Electromagnetic Levitation Control with Sensorless Large Air Gap Detection for Translational Motion Application Using Measured Current-Ripple Slope

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1 Electromagnetic Levitation Control with Sensorless Large Air Gap Detection for Translational Motion Application Using Measured Current-Ripple Slope Salman Ahmed, Van-Duc Doan, Takafumi Koseki Department of Electrical Engineering and Information Systems, The University of Tokyo, Japan Abstract In this paper, sensorless magnetic levitation for large air gap translational motion is investigated. Most of the research in this field has been for magnetic bearings whose system characteristics and control requirements are different from a translational moving vehicle. An obvious difference is the range of operational air gap. Furthermore, the quality of magnetic materials, in terms of laminations, saturation, etc. used is also different. Therefore, well developed sensorless methods for magnetic bearings cannot be directly applied to large air gap applications. Thus, a novel method is proposed for detecting air gap by injecting high frequency signal into the magnetic coil and measuring either rising or falling current-ripple slope (single slope detection) using high speed sampler. Resistance is simultaneously estimated as well. Finally, stable levitation using the detected air gap signal as control feedback is demonstrated experimentally. I. INTRODUCTION Electromagnetic Levitation (EML) is a type of magnetic suspension which uses only attractive force to obtain contact-less suspension. This results in an inherently unstable system and thus requires active control. Information about the controlling quantity, the air gap, is of utmost importance in stabilizing control of EML. Hence, displacement sensors are generally used along with current sensors to obtain stable suspension. Out of two, displacement sensors form a major portion in the cost of the over all system and removing it allows significant cost reduction. It also adds redundancy allowing operation in harsh conditions where optical displacement sensors fail. Therefore, sensorless EML holds considerable importance for industrial applications. Sensorless or self-sensing EML uses the current and voltage measurements as a virtual displacement sensor. The basics of sensorless EML is to estimate the system inductance by signal processing the current and voltage measurements and using accurate system model to detect air gap from the estimated inductance. The sensorless EML is divided into two types. Observer based estimation that considers the system as Linear Time Invariant (LTI) and uses the conventional Luenberger observer to estimate the air gap using current state as an output signal. The system is shown to be observable under such assumption of LTI system []. However, it lacks robustness and is sensitive to system parameter variations as theoretically shown in []. A deliberate voltage perturbation is injected into the system and the resulting coil current ripple is then measured and passed through signal processing filters to obtain an estimate of air gap. It was first demonstrated by Okada [3]. This was later modeled as Linear Time Periodic (LTP) process and was shown to have improved robustness in comparison to the Observer based method [4]. High frequency injection methods have been mostly researched upon due to its enhanced robustness. Industrial wide spread and high energy efficiency of switching amplifiers (Inverters/H-bridges) allow injection of high frequency without additional hardware. These methods are further categorized into two types. ) One method is to use demodulation techniques to filter the current ripple and extract out the air gap. An observer like method [5] was presented which estimates the air gap using error between Band Pass Filtered (BPF) measured current and real time simulated BPF current. Relatively newer approach demonstrates the compensation of duty cycle variation and magnetic non-linearity by utilizing BPF voltage and current measurements [6]. The demodulation approach has its demerits that it induces phase delays inherent in the High Pass Filter (HPF) and Low Pass Filter (LPF). Furthermore, the duty cycle variation is compensated using additional voltage sensor. ) The second category uses a simple approach of estimating the current-ripple slope directly. This was first time presented in [7]. However, the method used additional coil for detecting current change rate. Since then, as a consequence of the availability of low cost and high speed processing hardware, research has been