PRECISE positioning/tracking control is being studied

Transcription

1 Design of High Accuracy Tracking Sysems wih H Preview Conrol Anonio Moran Cardenas, Javier G. Rázuri, Isis Bone, Rahim Rahmani, and David Sundgren Absrac Posiioning and racking conrol sysems are an imporan componen of auonomous robo applicaions. This paper presens he design mehod of racking conrol sysems based on H preview conrol where he presen and fuure desired posiions of he robo are used o deermine he conrol acions o be applied so ha he robo describes he desired rajecory as close as possible. The performance improvemens achieved wih H preview conrol have been examined in he frequency and ime domains for differen ypes of reference signals when applied o a one-dimensional posiioning sysem. I was found ha preview conrol improves he racking performance by improving he phase response of he racking sysem. Index Terms Roboics, planning and scheduling, predicive conrol, H conrol, racking conrol, conrol H 2, frequencydomain analysis, ime-domain analysis. I. INTRODUCTION PRECISE posiioning/racking conrol is being sudied in many manufacuring fields in order o improve he accuracy and performance of manufacuring process and manipulaor driving sysems which every ime demand more precise, robus and efficien conrol sysems 1]. Cerain behavior is desired in posiioning/racking conrol sysems: fas response and convergence, zero racking error and robusness agains changes in he sysem iself and/or is environmen. The classical way o solve he racking conrol problem for linear ime-invarian sysems has been o design a one-degree-of-freedom, or beer, a wo-degrees-of-freedom conroller which will achieve he desired performance as close as possible. The inheren shorcoming of he classical approach is he overdesign ha is eniled in requiring a performance ha is independen of he specific measurable signal o be racked. In he case of sandard H conrol, echnique inroduced by Zames 2] a conroller is designed for a wors case reference signal which may be differen from he one which is acually encounered. Obviously such an approach may inroduce a considerable emphasis on he racking properies of he sysem which may lead, in some Manuscrip received on July 29, 2014, acceped for publicaion on Sepember 22, 2014, published on November 15, Anonio Moran Cardenas is wih Ponifical Caholic Universiy of Perú PUCP, Lima, Perú ( amoran@pucp.edu.pe). Isis Bone is wih he Anioquia School of Engineering (EIA GISMOC), Colombia ( pfibone@eia.edu.co). Javier G. Rázuri (corresponding auhor), Rahim Rahmani, and David Sundgren are wih he Sockholm Universiy (DSV), Deparmen of Compuer and Sysems Sciences, Sweden ( {javier, rahim, dsn}@dsv.su.se). cases, o insufficien disurbance aenuaion and robusness properies of he closed-loop sysem 3]. In order o avoid he shorcomings of classical conrol when he reference signal is known a priori (for some inerval of ime), he concep of preview racking conrol has been inroduced. According o his conrol sraegy, he conrol inpu is calculaed using no only informaion corresponding o he presen sae of he sysem bu also using he known fuure value of he reference signal. The advanages of preview conrol has been sudied by several auhors 4] 6]. An updaed review of preview conrol is presened in 7]. The mos of preview conrol mehods are based on classical LQ/LQG conrol heory where preview conrol is used for furher reducion of he minimum value achievable by he cos funcion being opimized 8] 10]. However, LQG conrol mehods presen he limiaion of how o include robusness specificaions for designing he conroller which are usually defined in erms of H norms. Several preview conrol mehods based on H conrol heory, which ake ino acoun he unknown disurbances o design conrol sysems wih beer performance measures, have been developed. In 11] 13], he preview performance