Document
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 o f 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 systemwas formulated in state vector format and the system of equations was completed with fully 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 fully unsteadyaerodynamics. The application of classical performance analyses including frequency response, poles and zeros, mean-squareresponse, and time response i n FCAP i n state vector format was discussed.
INTRODUCTION The integrated study of the interactive effects of the flight control system i n the active control of flexible aircraft has received considerable attention in recent years. In particular, Active Control Technology (ACT) i s being investigated for improving ride quality, decreas- ing structural deformation, extending the fatigue life 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 i n amodular 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 i t s 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.
l -
I SYMBOLS state-space matrices of ACT system and subsystems A,BjC,D matrix defined by Equation (17) Al matrix defined by Equation (16) F H matrix o f transfer functions i w imaginary part of complex frequency, s generalized mass/inertia matrix dynamic pressure Lagrangian generalized coordinates .
covariance matrix of ACT system outputs r 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 matrixof aerodynamic forces due to turbulence ’ covariance matrix of ACT system state variables X state vector Subscripts: A actuator aircraft displacement D controllogic L aircraft rate R S sensor Superscripts: T transpose ( O ) , (1) coefficients of power series expansion A tilda ( - ) over a symbol indicates that it 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 pilot or guidance systemcommands 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 i n similar format. In state spacemethods, the motion of a given dynamic system i s described by the following pair of matrix equations (ref. 3):
A = A x + Bu
(1 )
r = C x + D u
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
= statevector of thedisplacementvariables = [x,y,z,#,0,Y,q7, ... qN]
XD T
xR = statevector of the ratevariables = [u,v,w,p,q,r,~7, ... 4NJ
Also, MR = generalized mass matrix, A = Coriolis force/damping matrix, A = stiffness/ RR RD = generalized aerodynamic forces in uniform flow, and u' = forces gravity-force matrix, UR R due to turbulence.
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 D' R' R to write - 'D - 'SD XD 'SR XR + ' R where the coefficient matrices U and U i R are functions of the types of sensors and SD' 'SR' their location.
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 thestatevector form of Equation (1). In the following sections, let xx and x bethe state vectors for the sensors, logic, and actuators, respectively.
S f L A Sensors The state vector equations for sensor dynamics are given by
xs = A x + Bs us
ss s
and r s = Cs xs + DS us i s the input to the sensorsystem from the aircraft.
where u S Con tro I Logic 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 pilot and guidance L system ccmmands (see fig. 1 ) . 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 considerableadvantage 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. Thesubsonic 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 toactuator (i.e., control surface) deflections, In the time R' 'A domain, the unsteady aerodynamic forces are expressed as There URR, UR A. and U R ~ are operators corresponding to frequency-dependent matrices URR, URD, andDURA usua yknown as aerodynamic-influence-coefficient matrices.
Forsystem 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.
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 eachsubsystem 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 0 0 0 0 0 u ( ' ) 0 0 U ! L C * MR RR 0 0 0
' O O + q
F = - ' S k R 0 1 0 0 0 0 0 -'lDSU:R
0 0 1 ::j 0 0 0 0 : I
-'ADLDSUkR I 0 0 0 0 0 A~~ A~~ 0 0 Uf0) 0 0 A~ D 'RR A, =
0 0 + 1. 0 R R 0 0
'5 ' S D 'SUSR A S s 0 0 0 0 ' L ~ S ~ S D 'LDSUSR 'LCS A~~ I B ~ D ~ D ~ U ~ ~ ' A ~ L ~ S ~ S R ' A ~ L ~ S ' A ~ L A~~ 0 0 0
uk
u = BLUi
a~D~ui
The input u to the ACT system depends upon the gust forces, uh, and the pilot and guidance systemcommands, 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 initial 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 = i c u by computing the amplitude and phase from Equation (21) for a range of frequencies.
Polesand Zeros
-
Poles and zeros afe evaluated in FCAP using a different form of H than thatgiven in ka Equation (21). Note that Equation (21) may be rewritten as
-
The poles of H k a are the zeros of the denominator, i.e., the eigenvalues of the matrix,A.
Thezeros of Rka are the zeros of the numerator. The procedure to obtain them i s given in references 1 and 10.
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 matrixequation(ref. 1 1 , pp. 330-332)
AX@) + X(0) AT + BU(0) BT = 0
Time Response The time response of a closed-loop system with initial conditions, x(O), and an arbitrary input function can be determined in FCAP using a fourth-order Runge-Kutta numerical inte- gration algorithm. For thespecial case where the input fort > 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 fully 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 i n FCAP has also been presented. In particular, the application of classical frequency response, poles and zeros, mean-squareresponse,and time response i n 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.
REFERENCES 1. Noll, R. B.; and Morino, L.: FlightControlAnalysis 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.; andMorino, L.: FCAP - A New Tool for theEvaluation of Active
Control Technology. A I M Paper No. 75-1059, August 1975.
3. Schultz, D. G.; and Mefsa, J. L.: State FunctionsandLinear Control Systems.
McGraw-Hill BookCompany, Inc., New York, 1967.
4. Etkin, B.: Dynamics ofFlight. John Wiley and Sons, Inc., New York, 1959.
5 . Ashley, H.: EngineeringAnalysis ofFlight 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 AroundComplexConfigurations. 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 i n 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.; etal: DigitalFlightControl 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.: AppliedOptimalControl:Optimization, Estimation, and Control.BlaisdellPublishing Company, Waltham, MA, 1969.
12. Melsa, J. L.; and Jones, S. K .: Computer Programs forComputational Assistance in
the Study of Linear Control Theory. McGraw-Hill BookCompany, Inc., New York, 1973.
- U k
GUSTS e TURBULENCE
+
b 1 AERODYNAMICS
4 UR+ Uk
- 'D 4 XR
FLEXIBLE AIRCRAFT 'A 'D
-
-
DYNAMICS
-
"" - "" ""- " " " . " ",
v I
;"-
I I
CONTROL
4 - 4 =
ACTUATORS SENSORS
I
I LOGIC
- 'A 'L - uL I s
I
I
I
CONTROLSYSTEM
L _ _ _ - _ _ _ - - - - - ----------".- J
P I L O T & GUIDANCE COMMANDS
- U L
Figure 1 . ACT System.