degree of freedom idealization used to study the interaction between This model was developed directional stiffness and feedback control.
by Professor Weisshaar and has been implemented on the computer by Mr. Sallee. A two-mode flexible model could also be used. However, past experience with the semi-rigid model has been quite good. As a result, it is the choice for demonstration purposes.
An I d e a l i z e d Aeroelastic Model f o r Active Control S t u d i e s
--- --
Consider t h e i d e a l i z e d l i f t i n g s u r f a c e shorn i n Figures 1, 2, 3.
The s u r f a c e I t s e l f is r i g i d , but is a t t a c h e d t o a pivot on a wall; it h a s m s s uniformly d i s t r i b u t e d along the span. A reference a d s , t h e y-axis in Figure 2, is used f o r t h e determination of the equations of motfon: t h e r e f e r e n c e axis lies a d i s t a n c e ba aft of t h e midchord. The l i n e of aerodynamic c e n t e r s is l o c a t e d a t a d i s t a n c e ba ahead of the C a i r f o i l midchord and is shown i n Figure 3. For t h e p o s i t i o n shown, b(a-ac) i s a negative quantity. The chordwise o f f s e t of t h e l i n e of c e n t e r s of mass from t h e r e f e r e n c e a x i s is denoted as x t h i s latter a ' coordinate is p o s i t i v e when t h e s e c t i o n a l c e n t e r s of mass are located a f t .of the r e f e r e n c e 2xis.
The downward d e f l e c t i o n of t h e l i n e of c e n t e r s of mass is denoted .
as 2. This d e f l e c t i o n and the v e l o c i t y z are expressed i n terms of t h e t o r s i o n a l r o t a t i o n , 9, and "bending" r o t a t i o n , +, as: . .
g = ; = Xa9 - $y
The a i r f o i l has constant mass per u n i t l e n g t h , m, so t h a t the k i n e t i c T, may be w r i t t e n as: energy,
1 ' 2 1 1 '2
T =
{Ioe 11 f 7 1 mz dy
o r T = - (4) where Io I s t h e p i t c h mass moment of Inertia per u n i t length of each s e c t i o n along t h e wing surface, taken about t h e l i n e of c e n t e r s of mass.
The s t r a i n energy i n the spring supports due to deformations e and 9 is:
/
/
Figure 1 - Idealized airfoil, s h o m swept at an angle A to the airstream
and attached to a pivot on the wind-tunnel w a l l .
Figure 2 - Planform view of 2-D, idealized airfoil showing: rotational
deformations 8 and $; orientation of principal bending and torsion axes, a; and, effective root and tip approximations.
Figure 3 - Planform view of i d e a l i z e d a i r f o i l showing: aerodynamic
c e n t e r l r e f e r e n c e a x i s o f f s e t ; reference a x i s / c e n t e r of mass o f f s e t ; reference axis/milchord o f f s e t distance. ba; and. normal component of v e l o c i t y , V .
n Figure 4 - Full spar, control surface model.
U -
(5) K { e cos y - + sin y)
f e
K { + COS y + 8 sin y)
+ T
+
Prom Iagrange's equations the equations of motion for free vibration in the absence of airloads are found to be: To simplify the writing of these equations, elements of the matrices in Eqn. 6 are defined as follows.
Inertia terms
m = &x:+ 2 1 = m&t h e r e r2 = - IO
1 1 o m mI2 -- -m 4 {- x$,
(Note that .fl is the total mass of the idealized wing.)
Stiffness terms
kI2 = (KO - Ke)cosy siny
k22 = Kesin2 Y + K cos2 y
Aerodynamic Forces Aerodynamic. forces due to the motion of the idealized airfoil, can be related to the two degrees of freedom, 8 and $. For the present analysis these forces were computed from modified aerodynamic strip theory as outlined by Yates in Reference 1 . The pitching moment about the y-axis in Figure 3 is denoted as Me, while bending moment about the x-axis, l o c a t e d a t t h e root, is M 0' The aerodynamic load per u n i t l e n g t h along t h e swept y-axis its P .
m e load P is p o s i t i v e when it acts i n t h e upward d i r e c t i o n , out of t h e plane of t h e paper. In t h e p r e s e n t case, t h i s l o a d is [l]: a . .
