LATERAL AUTOPILOT DESIGN FOR A UAV USING COEFFICIENT DIAGRAM METHOD

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1 th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES LATERAL AUTOPILOT DESIGN FOR A UAV USING COEFFICIENT DIAGRAM METHOD Ru Hrokawa, Koch Sato MtsubshElectrcKamakuraWorks Keywords: Flght Control Systems,Unmanned Ar Vehcles, Coeffcent Dagram Method Abstract The lateral autoplot of an UAV s desgned usng Coeffcent Dagram Method (CDM). The CDM s an algebrac control system desgn approach and the desgn s performed by the coeffcent dagram drawng the coeffcent of system characterstc polynomal. The lateral control system s formulated as MIMO system and s dvded nto beta and bank control problems by control nput decouplng. The controllers are successfully desgned by CDM and the performance s confrmed by sx-degrees-of-freedom (6DOF) smulaton. Introducton By the recent advance of MEMS sensors and GPS technologes, the low-cost mnature UAV have become realsable. The flght controller of such UAV s requred to be smple and robust because of the CPU resource restrcton of the onboard mcro-controller. In ths paper, the lateral autoplot of an UAV havng smple structure and robustness s desgned usng Coeffcent Dagram Method (CDM). The CDM s an argebrac desgn approach for control system proposed by Shunj Manabe and the desgn s performed by the coeffcent dagram drawn the coeffcent of system characterstc polynomal [8. The stablty crtera of CDM s based on Lpatov s paper [, t s represented n the coeffcent dagram. The CDM s successfully appled to the varous applcatons such as atttude control of satellte [6 and robotc manpulator [. The CDM s very effectve desgn approach for SISO systems. Recently the CDM s also appled to MIMO systems. Manabe proposed the MIMO controller for longtudnal control of a hgh performance fghter usng decouplng approach, the result s compared wth the H method [7. The authors proposed an blended autoplot for a mssle wth reacton-jet and aerodynamc fn [. Km proposed the MIMO controller of hot rollng mll wth decouplng compensator, the performance compared wth the old conventonal desgn s confrmed by the numercal smulatons [. In ths paper, the basc formulaton of CDM s summarsed n secton. The lateral dynamcs of an UAV s formulated n secton, and the controller s desgned n secton. The performance of the system s evaluated by the numercal smulatons n secton. Formulaton based on CDM The plant and controller are defned as follows n CDM. Plant: { Ap x = B u (u + d) y = B p x Controller: A c u = B a y r B c (y + n) () Where, the states vector x, the outputs vector y, the reference nput vector y r, the control nputs vector u, the dsturbance vector d, the nose nput vector n are used. Standard block dagram of the CDM desgn s shown n Fg..

2 RUI HIROKAWA, KOICHI SATO y r dsturbance controller plant B a + u + + A - c B A - p Bp u B c Fg. Block dagram of control system x y + nose + Ä Å The closed loop system polynomal matrx s defned as, A = A c A p + B c B p () The characterstc polynomal P s defned as, P = A = a n s n a s + a = n l= a l s l () The stablty ndex γ and the tme constant τ are utlsed as desgn parameters n CDM and defned as, a γ = a + a ( =,...,n ) () τ = a /a () a s plotted n the coeffcent dagram as shown n Fg.. The stablty ndex s represented as convexty and the tme constant s represented as slope n the dagram. The lower bound of stablty ndex γ s defned as, γ = /γ + + /γ (γ = γ n = ) (6) The suffcent condton for the stablty [ s gven as, γ >.γ ( =,...,n ) (7) By the use of Eqs.(,), the coeffcent a s expressed by τ, a and γ. a = a τ γ γ...γ γ (8) In the CDM, followng stablty ndces are recommended [8. γ =.,γ =... = γ n = (9) In ths case, the step response has no overshoot, and the settlng tme s about...τ [8. ÁÄ ÁÃ ÁÂ γ γ Fg. A sample of coeffcent dagram Lateral dynamcs of UAV τ γτ The lateral dynamcs of UAV s represented as lnear system by sde-slp angle β, roll rate p, yaw rate r, bank angle φ. The control nput varables are aleron δ a and rudder δ r. The requrements are to follow the bank command and to suppress excessve sde-slp angle n the turnng. x = [ β p r φ, u = [ δ a δ r () y = [ β p r φ, y r = [ β c φ c () The system s represented as, A p = B u = s Y β snα cosα gcosα V L β s L p L r N β N p s N r tanα s Y δa L δa N δa Y δr L δr N δr () () The dynamcs of fn actuators are defned as

