µ Control of a High Speed Spindle Thrust Magnetic Bearing Roger L. Fittro* Lecturer Carl R. Knospe** Associate Professor * Aston University, Birmingham, England, ** University of Virginia, Department of Mechanical and Aerospace Engineering, Charlottesville, Virginia, 2293 r.l.fittro@aston.ac.uk, crk4y@virginia.edu Abstract Experimental results demonstrate that µ synthesis provides an excellent tool for increasing the dynamic stiffness of thrust magnetic thrust bearings. Accurate system modeling and appropriate uncertainty descriptions are critical to obtaining such results and are described in detail. Both complex µ and mixed µ synthesis were used to design controllers. The performance of the µ controllers is compared to that of an optimized PID controller. The dynamic stiffness of the spindle's thrust axis was increased by a factor of two over that obtained with optimized PID control. 1 Introduction Active magnetic bearings (AMB) have several advantages over conventional rolling element bearings for high speed machining including stiffer spindles, adaptive balancing, and reduced compliance via active control. Over the last several years, the authors have been developing a new high speed machining spindle controlled by active magnetic bearings. This spindle has been magnetically levitated via decentralized feedback control and calibrated. Multivariable controllers have been developed for both radial and axial directions via application of µ- synthesis [7]. Herein, we focus on the development of a system model with an appropriate uncertainty description for the thrust axis of the spindle and present theoretical and experimental results. The µ- synthesized controller achieves much better performance than that obtainable using conventional control methods. This result may seem surprising when one considers the SISO nature of this problem. 2 High Speed Spindle Several magnetic bearing machining spindles are presently in production or laboratory testing [1-4]. To further explore the potential of AMBs for spindle control, the authors have developed a high speed spindle [7]. Figure 1 shows a photograph of the spindle s rotor upon which the bearing journals, thrust disk, and motor rotor are carried. The objective is to control the flexible rotor such that motion at the cutting tool tip is minimized. Of particular importance here, the thrust disk is fabricated from non-laminated, high strength steel and is attached to the shaft via an interference fit. It is acted upon by opposing non-laminated thrust stators. The spindle is controlled by a parallel processing digital controller designed and built at the University of Virginia [5]. Four Texas Instruments TMS32C4 digital signal processors are employed for control algorithm computations. Controller computational power is sufficient to implement a 7 input, 7 output, 75th order controller with a throughput rate in excess of 12 khz. Figure 1: Spindle rotor 3 µ Control Since the cutting forces vary drastically over the range of spindle speeds, feedrates, cutting tools, and cutting conditions (e.g., slotting, face milling, end milling), and the quality of the cut is directly dependent on tool vibration, the control problem posed has an H performance requirement [6]. Plant uncertainties must be included in the design procedure since the rotor is very lightly damped. Fortunately, the uncertainties in this system are highly structured (often parametric). Therefore, controllers may be designed which sacrifice only a little performance to achieve the necessary robustness. The µ-synthesis procedure permits the design of multivariable controllers for complex, uncertain linear systems so as to minimize an H cost function. It is a rather natural augmentation of H control theory with the analysis techniques of the structured singular value µ.
4 Model, Uncertainty Representation, and Performance To model the AMB spindle, models of the magnetic actuators, rotor, substructure, amplifiers, sensors, antialiasing filters, and digital controller must all be obtained. Also, an uncertainty description is needed for each of these components. 4.1 Sensors Since the optical sensors used are linear and have very high bandwidth, they may be modeled as a gain. The thrust axis sensor was calibrated in situ repeatedly so as to determine its gain and the range of error that results from our calibration procedure. In this manner, the gain was obtained with a residual uncertainty of ±3%. This uncertainty is treated via a scalar multiplicative uncertainty block in the LFT system model. 4.2 Switching Amplifiers The commercial transconductance power amplifiers for the thrust actuators have a switching frequency of 22 khz and are rated for 3 amps peak and 15 amps continuous operation. A sine sweep test was conducted on each amplifier driving the bearing load with the rotor shimmed to its nominal gap. This experimental data was then fitted (in an H sense) with a fourth order low-pass filter. The frequency response of the model obtained matched the experimental data very well up to 1 khz, see Figure 2. A scalar, complex, multiplicative uncertainty of magnitude 2.5% covers the magnitude and phase differences between the data and the rational system provided by the fit. K i, axial loads were applied and then the perturbation current i p that resulted when the integral term returned the rotor to the centered position was recorded. The