Smart Combination of Sensorless Electromagnetic Levitation and Zero Power Control: A Complimentary Pair Enhancing Mutual Strengths

Transcription

1 Smart Combination of Sensorless Electromagnetic Levitation and Zero Power Control: A Complimentary Pair Enhancing Mutual Strengths Salman Ahmed, Takafumi Koseki Department of Electrical Engineering and Information Systems, The University of Tokyo, Japan s.ahmed@koseki.t.u-tokyo.ac.jp Abstract In this paper a unique combination of sensorless electromagnetic levitation stabilized by zero power control is experimentally demonstrated. The sensorless method includes high frequency voltage injection to electromagnetic coil by a switching amplifier and measurement of the resulting currentripple slopes. Inductance is estimated from a weighted contribution of both (positive/negative) slopes from which air gap is calculated. Zero power control forces average current to approach zero and consequently duty ratio to., which is an optimal case for least dynamic noise in the gap detection. Furthermore, since sensorless methods are often influenced by magnetic circuit modeling errors, instead of absolute and exact air gap, rather a proportional signal to air gap is estimated which has a possible offset. Zero power control being a reference current tracking control, thus does not require absolute information of air gap. These mutually enhancing characteristics show that such combination has potential of a simple, robust and low cost magnetically levitated carrier in industrial environment. I. INTRODUCTION Electromagnetic Magnetic Levitation [EML] allows suspension with no physical contact. However, this comes at expense of active feedback control which requires continuous monitoring of air gap and a continuous power consumption. Air gap sensors form a major part of total cost of an EML system and removing them allows for drastic cost reduction and hence the term sensorless or self-sensing EML. The basics of sensorless EML includes measurement of terminal voltages and currents to estimate a proportional signal to air gap using magnetic circuit modeling. Conventionally, these methods are divided into two categories. The first one uses state observers by considering current as output and estimating remaining states of air gap and velocity [], but it has been theoretically shown to lack robustness to parameter variations in []. The second type uses deliberate injection of high frequency to perturb the EML system and record its response, first demonstrated by Okada [3]. This method is more robust [] and has been researched upon in past two decades [] []. The injection of high frequency in most cases is achieved by widely used and industrially prevalent switching voltage amplifiers. Frequency injection methods are further divided into two types. One involves using series of signal processing filters to extract air gap signal [], []. The second type uses a more direct approach by measuring the response of high frequency voltage injection, which is a triangular current waveform, using fast Analog to Digital Converters (ADC) [7] []. As a consequence of the availability of low cost and high speed processing hardware, this method has been gaining acceptability. However, it is often influenced by varying duty ratio. This problem was solved by fixing the duty cycle range as in [7] or by having a dedicated sensing cycle as in []. But this reduces the available DC link voltage. We previously introduced a method having robustness to duty ratio variation with no restriction to its range by using weighted contributions of rising and falling current-ripple slopes to calculate inductance and consequently air gap []. In this paper, we explicitly demonstrate the goodness of weighted slopes method and show a logical flow to deciding the linear range for weights. The optimal duty ratio condition for least dynamic noise is also shown to be in vicinity of.. Of all the control methods applied for stabilization of EML systems, zero power control [] [] stands out as the best possible control law to be combined with sensorless methods. Zero power control not only minimizes power consumption but also enhances certain features of sensorless method. The requirement of. duty ratio for least dynamic noise is for most part fulfilled as zero power maintains a constant zero average coil current. Furthermore, due to slight modeling error in inductance vs air gap profile which is used to calculate air gap, an error offset is present. Zero power being a reference current