2 -.
P = WP? [ - 0 y + Vne - Vn+tanA - bae]
where Q IB the downwash v e l o c i t y a t t h e c o n t r o l point on t h e a i r f o i l .
The Theodorsen f u n c t i o n C(k) is v a l i d only when 41 and 8 are simple har- monic f u n c t i o n s of time. The downwash is: I n Eqn. 8 , t h e term a r e p r e s e n t s t h e d i s t a n c e , measured i n semi-chords, C
t h a t t h e static aerodynamic c e n t e r l i e s behind t h e wing -- mid-chord posi-
-
- t i o n . For subsonic flow t h i s term is negative. For incompressible flow
c o n d i t i o n s , a = - - s i n c e the aerodynamic c e n t e r w i l l be a t the air- C 3.'
f o i l quarter-chord position.
The aerodynamic p i t c h i n g moment, p e r u n i t l e n g t h , measured p o s i t i v e nose-up about t h e r e f e r e n c e a x i s is:
Ma = - n p b 4 ( 3 a 2 ) e ' - rpb 2 - Vn(+y+Vn$tanA)
.
- rpb 3 a ( + * * y+Vn$tanA) + apb 2 2 Vn8
c1
- 2rpVnb2t+ - (a-a C )C(k)$]Q ( 9 )
f o r t h e motion dependent a i r l o a d s , we d e f i n e To develop t h e equations t h e moments M e and M as follows 4I Equations 10 and 11 can be w r i t t e n as (12) When motion of the form I n Eqn. 12, o is a r e f e r e n c e frequency.
P can be constructed. These expres-
i s assumed, expressions for h , and
d s i o n s are w r i t t e n symbolically as follows: ] matrix a r e t h e apparent mass terms f o r t h i s air- Elements of the [M i j ] matrix r e p r e s e n t s t h e aerodynamic damping.
f o i l , while the [B [Aij] i j The elements of t h e s e t h r e e is t h e aerodynamic s t i f f n e s s matrix.
m a t r i c e s may be conveniently defined i n terms of a group of parameters.
These parameters are: -2
d = wa
1/2b ( A R ) * s t r u c t u r a l aspect r a t i o The matrix elements are then written as follows: 4(AR)' (22) ' 2 2 * 3d
CLc %
(23) *11=- m d
CL(AR) vn
BI2 = ( ( A R ) + a tanh - II 1
-
4 "n
B22 = (AR)(tanh + 7 cla (AR)C(k));i-
L I f -2 cL ' n
(-1 + -) ( 2 7 )
*I1 =r f
-2 CLtanh ' n
- (tanh - II
1 (28)
A12 -2-
-2
'*
( c1 ( AR) C(k) tanh) A22 = nd a The equations of motion are written as 1s 2 tmijl + ~ k i j ~ ~ ~ + rnlriwf#J= 0 2 2 Dividing by the factor mlr w gives the following equations: a P The parameter
-
xa - xa/b
while and = K11'K22 These matrix equations may be combined and written as where
[%I = bij1 + [Mijl
[%I = [kijl + [AjJ
This equation may also be written as follows: the problem in the following form: If the vector {rl) is defined as
where {n} represents a vector of system states, then
and damping in The eigenvalues of [A] determine the natural frequencic the system.
Eigenvalue S e n s i t i v i t y Derivatives The o b j e c t i v e of t h i s s e c t i o n is t o o u t l i n e a procedure f o r calcu- l a t i n g t h e f i r s t - o r d e r changes ( f i r s t d e r i v a t i v e s ) i n system eigenvalues t h e s e d e r i v a t i v e s w i l l due t o changes i n system parameters e , R and v.
be used t o estimate t h e e f f e c t s upon s t a b i l i t y of a change i n s t i f f n e s s cross-coupling ($), p r i m a r y s t i f f n e s s r a t i o (R) or a i r s p e e d (v>.
