Extremum Seeking Based on Atmospheric Turbulence for Aircraft Endurance

Transcription

1 JOURNAL OF GUIANE,ONTROL, AN YNAMIS Vol. 34, No. 6, Noveber eceber 20 Extreu Seeking Based on Atosheric Turbulence for Aircraft Endurance Jaes P. Krieger and Miroslav Krstic University of alifornia, San iego, La Jolla, alifornia OI: 0.254/ Traditional extreu seeking deends on adding a erturbation to the control inut, but it is untenable to continuously erturb the throttle in a controller eant to iniize fuel consution. Insired by a recent alication of extreu seeking to a fusion reactor where internal noneriodic erturbations were eloyed in the seeking rocess, a novel variant of extreu seeking is roosed that uses naturally occurring stochastic disturbances in lieu of the traditionally added erturbation signal. Relying on airseed erturbations fro atosheric turbulence to reveal the local sloe of the drag curve, the schee induces a gradient descent to the iniu drag seed. Using stochastic averaging, it is roven analytically that the extreu-seeking controller stabilizes airseed to the iniu drag seed, with an average offset roortional to the third derivative of the drag curve and the variance of the airseed. Brief siulation results illustrate the erforance of the algorith. Noenclature A = asect ratio a = stochastic disturbance ost-saturation scaling factor B = tiescale-shifted and scaled Brownian otion b = engine thrust roortionality constant = coefficient of drag i = induced drag coefficient 0 = zero-lift drag coefficient L = coefficient of lift 2, 4 = averaging constants = drag = recentered drag function e = Oswald efficiency nuber (also base of natural logarith) g = acceleration due to gravity J = Jacobian of a syste k ES = extreu-seeking gain k i = integral gain k = roortional gain L = lift L u, L v, L w = characteristic lengths of turbulence field (longitudinal, lateral, vertical) = ass of the aircraft n 0, n, n 2 = coefficients of the assued for of q = stochastic disturbance resaturation scaling factor S = reference area t = tie U 0 = noinal airseed for turbulence odel u = throttle osition V = airseed v = ground seed v = iniu drag seed ^v = estiate of iniu drag seed W = standard Brownian otion = turbulence tie constant Received 27 January 20; revision received 29 Aril 20; acceted for ublication May 20. oyright 20 by Jaes Krieger and Miroslav Krstic. Published by the Aerican Institute of Aeronautics and Astronautics, Inc., with erission. oies of this aer ay be ade for ersonal or internal use, on condition that the coier ay the $0.00 er-coy fee to the oyright learance enter, Inc., 222 Rosewood rive, anvers, MA 0923; include the code / and $0.00 in corresondence with the. Graduate Student, eartent of Mechanical and Aerosace Engineering, 9500 Gilan rive, Mail ode 04; jkrieger@ucsd.edu. Professor, eartent of Mechanical and Aerosace Engineering, 9500 Gilan rive, Mail ode 04; krstic@ucsd.edu. 876 = wind seed, resaturation d = invariant distribution of = air density = integrator value u, v, w = root-ean-square turbulence intensity (longitudinal, lateral, and vertical coonents) H = siulation high-ass-filter tie constant L = siulation low-ass-filter tie constant ug, vg, = turbulence sectra (longitudinal, lateral, vertical) wg = tiescale-shifted wind seed = satial fruency! = teoral fruency Subscrit = uilibriu value Suerscrits a = average syste variable e = error variable Introduction EXTREMUM seeking is traditionally erfored by adding a erturbation signal to the setoint of a syste. The erturbation signal is usually a sinusoid [ 3] but can also be nonsinusoidal [4] or stochastic [5,6]. It is also ossible to use naturally occurring disturbances in lieu of an added erturbation signal [7]. Here, this aroach is taken to otiize the seed of an aircraft for axiu endurance: that is, to axiize the length of tie an aircraft can stay aloft. onvergence to the otiu seed is roven analytically and shown in siulation. Over the ast decade, extreu seeking has been adated to any different alications, including antilock braking [8], article bea atching [9], axial coressors [0], lean reixed cobustion [,2], flow control [3], bioreactors [4], tokaak fusion devices [5], and foration flight [6]. Extreu seeking ay also be alied to an aircraft otiizing airseed for best ossible endurance. Aircraft wings have an otial angle of attack that rovides a axiu lift-to-drag ratio. All other factors being ual, a jet aircraft flying at the seed that achieves this angle of attack burns fuel ore slowly than when flying either slower or faster than this otial seed. This otial seed is calculated during the design of the aircraft based on wind-tunnel data; however, the otial seed for any articular aircraft varies fro the calculated value to soe degree. The otial seed varies based on any factors, fro