tracking control does not need absolute and exact air gap signal. A gap signal with error offset and low noise variance is sufficient to achieve stable levitation. Since exact position depends on the total weight of the levitating object, we are of opinion that to give up exact position for gaining low cost device using sensorless method is an efficient bargain. Therefore, we have proposed this smart combination of our sensorless method with zero power control and experimentally verified the claim. This has the potential of being a robust and low cost solution for magnetically levitated carriers in industrial applications. Lastly, according to authors best knowledge, such unique and smart combination for large air gap EML systems has never been explicitly discussed or demonstrated before. II. ELECTROMAGNETIC LEVITATION MODEL Electromagnetic levitation plant model is first derived. The free body diagram and its corresponding magnetic circuit is

2 () [A] ()[V] ()[m] Permanent Magnet () () H c Permanent Magnet H c L[H].... Simplified analytical model. Experimental data Fitted curve [N] [N] Fig.. Electromagnetic Levitation (EML) Model. Magnetic circuit with permanent magnets. Parameters m [kg] N H c [A/m] A [m ] l c [m] l m [m] μ r μ m [m] f mag [N] i(t) [A] v(t) [V] λ [Wb] TABLE I PHYSICAL PARAMETER S DESCRIPTION Description Mass of the mover Number of coil turns Coercivity of Permanent Magnet (PM) Area of pole/core/pm Magnetic path length Length of PM Relative permeability of core Recoil permeability of PM Air gap Magnetic force (Reluctance force) Coil current Applied voltage Flux linkage shown in Fig.. The physical parameters used are described in Table. I. Newton s and Faraday s law for Fig. are given as: v(t) =i(t)r dλ dt mẍ(t) = f mag mg () = i(t)r d(nφ(t)) dt The two equations represent the electrical and mechanical domains linked via the magnetic quantity, flux φ. Using Fig. where R c, R g, R f, R l,r m are magnetic reluctance associated with magnetic core, air gap, fringing effects, leakage flux and permanent magnet respectively, under the assumption of linear magnetic material or a dominant air gap, the magnetic reluctance force is given as: f mag = () φ μ A = μ ( ) A Ni(t)Hc l m (l m l (3) c ) μ r Where we have assumed μ m is linear and approximately equal to μ, and have ignored leakage and fringing effects with remaining reluctances as: R g = μ A R c = l c μ μ r A R m = l m μ m A x [m] Fig.. Inductance profile displaying fringing and leakage effects. For the same magnetic circuit, the inductance L(x) is calculated as: L(x) = λ(t) i(t) = μ AN l m l c μ r () Using (), f mag is expressed for convenience as: f mag = L(x) (i(t)i m) () Where the effect of PM has been represented as constant equivalent current i m = Hclc N. This facilitates identification of plant parameters by reducing experimental requirements to just L(x) profile and i m = mg. It is important to note L that at higher air gap values, () becomes invalid due to nonnegligible fringing and leakage effects as shown in Fig.. In order to design zero power control, the dynamic equations () and () are linearized at nominal operating point (x,v,i )= (x,, ). Assuming =x Δ and i(t) =i Δi(t), linearized () is given as: mδẍ(t) = k x Δ k i Δi(t) () where k x = L (i i m ), k i = L (i i m ) and f mag By substituting Nφ = L(x)i(t) and (x,v,i )=mg. taking L(x) = L ΔL(x) and v(t) = V Δv(t), () linearizes to: Δv(t) =Δi(t)R L Δ i(t) k i Δẋ(t) (7) N μ A Where V = i R and L = is nominal x l m lc μ r inductance. Using () and (7), EML plant model is represented [ as state feedback ] system with state vector x = T Δ Δẋ(t) Δi(t) as: Δẋ(t) Δẍ(t) = kx m ki Δ m Δẋ(t) Δv(t) Δi(t) k i L R L Δi(t) L (8)

3 () () t <. >. t time Fig. 3. Current response to high frequency voltage injection III. WEIGHTED CURRENT-RIPPLE SLOPES METHOD Sensorless methods employ switching amplifiers to inject high frequency voltage into the EML system. Output of such a switching amplifier is given as: { Vdc <t αt v(t) = pwm (9) V dc αt pwm <t<t pwm Where V dc is the DC link voltage, α [ ] is duty ratio and T pwm is switching amplifier time period. In response to this voltage, the resulting current-ripple waveform is shown in Fig. 3. Integrating () from t to t N : N t v(t)dt = N t N t N i(t)rdt L(x)di(t) t i(t) L(x) d () L Since at gap lengths x =[ 8], and average change in air gap within the interval t t N is relatively small, the third term on the right hand side of () is ignored: L(x) i(t N ) i(t ) t N t Let Δi = i(t N ) i(t ) t N t = V dc t N