Begin by d e f i n i n g t h e eigenvalue problem a t a given a i r s p e e d 7 .
-
s { d - [Al{n)
The s o l u t i o n to Eqn. 42 is w r i t t e n as: XiIeil = [Aijl{ei) (43) where X i s an eigenvalue corresponding t o the eigenvector {ei). The eigenvalue X and t h e vector {et) may be complex.
The parameters $, R and 7 may be represented, i n g e n e r a l , by t h e symbol p. L e t ' s d i f f e r e n t i a t e Eqn. 4 3 with respect t o p.
Equation 44 now may be w r i t t e n as: We a r e only i n t e r e s t e d i n the change i n X i , not the change i n the eigen- v e c t o r e To e l i m i n a t e ae,/ap from Eqq. 45, consider t h e so-called I' "left-hand" o r transpose eigenvalue problem defined as: T ( 4 6 ) [Aij] {r,) = a i { r i ~ Equation 46 d e f i n e s an eigenvalue problem f o r t h e matrix transpose of The eigenvalues of Eqn. 46 w i l l be the same as those found i n [Aij].
Eqn. 43, since t h e determinant of t h e matrix transpose 1s t h e same as As a r e s u l t , both Eqns. 43 and 46 have t h e determinant of t h e matrix.
t h e same c h a r a c t e r i s t i c equations. However, t h e eigenvectors {ri} and (e,) a s s o c i a t e d wlth X i are _I not i d e n t i c a l u n l e s s [A] = [ A I T , t h a t is, u n l e s s t h e [A] matrix i s symmetrical. Equation 46 i s important; taking its matrix transpose, it becomes: Notice t h e s i m i l a r i t y between t h e term i n Eqn. 47 and t h e l a s t term i n Eqn. 4 5 . Pre-multiplying Eqn. 45 by { r i l T , we g e t I The l a s t term i n Eqn. 48 i s zero, from Eqn. 47. This gives the follow- inr: r e s u l t f o r t h e change i n X with respect t o p.
where ( 5 0 )
c i - triJ{eiI + 0
Equation 49 i s an exact s o l u t i o n f o r t h e f i r s t - o r d e r ( f i r s t d e r i v a t i v e ) s e u s i t i v i t y of t h e eigenvalue X t o changes i n a system parameter, p, present i n t h e [ A ] matrix. Since t h e m a t r i x [ A I defined i n Eqn. 40 ij i j c o n s i s t s of a l g e b r a i c expressions, w e a l s o can d e r i v e a l g e b r a i c expres- aA 8iOnS f o r t h e elements A, as w i l l be i l l u s t r a t e d .
aP T W e must know { r 1 and {ei) before w e can c a r r y out t h e operation defined i n Eqn. 49. Since the transposed eigenvalue problem i s r e l a t e d t o the o r i g i n a l eigenvalue nroblem (Eqn. 4 3 ) , it can be shown t h a t :
[ R l - [El" ( 5 1 )
where t h e columns of t h e N x N modal matrix [E] are constructed by i n e e r t - i n g t h e N eigenvectors {ei} such t h a t As a r e s u l t , b and t o Since t h e node shapes (eigenvectors) of both problems are ort..ogona each o t h e r , the s e n s i t i v i t i e s of a l l eigenvalues can be computed i n a s i n g l e o p e r a t i o n , as follows.
L e t us d e f i n e a matrix [ D ] a s follows: i j Then a
-
( 5 6 ) = Dii aP Note t h a t t h e off-diagonal elements of D are not zero, nor are they i j meaningful.
The procedure for computing s e n s i t i v i t y d e r i v a t i v e s of o m eigen- values is now e a s i l y constructed.