in erms of H crierion was invesigaed. Some auhors have been presened soluions for discre-ime cases 14], 15], and ohers for coninuous-ime problems 16] 18]. This paper presens he soluion o he H preview racking conrol problem. The analyical formulaion of he H preview racking conrol law has been derived in he coninuous ime domain. Preview conrol laws have been derived for wo racking problems: (a) he reference signal is perfecly known in advance for he oal working ime and (b) he reference signal is previewed for a fixed inerval of ime. For boh cases he srucure of he preview conroller corresponds o a wo-degrees-of-freedom conroller wih feedback and feedforward (preview) pars. The benefis and limiaions of he proposed conrol sraegies are analyzed heoreically and experimenally in he ime and frequency domains for a wide range of operaing condiions and differen ypes of reference signals. This paper is srucured as follows: Secion II presens he srucure of he experimenal posiioning/racking sysem, while Secion III describes he Sae Equaion Model of he sysem. In Secion IV, we described he formulaion process relaed o he generalized plan. Experimenal resuls and discussion are presened in Secion V, while Secion VI presens some conclusions of he research. ; pp Polibis (50) 2014

2 Anonio Moran Cardenas, Javier G. Rázuri, Isis Bone, Rahim Rahmani, David Sundgren Fig. 1. Experimenal racking sysem Fig. 2. Vibraional model of experimenal sysem A. Experimenal Sysem II. EXPERIMENTAL SYSTEM Figure 1 shows he srucure of he experimenal posiioning/racking sysem. A DC moor roaes a ballscrew o longiudinally move a posiioning sage which is composed of a main sage and a sub-sage conneced hrough wo flexible plae springs. The conrol objecive is o place he sub-sage in an arbirary desired posiion or o follow any desired rajecory. To do ha, a laser posiion sensor measures he posiion of he sub-sage and sends he measured signal o a compuer which calculaes a conrol volage according o he algorihm of he H conroller. Using a power amplifier, he conrol signal is applied o he DC moor o conrol he roaional moion of he ballscrew in order o achieve he desired moion of he sub-sage. The DC moor and main sage are affeced by fricion orque and fricion forces, respecively. The sub-sage moves on a rough surface whose degree of roughness (fricion) may be varied o analyze he robusness of he sysem agains exernal fricion forces. B. Vibraional Model Figure 2 shows he equivalen vibraional model of he posiioning/racking mechanism. The flexible plaes are modeled as a linear spring wih siffness K sp. The roaional damping of he DC moor is represened by K cr and he fricion orque by T f. Exernal forces acuaing on he sub-sage are represened by linear damping K cl and fricion force F f. A. Nominal Model III. STATE EQUATION MODEL Neglecing he inducance of he DC moor, is dynamics can be described by he following equaion: ( ) Ra T = K pa u K e θ, (1) K where T and u are he orque and volage of he DC moor and θ is he angular velociy of he ballscrew. The roaional moion equaion of he driving mechanism (moor shaf and ballscrew) is given by T T f T qr = J θ + K cr θ, (2) where T f is he Coulomb fricion orque and T qr is an equivalen orque represening he roaional effec of he ineria of he main sage and is equal o T qr = K sp K bs (x m x s ), (3) where x m and x s represen he posiion of he main sage and sub-sage, respecively. The linear moion equaion of he sub-sage is given by (x m x s )K sp F f = Mẍ s + K cl ẋ s, (4) where F f is he fricion force affecing he sub-sage and M is he mass of he sub-sage. Defining he sae vecor: ] T x = θ, θ, ẋs, x s, (5) and combining Equaions 1 o 4, he following sae-space equaion of he nominal sysem P can be formulaed: x = Ā x + B 1 w + B 2 u, (6) where w = T f, F f ] T. The parameers specificaions are shown in Table I. TABLE I LIST OF PARAMETERS OF EXPERIMENTAL MODEL Name Simbol Value Power amplifier gain K pa 10 DC moor resisance R a 1.1 ohm DC moor orque consan K N.m/A Back elecromoive force consan K e x 10 2 v.s Momen of ineria J x 10 5 kg.m 2 Roaional damping facor K cr x 10 3 N.m.s Ballscrew ransducing coefficien K bs x 10 3 m/rad Spring siffness K sp 264 N/m Linear damping facor K cl N.s/m Mass of sub-sage M kg Polibis (50)

