JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 1, pp. 31-50, Warsaw 2006 PRACTICAL ASPECTS OF IDENTIFICATION OF THE AERODYNAMIC CHARACTERISTICS Jacek A. Goszczyński Korporacja polskie Stocznie S.A. (Polish Shipyards Corporation Company), Warsaw e-mail: jaglot@poczta.onet.pl Theproblemof identification of aircraft aerodynamic characteristicsper- formed by means of recording current flight parameters is presented in the paper. Basic concepts of fast identification algorithms; e.g. Non- Linear Filtering (NF) (based on the Lipcer and Sziriajev theory) and Estimation BeforeModelling (EBM) are presented as well. Tips on how to implement theEBMandNFmethods inpracticeare shown.Presented numerical results seem to be very interesting. Key words: dynamics of flight, non-linearmodel, flight simulation, aero- dynamic characteristics identification 1. Introduction Anaircraft is a complexdynamic systemthatmoves in real atmosphereand executes dynamic controlledmanoeuvres. Aerodynamic loads acting upon the aircraft as well as surrounding atmosphere (environmental conditions) exert fundamental influence on its behaviour and dynamic properties. One of the effective ways of determination of aerodynamic coefficients appearing in the formulae for aerodynamic forces and moments in the aircraft flight is the identification. Aerodynamic characteristics of an aircraft change according to velocity and flight altitude variations. It is, therefore, necessary to apply identification methods which could follow up those variations. The contemporary problem of system identification (assume shape as flying object), can be divided into three main parts (Fig.1): • Measurement equipment – a subsystem logging measurement data and recording them through appropriate on-board and on-ground equipment with respect to their ”quality” – knowledge of measurement errors. 32 J.A.Goszczyński • Flight test techniques – a subsystem selecting adequate test flight pro- grams of the flying object. Input signals are optimized in terms of their spectrum so that parameters of the object could be estimated. • Flight data analyzer – a subsystem based on a mathematical model of the flying object and estimation criteria to find a solution to the given computational identification algorithm from initial conditions or specified a priori estimates of unknown parameters and tominimize the error system response of the best estimate parameter. Fig. 1. Correlation in the process of identification of aerodynamic characteristics Identification of aerodynamic characteristics of aflyingobject (with control and stability derivates) depends on numerical solution of values based on test flights. In preparation of an identification method to practical use, we must assume its applicability in a step by stepmanner (Giergiel andUhl, 1990).We divide the development of the method into three phases (Fig.2): • Phase 1 – depends on numerical simulation of a tested object aimed at the identification of flight regimes, for example the problem of high angles of attack. • Phase 2 – depends on an identification algorithm determining the influ- ence of object control andmeasurement errors on recorded data proces- sing. Practical aspects of identification... 33 • Phase 3 – depends on practical use of Phase 1 and 2 which are applied to data processing recorded during flight tests. Fig. 2. Phases of practical identification of aerodynamic characteristics of a flying object (Giergiel and Uhl, 1990; Goszczyński, 2000) In the second phase, a selection of critical elements for the identification process are used for estimation of parameters verification of the formulated mathematical model. Requirements of the above phases indicate fundamental need for aerody- namic tunnel tests and knowledge of the flying object physics. As a matter of fact an aerodynamic model of a flying object in the de- terministic sense must reflect particularly strongly nonlinear components of aerodynamic forces andmoments. 