PIM Digital Redesign and Experiments of a Roll-Angle Controller for a VTOL-UAV

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1 1 roceeding of the International Conference on Information and Automation, December 15-1, 5, Colombo, Sri Lanka. IM Digital Redeign and Experiment of a Roll-Angle Controller for a VTOL-UAV Takahi Kahimura* and Noriyuki Hori + Digital Control Laboratory Graduate School of Sytem and Information Engineering Univerity of Tukuba Tennoudai, Tukuba, Ibaraki * takahik@edu.ey.tukuba.ac.jp Telephone: , Fax: hori@kz.tukuba.ac.jp Telephone: , Fax: Abtract Thi paper i concerned with the roll-angle control of a miniature, planar, Vertical Take-Off and Landing (VTOL), Unmanned Aerial Vehicle (UAV), which i actuated by air-jet upplied from the ground. A control ytem i deigned in analog domain uing a partial model matching method, dicretized uing the o-called the lant-input-mapping (IM) Method, and implemented uing a Digital Signal roceor (DS). It i found that the plant model identified accurately in open loop lead to teady-tate error in the IM control ytem, which can be removed by modifying the plant model. The experimental reult how that the performance of the IM controller i better than the widely ued Tutin controller at all the control rate teted; i.e., 5, and Hz. A I. INTRODUCTION miniature, planar, Vertical Take-Off and Landing (VTOL), Unmanned Aerial Vehicle (UAV) i contructed a a tet-bed for experimental verification of the o called the lant-input-mapping (IM) digital redeign method. The IM method i one of the digital redeign method for converting an analog control ytem into a digital one. By taking into account the cloed-loop characteritic of the analog control ytem in the form of the lant-input Tranfer-Function (ITF), the IM method can guarantee the tability for any non-pathological control rate, ha good performance even for low control rate, and i applicable to a variety of analog control method. The mot popular Tutin method doe not require a plant model, but require a fat control rate, wherea the IM method require a plant model, but give good performance with low control rate. Since a control ytem with leat proceing reource i deired for mall-ized UAV, the IM method i a good candidate for implementing digital control law. The VTOL-UAV are expected to find application in uch activitie a reconnaiance, urveillance, afety, fire fighting, and law-enforcement and a number of tudie have appeared for controlling planar VTOL vehicle [1] [3]. A real VTOL-UAV i uually elf-propelling, carrying it own propellant on board and thee paper eem to be eeking thi goal. The ytem conidered here i a mall-cale air-borne object whoe propellant i the compreed air provided from the air-upply on the ground. The UAV will, therefore, have to drag air-tube and mut be controlled with robut performance. At the moment, the ervo-valve are alo located on the ground, making the air-path from the valve to the nozzle long. Thi increae the time-delay, making control ytem deign more difficult. Compared with a UAV with rotating-wing, the UAV with air-jet are more agile, making the controller deign more demanding. The main goal of the preent tudy i to prove that the baic IM control law can be implemented on a DS for roll angle control of the UAV. II. EXERIMENTAL SETU A. Overall Sytem Decription Fig. 1 how the overall functional block of the experimental et-up. It conit of the UAV, which carrie enor and nozzle on board, the control unit, which include a DS, ADC, DAC, and a C, the air upply unit, which include an air compreor, a tank, a primary preure-regulator, and ervo-valve actuator. A hown in Fig., the VTOL-UAV i attached at the tip of a parallel-arm link that retrain the motion of the UAV in a urface within a limited range, which can be conidered quai-planar. Matlab/Simulink i ued for the analyi, ynthei, and imulation of variou control ytem. Once a control ytem i deigned, it i converted into c-code uing Real-Time Workhop and then down loaded to the DS. The DS calculate an appropriate control ignal to the ervo actuator, which in turn control the airflow to the nozzle.

