For review for publication in 1EEE Transactions on Aerospace amt Electronic Systems ROBUST DAMAGE-MITIGATING CONTROL OF AIRCRAFT FOR HIGH PERFORMANCE AND STRUCTURAL DURABILITY _ Jeffrey Caplin Asok Ray t, Senior Member, IEEE Mechanical Engineering Department The Pennsylvania State University University Park, PA 16802 Suresh M. Joshi, Fellow, IEEE NASA Langley Research Center Hampton, VA 23681-0001 "["Corresponding Author: Tel: (814) 865-6377; Email: axr2(,:t_psu.edu Kevwords: Damage-Tolerant Systems, Robust Control, Structural Durability ABSTRACT This paper presents the concept and a design methodology for robust damage-mitigating control (DMC) of aircraft. The goal of DMC is to simultaneously achieve high performance and structural durability. The controller design procedure involves consideration of damage at critical points of the structure, as well as the performance requirements of the aircraft. An aeroelastic model of the wings has been formulated and is incorporated into a nonlinear rigid-body model of aircraft flight-dynamics. Robust damage-mitigating controllers are then designed using the H_ -based structured singular value (g) synthesis method based on a linearized model of the aircraft. In addition to penalizing the error between the ideal performance and the actual performance of the aircraft, frequency- dependent weights are placed on the strain amplitude at the root of each wing. Using each controller in hun, the control system is put through an identical sequence of maneuvers, and the resulting (varying amplitude cyclic) stress profiles are analyzed using a fatigue crack growth model that incorporates the effects of stress overload. Comparisons are made to determine the impact of different weights on the resulting fatigue crack damage in the wings. The results of simulation experiments show significant savings in fatigue life of the wings while retaining the dynamic performance of the aircraft. :_The research work reported inthis paper has been supported in part by: NASA Langley Research Center under Grant No. NCC-1-249; National Science Foundation under Grant Nos. DMI-9424587 and CMS-9531835; National Academy of Sciences under a Research Fellowship award to the second author. 1 INTRODUCTION In the post-Cold War era, there is an increasing concern about high acquisition and maintenance cost associated with complex weapons systems. Fighter aircraft is a good example of such systems because the airframe undergoes significant cyclic stresses resulting in the need for frequent inspection and replacement of critical components. While airframe manufacturers are constantly updating the technology base to handle higher stress levels, these improvements often do not fully translate into an equivalent increase in component life due to the ever-increasing performance requirements of fighter aircraft. This is particularly true of the current generation of fighter aircraft, as the high thrust-to-weight ratio allows extreme flight maneuvers that were not possible earlier and thereby the airframe is often subjected to very high instantaneous and sustained stress levels. From an economic standpoint it is desirable to obtain the maximum amount of useful life from the most expensive (and hard to replace) components of the aircraft, as well as to reduce the number of maintenance inspections required to ensure structural integrity of critical components. This practice is also desirable from an operational viewpoint, since reductions in downtime for inspection and repair result in increased availability. However, since failure of certain components may result in loss of the aircraft, and more importantly, loss of human life, safety considerations mandate replacement of all critical components before a failure is likely to occur. This requirement is realized in the following way. A fighter aircraft that exceeds its design load factor during a flight is temporarily removed from service, and it must undergo a rigorous inspection to determine if any special maintenance is required prior to its return to flying status. (Note: Load factor is defined as the total lift force acting on the aircraft divided by the aircraft weight). Designers of flight control systems have recognized the possibility of actively reducing damage in certain aircraft structures, particularly the wing. The simplest concept that is currently employed on both F-16 and F/A-18 aircraft is the so-called "g-limiter". It serves to limit the aircraft's maximum load factor to a predeflned value. Transport aircraft have used the Gust Load Alleviation (GLA) system [Matsuzaki et al., 1989; Baldelli et al., 1993] that uses feedback from accelemmeters on the wing to drive special control surfaces in order to reduce the additional loads imposed by atmospheric disturbances. A similar concept, known as Maneuver Load Alleviation (MLA) [McLean, 1990] or Maneuver Load Control (MLC) [Thornton, 1993], has been proposed for high-performance aircraft. The aim of these systems is to shift the lift distribution inboard during high loading conditions to limit the bending moment at the wing root. Dynamic stresses