conducted in directly measuring the coil current rate and estimating air gap as in [] []. In [] high speed Analog to Digital Converter (ADC) is used to obtain multiple current samples and calculate the currentripple slope using least squares method. By using double slopes, the shift of resistance due to temperature change is mathematically compensated by using both voltage and current measurements. Similar results are presented in [] but with a dedicated sensing/control cycle. Since, the air gap detection relies strongly on current sensor measurement, significant amount of noise degrades the accuracy of the detection algorithm. Furthermore, in order to have an effective estimate of current-ripple slope, a threshold on the minimum number of current samples is always required /6/$3. 6 IEEE 475

2 This was done by fixing the duty cycle range in [] or by having a dedicated sensing cycle in [], which reduces the overall available DC voltage. To solve this, the proposed method maintains a minimum threshold on the number of current samples by estimating air gap from either rising or falling slope (single slope detection). However, this causes the air gap detection to be susceptible to resistance change due to temperature drift. To compensate it, resistance is estimated using intermittent injection of a dedicated double slope detection cycle at relatively a larger time period. Furthermore, the application of sensorless EML for large air gap translational motion requires different conditions than the magnetic bearings. Firstly, at large air gap, the sensitivity of inductance to the change in air gap is drastically reduced due to the leakage flux and fringing effects. This puts tighter constraints of noise on the air gap detection algorithm. Secondly, large eddy currents are induced in non-laminated iron rail, which leaves the portion of sampled current-ripple useless for slope estimation []. Keeping these issues in view, this paper also demonstrates, experimentally, proper working of our algorithm and successfully obtains stable levitation at [mm] by application of detected air gap signal as feedback. The paper is divided into six sections. In section II experimental model of EML is derived. Section III deals with derivation of the explicit relations for inductance and resistance. In section IV and V, our method is verified in simulations and experiment. In Section VI the research work is concluded and future works are presented. II. ELECTROMAGNETIC LEVITATION MODEL An accurate model of the Electromagnetic levitation system is essential for design of control and the air gap detection algorithm. Figure shows a typical EML system in which the upper part is fixed and the lower one can move vertically. Table I shows the description of the parameters used in the mathematical model of EML. Using Newton s second law and Faraday s law, we define the dynamic relations of the EML [A] [V] [m] system as: Fig.. Magnetic reluctance network. TABLE I PHYSICAL PARAMETER S DESCRIPTION Parameters Description m [kg] Mass of the mover N Number of coil turns A [mm ] Area of pole l c [m] Magnetic length µ r Relative permeability x(t) [m] air gap f mag [N] Magnetic force (Reluctance force) i(t) [A] Coil current v(t) [V] Applied voltage L(x) [H] Coil inductance λ [Wb] Flux linkage mẍ(t) = f mag mg () v(t) = i(t)r dλ () The above two equations represent the electrical and mechanical domains linked via the magnetic quantity, flux φ. This link is expressed using the following two equations [3]. f mag = φ µ A = L(x) x i(t) (3) λ = Nφ = L(x)i(t) (4) It should be noted the inductance L(x) is vital for accurate modeling. A simplified magnetic circuit of EML is shown in Fig. where R c, R g, R f, R l are reluctance associated with magnetic core, air gap, fringing effects and leakage flux respectively. Using the reluctance network, under the assumption of linear magnetic material or a dominant air gap, inductance is expressed as: [N] [N] Fig.. Electromagnetic Levitation (EML) plant L(x) = N R tot (5) Where, R tot is the total reluctance of Fig.. By solving the network, R tot is expressed in terms of individual reluctances. Compared to the overall geometry of the EML, if the air gap 476