3 Design of High Accuracy Tracking Sysems wih H Preview Conrol B. Generalized Plan wih Inegral Acion In order o design a conroller wih inegral acion o eliminae seady-sae racking errors, he generalized plan used for designing he conroller is srucured considering inegral acion. To do ha, Equaion 6 represening he nominal plan, is inegraed wih he equaion of he racking error e given by he expression: e = r x s, (7) where r is he reference signal o be racked. Defining he sae vecor T x = θ, θ, ẋ s, x s (r x s )d], (8) he sae-space equaion of he generalized plan wih inegral acion is: ẋ = Ax + B 1p w + B 2 u + B r r, (9) where i can be noed ha he reference signal r is no included in he disurbance vecor ŵ Marices A, B 1p, B 2 and B r are obained by combining Equaions 6 and 7. IV. H PREVIEW CONTROL LAW As i is well known, he firs sep for designing an H conroller is he formulaion of he generalized plan. By defining he conrolled oupu vecor z as T z = ρ 1 ẋ s, ρ 2 x s, ρ 3 (r x s )d, u], (10) and he measured oupu vecor y as: ] x y = s + η 1, (11) (r xs )d + η 2 he equaions describing he generalized plan P are: ẋ = Ax + B 1 w + B 2 u + B r r (12) z = C 1 x + D 12 u (13) y = C 2 x + D 21 w (14) where he disurbance vecor w = T f, F f, η 1, η 2 ] T and B 1 = B1p0 ] T T. Facors ρ1, ρ 2 and ρ 3 in Equaion 10 are weighing coefficiens seleced o ailor he performance and robusness specificaions. η 1 and η 2 in Equaion 11 represen measuremen noise which are assumed o be independen of he plan disurbances w. Given his assumpion, marices B 1 and D 21 urn o be orhogonal. Also, given he componens of he conrolled oupu z, marices C 1 and D 1 2 are orhogonal. The srucure of he generalized plan is shown in Figure 3 (o gain clariy, he conrolled oupu z is no shown). According o H conrol heory, given he generalized plan P, a conroller is designed so ha he H norm of he ransfer funcion T zw from disurbances w o conrolled oupu z is less han a given scalar γ: T zw < γ. (15) Fig. 3. Block diagram of generalized plan and preview conroller. Opimal H conrollers are designed for he minimum value of γ and H 2 conrollers are designed for γ =. By analogy wih Game Theory, he H x conrol problem can be expressed as he minimizaion of he following cos funcion: J = z T z γ 2 w T w ] d, (16) which can be expanded as: J = x T C1 T C 1 x + w T u T ] ] γ 2 w ] 0 ] d, (17) 0 R u 0 where R = D T 12D 12. The convenional approach o design H racking sysems has been o formulae a generalized plan which includes he reference signal as exernal disurbance and design a one or wo-degrees-of-freedom conroller which will achieve he desired specificaions. The inheren shorcoming of his approach is he overdesign ha is eniled when designing a conroller independenly of he reference signal o be racked which is known a priori. ŵ = T f, F f, η 1, η 2, r] T. In oher words, a conroller is designed for he wors reference signal r wors which is, for he mos of cases, differen o he desired reference signal r. To overcome he shorcoming of he classical H approach, his paper proposes H preview conrol where he known reference signal is used, as i is, for designing he conroller. The design of he H preview conroller has been divided in wo pars: firs, H preview conrol laws have been derived for he case of sae feedback, and aferwards an observer was designed considering he convenional approach o design H oupu feedback conrollers. The srucure of he H preview conroller is shown in Figure 3 where i can be noed he feedback par K f and preview par K p of he conroller. H conrol laws have been derived for wo cases: (1) he reference signal is known for he oal working ime and (2) he reference signal is previewed for a fixed inerval of ime shorer han he oal working ime. 23 Polibis (50) 2014

