Small perturbation equations of motion of a flexible aircraft with an active control technology (ACT) system were developed to evaluate the stability and performance of the controlled aircraft. The total aircraft system was formulated in state vector format and the system of equations was completed with fully unsteady and low frequency aerodynamics for arbitrary, complex configurations based on a potential aerodynamic method. The ACT system equations were incorporated in the digital computer program FCAP (Flight Control Analysis Program) which can be used for the analysis of complete aircraft configurations, including control system, with either low frequency or fully unsteady aerodynamics. The application of classical performance analyses including frequency response, poles and zeros, mean square response, and time response in FCAP in state vector format was discussed

Morino, L.

Noll, R. B.

English

NASA Technical Reports Server

PERFORMANCE ANALYSIS OF FLEXIBLE AIRCRAFT WITH ACTIVE CONTROL* 
Richard B. N o l l  
Aerospace Systems, Inc. 
Luigi Merino** 
Boston University 
SUMMARY 
The small-perturbation equations o f  motion of  a flexible  aircraft  with an active  control 
technology (ACT) system were  developed  to  evaluate the stalbility  and  performance of the 
controlled  aircraft. The total  aircraft system was formulated in state  vector format and the 
system of equations was completed with  fu l ly  unsteady and Iow-frequ.ency  aerodynamics for 
arbitrary, complex configurations based on a potential aerodynamic method. The ACT sys- 
tem equations have been incorporated i n  the  digital computer program FCAP (Flight  Control 
Analysis Program) which  can  be used for  the  analysis o f  complete  aircraft  configurations, 
including control system, with either low-frequency or fu l ly  unsteady aerodynamics. The 
application  of  classical  performance analyses including  frequency response, poles and zeros, 
mean-square response, and time response in  FCAP in  state vector format was discussed. 
INTRODUCTION 
The integrated study of  the  interactive effects of the flight  control system in  the  active 
control  of  flexible  aircraft has received  considerable  attention  in  recent years. In particular, 
Active  Control Technology (ACT) is  being investigated for improving ride quality, decreas- 
ing  structural  deformation,  extending  the  fatigue l i fe  of  the  aircraft,  relaxing  static  stability 
requirements, suppressing flutter, and reducing structural loads. 
A new computer program, Flight  Control  Analysis Program (FCAP) has been developed 
for NASA to analyze ACT systems (refs. 1 and 2). The program was designed in  a modular 
fashion to  incorporate  aircraft dynamics, aerodynamics for complex configurations, and 
sensor, actuator,  and  control  logic dynamics, as well as analysis methods for determining 
stability and performance of  the ACT system. The formulation o f  the  total  aircraft  dynamic 
system for FCAP was unified  by  casting  all the equations i n  state space format. This paper 
presents the  state-vector  formulation  of  the ACT system, and discusses its application  in 
FCAP for the  performance  analysis o f  ACT systems. 
*This paper was derived from  work conducted under NASA Contract NAS 1-1 3371 . 
**Consultant  to Aerospace Systems, Inc. 
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https://ntrs.nasa.gov/search.jsp?R=19770003436 2020-03-20T07:06:40+00:00Z
I 
SYMBOLS 
A,BjC,D 
Al 
F 
H 
i w  
r 
X 
Subscripts: 
state-space matrices of ACT system and subsystems 
matrix  defined  by Equation (17) 
matrix  defined  by Equation (16) 
matrix o f  transfer functions 
imaginary  part of complex frequency, s 
generalized mass/inertia matrix 
dynamic pressure 
Lagrangian generalized coordinates . 
covariance  matrix  of ACT system outputs 
output  vector for ACT system and subsystems 
complex  frequency 
covariance  matrix  of ACT system inputs 
generalized aerodynamic force  coefficients 
coefficient matrices of  aircraft dynamics equations at sensor locations 
input vector for ACT system and subsystems 
matrix  of  pilot and guidance system  commands 
generalized aerodynamic forces in uniform flow 
matrix of aerodynamic forces due to turbulence ’ 
covariance  matrix  of ACT system state  variables 