34 J.A.Goszczyński 2. Mathematical models Anaircraft is defined as a flying object (FO) considered in a flight configu- ration as a rigid body with movable control surfaces. A mathematical model of FO is defined in the FO body-fixed co-ordinate system (Hamel and Jatega- onkar, 1996; Main and Iliff, 1985; Maryniak, 1985), see Fig.3. Fig. 3. Assumed FO co-ordinate systems and displacements of control surfaces Within the framework of analytical mechanics, we arrive at the following equations of motion (Goszczyński, 2000; Goszczyński et al., 2000; Maryniak, 1985; Sibilski, 1998) ẋd =B −1(VωBxd+FM) (2.1) where xd – dynamical part of the state vector xd = [U,V,W,P,Q,R] > (2.2) B – matrix of inertia Vω – matrix of linear and angular velocities FM – vector of external forces andmoments FM = [ F M ] = [Fx,Fy,Fz,Mx,My,Mz] > (2.3) Practical aspects of identification... 35 and the kinematic relations ẋk =T(xk)xd (2.4) where T denotes the transformation matrix from the FO body-fixed axes to the earth-fixed co-ordinate system xk = [Φ,Θ,Ψ,x1,y1,z1] > and the vector FM is represented as a sumof gravity, thrust and aerodynamic forces andmoments FM =F G M +F T M +F A M (2.5) We assume that the gravity and thrust forces andmoments are known, while the aerodynamic forces andmoments FAM = [Px,Py,Pz,L,M,N] > (2.6) have to be estimated basing on recorded digital signals of FO motion with filtering and smoothing techniques used. These estimates are unknown poly- nomials of the state variables, control function andMach number.Their forms and coefficients are to be identified (Goszczyński et al., 2000). 2.1. A particular mathematical model of an aircraft In a simplified case, we can analyze a rigid and symmetrical aircraft mo- ving through atmosphere whichmoves with a uniform speed over a flat earth (Maryniak, 1985). Using the body-fixed reference frame Oxyz with the origin in the centre of gravity, are obtains equations of motion (2.3) as presented below X =m(U̇ +QW −RV )+mg sinΘ Y =m(V̇ +RU−PW)−mgcosΘ sinΦ (2.7) Z =m(Ẇ +PV −QU)−mgcosΘcosΦ L= IxṖ − (Iy − Iz)QR− Ixz(Ṙ+PQ)+ −ITiωTi(RsinϕTzi+QcosϕTzi sinϕTyi) M = IyQ̇− (Iz − Ix)RP − Izx(R 2−P2)+ +ITiωTi(RcosϕTzicosϕTyi+P cosϕTzi sinϕTyi) (2.8) N = IzṘ− (Ix− Iy)PQ− Izx(Ṗ −QR)+ −ITiωTi(QcosϕTzicosϕTyi−P sinϕTzi) 36 J.A.Goszczyński Equations (2.7) and (2.8) take the form of first order differential equations for, respectively for translational velocities, angular velocities and attitude angles in the body-fixed reference frame Oxyz. The forces X, Y ,Z represent components of the total aerodynamic force, including aerodynamic effects of propulsion systems. L, M, N denote the total aerodynamic moments (inclu- ding any aerodynamic effects of the propulsion system) about the body axes Oxyz.Both components (2.7) and (2.8) definea formof anaerodynamicvector (2.6). Completing equations (2.7) and (2.8) with kinematic relations (2.4) and components of the total aerodynamic force and moment (known also as the aerodynamic model) leads to the full set of aircraft dynamic equations of mo- tion. It is worth noting here that the ”physical” input variables such as displ- cements of control surfaces and engine thrust (or power changes) also serve as inputs to the above set of differential equations as they should appear as independent variables in the aerodynamic model of the flying object. In the written above kinematic model of an aircraft, the measured varia- bles, i.e. specific aerodynamic forces and body rotation rates appear as forcing functions. The specific force is defined here as an external non-gravitational field force per mass unit. The specific forces are variables measured by ”ideal” accelerometers in thebody’s centre of gravity, irrespective ofwhether thebody is influenced by the gravitational field or not (Mulder et al., 1994; Stalford, 1979). In flight tests, such ideal accelerometers would measure the specific aerodynamic forces according to X =Axm Y =Aym Z =Azm (2.9) in which Ax, Ay, Az denote the specific aerodynamic forces along the body reference axes Oxyz. Substitution of (2.9) into (2.7) and division by m leads to the following expressions U̇ =Ax−g sinΘ−QW +RV V̇ =Ay+gcosΘ sinΦ−RU+PW (2.10) Ẇ =Az +gcosΘcosΦ−PV +QU As the mass has been eliminated, equations (2.10) represent a set of what might be called kinematical relations. The two sets of equations, (2.9) and Practical aspects of identification... 