3 Lateral Autoplot Desgn for a UAV Usng Coeffcent Dagram Method the frst order lag as follows, δ a δ ac = s/ +, δ r δ rc = Controller desgn. Characterstcs of the plant s/ + () The aerodynamcs parameters of an UAV [ are used n ths desgn. The desgn s performed n velocty m/s, alttude m. The steady state flght condton s assumed, and the angle of attack n steady state α s.9deg. The aerodynamc coeffcents n ths condton are defned as n Tab.. Table aerodynamc coeffcents The denomnator and numerator of transfer functon are shown n Fg.. The negatve coeffcents are ndcated by the sgn. A negatve coeffcent of denomnator ndcates unstable spral mode. The order of controller should be mnmsed to realze the smple on-board controller. The structure desgn of feedback controller s performed usng ths dagram. b b a p b Y δa.7 Y δr.6 L δa 6. L δr.67 N δa.968 N δr -.6 Y β.67 L p -.77 L β.6 L r.8 N β 7.9 N p -.8 N r.9 b (a) aleron channel The pseudo control nput u s defned for decouplng between yaw and roll channel. [ E u Lδa L = u, E = δr () N δa N δr A p x = B u u = B u E u = B uu (6).7.6 B u = (7) By the use of Eq.(), the transfer functon s defned as, b b B p = b b b b (8) b b y = a p B pu (9) a p = A p (s/ + ) () b b b b (b) rudder channel a p Fg. Coeffcent dagram of lateral dynamcs. Yaw channel controller In the yaw channel, the proportonal plus ntegral feedback of sde-slp angle β and yaw rate r

4 RUI HIROKAWA, KOICHI SATO are used as feedback sgnal. The control nput s the pseudo rudder angle δ r. The frst order coeffcent of b has the large value. Whereas, the frst and second order coeffcent of b has almost the same value, and the zeroth order coeffcent has negatve value of nearly the same magntude. The desgn process on CDM s to fnd the best combnaton of the feedback sgnal to compose the reference coeffcent P defned n Eq.(). The feedback sgnal havng the large negatve value s not preferable to mantan the robustness. From ths consderaton, the proportonal and ntegral feedback of beta and proportonal feedback of yaw rate are selected as feedback sgnal. The controller and the system of the yaw channel are defned as, B p = [ b b, A p = a p () B c = [ k s + k k s, A c = s () B a = m () The reference coeffcents are defned as P of Fg. based on the recommended form of CDM n Eq.(8), and the feedback gan to realse these coeffcents are calculated as, k = 7.6, k = 66.8, k =. m = The coeffcents of feedback sgnals are also shown n the same fgure. In the coeffcent dagram of Fg., the characterstc polynomal P s defned as the summaton of the feedback sgnal as follows, P = sa p + k sb + k b + k b () The effectveness and contrbuton of the each feedback sgnal are easly understandable by the coeffcent dagram. Although P has the negatve coeffcent of the zeroth order and the slow unstable spral mode s stll exstng, t s easly compensated n the roll channel shown n the next secton. - 6 Fg. Coeffcent dagram of yaw channel. Roll channel controller The frst order coeffcent of b whch s also the zeroth order coeffcent of b has large value. The frst order coeffcent of denomnator a p has also large value. Consderng these characterstcs, the ntegral feedback of bank angle and the ntegral feedback of roll rate are selected. the roll rate feedback s not effectve because the second order coeffcent already has the large value. The controller and the system of the roll channel are defned as, B p = [ b b, A p = a p () B c = [ k k, Ac = s (6) B a = m (7) The feedback gan are derved n the same manner as yaw channel. k = 7., k =., m =.6 The coeffcents of transfer functons are shown n Fg.. The tme constant τ s selected as.7.