have been considered in the so-called Fatigue Reduction (FR) system [McLean, 1990] that seeks to minimize the amplitude and/or number of stress cycles experienced at the critical point(s). While these systems have shown tangible benefits, there is apparently a common weakness that may well prevent them from achieving their maximum potential. In all cases, the actual dynamics of the fatigue crack damage phenomenon in the structural material are not included in the analysis. It is simply assumed that, by limiting the peak stress at the critical points of the structure, life-savings could be maximized. Since transient stress overloads could result in a temporary retardation of crack growth [Anderson, 1995; Patankar et al., 1999], the frequency content of the applied stresses could be shaped by control actions to achieve larger fatigue life than the traditional approach of simply limiting the peak stress. This paper addresses the above issue focusing on fatigue damage mitigation in the wings of high performance aircraft that are usually instrumented for health monitoring and control. The thrust of the paper is on robust damage- mitigating control (DMC) where the goal is to achieve large gains in structural durability by manipulation of stress profiles with no significant loss of performance [Ray et al., 1994]. This concept of DMC has been investigated for reusable rocket engines [Dai and Ray, 1996; Holmes and Ray, 1998], rotorcraft [Rozak and Ray, 1997, 1998], and fossil fuel power plants [Kallappa et al., 1997]. In all cases, simulation results show a substantial increase in component life with no significant loss in system performance. Efficacy of the DMC concept has also been demonstrated by laboratory experimentation on test apparatuses [Tangirala et al., 1998; Zhang and Ray, 1998]. The paper is organized in five sections including the introduction. Section 2 describes model formulation for damage mitigating controller design. Section 3presents a procedure for synthesis of the robust damage-mitigating control law. Section 4 presents the overall simulation structure and the results of the aircraft performance and crack- growth damage for a family of robust controllers. Section 5 summarizes and concludes the paper with recommendations for future work. 2 MODEL FORMULATIONFORDAMAGE-MITIGATNG CONTROLLER DESIGN Although it has tong been recognized that controller design for highly flexible aircraft, such as transports, requires dynamic models that explicitly include structural flexibility [McLean, 1990], these effects are often ignored on aircraft that experience relatively small elastic deformation. This is particularly true when modern robust control techniques, such as H®-synthesis, are employed, since the effects of unmodeled dynamics due to flexibility can be included within the (unstructured) uncertainty model. However, explicit modeling of structural flexibility may provide a solution to another problem faced by flight control systems designers, namely that of control surface redundancy. High-performance aircraft require two (or more) different sets of control surfaces for roll control and, in some cases, have multiple controls for pitch and yaw as well. As these controls have different levels of effectiveness at different flight conditions, they appear to be redundant actuators in the controller synthesis process at agiven operating condition. Therefore, special measures are required to allocate commands between the various control surfaces. Methods that have been developed to date include the use of a nonlinear control selector [Buff'mgton et al., 1994], or the use of off-line constrained optimization procedures [Durham, 1992]. In both cases, the methods only examine the effects of the controls on the rigid-body motion of the aircraft. The damage mitigating control (DMC) takes advantage of the control surface redundancy by utilizing the elastic behavior of the aircraft structure as well as the rigid-body motion. The theme of DMC design is that different locations of control surfaces on the airframe may result in different effects on the elastic modes of the aircraft structure. Thus, the control systems designer can make use of all available control surfaces to simultaneously achieve the desired level of performance while mitigating the structural damage by reshaping the stress profile. So far the only application of DMC to aviation systems has been the work of Rozak and Ray (1997, 1998) who developed arobust controller for rotorcraft with the objective of reducing damage to the control horn of the main rotor. Since the control horn does not directly experience any significant aerodynamic forces, the loading is of a purely mechanical nature. In contrast, the loads (and hence stresses) acting on an aircraft wing occur due to aeroelasticity, which deals with interactions of aerodynamic forces with flexible structures. The aerodynamic forces play a dominant role in determining the dynamics of the aircraft that, in turn, lead to deformations and stresses in the critical structures. Therefore, changes in aerodynamic forces due to structural deformation have been included in the plant dynamic model in combination with the rigid body model. 