state vector 
A 
D 
L 
R 
S 
actuator 
aircraft displacement 
control logic 
aircraft  rate 
sensor 
1704 
Superscripts: 
T transpose 
(O), (1) coefficients  of power series expansion 
A tilda ( - )  over a symbol indicates that i t  i s  designated i n  the Laplace domain. A dot 
over  a variable  indicates  time  differentiation. 
ACT ANALYSIS 
The ACT system formulated i n  FCAP i s  shown in Figure 1 .  External disturbances to the 
ACT system are seen to  be atmospheric turbulence and gusts contributing  to the aerodynamic 
forces and moments, and pi lot  or guidance system commands introduced  through the control 
logic. The aircraft dynamic system includes both rigid-body and flexible-body dynamics. 
The analysis of ACT systems i s  unified  by casting a'll system equations in either  the  time 
domain or the frequency domain, and in  similar format. In state space methods, the motion 
of a given dynamic system i s  described by the following  pair  of  matrix equations (ref. 3): 
A = A x  + Bu 
r = C x  + D u  1 (1 ) 
where x i s  the state vector, u i s  the input (or control) vector, r i s  the output vector, and A, 
B, C, and D are the matrix coefficients. The equations for the dynamics of the state vari- 
ables, and for the outputs of the aircraft dynamics, sensors, logic, and actuators are given 
in the following sections. The equations are then combined with equations for aerodynamics 
of the aircraft, and the total system matrix equations are formulated using the compatibility 
relationships among the  dynamic systems. 
Aircraft Dynamics 
The FCAP aircraft dynamics equations for N,degrees of freedom (six  rigid-body and 
(N-6) flexible-body degrees of freedom) are  restricted to small perturbations  which reduce 
the equations to  linear form. This i s  a reasonable approximation for ACT studies (e .g., refs. 
4 and 5). The aircraft equation of  motion expressed in state vector form i s  (ref. 1): 
where 
T 
XD = state vector of the displacement variables = [x,y,z,#,0,Y,q7, ... qN] 
T 
xR = state vector of  the rate variables = [u,v,w,p,q,r,~7, ... 4NJ 
1705 
Also, MR = generalized mass matrix, A = Coriolis force/damping matrix, A = stiffness/ 
gravity-force matrix, = generalized aerodynamic forces in uniform flow, and u' = forces 
due to turbulence. 
RR RD 
UR R 
The output  of  the  a,ircraft dynamics at the sensor locations may be expressed linearly  in 
terms o f  displacements,x  rates, x  and  accelerations, i , ,and, therefore, i t  i s  possible 
to  write D' R' R 
- 
'D - 'SD XD 'SR XR + 'R 
where the coefficient matrices U and U i R  are  functions  of  the types of sensors and 
their location. SD' 'SR' 
Control System Dynamics 
The control system i s  defined as consisting of sensors, control  logic, and actuators for 
FCAP. Classically, control system dynamics are expressed in the form of a transfer function 
which can be redefined in the state vector form of Equation (1). In the following sections, 
let x x and x be the state vectors for the sensors, logic, and actuators, respectively. 
Sensors 
S f  L A 
The state vector equations for sensor dynamics are  given  by 
xs = A x + Bs us ss s 
and 
rs  = Cs xs + DS us 
where u i s  the input  to the sensor system from the aircraft. 
Con  tro I Logic 
S 
The state vector equations for the control  logic dynamics are expressed 
and 
rL = CL xL + DL (uL + u I )  (7) 
where u i s  the input to the control logic from the sensors, and u' i s  p i lot  and guidance 
system  ccmmands  (see fig. 1) .  L L 
17Q6 
Actuators 
The equations for the dynamics of  the  actuators in FCAP are given  by 
"A - A~~ "A + 0~ U~ 
- 
and 
where uA i s  the input  to the  actuators. 
Note  that a term of the type DA uA i s  absent in Equation (9). This implies that i n  the 
transfer function of the  actuators  the  degree of  the numerator. i s  lower than the  degree of 
the denominator. This yields considerable advantage i n  expressing the low-frequency- 
aerodynamics closed-loop system. 
Aerodynamics 
The potential aerodynamic method developed in references 6 to 9 provides  a unified 
approach  for both steady and unsteady subsonic  and  supersonic aerodynamics around complex, 
three-dimensional configurations. The subsonic portion of this method i s  incorporated into 