37 (2.10), may be solved numerically if the specific forces Ax, Ay, Az and the angular rates P ,Q, R are taken as input variables. We can interpret (2.4), (2.10) as to represent the dynamical system and define the state vector xa = [U,V,W,Φ,Θ,Ψ,x1,y1,z1] > (2.11) as well as the input vector u= [Ax,Ay,Az,P,Q,R] > (2.12) The equation of the system state may be written as ẋa =f(xd,u) (2.13) where f denotes a non-linear vector function of xa and u. While the accelerometers and gyroscopes serve to measure components of the input vector, the barometric and other sensors may be used to measure components of the observation vector. 2.2. The aerodynamic model of a flying object Aerodynamic models are defined as mathematical models of aerodynamic forces andmoments in the body-fixed Oxyz or wind-fixed Axayaza reference frames. Development of aerodynamicmodels fromdynamicflight test data requires an initial ”guess” of themathematical structure of themodel.The initial guess is referredas ana priorimodel, indicating that noflight datawasused to build themodel.Apriorimodels canbebasedonphysical knowledge, semi-empirical databases, results found fromComputational FluidDynamics (CFD) orWind Tunnel measurements. A generalized aerodynamic force (2.6) may be written as follows (Main and Iliff, 1985, 1986; Mulder et al., 1994) PA =PS(α,β)+ ∑ n PδnA (α,β)δn+P p A(α,β)P +P q A(α,β)Q+P r A(α,β)R+ +P pq A (α,β)PQ+P p2 A (α,β)P 2+P pr A (α,β)PR+P q2 A (α,β)Q 2+ (2.14) +P qr A (α,β)QR+Pr 2 A (α,β)R 2 where PS(α,β) is a part of the aerodynamic force depending on the angle of attack and angle of sideslip, P p A , P q A , PrA, P pq A , P p2 A , P pr A , P q2 A , P qr A , Pr 2 A are 38 J.A.Goszczyński parts of the aerodynamic force in function of the roll, pitch and yaw rate, Pδn A are parts of the aerodynamic force depending on aileron, elevator and rudder (δn) deflections. As estimated parameters are likely to be comparedwith results determined fromwind tunnel experiments (or CFD), a standardway of systemmodelling through Taylor series of dimensionless aerodynamic coefficients (Maryniak, 1985) should be used CD =CD0+CDαα+CDα2α 2+CDq qca V +CDδeδe CY =CY0+CYββ+CYp Pb 2V +CYr Rb 2V +CYδaδa+CYδrδr CL =CL0+CLαα+CLu u V +CLq qca V +CLδeδe (2.15) Cl =Cl0+Clββ+Clp Pb 2V +Clr Rb 2V +Clδaδa+Clδrδr Cm =Cm0+Cmαα+Cmq qca V +Cmδeδe Cn =Cn0+Cnββ+Cnp Pb 2V +Cnr Rb 2V +Cnδaδa+Cnδrδr where α and β denote the angle of attack and sideslip, P , Q,R are the roll, pitch and yaw rates, δa, δe, δr are aileron, elevator and rudder deflections, CD, . . . ,Cn are dimensionless aerodynamic coefficients, CDα,CLα, . . . are ae- rodynamicparameterswhichdenote partial derivatives ∂CD/∂α,∂CL/∂α,. . .. 3. Identification algorithms 3.1. Non-Linear Filtering (FN) method The FN theory formulated by Lipcer and Sziriajev (Anderson andMoore, 1984; Lipcer and Sziriajev, 1981; Ocone, 1981) consists in finding a pair of stochastic processes in a non-linear form of Stochastic Differential Equations (SDE) dxt = [a(t,y)+ b(t,y)xt]dt+ c(t,y)dut xt=0 =x0 dyt = [A(t,y)+B(t,y)xt]dt+C(t,y)dwt yt=0 = y0 (3.1) where only the process yt is observed, whereas ut and wt are independent Wiener processes. Practical aspects of identification... 