Event based Kalman filter observer for rotary high speed on/off valve

Transcription

1 28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 28 WeC9.6 Event base Kalman filter observer for rotary high spee on/off valve Meng Wang, Perry Y. Li ERC for Compact an Efficient Flui Power Department of Mechanical Engineering University of Minnesota {wang134, Abstract A novel hyraulic rotary self-spinning high spee on/off valve is being evelope to enable hyraulic systems to be controlle in a more efficient throttle-less manner via pulsewith-moulation (PWM). The valve is esigne to operate at a spool frequency of 2Hz to 4Hz. A coarse non-contacting optical encoer is propose to measure the angular position of the valve spool. Measurement events in the form of encoer count changes are obtaine at irregular times an infrequently. An event-base Kalman filter is evelope to improve on the resolution an to provie continuous estimation of the spool angular position an velocity. Simulation an experimental results show that the event base Kalman filter can provie accurate continuous estimation for both spool angular position an velocity. I. INTRODUCTION A Kalman filter is an optimal recursive filter that estimates the state of a ynamic system from a series of incomplete an noisy measurements. Discrete-time Kalman filters an continuous-time Kalman filters have been wiely use as an estimating approach in the presence of moel uncertainty an measurement noise. However, with irregular measurements, traitional Kalman filter structure must be aapte to operate as an event-base estimator. In this paper, such an alternative is evelope for use with a novel type of hyraulic rotary high spee on/off valve. Traitional metho of controlling hyraulic systems makes use of a fixe isplacement pump, a pressure regulating relief valve an a throttling valve. In this case, unneee flow is ble off at high pressure resulting in high power loss, an significant energy is lost ue to pressure rop across the throttling valve. As an alternative to throttling control, a high spee on/off valve can be pulse with moulate (PWM) to control the average flow. In either the on or the off state, the power loss of the valve is minimal, because either the pressure rop across the valve is very small, or the flow rate through the valve is zero. However, fast on/off transition is critical to minimize throttling loss uring transition an high PWM frequency is neee to reuce flow or pressure ripple. To this en, a new type of rotary high spee self-spinning on/off valve has been propose in [1]. The continuous uniirectional rotary motion overcomes the challenge of linear on/off valves which require significant power to repeately accelerate an ecelerate the spool. In aition, the rotary motion is actuate by scavenging energy from the flui flow irectly. In one configuration, the rotary valve is use together with a small accumulator, which enables a fixe isplacement pump to realize the function of a variable isplacement pump [2] [3]. One key feature of the valve is that the spool rotates an travels axially insie a stationary sleeve, as shown in Fig.1. Knowlege of the axial an rotary states are useful for controlling the function of the valve effectively. Laser moule M DC motor Gerotor Pump Polycarbonate Plate Fig. 1. LED System schematic Coewheel Photoioe The axial motion of the spool is actuate hyrostatically using a small gerotor pump. Both axial an rotary sensing are achieve using non-contact methos, which simplify the spool structure an the sealing structure. The axial position is measure using an optical sensor consisting of LEDs an a photoioe [1]. The intensity of the LED light reflecte back from the spool en is a monotonic function of the istance between the spool an the sensor, an can thus be use as an axial position sensor. The rotary states of the spool are also measure using a non-contacting optical sensor, which consists of a laser moule, a photoioe an a rotary encoer coe wheel. The light emitte from the laser is reflecte from the coe wheel. The photoioe response epens on whether the laser is reflecte off of a black or white sector on the coewheel, an the changes can be use to sense the rotary position. Due to the small spool iameter (2.5cm) /8/$ AACC Authorize license use limite to: University of Minnesota. Downloae on April 7, 29 at 14:22 from IEEE Xplore. Restrictions apply.

