Gain-Scheduling Approaches for Active Damping of a Milling Spindle with Speed-Dependent Dynamics

Transcription

1 Gain-Scheduling Approaches for Active Damping of a Milling Spindle with Speed-Dependent Dynamics Simon Kern, Andreas Schwung and Rainer Nordmann Abstract Chatter vibrations in high-speed-milling operations can be avoided by increasing the damping of the spindle. Using an electromagnetic actuator active damping is implemented in a prototype of an adaptronic spindle and via µ-synthesis robust controllers for two extreme spindle speeds are designed. To apply active damping in the whole speed range gain scheduling techniques have to be used, since the spindle s ball bearings show speed-dependent radial stiffness. Three approaches for such active control are presented and tested at the prototype: switching of controllers with focus on bumpless transfer using an observer-like structure, fuzzy gain scheduling and gain scheduling of controllers poles, zeros, and gain. 1 Introduction In high speed cutting the maximum material removal rate is mostly limited by either spindle power or process stability. The latter mainly depends on several factors like the spindle speed, the dynamics of the spindle, the tool, and the depth of cut. In case of instability self-excited oscillations between tool and work piece, called chatter vibrations, occur. One approach to increase the process stability is to improve the damping of the spindle [4]. To reach this aim, an adaptronic hybrid bearing spindle is developed which allows active vibration control via an electromagnetic actuator. This actuator is an active magnetic bearing (AMB) that is mounted to the front of a conventional motor spindle with angular contact ball bearings (see Fig. 1). The bearing allows Simon Kern Rainer Nordmann Technische Universität Darmstadt, Mechatronics in Mechanical Engineering, Darmstadt, Germany, kern@mim.tu-darmstadt.de, nordmann@mim.tu-darmstadt.de Andreas Schwung Technische Universität Darmstadt, Institute for Automatic Control, Darmstadt, Germany aschwung@rtr.tu-darmstadt.de 1

2 2 Simon Kern, Andreas Schwung and Rainer Nordmann to apply radial forces to the rotating spindle contactlessly and eddy current position sensors measure the lateral displacement of the rotor. This hybrid approach combines the advantages of both technologies: the stiffness and reliability of the ball bearings and the possibility of active vibration control and identification via the active magnetic bearing. Like in many conventional motor spindles, the spring prestressed angular contact ball bearings of the existing adaptronic spindle prototype have speed-dependent radial stiffness what results in speed-dependent characteristics of the control loop. Based on two existing linear time-invariant robust controllers, that are only suitable for limited speed ranges, one switching and two gainscheduling approaches are presented to apply active damping in the whole speed range of the spindle. 2 System Setup and Modeling A schematic of the prototype spindle and its periphery is shown in Fig. 1. The spindle is supported by four hybrid angular contact ball bearings. To the front of the spindle s housing the stator of a 8-pole active magnetic bearing (AMB) is mounted. The sheet-metal rotor of the AMB is connected to the spindle via the spindle s tool adaptor and it offers a mount for the milling tool at its tip. The amplifiers that drive the AMB have a maximum current of 5 A which induces static radial forces of approx. 6 N on the rotor. Eddy current sensors located in front of the stator measure the spindle s lateral position with a resolution of about 2 µm. The signals are sampled by a dspace real time system running at 1 khz sampling rate without additional anti-aliasing filter. Fig. 1 test rig with components (left), schematic of the motor spindle in bearings with speed dependend radial stiffness (right)

