Gain margin and phase margin are important specifications in the frequency domain for
the analysis and design of practical control systems and have served as important measures of
robustness analysis which is always of primary concern. This is because the models used are
usually imprecise and the parameters of all physical systems vary with the operating
conditions and time. They are usually obtained numerically or graphically by the use of
system frequency response like Bode plots. Studying for controller design to satisfy GM, PM
or sensitivity conditions was proposed by several articles such as in [1]-[6], There are also
many design methods to determine the parameters to meet different objectives [7]-[9].
Designing a controller for a fixed and exact control plant is not usually practical in the natural
environments. Due to the simplified models or the factors resulting from the changing
environments, the uncertainties in system parameters can always occur. Uncertain parameters
in a linear control system can be robustly analyzed by the parameter plane method or the
stability of perturbed interval polynomials, is to guarantee if all the polynomials have the
roots in the left-half plane [17]. The perturbed parameters will result in root-clusters, within
which the roots of the perturbed polynomials will be located. Usually, a change in a physical
quantity typically appears in more than one coefficient of the characteristic equation. Robust
Gamma-stability analysis for a perturbed vehicle plant was also studied [18]. The methods of
analyzing the gain-phase margin of a linear control system with adjustable parameters have
been developed [19]-[21]. Strictly speaking, the majority of the researches mentioned above
are not concentrated on the controller design for perturbed systems. Sensitivity functions are
usually used as a design specification to indicate the robustness of a system. In [6] and [8],
Yaniv and Nagurka proposed a robust controller design method satisfying GM, PM and
sensitivity constraints on the perturbed systems, not with the system parameters in uncertain
continuous intervals, but with the system uncertainties in the finite discrete set of gains and
pole locations.
Undesirable oscillation phenomena due to nonlinearities in a feedback closed system
have been studied by many publications [22]-[26] and it is important for the designer to
predict the limit cycle behavior of a perturbed vehicle system with nonlinearities. It is of
interest to know the frequency, amplitude, stability and instability of the limit cycle occurred.
The describing function technique is mainly employed to predict the existence of constant
amplitude oscillations of closed nonlinear systems and has been successfully used in many
applications although some limitations exist in the systems which don’t satisfy the assumption
of filtering out the higher order harmonics [27]-[30].
In addition, some researchers have developed the experimental and analytic describing
functions of fuzzy controller in order to analyze the stability of fuzzy control systems [31-32].
Furthermore, the describing function technique to design a fuzzy controller for switching
DC-DC regulators was proposed by Gomariz et al [33]. The describing function was also
applied to find the bounds for the neural network parameters to have a stable system response
and generate limit cycles [34]. The results in [32] and [33] are extended to analyze the
stability of a fuzzy vehicle steering control system under the effects of system parameters and
gain-phase margin by the use of methods of describing function, parameter plane and a
gain-phase margin tester. A simple vehicle steering control model with perturbed parameters
is cited to verify the design procedure.
On the other hand, there are a large number of studies concentrated on the subject of
phase-locked loops (PLL) in the latest decades. The theoretical description of PLL was well
proposed [35]-[39]. A PLL is essentially a circuit that has a particular system lock its
frequency as well as the phase to those of the input applied to it. When the phase error is built
up in the locked state, a feedback mechanism acts on an oscillator called VCO so that the
error is reduced to a minimum and a phase output of VCO is really locked to the reference
input. There are a considerable number of applications in many areas. A technique using PLL
was established on motor speed control [40]. In the design of Global Positioning System
receivers, PLL is very useful especially in a noisy environment [41]. PLL was also applied in
the design of frequency synthesizer [42].
In this thesis, GM and PM performances are defined for a perturbed system with
uncertain continuous interval parameters and shown here graphically in the system parameter
space. By the use of parameter space method and robustness stability criteria, stability
boundary curves corresponding to specific GM and PM constraints are generated. Owing to
the complexity of the controller design for perturbed control systems, it is not an easy job to
find out a qualified controller together with the system plant with uncertain interval
parameters so that the whole closed system at every point in the perturbed system parameter
region satisfies all the three specifications of GM, PM and sensitivity. The main concern in
the controller design is to find a desired region in the controller coefficient plane so that the
performance of the whole system with uncertain parameters inside a perturbed space satisfies
given specifications. The desired controller will be determined graphically from a figure in
which a qualified controller coefficient area is to be found out. With the help of stability
boundary curves in the controller coefficient space, the objective of designing a suitable
controller meeting the specified requirements is achieved.