resulting data for various applied loads is shown in Figure 3. The scatter in the data results from the significant hysteresis of the thrust disk's material, high strength steel. A least squares fit of the data provides the line also shown in the figure, the slope of which is the nominal value of K i. Given the data, how should one assign the uncertainty in this value? We used an empirical approach by answering the question: if half the data points are used to determine a new least squares fit, how much can the slope of this differ from the nominal slope found using all the data? This yielded an answer of 11% for the data shown. Force (lbf) 8 7 6 5 4 3 2 1-1 8.2 8.4 8.6 8.8 9 Current (Amps) Figure 3: Thrust K i Calibration -5-1 -15 1 4 1 4 Figure 2: Amplifier Experimental Data (solid) and Model (dashed) 4.3 Actuator For a low frequency description of the actuator, two parameters must be determined: the actuator gain K i (lbf/amp) and the open loop stiffness K x (lbf/inch). These were ascertained for the thrust actuator via an experimental calibration procedure with the rotor in support under a nominal PID controller. To determine The actuator open loop stiffness K x can be shown to be related to K i by i p K x = Ki (1) x x= i p = The partial derivative here may be determined by adding a perturbation signal to the thrust sensor signal so that integral term causes the spindle to translate. Then, the perturbation current employed with the rotor stabilized in this off-center position is recorded. The resulting data is highly repeatable and quite linear. The desired partial derivative is the slope of this line at zero displacement. From this the value of K x can be determined using Eqn. 1. Note that the uncertainty in the value of K i results in an uncertainty in the value of K x. Thus, a multiplicative uncertainty of 11.5% was dictated for the system description, resulting largely from the uncertainty in K i. Since the thrust actuator (both disk and stators) is not laminated, significant eddy currents will arise which have a profound effect on the actuator s transfer function (between currents applied and force produced). These
effects will occur at higher frequencies and do not degrade the accuracy of K i andk x as determined above. To determine the transfer function of these eddy current effects, the rotor was magnetically levitated under a PID control and a sine sweep test up to 1 khz was conducted. Both the actuator coil current applied and the resulting position were then used to determine the open loop transfer function of the actuator-rotor-sensor system. Since the rotor and sensor are well known, the actuator transfer function can be determined on a frequency-byfrequency basis. This experimental data was then fitted (in an H sense) with a 3rd order transfer function. The experimental frequency response and the fit are shown in Figure 4. The error between this fit and the experimental data was then covered with a complex additive uncertainty of magnitude.55 (compared to the unity DC gain of the fit). This magnitude of additive uncertainty not only covers the discrepancies between model and experimental data below 1 khz, but will also cover the 45 of high frequency phase inaccuracy that any finite dimensional eddy current model will possess. The large deviation at 3 Hz shown in Figure 4 is due to a substructure resonance (see discussion below). -1-2 -3-4 -5 Figure 4: Thrust actuator frequency response and model (w/o substructure model) 4.4 ZOH and Delay Since the analog controller designed via µ-synthesis will be discretized and implemented on the digital controller as a difference equation, models of the infinite dimensional zero-order-hold and throughput delay of the controller must be included in the synthesis model to properly account for their magnitude and phase contributions. Both of these were modeled by 3rd order rational-polynomial transfer functions. The residual error between these models and the actual elements may be covered by a 2% multiplicative uncertainty. This was experimentally verified by conducting an open loop sine sweep test on the digital controller. 4.5 Anti-Aliasing Filter The eighth order Cauer elliptic filter used for antialiasing was chosen so as to have minimum phase lag over the control bandwidth. The frequency response of an analytic model of this component matched experimental data to within 1% at low frequency climbing to 4% by the cut-off frequency. This error was represented in the synthesis model as a multiplicative uncertainty at the anti-aliasing filter input with a frequency dependent weight. 4.5 Rotor Usually modeling a rotor in the axial direction as a rigid mass is adequate for AMB control system design and for our initial designs this model was effective. However, as high gain, high performance controllers were tried on the spindle, a 3.8 khz limit cycle appeared which appeared to be a thrust disk mode. From the selfexcitation current and sensor signals during this limit cycle an open loop plant frequency response for the high frequency range was found. This data was combined with low frequency sine sweep data, and the frequency response of the actuator's transfer function was then removed, resulting in an open loop frequency response for the rotor's thrust axis. This data was then fitted with a linear system to provide a more accurate rotor axial model. Both the rotor frequency response data and the fitted model's frequency response are shown in Figure 5. As the figure indicates, there are quite a few resonant modes above 1 khz. These modes are believed to attributable to either the thrust disk, the thrust stator mounting, or the sensor ring mounting. 1-2 2-2 -4-6 -8 Figure 5: Open Loop Axial Rotor Experimental Data (solid) vs. Theory (dashed) In our first attempts at µ-synthesis, no uncertainty was included for this component in the synthesis model. When the resulting controllers were experimentally tested, a small 3.8 khz limit cycle was present. This could only be removed by severely limiting the controller bandwidth, resulting in a significant degradation in achievable performance. To induce additional robustness to this mode without paying such a great performance