2 15 roceeding of the International Conference on Information and Automation, December 15-1, 5, Colombo, Sri Lanka. C Air Compreor rimary reure- Regulator The Control Unit The Air Supply Unit Input Signal DS Servo-Valve The Actuator VTOL-UAV Rate Gyrocope Senor Nozzle Fig. 1. The functional block of the experimental et-up. The control input that the controller generate are the voltage ignal to the ervo valve, which control the thrut. For modeling purpoe, u 1 [volt] in Fig. 3 repreent the tranlational, common-mode input and u [volt] repreent the rotational, differential-mode input, to the UAV. Since only the roll angle i of concern in the preent tage of the project, u 1 i fixed to a contant. C. Analog Control Sytem Shown in Fig. i the analog I-D control ytem that i ued a a bae for digital redeign, where r i the reference input. r( t ) u ( t) φ ( t) φ ( t) K + / G ( ) + KI φ K D Fig.. The analog I-D control ytem. Fig.. The VTOL-UAV attached at the top of a parallel-arm link. B. VTOL-UAV lant Fig. 3 how the chematic of the VTOL-UAV, the ytem to be controlled. It ha one nozzle at each end, pointing downward, through which the compreed air i releaed into the atmophere. The thrut thu generated can create the tranlational force and rotational torque to actuate the UAV motion. An angular velocity enor (rate gyrocope) i attached at the center of rotation and meaure the roll angular velocity φ [deg/]. u Nozzle u 1 φ Nozzel Rate Gyrocope Senor Fig. 3. The chematic of the VTOL-UAV. The parameter of the tranfer function G ( ) φ from the input u to the angular velocity φ are identified from experimental data uing the tandard leat-quare method in the hift operator form at 1kHz. The obtained tranfer function i converted into one in Euler operator [], which i defined a z 1 ε = (1) T where z i the uual zee operator and T i the ampling interval. The Euler operator i ued here for better numerical propertie in digital control implementation and eae of relating dicrete-time reult to continuou-time counterpart []. The reaon for converting the tranfer function in z into that in Euler i the following: Uing the ADC and DAC, the plant model i that of the o-called Step-Invariant-Model [5], which ha an added ampling zero. Although it i difficult to ditinguih thi zero from the plant itelf uing z, it i eay uing the Euler form; the ampling zero can be removed by ignoring the term that are unneceary in the numerator []. Furthermore, a the ampling rate increae, the coefficient appearing in the tranfer function expreed in Euler operator approach thoe correponding coefficient in [5]. The analog control ytem i deigned baed on the partial model-matching method [7]. For the econd order plant of the form b G( ) =, () a + a1+ a the cloed-loop characteritic polynomial i given by

3 1 roceeding of the International Conference on Information and Automation, December 15-1, 5, Colombo, Sri Lanka. a + K b a + K b a (3) K b K b K b 1 D 3 1 I I I Thi i et equal to the deired polynomial given by α + α ( σ) + α ( σ) + α ( σ), () where α = 1, α1 = 1, α = 3 /, α3 = 1/1 []. The time-cale parameter σ i determined experimentally to be σ =.. The controller parameter K, K, and I K are determined D by equating (3) and (), and fine-tuned experimentally to be 3 K = 3.9 K I = 1.9 (5) K D =.7. The IM and Tutin digital control ytem are deigned to aimilate the performance of thi analog control ytem. III. THE IM METHOD A. The General IM Deign In thi ection, a general IM deign method [9] i reviewed briefly and applied later for the VTOL-UAV. Conider the analog control ytem repreented in Fig. 5. Aume that the continuou-time plant i linear, time-invariant, and trictly proper, and i denoted a ng() G () = dg(). () Aume alo that the analog control ytem i internally table, atifie all the deign pecification, and i realized with proper tranfer function given a na() nb() nc() A () =, B () =, C () = (7) d A() d B() dc() r() M () + u () A() C () B() G () y () Fig. 5. Continuou-time control ytem to be digitally redeigned. In the IM method, both the cloed-loop characteritic and plant information are ued in the dicretization proce in the name of the lant-input-tranfer Function (ITF). The ITF i the tranfer function from the reference input to the plant input and i given by u () AC () () M() = r () 1 + BCG () () (). () It i known [] that, in general, the denominator of the ITF contain the cloed-loop characteritic polynomial and the numerator contain the plant characteritic polynomial. Rather than dicretize each of the analog controller block a in Tutin method, only the ITF i dicretized in the IM method. Thi i carried out uing the Matched-ole-Zero method [11] and the reulting dicrete-time model become the target ITF. Once thi dicrete-time ITF i obtained, thi mut be realized in a cloed-loop configuration, uch a one hown in Fig.. Firt, the plant i dicretized uing the Step-Invariant-Model [5] and i expreed a ng ( ε ) G( ε ) =. (9) d ( ε ) The target ITF can be written, then, a * nm( ε ) dg( ε ) M ( ε ) =, () d ( ε ) which contain the denominator of the Step-Invariant-Model of the plant. Chooing the dicrete-time controller block [] a G na( ε ) nb( ε ) λ( ε ) A( ε) =, B( ε) =, C( ε) = (11) λ( ε) λ( ε) d ( ε) the dicrete-time control ytem can be implemented a hown in Fig., where λ( ε ) i an arbitrary, table polynomial of appropriate degree [9]. The controller deign i now reduced to the determination of na() ε, nb () ε, and d ( ) C ε. The actual ITF of thi control ytem i given by M na( ε ) dg( ε ) M ( ε ) =. (1) n ( ε ) n ( ε) + d ( ε) d ( ε) B G C G In the above, ng ( ε ) and dg( ε ) are known from of the plant (9). By equating the target and the actual ITF, it can be een that the polynomial na() ε mut be nm () ε, wherea nb () ε and d ( ) C ε mut be determined by olving the following Diophantine equation: n ( ε ) n ( ε) + d ( ε) d ( ε) = d ( ε) (13) B G C G M under appropriate degree condition [9]. Thi can be olved uing the eliminant matrix or a tate pace formulation [,9]. C