FCAP. The aerodynamic method of reference 6 i s  compatible with FCAP in that  the gener- 
alized aerodynamic forces are proportional to the aircraft dynamic generalized coordinates, 
xD, and rates, x and to actuator (i.e., control surface) deflections, In the time 
domain, the unsteady aerodynamic forces are expressed as R' 'A 
There URR, UR A. and U R ~  are operators corresponding to frequency-dependent matrices 
URR, URD, andDURA usua y known as aerodynamic-influence-coefficient matrices. 
For system stability and performance analyses, low-frequency aerodynamics i s  often 
adequate. Therefore, in the rest of this paper only low-frequency aerodynamics i s  con- 
sidered. In this case, the equations for the aerodynamics become linear with constant 
coefficients. The low-frequency aerodynamics equations are expressed in the time domain as 
where, for example, Ukvand  U(')are the first two terms of the  Maclaurin-Taylor series 
of uRR. AI I of these coefficienp$are frequency-independent. 
1707 
ACT System 
The state vector equations for each of the ACT subsystems are  given  by Equations (2) to 
(9). Using the low-frequency aerodynamics given by Equation (11) and recognizing from 
Figure 1 that the input for each subsystem i s  the output of the previous subsystem, the ACT 
system can be cast in the form of Equation (1) as 
where 
T x = [ x x x x x J  D R S L A  
with 
F =  
A, = 
u =  
0 
MR 
-'S k R  
-'lDSU:R 
-'ADLDSUkR 
' O O + q  
0 1 0  
0 0 1  ::j
I 
A~~ 
A~ D 
'5 'S D 
' L ~ S ~ S D  
I B ~ D ~ D ~ U ~ ~  
uk 
0 
BLUi 
a~D~ui 
A~~ 
'RR 
'SUSR 
'LDSUSR 
' A ~ L ~ S ~ S R  
0 
0 
AS s 
'LCS 
' A ~ L ~ S  
0 1 0 0 0  u(') 0 0 RR 0 0 0  0 0 0  0 0 0  0 0 : I  U!LC* 
0 
0 
0 
A~~ 
' A ~  L 
0 + 1. 0 R R  0 0  0 0 0  
0 0 0 0  
Uf0) 0 0 
A~~ 0 0 0  
The input u to the ACT system depends upon the gust forces, uh, and  the pi lot  and guidance 
system commands, u;. 
.FCAP PERFORMANCE ANALYSIS 
Performance analysis routines  are available  in FCAP to compute frequency response, 
transfer function poles and zeros, mean-square response to random inputs and time response. 
The theoretical basis for each of these techniques i s  well founded in the literature; there- 
fore, the following discussions w i l l  emphasize the  nature of  the technique as applied  to 
FCAP equations . 
Frequency Response 
Since  the low-frequency-aerodynamics ACT system dynamics are expressed in state 
vector format i n  terms of the (constant) A, B, C, D matrices, the frequency response of the 
kth output  to the ath input can be obtained (in the s-plane with zero init ial  conditions) from 
and 
Solving Equation (19) for and substituting into Equation (20) yields 
Classical frequency response i s  obtained for the special case where s = icu by computing the 
amplitude and phase from Equation (21) for  a  range of frequencies. 
Poles and Zeros 
Poles and zeros afe  evaluated in FCAP using a different 
Equation (21). Note that Equation (21) may be rewritten as 
form of H than that given in 
- 
ka 
The poles of Hka are  the zeros of the denominator, i.e., the  eigenvalues of the  matrix,A. 
The zeros of  Rka are the zeros of the numerator. The procedure to obtain them i s  given  in 
references 1 and 10. 
- 
1709 
I -  I l l l l l I l l l l l l l  l l l l l  
Mean-Square Response to Random Inputs 
The response of a flexible  aircraft to stationary randgm inputs may be  described in the 
frequency domain in terms of the spectral density matrix R(w). The definition used for the 
outputs are  also valid for  the inputs and the  state  variables. 
WheJe the inputs  are represented as zero-mean white noise with constant  spectral  density 
matrix, U, the  covariance  matrix  of the outputs R(0) i s  given  (for D = 0) by 
R(0) = CX(0) CT 
where the  covariance  matrix  of the state variables, X(()), i s  determined from the linear 
matrix equation (ref. 11,  pp. 330-332) 
AX@) + X(0) AT + BU(0) BT = 0 
Time Response 
The time response of a closed-loop system with  ini t ial  conditions, x(O), and an arbitrary 
input  function can be determined in FCAP using a fourth-order Runge-Kutta numerical  inte- 
gration algorithm. For the special case where the input for t > 0 has a rational Laplace 
transform (i.e., the input  can be described as a transfer function), the solution technique 
described in reference 12 i s  used. 