39 Finding a solution to the filtering problem is possible on the following assumptions: a) The right-hand side of SDE (3.1)2 depends linearly on the Unknown Parameters Vector (UPV),which is independentof stochastic excitations (this vector describes the FO in flight, while stochastic terms represent external disturbances). b) The a priori distribution of the UPV is normal. Unknown parameters have often physical or technical meaning, thuswe can determine their li- miting values.However, if it is impossible to determine the range of those parameters, it is reasonable to make the aforementioned assumption. c) The UPV is stochastically independent of theWiener process wt. d) There exists an inverse to thematrix [C>(t,y)C(t,y)]−1, i.e. the stocha- stic disturbances must affect the FO adequately. e) The right-hand side of Eq. (3.1)2 has a strong solution, which imposes the requirement for existence anduniquenessof the classic solution to the ordinary differential equation resulting from Eq. (2.10) when neglecting the noises. On the above assumptions, it can be proved that the conditional expected value is the bestmean square estimator of the non-observed stochastic process (SP) x when observing the process y in the time interval [0, t]. The optimal estimator andminimal error are given by a finite system, i.e. • Filtration tasks have finite dimensions and, therefore, can be realised technically. • The optimal estimator is directly represented by dynamics of the pro- cesses x and y. • The optimal estimation at the instant t+dt results from the optimal es- timation at the instant t, suppliedwith a newobservation in the interval [t,t+dt], which allows for construction of a recursive filter. • The solution is of the on-line type. • When using fast computer systems, it is possible to reach the real-time solution. So as to properly formulate the parameter estimation in terms of the filte- ring problem, the stochastic process x should be stationary and represented by the sameUPV. That directly leads to formulation of a filtering problem in a specific form (Goszczyński, 2000). 40 J.A.Goszczyński 3.1.1. Requirements imposed on the state and output (measurement) vectors For the UPV estimation purposes by means of the NF method, the equ- ation of motion of the FO in flight should be represented in terms of the me- asurement vector, since this is the only information about the real FOmotion we are provided with. Equation (3.1)2 should therefore satisfy the following conditions: • Noises encountered in the course of the state vector measurement are negligible when compared to the external stochastic disturbances affec- ting the FO in flight. If the noises arise also in themeasurement process, the estimation task of both the state vector andUPV are infinitemulti- dimensional (Goszczyński, 2000). • The relation between the state and measurement vectors has the follo- wing linear form y=Hx (3.2) where H is a constant or time-dependent matrix and det(H>H) 6=0 (3.3) Thus, we can rewrite Eq. (3.2) as follows x=(H>H)−1H>y (3.4) By virtue of Eq. (3.4), equation of motion (2.3), representing evolution of the process x, may be presented in terms of the measurement vector. 3.1.2. Application Themodel of a controlled aircraft in 3D-flight (2.3) within the framework of non-linear filtering theory (FN) can be represented in the form ẋd =B −1g(xd, t)+B −1f(xd, t) (3.5) where B – inertial matrix g – gravity and thrust forces vector f – aerodynamic forces vector. Thevector of aerodynamic forces has the following linear formwith respect to the unknown parameters FAM =f(xd, t)=X(xd, t)p (3.6) Practical aspects of identification... 