3 Gain-Scheduling Approaches for Active Damping 3 Fig. 2 Measured frequency response functions and identified parametric models for min -1 and 22 min -1. The peak at 367 Hz results from speed synchronous sensor runout. Magnitude (db) Phase (deg) Bode Diagram of control plant measurement (n) 18 parametric model (n) 27 measurement (n22) parametric model (n22) Frequency (Hz) The control plants input (i.e. actuating variable) is the reference value for the current controlled power amplifiers which can be modeled as a PT1 element (time constant T A 2.4 ms). The control variable is the measured deflection of the AMBrotor. For the design of the control algorithms only one lateral direction of the AMB is considered because coupling effects between the two lateral directions are small and therefore neglected. The same controller structure and the same parameters are used for both lateral directions. Fig. 2 shows experimentally identified frequency response functions of the control plant (measured position of the spindle to sine-sweep excited reference current). The first and second resonance frequencies of the system can be seen at 56 Hz and 87 Hz for standstill. They represent the first two bending modes of the spindle. Caused by increasing centrifugal forces the radial stiffness of the angular contact ball bearings decreases and the resonance frequencies of the system change [2]. This results in decreasing natural frequencies of the system with rising rotational speeds, e.g. 51 Hz and 71 Hz for rotational speed n = 22 min -1. The well damped transfer behavior of the unfortunately not well suited current amplifiers causes a phase loss of about 175 at the first resonance frequency. 3 Approaches for Speed Dependent Gain-Scheduling Due to the significant changes in the system dynamics described above, no linear time-invariant controller could be found, that shows satisfactory damping performance for all spindle speeds. To achieve high damping of the resonances at all spindle speeds, gain scheduling approaches based on two linear robust µ-synthesis controllers are proposed. Without describing the design process of the robust controllers for the two extreme spindle speeds in detail, three ways to combine these to a nonlinear gain-scheduling controller are presented: switching between controllers connected in an observer-like structure for bumpless transfer, fuzzy gain-scheduling, and gain-scheduling via interpolation of the controllers poles, zeros, and gain. All three approaches use the spindle s speed as gain-scheduling (or switching) variable

4 4 Simon Kern, Andreas Schwung and Rainer Nordmann since it has well-defined influence on the system and is known from the machine control. Focus of the efforts is not only stable closed loop characteristics but also speed-independent damping performance. In contrast to analytically derived gainscheduling laws with the help of parametric nonlinear models, e.g. [12] and [1], here a more pragmatic approach based on measured frequency response functions and identified parametric models is followed. 3.1 Robust Control for Selected Speeds Although the sensor and actuator are nearly collocated, simple PD-control for active damping of the resonances is not possible. The limited current control bandwidth results in noticeable phase loss and PD-control destabilizes the systems resonances. Model-based control design and especially robust control design proved to be more systematic and practicable for the system at hand than time consuming manual tuning of higher order controllers. Robust controllers are designed for the two extreme points standstill (C for min -1 ) and maximum speed (C 22 for 22 min -1 ). They are designed via µ- synthesis using the µ-synthesis framework proposed by Schönhoff in [11]. Basis of the design process are parametric models of the control plant. Frequency response functions are measured at the two spindle speeds and parametric models are identified using the identification algorithms described in [1] and provided within de Callafons FREQID-Toolbox for Matlab. The models are all of 9 th order. In Fig. 2 the bode plots of the measured frequency response functions and the identified models are depicted. These models are augmented by modal uncertainty to take into account measurement and identification errors and the influence of the milling process. Additive uncertainty is used to account for unmodelled higher dynamics. Via bounding functions for the closed loop s transfer functions the desired control performance is defined and the controllers are synthesized using well-known DK-iteration. Additional controller reduction yields controllers of 12 th order in state-space representation. Their bode diagrams are shown in Fig. 3 as well as the measured frequency response functions of the system with and without control at the corresponding spindle speeds. For a more detailed description of the µ-synthesis of similar controllers for the same prototype it is referred to [7]. Both controllers only work satisfying in defined speed ranges. Controller C works well up to 13 min -1, decreases damping for speeds greater than 13 min -1 and becomes instable for speeds above 18 min -1. C 22 works well for speeds between 11 min -1 and 22 min -1 decreases damping of the first resonance below these speeds. So the controllers work in overlapping parameter-ranges of the gain-scheduling variable which is necessary for switching of the controllers.

