Model Validation for a Self-bearing Motor Prototype

Transcription

1 Model Validation for a Self-bearing Motor Prototype José D.G. Garcia 1, Afonso C. N. Gomes 2 and Richard M. Stephan 2 1 Instituto Federal do Rio de Janeiro, Nilópolis, BR, 2 Universidade Federal do Rio de Janeiro/COPPE, Rio de Janeiro, BR. jose.garcia@ifrj.edu.br, nero@coep.ufrj.br, richard@dee.ufrj.br Abstract Experimental models for active magnetic bearings, which are intrinsically unstable, can only be obtained by applying closed loop system identification techniques. In this paper, initially, a brief discussion on the difficulties associated with closed loop identification is presented. Using the knowledge of system parameters and analytical equations, the physical model for a laboratory self-bearing motor is established. Based on such model, a stabilizing LQR controller is designed. Once the plant is stabilized, system identification techniques are applied to process the experimental data, and parametric models are obtained for three sample datasets and two model development techniques (PEM and N4SID). Consistency among models developed using the two identification algorithms, as well as for the three different datasets used, can be observed. 1 Introduction Due to its peculiar characteristics, active magnetic bearings have experienced an increasing acceptance in applications with more restrictive requirements, like high rotational speeds ultracentrifuges and machine-tools, high vacuum turbo-molecular pumps, or applications with difficult access for maintenance submersible pumps or artificial heart pumps (Beams, 1954), (He, Shinshi, Zhang, Yuzawa, & Sato, 2010). As discussed later, magnetic bearings, and its counterpart self-bearing motors, are inherently unstable, requiring a closed loop control system implementation. The development, or optimization, of control systems, requires plant mathematical models, as close as possible to the actual plant. For that reason, it is desirable to obtain the experimental model for the plant. On the other hand, experimental models are usually obtained testing the open loop plant, which is not possible for magnetic bearings and self-bearing motors, due to instability. This paper discusses results reached while developing an experimental model for a self-bearing motor prototype at UFRJ Laboratory of Applied Superconductivity (LASUP). Firstly, a brief review on physical modeling of magnetic bearings is presented. Later, the methodology for closed loop identification is discussed. Finally, the results obtained in the experimental development of a plant model are presented and applicable comments are considered.

2 2 Active Magnetic Bearing Dynamics The principle underlying the operation of magnetic bearings is magnetic levitation, as will be briefly discussed in the text that follows. A simple magnetic circuit is presented in figure 2.1 Figure 2.1 For the magnetic circuit displayed above, gap energy may be expressed as (Schweitzer, 2009): where B a is the magnetic induction, H a is the magnetic field, V a is the air gap volume, A a is the nucleus cross section and d is gap distance. Considering the principle of virtual displacement, the following expression is reached for the reluctance force, considering μ 0 the magnetic permeability, n the number of coil turns and i the coil current (1) (2) The result obtained above reflects the fact that the reluctance force has a non-linear dependency on the applied current, as well as on the gap dimension. Supposing that the system is operating around an operating point (i 0,d 0 ), this expression may be expressed as a Taylor series, allowing the linearization of the reluctance force. The result obtained is: with: (3)

3 and This magnetic force is then applied to the bearing in order to counterbalance the disturbance efforts that may take the rotor away from its reference position. Now, in order to continue the physical modeling of the magnetic bearing, rotor dynamics should be analyzed, as follows. Figure 2.2, representing the geometry of the motor being studied, shall be considered. Figure 2.2 Considering the geometry shown, letting I x, I y, I z represent the moments of inertia with respect to the CM, m the rotor mass and ω r the rotor speed, moreover supposing the center of mass as the reference origin, rotor dynamics equations, may be written as (Schweitzer, 2009) (4) with: ; ; and: ; where subscript b means measurement taken at bearings, A and B subscripts mean upper and lowering bearings. Nevertheless, for the prototype being studied, the upper bearing will be implemented by a magnetic actuator, while the lower bearing will be based on a ball bearing. Taking that into