2 an the relatively large beam size, the encoer resolution is relatively poor (about 8 sectors per revolution). Improving the quality of the laser source an improving the resolution of the coe wheel may improve the sensor resolution, however, they are not easy or costly to realize. Because of the coarse encoer resolution, it is beneficial to estimate the angular position an velocity between transitions of the encoer counts. Since the measurement of black-white transition events can occur at irregular time, we propose an event base Kalman filter observer for this purpose. The transition events are subect to uncertainty ue to finite sampling interval, an the fact that optical signal changes graually an the threshol for istinguishing a white or black sector is uncertain. Continuous time varying Kalman filter theory is aapte to accommoate the uncertain event base measurements. The resulting algorithm is such that between events, the Kalman filter operates in an open loop manner; when a transition occurs, both the Kalman filter gain an the state estimate are upate iscretely. Two approaches are frequently use to estimate angular velocity from encoer measurement: finite-ifference an inverse-time. In finite ifference metho, angular velocity is calculate by counting the number of pulses within a fixe time interval, converting the counts to angle, an iviing the angle by the time interval. In inverse-time metho, angular velocity is calculate by iviing the sector angle by the time between successive pulses. In both cases, measurement precision relies heavily on the coewheel resolution. Gla an Lung [6] presente a Kalman filter to o velocity estimation base on position measurements obtaine at irregular time instants. Measurement noise ue to occurrence time uncertainty was consiere, an simulation results showe that in the presence of high noise this Kalman filter is superior to the previous two methos. Position estimation was not iscusse is the methos above. In the next section, the position measurement approach an the ynamic moel of the system will be presente. In section III, the esign of the event-base Kalman filter will be escribe. Simulation results an experimental results will be presente in section IV. Concluing remarks an future research plans will be iscusse in section V. II. SYSTEM DESCRIPTION AND MODELING A. Rotary On/off Valve Working Principle The high spee rotary on/off valve consists of a spool that rotates insie a stationary sleeve (Fig. 2). The spool has a center PWM section an two outlet turbines. The center section is compose of helical barriers, which act as turbine blaes, so that when high velocity flui is transferre from the inlet nozzles to the valve spool, the flui momentum is capture an transferre into the angular momentum of the spool. When the flow leaves the spool to either application or tank, the outlet turbines reverse the irection of the flow relative to the inlet, an a reaction force is generate on the spool. As the spool rotates, flow is apportione to application (on) or tank (off) by the helical barriers. The uty ratio, which is efine as the ratio of valve on-time to the PWM Outlet turbine blaes Tangential rhombus inlet nozzle Helical barriers/ inlet turbine blaes Decrease s (more flow to tank) To Application s=1 (1% flow to application) s=.75 s=.5 (5% flow to application) s=.25 s= (Flow fully iverte) Increase s (more flow to application) Fig. 2. Rotary on/off valve principle To Tank perio, is controlle by controlling the axial position of the center part relative to the rhombus inlet nozzle [1]. Flow is continuously fe into the spool regarless of the uty ratio, an the valve is esigne to rotate at between 2Hz to 4Hz. Since three PWM cycles occur per revolution of the spool, this correspons to a PWM frequency of 6Hz-12Hz, B. Non-contact Rotary Sensing A sensing system for the rotating spool is shown in Fig. 1. The rotary optical sensor consists of a laser ioe moule light source, a photoioe an a coe wheel printe on a piece of transparent meia. The coe wheel is attache to one en of the spool. A low power laser moule an a photoioe are mounte next to each other on one en of the sleeve. The coe wheel is esigne to inclue a small number of sectors an an inex. As the spool rotates, the laser beam reflects off of either a black or a white (metal) sector, causing a measurable alternating signal from the photoioe. The inex is foun by calculating the light responses each sector receives, an calculating which sector has a longer uration than the previous one. The output voltage from the photoioe is amplifie using an op-amp. The reaing is iscretize by comparing with a threshol value (Fig. 3). We are intereste in the response of the photoioe when the center of the light spot is at the bounary between the white an black sectors, Fig.4). That threshol value is calculate by averaging the transition response of the photoioe over one revolution. After iscretizing the photoioe response, a counter is use to etect the inex an the eges of each sector on the coe wheel, which represent the spool position. For example, as shown in Fig.5, the ege between the inex an the sector next to it is assigne to be zero. Following the inex, each ege represents another. The counter value is incremente whenever a iscretize signal changes values Authorize license use limite to: University of Minnesota. 