2 KRIEGER AN KRSTI 877 anufacturing differences to the condition of the wing. Accuulated bugs, nicks and dents can all change the otial angle of attack, which changes the airseed for otial endurance. Extreu seeking ay be used to find the otial airseed for the current condition of each individual aircraft as it flies. A recent survey of extreu seeking is given in [7]. While traditional, erturbation-based extreu seeking is a ossibility, there are otential disadvantages to using this technique. The first is the ossibility that the act of introducing the airseed erturbation would decrease endurance. Periodically changing the coanded airseed would cause the throttle coand to oscillate, which quite ossibly would use ore fuel than a ore steady throttle coand. While finding the otial airseed would decrease drag and irove endurance, an oscillating throttle coand could use ore fuel and hurt endurance. The second disadvantage to using traditional extreu seeking has less to do with technical erforance than aviation adinistration. It ay be seen as undesirable for the seed of an aircraft to be continuously varying. Traditional extreu seeking alters the observable otion of the aircraft, which ay act as an iedient to its ileentation. Used herein is a for of extreu seeking that relies on naturally occurring disturbances, rather than anually added erturbations, to avoid these disadvantages. Aircraft in flight on occasion encounter turbulence, which acts as a stochastic disturbance in airseed. By taking advantage of this, a turbulence-based extreu-seeking algorith can avoid the disadvantages of traditional extreu seeking. If the throttle is used to control airseed in resonse to turbulence, no objection is raised; it is exected that airseed will be controlled. oing so does not use any additional fuel. The second disadvantage can be answered siilarly; since the aircraft s seed is only being erturbed by turbulence, the observable otion of the aircraft is not altered. These two reasons ay ake turbulence-based extreu seeking a good fit for this alication. It should be entioned that steady level flight is not necessarily otial, but is the only for of flight considered in this aer. Periodic flight consisting of alternating higher-seed clibs and lower-seed glides can achieve endurance suerior to steady flight [8 2]. Since such eriodic flight would alter the observable otion of the aircraft even ore than traditional extreu seeking, only the case of steady flight at a given altitude is considered. This aer is organized as follows. First, the relevant aerodynaics are reviewed. Next, the dynaic aircraft odel is develoed and the extreu-seeking control law is designed. An analysis of the stability of the syste is then resented, followed by siulation results. A discussion is given and then lastly, soe concluding rearks. 0 i (3) Induced drag is coonly aroxiated as having a quadratic deendence on the coefficient of lift [23]: i 2 L Ae Level Flight Using these exressions for lift and drag, the case of level flight is analyzed. To aintain a constant altitude, the lift roduced by the aircraft ust ual the weight of the aircraft. For a given altitude and airseed, this ilies a ruired value of L. This value of L is given by L g 2 V2 S So, the L ruired for level flight is a function of seed. Fro Eqs. (3) and (4), it is aarent that this ilies a certain coefficient of drag. Substituting this coefficient of drag into Eq. (2) gives the drag for level flight as a function of seed: 0 2 V2 S g2 Ae 2 V2 S Here, the first ter reresents the zero-lift drag (also called arasite drag) and the second ter is the induced drag. These two ters are lotted along with the total drag in Fig.. Turbulence Atosheric turbulence is tyically odeled as filtered Gaussian white noise. The three coonents of turbulence (i.e., longitudinal, lateral and vertical) are odeled indeendently. The ryden turbulence odel is one coonly used atosheric turbulence odel [24]. It secifies the sectra of the three coonents of turbulence as follows: ug 2 u vg 2 v wg 2 w 2L u L v L w (4) (5) (6) L u 2 (7a) 3L v 2 L v 2 2 3L w 2 L w 2 2 (7b) (7c) Aerodynaics This section resents the silified aerodynaic odel that is used in the following analysis. Additional background can be found in [22]. Readers cofortable with aerodynaics and turbulence odeling ay roceed to the control design below, noting that for silicity the vertical coonent of turbulence is ignored total drag arasite drag induced drag Tycial rag urve Lift and rag Lift is a function of air density, airseed, a coefficient of lift, and a reference area. The exression for lift is L L 2 V2 S () The coefficient of lift is a function of angle of attack, and is roughly linear for sall angles. rag is calculated siilarly, only using a drag coefficient. rag is calculated as 2 V2 S (2) The coefficient of drag is coonly calculated as the su of two arts: the zero-lift drag coefficient and the induced drag coefficient: rag Force ounds Airseed feet/sec Fig. rag V in level flight.