4 consideration, the moments of inertia I x and I y should be reflected to the lower ball bearing position, considering the rotor symmetric and balanced (J x = J y = J). Then, the following system may be written where ; ; (5) and ; ; 2.1 LASUP Self-bearing Motor Model The existing LASUP self-bearing motor prototype has been built based on a two-phase 4 poles induction motor, and has an upper rotor and a lower rotor. As for the experimental data developed for this paper, just the upper rotor was implemented, and the lower bearing was implemented with a roller ball bearing, both for radial as well as for axial action. Figure 2.3 Self-bearing motor prototype parameters are listed in table 2.1 (Kauss, 2008). Table 2.1 Item Symbol Value (SI) Reference 1 A 68,9 x 10-3 m Rotor Geometry 2 B -82,8 x 10-3 m Rotor Geometry 3 C 148,0 x 10-3 m Rotor Geometry 4 D -163,0 x 10-3 m Rotor Geometry 5 G 218,0 x 10-3 m Rotor Geometry 6 M 4,42 kg Rotor Mass 7 I x 50,3 x 10-3 kgm 2 Momentum of Inertia 8 I y 50,3 x 10-3 kgm 2 Momentum of Inertia 9 I z 2,17 x 10-3 kgm 2 Momentum of Inertia

5 Item Symbol Value (SI) Reference 10 k h 1368 N/m Superconductor Bearing 11 c h 2,89 Ns/m Superconductor Bearing 12 n e 99 Number of coil turns 13 H 0,4 x 10-3 m Gap Geometry 14 a g 3,734 x 10-3 m 2 Gap Geometry 15 I 0 1,17 A Motor biasing current 16 µ 0 4П x 10-7 N/A 2 Magnetic Permeability 17 L 2 34,24 x 10-3 H Electric Parameter 18 L m 350,43 x 10-3 H Electric Parameter 19 R 2 13,97 Ω Electric Parameter 20 Γ 9,81 m/s 2 Gravity acceleration 21 P 4 Motor number of poles 22 0,05 Motor slip Based on those parameters, a physical model was developed for the self-bearing motor, considering 120 Hz as frequency of operation: This physical model has been the base, initially, for the development and implementation of a PID controller (Gomes, 2007), and later for an LQR controller (Kauss, 2008). Both controllers have shown a dynamic performance acceptable for the self-bearing motor. Upon reaching system stability, it is possible to validate the physical model by identifying, in closed loop, the experimental plant model. 3 Closed Loop System Identification Closed loop system identification has called the attention of the scientific community, and has been discussed by many authors in many papers, like (Gustavsson, Ljung, & Soderstrom, 1977), (Ng, Goodwin, & Anderson, 1977) (Ljung & MacKelvey, 1996) (Lakshminarayanan & al, 2001), (Zhu & Butoyi, 2002).

6 A first comment to be made is that it cannot be assured that closed loop identification is feasible (Gustavsson, Ljung, & Soderstrom, 1977), (Ng, Goodwin, & Anderson, 1977). Many different approaches have been suggested to overcome the existing difficulties, and to allow a consistent identification. Aspects to be considered in this task are listed below (Gustavsson, Ljung, & Soderstrom, 1977): Identifiability and identification precision Choice of identification method Model structures impact on identification Impact on identifiability and identification precision with the use of variable structure control Gustavsson (Gustavsson, Ljung, & Soderstrom, 1977) calls the attention to the fact that there are four aspects to be considered in the closed loop identification process: The system to be identified, which dynamic may present larger or smaller difficulties for identification Model structure and/or/ its parameterization Identification method to be employed Experimental conditions Different ways of applying identification methods to closed loops systems are possible. Application of identification methods to input/output data, as if the system operated in open loop, is called direct identification. In case the system been identified has a linear, time invariant, noiseless controller, or in the case there is an alternation between different controllers, an indirect identification method may be used, by first identifying the closed loop dynamics, and then obtaining the open loop dynamics using the knowledge of both the closed loop dynamics and the controller dynamics. One way or another, different approaches may be taken, like applying spectral analysis or correlation, or parametric identification methods. On the other hand, Lakshminarayanan (Lakshminarayanan & al, 2001) calls the attention to the fact that closed loop identification based on experimental data of signal applied to plant input and measurement of plant ouput signal, applying correlation or spectral analysis, without taking into consideration the causal dependency of input/output, may lead to the identification of the controller dynamics, instead of the identification of plant dynamics. In this sense, it is suggested that, to allow experimental plant identification in closed loop, one of the listed approaches should be taken: Apply changes in the setpoint Apply a high frequency disturbance signal Commute two different controllers 3.1 Identification Methods Application to Self-bearing Motor Simulated Model The physical model developed in paragraph 2.1 has been simulated with MATLAB. Figure 3.1 shows the simulation diagram for referred model.