5 Gain-Scheduling Approaches for Active Damping 5 Magnitude (db) Phase (deg) Bode Diagram of the controllers 36 C 18 C Frequency (Hz) Phase (deg) Magnitude (db) Bode Diagram of measured FRFs 2 1 without control (n) with control (n) 1 without control (n22) 2 with control (n22) Frequency (Hz) Fig. 3 µ-synthesis controllers (left) and measured frequency response functions with and without control (right) 3.2 Bumpless Transfer Bumpless Transfer of µ-controllers has already been presented in [8]. The concept is based on a switching between controllers designed for a small speed range where the switching supervisory control is extended by a bumpless transfer scheme to avoid bumps and transient effects in the control signal [5]. Different to [8], where a Kalman Filter is used for state estimation of the inactive controller, here the inactive controller is coupled to the active controller in an observer-like structure (Fig. 4). Unlike to a standard observer setup, the state-space matrices are different, since the two controllers are different inherently. However, the basic principle of estimating the latent controller s states is the same. Condition is the same order and basically similar behavior of the two controllers. This setup reduces transient effects significantly. Due to the switching of the feedback signal (i.e. the output of L), small bumps in the control signal can still appear. These are further reduced by time controlled blending of the controllers instead of hard switching of the signals. The appropriate switching speed is calculated using the identified transfer functions of the system G n for n N = [,1,...,22 min -1 ]. The switching speed n s is chosen to maximize damping performance. Therefore the optimization problem ( min max GS n n ). (1) s n is defined and solved numerically. Here GS n is the closed loop disturbance transfer function constructed from model G n at speed n and the controller C: { GS n = G n C n < n s with C =. (2) 1+G n C C 22 n n s G denotes the H norm for dynamic systems defined as

6 6 Simon Kern, Andreas Schwung and Rainer Nordmann Fig. 4 The inactive controller is connected to the active one in an observer-like structure to guarantee bumpless transfer when switching between controllers. G := sup σ [G(ω)] (3) ω R where σ (G(ω)) is the largest singular value of G(ω). This way the maximum gain of the disturbance transfer functions is minimized over all speeds. The feedback matrix L is calculated via pole placement similar to the design of a standard Luenberger observer. To obtain good tracking performance of the inactive controller, the poles are moved to small real values (R{p i } < 1 5 ). 3.3 Fuzzy Gain Scheduling Fuzzy-gain-scheduling is basically a blending between single controllers, each designed for a small speed range. According to Fig. 5, each controller is weighted by a speed-dependent membership-function. The concept is strongly related to takagisugeno fuzzy models [13], which are already used for modelling and control of active magnetic bearings [6]. Unlike in [1] here the membership-functions are not defined analytically based on a nonlinear model, but on identified models of the plant at the different spindle speeds n. According to the concept of parallel distributed compensation [14], the controllers designed at the bounds of the speed-range are weighted by the same membership-function (Fig. 5, left) which is determined for the corresponding models. To determine this membership-function the identified models at the bounds of the speed range ( min -1 to 22 min -1 ) are combined according to the optimization problem min λ n [λ n G +(1 λ n )G 22 G n ] n N. (4) This yields pairs of (n, λ n )-values for all n N. Finally a cubic interpolation between these pairs of values gives the continuous membership-function (see Fig. 5, right) which describes the change of the models over n and is used for the blending of the

