Carl R. Knospe

Feedback linearization of an active magnetic bearing with voltage control

A feedback linearization controller is presented for a single-degree-of-freedom (DOF) magnetic bearing test rig. The feedback linearization controller is derived from a detailed nonlinear electromagnet model using both analytic relationships and experimental calibration data. The controller is implemented on the test rig in voltage mode, and measured open-loop transfer functions are used to demonstrate the effectiveness of the feedback linearization controller in transforming the nonlinear system to a linear plant. Next, a high-performance controller for the feedback linearized plant is designed with /spl mu/-synthesis to guarantee a beam compliance performance specification. A common problem associated with voltage control in magnetic suspensions and coil resistance variation, is successfully handled by augmenting the feedback linearized plant with a structured uncertainty. Experimental results demonstrate that the /spl mu/ controller with feedback linearization achieves the performance specified during design for the nonlinear plant independent of the disturbance force level or displacement incurred.

Experiments in the control of unbalance response using magnetic bearings

Abstract Unbalance response is a common vibration problem associated with rotating machinery. For several years, researchers have demonstrated that this vibration could be greatly alleviated for machines using active magnetic bearings through active control. Many of the control strategies employed fall into a class which the authors have termed adaptive open loop control. In this paper, three algorithms in this class are presented and their performances are examined experimentally. These algorithms are (1) a non-recursive control law with simultaneous estimation, (2) a recursive control law with simultaneous estimation, and (3) a recursive control law with gain scheduling according to operating speed. Each algorithm was coded in C and executed on a high-speed, multi-tasking digital controller. The advantages and disadvantages of each algorithm are illustrated by examining experimental results from a laboratory magnetic bearing rotor rig. These results clearly demonstrate the high degree of synchronous vibration attenuation (over 30 dB) which can be achieved with adaptive open loop methods. The response of these algorithms to a sudden change in “simulated imbalance” is used to evaluate their relative transient performances. These results indicate the benefits of recursive control laws in adapting the synchronous open loop control currents to cancel the vibration. The ability of each of the algorithms to adapt the open loop control during changes in rotor speed is also examined. On this test, the recursive gain scheduled algorithm shows superior performance: rotor midspan vibration is almost completely eliminated over the operating speed range. However, surprisingly, the non-recursive control law shows better performance than the recursive law with simultaneous estimation. This result is explained in terms of the stability of the adaptation process.

Stability analysis of LPV time-delayed systems

This work provides one of the first attempts to derive computationally tractable criteria for analysing the stability of linear parameter varying (LPV) time-delayed systems. We present both delay-independent and delay-dependent stability conditions, which are derived using appropriately selected Lyapunov-Krasovskii functionals. According to the system parameter dependence, these functionals can be selected to obtain increasingly non-conservative results. Gridding techniques may be used to cast these tests as linear matrix inequalities (LMIs). In cases when the system matrices depend affinely or quadratically on the parameter, gridding may be avoided. These LMIs can be solved efficiently using available software. A numerical example of a time-delayed system motivated by a metal removal process is used to demonstrate the theoretical results.

Robustness of Adaptive Unbalance Control of Rotors with Magnetic Bearings

Rotor unbalance in the primary cause of unacceptable vibration in rotating machinery. Over the last decade, researchers have explored different methods of taking advantage of the active nature of magnetic bearings to attenuate unbalance response including both feedback and adaptive open loop methods. An important issue in the application of this technology to industrial machines is the robustness of the unbalance control algorithm. The stability and performance robustness of a promising adaptive open loop control algorithm is examined. Expressions are derived for a number of unstructured uncertainties. Experimental results are then presented, which evaluate the algorithms robustness with respect to three variations: gain schedule errors, random additive errors, and feedback loop gain. The robustness exhibited in these tests was quite good and, along with the excellent vibration attenuation obtained, recommend the algorithm for further testing and industrial application. The experimental results indicate that the theoretical robustness expressions do provide an upper bound on actual performance, however this bound is not tight. Although the conservatism in the results is partly due to the variations considered and the worst-case nature of the performance robustness guarantees, the results also indicate that further research is needed on unstructured performance robustness for this method of rotor vibration control.

