A Unified Approach to High-Gain Adaptive Controllers

It has been known for some time that proportional output feedback will stabilize MIMO, minimum-phase, linear time-invariant systems if the feedback gain is sufficiently large. High-gain adaptive controllers achieve stability by automatically driving up the feedback gain monotonically. More recently, it was demonstrated that sample-and-hold implementations of the high-gain adaptive controller also require adaptation of the sampling rate. In this paper, we use recent advances in the mathematical field of dynamic equations on time scales to unify and generalize the discrete and continuous versions of the high-gain adaptive controller. We prove the stability of high-gain adaptive controllers on a wide class of time scales

Asymptotic regulation of a one-section continuum manipulator

Abstract Continuum manipulators are robotic manipulators built wing one continuous, elastic, and highly deformable %ackbone" instead of multiple rigid links connected by joints. This paper eztends a previous control result for planar continuum robots by proposing a new asymptotic wnvergence argument for a PD-plw-feedforward controller. The benefit of the asymptotic a w m e n t s is that the backbone bending stifness can be adpatively updated by the controller if it is not known a priori

Controllability, Observability, Realizability, and Stability of Dynamic Linear Systems

We develop a linear systems theory that coincides with the existing theories for continuous and discrete dynamical systems, but that also extends to linear systems defined on nonuniform time domains. The approach here is based on generalized Laplace transform methods (e.g. shifts and convolution) from our recent work \cite{DaGrJaMaRa}. We study controllability in terms of the controllability Gramian and various rank conditions (including Kalman's) in both the time invariant and time varying settings and compare the results. We also explore observability in terms of both Gramian and rank conditions as well as realizability results. We conclude by applying this systems theory to connect exponential and BIBO stability problems in this general setting. Numerous examples are included to show the utility of these results.Comment: typos corrected; current form is as accepted in EJD

Time Scale Discrete Fourier Transforms

Abstract-The discrete and continuous Fourier transforms are applicable to discrete and continuous time signals respectively. Time scales allows generalization to to any closed set of points on the real line. Discrete and continuous time are special cases. Using the Hilger exponential from time scale calculus, the discrete Fourier transform (DFT) is extended to signals on a set of points with arbitrary spacing. A time scale consisting of points in time is shown to impose a time scale (more appropriately dubbed a frequency scale), , in the Fourier domain The time scale DFT's (TS-DFT's) are shown to share familiar properties of the DFT, including the derivative theorem and the power theorem. Shifting on a time scale is accomplished through a boxminus and boxplus operators. The shifting allows formulation of time scale convolution and correlation which, as is the case with the DFT, correspond to multiplication in the frequency domain

Nonregressivity

in switched linear circuits and mechanical systems