5 Lateral Autoplot Desgn for a UAV Usng Coeffcent Dagram Method and lower than degree. The transent response s mproved by the cross-feed compensaton. The bank response has no overshoot and ts settlng tme s about τ. β [deg.. wthout cross-feed wth cross-feed p [deg/s 6. r [deg/s φ [deg - 6 Fg. Coeffcent dagram of roll channel δ a [deg tme [s δ r [deg 6 tme [s. Cross-feed compensaton and total system The nter-channel cross-feed compensaton k s ntroduced to compensate the knematc couplng term psnα on the sde slp dynamcs. k s selected as. for proper tme response. The controller of the total system s defned as, [ k B c = k (8) k s + k k k s [ s A c = (9) s [ m B a = () m Desgn confrmaton by the numercal smulatons Some nonlnear 6DOF smulatons [ are performed to confrm the performance of the control system. The step response of degrees bank command s shown n Fg. 6. To evaluate the effectveness of the cross-feed compensaton, the result wthout the cross-feed compensaton (k = ) s also shown n the same fgure. The maxmum sde slp angle n the transent s small Fg. 6 6DOF smulaton of roll step response 6 Concluson The MIMO lateral autoplot for an UAV s desgned usng CDM. The control nputs are decoupled by ntroducng the pseudo-control nput. The yaw and roll channel controller s separately desgned for the decoupled system. The nterference between yaw and roll channel s compensated by the cross-feed compensator. The controllers are successfully desgned by CDM and the performance s confrmed by sx-degrees-offreedom (6DOF) smulaton. Acknowledgements The authors would lke to thank Shunj Manabe for hs extensve support for ths research.

6 RUI HIROKAWA, KOICHI SATO References [ A.Ucar and S.E.Hamamc. Controller based on coeffcent dagram method for the robotc manpulators. In The 7th IEEE Internatonal Conference on Electroncs, Crcuts and Systems, pages , Kaslk, Lebanon, December. [ Ru Hrokawa, Koch Sato, and Shunj Manabe. Autoplot desgn for a mssle wth reactonjet usng coeffcent dagram method. In Proceedngs of AIAA Gudance, Navgaton, and Control Conference, number AIAA -6, Montreal, Canada, August. [ Young-Chol Km and Myung-Joon Hur. Applcaton of CDM to MIMO systems: Control of hot rollng mll. In ICASE: The Insttute of Control, Automaton and Systems Engneers, volume, pages 6, Korea, December. [ A. V. Lpatov and N.I.Sokolov. Some suffcent condtons for stablty and nstablty of contnuous lnear statonay systems. Automatc Remote Control, pages 8 9, 979. translated from Automatka Telemekhanka, No.9,pp. 7, 978. [ Aerosonde Ltd. The aerosonde system. Nottng Hll, Australa. [6 S. Manabe. Coeffcent dagram method as appled to the atttude control of controlled-basmomentum satellte. In th IFAC Symposum on Automatc Control n Aerospace, pages 7, Sept 99. [7 S. Manabe. Applcaton of coeffcent dagram method to mmo system. In th Workshop on Astrodynamcs and Flght Mechancs. ISAS, July. [8 Sunj Manabe. The coeffcent dagram method. In th IFAC Symposum on Automatc Control n Aerospace, Seoul, Korea, August 998. [9 Sunj Manabe. Comparson of h-nf and coeffcent dagram method n aerospace. In Proceedngs of 9th Gudance Control Symposum, Yokohama,Japan, November. [ Unmanned Dynamcs, Hood Rver OR. AeroSm User s Gude,. 6