penalty, a filter with a resonance at 3.8 khz and unity gain elsewhere was appended to the multiplicative uncertainty weight used for the amplifier. This forces the synthesized controller to gain stabilize this rotor mode. 4.6 Substructure The open loop data collected to determine the thrust actuator transfer function (above) also displayed a significant substructure (i.e., test stand) mode at 3 Hz. Substructure vibration is a difficult problem for this application as the spindle may be installed in a variety of machining centers, each with its own particular dynamics. To account for this mode, a simple 2 nd order substructure model (spring-mass-damper) was appended in parallel to the rotor model. The mass associated with this substructure model was approximated as that of the spindle housing and the stiffness and damping terms were optimized so as to match the experimental open loop frequency response. A very accurate match of the experimental data was obtained as shown in Figure 6. No uncertainty was attributed to this component in the synthesis model. accuracy at the resonance peaks (Figure 7b) indicating the correctness of the hypothesis. Since the error in the eddy current model's fit was covered with additive uncertainty, it was not viewed as necessary to derive a more accurate model of the eddy current transfer function. Certainly, this direction could be pursued and the magnitude of the required additive uncertainty reduced at the expense of greater synthesis model order. 4.8 Performance The cutting force depends upon many factors which are specific to the machine tool and type of cut. Description of such factors is beyond the scope of this paper. In short, the general control problem for the high speed spindle is that of minimizing the H norm of the cutting tool dynamic compliance while avoiding high controller gain which may cause self-excitation [6,7]. Two fictitious uncertainties were included in the synthesis model for the dynamic compliance and controller gain performance requirements. These have weights of W p, a constant gain we wish to maximize to reduce compliance, and W c, a dynamic system whose inverse frequency response is shown in Figure 9. 1-2 2 18 16 14 12 1 1-2 1-2 (a) with eddy current model 1 1 1 1 1 2 1 2 1 3 1 3 Frequency (Hz) Figure 6: Axial Open Loop Experimental Data (solid) and Rotor/Substructure Model (dashed) 4.7 Closed Loop System Model Validation Closed loop frequency response data was taken via sine sweep testing with the spindle in support under a nominal PID controller. This was compared to the frequency response of the nominal closed loop system model (with all components determined as described previously). The match was quite good, but some discrepancies in the closed loop resonance peaks were apparent, see Figure 7a. This was believed to be due to the accuracy of the eddy current transfer function's fit to the data (Figure 4). To test this hypothesis, the closed loop frequency response was calculated using the nominal system model and the backed-out eddy current frequency response data. This matched the experimental data with much greater 1-2 Frequency (b) with eddy Response current Data data Figure 7: Closed Loop Frequency Response - Experimental (solid) and Model (dashed) 6 Controller Synthesis 6.1 Synthesis Model The block diagram of the augmented plant for the thrust axis is shown in Figure 8. Since the feedback loop is SISO, the multiplicative uncertainties in switching amplifiers and ZOH/delay may be combined (with weight W in ). Both complex µ and mixed µ synthesis (D- K and D-G-K algorithms respectively) were used for controller design. For mixed µ synthesis, the uncertainties in K i, K x, and the sensor gain were treated as real. For complex µ synthesis, the sensor and anti-aliasing filter uncertainties were combined to form a single output multiplicative uncertainty. In this case, the switching amplifier, actuator gain, and ZOH / delay
