Lyapunov Functions in Piecewise Linear Systems: From Fixed Point to Limit Cycle

This paper provides a first example of constructing Lyapunov functions in a class of piecewise linear systems with limit cycles. The method of construction helps analyze and control complex oscillating systems through novel geometric means. Special attention is stressed upon a problem not formerly solved: to impose consistent boundary conditions on the Lyapunov function in each linear region. By successfully solving the problem, the authors construct continuous Lyapunov functions in the whole state space. It is further demonstrated that the Lyapunov functions constructed explain for the different bifurcations leading to the emergence of limit cycle oscillation

Thermal contributions to the structural phase transitions of crystalline aluminum under ultrahigh pressures

The \emph{ab initio} computations of thermodynamic properties of condensed matters under extreme conditions is a long-time pursuit. In this work, the pressure induced structural phase transitions of crystalline aluminum up to $600$ GPa at room temperature is investigated by the criterion of free energy derived directly from the partition function formulated in ensemble theory with the interatomic interactions characterized by density functional theory computations. The transition pressures of the FCC$\rightarrow$HCP$\rightarrow$BCC phase transition are determined at $194$ and $361$ GPa, the axial ratio of the HCP structure is found to be equal to $1.62$ and the discontinuities in the equations of states are confirmed to be associated with $-0.674\%$ and $-0.897\%$ volume changes, which all validate the accuracy of measurements by one of the recent experiments but disagree with the results by others. Compared with the results obtained by the widely used criterion of enthalpy at $0$K, this work shows that the thermal contributions play a nontrivial role for the structural stability of aluminum even under a room-temperature circumstance.Comment: 6 pages, 5 figure

Autonomous navigation with constrained consistency for C-Ranger

Autonomous underwater vehicles (AUVs) have become the most widely used tools for undertaking complex exploration tasks in marine environments. Their synthetic ability to carry out localization autonomously and build an environmental map concurrently, in other words, simultaneous localization and mapping (SLAM), are considered to be pivotal requirements for AUVs to have truly autonomous navigation. However, the consistency problem of the SLAM system has been greatly ignored during the past decades. In this paper, a consistency constrained extended Kalman filter (EKF) SLAM algorithm, applying the idea of local consistency, is proposed and applied to the autonomous navigation of the C-Ranger AUV, which is developed as our experimental platform. The concept of local consistency (LC) is introduced after an explicit theoretical derivation of the EKF-SLAM system. Then, we present a locally consistency-constrained EKF-SLAM design, LC-EKF, in which the landmark estimates used for linearization are fixed at the beginning of each local time period, rather than evaluated at the latest landmark estimates. Finally, our proposed LC-EKF algorithm is experimentally verified, both in simulations and sea trials. The experimental results show that the LC-EKF performs well with regard to consistency, accuracy and computational efficiency