Control Approaches to the Suppression of Machining Chatter Using Active Magnetic Bearings

Several control approaches to the active suppression of machining chatter, a self-excited vibration that limits metal removal rate, are examined using a specially constructed turning experiment. The experiment employs a magnetic bearing for actuation and mimics the dynamics of a flexible rotor. Control forces are applied and vibration measurements taken at a location along this structure that is not collocated with the tool. Three control approaches are considered: speed-independent control, speed-specified control, and speed-interval control. Experimental results with these are compared to those obtained using proportional-integral-derivative (PID) control, a standard approach in the magnetic bearing industry today. Significant improvements over PID in machining stability lobes are obtained and the capability to highly tailor the cutting tool compliance so as to inhibit the onset of chatter is demonstrated. Cutting tests are also presented which demonstrate the significant improvements in chatter-free chip width that may be obtained with advanced control methods

Feedback linearization of active magnetic bearings: current-mode implementation

Feedback linearization is a promising approach to the nonlinear control problem posed by active magnetic bearing systems. In this paper, feedback linearization is employed in combination with robust control techniques for the regulation of a single axis test rig actuated by a multiple pole magnetic bearing. To this end, a nonlinear polynomial model of the magnetic actuator was developed based on its experimental calibration. The effect of the amplifier and measurement system dynamics on the feedback linearization performance, was also examined, and compensation filters were developed. Finally, an uncertainty framework was proposed for the linearized plant, and a robust controller was designed via /spl mu/ synthesis. Experimental results demonstrate that the feedback-linearized active magnetic bearing system can achieve stability and the specified performance over the entire range of bearing clearance. The introduction of compensation filters is shown to be essential to this result.

Modeling of Nonlaminated Electromagnetic Suspension Systems

Eddy currents induced within nonlaminated electromagnetic actuators by time-varying magnetic fields have a strong effect on the dynamics and control of electromagnetic suspension systems. This paper examines the modeling of these suspension systems and resolves two important problems: 1) the effect of time-varying flotor position on electromagnetic force production and 2) the proper manner in which to model voltage-mode operation of the suspension. The models developed are explicit functions of actuator material and geometric properties. The investigation focuses on axisymmetric cylindrical electromagnetic actuators. Similar results are provided for nonlaminated actuators with C-core stators. Experimental results are presented that demonstrate the accuracy of the modeling approach.

Stability of linear time-delay systems: a delay-dependent criterion with a tight conservatism bound

The stability of linear time-delay systems is investigated via the robustness analysis of a related delay-free comparison system with an uncertain real parameter. By exploiting its phase properties, the delay element is removed from the system via a parameter-dependent Pade approximation. We then present a simple yet rigorous condition for delay-dependent stability of the original time-delay system. The novelty of this result is that it explicitly provides an a priori upper bound of how conservative this condition can be, and this bound depends only on the order of Pade approximation and can be reduced to any desired degree. Furthermore, the delay margin provided by this condition can be computed explicitly without incurring any additional conservatism for the single delay case. This condition can also be checked with some (typically small) additional conservatism by reducing it to finite-dimensional linear matrix inequalities (LMIs). Finally, several numerical examples demonstrate that this simplified LMI criterion can be significantly less conservative than those in the literature.

A multitasking DSP implementation of adaptive magnetic bearing control

Several adaptive on-line synchronous vibration control algorithms are demonstrated on a laboratory rotor supported in magnetic bearings with a high-speed digital controller. Central to the controller hardware developed for this application is a floating-point digital signal processor. The algorithms are executed under a multitasking real-time operating system specifically written for the hardware platform. The multitasking operating system allows the feedback control algorithm and the adaptive vibration control algorithm to be written and executed as separate tasks. This greatly simplifies programming, code verification, and algorithm development. Tests conducted on the laboratory rotor system indicate that greater than 98% attenuation in unbalance response can be achieved with the adaptive control algorithms.

Analytic model for a nonlaminated cylindrical magnetic actuator including eddy currents

The eddy currents induced within a nonlaminated cylindrical magnetic actuator by a changing field have a fundamental influence on the actuators performance. Understanding of these dynamics is essential in designing high-performance actuators and developing control algorithms for them. This paper presents an analytical approach to modeling the relationship between applied magnetomotive force and mechanical force. The approach is based on dividing the actuator into elements according to the flux distribution inside the actuator and finding the frequency-dependent reluctance of the flux paths of each element. An analytic model and its half-order simplification are derived, both of which are explicitly dependent on actuator material and geometric properties. Performance predictions from both analytic models are compared with finite-element analysis, demonstrating the accuracy of the models.