Pulsewidth pulsefrequency (PWPF) modulators are widely used in spacecraft thruster control. Their dynamic characteristic is still lack of effective analysis tools. This paper presents a fractional describing function method to describe the frequency characteristics of PWPF. A frequencydependent gain and phase shift are clearly described by fractionalorder expression, and the fractionalorder behaviors depict the nonlinear properties of PWPF modulators. This fractional describing function method can also be applied to other kinds of modulators.
Spacecrafts commonly deploy thrusters as actuators for attitude control [
Describing function method is a wellknown analysis tool for a kind of nonlinear system with certain structure. If the output signal of a nonlinear device can be approximated by the fundamental harmonics, the fundamental harmonics can be used to define the frequency characteristics of the nonlinear element and is called a describing function [
Recently, fractional calculus has been increasingly applied to mechanical systems, electricity, and bioengineering [
Pulse modulators are commonly used in thruster control of fuel valves. There are various kinds of pulse modulators, such as pulsewidth modulator, pulsefrequency modulator, pseudorate modulator, and pulsewidth pulsefrequency (PWPF) modulator [
Structure of PWPF modulator.
With a constant input
Static characteristic variables of PWPF modulator.
On time 

Off time 

Modulator frequency 

Duty cycle 

Equivalent internal deadband 

Equivalent internal saturation level 

The static analysis of PWPF modulator shows that it operates near linear to the constant input over a large range between the deadband
Describing function method is an approximation method for analyzing nonlinear dynamics, because only the first harmonic of the output of a nonlinear element is considered. But here describing function method is beneficial for PWPF modulator analysis for the following reasons. Firstly, the firstorder filter which is in series with the nonlinear element serves as a lowpass filter in PWPF modulator. Secondly, the attitude control system is a highorder system, where high harmonics are attenuated substantially. In this section, the PWPF modulator is considered as a single device and its fractional describing function is developed.
A nonlinear element to a sinusoidal input
With the assumption that the amplitude of fundamental harmonic is much larger than the amplitude of other harmonics, the describing function is defined as the complex ratio of the fundamental harmonic component of the output and the input, that is,
Imposing this harmonic balance principle,
The describing function defined above is based on fundamental harmonic equivalence. The influence of high order harmonics can be considered as system uncertainty.
The PWPF modulator is a unit in the attitude control system. Considering, the frequency characteristics of PWPF as a whole is convenient for analysis and design of the attitude control system. In the structure of Figure
Then the frequency characteristic of the PWPF closedloop system is
If the describing function is independent of frequency
Now, a fractional describing function is introduced to give a direct relationship between the nonlinear characteristic and frequency. For illustration purpose, set the parameters of PWPF as follows:
Nichols plot of
To reveal the relationship between the describing function and the frequency of interest, the real part and imaginary part of
Loglog plots of
Loglog plots of
The fractionalorder behavior is described by power functions, so it is called fractional describing function. The new function can be written as
Figure
Bode diagram of
The deduced fractional describing function is useful for system stability analysis for it is considered around the crossover frequency.
PWPF modulator is a nonlinear actuator in spacecraft attitude control system. The nonlinear dynamic behavior of PWPF modulator is investigated by the fractional describing function in this paper. The nonlinear element of PWPF is a Schmitt trigger, and its frequency characteristic can be described by describing function, and the fractional behaviors are caused by nonlinear element in PWPF. The frequency characteristic of the actuator is frequency dependent. The loglog plots of the real part and imaginary part of a modulator clearly reveal the fractionalorder behavior. The imaginary component is described by fractionalorder power function over a certain frequency range. With fractional calculus, these frequencydependent gain and phase information can be plotted in Bode diagram and used for control system design. Furthermore, the fractional describing function method should be an effective tool for other kinds of modulators.
This work is funded in part by the China Scholarship Council (CSC) and in part under Project Agreement no. DSOCL10004 with DSO National Laboratories, Singapore.