4 17 roceeding of the International Conference on Information and Automation, December 15-1, 5, Colombo, Sri Lanka. r() r( ε ) M ( ε ) + u( ε ) u() y () A( ε ) B( ε) ZOH G () y( ε ) C( ε ) Fig.. Dicrete-time control ytem redeigned uing the IM method. IV. EXERIMENTS The following apply to all the experiment conducted in thi tudy: The angular velocity i ampled at Hz and integrated numerically to obtain the roll angle. However, thi information i taken into the controller at the ame rate a the control rate (,, and 5Hz). The analog controller i not phyically built uing the analog component, but rather implemented digitally with all controller parameter unchanged and run at the fat ampling rate of 1kHz. The common-mode input i fixed at u 1 =.5 volt. A. IM Redeign of the Roll-Angle Control Sytem Uing the procedure explained earlier, with the ampling rate of 1kHz and the truncation of the ampling zero, the continuou-time tranfer function from the voltage input u to the angular velocity φ i obtained a 79 G ( ) = φ , (1) whoe pole are located at =.5 ±.75 j. Simulation reult obtained uing the UAV model (1) and the controller hown in Fig. with parameter (5) confirm that the deired analog controller performance i indeed achieved. For thi roll-angle control ytem, the IM method can be applied by conidering the VTOL-UAV including roll-rate feedback a the plant. The ITF to thi plant i dicretized uing the Matched-ole-Zero method and realized in cloed-form a the dicrete-time roll-angle control ytem hown in Fig. 7. Fig. (a)-(c) how the experimental reult obtained uing the control rate of 5Hz, Hz, and Hz, repectively. The reference input i the tep ignal occurring at t=1. econd with the amplitude of degree. It can be een from thee plot that at the control rate of 5Hz, the performance of Tutin and IM controller are cloe to the analog controller, although there i a mall teady-tate error with the IM controller. At the Hz rate, the Tutin deign ha the overhoot approaching % with continuing ocillation for more than econd. The IM repone ha about % teady-tate error with a maller ocillation than the Tutin during tranient. At the Hz control rate, both the Tutin and IM method are unatifactory, ince the Tutin controller induce violent ocillation and the IM produce about % teady-tate error r( t ) u ( t) φ () t φ ( t) K + / G ( ) + KI φ K D Fig. 7. The IM roll-angle control ytem for the VTOL-UAV. 1 Tutin (T=.) IM (T=.) - Time[ec] Fig. (a). Roll-angle repone to -degree tep reference change at the control rate of 5Hz. 1 Tutin (=.5) IM (T=.5) - Time[ec] Fig. (b). Roll-angle repone to -degree tep reference change at the control rate of Hz.

5 1 roceeding of the International Conference on Information and Automation, December 15-1, 5, Colombo, Sri Lanka. 1 Tutin (T=.1) - IM (T=.1) Time[ec] Fig. (c). Roll-angle repone to -degree tep reference change at the control rate of Hz. B. Modification of the lant Model Since the performance of the IM controller wa not a good a expected, the plant model wa modified and the controller wa recalculated. The reulting controller wa evaluated experimentally, until a good tranient repone with no teady-tate error i obtained. Thi modification i carried out baically by trial and error, but i eaier to do uing the knowledge on the plant dynamic than adjuting the controller parameter directly. The parameter appearing in thi model are not o enitive to the overall performance and can be ballpark figure. It i found that fine-tuning of the plant model i eay to perform, with the low frequency gain adjutment being the mot important. One uch model obtained in thi manner i given by 3 G ( ) = φ (15) whoe pole are at =.35 ± 3.5 j. Fig. 9 (a)-(c) how the repone of the roll-angle to the tep reference-angle change from to degree occurring at about 1 econd. At the control rate of 5Hz, the IM controller perform very well and inditinguihable from the analog control repone, with the overhoot of about 5% and the ettling time about econd. At the Hz rate, the repone with the IM method ha little ocillation, ha no ignificant overhoot, and ettle fater than the analog controller in thi cae, which i not intended. At the control rate of Hz, while the Tutin cae i unacceptable, the IM repone i till repectable. Although the IM repone ha a teady-tate ocillation with around.5-degree peak-to-peak amplitude, they are not really noticeable viually. 1 Tutin (T=.) IM (T=.) - Time[ec] Fig. 9(a). Roll-angle repone to -degree tep reference change at the control rate of 5Hz. 1 Tutin (T=.5) IM (T=.5) - Time[ec] Fig. 9(b). Roll-angle repone to -degree tep reference change at the control rate of Hz. 1 Tutin (T=.1) - IM (T=.1) Time[ec] Fig. 9(c). Roll-angle repone to -degree tep reference change at the control rate of Hz.