3 878 KRIEGER AN KRSTI The sectra are given in ters of satial fruency, which is converted to teoral fruency! by ultilying by the seed of the aircraft:! U 0 (8) At ediu to high altitudes (above 2000 ft) the turbulence is assued to be isotroic. The characteristic lengths and the intensities in each direction are ual to each other. A tyical characteristic length is 750 ft. Intensities are charted as a function of altitude. Moderate turbulence has a root-ean-square intensity of about 0 ft=s at 2000 ft, decreasing roughly linearly to near zero at 60,000 ft. Whereas lateral turbulence has little effect on the seed of an aircraft, longitudinal turbulence has a direct effect on airseed. Longitudinal turbulence with a sectru atching that given in Eq. (7a) can be obtained by assing white noise through a filter of the for s 2L u u L U u (9) 0 U 0 s Vertical turbulence has an indirect effect on airseed, but for this analysis it is ignored. ontrol esign ynaic Model Based on the aerodynaic odel resented above, a sile dynaic odel is considered: a scalar syste erturbed by turbulence. The syste is of the for dv Vbu d qdw (0a) (0b) The wind seed is defined as a headwind being ositive, so that airseed is the su of groundseed and wind seed. More recisely, airseed is considered to be given by V v asat () where asat is the wind seed. The saturation function ( if << sat if if (2) is introduced for atheatical convenience. Its sole urose is to ensure that the wind seed is bounded, which is a ruireent for the stochastic averaging results used below. However, it is quite reasonable to bound the wind seed because it is not hysically ossible for the wind seed to be unbounded. The effect of the sat function can be ade negligible by choosing q sall with resect to one. The constant a can then be chosen to give the desired wind alitude. The wind seed is odeled as filtered Gaussian white noise, as er the ryden longitudinal turbulence odel. The tiescale of the turbulence ( L u =U 0 ) is taken to be a constant, using an airseed reresentative of the range of airseeds exected to be encountered. Without loss of generality, the wind seed is taken as zero ean. If there were a steady-state coonent to the wind, then v would reresent the ground seed lus the steady-state wind coonent, but the dynaics of the syste would reain the sae. The rate of change of v is deterined fro total drag, engine thrust, and the ass of the aircraft. Engine thrust is odeled as roortional to the control inut, naely throttle osition. The drag function is taken as a general convex a with a iniu at seed v. It is assued that a roortional integral control law is used to control airseed to a setoint ^v. The control law is written as u k ^v Vk i d ^v V (3a) (3b) obining Eqs. (0) and (3), the aircraft odel is written as follows: dv Vbk ^v Vk i d ^v V d qdw (4a) (4b) (4c) In this odel, a, b, k i, k,, q, ^v, and are ositive. The airseed V can be easured by a itot-static syste. It is considered that dv=, the acceleration of the aircraft, can also be easured. Further, it is assued that the thrust roduced by the engine (i.e., bu) and the ass of the aircraft are known. In the control design below, this is used. The aircraft odel in Eq. (4) reresents the quasi-steady dynaics of the aircraft with altitude tightly controlled. The underlying assutions of this odel are that altitude is being controlled with the elevator, and that the altitude resonse is uch faster than the airseed resonse. The assutions are chosen to be consistent with a jet aircraft in slow flight. Using elevator to control the airseed of a jet in slow flight can result in substantial loss of altitude, so altitude is aintained with elevator and throttle is used to control airseed. The throttle resonse of a jet aircraft, esecially at a low throttle setting, is slow. Because of this, the airseed controller is assued to have