2.1 The Rigid Body Model The model describes the motion of a rigid-aircraft of fixed mass flying through a stationary atmosphere over a flat, non-rotating earth. The aircraft is a generic, twin engine, high-performance fighter similar to the F-15. Three coordinate systems that are used together to define the position and orientation of the aircraft are the inertial axes, the vehicle-carried vertical axes, and the body-fixed axes. As the name implies, the inertial axes are considered fixed in space, with the X-axis directed north, the Y-axis directed east, and the Z-axis directed down. The vehicle carried vertical axes (denoted by the subscript 'v') have the same orientation as the inertial axes, but their origin lies at the aircraft center of gravity, and translates with the aircraft. Three state variables are needed to define the position of the vehicle-carried vertical axes with respect to the inertial axes. These are the displacement north, displacement east, and the altitude, denoted as x,y, and h, respectively. The body-fixed axes (denoted by the subscript 'b') also have their origin at the aircraft center of gravity; however, they rotate as well as translate with the body. They are oriented such that the xb-axis is positive towards the nose of the aircraft, the yb-axis is positive towards the right wing tip, and the zb-axis is positive towards the bottom of the aircraft. The orientation of the body-fixed axes (3-2-1 rotation) with respect to the vehicle-carried vertical axes is defined by the state variables ur',®, and _, known respectively as the yaw angle, pitch angle, and roll (or bank) angle. The order in which these rotations are carried out is demonstrated in Figure 1. First, the vehicle- carried vertical axes are rotated about the zv-axis through the angle T to obtain an intermediate set of axes designated (xby_,z_). This axis system is then rotated about the y_-axis through the angle ® to obtain the (x2,y2,z2) axes. Finally, this system is rotated about the x2-axis through the angle • to obtain the body-fixed axes. Two coordinate systems are used to define the relative motion between the aircraft and the atmosphere. The stability axes (denoted by the subscript 's') are obtained by rotating the body-fixed axes through the angle of attack a, which isthe angle between the xb-axis and the projection of the velocity vector onto the Xb-Zbplane. The pitching behavior of the body-fixed axes with respect to the stability axes is used to characterize the longitudinal stability of the aircraft, hence the name "stability axes." The wind axes (denoted by the subscript 'w') are obtained by rotating the z_-axis through the sideslip angle 13, which brings the Xw-axis in line with the velocity vector. The primary advantage of using the wind axes, and choosing the magnitude of the total velocity vector as one of the state variables, is that this approach permits the explicit computation of derivatives of ccand 13in terms of other state variables. An alternate formulation, which used the velocity components along the body-fixed axes as states, would result in equations for the derivatives of c_ 4 and[3intermsofboththeremaininsgtateasndtheirderivative[Bslakelock1,99]1. Figure2showtsherelationship betweetnhebody-fixeds,tabilitya,ndwindaxes.Thelastthreestatevariabledsefinetherotationaml otionofthe aircraft.Theroll,pitch,andyawratesd,enoteadsp,q,andr,respectivealyr,etherotationaraltesabouttheXb-axis, yb-axisa,ndzb-axisre,spectivelIyt.shouldbenotedthatwhilethedirectionosftheserotationaral tesaredefinedby theorientatioonfthebody-fixeadxest,heirmagnitudaersemeasureindtheinertiarleferencfreame. Therigid-bodfylightdynamimcodeilnthispapearresimilartothecorrespondmingodedlevelopefdorthe AIAAControlDsesignChalleng[Berumbaug1h9,91a]ndtherefortehesemodeelquationnsotrepeatehdere.The controslurfaceinscludeleftandrightaileronsle,ftandrightstabilatorasn,dasinglerudder.Althoughtheaircraft hasfivecontroslurfaceos,nlyfourvariableasrerequiretdospecifytheirpositionsth:eailerondeflection8A;the ruddedreflection8R,thesymmetrsictabilatodreflection8H;andthedifferentiasltabilatodreflection8D' The positionosftheindividuaclontroslurfaceasredetermineadsfollows: left aileron position = 0.58A."right aileron position = -0.5 8¢_ left stabilator position =8H +0.5 8D; right stabilator position =8H - 0.5 8D rudder position =8R In the original model of Brumbaugh (1991), all actuator dynamics are represented as first order lags with a time constant of 50 ms that has been replaced with more detailed dynamics [Adams et al., 1994]. The transfer functions and rate limits of the actuators are given in Table I. Linearization of the equations of motion for a rigid, fixed-wing aircraft yields two uncoupled sets of equations. One set governs the longitudinal dynamics of the aircraft while the other governs the lateral dynamics. We have followed the standard practice to design separate controllers for lateral and longitudinal dynamics based on the uncoupled linearized models and then evaluate the control