t and ī = t N t N t N t i(t)rdt i(t)dt. L est = V dc ī R () Δi Where superscript denotes values during positive current slope and L est=l(x) with a fair assumption of constant inductance within one T pwm valid for T pwm << τ LR = L R. For fixed V dc and R, inductance can be estimated from current and its slope ( i, Δi ). Two current samples are mathematically sufficient to calculate current-ripple slope but due to measurement noise of current sensor, small deviations of samples cause large noise variance in inductance. For this reason, current is sampled multiple times and using statistical concept of Least Squares Method (LSM), slope is calculated with low noise variance as: i =[i,i,i 3, i N ] t =[t,t,t 3 t N ] Δi N = k= (i k ī )(t k t) N k= (t k () t) w i.8... δd w w. Duty Ratio. Duty Ratio Fig.. Piece wise linear weight function. Possible shapes of Piece wise weight function (. <δd<.). w.8... δd=. δd=. δd=.3 δd=. δd=. δd=. Where t is the mean of the array t. Similarly, using negative slope, inductance is calculated as: L est = V dc ī R (3) Δi Where superscript denotes values during negative current slope. Division of current samples into positive i and negative i category is achieved by exploiting the fact that duty ratio of successive switching period is known once control law is executed as: α = V avg. () V dc Where V avg is controller average voltage. Successive time period s α allows a pre-division of T pwm into rising t and falling t intervals, and current samples are sorted accordingly as explained in [] in detail. The noise variance in L est and L est depends on the size of i and i respectively. But since control law is executed continuously, the noise cannot be controlled. For extreme values of α, one slope has less noise while other one has large noise variance and vice versa. Therefore, a weighted contribution of both slopes is used to calculate inductance where the weights are functions of duty ratio. The inductance thus is given as: L est = w L est w L est () Where w and w = w are respective weight functions of duty ratio. Weight functions can be of different categories []. We have used a piece wise function as shown in Fig. and given as: w = α<(. δd ) (α.) δd. (. δd ) α (. δd ) (. δd ) <α () Where δd is the width of linear range. Lastly air gap is calculated from L est. It is noted here that instead of using () which is invalid for large air gaps, an experimentally obtained data base, shown in Fig., fitted to a rational curve = f(l est ) is used. Statistical goodness of the weighted slopes method over previously researched double slope ( [7], []) and single slope method [] is experimentally verified by comparing the quality of estimated inductance L est in each case. Position of

4 Normalized estimation noise [%] Normalized estimation noise [%] Duty ratio Duty ratio Normalized estimation noise [%] Duty ratio (c) (d) Fig.. Normalized estimated inductance noise: Weighted slopes method. Double slope method. (c) Single slope method. (d) Standard deviation σ of normalized estimated inductance noise. levitated object is fixed at about 8[mm] which corresponds to L(x) =. 3 [H]. Electromagnetic coil is excited with simultaneous execution of double, single and weighted slopes detection methods, outputs of which are L est, L est and L est respectively. The α is varied from. to.8 and estimated inductances are logged. At each value of α, inductance is calculated times allowing calculation of noise spread. For V dc =[V] and α =.8, the steady state current exceeds the maximum allowable coil current, therefore for this analysis V dc =[V] is applied. Consequence of such change results in increased estimation noise. But since it is a comparison analysis, all three cases are equally affected. The noise spread of normalized estimated inductance is shown in Fig.. It is evident from Fig. that double slope method has least noise around α =.. But it increases exponentially as α approaches its extreme value. This is due to the decrease in number of current-ripple samples for the falling interval. The single slope method shows robustness towards α change. However, its noise spread is larger than double slope method for α near.. Secondly, single slope is also susceptible to resistance change which we have not considered in this analysis. Weighted slopes method combines the advantages of both as shown in Fig.. Weights depending on α are calculated using the piece-wise function (Fig. ) which exploits the low noise characteristic of double slope method at α. and robustness