a uch lower bandwih than the altitude controller. Turbulence-Based Extreu Seeking The goal is to otiize endurance. It is assued that fuel consution is an increasing function of thrust and that the thrust vector is level with the flight ath. Then, otiizing endurance is uivalent to iniizing throttle osition. That is, the cost to be iniized is the control inut u. Norally, this would ruire adding a erturbation to the setoint of the syste, ^v. By odulating the cost with this erturbation signal, the rate of change of the cost with resect to ^v would be estiated. The setoint of the syste would then be udated using an integral control law with a gain roortional to the estiated gradient. It is desired to do the sae thing, but without adding a erturbation signal. Observe that the goal of iniizing u is, ore secifically, the goal of iniizing u in steady flight. Note that in steady flight Eq. (0a) reduces to Vbu, so iniizing u in steady flight is uivalent to iniizing V. Also, in steady flight Eq. (3b) reduces to ^v V. The goal can then be stated as choosing ^v such that ^v is iniized. While drag is not directly easurable, fro Eq. (0a) it is seen that drag is indirectly easurable. Using knowledge of vehicle acceleration, engine thrust, and ass, drag V is calculated as Vbu dv (5) It is desired to deodulate this signal with the erturbation velocity. If the aircraft has a easureent of ground seed, say fro an inertial navigation syste, and a easureent of airseed, the difference of the two could serve as a easureent of the erturbation. If the aircraft is not so uied, it is ossible to obtain an aroxiation of the airseed erturbation fro the error signal feeding the control law ( ^v V). Here, the latter aroach is taken. As in traditional extreu seeking, an integral control law is used to udate the syste and a gain roortional to this deodulated signal is chosen:

4 KRIEGER AN KRSTI 879 Fig. 2 Block diagra of syste and extreu-seeking feedback. The algorith uses easureents of dv=, bu,, and v asat, but not of v and alone. d^v k ES ^v V bu dv (6) The syste and extreu-seeking control law are shown in Fig. 2. Analysis In this section, analysis of the closed loo syste fored by cobining the aircraft odel (4) with the extreu-seeking control law (6) is erfored. For analysis, the exression (bu dv=) in the control law is written sily as V. The exression for V fro Eq. () is also substituted throughout. This roduces the following set of uations: dv v asatbk ^v v asat k i d^v d ^v v asat k ES ^v v asatv asat d qdw (7a) (7b) (7c) (7d) Theore : onsider syste (7) coosed of the aircraft odel (0 3) and araeter udate law (6), where is a convex a with a iniu at v, all constants are ositive, and K ES 20; bk =, where is a known uer bound on iniu drag, v. Let constants 2 and 4 be defined by 2 q q2 2 erf q q e q 2 erf q 4 q 3 4 q4 erf q q e q q2 erf q (8a) (8b) Suose is three ties differentiable at v. Then there exists a constant a such that for any 0 <a<a there exist constants r>0, c>0, and >0 and a function T: 0; 0!N with the roerty li!0 T, such that for any initial condition j ;a 0j <r, and any >0, li!0 infft 0: j ;a tj >cj ;a 0je t Oa 3 g; a:s: and where li!0 Pfj ;a tj cj ;a 0je t Oa 3 (9) 8 t 20;Tg (20) 0 0 vt v 6 ;a B v t A B i ^v t v v 4 00 v 2 a 2 00 v 2 2 a v 4 00 v 2 a 2 Theore roughly states that choosing the gain K ES as a sall ositive nuber causes the average airseed to converge (both alost surely and in robability) to the iniu drag seed, with a sall bias. The conditions for this are that the turbulence alitude is sall and the initial state of the aircraft is sufficiently close to the final average uilibriu. The reainder of this section is dedicated to the roof of Theore. These uations are transfored into error variables that are exected to converge to near zero. Stochastic averaging is used on this error syste to find the average syste. Then the uilibriu of the average syste is found and the stability of the uilibriu is tested. Throughout the analysis, drag is treated as a general convex a. No assutions are ade about the exact nature of the nonlinearity. Error Variables efine the error variables v e v v e v ^v e ^v v A (2a) (2b) (2c) The fraction v = is the uilibriu value of the integrator when the aircraft is flying at the iniu drag seed with no disturbances. Also define tt (22a)