system based on the simulation of the nonlinear model. The stick-fixed longitudinal motion of a rigid aircraft disturbed from equilibrium flight is described by two oscillatory modes of motion: the short period mode and the long period (or phugoid) mode. The short period mode typically has a period on the order of a few seconds, with motion characterized by changes in angle of attack, pitch angle, and altitude, while the flight velocity remains practically constant. The phugoid mode has a much longer period, on the order of tens or hundreds of seconds, with motion characterized by changes in velocity, pitch angle, and altitude, with angle of attack remaining approximately constant. Because of the slow dynamics associated with the phugoid mode, this mode has been ignored during the synthesis of the damage-mitigating control laws for manual flight. The pitch rate q, and angle of attack cc are the state variables of interest for longitudinal rigid body motion. However, for the design of an autopilot, the phugoid motion becomes the primary mode of interest. The desired short period response of the aircraft is that of a second order system. The natural frequency is a function of the acceleration sensitivity of the aircraft, which is the change in load factor per unit change in angle of attack. The acceleration sensitivity is determined by the aerodynamics of the aircraft and the particular flight condition under consideration. Thus, the desired natural frequency also varies with different flight conditions. The dampinrgatiooftheshortperiodmodeisrequiretdobebetwee0n.35and1.3forallflightconditionbsasedon MIL-F-8785sCpecificatio[nDsoD,1980]. The roll rate p, yaw rate r, and sideslip angle [3 are the state variables of interest for lateral rigid body motion. The stick-fixed lateral motion of a rigid aircraft is described by three natural modes: (i) the Dutch roll mode consisting of lightly damped, oscillatory, out-of-phase roll, yaw, and sideslip motions; (ii) the roll mode consisting of a non-oscillatory, highly convergent mode describing the rolling characteristics of the aircraft; and (iii) the spiral mode consisting of non-oscillatory, convergent or divergent motion following a sideslip disturbance. Note that an unstable spiral mode may cause the aircraft to go into a turn that becomes increasingly tighter with time. The handling qualities requirements specify that the Dutch roll mode should have a frequency of at least 1 radian per second. The damping ratio must be greater than or equal to 0.4, or the product of the frequency and damping ratio should be greater than or equal to 0.4 radians per second, whichever results in the larger value for the required damping. The roll mode requirement states that the roll time constant must be less than or equal to 1.0 second. This requirement is actually conservative with regards to modem fighters, which typically have roll time constants in the range of 0.33 to 0.5 seconds [Adams et al., 1994]. The spiral mode requirement specifies that the minimum time to reach a 40° bank angle following a 20° bank angle disturbance must be greater than or equal to 12 seconds. Because of the slower dynamics of the spiral mode, it is typically ignored during controller synthesis. 2.2 Atmospheric Model Atmospheric properties are based on the US Standard Atmosphere (1962). The model outputs values for temperature, static pressure, density, and speed of sound as functions of altitude. While not strictly an atmospheric property, the model also includes tabulated values for the acceleration due to gravity, also as a function of altitude. The atmospheric model in this paper is similar to the corresponding model developed for the AIAA Controls Design Challenge [Brumbaugh, 1991]and is not repeated here. 2.3 Aeroelastic Model Aerodynamic forces acting on abody depend on the time history of the body's motion [Etkin, 1972]. When the dynamics of a rigid aircraft are the subject of interest, particularly if the aircraft does not perform any severe maneuvers, the much faster dynamics of the fiowfield are ignored based on the principle of singular perturbation. For aircraft that are required to perform extreme maneuvers, it is often sufficient to introduce approximate correction factors into the equations of motion to account for any unsteady aerodynamic forces. These correction factors generally depend on the time derivatives of angle of attack and sideslip angle. However, when the flexible structures of aircraft (that have a faster time scale than the rigid-body dynamics) are of interest, itbecomes necessary to explicitly model the dynamics of the flow-field to identify any potential instability due to fluid-structure interactions, known as flutter. Since wing flutter results in catastrophic failure of the aircraft, it is prevented either through the design of the wing, or through the use of Active Flutter Suppression (AFS). Therefore, an aeroelastic model of the critical structure (i.e., the wings) is required for the DMC design for the following reasons: • To shape the profiles of transient stresses for fatigue damage reduction; and • To ensure that the control system does not adversely affect the flutter characteristics. 