Multi-rate Digital Design for Sliding-Mode- Observer-based Feedback Control

Transcription

1 005 American Control Conference June Portlan OR USA ThB05.1 Multi-rate Digital Deign for Sliing-Moe- Oberver-bae Feeback Control I. A. Shkolnikov Y. B. Shteel Senior Member IEEE an S. V. Plekhanov Abtract Implementation of Sliing-Moe-Oberver-bae feeback control i treate a a multi-rate igital ignal proceing problem. The overample elta-igma moulation technique i ue to eign a ecimation filter that integrate high-frequency witching output of the liing moe oberver into the feeback control which operate at much lower preetermine control upate rate. Application of a multi-rate liing moe oberver in the function of a nonlinear uncertain ytem etimator in the terminal guiance loop i emontrate. It hown in thi application that the correctly eigne ecimation filter for a multi-rate ytem eliver cloe-loop ytem performance uperior to that of traitionally propoe low-pa filter of autoregreive type which etimate equivalent control component in the igital liing moe oberver in the manner equivalent to their continuou-time analog implementation. D I. INTRODUCTION ynamic ytem tate etimator i.e. ynamic oberver have become an eential part of the integrate navigation-guiance-control (GNC) olution for many aeropace flight control problem [1011]. Stochatic eign of a plant-moel-bae etimator which i preente in numerou variation of Kalman filtering technique i one of the popular choice for GNC integration. Another potentially powerful alternative or complementary technique to Kalman filtering in the omain of ynamic-oberver-bae control i the liing moe technique [1-3]. However one cannot eny that Kalman filter (KF) ominate the contemporary engineering olution for GNC application an it can be attribute to a number of particularly attractive feature of it igital eign. Beie the fact that igital eign of KF i abolutely rigorou KF algorithm can eaily accommoate ignal input ource upplie at ifferent rate can utilize tatitical reunancy not only from time- I. A. Shkolnikov i with the Z/I Imaging Corporation. an Intergraph Company 30 Buine Park Blv. Maion AL USA ( USA ( ihkolnikov@ziimaging.com). Y. B. Shteel i with the Department of Electrical an Computer Engineering The Univerity of Alabama in Huntville Huntville AL (a correponing author phone: (56) fax: (56) hteel@ece.uah.eu) S. V. Plekhanov i with the Department of Electrical an Computer Engineering The Univerity of Alabama in Huntville Huntville AL ( plekha@ece.uah.eu). correlation within one ignal ource but alo from fuion of multiple meaurement of the ame ignal. While proceing rate in navigational KF [10] can match that in feeback control that provie tracking of guiance comman compute from navigational ata liing moe oberver mut have proceing rate much higher than control upate rate. Therefore the ownampling problem i eential in igital implementation of liing moe oberver an i overlooke in the available literature on liing moe oberver bae controller [1-3]. The lack of practical guieline an tuie of igital implementation of liing moe oberver in a multiple-rate controlobervation architecture ownplay application of thi potentially powerful metho in ome area incluing one where the cloe-loop performance of the oberver-bae control rather than ignal-to-noie ratio or RMS of the oberver etimation error erve a the ultimate performance criterion for the complete integrate GNC ytem. Thu thi work concentrate on the problem of multi-rate igital proceing in liing moe oberver i.e. their igital implementation namely ecimation filter eign that provie (peaking in igital ignal proceing term) elta-igma emoulation [4-6] an ownampling for high-frequency witching component of the liing moe oberver which retrieve the uncertain part of the equivalent control for the liing moe exiting in the oberver. The ue of the contemporary overample-eltaigma-moulation technique [4-6] i propoe to eign the ecimation filter. The inherent feature of any practical implementation of liing moe i.e. finite frequency witching moe the overampling elta-igma moulation of the uncertain component of the equivalent control put an emphai on eigning the ecimation filter that integrate the oberver output to the feeback control operating at much lower control upate rate. The goal of thi ecimator/emoulator i to achieve low-pa filtering an ownampling to a preetermine ampling rate of the input one-bit ignal avoiing aliaing an elivering maximally efficient bit reolution of the output ignal. The ultimate performance criterion of thi ecimation filter i in achieving accurate robut tracking performance for the cloe-loop ytem. Therefore only the cloe-loop ytem behavior rather than RMS or pectrum of etimation error houl be coniere /05/$ AACC 47