5 880 KRIEGER AN KRSTI Then the error syste is Bt Wt dve ve v asatt= b k ^v e asatt= k i e v d e ^v e asatt= (22b) (23a) (23b) Z dve;a d^ asatdbk ^ k i e;a d e;a ^ (29a) (29b) k ES ^ asat asatv d asatdbk ^ k i e;a (29c) d ^v e d ^v dv k ES^v e asatt=v e v asatt= v e v asatt= b k ^v e asatt= k i e v d tt qdbt (23c) (23d) Stochastic Averaging Using stochastic averaging [5], the average syste is then Z dve;a v asatd Z b k ^ asatd k i e;a v (24a) d e;a ^ asatd d ^ k ES ^ asat v asatd v asatd b k ^ (24b) asatd k i e;a v (24c) where the invariant distribution of is given by d e 2 q 2 d (25) q To silify these uations, a function is introduced. This function is the sae as, but recentered around the iniu drag oint: asat v asatv (26) This new drag function is zero when its arguent is zero: 00 (27) The average syste (24) is silified using and noting that asatd0 (28) This can be seen since sat is an odd function and is an even function, aking the integrand odd; the integral fro to 0 cancels the integral fro 0 to. The average syste becoes the following: Equilibriu of the Average Syste To find the uilibriu of the average syste, first observe fro Eq. (29b) that the uilibriu value of ^ is zero, that is, ^ 0 (30) Next, fro Eq. (29c) it is seen that at uilibriu, k ES ^ asat asat dv e;a v d 0 (3) Here, it has been noted that the second ter in Eq. (29c) is just d =, which ust be zero at uilibriu. After silification, which eloys Eqs. (28) and (30), the exression (3) reduces to sat asatd0 (32) To solve this uation for, the rocess is siilar to the roof of stability for a general nonlinear dynaic syste resented in [5]. Two exansions are used: one for in a and one for in in the for. Take n 0 n a n 2 a 2 n 3 a 3 Oa 4 (33) Also use an exansion of the drag function in ters of owers of its arguent. enter the exansion around n 0. v n 0 0 n 0 v n 0 00 n 0 v n n 0 v n 3! 0 3 Ov n 0 4 (34) Using these exansions for and, the condition for uilibriu in Eq. (32) becoes sat asatd sat n 0 0 n 0 n a n 2 a 2 n 3 a 3 Oa 4 asat 00 n 0 n a n 2 a 2 n 3 a 3 Oa 4 asat n 0 n 3! a n 2 a 2 n 3 a 3 Oa 4 asat 3 On a n 2 a 2 n 3 a 3 Oa 4 asat 4 d 0 n 0 2 a 00 n 0 2n 2 a 3 2n 2 a n 0 3n 2 3! 2 4 a 3 Oa 4 0 (35)

6 KRIEGER AN KRSTI 88 Here, the following integrals have been used: sat 2k d0; where k 0; ; 2;... (36a) sat 2 d q2 2 erf q q e q 2 erf q 2q (36b) sat 4 d 3 4 q4 erf q q e q q2 erf q 4q (36c) To hel understand the exressions for 2 and 4, note that 2 and 4 can be aroxiated by 2 q q2 2 4 q 3 4 q4 (37a) (37b) for sall q. For q saller than 0.5, the aroxiation for 2 is accurate to within % and the aroxiation for 4 is accurate to within 3%. Both 2 and 4 are zero when q is zero, and onotonically aroach as q grows without bound. Because the exression in Eq. (35) is a olynoial in a, and the olynoial is ual to zero, each of the coefficients of a ust be zero. This gives the following syste of uations: a : 0 n a 2 : 00 n 0 n 2 0 a 3 : 00 n 0 n n 0 3n 2 3! (38a) (38b) (38c) Since is assued convex, its first derivative can only be zero when evaluated at a iniu oint. Noting this, and that has been defined with its iniu at zero, the solution to this set of uations is so an Oa 3 exression for 6 n 0 0 n 0 n is 4 2 (39a) (39b) (39c) q q a2 Oa 3 (40) Recalling Eq. (37), it is noted that for sall q the following is obtained: q2 a 2 Next, using the exression (40) for, Eq. (29a) is set ual to zero and solved for e;a. This tie