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The variable sampling rate system is encountered in many applications. When the speed information is derived from the position marks along the trajectory, one would have a speed dependent sampling rate system. The conventional fixed or multisampling rate system theory may not work in these cases because the system dynamics include the uncertainties which resulted from the variable sampling rate. This paper derived a convenient expression for the speed dependent sampling rate system. The varying sampling rate effect is then translated into multiplicative uncertainties to the system. The design then uses the popular

The brushless DC (BLDC) motor applications are getting popular due to their efficient operation and high power density characteristics. Because of its high efficiency and easy maintenance, the application range has been found from large system like electrical vehicles to small systems like computer peripherals. The BLDC motor usually uses hall-effect sensors to determine the commutation timing among the armature windings. The same hall sensor signal is also used to measure motor speed. There are many commercial industrial driver ICs which apply such kind of design.

BLDC motors may be standard application, but ones still encountered many difficulties when engineers try to implement in precision speed control. As mentioned before, the typical BLDC motor feedback signal is from the few hall-effect sensor signals which originally are designed for commutation purpose. The time interval between the consecutive feedback signals depends on the motor speed. Notice that BLDC motor is used in many high performance industrial applications like computer hard disk drive and the optical drive servosystems. High performance controller design is necessary. The BLDC motor system design maintains high resolution speed measurement through high frequency clock. The traditional driver design uses current transducer to achieve constant current sampling for the acceleration loop; however, the velocity loop has to be based on the speed dependent sampling, and some sort of robust variable sampling rate controller design is thus desirable.

Most previous researches on multirate sampling systems in the literatures have focused on multiple but fixed sampling rate problems [

In the preceding work [

In this paper, the authors realized that the variable sampling rate system model derived for the observer can be reduced to describe the variable sampling rate system itself. Instead of treating the system with a prespecified sampling rate, one can look at the system only when the feedback is available and lump the effects of the variable sampling time into system uncertainties. It is now possible to describe the system with the standard linear fractional transformation. The

Consider two time scales, the underlying system sampling rate is denoted by

The variable rate time frames.

If the basic sampling time for the system is

For the design purpose, assume that the system model is precisely known, and the manipulated input

Consider that “

Thus, for every instance

The uncertainty representation of variable sampling rate systems.

Note again that

The system in Figure

To facilitate the

The control system can be represented with the familiar linear fractional transformation (LFT) as in Figure

LFT representation of the system.

The uncertainties

This experiment uses a BLDC motor from Troy Co., Taiwan, and the TMS320F243 from Texas Instrument for the controller implementation. The driver uses a simple protection circuit to read the hall sensor signal and determine the rotor angle and the rotor speed. Another protection circuit is used to drive the six MOSFET switches for the three-phase winding. The controller determines the proper on-off sequence and the controlled PWM duty cycle for the MOSFET to achieve the control purpose. The schematics of the control circuit and the physical setup are shown in Figure

The BLDC system setup.

The capture unit built in the TMS320F243 is used to detect the hall sensor signals and determine the rotor angle. The proposed driver also uses these signals to determine the proper MOSFET to turn it on. There are shunt resistors for constant rate sampling of the phase currents, but the motor speed measurement still depends on the hall sensor signals. The time interval between two consecutive hall sensor signals is used to determine the rotor speed. The sampling rate of the feedback system is thus speed dependent. In this experiment, the underlying sampling frequency for the TMS320F243 is set at 4 KHz. And the PWM module built in TMS320F243 is set to generate PWM waveform whose carrier is also set at 4 KHz. In an other site, there are 12 hall sensor signal updates per revolution for a 4-pole permanent magnet arrangement. Therefore there would be 66 samples between measurements when the motor runs at 3000 rpm.

Figure

The frequency spectrum of the system response.

The filtered signal can serve as the bases for system identification and for the control feedback. The identified system transfer function is

Again, there are 12 measurement updates per cycle. For the motor operating at 300~3000 rpm, the sampling frequency would have range of 60~600 Hz. The sampling frequency for the underlying control is 4 KHz. Therefore the number of sampling periods,

The experiment then uses the MATLAB

The performance specification for the control synthesis.

The synthesis resulted in a 6th order controller with the following form:

Figure

The theoretical system response to the different reference commands.

The controller calculates a PWM duty cycle in the actual system. Inherently, there is no negative command from the duty cycle. There is a separate logic for the reversed rotation, but it will not be discussed here. The lack of negative command would result in slower recovery from the overshoot. The MOSFET switches also impose limits on the output current. Figure

Simulation response of the system under different reference commands.

The TMS320F243 is a 16-bit fixed-point digital signal processor. There are special considerations when implementing high order control algorithm. The compensator is first decomposed into fractional expansion in Figure

The first two terms in Figure

The hard implementation and the controller flow chart.

Figure

The experimental response of system under different reference commands.

Detailed response.

Saturation effect and the effect of nonnegative efforts.

The fractional expansion of the controller.

This paper proposed a controller design for systems with variable sampling rate. By variable sampling rate, the authors mean a system with undetermined sampling frequency or speed dependent sampling frequency. The paper first presented the system modeling for the variable sampling rate system. The changing sampling rate was then translated into system uncertainties. The uncertainties appear in different locations in the system with fixed structure; therefore, the popular

This project is sponsored in part by the National Science Council, Taiwan, under Contract no. NSC 102-2221-E-224-028.