2.3.1 Structural Model Although composite wing structures have been used in recent aircraft, most fighter aircraft have wings that are at least partially, and in most cases solely, built from ductile alloys. A typical wing structure contains at least two spars that run the length of the wing semi-span to bear the majority of the bending loads. The spars and the skin together form several torsion boxes to resist twisting deformation of the wings. The wings also contain many lesser structural members whose primary function is to maintain the shape of the skin. In this paper, the structural model is formulated as a pair of Euler beams to represent the important structural behavior of the wings. Each beam model is aligned with the elastic axis (i.e. the line through which loads applied normal to the plane of the wing result in pure bending). The center portion of the model, where the beams meet, is assigned proportionately higher values of bending and torsional stiffness in order to represent the fuselage. The model is spatially discretized and cast in the finite element setting. While the details of the finite-element model, including the element type and shape functions, are reported by Caplin (1998), its basic features and dominant mode shapes are presented below: The generalized displacement vector {(t) is obtained by orthogonal transformation of the physical displacement vector 0(t), i.e., _(t) =_r0 (t) where • is the orthogonal matrix whose columns are the individual mode shapes. The governing equation for { (t) is obtained in the transformed coordinates as: M_'(t) + C_(t) + X_(t) = f(t) (1) where the transformed "modal mass" and "modal stiffness" matrices, M and K, are diagonal; C is the "modal damping" matrix representing energy dissipation; and f(t) the total generalized force vector which is obtained by orthogonal transformation of the applied nodal force vector that is a linear combination of aerodynamic force due to both vibratory motion of flexible modes and rigid-body motion. Accordingly, the total generalized force vector is expressed as: f(t) = f fl_ (t) + frigid (t) (2) where fflex is the generalized aerodynamic force vector acting on the flexible modes and frigid is the generalized force vector due to rigid-body motion. For the specific wing structure considered in this paper, we have used the six lowest modes, of which three are symmetric and three are antisymmetric with respect to the aircraft body-fixed x-axis. Figures 3 and 4 show the symmetric mode shapes and the antisymmetric mode shapes for both linear displacement and angular twist. It is necessary to make the distinction between symmetric and antisymmetric modes because the dynamics of the two sets of modes are uncoupled from each other. Within each set, however, the dynamics are coupled due to the aerodynamic forces generated by the deformation of the wing. 2.3.2 Integrated Model of Unsteady Aerodynamics and Structural Dynamics Although the current state-of-the-art in Computational Fluid Dynamics (CFD) allows time-domain solutions to the Euler equations (for inviscid compressible flow) or the Navier-Stokes equations (for viscous compressible flow) for many flows of practical interest, the use of these techniques is still rather limited due to high computational cost. Within the aerospace industry, a vast majority of unsteady flow applications, such as flutter analysis, rely on computational techniques that have been developed for the restricted case of thin wings undergoing simple harmonic motion for unsteady subsonic potential flow [Dowel et al., 1995]. The Doublet-Lattice Method [Albano and Rodden, 1969] has been adopted in this paper for aerodynamic analysis. In this method, the wing is divided into a finite number of trapezoidal panels. The lifting force acting on each panel is assumed to be concentrated along the one-quarter chord line of the panel, where a line of acceleration potential doublets is placed. The strength of the doublets is assumed to be uniform within each panel and is allowed to vary from one panel to the next. A control point is placed at the three-quarter-chord point of the mid-span of each panel. Details are reported by Caplin (1998). The transfer matrix Q from the generalized displacement vector { to the flexible part fflex of the total generalized force vector f is approximated as a function of the dimensionless Laplace transform variable .7 and dynamic pressure _" {Karpel, 1990]: Q(s) _(A 0+_'A 1+_2A 2 sb +_'D(}'I-R)-IE q; S-U • _'-19oo(U_o) 2 (3) where Uo_ and P_o are the undisturbed free-stream air flow speed and density, respectively; b is the wing semi- span; the matrix A0 is obtained from the steady-sate response of experimental or CFD simulation data; other matrices, AI, A2, D, and E, are obtained by frequency-domain system identification based on experimental or CFD simulation data; and R is a diagonal matrix whose elements are chosen to be the poles of additional aerodynamic states within the frequency range of interest. Remark 2.1: The terms involving