2 to jutify a particular eign for a liing moe oberver with choen ecimator/emoulator. Becaue of practical emphai of thi work only SISO cae i coniere while MIMO cae with ytem tructure reuce to vector relative egree [11 1] or [ ] are traightforwar an o not contribute any eential feature that are worth coniering from the tanpoint of the choen ownampling problem. II. PROBLEM FORMULATION Conier n th orer nonlinear SISO plant uner calar control an uncertain iturbance input. State flow i hyperbolic at the origin globally Liphitz an atifie all nice propertie of input-output feeback linearizable ytem. For a given meaure liing variable y of the plant the input-output ynamic i of the firt orer x f ( x) u( ( (1) y x where y x i the output an the tate of the firt orer input-output liing variable ynamic (1) f (x) i known nonlinear term u ( i known input to ytem (1) u( ZOH ( u[ n]) u[ n] u( to nt ) cont with ZOH taning for zero-orer hol an ( i uncertain boune an low-pa ban-limite iturbance input to be etimate. Conventional liing moe oberver of a continuou form i a tracking ytem that follow ytem (1) output xˆ y xˆ f ( x) u( () u where 0 i a contant that atifie inequality max ( max f ( xˆ(0)) f ( x(0)) or aaptive feeback gain. The witching control function u i a ignum of liing quantity u e y xˆ ign (3) which i equal to tracking error for oberver (). Stability of ytem () an the exitence conition of liing moe 0 i well known [1]. Accoring to [1] equivalent control for ytem () in the liing moe 0 i obtaine from conition 0 u (. (4) eq In the ieal liing moe the left part in (4) can be coniere a the etimate of uncertain input ( to the ytem (1). Thi etimate i obtaine for the ieal liing moe which i featuring infinite frequency witching of (3) while the witching frequency in any practical igital ytem i limite to proceing cycle frequency. Now we procee to formulating the problem coniere in thi work. One poible igital implementation of oberver () employing Euler igital integration i ue xˆ[ k] xˆ[ k 1] T f ( xˆ[ k 1]) u[ k 1] T u[ k 1] (5) u [ k 1] ign y[ k 1] xˆ[ k 1] where k i ample number T i ampling/proceing time of oberver (5). It i aume that upate rate of the known ignal u ( i much lower than that for oberver (5) proceing rate therefore u [k] i actually equal to a contant u [n] for many proceing cycle k where n i control upate ample number correponing to upate time t to nt T T for control upate in ytem (1). Alo ampling rate of meaurement input i uually lower than proceing rate 1 /T. Thu proceing in (5) eliver ignificant overampling with repect to the input u an y. An etimate of the iturbance ( in (1) i euce from the previou icuion a ˆ LPE T u (6) where LPE tan for low-pa equivalent of the agrument. In orer to complete igital eign of the iturbance etimating ytem oberver (5) the following problem i formulate. The problem i in eigning a igital linear filterecimator that extract a given low-pa-ban component f 0 f ] with a given output rate [ B 1 /T from the binary wave form [ k] k given in real time cale by the factor upplie at the rate 1 /T 1/ T 1/ T. Remark: u T an 1) Linearity of the filter i require to provie linear phae-hift in the output which coul be a typical requirement in many practical ytem an the poibility to carry out a multiplicative caling term while working with only one-bit number u [k] which i really hany for circuit-level eign. ) The objective of thi ecimator/emoulator i to achieve low-pa filtering an ownampling to a preetermine ampling rate of the input one-bit ignal avoiing aliaing an elivering maximum to the effective bit reolution of the output ignal. III. OVERSAMPLING DELTA-SIGMA MODULATION AND DIGITAL IMPLEMENTATION OF SLIDING MODES The cloet reemblance can be oberve between the liing-moe oberver ecribe in Section an the pule moulator calle elta ( ) moulator an elta-igma ( ) moulator [4-7]. Pule moulation communication 48