of single slope method for large values of α. The above information is represented in a more compact form by calculating the standard deviation σ for each case as shown in Fig. (d). The piece wise function s parameter δd is set by choosing the value of α where L est and L est s trend intersect each other which in this case is.7 resulting in δd =.. Standard Deviation ( σ) [%] Lest L est L est Duty ratio IV. SENSORLESS EML WITH ZERO POWER CONTROL The proposed combination is shown in Fig.. The only input to the digital computer using sensor is coil current. Furthermore, due to zero power control the power consumption is minimal. Hence this combination produces a cost effect EML system with low initial and operating cost. In the design of the controller, air gap detector is taken as a noisy sensor and conventional method of designing state feedback controller is used. Pole placement is done by using the Manabe s canonical form to ensure stable closed loop system []. For zero power control the input to the plant is: v(t) = Kx k e x e (7) Where x e = i ref Δi(t), K = [k,k,k 3 ] and k e is the integral gain. The closed loop system is derived as : [ ] [ ] ẋ = kx m ki m x x e k k i L L k L R L k3 k e L L x e i ref A Solving det si A =, the characteristic polynomial obtained is: ( R s k ) ( 3 k s 3 i k ik k x L L ml ml m k ) e s L ( kx R k xk 3 k ) ik s k ek x = ml ml ml ml (8) From Manabe s method [], for a th order polynomial With s a 3 s 3 a s a s a = (9) a a = τ z and a i a i a i = λ i Where τ z is the equivalent time constant and λ i are stability index which for Manabe s case are λ =. and λ i =for i>. The gain matrix K and k e are calculated using (8) and (9) as: ( k = k x R k x k 3 (.3 )( )() ) τz 3 ml k i ( k k = i k x ml m k e (. )( )() ) ml L τz k i k 3 = (.)()() L R τ z k e = (.3 )( )()ml τz k x V. EXPERIMENTAL VERIFICATION The benefits of this smart combination are experimentally validated on a test bench shown in Fig.. It consists of two iron C-cores with upper one fixed and lower one movable vertically. Other degree of freedoms (pitch, yaw) are

5 T T T3 T Electromagnet Air gap sensor Linear guide Levitating object Permanent magnets Inside digital computer Loading compartment Fig.. Sensorless EML with zero power control. Experimental test bench. TABLE II PARAMETER VALUES USED IN EXPERIMENTS Parameters Value Parameters Value m [kg]. T s[sec]. V dc [V] T pwm [sec] 3 R [Ω].7 x [m]. 3 i [A] i m[a]. l m[m] 3 L [H]. 3 k i [N/A].3 k x [N/m].8 3 τ z [sec].7 k [A/m] 37 k [As/m]. k 3 [V/A]. k e[v/as] δd. restricted by using linear guides. The digital computer used is Texas Instrument s micro-controller TMS3F8377S. The current sensor used is LEM LTSR -NP with a sensitivity of [mv/a]. Air gap sensor, Omron ZW-VR LED Displacement Sensor, is only used to log measured air gap for comparison. The parameters of test bench and controller are given in Table II. Experimental results are presented in Fig. 7. An additional weight of.88[kg] is added (Fig. 7) and removed (Fig. 7) to demonstrate the change of position for a zero average coil current. There remains a constant offset error between and, attributed to the modeling error in inductance profile, but since the objective of zero power control is to maintain zero average current i.e reference current tracking, exact and absolute air gap signal seems redundant. It is emphasized here, that a signal having an error offset with low noise variance such as is sufficient for stable levitation. Since exact position depends on the total weight of the levitating object, the authors feel that it is an efficient bargain to give up exact position for gaining low cost device using sensorless method. Furthermore, Fig. 7(c) and Fig. 7(d) show that for most part, α. which ensures minimal noise variance for our air gap detection method. However, addition of PM into magnetic path results in increase of total magnetic reluctance. PM consumes the available physical gap length between fixed core and mover. This shifts the L(x) vs profile towards left as shown in Fig. 8. Since sensorless method s output noise strongly depends on sensitivity of inductance to air gap L, the decrease Duty Ratio α x (c). Duty ratio α. 3. x (e) (f) Fig. 7. Weight loading: Position. (c) Duty ratio α. (e) Average coil current. Weight unloading: Position. (d) Duty ratio α. (f) Average coil current. in L B for the operating point x in zero power control produces high variance noise, compared to the case ( L A ) when there is no PM []. Stable levitation profile for two cases (x =.[mm], x =8[mm]) is shown in Fig. 9. The levitation profile for x =8[mm] is oscillatory owing to the aforementioned reasons. Furthermore, loading and unloading results for x =8[mm] are also shown in Fig.. To obtain.. (d)