41 Itdetermines the structureofboththevector p, andmatrix X(xd, t), unknown at the moment. Having thematrix X(xd, t) determined, after substitution of Eq. (3.6) into Eq. (3.5), and introducing the formulae for external stochastic disturbances in flight, we arrive at the stochastic equation of motion dxdt = [B −1g(xdt, t)+B −1 X(xdt, t)p] dt+D dωt (3.7) whichwe consider as the observation equation (in theNF theory sense), where ωt ­ 0 is the 6DWiener process representing the influence of stochastic factors on the aerodynamic forces andmoments. Fig. 4. Identification of the lift coefficient CL (FN) 3.2. Estimation Before Modeling (EBM) The EBM consists of the following two-steps (Goszczyński et al., 2000)[7]: Step 1 – estimation of the state vector using a filter; Step 2 – modelling itself, e.g. bymeans of the regressionmethod ẑ=Ap̂+ ε̂ (3.8) where 42 J.A.Goszczyński Fig. 5. Identification of the pitching moment coefficient Cm (FN) ẑ – estimation of the output vector (resulting from the filter) A – estimation matrix of the vector x (cf. the observation ma- trix X inMańczak and Nahorski (1983)) ε̂ – vector of errors with zero mean values and a constant cova- riance matrix. The problem of the model parameters identification is schematically pre- sented in Fig.6. The EBMmethod is one of the equation error methods, with its name adequately representing the order of operations to be performed (Goszczyński et al., 2000). Fig. 6. Overwiew of the Estimation BeforeModelling technique (Stalford, 1979) Practical aspects of identification... 43 Fig. 7. Concept of the EBMmethod (Goszczyński et al., 2000) Acrucial role in theEBMplays the aerodynamicmodelling in terms of the state equation, for the requirements of Kalman’s filter theory to be met. To this end, each component of the vector of aerodynamic forces andmoments is represented in the form of the Gauss-Markov process (i=1, . . . ,6) ẋdi(t)=Ki(t)xdi(t)+Giζi(t) xdi(0)=xdi0 (3.9) where ζi(t) – white (gaussian) noise Gi – output matrix xdi – state vector Ki – state matrix in the form Ki =   0 1 0 0 0 1 0 0 0   (3.10) The state estimates obtained in the first step of the EBM method are the input data for the second step.Therefore, the identification problem is addres- sed in a completely differentway, in contrast to a typical identification process of parameters. In theEBMmethod, a structural identification is performed as well. Selection of the aerodynamic model structure is of crucial importance. Usually, the linear regression technique is used, in which n parameters (N ­ n) are determined from N measurements, and a simple parametrical model in the following form (corresponding to Eq. (3.5)) is assumed yi =Xipi+ei i=1, . . . ,6 (3.11) where 44 J.A.Goszczyński yi – vector of aerodynamic forces or moments of the Nth order Xi – matrix of independent variables of the (N ×n)th order pi – - vector of unknown parameters of the nth order ei – - error vector of the Nth order. Applying the least square method, by virtue of Eq. (3.3) (the rela- tion between the state and measurement vectors is linear) we arrive at the equation p̂i =(X > i Xi) −1 X > i yi (3.12) representing explicitly the identification process. Usually, at high angles of attack, aerodynamic characteristics are strongly non-linear depending on the state and control vectors (2.3) in an unknown way.The function Xi(xd, t) is represented in the formof splines or polynomials with unknown coefficients pi. Basing on the dynamical limitations imposed on all degrees of freedom (flightmodelling), it is possible to estimate xdi0 and the coefficients pi, which completes the first step of the EBM identification method – the state estimation. 3.2.1. State estimation In the first step, realised bymeans of the filtering technique, the extended Kalman filter is applied (Goszczyński, 2000; Goszczyński et al., 2000). The loading introduced this way can be reduced by means of linear smoothing, e.g. by employing the modified Bryson-Frazier filter. An alternative approach consists in application of the smoothing with a constant delay, which may occur to be simpler and less time-consuming, giving at the same time both the smoothing and estimation of the state variable derivatives. 