a lower-order exansion for is used: v 0 0 0v 00 0 v 2 Ov 3 (4) The function has been defined so that it has a iniu at the origin, so the first two ters of the exansion are zero and the exression is silified to v 00 0 v 2 Ov 3 (42) Using these exressions for and, the condition for uilibriu in Eq. (29a) becoes asatdbk ^ k i e;a 00 0 n a 2 Oa 3 asat 2 On 2 a 2 Oa 3 asat 3 d e;a 00 0 a 2 Oa 3 e;a 0 (43) which gives e;a a 2 Oa 3 (44) So an uilibriu of the average syste in ters of the error variables has been found. ollecting the exressions in Eqs. (30), (40), and (44), the uilibriu is exressed as the following: 6 e;a a 2 Oa a 2 Oa 3 ^ 0 (45a) (45b) (45c) onverting back to the original variables, the uilibriu of the average syste is the following: v a v 000 v a 2 Oa 3 v 2 a v 00 v 2 ^v a v a 2 a 2 Oa 3 (46a) (46b) (46c) For sall disturbance agnitudes (i.e., sall a) the average seed of the syste has an uilibriu oint at the iniu drag seed, v. The deviation fro the iniu drag seed is roortional to the third derivative of the drag curve at the iniu drag seed and decreases with a 2. If the drag curve is asyetric, the seed bias is to the flatter side of the drag curve relative to the iniu drag oint. Stability of the Equilibriu To deterine the stability of the uilibriu, the average syste (29) is linearized around the uilibriu oint and the linearized syste is tested for stability. The Jacobian of the average syste (29) at the uilibriu oint in ters of the error variables ; e;a ; ^ is 0 J ; bk 0 0 J 3; J 3;3 where J ;, J 3;, and J 3;3 are given by A (47a)

7 882 KRIEGER AN KRSTI J 3; J 3;3 J ; 0 asatd (47b) k ES ^ asat 0 asatd 0 asatd (47c) k ES asatv d bk (47d) The characteristic uation of the syste is given by detsi J0, or s J ; bk 0 s bk J i 3; s J 0 (48) 3;3 Writing this as a olynoial, the characteristic uation is as follows. s 3 J ; J 3;3 s 2 J ; J 3;3 bk J 3; bk i s J ; J 3; 0 (49) By alying Routh s criterion, it is seen that the syste is stable if each of the coefficients in the characteristic olynoial are ositive and the roduct of the s 2 and s coefficients is greater than the roduct of the s 3 and s 0 coefficients. To test this exressions for J ;, J 3;, and J 3;3 are ruired. Exressions accurate to Oa 3 are found for each. First, J ; is found. Using the exressions n 2 a 2 Oa 3 and 0 v 00 0v v 2 Ov 3 the following is obtained: J ; 0 asatd 00 0n 2 a 2 Oa 3 asat n a 2 Oa 3 asat 2 On 2 a 2 Oa 3 asat 3 d 00 0n a 2 Oa 3 (50) Next, using the sae exression for and 0 v 00 0v Ov 2 the exression for J 3; is found. Note that the second ter in the original exression for J 3; is the negative of J ; : J 3; k ES ^ asat 0 asatd 0 asatd k ES asat 00 0n 2 a 2 Oa 3 asat On 2 a 2 Oa 3 asat 2 dj ; k ES n a 2 Oa 3 (5) Finally, an exression for J 3;3 is found. Here, the aroxiations 0 Oa 2 and v 00 0 v 2 Ov 3 are used. Then J 3;3 can be exressed as J 3;3 k ES asatv d bk 00 0 k ES Oa 2 asat 2 OOa 2 asat 3 v d bk k ES v bk k ES 00 0 a 2 Oa 3 (52) Now, using these exressions for the coonents of the Jacobian of the syste, it is ossible to calculate the coefficients in the characteristic uation and test for stability: s 2 : J ; J 3;3 00 0n k ES v bk bk k ESv k ES s : J ; J 3;3 bk J 3; 2 a 2 Oa 3 k ES 00 0 a 2 Oa n a 2 Oa 3 (53) 00 0n k ES v bk bk k ES Oa 3 bk k ES s 0 : J ; J 3; bk i 00 0n bk i k ES Oa 3 2 a 2 Oa k ES a 2 Oa n a 2 k ESv 00 0n a 2 Oa 3 (54) 2 a 2 Oa n a 2 k ES a 2 Oa 3 (55) So the characteristic uation of the average syste can be written as bk s 3 k ESv 00 0n k ES a 2 Oa 3 s 2 bki k ESv 00 0n bk k ES a 2 Oa 3 s bki k ES a 2 Oa 3 0 (56)