6 L(x) Without PM With PM Air gap [mm] Fig. 8. Inductance shift due to insertion of PM in magnetic path resulting in decrease in sensitivity to air gap.. x x 3 7 Fig. 9. Stable levitation profile: At i ref =[A],.[mm]. At i ref =[A], 8[mm]. i(t) =at large air gap requires a new design of PM and modification to the test bench accordingly. Therefore, for testing purpose only, x =8[mm] is achieved by setting i ref =[A] emulating a stronger PM with increased i m =(. )[A] x x 3 (c) (d) Fig.. Weight loading: Position. (c) Average coil current. Weight unloading: Position. (d) Average coil current..... VI. CONCLUSION A simple, robust and low cost solution for magnetically levitated carriers for industrial environment has been experimentally demonstrated. The weighted current-ripple slope measurement method has been experimentally shown to possess low noise and robustness to duty ratio variation. Lastly, smart combination of sensorless method with zero power control is shown to be mutually enhancing each other s performances. It has been pointed out that inclusion of PM results in increased magnetic path reluctance and consequently decrease in sensitivity of inductance to air gap which is pivotal to sensorless method at large air gap. Optimization of magnetic path so as to achieve minimum increase in magnetic reluctance due to PM is an important next step which will be dealt in future. REFERENCES [] D. Vischer and H. Bleuler, Self-sensing active magnetic levitation, Magnetics, IEEE Transactions on, vol. 9, no., pp. 7 8, Mar 993. [] N. Morse et al., Position sensed and self-sensing magnetic bearing configurations and associated robustness limitations, in Decision and Control, 998. Proceedings of the 37th IEEE Conference on, vol. 3, 998, pp. 99 vol.3. [3] Y. Okada et al., Sensorless magnetic levitation control by measuring the pwm carrier frequency component, in International Symbosium on Magnetic Bearings, 99. [] E. H. Maslen et al., Robustness limitations in self-sensing magnetic bearings, Journal of Dynamic Systems, Measurement, and Control,. [] M. Noh and E. H. Maslen, Self-sensing magnetic bearings using parameter estimation, Instrumentation and Measurement, IEEE Transactions on, vol., no., pp., Feb 997. [] A. Schammass et al., New results for self-sensing active magnetic bearings using modulation approach, Control Systems Technology, IEEE Transactions on, vol. 3, no., pp. 9, July. [7] T. Gluck et al., A novel robust position estimator for self-sensing magnetic levitation systems based on least squares identification, Control Engineering Practice, vol. 9, no., pp. 7,. [8] A. Ranjbar et al., Estimation of airgap length in magnetically levitated systems, Industry Applications, IEEE Transactions on, vol. 8, no., pp. 73 8, Nov. [9] A. C. Niemann et al., A self-sensing active magnetic bearing based on a direct current measurement approach, Sensors, vol. 3, no. 9, p. 9, 3. [] J. Wang and Binder, Position estimation for self-sensing magnetic bearings based on double detection of current slopes, th International symposium on Magnetic Bearings,. [] S. Ahmed and T. Koseki, Noise suppression strategy for sensorless electromagnetic levitation using current-ripple measurement method with extended operational range of duty ratio, in Electrical Machines and Systems, 9th IEEE International Conference on, Nov. [] M. Morishita et al., A new maglev system for magnetically levitated carrier system, IEEE Transactions on Vehicular Technology, vol. 38, no., pp. 3 3, Nov 989. [3] J. Liu and T. Koseki, 3 degrees of freedom control of semi-zeropower magnetic levitation suitable for two-dimensional linear motor, in Electrical Machines and Systems. Proceedings of the Fifth IEEE International Conference on, vol., Aug, pp vol.. [] T. Mizuno and Y. Takemori, A transfer-function approach to the analysis and design of zero-power controllers for magnetic suspension systems, Electrical Engineering in Japan, vol., no., pp. 7 7,. [] S. Ahmed et al., Electromagnetic levitation control with sensorless large air gap detection for translational motion application using measured current-ripple slope, in Industrial Electronics Society, IECON - st Annual Conference of the IEEE, Oct. [] S. Manabe, The coefficient diagram method, in th IFAC Symposium on Automatic Control in Aerospace, 998.