3 L[H] Simplified analytical model. Experimental model Fitted curve.5. x [m] Fig. 3. Leakage and fringing flux effects to inductance. T T3 in the current-ripple which is exploited to detect the air gap. Using () and (4) we proceed as: v(t) = i(t)r dl(x)i(t) v(t) = i(t)r L(x) di(t) i(t) L(x) dx(t) x The dynamic frequency range of a levitating vehicle is much less than the switching frequency of the amplifier. In magnetic bearings the operational range is. mm and sensitivity of inductance L(x) x is high. Unlike that, at mm L(x) x is quite low as shown in Fig. 3. Furthermore dx(t) is around.5 m/s in our experiment. Thus, third term in () is negligible and is ignored. Integrating modified () from t to (Fig. 5) gives: t N () () V dc T4 T4 R L(x) N t v(t) = L(x) i(t N ) i(t ) t N t N t N i(t)r L(x)di(t) () t = V dc t N t N t i(t)r () Fig. 4. Switching amplifier and gate signals configuration is sufficiently small, R f and R l are ignored. The inductance is thus defined as: L(x) = N µa x(t) lc µ r (6) With R g = x(t) µ A and R c = l c µ µ r A Under the assumptions of negligible fringing effects and leakage flux, (6) is valid. However, at large air gap, the above stated assumptions are no longer valid and inductance cannot be expressed by (6). This is shown in Fig. 3. It is for this reason that in the next section, an explicit relation of inductance is derived and air gap is estimated by an experimentally obtained fitted curve as a function of inductance. As stated in section I, Fig. 3 shows that the sensitivity of inductance to air gap drastically decreases at such gap length. Therefore, tighter constraints are put on the estimation error. It must be ensured that the noise from current sensor is sufficiently less to allow acceptable estimation of inductance with low variance. III. AIR GAP DETECTION ALGORITHM Due to the industrial wide spread of switching amplifiers, they are used as source of high frequency injection as well. Figure 4 shows a typical switching amplifier which is employed in our experiments. The switching amplifier generates a pulsating voltage given as: { Vdc < t < αt v(t) = pwm (7) V dc αt pwm < t < T pwm In response to (7), electromagnetic coil current response is shown in Fig. 5. Information about inductance is embedded Let i = i(t N ) i(t ) t N t and ī = t N t L(x) = V dc ī R i N t i(t). () Within a switching period, only two measurements of the rising part of current-ripple are sufficient to calculate its slope i. But due to measurement noise, the slope calculation has large variance. To provide sufficient accuracy, multiple current samples are used and current-ripple is modeled as a straight line by employing the Least Squares Method (LSM). This approximation of the current-ripple as a straight line is valid only if the time constant of LR circuit is sufficiently larger than the switching period T pwm so that the slope remains nearly constant. The sampled data and the estimated current-ripple slope using LMS is given as: i = [i, i, i 3, i N ] t = [t, t, t 3 t N ] i N = k= (i k ī )(t k t) N k= (t k (3) t) Where t is the mean of the array t.the same procedure can be applied to obtain the inductance by negative slope calculation. L(x) = V dc ī R i (4) Once the inductance is correctly calculated, an experimentally obtained data set, fitted to a rational polynomial x est = f(l), between air gap and inductance is used to obtain the estimated air gap signal. A. Single Slope Detection Algorithm Since the duty ratio α continuously varies when the control law is being executed, the time duration of positive and 477

4 V dc v(t) <.5 >.5 R Update (Double Slope) L(x) (Single Slope) R Update (Double Slope) L(x) (Single Slope) R Update (Double Slope) i(t) time T pwm time V dc T pwm t t t t N t t N T s T pwm T R T R Fig. 6. Resistance detection by intermittent double slope cycles with time period T R Fig. 5. Timing diagram for single slope detection negative slope changes. Consequently, the number of corresponding current samples (N and N ) vary as well. To ensure a threshold on maximum error in slope calculation, number of current samples should not be reduced below a specific number. Otherwise slope estimation is erroneous. To prevent that, in previous works α is restricted to.5.75, limiting the available DC voltage. Having % of the DC bus voltage allows for a larger range of control effort and improved current control. The total number of current samples remain constant in a switching period. For α >.5, rising duration of currentripple is more than the falling duration and vice versa for α <.5 as shown in Fig. 5. Therefore exploiting the fact that it depends on geometrical parameters, inductance is estimated using either positive slope or negative slope, whichever has higher number of current samples. This ensures that there are always enough samples to obtain an estimation within desired error bounds. Lastly, current samples are categorized into rising or falling at real time. Instead of using a voltage sensor, the algorithm makes use of the fact that the successive switching period