7 Gain-Scheduling Approaches for Active Damping 7 1 λ λ 22.8 membership spindle speed in min 1 x 1 4 Fig. 5 Fuzzy gain scheduling controller and optimized membership-functions. controllers. For the system at hand, the resulting membership is nearly proportional to the change of the first resonance frequency. This approach does not need any switching logic and is easy to implement but both controller outputs have to be calculated what limits the maximum sampling rate on the real time system. 3.4 Gain Scheduling of Controller Poles and Zeros The second gain-scheduling approach discussed in this paper is based on the variation of all controller parameters. It has already been successfully applied e.g. to flight control problems [9]. Starting point are again the two controllers of the same order designed at the bounds of the speed range. At first the poles and zeros of both controllers are assigned to each other to obtain pairs of corresponding poles and zeros. The changing stiffness of the bearings results in continuous change of the system dynamics and it is easily assumed that the poles and zeros of the controller also change continuously over speed. The next step is to interpolate between these pairs of poles and zeros. The location of the pole p i depends on the corresponding poles in C and in C 22 : p i = p,i + g(n) (p 22,i p,i ) (5) with g(n) [..1]. The zeros and gain are interpolated equally. This means illustratively, that the poles and zeros of the controller C migrate to the corresponding poles and zeros of the controller designed for 22 min -1 as the spindle speed increases. The speed of migration depends on the variation of the resonance frequencies and an interpolation function g(n) is chosen similar to the membershipfunctions in Fig. 5. If the gradient of the resonance frequency with respect to the spindle speed is high, then also migration speed is high; if the gradient of the resonance frequency with respect to the spindle speed is low, then also migration speed is low.

8 8 Simon Kern, Andreas Schwung and Rainer Nordmann The controller is transformed to controllable canonical form and implemented as a series of integrators with scalar feedback and output gains. These gains are retrieved from offline calculated look-up-tables. This approach has the advantage, that only one controller output has to be calculated in real-time and the scheduling via the lookup-tables can be performed at a lower sampling rate. 4 Results The three approaches are implemented and tested at the prototype. For different spindle speeds the system is excited by sine-sweep signals via the AMB-current and the deflection of the spindle is measured. Fig. 6 shows FRFs identified from these signals for the speed of 2 min -1. This speed proved to be the worst-case scenario for the given controllers. It can be seen, that in the range of the first resonance the controlled systems gains are always below that of the passive system, which is the initial goal of the controller design. All three controllers show good performance over the whole speed range. To quantify the performance the worst case amplification of the disturbance transfer functions over all speeds is calculated. Table 1 shows the closed loop s maximum gain γ = max GS n n (6) with GS n = G n 1+G n C and C {C,C 22,C fuzzy,c bt,c gs }. The switching speed for the bumpless transfer algorithm is n = 115 min -1. So the result with controller C 22 is equal to the one with bumpless transfer at n = 2 min -1. In terms of implementation issues, the pole/zero gain scheduling has advantages to the other schemes, since only one controller-output has to be calculated in realtime. If the spindle only works at constant speeds and transient effects are no issue, a switching scheme is probably advantageous. For the continuous adaptation schemes it is possible to prove stability by using parameter varying Lyapunov-functions [3]. However due to the slowly varying spindle speed with respect to the system s dominant eigenfrequencies, it is sufficient to prove stability for each operating point. Such investigations showed stability for the proposed approaches. 5 Conclusion For the active damping of a milling spindle with speed-dependent bearing-stiffness, gain scheduling depending on spindle-speed gives good performance. Three ap-

9 Gain-Scheduling Approaches for Active Damping 9 2 Magnitude (db) 1 1 without control with controller C22 with pole/zero gain scheduling with fuzzy gain scheduling Phase (deg) Frequency (Hz) Fig. 6 Measured frequency response functions with different controllers at spindel speed 2 min -1. The peaks at 333 Hz are disturbances due to sensor runout and induced by the control. Table 1 Maximum gains of the closed loop over all spindle speeds in steps of 1 min -1. The gains are relative to the gain of the system without control. Controller max. gain γ no control 1 % controller for n = min -1 (C ) - a controller for n = 22 min -1 (C 22 ) 116 % fuzzy gain scheduling (C fuzzy ) 57 % bumpless transfer (C bt ) 5 % pole/zero gain scheduling (C gs ) 56 % a system is unstable for certain speeds. proaches to combine two or more linear time-invariant robust controllers to a nonlinear gain scheduling controller are presented. One switching scheme guaranteeing bumpless transfer and two continuous approaches, fuzzy gain scheduling and gain scheduling of controller poles and zeros, are implemented at the prototype of an adaptronic motor spindle, which allows active vibration control via an electromagnetic actuator. Experimental results show that all three algorithms work satisfying for the whole speed range. Although none of the proposed methods can be considered better to the others in general they might be chosen according to the following properties: If transient behaviour is not an issue or the adaptation variable does not change continuously, simple switching between controllers with only small effort to bumpless transfer via time-based blending of the controllers might be preferable. Continuous adaptation can offer good performance for systems with continuosly changing adaptation variable. Fuzzy-Adaptation is easier to implement than the proposed gain-scheduling of