3 ytem ( -moulator) an feeback relay control ytem operate in high-frequency auto-ocillation moe calle the liing moe. In the fiel of communication ytem the principle of a feeback proceing [8] an utilization of robut limit cycle ocillation (autoocillation) [9] for pule moulation technique have been tuie ince the beginning of the ixtie. A pulemoulation ytem a a ample-ata ytem relie on the uniform ampling theorem for low-pa ignal an it immeiate corollary that continuou ban-limite ignal can be encoe by a equence of icrete ample an recontructe from it with negligible error provie the ampling rate i ufficiently high [4-6]. Thi property of ignal recontruction an the tructure of the feeback loop are common to -moulator an liing moe oberver; therefore they can benefit from each other in eign olution. One equivalent ample-ata repreentation of the overampling elta-igma moulator can be given a xˆ[ k] xˆ[ k 1] T m[ k 1] T u u [ k 1] ign 0 xˆ[ k 1] [ k 1] where k i ample number max m( T i ampling/proceing time an overampling mean that 1 /T i ignificantly higher that Nyquit rate for the meage ignal m ( encoe by a binary ignal u [k]. Delta-igma moulation i a popular metho of analog-toigital converion where the proceing rate of moern ytem reache GHz ban which eliver ignificant overampling to many phyical ignal ource. The problem of igital emoulation of binary encoe ignal i equivalent to the problem formulate in Section for liing moe iturbance oberver which in thi cae i to convert elta-igma moulate ignal back to the igital form ample at Nyquit rate with maximal poible bit reolution in magnitue for a given ynamic range. A we have mentione the tenency to increae the overampling rate in elta-igma moulator an liing moe oberver i motivate by the immeiate corollary from the uniform ampling theorem for low-pa ignal: the higher i the ampling rate the lower i the ignal recontruction error which i vali for both the known meage ignal to be encoe or the unknown iturbance ignal to be etimate in the ynamic oberver. Effectively elta-igma moulation technique trae reolution in time for reolution in amplitue. However the emoulation proce i.e. abrupt low-pa filtering an ecimation at elevate ampling rate preent a ignificant circuit-level eign challenge an ae cot while imple eign olution alway carry ignificant noie penalty becaue of aliaing. Fortunately the icue problem for elta-igma moulation ha been in the focu of reearch community for many year [4-6] an practical eign olution have been evelope an implemente in igital electronic. (7) Therefore the olution of the problem in Section formulate for liing moe oberver (5) an (6) can irectly benefit from it communication ytem counterpart. One of the proven eign olution multitage ecimation will be preente an applie to liing moe oberver in Section 4. The imulation an analyi of Section 5 will gauge the benefit of the electe eign for cloe-loop performance of the oberver-bae control ytem. IV. MULTISTAGE DECIMATION FOR SLIDING MODE OBSERVER One-bit coe u [ k] k in oberver (5) i a repreentative of elta-igma moulation of the uncertain iturbance ( acting on ytem (1). However ue to meaurement noie in the ene ytem (1) output y an moeling error in oberver (5) ignal u [k] alo repreent thee component plu quantization noie. Therefore the problem formulate in Section inclue the ban limit f for etimating the low-frequency iturbance B ( while all out-of-ban component houl be eliminate. The ieal olution to the problem i to apply ieal low-pa filter followe by ecimator to the require frequency. Practical low-pa filter with abrupt characteritic for elevate ampling rate woul be of very high orer. Therefore ecimation an filtering i uually performe a a two-tage proceure [4]. Accoring to one eign approach in [4] the firt tage ecimate from high proceing rate to ome intermeiate ampling rate f D. The filter for thi tage i eigne to attenuate noie component in the vicinity of f D an it harmonic which fol into the ignal ban. One well-uite igital filter having neceary characteritic i ync filter 1 H ( z) N 1 z 1 z N 1 (8) where N f / f D i the ecimation ratio of the firt tage. It can be implemente a a cacae of two accumulator proviing 1/(1 z 1 ) function followe by reampling at rate f D an followe by a cacae of two igital ifferentiator working at thi lower rate f D an generating total ignal tranformation accoring to (8). The econ tage conit in the low-pa filter eigne to meet pecification for ban limit f B followe by the accumulate-an-ump block ecimating from f D to f B. Thu the utility an benefit of thi approach conit in the fact that only one witching element an a number of igital accumulator houl work at elevate ampling rate in a eicate but very imple harware moule while lowpa filter/emoulator of any complexity work at regular 49