s duty cycle is known once the control law is executed. The successive α is calculated by (5), where V avg is the output of the control law. α = V av V dc.5 (5) The firing instants of gate signals T, T, T 3, T 4 in Fig. 4 are decided for the successive period. A counter t count emulates the time steps of the switching period. By this, the digital computer is at all times aware of the output voltage sign and hence the slope direction, and accordingly stores the current samples in corresponding category (i or i ). Systematic flow of the algorithm is given as follows: ) Calculate α ) Update the gate signals as: If ( < t count αt pwm ) T,T On If (αt pwm < t count < T pwm ) T3,T4 ON 3) Update the sample array as When(T,T = ON) Store in i When(T3,T4 = ON) Store in i 4) When (t count == T pwm ) If α.5 Use i If α <.5 Use i B. Resistance Detection By Double Slope The single slope detection is susceptible to resistance change. The estimated value from positive and negative slope is equal only if the resistance is accurately known. In practical systems, the resistance changes due to temperature drifts. Therefore, for proper working of single slope detection algorithm, accurate resistance value is required. In this section, resistance estimation is proposed by using both positive and negative slopes within one switching period. But since the time constant for resistance change due to temperature is much larger than the EML time constant, a deliberate cycle of fixed duty ratio (.45.55) to allow for sufficient time (samples) for rising and falling interval (double slope cycle) is injected at intermittent instants with time period T R as shown in Fig. 6. The EML control and air gap estimation proceeds unaffected and the available DC voltage also does not decrease. Equating () and (4), and with some algebraic manipulation estimated resistance is expressed as: R est = V dc( i i ) ī i ī i (6) Where R is replaced by R est. It is observed, later in experiments, that resistance update after every T R causes sudden changes in air gap estimation. This is associated to the fact that inductance changes suddenly once resistance updates, causing a sudden change in detected air gap due to its low sensitivity. To compensate it, estimated resistance is passed through a simple first order low pass filter with a high time constant τ R. This smooths the detected resistance value. In addition, (6) is invalid for cases when the denominator is zero. Physically, it is equivalent to saying i avg =, where i avg is the average current of one switching period. Practically, it has no fatal effects since, when i avg =, the effect of the term ī R in () is minimal. This is avoided by a coding routine which uses nominal R at i avg =. 47

5 TABLE II N UMERICAL VALUES USED IN SIMULATIONS / EXPERIMENTS Value Parameters N Tpwm [sec] TR [sec] R [Ω] i [A] Value Electromagnet Air Gap Sensor Linear Guide Levitating Object PI Control PD Control T Voltageto T Gate-signal T3 T4 converter Electromagnetic Levitation System Switching PowerAmplifier DH- bridgem The proposed algorithm is verified first by a numerical case study. Matlab/Simulink is used for simulations. EML plant is modeled in Simulink by () and (). We have used the experimentally obtained relation between L(x) and x(t), and consequently L(x) x for fmag model, thus all un-modeled dynamics such as leakage flux, fringing etc in the magnetic circuit are accounted for. Table II shows the parameter values used in numerical case study as well as experiment. Two feedback control loops are employed. Since EML system is contact less and thus has no natural damping, necessary artificial damping is required for stabilization. This can be achieved with the most basic derivative (D) control. Thus outer loop constitutes only a Proportional-Derivative (PD) law. Inner Proportional-Integral (PI) law is used to control coil current. This allows independent design for each loop. The controller gains are obtained by polynomial design approach using Manabe s Canonical [4] form solved for the respective plant s transfer functions. The schematic of the feedback loop and the proposed estimated algorithm is shown Fig. 7. Simulation results for the proposed algorithm are shown in Fig. with clean and a noisy current signal. In both cases, the estimated signal stabilizes the plant. However, it is observed that xest estimation error is strongly susceptible to current measurement noise. 3 3 x Air gap [m] Air gap [m] x R IV. S IMULATION C ASE STUDY. 3 x Proposed Method Fig. 7. Schematic of the feedback control loop and proposed estimation method.. Fig.. Experimental test bench. Measured current-ripple for Tpwm..5.5 Fig.. Simulation results of air gap estimation. With clean current signal. Noisy(σ = 3 ) current signal. 