Nonparametric Identification and Robust H Controller Synthesis for a Rotational/Translational Actuator

Transcription

1 Proceedings of the 6 IEEE International Conference on Control Applications Mnich, Germany, October 4-6, 6 WeB16 Nonparametric Identification and Robst H Controller Synthesis for a Rotational/Translational Actator FBagheri, S Prazarm, and H D Taghirad 1, Member, IEEE Abstract In this paper an H methodology based on nonparametric identification is developed for RTAC benchmark problem In this nonlinear system, it is reqired to design a controller to satisfy stabilization and distrbance rejection objectives in spite of limited control effort In order to design an H controller, first, the nonlinear system is estimated as a nominal linear system Moreover, the deviation of the system from the model, which involves nonlinearities, ncertainties, and distrbances, is encapslated by a mltiplicative ncertainty Then, a robst controller is designed by developing a mixed-sensitivity problem to satisfy all performance reqirements Simlation reslts illstrate the tracking performance achievements I INTRODUCTION HE Rotational/Translational Actator experiment has Toriginally been stdied as a simplified model of a dalspin spacecraft to investigate the resonance captre phenomena [1] It is shown that this system is mathematically and qalitatively eqivalent to a dal-spin spacecraft, ie they have similar averaged behavior and exhibit similar dynamic behaviors The RTAC system has been stdied to investigate the seflness of a rotational proof mass for stabilizing translational motion [] Consider the nbalanced oscillator shown in Figre 1 The RTAC is bilt with a card (weigh M) fixed to the wall by a linear spring (stiffness k) and constrained to have one dimensional travel An embedded proof mass actator (mass m, moment of inertia I) is attached to the center of the card and can rotate in the horizontal plane The radis of rotation is e The card is sbmitted to the distrbance F and a control torqe N is applied to the proof mass The control objectives sch as internal stability, fast settling time for a class of initial condition and good distrbance rejection for some signals in spite of limited control effort mst be satisfied by designing a controller Several approaches of the RTAC problem have been proposed in the past few years Varios stabilization strategies are compared in [3] Partial feedback linearization and integrator back-stepping are stdied in [4] Adeqate Manscript received on Febrary 1, 6 Athors are with the KN Toosi University of Technology, Electrical Engineering Department, Advanced Robotics and Atomated System (ARAS), PO Box , Tehran, IRAN (Phone: ; fax: ; Taghirad@kntacir) Figre1 Model of the RTAC Experiment Lyapnov fnctions lead to a back-stepping controller in [5] and to a passive nonlinear in [6] State-feedback nonlinear controller is obtained in [7] Most of these techniqes do not address mlti-objective problems directly Frthermore, they may lead to eqations that are hard to solve nmerically, in particlar when otpt-feedback robst synthesis is involved Finally, most of these approaches do not take into accont ncertainty, or mltiple specifications, explicitly An H methodology proposed here, that garantees the varios design constraints simltaneosly II SYSTEM MODELING Let and denote the anglar position and velocity of the proof mass, and let q and q denote the translational position from its eqilibrim position and the velocity of the card The nonlinear eqations of the motion are given by: M mq kqme F ( I me ) meqcos N ( ) ( cos sin ) With following transformation [8] M m k M m q, t, v N I me M m ) k( I me 1 M m me D F, () k I me ( I me )( M m) the normalized nonlinear eqations of motion of the card are b c ( sin cos ) D km ( m) (3) cos V Vales of the parameters for the RTAC eqations are given in Table I (1) /6/$ 6 IEEE 715

2 TABLE I: RTAC PHYSICAL PARAMETERS Cart Mass M 1Kg Arm Mass m 1Kg Spring Constant k 186N / m Eccentricity e 6m Arm Inertia I e 4 Viscos friction constant b c 1 III NON-PARAMETRIC SYSTEM IDENTIFICATION Several approaches of nonlinear control design were developed for RTAC problem based on the ideal nondissipative model of the nonlinear system given by Eqation (1) and () The basic problem we consider in this paper is to find a linear controller for the given nonlinear system sch that the closed-loop system meets a nmber of specifications We design a linear H controller while the nonlinear system is estimated as a nominal linear system and nmodeled dynamics and derivations from this nonlinear model are taken into accont as an ncertainty model The basic techniqe is to model the plant as belonging to an nstrctred set of linear plant Unstrctred set is sed for two reasons First we believe that all models sed in feedback design shold inclde some nstrctred ncertainty to cover nmodeled dynamics, particlarly at high freqency Secondly, for a specific type of nstrctred ncertainty, we can develop a simple and general analysis method Sppose that the nominal plant transfer fnction is P and consider the pertrbed plant transfer fnctions of the form P P (1 ( W ), where W is a fixed stable transfer fnction and is a variable transfer fnction satisfying 1 Frthermore, it is assmed that no nstable poles of P are canceled in forming P Hence if 1, then j) 1 W ( j), P ( j) j) By plotting the system variations 1, for all P ( j) experimental freqency response estimates of the system P ( j), and estimate an pper bond to those variations as a transfer fnction, the mltiplicative ncertainty weighting fnction W ( is obtained If a set of identification experiments is performed on the nonlinear system, nder different operating condition, the linear freqency response estimate reslts P ( j), will form the set By finding a nominal fit P ( j) throgh these freqency response estimates, the nonlinearities, distrbances and ncertainties of the system will be Figre -a Freqency responses of the estimated system Figre -b The ncertainty profile and the ncertainty weighting fnction condensed into the pertrbation W It shold be noted that the selected nominal system shold attain the minimm ncertainty weighting fnction at least in lower freqencies For identification tests, a chirp fnction generator is sed as the excitation inpt The reason is that a chirp signal demonstrates an almost flat spectrm over the freqency range of interest It is also highly persistent excitation Identification tests are done for different initial arm angles sch that the estimated transfer fnctions are valid representatives of the nonlinear system all over its operating range The amplitdes of the chirp range from 5% to 1% of their maximm vale To obtain a lower ncertainty weighting fnction, it is necessary to control the anglar position dring the identification experiments, therefore, a simple negative PD feedback of is sed The nominal linear model has been obtained throgh freqency response estimates depicted in Figre -a The nominal transfer fnction / V is given by P ( s 658)( s 36)( s 67)( s ) ( s 354)( s 173)( s 86s 1) 716