4 ampling rate of main guiance computer that carrie on all the other GNC proceing. Without the propoe multi-rate eign liing moe oberver in a GNC ytem ha two choice: either working at the available proceing rate that ha been etablihe for ay Kalman filter an efinitely failing the competition or requiring elevate proceing rate for the whole ytem which i not cot effective. The preente important tructural olution bae on proven technique in communication fiel an applie to igital liing moe oberver may in our opinion bring robut technique of liing moe control theory cloe to the poition competing againt Kalman filter bae oberver or complementing them in a hybri eign for contemporary GNC ytem. V. CASE STUDY: HOMING GUIDANCE SYSTEM The following example of a real-worl cloe-loop control problem terminal (homing) guiance loop uner Augmente Proportional Navigation (APN) law [1011] i electe to illutrate irect benefit in applying multi-rate igital liing moe oberver for the purpoe of etimating target acceleration a a component of APN guiance. Coniering only planar motion without account for gravity one can erive planar engagement kinematic in polar coorinate ytem where the relative poition i preente by R ( r ) r = range along line-of-ight (LOS) an = LOS angle a r r at r r a r am r a T M or accepting range rate a V r LOS rate a an miile normal acceleration a an r Vr V r r at r in( M ) an (10) 1 V r at co( M ) an r where M i the irection of miile velocity in thi - imenional ytem; let it be for example vertical plane an M angle i miile flight path angle. Acceleration projection for target an miile a T r am r are along LOS an a T am are tranveral to LOS. Miile airframe compenate ynamic a a tranfer function for normal acceleration will be preente by a firt-orer lag block with time contant 0.5[ec] rate aturation a n 100[ g / ec] an magnitue aturation for APN guiance comman an c 30[ g]. (9) Augmente Proportional Navigation APN which i optimal for zero-lag guiance ytem in preence of tepcontant target maneuver i compute a c a 1 M N' Vc a T (11) where N i the navigation gain Vc V r i cloing c velocity. Guiance comman am will be upplie to the input of flight control ytem (11) a a piecewie contant ignal with 50[Hz] upate rate. Miile eeker (target tracking hea) a a LOS rate etimator i aume to have a ingle-lag low-pa filter for LOS ervo-tracking error meaurement uch that LOS rate meaurement proce i generate a 1 (1) T where T 0.1[ ec] input noie to the filter i uncorrelate zero-centere Gauian proce with 0.01[ ra / ec]. Meaurement of LOS rate are available to the guiance computer at the rate of 500[ Hz ]. Cloing velocity range to the target an the actuate miile acceleration orthogonal to LOS are aume to be known an the only quantity to be etimate i target acceleration a T that i ue in APN guiance law (11). Taking a meaurement of LOS rate an omitting fat eeker ynamic the following firt-orer liing moe oberver i eigne uing the lat equation in ytem (10) T ˆ [ k] ˆ [ k 1] ( (V r[ k 1] a [ k 1]) u [ k 1] u [ k 1] ign [ k 1] ˆ [ k 1] M r [ k 1] ˆ [ k 1] (13) where T i ampling time. Parameter if it i not aaptive but contant houl be equal to the known limit of a T. Etimate of target acceleration a ˆT i obtaine by low-pa filtering of binary equence u [k] with caling by T. We elect n orer low-pa Butterworth filter with cut-off frequency of 0.15[ ra / ec] implemente in igital form for each correponing ampling rate. APN guiance comman i compute uing c thee etimate a a 1 M 4 Vc ˆ aˆ T an ecimate from proceing rate 1/T to the rate 50[Hz]. One particular engagement cenario ha been imulate an reulte in a irect hit where the range etimate of cloet approach point i le than the itance travele in ample time of meaurement upate. Engagement trajectory i given in Fig

