The optimal control law that minimizes the gust responses of an airplane's longitudinal motion is obtained, making assumptions that the airplane is approximated as a point and that aeroelastic problem is ignored. The airplane gust-alleviation problem has been treated mainly in the frequency domain because of the simplicity of the input-output relations for the power spectra. But in the optimal control problem, the approach in the time domain, applying the optimal stochastic control theory, seems to have more advantages for investigating such a complex control as gust-alleviation. The system state equations consist of the short-period equation of an airplane and the gust shaping filter. The optimal linear control law is derived by applying the Matrix Minimum Principle which minimizes the R.M.S. values of the normal acceleration and the pitch rate at the center of gravity. The results of the numerical calculation for two types of control systems, one being the linkage-control system and the other the independent-control system, are shown in the case of a conventional transport. The latter system is ascertained to have a fairly better performance. The optimal system is also ascertained to have very low sensitivity to the change of the scale of turbulence

KOBAYAKAWA, Makoto

MAEDA, Hiroshi

OKUBO, Hiroshi

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:An Application of StochasticControl Theory to the Gust-Alleviation System for a TransportAirplaneOKUBO, Hiroshi; KOBAYAKAWA, Makoto; MAEDA,HiroshiOKUBO, Hiroshi ...[et al]. An Application of Stochastic Control Theory to the Gust-Alleviation System for a Transport Airplane. Memoirs of the Faculty of Engineering, KyotoUniversity 1977, 39(3): 375-3881977-09-21http://hdl.handle.net/2433/281043An Application of Stochastic Control Theory to the Gust-Alleviation System for a Transport Airplane By Hiroshi 0KuBo*, Makoto KoBAYAKAWA** and Hiroshi MAEDA** (Received March 30, 1977) Abstract The optimal control law that minimizes the gust responses of an airplane's longi-tudinal motion is obtained, making assumptions that the airplane is approximated as a point and that aeroelastic problem is ignored. The airplane gust-alleviation pro-blem has been treated mainly in the frequency domain because of the simplicity of the input-output relations for the power spectra. But in the optimal control problem, the approach in the time domain, applying the optimal stochastic control theory, seems to have more advantages for investigating such a complex control as gust-alleviation. The system state equations consist of the short-period equation of an airplane and the gust shaping filter. The optimal linear control law is derived by applying the Matrix Minimum Principle which minimizes the R.M.S. values of the normal acceleration and the pitch rate at the center of gravity. The results of the numerical calculation for two types of control systems, one being the linkage-control system and the other the in-dependent-control system, are shown in the case of a conventional transport. The latter system is ascertained to have a fairly better performance. The optimal system is also ascertained to have very low sensitivity to the change of the scale of turbulence. Nomenclature u0 flgit velocity c mean aerodynamic chord m mass of airplane S wing area 18 moment of inertia of airplane about y-axis p air density g acceleration of gravity µ m pS(c/2) relative mass parameter 375 * Assistant, Department of Aeronautical Engineering, University of Osaka Prefecture, Sakai, Osaka, Japan. ** Department of Aeronautical Engineering 376 Hiroshi OKUBO, Makoto KoBAYAKAWA and Hiroshi MAEDA pS(r/2) 3 non-dimensional moment of inertia t t*=r/2µ t=t/t* : a n, L suffix g: </) X u V time non-dimensional time angle of attack pitch angle pitch rate flap deflection angle elevator deflection angle normal acceleration factor vertical gust velocity circular frequency reduced frequency representative scale of atmospheric turbulence R.M.S. value of the gust velocity non-dimensional quantity quantities due to gusts ,power spectral density state vector control vector white noise d response vector F, G, C, A, B : coefficient matrices J W,Z K X p H T criterion functional (performance index) weighting matrices feed-back gain matrix state second-moment