8 KRIEGER AN KRSTI 883 or, ore sily, as bk s 3 k ESv Oa 2 s 2 bki Oa2 s bki k ES a 2 Oa 3 0 (57) The first ste in checking for stability is to ensure that all of the coefficients of this olynoial are ositive. Since all of the constants in this olynoial are assued ositive, for sall a the coefficients are all ositive if 0 <k ES < bk v (58) It ust also be ensured that the roduct of the s 2 and s coefficients is greater than the roduct of the s 3 and s 0 coefficients: bk k ESv > Oa 2 bki bki k ES a 2 Oa 3 Oa2 By aking an Oa 2 aroxiation, this reduces to (59) bk k ESv >Oa 2 (60) which is true for sall a if Eq. (58) is satisfied. So, by Routh s criterion, the linearized average syste is stable for sall a if k ES is chosen sall enough to satisfy Eq. (58). The stability of the linearized average syste ilies stability of the nonlinear erturbed syste. Because the linearized average syste is stable, the nonlinear average syste (29) is locally exonentially stable (orollary 4.3 in [25]). Fro the stochastic averaging results in [5], exonential stability of the average syste ilies that the syste is weakly exonentially stable under the rando erturbation. This result is foralized in Theore. The stability analysis concludes that turbulence-based extreu seeking stabilizes the aircraft to the the seed for otial endurance. Siulations A siulation run using Siulink is resented here. The siulation follows Eq. (7) closely, with the addition of a high-ass filter and a low-ass filter. Although not necessary for stability, the use of a highass filter has been shown to irove the rate of convergence in other extreu-seeking alications [5]. Here, the high-ass filter is alied to the drag signal in the extreu-seeking loo. A low-ass filter is also added to sooth the controller. The filters are shown in Fig. 3. The araeters used in siulation are listed in Table. A lot of the siulation results is shown in Fig. 4. For a direct coarison with the above analysis, a siulation without the high-ass and low-ass filters is resented in Fig. 5. The siulation araeters are the sae as in Table, excet that the filter tie constants are not alicable and k ES ust be chosen uch saller. The stability liit for k ES can be calculated using Theore and the siulation araeters in Table. For the siulation k ES was chosen to be one-eighth of its stability liit. Nuerical values for the stability liit of k ES and k ES itself are shown in Table 2. Also shown in Table 2 are the redicted uilibriu oint and Jacobian atrix of the average syste, corresonding to Eqs. (46) and (47), resectively. The Jacobian eleents J ; J 3; and J 3;3 are calculated using the Oa 3 aroxiations given in Eqs. (50 52). iscussion The turbulence-based extreu-seeking algorith successfully stabilizes the aircraft odel to the airseed for otial endurance, with an average bias roortional to the third derivative of the drag curve and the square of turbulence intensity. The extreu-seeking controller does this without adding a erturbation to the setoint of the syste. Siulations conducted using a quadratic drag olar show little bias fro the true iniu. The drag curve used in siulation is the sae shown in Fig.. Note that while a quadratic drag olar is used, the drag curve is a nonolynoial function of airseed and contains higher order ters sufficient to investigate the bias of the estiated iniu drag seed redicted by Theore. The siulation deonstrates erforance in turbulence with a root-ean-square alitude of 3 ft= sec, which reresents light to oderate turbulence at ost altitudes. This raises the question of how the controller will erfor in other turbulence intensities. As shown in analysis, the steady-state bias iroves as the turbulence intensity decreases. This is liited only by the fidelity of the acceleroeter reading and the accuracy of the thrust and weight estiates. However, for any given k ES the rate of convergence to the iniu drag seed also decreases with the turbulence intensity. Because encounters with turbulence are tyically fairly short, on the order of inutes, it is desirable to increase the rate of convergence as Fig. 3 Block diagra of the control syste as siulated, including added filters.