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3 photoioe response (volt) iscretize photoioe response Fig Raw (top) an iscretize (bottom) signal from the photoioe. As shown in Fig.6, error in the threshol value setting will introuce measurement noise. The secon line shows the case when the threshol value is lower than the true value, while the thir line showe the case when the threshol value is higher than the true value. At a given time, the error in the measure position ue to error in the threshol value will be biase; negative if the threshol is too high, an positive if it is too low. However, since the error in the threshol value is assume to have zero mean, an the shape of the black to white an the white to black transitions are assume to be the same, the position error shoul be a zero-mean. C. System Moeling The angular velocity of the spool is mainly a function of the inlet flow rate [1] which shoul be a constant. Thus, we moel the angular velocity to be constant except for ranom effects ue to seconary flui flow forces. Since the effect of the flui on the angular velocity of the spool is ifficult to preict, a process noise term is ae to the spool ynamic moel: coe wheel light spot Fig. 4. Fining threshol value θ(t) = ω(t) ω(t) = (t) (1) where θ is the angular position of the spool, ω is the angular velocity of the spool, an is the angular acceleration of the spool which is assume to be ranom. Let the transition counter output at time t be count(t). Let the sampling perio be T an the sampling times be t k = k T. A change in the counter value signifies that either a rising or a falling transition ege has occurre between the current sampling instant an the previous one. We say that an transition event is etecte at t k if: count(t k ) count(t k 1 ) 1. (2) Fig Coe wheel interpretation / 36 / Because the coe wheel has a limite number of sectors ( 8), for the range of angular velocities (2-4Hz) uner consieration, the sampling rate is sufficiently fast to safely assume that at most one transition ege has occurre between time samples. Here we have assume that counter overflow has been properly auste. The counter reaings can provie position measurement which is accurate up to the resolution of the encoer. For example, with an 8 sector encoer, the accuracy is 2π/8 =.73ra. Our goal is to erive a Kalman filter so that it can utilize the transition event signal to increase the resolution of the sensing system. Fig. 6. error error Effect of threshol value setting error on measurement III. EVENT-BASED KALMAN FILTER A. Kalman filter problem Consier the timing iagram in Fig. Fig.7. Angular position information is provie when the light spot transitions from a black to a white sector an vice versa. The time when the th, = 1,2,..., transition of this type happens is enote by τ an the corresponing spool s angular position be enote by x p (count). Noticethatτ occur irregularly epening on the spool angular velocity. On the other han, 1548 Authorize license use limite to: University of Minnesota. Downloae on April 7, 29 at 14:22 from IEEE Xplore. Restrictions apply.

4 * 1 t 1 k m 1 tk m k 1 Fig. 7. t Time enotations the computational sampling times, t k = k T, k =,1,...,o occur regularly. A transition is etecte at t k if count(t k ) count(t k 1 )=1. Then, τ woul be uniformly istribute on (t k 1,t k ]. Therefore if we estimate the transition event to happen at τ :=(t k + t k 1 )/2, the uncertainty on transition event occurrence time can be converte to a zero mean ranom measurement noise on rotary position. We use n 1 to represent this measurement noise, which is uniformly istribute, an is boune as n 1.5ω T ; the variance of this variable is (.5ω T) 2 /3. This measurement noise is linearly affecte by the sampling time T. Another measurement noise comes from the threshol value setting, as shown in section II-B. This kin of noise is enote by n 2. The transition time from black to white or from white to black is enote by t tran. We assume this variable is a normal istribute variable with 2σ =.5ωt tran, an therefore, the variance of n 1 is ((.5ωt tran )/2) 2. Since n 1 an n 2 are inepenent, we efine the measurement noise term n := n 1 + n 2, which can be expresse as: y(τ ) = x p (count(t k ))+n * tk E[n(t)n (t)] = R(t)δ(t τ) E[w(t)] R(t) = 1 3 (.5ω T)2 +( 1 2 (.5ωt tran)) 2 (3) Define system states as X =(θ ω) T, the system moel is represente as: ( ) ( ) 1 t X(t) = X(t)+ (t) 1 }{{} F { ( 1 ) X(t)+n if t = τ y(t) = not avaliable if t τ E[(t) (t)] = q(t)δ(t τ) E[(t)] Where q(t) is assume to be a Gaussian istribute noise with zero mean an stanar eviation σ. B. Event base Kalman Filter algorithm 1 Fig. 8. A tk 1 Event-base Kalman filter time line B A t k -th event etecte time Base on the moel iscusse in the previous section, An event-base Kalman filter was esigne to continuously estimate the angular position an velocity of the spool. Time flow is shown in Fig. 8. t k is the sampling time. τ is the transition event time. Open loop manner estimation is enote by A, an measurement upate action is enote by B. The algorithm is explaine along the time line. We use the notation ˆX(t ) to enote the estimate of X(t) if the first events have been etecte, an P(t ) being the covariance of the estimate. The algorithm is initialize with ˆX(t = t = ) being the initial a-priori estimate of the the state; an P(t = t = ) being the corresponing estimation