1. Compute t h e N eigenvalues and e i g e n v e c t o r s of t h e problem 2. Construct t h e matrix [E 1 ij 3. I n v e r t [E ] t o f i n d [R ] i j i j
4 . Construct t h e matrix 1 2 1 -3r t h e parameter o
i n t e r e s t .
5 . ,Compute [ D ] = [R]
a xi
6 . -E Dii aP
Now, let's t u r n t o the a c t u a l computation of t h e m a t r i x 1 2 1 f a r a few
parameters. F i r s t consider t h e s t i f f n e s s cross-coupling parameter JI.
The element Theref o r e Next, consider changes with respect t o the primary s t i f f n e s s r a t i o , R = K11’K22’ The element8 of [a are: r 2\ lii: (61)
1 -1 /R
L Therefore : I t i s a l s o important t o predict how the eigenvalues change with Let’s com- airspeed s i n c e the eigenvalues determine systen s t a b i l i t y .
pute [ + I , a s follows: and ..
0 0
[ S j
avn
-
J
L e t us represent t h e matrix of changes In
yr as
and change6 in B as t j L , I Elements of thewe matrices are: 2Tn
CL
K , (1,l) = d (-1 + -) 1
' n Elements of the changes in the aerodynamic d a p i n g matrix are: ' n
B-(1,2) = (AR + a tan A - CL(AR)/r)/d
vn
- The -Mdition -- of a Control Surface t o the I d e a l i z a t i o n
I f a full-span, t r a i l i n g edge c o n t r o l is attached as shown in pig- If the con- u r e 4 , t h e equations of moticn (Eqn. 36) w i l l b e modified.
t r o l d e f l e c t i o n is denoted as 60 and t h e c o n t r o l is i r r e v e r s i b l e , then p i t c h i n g and bending moments at t h e a i r f o i l r o o t may be w r i t t e n as: where and
-
e r a - a .
(77) C Equation 36 now becomes where
2vt -
+ S C )
C 8 & = -
( 7 9 ) 2 1 6 -2 fur,, and I n terms of time d e r i v a t i v e s of 0 and 9, Eqn. 78 is w r i t t e n as: 1s Equation 81 m a y be w r i t t e n in state-spacc form as: where (83) Note t h a t t h e symlsl [B] a l s o has been used previously t o denote f o r
aerodynamic damping. However, --- i n a l l that follows, t h e symbol B w i l l
r e f e r t o t h e c o n t r o l matrix, i n t h i s case, a v e c t o r q u a n t i t y defined in Eqn. 83.
Modal C o f i t r o l l a b l l i t y The a e r o e l a s t i c response problem is now cast i n terms of the state v e c t o r {n) (defined i n Eqn, 4 1 ) as follows: q = A q + B 6 .
Again, t h e eigenvectors of the problem rl = AII are {e,) and, as before, can be arranged t o form an NxN modal matrix [E defined i n Eqn. 51.
i j W e can use [E ] t o transform t h e n coordinates t o a new s e t of coordl- i j n a t e s { E , ) , defined as: so t h a t { C ) = [Eijl-'{n? = I R l l n ?
Our equation of motion, including t h e c o n t r o l , then becomes: or The matrix RAE = J , is a diagonal matrix composed of the N eigenvalues of the matrix [ A ] . The matrix [ . I ] is called the Jordan canonical form of [A]. Let us define a column matrix {P} as ( 8 9 ) so that now we have the equation of motion written as: .
{ E } = [Jl{C} + {P}60 (90)
The matrix {PI is called the mode-controllability matrix of the system and has some interesting characteristics.
Since [J] is diagonal and {P) is a vector, the equations of motion in terms of 5 are uncoupled and have the general form: i
(i = 1, 2 , ..., N) (91)
5 , = \$Ei + Pido
where X is the ith eigenvalue and 5 is the generalized coordinate i i corresponding to the ith mode of the system. From this relationship, it is seen that the ith mode is controllable bj the control surface only if p is unequal to zero. The entire system is controllable only if all i.
modes are controllable.