7 Figure 3.1 It should be noticed that an excitation random signal is applied to the plant input, in order to observe system dynamics, according to suggested approaches presented earlier, namely to apply a high frequency disturbance signal in order to increase system identifiability. The operation of the system was simulated using this diagram, and the input and output signals obtained are shown in figures 3.2 and 3.3, on the next page. 3.2 Identification Technique Applied to Self-bearing Motor Simulated Model Data obtained in simulation were submitted to Prediction Error Method algorithms available in MATLAB. An identification model was developed supposing the system to have order 4, and the discrete model obtained was, then, converted to continuous model. Matrices obtained in this process are listed in Annex I. System poles for identified model are shown on table 3.1 By the same token, simulation data was submitted to N4SID (subspace identification) algorithm, available in MATLAB, considering, again, system order to be 4. As described above, discrete model was converted to continuous model, and the matrices obtained are also shown in Annex I. Also, like done above, systems poles were calculated, and are displayed in table 3.1

8 Figure 3.2 Input Figure 3.3 Output Finally, system physical model poles, both for open loop and for closed loop, were calculated, and are also listed in table 3.1. Table 3.1 PEM N4SID Physical model Closed loop P i i P , ,41 i i P i -72, i P i i Analyzing table 3.1, one may observe that both methods have detected that the system has unstable poles, but none of the identification methods have resulted neither in observing the complex nature of the poles, nor its precise values. Also it can be observed that pole P1 has a real part close to the open loop physical model, but values very far from the closed loop poles. On the other hand, pole P3 obtained by PEM is very close to the absolute value of the real part of closed loop pole. It should be recalled that comments have been made that, in such identification of closed loop systems,

9 identification methods could result in observing partly the open loop dynamics, an partly the closed loop dynamics. Nevertheless, it should be noted that there is reasonable consistency for the poles obtained by PEM and N4SID identification algorithms. 4 System Identification Based on Experimental Data Having worked on simulated system data, experimental data was next obtained for the LASUP self-bearing motor prototype. In order to generate an excitation signal to produce prototype response, a routine for the generation of pseudo-random binary signal (PRBS) was implemented in the Digital Signal Processor (DSP), which is been used to control the self-bearing motor. This testing signal was added to the position controller output signal. This same signal was applied to a DSP digital output, in order to allow its observation and measurement. Many series of experimental data were obtained, testing various amplitude for the random testing signal, and other stimulation techniques suggested were also applied, in the effort to create the best conditions for the identification, like the commutation of controllers proposed by Gustavsson (Gustavsson, Ljung, & Soderstrom, 1977). The effect of different sampling frequencies was also tested. Experimental data was collected for the self-bearing motor running with a power signal of 40 Hz. Since the controllers were designed to operate under a frequency of 120 Hz, system matrices were recalculated for this frequency of 40 Hz, as well as the system poles. New system dynamics poles obtained for this frequency are shown below in table 4.1 Table 4.1 Physical Model Closed Loop P ,008 i ,62 i P ,008 i ,62 i P ,008 i ,008 i P ,008 i -59-0,008 i Although many different experimental data series have been obtained, only the results for the sampling frequency of 5 khz will be discussed, sampling frequency that is within the range of frequencies suggested by René Larsonneur (Schweitzer, 2009) 4.1 System Identification Using Parametric Methods Experimental data was obtained for the self-bearing motor operating in closed loop, using the LQR controller implemented in the prototype DSP. As mentioned before, the testing signal used was a PRBS added to the controller output, and observed in a DSP digital output. Three sets of experimental data were gathered, for the testing signal and corresponding x and y position signals. Such data was submitted to both methods studied in the preceding paragraph. Decision was made to develop, for the plant, models of order 6, both for the PEM algorithm and for the N4SID algorithm, since this order showed a significant improvement over the order 4 model. Annex I shows system matrices obtained for one of the experimental series, obtained by both methods. Bode diagrams were plotted for both models obtained, an are displayed in figure 4.1, Annex II. It should be noted that both diagrams are consistent with the models developed for the three experimental data series, as well as for both identification methods.