4 Anonio Moran Cardenas, Javier G. Rázuri, Isis Bone, Rahim Rahmani, David Sundgren A. Reference signal known for he oal working ime When he reference signal is known for he oal working ime, + T ], he H preview conroller is designed considering he generalized plan of Equaions 12 and 13 and he following cos funcion: J = +T z T z γ 2 w T w ] d (18) Using he Hamilonian approach of Opimal Conrol Theory 19], he preview conrol law which minimizes he cos funcion of Equaion 18 subjec o he consrain of Equaion 12 is given by: u() = R 1 B T 2 P ()x() + q()], (19) where P () is he soluion of he following differenial Riccai equaion: P (τ) = P (τ)a P (τ)(b 2R 1 B2 T γ 2 B 1B1 T )P (τ)+ (20) +A T P () + C1 T C 1, wih r +T and P (τ) saisfying he erminal consrain: P ( + T ) = 0; q() in Equaion 12 is he soluion of he following differenial equaion: q() = (A T P (τ)(b 2 R 1 B T 2 saisfying he erminal consrain: q( + T ) = 0. γ 2 B 1B T 1 ))q(τ)+ +P B rr(τ), (21) Since o compue q() and P (), Equaions 20 and 21 should be inegraed backwards in ime, he compuaional amoun could urn cumbersome and impracical so ha an approximaed simplificaion easy o compue is necessary. Assuming he working ime T is long enough so ha P ) = 0 and P = cons, q() can be calculaed from he following equaion: where and ] q() = exp (A T cl + M) x +T ] ] x exp (A T cl + M)τ P B rr(τ) dτ, (22) A cl = A B 2 R 1 B T 2 P (23) M = γ 2 B 1 B T 1 P. (24) A cl in Equaion 23 represens he sae marix of he closed-loop sysem for he case of only feedback conrol. I is imporan o noe ha he conrol law of Equaion 19 has he srucure of wo-degrees-of-freedom conrollers wih feedback and feedforward (preview) pars so ha preview conrol improves he racking performance of he sysem wihou affecing is feedback characerisics (sabiliy, disurbance aenuaion, ec.). B. Reference signal previewed for a fixed inerval of ime In his case i is assumed he reference signal is known for a fixed inerval of ime T which is much shorer han he oal working ime assumed o be long enough. In his case, he cos funcion o be opimized is J = z T z γ 2 w T w ] dτ, (25) which can be decomposed in wo pars: where J 1 = J = +T + +T = +T z T z γ 2 w T w ] dτ+ z T z γ 2 ŵ T ŵ ] dτ, z T z γ 2 w T w ] dτ and J 2 = +T z T z γ 2 ŵ T ŵ ] dτ (26) Since he reference signal r is only known for he inerval, + T ], i is no included as a componen of he disurbance vecor w for his inerval (cos funcion J 1 ), bu i is included in w = ŵ for he inerval + T, ]. Since in his inerval r is no known i should be replaced by r wors as in sandard H racking conrol. From Linear Opimal Conrol Theory i is well known ha he minimum value of he cos funcion J 2 is J 2 = x T ( + T ) ˆP x( + T ), (27) where ˆP is he soluion of he following algebraic Riccai equaion ˆP A ˆP (B 2 R 1 B2 T γ 2 (B 1 B1 T + B r Br T )) ˆP + (28) +A T ˆP + C T 1 C 1 = 0 From Equaions 26 and 27, he cos funcion J can be expressed as: J = x( + T ) ˆP x( + T ) + +T z T z γ 2 w T w ] d (29) Since he cos funcions of Equaions 29 and 18 are similar wih he only difference being he erm x( + T ) ˆP x( + T ) in Equaion 29, he conrol law which minimizes he cos funcion J of Equaion 29 subjec o he consrain of Equaion 12, is also given by Equaion 19 wih P () calculaed from Equaion 20 and q() calculaed from Equaion 21. The only difference is he erminal value of P which for his case is P ( + T ) = ˆP, where ˆP is already known from he Riccai equaion 28. Since he compuaion of q() is cumbersome when P varies wih ime, an approximaion for he easy compuaion of q is necessary. Similarly as in subsecion IV-A, assuming ha P = 0 hen P = cons and akes he value for ime + T, e.g., P = ˆP. Wih his simplificaion, he conrol signal u can be compued easily a every sage of conrol. Polibis (50)