10 1 Simon Kern, Andreas Schwung and Rainer Nordmann controller poles and zeros. But it needs higher real-time computational power since all single controller outputs have to be calculated simultaneously. Acknowledgements This research is supported by the Deutsche Forschungsgemeinschaft (DFG) within the priority program 1156 Adaptronik für Werkzeugmaschinen. References 1. R. de Callafon, D. de Roover, and P. Van den Hof. Multivariable least squares frequency domain identification using polynomial matrix fraction descriptions. In Decision and Control, 1996., Proceedings of the 35th IEEE, volume Vol. 2, pages pp23 235, Dec U. Fiedler. Prozesssicherheit beim HSC-Fräsen von Aluminium-Knetlegierungen. PhD thesis, TU Darmstadt, Shaker Verlag, Aachen, P. Gahinet, P. Apkarian, and M. Chilali. Affine parameter-dependent lyapunov functions and real parametric uncertainty. IEEE Transactions on Automatic Control, Vol. 41:pp , A. Ganguli. Chatter reduction through active vibration damping. PhD thesis, Universite Libre De Bruxelles, S. Graebe and A. Ahlen. The Control Handbook, chapter Bumpless Transfer, pages pp CRC Press, S.-J. Huang and L.-C. Lin. Fuzzy dynamic output feedback control with adaptive rotor imbalance compensation for magnetic bearing systems. IEEE Transactions on Systems, Man, and Cybernetics, Vol. 34(4):pp , August S. Kern, C. Ehmann, R. Nordmann, M. Roth, A. Schiffler, and E. Abele. Active damping of chatter vibrations with an active magnetic bearing in a motor spindle using mu-synthesis and an adaptive filter. In The 8th International Conference on Motion and Vibration Control MOVIC, Daejeon, Korea, G. Li, Z. Lin, P. E. Allaire, B. Huang, W. Jiang, E. S. Zori, and R. O. Bartlett. Stabilization of a high speed rotor with active magnetic bearings by a piecewise mu-synthesis controller. In 6th Symposium on Magnetic Suspension Technology, Turin, Italy, R. Nichols, R. Reichert, and W. Rugh. Gain scheduling for h-infinity controllers: A flight control example. IEEE Transactions on Control Systems Technology, Vol. 1(no. 2), W. Rugh. Analytical framework for gain scheduling. Control Systems Magazine, IEEE, Vol. 11(no. 1):pp79 84, Jan U. Schönhoff. Practical robust control of mechatronic systems with structural flexibilities. Forschungsberichte Mechatronik & Maschinenakustik. Aachen. Shaker, D. Stilwell and W. Rugh. Interpolation methods for gain scheduling. In Decision and Control, Proceedings of the 37th IEEE Conference on, volume Vol. 3, pages pp33 38, Dec T. Takagi and M. Sugeno. Fuzzy identification of systems and its application to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, Vol. 15:pp , H. Wang, K. Tanaka, and M. Griffin. An approach to fuzzy control of nonlinear systems: Stability and design issues. IEEE Transactions on Fuzzy Systems, Vol. 4(1), 1996.