3 which is a stable and non-minimm phase transfer fnction It has two slow modes and two fast modes Having these two types of modes is one of the difficlties of the control design Another difficlty is the presence of nstable zeroes j) The ncertainty profile 1 and the ncertainty P ( j) weighting fnction of this system are depicted in Figre -b It shows that the selected fnction falls a little lower than the ncertainty level over low freqency W ( is s 11 W s 115 The nominal transfer fnction and ncertainty weighting fnction of have been obtained as follows D P d W d ( s 1) 1 5 ( s686)( s81)( ( s66) ( s1961)( s857)( s 86s 1) IV ROBUST H CONTROL The control objectives can be defined as robst stability of the closed loop system as well as a satisfactory distrbance rejection and oscillation sppression of the otpts prone to the dominant oscillatory modes while avoiding actator satration Robst controller C (, will be designed by developing the so-called mixed-sensitivity problem: Ws S opt min WT T 1 W pd P where S and T 1 p 1 P Figre 3 illstrates the block diagram of the setp sing mltiplicative ncertainty representation, which formlates the abovementioned problem as a standard H problem Replacing P d, sensitivity fnction will change into S 1/(1 p C) which is desired, therefore ( shold be chosen in the form ofw 1 Pd More specifically, referring to Figre 3, we wold like to design a controller to minimize the following norm: WT T opt min W1 Pd S W where W T is the ncertainty weighting fnction The robst stability is garanteed by minimizing the infinity norm of weighted transfer fnction from d to z3, which is eqivalent to the weighted complementary sensitivity fnction W T T 1 (Small-gain theorem) Weighting fnctions Ws W s Figre 3 Closed-loop system as a generalized reglator problem and W are also considered to normalized and assign freqency content of the performance objectives Response speed and distrbance attenation is obtained by minimizing the infinity norm of d to z1, or the weighted sensitivity fnction W S S 1 Finally, the infinity norm of d to z, or weighted control effort transfer fnction penalizes controllers with high control effort, and provides a media in or optimization to inclde directly the control effort limitations into the controller synthesis Hence, by simltaneosly optimization of the infinity norm of the opt 1, all the objectives are satisfied The problem has been solved for the represented RTAC with an pper bond for control effort corresponding to W The sensitivity weighting fnction is determined in order to have soltion for the mixed sensitivity problem as well as to have maximm reachable bandwidth as following: 3Pd WS s 1 LMI approach has been employed in the design of the H soltion in this problem Figre 4 shows magnitde diagram of the sensitivity fnction by implementing H controller Since the system dominant oscillatory modes are at 1rad / sec, by decreasing the sensitivity fnction magnitde in that freqency, the settling performance has been improved To analyze the performance of the closed-loop system, the nonlinear model is sed in simlations Closed-loop response is illstrated as dashed line in Figre 6 The reslt illstrates the effectiveness of the proposed H controller to provide a fast settling time Figre 7 shows the control torqe Dashed line illstrates large picks in control effort that can damage actators We wold like to design a controller to trade off between increasing the settling time and decreasing the control effort As mentioned before, the system dominant oscillatory modes are at 1rad / sec Therefore it is enogh to have maximm control effort only arond this range By shaping W in order to minimize this transfer fnction arond 1rad / sec effort can be eqal to freqencies, the maximm magnitde of the control 1 W and it can be limited at the other W is determined according to 717

