Wang, Yuan

One of the central issues in stability analysis for control systems is how robust a stability property is when external disturbances are presented. This is even more critical when a system is affected by time delay. Systems affected by time delays are ubiquitous in applications. Time delays add more challenges to the task of stability analysis, mainly due to the fact that the state space of a delay system is not a finite-dimensional Euclidean space anymore, but rather an infinite dimensional space of continuous functions defined on the delay interval. In this work, we investigate robust output stability properties for nonlinear systems affected by time delays and external disturbances. Frequently in applications, the requirement of stability properties imposed on the full set of state variables can be too strenuous or even unrealistic. This motivates one to consider robust output stability properties which are related to partial stability analysis in the classic literature.
We start by formulating several notions on integral input-to-output stability and illustrate how these notions are related. We then continue to develop Lyapunov-Krasovskii type of results for such stability properties. As in the other context of Lyapunov stability analysis such as global asymptotic stability and input-to-state stability, a Lyapunov-Krasovskii functional is required to have a decay rate proportional to the magnitudes of the state variables or output variables on the whole delayed interval. This is a difficult feature when trying to construct a Lyapunov-Krasovskii functional. For this issue, we turn our efforts to Lyapunov-Krasovskii functional with a decay rate depending only on the current values of state variables or output variables. Our results lead to a type of Lyapunov-Krasovskii functionals that are more flexible regarding the decay rate, thereby leading to more efficient results for applications.

This paper presents a converse Lyapunov theorem for robust global asymptotic stability. The main
result extends previous converse Lyapunov theorems to systems with disturbances taking bounded
values in an arbitrary Banach space.

Systems with time delays have a broad range of applications not only in control
systems but also in many other disciplines such as mathematical biology, financial
economics, etc. The time delays cause more complex behaviours of the systems. It
requires more sophisticated analysis due to the infinite dimensional structure of the
space spaces. In this thesis we investigate stability properties associated with output
functions of delay systems.
Our primary target is the equivalent Lyapunov characterization of input-tooutput
stability (ios). A main approach used in this work is the Lyapuno Krasovskii
functional method. The Lyapunov characterization of the so called output-Lagrange
stability is technically the backbone of this work, as it induces a Lyapunov description
for all the other output stability properties, in particular for ios. In the study, we
consider two types of output functions. The first type is defined in between Banach
spaces, whereas the second type is defined between Euclidean spaces. The Lyapunov
characterization for the first type of output maps provides equivalence between the
stability properties and the existence of the Lyapunov-Krasovskii functionals. On the
other hand, as a special case of the first type, the second type output renders flexible Lyapunov descriptions that are more efficient in applications. In the special case
when the output variables represent the complete collection of the state variables,
our Lyapunov work lead to Lyapunov characterizations of iss, complementing the
current iss theory with some novel results.
We also aim at understanding how output stability are affected by the initial
data and the external signals. Since the output variables are in general not a full
collection of the state variables, the overshoots and decay properties may be affected
in different ways by the initial data of either the state variables or just only the output
variables. Accordingly, there are different ways of defining notions on output stability,
making them mathematically precisely. After presenting the definitions, we explore
the connections of these notions. Understanding the relation among the notions is
not only mathematically necessary, it also provides guidelines in system control and
design.

Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly
perturbed systems, yet most of the work so far focused on systems without
delays. In this thesis, we provide a new set of tools for the stability analysis for
singularly perturbed control systems with time delays.

In this thesis, we study the input-to-state stability (scISS) property and related characterizations for discrete-time nonlinear systems. Variations of scISS property were employed in solving particular control problems. The main contribution of this work is to provide a detailed analysis on the relations among various types of notations related to system stability and show that most scISS results for continuous-time nonlinear system can be extended to discrete-time case.

In certain applications, one needs to control physical plants that operate in hazardous conditions. In such situations, it is necessary to acquire access to the controller from a different (remote) location through data communication networks, in order to interconnect the remote location and the controller. The use of such network linking between the plant and the controller may introduce network delays, which would affect adversely the performance of the process control. The main theoretical contribution of this thesis is to answer the following question: How large can a network delay be tolerated such that the delayed closed-loop system is locally asymptotically stable? An explicit time-independent bound for the delay is derived. In addition, various practical realizations for the remote control tasks are presented, utilizing a set of predefined classes for serial communication, data-acquisition modules and stream-based sockets. Due to the presence of a network, implementing an efficient control scheme is a not trivial problem. Hence, two practical frameworks for Internet-based control are illustrated in this thesis. Related implementation issues are addressed in detail. Examples and case studies are provided to demonstrate the effectiveness of the proposal approach.

Underactuated mechanical systems are those possessing fewer actuators than degrees of freedom, making the class a rich one from a control standpoint. The double inverted pendulum is a particular underactuated system and a well-known benchmark case for which many solutions have been offered in the literature. The control objective is to bring the system to its unstable top equilibrium point. The underactuated horizontal double pendulum is a two-link planar robot with only one actuator either at the shoulder or the elbow. Almost no work was done on the underactuated horizontal pendulum, mainly due to the lack of controllability of such a system. The fundamental difference between a double inverted pendulum and an underactuated horizontal double pendulum is that in the latter gravity effects do not exist. Gravity is important to the controllability of the system. Thus, in search for a "gravity substitute," we added springs in the underactuated horizontal double pendulum in order to create a source of potential energy. Two different types of such systems are analyzed: spring coupled underactuated horizontal double pendulums and underactuated horizontal double pendulums with spring-loaded sliding bar constraint. The main contribution of the thesis is in proving that the zero state of the spring coupled systems is globally asymptotically stabilizable. Explicit control laws were developed.