Reachability analysis aims at detecting whether a hybrid system will reach a final set of states or satisfy a temporal logic formula starting from a given initial set. It is closely related to safety analysis, which is the problem of determining the safety condition such that the system never reaches an unsafe state configuration, and formal verification, which is to prove that the system performs as expected. This study is significant, for example, in the prevention of crashes in multi-agent systems, in unwanted parts be processed by a robot, in the avoidance of a wrong heading taken by an aircraft, and in the growth of unwanted cells in biological systems. Although the wide applicability of hybrid systems has inspired a great deal of research from both control theory and theoretical computer science, the applicability of state-of-the-art analysis and design techniques for hybrid systems has been limited to examples of small size due to complexity. For this reason, the first step of research will be on the reachability problem for piecewise affine (PWA) hybrid systems on simplices as many engineering systems can in a first approximation be described by a piecewise affine hybrid system. The computational and complexity issues of this class of systems seem comparatively simple. Therefore this class merits attention for the development of control theory.
A system composed of many simple entities, each obeying the
same simple rules of interaction, often displays complex
collective behaviors. Such emergent behaviors have been
observed in a multitude of physical, biological, and social
systems, (e.g., social aggregation, swarming, and
synchronization), which may serve as inspirations for the
design of autonomous multi-agent systems (MAS). Theoretical
studies of these systems are valuable because they help us
understand the mechanisms of how global system behavior
emerges from local interaction in coordination and control
applications, despite the absence of centralized coordination
and global information exchange. In studying these systems,
one of the central points has been the study of coupling
structures (both fixed and dynamic) among subsystems (agents),
together with individual dynamical aspects related to the
cooperative behavior. The aim of this research is to use and
further develop the underlying theory to deepen the
functionality of systems for desirable overall behavior, which
emerges from the individual agent's behaviors, and to provide
a context for developing dynamic and flexible systems for
coordination and control.
In the case of environments with discontinuous ground support, such as a rocky slope, a flight of stairs, or the rungs of a ladder, it is arguable that the most appropriate and versatile means for locomotion is legs. Legs enable the avoidance of support discontinuities in the environment by stepping over them. Moreover, legs are an obvious choice for locomotion in environments designed for human walking, running, and climbing. An impressive amount of technology has been amassed and specifically developed to build walking robot prototypes. Nevertheless, conceptual control breakthroughs have not kept pace with the technological developments, especially for bipedal robots --- a subclass of legged robots. A canonical problem in bipedal robots is how to design a controller that generates closed-loop motions, such as walking or running, that are periodic and stable (i.e., stable limit cycles). There is a huge deficit in fundamental control design concepts in comparison to the number of bipedal prototypes. The state-of-the-art is characterized by a heavy reliance on heuristics or on principles such as the zero moment point (ZMP) criterion that do not ensure stability.
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