5 lo of kinetic energy an iminih miile ability to perform intercept. Fig.1 Engagement Trajectory: Incoming near-orthogonal intercept Three imulation run ha been performe with ifferent configuration of liing moe oberver for etimating target acceleration a ˆT. All three engagement reulte in the target intercept an the following target acceleration tracking performance ha been emontrate. In the firt run oberver (13) an the following Butterworth filter ha been working at the rate 100[kHz] with following ownampling to 50[Hz]; target etimation i hown in Fig.. Fig. 3 Target acceleration etimation at proceing rate of 1[kHz] For the thir run multi-rate eign to oberver (13) ha been applie. The witching frequency of liing moe in oberver (13) ha been 100[kHz] binary output u [k] pae through ync ecimating filter (8) an cale by T. Output rate ha been 50[Hz] an the following igital Butterworth filter ha been working at the rate 50[Hz]. Target etimation for multi-rate oberver i hown in Fig.4. Fig. Target acceleration etimation at proceing rate of 100[kHz] In the econ run oberver (13) an the following Butterworth filter ha been working at the rate 1[kHz] with following ownampling to 50[Hz]; target etimation i hown in Fig. 3. One can oberve in Fig. 3 that low witching frequency of 1[kHz] in oberver (13) le to high quantization noie plu aliaing of the high frequency noie component in the input to the bae-ban of target etimate a ˆT. A a reult error in a ˆT (for the firt 5[ec] of flight time in abence of target maneuver) i about 1[g]. Although the econ engagement ha been ucceful the APN guiance comman that inclue a ˆT term ha caue unneceary lateral motion of the miile uring firt 5[ec] of engagement which coul lea to Fig. 4 Target acceleration etimation in multirate liing moe oberver One can notice that target etimation performance in Fig. 3 an 4 i almot ientical which jutifie the preente multi-rate eign with application of elevate ampling rate only for elta-igma moulation of target acceleration in oberver (13) followe by correctly eigne ecimation filter to provie all ubequent proceing at preetermine output ampling rate. VII. CONCLUSIONS The preente work carrie two main iea. The firt one i that elta-igma moulation technique in communication fiel an liing moe technique in 431

6 control fiel may benefit each other in great eal many more eign olution can be borrowe in each irection. The other iea i that the practical ucce of liing moe oberver in moern igital worl may be achieve only in multi-rate eign olution. The both iea are illutrate on a practical example: terminal (homing) guiance ytem bae on Augmente Proportional Navigation. Future work i to be eicate to rigorou tuy of multi-rate igital implementation of liing moe oberver incluing ue of high orer liing moe. REFERENCES [1] V.Utkin J. Gulner an J. Shi Sliing Moe in Electromechanical Sytem Taylor an Franci Lonon [] C. Ewar an S. Spurgeon Sliing Moe Control Taylor & Franci Britol PA [3] J.-J. E. Slotine J. K. Herick an E. A. Miawa On liing oberver for nonlinear ytem ASME Journal of Dynamical Sytem Meaurement an Control vol. 109 pp [4] J. C. Cany an G. C. Gabor Overampling Delta- Sigma Data Converter: Theory Deign an Simulation IEEE Pre 199. [5] S. V. Plekhanov I. A. Shkolnikov an Y. B. Shteel High Orer Sigma-Delta Moulator Deign via Sliing Moe Control Proceeing of the American Control Conference Denver CO June 003. [6] I. A. Shkolnikov Y. B. Shteel an S. V. Plekhanov Analog-to-Digital Converter: Sliing Moe Oberver a a Pule Moulator Proceeing of the Conference on Deciion an Control Orlano FL December 001. [7] R. E. Ziemer an W. H. Tranter Principle of Communication. Sytem Moulation an Noie 4th e. New York: John Wiley & Son [8] H. Inoe an Y. Yaua A unity bit coing metho by negative feeback Proc. IEEE vol. 51 pp [9] J. C. Cany A ue of limit cycle ocillation to obtain robut analog-to-igital converter IEEE Tranaction on Communication vol. pp [10] Zarchan P. Tactical an Strategic Miile Guiance vol.176 Progre in Atronautic an Aeronautic AIAA [11] Shkolnikov I. A. Shteel Y. B. Zarchan P. an Liano D. P. "Simulation Stuy of the Homing Interceptor Guiance Loop With Sliing Moe Oberver veru Kalman Filter" Proceeing of AIAA Guiance Navigation an Control Conference AIAA Paper # Montreal Canaa Augut