CONCLUDING REMARKS 
The equations'of  motion for a flexible  aircraft have been presented in  matrix format and 
incorporated into a digital computer program FCAP. The objective  in the development of  
FCAP was to model realistically those factors that  significantly  affect the stability and re- 
sponse of a flexible, ACT-configured vehicle. It should be noted, however, that FCAP i s  
intended primarily for use in the analysis of the  performance of an ACT system, rather than 
in the synthesis of the control system. 
The small-perturbation equations of motion  for the ACT system were obtained in state- 
vector  format and were  completed by the addition  of aerodynamics of arbitrary,  complex 
aircraft configurations. Both fu l ly  unsteady and low-frequency aerodynamic equations were 
presented; however, for performance analyses, low-frequency aerodynamics i s  usually 
adequate and, therefore, the ACT system equations were presented for low-frequency aero- 
dynamics only. Program FCAP, however, allows the analysis of. complete aircraft configu- 
rations, including  control system, with either low-frequency or unsteady aerodynamics. 
The analysis of ACT system performance in  FCAP has also been presented. In particular, 
the application  of classical frequency response, poles and zeros, mean-square response, and 
time response in  FCAP in state-vector  format has been discussed. 
Program FCAP provides  a  computerized method of  integrating  multiple systems into a 
matrix format, and then provides the means for obtaining desired solutions through classical 
analysis techniques. The program is currently  in the final stages of checkout and has been 
used to solve textbook examples of  control system problems. The program thus far has proven 
to be simple to use and requires a minimum of  input. 
1710 
REFERENCES 
1. Noll,  R. B.; and Morino, L.: Flight Control Analysis Program (FCAP) for Arbitrary- 
Configuration  Flexible  Aircraft With Active Control; Vof . I :  Theoretical Analysis. 
Aerospace Systems, Inc., ASI-TR-75-23, October 1975. 
2. Noll, R. B.; and Morino, L.: FCAP - A New Tool for the Evaluation of Active 
Control Technology. A I M  Paper No. 75-1059, August 1975. 
3. Schultz, D. G.; and Mefsa, J. L.: State Functions and Linear Control Systems. 
McGraw-Hill Book Company, Inc., New York, 1967. 
4. Etkin, B.: Dynamics of Flight. John Wiley and Sons, Inc., New York, 1959. 
5. Ashley, H.: Engineering  Analysis of Flight Vehicles. Addison-Wesley Publishing 
Company, Inc., Reading, MA, 1974. 
6. Morino, L.: A General Theory of Unsteady Compressible Potential Aerodynamics. 
NASA CR-2464, December 1974. 
7. Morino, L.; Chen, L. T.; and Suciu, E. 0.: Steady and Oscillatory Subsonic and 
Supersonic Aerodynamics Around Complex Configurations. A I M  Journal, Vol. 13, 
no. 3, March 1975, pp. 368-374. 
8. Morino, L.; and Chen, L. T.: Indicia1 Compressible Potential Aerodynamics Around 
Complex Aircraft Configurations. Published in  Aerodynamic Analyses Requiring 
Advanced Computers, Vol. II, NASA SP-347, 1975, pp. 1067-1110. 
9 .  Tseng, K.  D.; and Mocino, L.: Fully Unsteady Subsonic  and  Supersonic Potential 
Aerodynamics of Complex Aircraft Configurations for Flutter Application. Presented 
at'AIAA/ASME/SAE 17th Structures, Structural Dynamics, and Materials Con- 
ference (King of Prussia, PA), May 5-7, 1976. 
10. Konar, A. F.; et al: Digital Flight Control Systems for Tactical Fighters; Vol. I: 
Digital Flight Control Systems Analysis. AFFDL-TR-73-119, Vol. I, December 1973. 
11. Bryson, A. E.; and Ho, Y. C.: Applied Optimal Control: Optimization, Estimation, 
and Control. Blaisdell Publishing Company, Waltham, MA, 1969. 
12. Melsa, J. L.; and Jones, S. K .: Computer Programs for Computational Assistance in 
the Study of  Linear Control Theory. McGraw-Hill Book Company, Inc., New 
York, 1973. 
1711 
- U k  
+ GUSTS e TURBULENCE 
b 1 AERODYNAMICS 
4 UR+ Uk - 'D 4 XR 
FLEXIBLE 
AIRCRAFT 
DYNAMICS 
'A 'D - - 
0 
- 
;"- 
"" - "" ""- "." ", 
v I 
I CONTROL 
I 
I 
I 
I 
I 
I L ___-___-- - - -  ----------".- J 
ACTUATORS 4 - LOGIC - 'A 'L - uL I s  4 =  SENSORS 
CONTROL  SYSTEM 
PILOT & GUIDANCE 
COMMANDS 
- U L  
Figure 1 .  ACT System. 
1712 


Performance analysis of flexible aircraft with active control

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