Amp W ki K i Eddy Current - W eddy K x Rotor W kx W in Digital Controller Anti-alias Filter W out Sensor W sens Figure 8: Synthesis model for thrust axis uncertainties were also combined to form a single input uncertainty. In both complex µ and mixed µ synthesis results presented here, the amplifier multiplicative uncertainty was increased by 5% to provide for an increased robustness margin. Synthesis was also carried out without this additional margin, but insignificant performance improvements resulted. 6.2 µ Controllers Two controller will be discussed herein: one designed using the D-K iteration algorithm from the µ Analysis and Synthesis Toolbox (labeled µ1) and the other using a beta version of D-G-K algorithm provided by Professor Peter Young (labeled mixed µ) [8]. For the complex µ-controller, three D-K iterations were required to obtain µ=1.65 with a specified compliance performance weight of 6, lbf/in. The resulting controller was 76th order and was reduced to 22nd order by a balance-and-truncate procedure with µ checked at each level of reduction, resulting in a µ of 1.8. The frequency response of this controller is shown in Figure 9 along with the inverse of the controller gain weight. Note the sharp notch at 3.8 khz that the synthesis algorithm inserted to gain stabilize the thrust disk flexible mode. 1 2 1 1 1-1 1 4 Figure 9: µ1 Controller (solid) and W c -1 (dashed) The mixed µ controller required 3 D-G-K iterations to achieve µ=1.73 when the compliance performance weight was specified to be 62,5 lbf/in. The resulting controller had 154 states and was reduced to 24 th order with a final µ of 1.63. Thus, treating the uncertainties as mixed resulted in a very small improvement in predicted worst case dynamic compliance. 6.3 PID Controller For comparison, an optimized PID control was designed for the thrust axis by an exhaustive, numerated search in combination with many experiments. Optimality was measured via the nominal system's maximum dynamic compliance (H norm). The theoretically achieved nominal compliance translates to a dynamic stiffness of 45, lbf/in. 6.4 Performance Robustness Comparison At each frequency, the maximum compliance that could result given the uncertainties specified in the system model was calculated for each of the three controllers. The result is shown in Figure 1. Since the PID controller was optimized on the basis of the nominal model, it can have very poor robustness to the model uncertainties as shown. On the other hand, the µ controllers' worst case compliances were less sensitive to the system uncertainties. In fact, the µ controllers' peak worst case compliances were 25% lower than that obtained by the PID controller operating on the nominal system model. 7 Experimental Results 7.1 Hardware The parallel processing digital controller was programmed in assembly language to implement the feedback control algorithms as discrete time, state space systems. The x radial plane, y radial plane, and z (thrust) axis were each controlled by feedback algorithms residing on separate TMS32C4 processors. (Each radial plane has three actuators and three sensors.) In the experiments presented herein, the radial bearings were controlled by decentralized PID controllers.
lbf) 7 6 5 PID lbf) 15 4 1 Mixed µ 3 µ 1 2 5 1 Mixed µ µ 1 Figure 1: Worst case compliance for PID, µ1 (D-K), and mixed µ (D-G-K) controllers (theory) Figure 12: Dynamic compliance of the thrust axis with µ1 and mixed µ controllers (experiment) The µ and PID controllers were executed with a sampling rate of 12 khz. Dynamic compliance was measured using an instrumented hammer and the spindle's axial optical sensor. Figure 11 shows the dynamic compliance of the spindle with the optimized PID and µ1 controllers. Both compliance curves are close to those predicted from the system model (not shown - see Reference 7) and less than the worst case analysis shown in Figure 1. This indicates the fidelity of the nominal system model and appropriateness of the uncertainty description used. The µ1 controller produced far lower dynamic compliance than the optimized PID controller as expected. The experimental results demonstrate a 54% reduction in the maximum dynamic compliance over that achievable with the standard AMB control technique. Figure 12 compares the performance of the complex µ and mixed µ controllers. An insignificant difference is evident. lbf) 35 3 25 2 15 1 5 µ 1 PID Figure 11: Dynamic compliance of the thrust axis with optimized PID and µ1 controllers (experiment) 8 Conclusions Significant improvements in the axial dynamic stiffness of magnetic bearing supported rotors can be obtained using µ synthesis. The key to obtaining these performance improvements is developing an accurate system model with an appropriately characterized uncertainty description. 9 References [1] Machine Design, "Magnetic Bearings Holds Spindle For Milling", November 9, 1989, pg. 56 [2] American Machinist, "Magnetically Levitated Spindle to Debut, Delivers Up to 52 kw at 4, RPM", August 1989, pp. 78-79. [3] Nonami, K., et. al., "H Control of Milling AMB Spindle", Fourth International Symposium on Magnetic Bearings, 1994, Zurich, pp. 531-536 [4] Brunet, M., and Wagner, B., "Analysis of the Performance of an AMB Spindle in Creep Feed Grinding", Fourth International Symposium on Magnetic Bearings, 1994, Zurich, pp. 519-524 [5] Fedigan, S.J., Williams, R.D., Shen, F., and Ross, R.A., Design and Implementation of a Fault Tolerant Magnetic Bearing Controller, 5th International Symposium on Magnetic Bearings, Kanazawa, Japan, August 28-3, 1996. [6] Stephens, L.S. and Knospe, C.R., µ-synthesis Based, Robust Controller Design for AMB Machining Spindles, 5th Int. Symp. on Magnetic Bearings, Kanazawa, Japan, August 1996. [7] Fittro, R.L. A High Speed Machining Spindle with Active Magnetic Bearings: Control Theory, Design and Application, Ph.D. Dissertation, University of Virginia, August 1998. [8] Young, P.M. Robustness with Parametric and Dynamic Uncertainty, Ph.D. Dissertation, California Institute of Technology, 1993.