error covariance. In section A, no measurement is etecte. The Kalman filter works in an open loop manner. Estimate of the states is calculate base on the knowlege of the system moel an the most recent measurement. The estimate of the states is enote by ˆX :=(ˆθ ˆω): t ˆX(t 1)=F ˆX(t 1) (4) Since no measurement is available in this perio of time, continuous-time Kalman filter is aapte to upate the covariance of the estimation error only with the knowlege of system moel. Uner this conition, the Riccati equation for calculating the corresponing covariance of the estimation error P(t 1) is simplifie into the form: t P(t 1) = P(t 1)FT + FP(t 1)+Q(t) ( ) Q(t) = (5) q(t) An analytical solution is evelope for the equation as: P 11 (t 1)= 1 3 q (t τ 1) 3 + P 22 (τ 1 1) (t τ 1 ) 2 + P 12 (τ 1 1) (t τ 1 ) + P 21 (τ 1 1) (t τ 1 )+P 11 (τ 1 1) P 12 (t 1)=P 21 (t 1) = 1 2 q (t τ 1) 2 + P 22 (τ 1 1) (t τ 1 ) + P 12 (τ 1 1) P 22 (t 1)=q (t τ 1 )+P 22 (τ 1 1) (6) Open loop manner estimation continues until a measurement is etecte at t = t k, an this information is use to upate the estimation of the states an the covariance of the estimation error at t = τ. This is step B in Fig. 8. The posteriori state estimate is enote as ˆX(τ ), which is calculate as: ˆX(τ ) = ˆX(τ 1)+K(τ )(y(τ ) H T ˆX(τ 1)) H = ( 1 ) T K(τ ) = P(τ 1)H (7) Similarly for estimation error covariance matrix at this instant, we enote a priori estimate by P(τ 1) an a 1549 Authorize license use limite to: University of Minnesota. Downloae on April 7, 29 at 14:22 from IEEE Xplore. Restrictions apply.

5 angular position estimation error (ra) estimation of spool angular velocity Fig. 9. Angular position estimatoin error when t = 1ms Fig. 11. Angular velocity estimation when t = 1ms angular velocity estimation error (ra/sec) Fig. 1. Angular velocity estimation error when t = 1ms posteriori estimate by P(τ ) P(τ )=[(P(τ 1)) 1 + H T (τ )R 1 (τ )H] 1 (8) After the iscrete upate, open loop estimation is use again to estimate the spool states an estimation error covariace from t = τ to t = t k. But now it is estimate with the information up to : t ˆX(t ) = F ˆX(t ) t P(t ) = P(t )FT + FP(t )+Q(t) (9) This information is use with step A metho until the next measurement is obtaine, an the the above process is repeate. IV. SIMULATION AND EXPERIMENTAL RESULTS The system escribe in section II-C was simulate using Simulink with certain parameters selecte base on an experimental setup. In the simulation moel, the angular velocity of the spool was assume to be 25Hz, orω = 157.1ra/s. The encoer is assume to have the resolution of.785ra. p was set to be 1. Assume the black-white transition time is smaller than 1.5ms, set R = ra 2 /s 4, q =.25ra 2. Initial conitions for the states are selecte to be [θ ω] T =[ 5] T : The sampling time was selecte to be 1ms. Asshownin Fig.9 an Fig.1, it takes the estimation error of angular position less than.1sec to converge to less than.6ra; an it takes the estimation error of angular velocity the same amount of time to converge to less than.3ra/sec. Compare with the measurement noise boun of.36ra, the event base Kalman filter significantly improves the estimation precision. A zoom in look on the spool angular velocity estimation on the perio circle in Fig.1 is shown in Fig.11. The estimation of angular velocity will not change between measurements were etecte, an will change when an transition event is etecte. Initially, the estimation relies only on moel ynamics an the initial conition of the current open loop perio. This can explain the huge estimation error on rotary position for the first.1sec, where the estimation is running with a poor initial conition, i.e: 5ra/sec compare with the true value of 157.1ra/sec. The experimental set up ran at a sampling time of.25ms in orer to capture the valve pressure profile uring transition, an the moel was simulate at a sampling time of.25ms as well. Since the ata aquasition time is lower, measurement noise n 1 is significantly smaller, while the noise ue to threshol value setting shou be almost the same. It takes the angular position estimation error less than.11sec to converge to aroun.4ra, an the angular velocity estimation error to converge to less than.8ra/sec within the same amount of time. Compare with the simulation results with a sampling time of 1ms, estimation precision is improve at the cost of a higher computation cost. This Kalman filter is use to estimate the rotary position an angular velocity of the spool experimentally. The coewheel has a iameter of m. As shown in Fig.1, incluing the inex, the coe wheel has 8 sectors. The system is operate so that the spool rotates at two ifferent angular velocities. At each angular velocity, the system was unergoing external austment uring the initial phase. When no external austment was applie (last 1sec of each section, the estimations of angular velocities were at aroun 114.5ra/sec an 173.5ra/sec with a variation less than 1ra/sec. Fig.12 an Fig.13 both show a fast transition response. In both the starting section an the step section, the transition perio are less than.1sec. The Kalman filter reflects the ynamical change on angular velocity as well. Inlet pressure is plote with the rotary position estimation. 155 Authorize license use limite to: University of Minnesota. Downloae on April 7, 29 at 14:22 from IEEE Xplore. Restrictions apply.