5 Design of High Accuracy Tracking Sysems wih H Preview Conrol V. RESULTS AND DISCUSSION In order o evaluae he racking performance of he H preview conroller, he response of he sysem has been examined in he ime and frequency domains for differen ypes of reference signals. The response of he sysem wih H preview conrol has been compared wih he response for H feedback conroller wih 1.5 degrees-of-freedom (DOF) and H 2 preview conroller 20]. The H preview conroller has been designed for he case when he reference signal is previewed for a fixed inerval of ime shorer han he oal working ime which is assumed o be long enough (sub-secion IV-B). The preview ime is chosen o be 1 second since longer imes do no yield significan performance improvemens. A. Frequency Response In order o compue he frequency response of he sysem wih preview conrol, he Laplace ransform Q(s) of he preview par q() of he conrol law of Equaion 19 is required o be known. Using Equaion 22 and recalling he definiion of Laplace ransform, where R(s) is he Laplace ransform of he reference signal r() and Q(s) is calculaed as: Q(s) = Q 1 (s) Q 2 (s)] R(s), (30) Q 1 (s) = exp T s] si + A T cl + M ] 1 exp (A T cl + M)T ] P B r, (31) Q 2 (s) = si + A T cl + M ] 1 P Br. (32) Figure 4 shows he gain and phase of he frequency response of he racking sysem for H 2 preview conrol (solid line) and 1.5DOF H 2 feedback conrol (broken line). I can be noed ha alhough he gain of he response is higher a high frequencies for 1.5DOF H 2 feedback conrol, he phase of he response for H 2 preview conrol is zero for frequencies up o 10 Hz. Zero phase response is desired in racking sysems since i is always desirable ha he sysem responds o he reference inpu wihou delays. Figure 5 compares he gain and phase of he frequency response for H preview conrol (solid line) and 1.5DOF H feedback conrol (broken line). Similarly as for H 2 conrol, he gain of he response is higher for 1.5DOF H feedback conrol bu he phase of he response for H preview conrol is close o zero for frequencies up o 1 Hz and is posiive for he medium frequency range up o 10 Hz. This posiive phase characerisics indicae ha he sysem wih H preview conrol responds in advance o he reference inpu. Since in he medium frequency range he gain of he frequency response for H preview conrol is low, he conroller ries o compensae his deficiency by responding in advance wih posiive phase. Figure 6 shows he gain and phase of he frequency response for H preview conrol (solid line) and H 2 preview conrol (broken line). I can be noed ha he gain of he response is higher for H preview conrol han for H 2 preview conrol. The phase response shows he zero-phase characerisics of H 2 preview conrol and he posiive characerisics of H preview conrol discussed before. B. Time Response The ime response of he racking sysem wih H preview conrol has been analyzed for sinusoidal and sep reference inpus. a) Sinusoidal Response Figure 7 shows he ime response of he sysem for a sinusoidal reference inpu of 0.5 Hz. Figure 7 (a) corresponds o 1.5DOF H feedback conrol and Figure 7 (b) o H preview conrol. In Figure 7 (a) i is noed ha alhough he ampliude of he response of he sysem (solid line) is almos he same as he reference inpu (broken line), he sysem responds wih delay and is unable o rack he reference inpu wih zero error. In Figure 7 (b) i can be noed ha he sinusoidal reference inpu and he response for H preview conrol pracically overlap from he iniial ime. I is clear ha he racking error is almos zero for he oal working ime. b) Sep Response I is usually said ha preview conrol improves he racking performance