matrix co-state matrix associated with X Hamiltonian function transpose of vectors and matrices I. Introduction The purpose of a gust-alleviation system for an airplane is the reduction of oscillating motion which is generated by gusty air; and it is important from the viewpoint of passenger comfort and structual fatigue. Since atmospheric turbulence affects the airplane as a continuous random disturbance, this problem An Application of Stochastic Control Theory to the Gust-Alleviation System for a Transport Airplane 377 should be treated with methods of statistical reasoning. There have been many analytical investigations of this problem. For exa-mple, K. Nakagawa et al. obtained the performances of several gust-alleviation systems for a typical conventional transport, analyzed by the classical control theory and Chan's optimal control theory in the frequency domain1•2>. Also, R.A. Hess advanced the analysis in the time domain and investigated the feasibility of the optimal gust-alleviation system3>. In this paper, the gust is considered as the stationary random disturbance to an airplane; and a gust-alleviation system for the airplane's longitudinal motion is designed by using the stochastic control theory. II. Gust Response of an Airplane In order to make the analysis simple, the following assumptions and approxi-mations are introduced. i) The airplane is rigid. And the aeroelastic effects on the airplane are ignored. ii) The gust is normal and one-dimensional, so that the span-wise change of the gust is neglected. Under these assumptions, we can derive the longitudinal equations of the motion of an airplane in a horizontal flight condition4>. Between the two kinds of modes which occur in a longitudinal motion, it may be considered that the phugoid mode is not so important for the mean square gust response. Therefore, we will treat only the short-period mode here-after. In addition to the elevator system, the wing-flaps should also be used for control, because it is obvious that the elevator system alone cannot be expected to control the airplane response sufficiently. In other words, to alleviate the oscillat-ing motion induced by gust, a more powerful direct lift control system is required. For the analysis of the gust response of an airplane, some additional assump-tions and approximations are made as follows: If the wave-length of the gust, )., is considerably large relative to the length of airplane, (i.e. ).~8/ or the reduced frequency k-;;;,k0=11:1:/8l, r:; mean aerody-namic chord l; airplane length) the wave-form of the gust can be approximated by the linear segment. Then, the aerodynamic effect of gust velocity wg is equi-valent to the change of angle of attack ag, and the gradient in wg is equivalent to the change of the non-dimensional pitch-rate </.g, ( 1 ) 378 Hiroshi OKUBO, Makoto KoBAYAKAWA and Hiroshi MAEDA ( 2) Furthermore, it is assumed that the gust field is "frozen", i.e. the statistical characteri.stics of the turbulence do not depend on time during the flight through this patch of turbulence. Accordingly, the space derivative may be transformed into the time derivative as follows; q' = _ 8ag = -Da g 8t g ( 3 ) The effective angle of attack (a+ag) and the effective pitch-rate (q+qg) are used to calculate the aerodynamic forces and moments. Then, in the case of the short-period approximation, the equations of an airplane's motion induced by the gust are given as follows: (2µD+C,<»)a -2µq = C,<»ag+C,6161 +C,o,o, -(Cm~D+Cm<»)a+(i8 D-Cm.)q = (Cm~D-Cmp+cm<»)a8 ( 4) +cm6t°t+cm6,0e where C,~ and c,. are assumed to be negligible. Next, since the gust is assumed to be one-dimensional, the power spectrum of gust given by Dryden is available. . However in this analysis, for simplicity the following approximate equation is used instead of the original Dryden modeI1l. where H 2 = 3a~ /411:L I= 0.725t/L and a 8 is the intensity of gust and L is the scale of turbulence. ( 5) In order to advance the analysis in time-domain and apply the stochastic control theory to this problem, "shaping filter" should be introduced (Fig. 1). The gust velocity wg with assigned power spectrum, Eq. (5), can be generated by a white noise through this filter. The transfer function of such a filter is expre-White Noise Shaping Gusts ..... Fi I ter .... H2 ~i?i,=H2 , .,,. G(s) ~ 9= 12+ k2 Fig. I. Shaping filter. An Application of Stochastic Control Theory to the Gust-Alleviation System for a Transport Airp,lane 379 ssed by the following form, if the power-spectral density of the input white noise is H 2• l G(s) =-I+s Therefore, the differential equation which ag satisfies 1s given as follows: where vis a white noise with the co-variance E{v(t)v(T)} =(H2fuii)o(t-T). ( 6) ( 7) Putting the above equations in matrix form, the state equation of the gust-alleviation system is expressed as follows: x = Fx+Gu+Cv ( 8) where x= [ a, 0, q, agF is a state vector and u= [o 1, o,F is a control vector. The matrices F, G and C are given by c,,./2µ 0 l c,,./2µ 0 0 0 F= Cm;,,C,,./2µ+Cm<» 0 Cm;,,+cm. Cm;,,C,l»/2µ+Cm-l(Cm;.-Cmq) ' Zs Zs Zs 0 0 0 -I C,81/2µ C,8)2µ 0 0 0 0 G= Cm;.C,s,f2µ+Cm8f Cm;,.C,s,f2µ+Cms, 'C= Cm;.-Cmq Zs Zs Zs 0 0 0 The normal acceleration and the pitch-rat~ at the center of gravity of the airplane are adopted to assess the performance of gust-alleviation systems. The rea-son is because these quantities are the principal factors which affect passenger comfort or structual fatigue. In order to non-dimensionalize these variables, the normal acceleration factor n, is introduced as follows: n = z normal acceleration l:lt C.G. g ( 9) According to the random process theory for a linear system, the power-spectra of a normal acceleration factor and a non-dimensional pitch-rate, i.e. n, and q 380 Hiroshi OKUBO, Makoto KoBAYAKAWA and Hiroshi MAEDA can be related with the input, or the power spectrum of gust as follows: <l>n,(k) =I:: (jk) r<l>g(k) <I>a(k) =\lg Uk) 12<1>g(k) (10) ( 11) The mean square values of response can be obtained by integrating the power-spectra with respect to k throughout frequency range. III. Gust-Alleviation by Stochastic Control Theory Fig. 2 shows the block diagram of the optimal stochastic control system. Since the gust is random, the state variable X is also a random one. Therefore, the state of an airplane should be estimated by several observed quantities. Howe-ver, in order to foucs our attention on the optimal control problem, it will be assumed that the instantaneous and exact measurements of the state are possible. Random Gust x· ~lfRJ~~ i---vi Estimation ...___U_,,o :...;...t --10ptimum Controller Fig. 2, Brock diagram of a stochastic optimal linear control system. Then, let the performance index be given by W = [WI O ], z = [d O] 0 w~ 0 T~ (12) where E { •} means expected values and W and Z are positive definite weighting matrices. Furthermore, putting d=[n., q]T, this is the response vector given by d=Ax+Bu where [-(2u~) C,,. A= gr 2µ 0 0 0 0 1 -(2µ~) c.,.l gr 2µ , 0 (13) .411 Application of Stochastic Control Theory to the Gust-Alleviation Systemfor a Transport Airplane 381 [- (2~g) C,s1 _ (2ug) C,s, l B = gc 2µ gr: 2µ • 0 0 Accordingly, the optimal control problem can be treated as follows: "Given the state equation of the system expressed by Eq. (8), and assuming that the states may be observed exactly, determine the optimal linear feed-back con-trol law u=Kx which minimizes the criterion functional J given by Eq. (12)." By introducing the state second-moment matrix X; X=E{x•xT}, the co-variance equation of the state equation, Eq.(8), can be derived as follows: (14) Exchanging the order of integration and expectation, the performance index J may be transformed into the following form after some manipulations. J = tr. [ {(A+BK) W(A+BK)T +KzKr}X] (15) In order to solve this problem, the Matrix Minimum Principle by Athans is applied as follows :6l The Hamiltonian function H is introduced into this problem as follows: H = tr. [(A+BK) W(A+BK)TX+KZKTX]+tr. [XP] (16) where Pis the co-state matrix associated with the state second-moment matrix X. If K* denotes the optimal feed-back gain matrix which minimizes J, and if X* is the state matrix which corresponds to K*, there exists a co-state matrix P* that sa-tisfies the following canonical equations and minimizes the Hamiltonian function H. Canonical equations are given as follows: BH I = X* = (F+GK*)X*+X*(F+GK*Y +(H2/u5)CCT = 0 (17) ap * BH I = -P* = (A+BK*)TW(A+BK*)+K*TP*K* ax* +(F+GK*)T P*+P*(F+GK*) = 0 (18) If it is assumed that K is not constrained, then the minimization of the Hamiltonian function H