6 12 1 angular velocity estimation (ra/sec) Fig. 12. Transition response when Kalman filter starts inlet pressure (psi) inlet pressure (psi) spool angular position estimation (ra) angular velocity estimation (ra/sec) Fig. 13. Transition response to step change in angular velocity As escribe in section II-A, uring one revolution, the pressure shows three repetitive pattern. Since the pressure at a fixe angular position shoul be more or less the same, if the angular position estimation is correct, the pressure v.s. estimation position over one revolution shoul be repetitive as well. As shown in Fig.14, the first line showe the pressure profile for three successive revolutions. The secon line showe ata for t = 29sec to t = 3sec, which inclues about 18 groups of ata pile together; A similar pressure profile as shown in the first line is reflecte in this plot, an the estimation variation (peak to peak) for a same pressure is smaller than.3ra. Since pressure profile varies from revolution to revolution, which will cause the profile unable to overlap with one another, we believe the Kalman filter offers a satisfactory angular positon estimation. V. CONCLUSIONS AND FUTURE WORK In this paper, a non-contacting optical sensing metho was escribe to measure the rotary position of a valve spool. Base on the measurement an spool ynamics, an event base Kalman filter was evelope to estimate both the spool rotary position an angular velocity. A system moel an an event base Kalman filter have been simulate with promising results in Matlab. The simulation results show that the moifie Kalman filter can provie continuous estimation of the rotary position an the angular velocity of the spool. Sampling time is a key factor in minimizing the estimation error. However, ecreasing sampling time will lea to a higher computational cost. Fig. 14. presure profile v.s. time an angular position estimation Increasing the coe wheel resolution can also ecease the estimation error; however, it is physically limite by the spool iameter an the laser spolt size. This Kalman filter works well experimentally, an it can reflect the ynamical change on spool states. New experimental set up is being built to test the estimation accuracy instea of the pressure profile as a robust reference. Contamination, air entrainment an oil temperature may change the optical properties of the oil. These issues will be investigate more in the future. VI. ACKNOWLEDGMENTS This material is base upon work supporte by the National Science Founation uner grant numbers ENG/CMS an EEC We thank Mike Rannow an Haink Tu from the University of Minnesota for their helpful comments an iscussion. REFERENCES [1] H. Tu, M. Rannow, J. Van e ven, M. Wang, P. Li an T. Chase, High Spee Rotary Pulse With Moulate On/off Valve Proceeings of the 27 ASME-IMECE, Paper No. IMECE27-559, 27. [2] M. Rannow, H. Tu, P. Li an T. Chase, Software Enable Variable Displacement Pumps - Experimental Stuies Proceeings of the 26 ASME-IMECE, Paper No. IMECE , 26. [3] P. Li, C. Li an T. Chase, Software Enable Variable Displacement Pumps Proceeings of the 25 ASME-IMECE, Paper No. IMECE , 25. [4] B. Anerson an J. Moore, Optimal Control Linear Quaratic Methos, Englewoo Cliffs, NJ:Prentice-Hall, Inc., 1989 [5] D. Simon, Optimal State Estimation, Hoboken, NJ:John Wiley & Sons, Inc., 26. [6] Gla,T. an L. Lung, Velocity estimation from irregular, noisy position measurements, Proceeings of the IFAC 9th Worl Congress, Buapest, Authorize license use limite to: University of Minnesota. Downloae on April 7, 29 at 14:22 from IEEE Xplore. Restrictions apply.