47 Average Estimated Air Gap[m] Coil Current [A] Parameters m [kg] Vdc [V] Ts [sec] τr [sec] x [m] L [H] A A 3A 4A Actual Air Gap [m] Fig.. Linearity range at different current values. V. E XPERIMENTAL V ERIFICATION Experimental verification of the proposed method is carried on an indigenously built test bench in our laboratory as shown in Fig. a. The control and estimation algorithms are implemented in a Texas Instrument s micro-controller TMS3F377S. The Micro-controller outputs a PWM to a switching power amplifier that drives the electromagnetic coil of the plant. The current sensor used in our experiment is LEM LTSR 6-NP with a sensitivity of 6 [mv/a]. Air gap sensor, Omron Z4W-V5R LED Displacement Sensor, is only used to log measured air gap for comparison purposes. Figure b shows the measured current ripple for one switching period. The noise in the measurement causes slope estimation to have large variance which does not produce air gap signal within acceptable error bounds. Furthermore, rail core used for translation motion vehicle has significant amount of eddy currents. In our system alike, the instants right after switching are influenced greatly by the eddy currents, dying out after approx 5[µsec]. Thus, rendering those instants useless in slope calculation. Remaining current-ripple region produces accurate slope estimation only if noise is small enough or there are sufficiently large number of current samples. In our experiment, the noise from current sensor, though not obviously visible, caused poor estimation and therefore to allow for good estimation, a switching period Tpwm of 3 [sec] is used. With ADC sampling time Ts =.5 6 [sec], it allows 6 samples in one period with minimum samples of for each slope calculation. This

6 Powered by TCPDF ( Air Gap [m] x Air Gap [m].5 x Fig.. Stable levitation with x est as feedback. Stable levitation. Step response. Resistance [Ω]....7 Nominal Air Gap [m] x x est (R compensated) x est (R constant(nominal)) x measured Fig.. Estimated resistance. Comparision of air gap estimation with and without R compensation. choice for T pwm and T s produces acceptable error for stable levitation. An initial test to determine the range of operation is conducted. Fig. shows a fairly linear relation between actual and estimated air gap under non-saturation condition of core. High current and small air gap results in near saturation values of magnetic field causing error offset in estimation. This puts a strict limitation on operation at only linear region of BH curve. Using the controller gains from numerical case study (with slight tuning), the results are shown in Fig.. We explicitly state here that x est is used as a feedback signal to the PD controller. Figure a shows that at stand still, the levitating object is stabilized at the nominal gap length. A small offset in the estimated and measured signal is present. Since around [mm], the sensitivity is low, therefore a small mismatch between actual value and the experimental curve x est = f(l(x)) results in an offset error. The aim is to achieve stable levitation and not reference tracking, thus we choose to ignore this offset. Experimental verification of resistance detection algorithm is shown in Fig.. By levitating at.5[mm], increased current causes temperature rise and consequently resistance change. Fig. a shows the estimated resistance. With a change of.3[ω], x est, without R compensation, drifts down to [mm] from.3[mm] as shown in Fig. b. On the contrary, by using the proposed R detection algorithm, x est remains around the nominal value demonstrating the practical goodness of our method. VI. CONCLUSION A novel air gap detection method with resistance compensation is proposed and verified experimentally. The proposed method ensures enough current samples for both slope estimations thus allowing greater range for duty cycle and consequently entire DC link voltage. Being susceptible to resistance change, R detection method is also proposed. In our experiments, choice of proposed method s parameters such as T pwm, T s, T R, τ R were chosen based on trial and error evaluation of current noise and output response. Appropriate choice could be obtain by detailed mathematical analysis. Also the straight line approximation for current ripple in our experiment is weak, since T pwm is in comparison to the system bandwih. By using a low noise current sensor or a fast ADC, T pwm can be reduced. 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