implies the following necessary condition. ( 19) 382 Hiroshi OKUBO, Makoto KoBAYAKAWA and Hiroshi MAEDA From this equation, K* can be solved as follows: (20) In the above equation, the co-state matrix P* is the solution of the following Ric-cati equation of matrix form. [F-G(BrwB+z)- 1Brw AFP*+P*[F-G(BTWB+z)- 1Brw AJ -P*G(BrwB+z)- 1GrP*+ArW[I-B(BTWB+z)- 1BrW]A = o where / is a unit matrix. IV. Examples of Numerical Calculations (21) In order to illustrate the performance of the optimal gust-alleviation system, let us calculate a numerical example of a conventional transport. Flight conditions and stability derivatives of this transport are shown in Table 1. It is assumed that the scale of turbulence L is 300 m and the intensity of gust a g is 2. 7 m/sec. w s In l u Table I. Aerodynamic data of the model transport Dimensions of the Model Transport Airplane weight Wing area Mean aerodynamic chord Moment of inertia of airplane about y-axis Total length Flight velocity Cruising altitude c.,,, = -5.82 Cm,,,= -9.50 Cma1= -0.30 Aerodynamic Derivatives Cmq = -34.8 C,a1= -1.65 Cma,= -1.74 21,800 Kg 94.8 m2 3.2m 50,300 Kgm·sec2 26.3m 459Km/h (123 m/sec) 6,100m Cm.,= -1.83 c.a, = -0.40 In this paper, we calculated two types of control systems. The first one is the linkage-control system (Fig. 3(a)), and in this system it is assumed that the elevator and the wing-flaps are linked with a fixed operation ratio r. The value r should be determined to satisfy the static equilibrium condition of the airplane as follows :1> (22) An Application of Stochastic Control Theory to the Gust-Alleviation System/or a Transport Aiprlane 383 Gusts Airplane Dynamics k X .. _) V U= [t] Gusts I -1, Airplane X Dynamics ,t K / "' Fig. 3 (a). The Linkage-control system. Fig. 3 (b). The Independent-control system. The next one is called the independent-control system (Fig. 3(b)), in which two kinds of controllers, that is, elevator and flap, can be operated independently. The latter is the "exact optimal" system, but the former has the advantage of simplicity in calculation and realization. 103~-----------,-------, N~ u - ------! 2 Zn's Formula 10 en I-Bl 0 (/) a.. ti 10 ::, <..9 10-2 Reduced Frequency k Fig. 4. Power spectral density of gusts; L=300 m, Gg =2. 7 m/sec. 384 Hiroshi OKUBO, Makoto KoBAYAKAWA and Hiroshi MAEDA Fig. 4 shows the power-spectral density of the gust. In this figure it is shown the approximate equation previously mentioned agrees well with Dryden's spectral density for the frequency range of a short-period mode. It should be noted that k0 inidicates the limit point of point approximation. Applying the analyses given above to this example, the optimal control laws can be determined, and the opwer-spectral densities as well as the R.M.S. values of the responses can be calculated for the controlled airplane. First, in the case of the linkage-control system, the power-spectral densities of the normal acceleration factor and those of pitch-rate are illustrated in Figs. 5(a) and 5(b) respectively. From these figures it is found that the optimal gust-alleviation system is remarkably effective for this transport. It is also ob-vious that the power-spectral densities of the responses are decreased with the reduction of r, which is the weighting factor for the control term in the performance index. Fig. 6 shows the relation between the R.M.S. values of gust responses and those of controller (i.e. elevator and flap) deflections. Next, Figs. 7(a) and 7(b) shows the R.M.S. responses of the independent-control system. Compared with the linkage-control system, the effectiveness of this system seems fairly better, which is indicated in the figures. In order to examine the sensitivity of the controlled systems to the change of 0.4-------------~ 0.2 0.1 0 0.02 0.04 ko 0.06 k Fig. 5 (a). Power spectral density of the normal acceleration factor (the Linkage-control system). An Application <if Stochastic Control Theory to the Gust-Alleviation System for a Transport Airplane 385 Q0'.)4,--------------, 0.003 0.002 0.001 0 0.02 0.04 ko 0.06 k Fig. 5 (b). Power spectral density of the pitch ratt> (the Linkage-control system). ----. ~ 0.20 ,--------------, 10. -o 0.16 ~ 0.12 ~ ID ti) C ~ & 0.08 0.04 0 L...,_ _ ......_ _ _,_ __ ~---0 0.5 1.0 1.5 2.0 1 /Y ..__,. 