10 4.2 Comparing Physical Model and Experimental Model Data As mentioned before, applied LQR controller was designed using physical model developed for an power frequency of 120 Hz. Since experimental data was obtained while self-bearing motor was operated with a frequency of 40 Hz, physical model was recalculated for such frequency, resulting in the following matrices: On the other hand, the controller designed by Kauss (Kauss, 2008) has a feedback matrix given by: Prototype physical model was then calculated using MATLAB function feedback, and later its Bode diagram was plotted together with experimental data models obtained in preceding paragraph. Such diagram is shown in figure 4.2 of Annex II. Adherence of experimental data models and physical model can be observed on such diagrams, validating, in such way, the model used to develop controllers for the prototype. 5 Conclusion The objective of validating the physical model used in the design process of controllers to be applied to the LASUP self-bearing motor prototype led to the consideration of different closed loop system identification techniques. It could be noted that parametric models development, followed by frequency domain analysis, resulted in consistent models, and that such models validated, experimentally, system physical model. Acknowledgements UFRJ and CNPQ should be acknowledge for the support and for having made available the required resources. References Gomes, R. R. (2007). Motor mancal com controle implementado em um DSP. Rio de Janeiro: COPPE/UFRJ.

11 Gustavsson, I., Ljung, L., & Soderstrom, T. (1977). Identification of Process in Closed Loop Identifiability and Accuracy Aspects. Automatica, V Kauss, W. L. (2008). Motor mancal com controle ótimo implementado em um DSP. Rio de Janeiro: COPPE/UFRJ. Lakshminarayanan, S., & al, e. (2001). Closed Loop Identification and Control Loop Reconfiguration: an Industrial Case Study. Journal of Process Control, V.11, pp Ljung, L., & MacKelvey, T. (1996). Subspace Identification from Closed Loop Data. Signal Processing, V.52, pp Ng, T., Goodwin, G., & Anderson, B. (1977). Identifiability of MIMO Linear Dynamic Systems Operating in Closed Loop. Automatica, V.13, pp Schweitzer, G. e. (2009). Magnetic Bearings - Theory, Design an Application to Rotating Machinery. Springer-Verlag. Zhu, Y., & Butoyi, F. (2002). Case studies on Closed-Loop Identification for MPC. Control Engineering Practice, V.10, pp

12 Annex I Simulated Plant Model PEM Algorithm N4SID Algorithm Experimental Plant Model PEM Algorithm

13 N4SID Algorithm

14 Phase (degrees) Amplitude Annex II Legend PEM 1 PEM 2 PEM 3 N4SID 1 N4SID 2 N4SID 3 Simulation 10 0 From u1 to y Frequency (rad/s) Figure 4.1a Bode diagram for transfer function of displacements on the x direction

15 Phase (degrees) Amplitude Phase (degrees) Amplitude 10 0 From u2 to y Frequency (rad/s) Figure 4.1b Bode diagram for transfer function of displacements on the y direction 10 0 From u1 to y Frequency (rad/s) Figure 4.2a Bode diagram for transfer function of displacements on the x direction

16 Phase (degrees) Amplitude 10 0 From u2 to y Frequency (rad/s) Figure 4.2b Bode diagram for transfer function of displacements on the y direction