6 19 roceeding of the International Conference on Information and Automation, December 15-1, 5, Colombo, Sri Lanka. V. CONCLUSION A quai-planar VTOL-UAV wa contructed and a IM digital control ytem deigned and teted to aimilate the analog control ytem, which wa deigned bae on a partial model matching method. When the plant model that i identified in open loop wa ued, there wa a teady-tate error uing the IM controller. Thi error increaed a the control rate wa reduced but could be removed by modifying the plant model. Fine-tuning of the plant model i eay to perform, with the low frequency gain adjutment being the mot effective. Further invetigation into thi apect i deired. There are poibly two reaon for the occurrence of thi error. One i that the tandard IM deign take into account only the ITF from the reference input to the plant input. Since there are diturbance, uch a thoe due to air tube, the tranfer function from the diturbance to the plant input hould alo be conidered [1]. The other i the nonlinearity that i ignored in the plant modeling uing the ampled input-output data. The dicretization of uch a nonlinear ytem hould be invetigated and exact dicretization [13] may pave the way to overcoming thi iue. Since the roll-angle controller ha a atifactory performance, it will be combined with other control, uch a altitude control, to carry out experiment in planar motion in the next phae of the project. [7] T. Kitamori and R. Kuwata, rinciple, utility, and ynthei for the ID control method, Meaurement and Control, SICE Magazine, vol. 37, pp. 1-, 199. [] K. Fujii, T. Yamamoto, and M. Kaneda, A deign of elf-tuning ID control ytem baed on a partial model matching cheme, roc. SICE Annual Meeting, Chiba, pp , July 199. [9] A. H. D. Markazi and N. Hori, A new method with guaranteed tability for dicretization of continuou time control ytem, roc. American Control Conference, vol., pp , Chicago, 199. [] M. K. Sain and C. B. Shrader, The role of zero in the performance of multiinput, multioutput feedback ytem, IEEE Tran. On Education, vol. 33, pp. -57, 199. [11] N. Hori, R. Cormier, Jr., and K. Kanai, On matched pole-zero dicrete-time model, IEE roc. art-d, Vol , pp. 73-7, 199. [1] H. Shimamura and N. Hori, Digital redeign of a tepping-motor driver in the preence of computational delay and diturbance, Tran. CSME, vol.9, pp , 5. [13] N. Hori and C. A. Rabbath, Application of digital control theory to exact dicretization of a logitic equation with a contant term, roc. IEEE Conf. on Control Application, Toronto, pp , Augut 5. ACKNOWLEDGMENT The work preented in thi paper wa upported by JSS grant #175. REFERENCES [1] L. Benvenuti,. D. Giamberardino, and L. Farina, Trajectory tracking for a VTOL aircraft: a comparative analyi, roc. IEEE Conference on Deciion and Control, pp , 199. [] F. Lin, W. Zhang, and R. D. Brandt, Robut hovering control of a VTOL aircraft, IEEE Tran. Control Sytem Technology, vol. 7, pp , [3] A. alomino,. Catillo, I. Fantoni, R. Lozano, and C. egard, Control trategy uing viion for the tabilization of an experimental VTOL aircraft etup, roc. Conference on Deciion and Control, pp. 7-9, 3. [] R. H. Middleton and G. C. Goodwin, Digital Control and Etimation, rentice-hall, Englewood Cliff, 199 [5] N. Hori, T. Mori and. N. Nikiforuk, A new perpective for dicrete-time model of a continuou-time ytem, IEEE Tran. on Automatic Control, vol. 37-7, pp , 199. [] N. Hori, A. S. annala,. R. Ukrainetz and. N. Nikiforuk, Deign of an electrohydraulic poitioning ytem uing a novel model reference control cheme, ASME J. Dynamic Sytem, Meaurement and Control, vol. 111, pp. 9-9, 199.