8.; Fig. 6. R.M.S. values of the responses and those of the controller deflec-tions (the Linkage-control system). 386 Hiroshi OKUBO, Makoto KoBAYAKAWA and Hiroshi MAEDA 0.2 -----~-------,-------, 0.1 0 0 2 4 6 ✓(6~) +{E!) (deg.) Fig. 7 (a). R.M.S. values of the normal acceleration factor (the Indepen-dent-control system). 0.02 .------~-----,-------, 0.01 0 ' ' 0 ' ' ' ', ', ', ', ___ ,_',+--------l--------i r'-------~i~~~ontrolSystem I 2 4 6 V{o:) + (6!) (deg.) Fig. 7 (b). R.M.S. values of the pitch rate (the Independentcontrol system). gust wave-length, the effects of scale of turbulence L to the responses are indicated in Figs. 8(a) and 8(b). It is found that the responses of optmally controlled sys-tems have not only lower levels at the nominal value of L (i.e. L0=300 m), but also have a lower sensitivity for the change of the scale of turbulence. V. Conculsion Applying the stochastic control theory, we obtained the optimal control law that minimizes gust responses of an airplane under some assumptions and approxi-mations. An Application ef Stochastic Control Theory to the Gust-Alleviation System for a Transport Airplane 387 0.3 .---------,.-------, 0.1 1--------+-----------l 0 -50 ~ = 5.0 3.0 0 (L- L.)/L, x10" (°lo) Lo= 300.0 m/sec. +50 Fig. 8 (a). Sensitivity of the gust-alleviation systems to the change of the scale of turbulence; normal acceleration factor. 0.03 .-----------.-------, 0.01 l--------+--------1 ~=5.o 0 ( L - Lo )/L. x 1 0.. ( °lo) L.=300.0 m/sec. +50 Fig. 8 (b). Sensitivity of the gust-alleviation systems to the change of the scale of turbulence; pitch rate Two kinds of control systems are investigated, 1.e. one 1s the linkage-control system, and the other is the independent-control system. In the case of the conven-tional transport used in the numerical calculation, the former system shows about 70 % reduction of the normal acceleration factor in the R.M.S. value as compared with an unalleviated airplane. This performance of alleviation has the same order as those investigated by Nakagawa.1l On the other hand, utilizing the latter type of system (i.e. the independent-control system), more than 92 % reduction in the normal acceleration factor is possible, even if th~ flap deflection is restricted within 4 degrees in the R.M.S. value. However, the practical realization of this system seems to be difficult, because the on-line measurement of gust velocity and the exact determination of the power-spectral density of gust are necessary. To overcome the first difficulty, an alterna-tive sub-optimal system was suggested by Hess. In this system, instead of a direct physical measurement of gust, Eq. 9 is utilized to construct ax indirectly by mea-suring n,, a, o, and o 1 . 388 Hiroshi OKUBO, Makoto KoBAYAKAWA and Hiroshi MAEDA The scale of turbulence, L, is also a quantity difficult to measure. However, in this paper it is shown that the change of L scarcely influences the performance of gqet-alleviation for the present optimal system. Acknowlegements The authors are deeply indebted to Prof. K. Nakagawa for his kind discussions and advice. References 1) Nakagawa, K., "An Analysis of Gust-Alleviation System for Transport Airplanes", Transactions of the Japan Socie!)! for Aeronautical and Space Sciences, Vol. 7, No. 10, 1964, pp. 20-29. 2) Nakagawa, K., Murotsu, Y., Tsumura, T., and Fujiwara, N., "On the Most Feasible Configura-tion for Airplane Gust-Alleviation System", Journal of the Japan Socie!)! for Aeronautical and Space Sciences, Vol. 18, No. 197, June 1970, pp. 214-222. 3) Hess, R.A., "An Application of Optimal Stochastic Control Theory to the Problem of Aircraft Gust Alleviation", Ph. D. dissertation, Aug. 1970, University of Cincinnati, Cincinnati, Ohio. 4) Etkin, B., Dynamics of Flight, Ch. 10, John Wiley & Sons, 1959, pp. 310-340. 5) Lee, Y.W., Statistical Theory ef Commwzication, Ch. 13, John Wiley & Sons, 1960, pp. 323-354. 6) Athans, M., "The Matrix Minimum Principle", Information and Control, Vol. 11, 1968, pp. 592-606. 

An Application of Stochastic Control Theory to the Gust-Alleviation System for a Transport Airplane

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An Application of Stochastic Control Theory to the Gust-Alleviation System for a Transport Airplane

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