3.2.2. Estimation of parameters The second step of theEBMmethod is reasonably called ”modelling”.This approach gives an insight into mechanical models of flight being currently in use (Goszczyński et al., 2000).Whenever an identification is to bemadewithin the area of substantial changes in values of physical quantities, which of course strongly affect values of parameters, it must be preceded by a proper selec- tion of subdomains. In each subdomain, a separate identification is realised (Batterson andKlein, 1989). Practical aspects of identification... 45 The selection of themodel structure consists inmultiple application of the linear regression technique (3.5) (Goszczyński et al., 2000). It results from the step-by-step introduction and removal of independent variables. An indepen- dent variable, which might be the best single variable at the previous step, could be needless in the next step, which can be checked by using the Fisher- Snedecor test (test F) (Draper and Smith, 1973). The EBMmethod can bemost efficient for determination of aerodynamic characteristics at high angles of attack (Mulder et al., 1994; Stalford, 1979, 1981; Stalford et al., 1977). Several advantages should bementioned (Sibilski, 1998): • A priori estimation of aerodynamic characteristics before modelling al- lows for more accurate determination of input data at the modelling step. • Estimation and identification of aerodynamic derivatives do not require construction of models depending on state parameters. • Simultaneous reconstruction of many manoeuvres leads to better preci- sion in the identification of aerodynamic derivatives. Themostadvantageous featureof theEBMmethodconsists in the fact that the model structure is constructed basing on the measurement of dynamical parameters of the aircraft. Fig. 8. History of the sideslip angle β (EBM) 46 J.A.Goszczyński Fig. 9. History of the pitch angle Θ (EBM) Fig. 10. History of the pitch angular velocity Q (EBM) Fig. 11. Estimation history of the aerodynamic drag coefficient CD = f(t) (EBM) Practical aspects of identification... 47 Fig. 12. Estimation history of the aerodynamic lift coefficient CL = f(t) (EBM) Fig. 13. Estimation of the aerodynamic lift coefficient CL = f(α) (EBM) 4. Conclusions The results of numerical tests of the presented methods are promi- sing. A good convergence of the numerical algorithms and low sensiti- vity to initial errors has been found. These features are hopeful, parti- cularly for aerodynamic characteristics the values of which can be pre- cisely a priori estimated. Investigations of the application of the pre- sented methods to the problem of a six-degree-of-freedom aircraft are being conducted (Goszczyński et al., 2000) based on real flight data records. 48 J.A.Goszczyński References 1. AndersonB.D.O.,MooreJ.B., 1984,Filtracja optymalna,WNT,Warszawa 2. 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Stalford H.L., 1981, High-alpha aerodynamic model identification of T-2C aircraft using the EBMmethod, Journal of Aircraft, 18, 10, 801-809 29. Stalford H.L, Ramachandram S., Schneider H., Mason J.D., 1977, Identification of aircraft aerodynamic characteristics at high angles of attack and sideslip using the Estimation Before Modeling (EBM) technique, AIAA Atmospheric Flight Conference Proceedings, Holywood, FL 50 J.A.Goszczyński Praktyczne uwagi w identyfikacji charakterystyk aerodynamicznych Streszczenie W pracy przedstawiono metodę estymacji przed modelowaniem (EBM), znaną również pod nazwąmetody dwu etapowej identyfikacji charakterystyk aerodynamicz- nych (i ichpochodnych).Przedstawionatechnika jest szczególnieprzydatnado identy- fikacji charakterystyk samolotu poruszającego się na dużych kątach natarcia i ślizgu. Wpracyprzedstawionopodstawowecechy i zależnościmetody.Uzyskanewyniki,wraz z posiadaną wiedzą o zakończonych badaniach innych zespołów, pozwalają określić przedstawioną technikę jako potencjalnie integralną część badań rozwojowych i oceny każdego samolotu. Manuscript received August 4, 2005; accepted for print September 26, 2005