9 884 KRIEGER AN KRSTI Table Siulation araeters Table 2 Paraeters for siulation without filters Paraeter Value Units u 3 ft=s L u 750 ft U 0 42 ft=s a 49 ft=s q s = s g 4,300 lb 444 slugs b 00 lbf= deg k 2.22 deg =ft=s k i 0.0 deg =ft=ss k ES ft=s=s=ft=slbf H 2 s L 5 s V 5:7 0 6 V 2 0:026=V 2 lbf (as in Fig. ) v 42.2 ft=s uch as ossible. This can be accolished by increasing k ES. are should be taken, though, not to increase k ES too uch. The analytical stability result fro Theore is stated in ters of the liiting case as a tends to zero. It is exected that the uer stability bound on k ES will decrease as a increases, ilying that k ES should be chosen conservatively to revent the syste fro becoing unstable in severe turbulence. The issue of rate of convergence also deonstrates the necessity of using the high-ass and low-ass filters. The filters irove the rate of convergence by orders of agnitude. Indeed without the, the control design is not ractical. The analysis above considers the siler control design for iroved clarity in the analysis. Oitting the filters kees the basic style of analysis and functionality of the controller fro being lost in the details. A liitation of this work is that vertical gusts have been ignored. The results in [26], which show the otential robustness of extreu-seeking control to stochastic disturbances, suggest that vertical gusts ay be handled well by this control schee. Including vertical gusts in future analysis and siulations would increase confidence that the turbulence-based extreu-seeking controller will function as desired in real turbulence. A related liitation is that longitudinal aircraft dynaics are not odeled in siulation. This revents the siulation fro testing the validity of neglecting altitude dynaics. The advantage of a turbulence-based aroach, that it oerates without using a erturbation signal, is also its disadvantage. Particularly at higher altitudes, an aircraft ay not exerience turbulence Seed [feet / sec] Endurance Seed Otiization airseed estiate of otial seed otial seed Tie [sec] Fig. 4 Siulation results of endurance seed otiization. Paraeter Value Units bk =v 9: ft=s=s=ft=slbf k ES : ft=s=s=ft=slbf v a 42.3 ft=s a ft=ss ^v a 42.3 ft=s J ; 2: s J ; s 2 J ;3 0.5 s J 2; 0 None J 2;2 0 s J 2;3 None J 3; :2 0 4 s J 3;2 0:0025 s 2 J 3;3 0:4374 s often. Furtherore, the otial airseed varies with the weight of the aircraft, and ossibly also with other flight araeters. Because of this, the current otial airseed ay not be the sae as the otial airseed found during the last encounter with turbulence. To address these issues, an alication of turbulence-based extreu seeking would need to rovide eans of recording otial seeds found during encounters with turbulence and extraolating the recorded seeds to the current flight condition. Another facet of this alication that ay rove colicated involves the varying curvature of the drag rofile with altitude and weight. To achieve raid convergence to the otial seed across the flight enveloe would ruire scheduling the extreu-seeking gain. Alternately, an extreu-seeking control law could be develoed that was insensitive to the curvature of the drag rofile. Such a ethod is develoed in [27] for traditional sinusoidalerturbation-based extreu seeking. Adating this ethod for stochastic extreu-seeking schees is a subject of current research. Finally, it is noted that while the assutions ade in this aer aly to a jet aircraft, it ay be ossible to odify the controller for roeller aircraft. For jet aircraft, flight at the best lift-to-drag ratio results in axiu endurance, but for roeller aircraft axiu endurance is obtained at the iniu ower seed. An extreuseeking controller for roeller aircraft ay be created by substituting an estiate of ruired ower for the estiated drag. This can be accolished by ultilying the drag estiate in the extreu-seeking controller by the easured airseed. Seed [feet / sec] Endurance Seed Otiization airseed estiate of otial seed otial seed Tie [hours] Fig. 5 Siulation results without added filters. By coarison with Fig. 4 it is clear that convergence can be sed u by several orders of agnitude by the addition of the filters.

10 KRIEGER AN KRSTI 885 onclusions A turbulence-based for of extreu seeking has been develoed for otiizing the seed of an aircraft for axiu endurance. The turbulence-based aroach allows extreu seeking to be erfored without introducing an external erturbation. Assuing longitudinal turbulence in level flight and a general convex drag curve, analysis shows weak exonential stability to the iniu drag seed. Siulations show siilar behavior with highass and low-ass filters added to the extreu-seeking loo. References [] Ariyur, K. B., and Krstic, M., SISO Schee and Linear Analysis, Real-Tie Otiization by Extreu-Seeking ontrol, Wiley, Hoboken, NJ, [2] Wang, H.-H., and Krstic, M., Extreu Seeking for Liit ycle Miniization, IEEE Transactions on Autoatic ontrol, Vol. 45, No. 2, ec. 2000, doi:0.09/ [3] hoi, J.-Y., Krstic, M., Ariyur, K. B., and Lee, J. 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