Mode-observability -- of the System
-
The measured "output" of the system can be expressed in terms of the system states as: where [C 1 represents the system output matrix. In terms of the ij transformed coordinates {Ei}, the output equation becomes: (93) where
[TI - [EI,lTICI (94)
The matrix [TI i r : called the mode-observabillty matrix for the system.
Modal Control Only a s i n g l e c o n t r o l i n p u t , b,, c o n t r o l s t h e aerrelastic system; let's measure t h e system states and feedback a signal, f , defined as:
f(t) - { V I T { ? ) (95)
The matrix { v } is a matrix of t r a n s d u c e r o u t p u t s from each state. This v e c t o r {v} is c a l l e d t h e measurement vector. The signal f(t) can be amplified by a p r o p o r t i o n a l c o n t r o l l e r having a gain, K . I n t h i s case, t h e c o n t r o l l e r is 4 so t h a t we move t h e c o n t r o l s u r f a c e an amount: O S Combining t h i s with our equation of motion, we now have t h e following prohl em.
Let Then Depending upon t h e t h e eigenvalues of [A,,J w i l l choice of K and { p ) d i f f e r from those of t h e o r i g i n a l p l a n t matrix [ A ] . How one chooses K and {p} depends upon t h e o b j e c t i v e of t h e c o n t r o l .
L e t u s suppose t h a t t h e o b j e c t i v e is t o modify a s i n g l e eigenvalue of t h e o r i g i n a l p l a n t .
Let us have as our o b j e c t i v e t h e changing of t h e jth eigenvalue, X t o a value y while l e a v i n g a l l o t h e r v a l u e s of Xi (i f j) of t h e j' j open-loop system unchanged. Here is one way t h a t t h i s may be done.
(Note t h a t what follows i s a g r e a t l y s i m p l i f i e d approach t o a e r o e l a s t i c
control . )
Let {u} = {rj) , the jth eigenvector of the transposed eigenvalue
problem. In this case Now, post-multiply by the kth eigenvector (k f j) of the open-loop sys- tem, {ek}, to get: [+I = [AI { e $ + K W I r jlT{ek} (102) T Because of eigenvector orthogonality, {r } { e , } = 0. This leaves
J
[\]{e,} = LAIlek} (103) Since, by definition, [ A ] { e , } = \{ek}, Eqn. 103 becomes: [\J{ekl x { e , l (k f j) (104) This last result in Eqn. 104 means that, for this selection of { p } , the eigenvectors and eigenvalues of the closed-loop system (represented by A $ are identical to those of the uncontrolled, open loop system [A], Let’s -- with the exception -- of the jth eigenvalue/eigenvector combination.
look at this latter combination.
If we now post-multiply Eqn. 1 0 1 by {e 1 then j [%]lej) = [Al{ej} + K{B}{rj}Tlej} or (106) [$]{ej} = [Al{ej} + K ( B 1 = AjIej} + K @ } Equation 106 is valid because of orthonormality of the vectors {r } and {e } in Eqn. 105. This result implies that, due to feedback control, X 1 9 is not an eigenvalile of [A$, nor is (e } an eigenvector of [\,ID
Since t h e vector {u} has been s p e c i f i e d , t h e only unknown i n the
c o n t r o l l a w is K, t h e gain. To determine t h e value of K necessary t o i s t h e element of modify X i by a certain amount, first remember t h a t p j t h e c o n t r o l l a b i l i t y m a t r i x for t h e jth mode. It can be shown t h a t , t o t h e g a i n K a c h i e v e our o b j e c t i v e of eigenvalue m o d i f i c a t i o n , we can set t o be:
K = ( p j - a j ) / p j (107)
t h where p is the element of t h e c o n t r o l l a b i l i t y m a t r i x r e l a t e d t o t h e j j mode and p is the new eigenvalue. Notice t h a t t h e g a i n K may be com- j plex. This modification procedure is s t r i c t l y v a l i d only when changing a real eigenvalue X to a r e a l value p If Xi is complex, w e need t o
. j j '
add an a d d i t i o n a l s t e p .