of he sysem especially in siuaions where he signal o be racked varies wih ime, and no significan improvemen can be achieved for sep reference signals for siuaions when he conrol acion sars jus when he reference signal is applied. However and as i is shown in Figure 8, i has been found ha alhough he sep response for he sysem wih 1.5DOF H feedback conrol (broken line) and H preview conrol (solid line) are almos he same, Figure 8 (a), he conrol signal u is differen for boh conrollers, Figure 8 (b). Preview conrol demands lower conrol signals and herefore requires less conrol energy o achieve he same racking performance han 1.5DOF H feedback conrol. Calculaed resuls show ha he conrol energy can be reduced even by 10% when using H preview conrol. VI. CONCLUSIONS This paper has presened a novel design mehod of posiioning/racking sysems based on H preview conrol using he known fuure value of he reference inpu. Analysis of he frequency response shows ha preview conrol improves he racking performance by improving he phase response of he racking sysem so ha he sysem responds o he reference inpu wihou delay. The sep response of he racking sysem shows ha H preview conrol requires less conrol energy han 1.5DOF H feedback conrol o achieve he same posiioning performance. ACKNOWLEDGMENT The auhors grealy appreciae he financial suppor provided by he insiuion VINNOVA Swedish Governmenal Agency for Innovaion Sysems hrough he ICT 25 Polibis (50) 2014

6 Anonio Moran Cardenas, Javier G. Rázuri, Isis Bone, Rahim Rahmani, David Sundgren Fig. 4. Frequency response of H 2 preview conrol Fig. 5. Frequency response of H preview conrol Fig. 6. Frequency response of H 2 and H preview conrol Polibis (50)

7 Design of High Accuracy Tracking Sysems wih H Preview Conrol Fig. 8. Sep response for H preview conrol REFERENCES Fig. 7. Time response for sinusoidal inpu (0.5Hz) projec The Nex Generaion (TNG). We also graeful o Ponifical Caholic Universiy (Perú) and Anioquia School of Engineering EIA - GISMOC (Colombia) in a join effor for collaboraive research. 1] B. Chen, Robus and perfec racking of coninuous-ime sysems, in Robus and H Conrol, ser. Communicaions and Conrol Engineering. Springer London, 2000, pp Online]. Available: hp://dx.doi.org/ / ] G. Zames, Feedback and opimal sensiiviy: Model reference ransformaions, muliplicaive seminorms, and approximae inverses, IEEE Transacions on Auomaic Conrol, vol. 26, no. 2, pp , ] B. M. Chen, Robus and H conrol. Springer, ] M. Tomizuka, Opimal coninuous finie preview problem, IEEE Transacions on Auomaic Conrol, vol. 20, no. 3, pp , Jun ] A. Hac, Opimal linear preview conrol of acive vehicle suspension, in Proceedings of he 29h IEEE Conference on Decision and Conrol, Dec 1990, pp vol.5. 6] A. J. Hazell and D. J. N. Limebeer, A framework for discree-ime H 2 preview conrol, Journal of dynamic sysems, measuremen, and conrol, vol. 132, no. 3, p , ] N. Birla and A. Swarup, Opimal preview conrol: A review, Opimal Conrol Applicaions and Mehods, pp. n/a n/a, Online]. Available: hp://dx.doi.org/ /oca ] M. O. Cole and T. Wongraanaphisan, Opimal {lq} feedforward racking wih preview: Pracical design for rigid body moion conrol, Conrol Engineering Pracice, vol. 26, no. 0, pp , Online]. Available: hp:// pii/s ] I. Youn, R. Tchamna, S. Lee, N. Uddin, S. Lyu, and M. Tomizuka, Preview suspension conrol for a full racked vehicle, Inernaional Journal of Auomoive Technology, vol. 15, no. 3, pp , Online]. Available: hp://dx.doi.org/ /s ] C.-C. Luo, R.-F. Liu, C.-D. Yang, and Y.-H. Chang, Helicoper H conrol design wih robus flying qualiy, Aerospace science and echnology, vol. 7, no. 2, pp , ] A. Kojima and S. Ishijima, H performance of preview conrol sysems, Auomaica, vol. 39, no. 4, pp , Online]. Available: hp:// S Polibis (50) 2014

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