4 1 Singlar Vales Solid: ncontrolled, dotted:controlled (W1), dash-dot:controlled (W) Singlar Vales (db) -5-1 Cart position Freqency (rad/sec) Figre 4 Singlar vale plot of sensitivity fnction Singlar Vales Ws*S WT*T Figre 6 Free responses of open-loop system (solid), closed-loop system with W 1 (dotted) and closed-loop system withw (dash-dot) 3 Dotted:control signal (W1), solid:control signal ( W) -4 Singlar Vales (db) Control signal Freqency (rad/sec) Figre 5 Singlar vale plot of WT T, WS S 3 347s 187s 3554s 19 W 3 s 594s 1368s 4 Figre 8 shows the magnitde diagram of W Closed-loop response is illstrated as solid line in Figre 6 and control torqe is illstrated as solid line in Figre 7 It shows a smooth pattern in control torqe in spite of a slow settling time in system response These reslts can be obtained ot of Table II It gives the -norm of the control effort and settling time of two closed-loop systems which are discssed above TABLE II: TWO-NORM OF CONTROL EFFORT AND SETTLING TIME Controller Settling time (sec) First ( W 1 ) Second ( W ) Figre 7 Control torqe when W1 is sed (dotted), control torqe when W is sed (solid) Now consider the closed-loop systems in presence of a random measrement noise Figre 9 shows the closed- loop response in this sitation Althogh noise does not inflence otpt response, it affects control effort considerably Table III compares the -norm of the control effort in both closedloop systems TABLE III: TWO-NORM OF CONTROL EFFORT AND SETTLING TIME Controller IN PRESENCE OF NOISE Settling time (sec) First ( W 1 ) Second ( W ) 4 It is evident that a reasonable choice forw, effectively limits control effort in presence of measrement noise Satisfaction of the performance reqirement by the 718

5 cart position Magnitde (db) Freqency(rad/sec) Figre 8 Magnitde diagram of W solid:controlled(w1),dashed:controlled(w) Figre 9 Closed-loop system response in presence of noise conseqent H controller is illstrated in freqency and time domain simlations According to this simlation stdy, the robst control design is entirely sccessfl in satisfying the reqired control objectives with a limited control effort V CONCLUSIONS Several nonlinear control design approaches for the RTAC problem have been proposed Most of these techniqes do not address mlti-objective soltions directly Frthermore, they may lead to eqations that are hard to solve nmerically, and most of these approaches do not explicitly take into accont modeling ncertainty In this paper an H controller was developed for this nonlinear problem to tackle the above objectives Use of a rotational proof mass actator for stabilizing translational motion was investigated by implementing an H controller The basic method proposed in this paper is how to find a linear controller for the given nonlinear system sch that the closed-loop system meets a nmber of specifications Then an appropriate controller was designed, while the nonlinear system was estimated as a nominal linear system and nmodeled dynamics and derivations of the nonlinear system from the nominal model were encapslated in the model ncertainty The nominal model was obtained throgh an nstrctred set of linear freqency response estimates by performing identification experiments on the nonlinear system nder different operating condition This representation provided sfficient information to bild a robst controller for the RTAC system The H controller was designed by solving the mixed-sensitivity problem for a fast settling time and distrbance attenation objectives, respecting the actator satration limits The closed loop performance of the system in freqency and time domain simlations were examined and the performance of the closed-loop system was evalated It was illstrated that the closed-loop system retains robst stability, while improving the desirable performance exceptionally well REFERENCES [1] R J Kinsey, D L Mingori, and R H Rand, Nonlinear Controller to Redce Resonance Effect of a Dal Spin Spacecraft throgh recession Phase Lock, in proceedings of the 31 st Conference on Decision and Control, 199, pp 35-3 [] M Tavakoli, H D Taghirad and M Abrishamchian, Parametric and non-parametric identification and robst control of a rotation/translational actator, The Forth International Conference on Control and Atomation (ICCA 3), Montreal - Canada, 1-1 Jne 3 [3] Abed, E, Y S Cho, A Gran and A Tits, Nonlinear stabilization and parametric optimization in the benchmark nonlinear control design problem, Proc IEEE American Control Conf, Seattle, WA, Jne 1995, pp [4] Wan, CJ, D Bernstein and V Coppola, Global stabilization of the oscillating eccentric rotor, Proc IEEE Conf Decision and Control, Orlando, FL, December 1994, pp [5] Kanallakopolos, I and J Zhao, Tracking and distrbance rejection for the benchmark nonlinear control problem, Proc IEEE American Control Conf, Seattle, WA, Jne 1995, pp [6] Bpp, R, D Bernstein and V Coppola, A benchmark problem for nonlinear control design: problem state, experimental test bed, and passive nonlinear compensation, Proc IEEE American Control Conf, Seattle, WA, Jne 1995, pp [7] Tsiotras, P, M Corless and M Rotea, An L distrbance attenation approach to the nonlinear benchmark problem, Proc IEEE American Control Conf, Seattle, WA, Jne 1995, pp [8] C-J-Wan, D S Bernstein, and V T Coppola, Global Stabilization of the Oscillating Eccentric Rotor, Nolin Dyn, Vol 1, pp 49-6, 1996 [9] John Doyle, Brse Francis, Allen Tanennbam, Feedback Control Theory Macmillan Pblishing Co,