the matrices Ao, AI, and A2 in Eq. (3) capture the dependence of the aerodynamic forces on the displacement, velocity, and acceleration, respectively, of the wing mode shapes. The remaining term, involving the matrices D, R, and E on the right hand side of Eq. (3), account for the lag in aerodynamic forces. * Making the transformation of Eq. (3) into the time domain and substituting the resulting expression into Eqs. (1) and (2) yield the following set of ordinary differential equations: M_'(t)+C_(t)+K_(t)= A0_(t)+ AlE(t)+ A2_'(t)+Dxa(t) q+ frigid(t) (4) 2a(t )=E_(t)+U°° R Xa(t ) b where xa is the vector of selected states to represent the aerodynamic lag; and frigid is the part of the total generalized force vector contributed by the rigid-body motion as defined in Eq. (2).. The aeroelastic model in Eq. (4) is rewritten in the state space setting for synthesis of damage-mitigating controllers as: frigie(t)) (5) _7(Xa(t))= E_'(t)+ U°_ R Xa(t) b by introducing the following definitions: M'-M- _A2; _-C-u--_'AI; K=_K-_A 0 (6) The aeroelastic model in Eq. (5) forms two uncoupled sets of equations, one for the symmetric modes and the other for the antisymmetric modes. Additional aerodynamic terms representing frigid in Eq. (5) must be added to each model to account for the relevant rigid-body motions. Furthermore, the perturbations in the rigid-body aerodynamic coefficients are different for each model. These considerations are addressed in the next two sections. 2.3.2.1 Symmetric Aeroelastic Model In general, the largest contribution to the aerodynamic loads acting on the wing is due to the rigid-body angle of attack. Strictly speaking, this requires the addition of three terms to Eq. (5); one for the angle of attack, and one each for its first two derivatives. However, it is generally recognized that terms involving the second derivative of the rigid body motion may be neglected for characteristic frequencies below 2Hz [Pierce (1988)]. Since angle of attack ct does not have any significant frequency content above 2 Hz, the term involving the acceleration d/ has not been included in the complete aeroelastic model. Two inertial terms, Qaz az(t) and Qq el(t), which are proportional to normal acceleration and pitch acceleration of the aircraft, respectively, are added to the model when longitudinal rigid-body motion is considered. Substituting these inertial terms along with additional aerodynamic terms into Eq. (5) yields the following set of equations for the symmetric aeroelastic model: d(xa):E_(t) + Uo_ Rxa(t)+E 6 d(t) b The main effect of symmetric deformation of the wing on the overall dynamics of the aircraft is assumed to be due to the change in lift coefficient. The transfer function from generalized displacement to change in lift coefficienistobtaineidnthesamemannearsthetransfefrunctionfromgeneralizeddisplacemetnotgeneralized forceT. hustheunsteadpyerturbatioinnliftcoefficienistobtaineidnthetime-domaains: ACL(t)=AOc{_(t)+-_ AlcL_(t)+ A2C_L'(t)+ DcL xa(t ) (8) 2.3.2.2 Antisymmetric Aeroelastic Model Two additional sets of aerodynamic terms areadded to the antisymmetric model. First, deflection 6A(t) of the ailerons is viewed as an additional deformation mode of the wing, and thus additional aerodynamic terms involving aileron position, rate, and acceleration must be included. Second, terms involving the rigid-body roll rate and roll acceleration are required although roll angle has no effects. Only one inertial term, Q/,/5(0, which is proportional to roll acceleration of the aircraft, is added to the model when lateral rigid-body motion is considered. The antisymmetric aeroelastic model thus takes the following form: b (9) _'-I 6A(t)+ (/)+ _'A(t)+ b-_App(t)+ App(t) "q + _'" wAs'ISA W AgA V_ d y Uoo _ (t)+Ep p(t) -_t( a)= Eri(t) +--_ Rya (t) + ES"A where the symbols r1 and Ya have been used for the generalized displacement vector and the aerodynamic state, respectively, in order to distinguish the antisymmetric states from the symmetric states { and xa . The main effect of antisymmetric deformation of the wing on the overall dynamics of the aircraft isdue to the change in roll moment coefficient. Its transfer function with respect to generalized displacement is obtained similar to Eq. (8). Thus the unsteady perturbation in the roll moment coefficient is obtained in the time-domain as: ACl(t)=AoQrl(t)+b Alctr_(t)+ __ A2cil'(t)+DQYa(t ) (10) Remark 2.2: The major effect of antisymmetric deformation of the wing on the overall dynamics of the aircraft is due to the change in roll moment coefficient. * 2.4 Propulsion System Model The propulsion system model is based on the data for an F-100 turbofan engine installed in the F-15. The steady-state values for idle, military, and maximum afterbumer thrust are tabulated as functions of Mach number, altitude, and power lever angle (PLA). The model includes first-order core dynamics, with a time constant computed via linear interpolation as a function of percent military thrust, Mach number, and altitude. The afterburner model includes sequencing logic to handle the transitions between afterburner stages. I0