Dylan Poulsen

Time scale-based observer design for battery state-of-charge estimation

We propose a novel time scale-based observer to estimate the state-of-charge (SoC) of a Lithium-ion battery. The design of the observer promises eventual hardware instantiations that require less power to operate than standard observer designs, while still increasing the accuracy of SoC estimation relative to Coulomb counting methods.1

Observer based feedback controllers on stochastic time scales

We study the problem of stabilization via state feedback where the time between sampling points is not known a priori, but has known statistical properties. In particular, we assume that the distance between sampling points is an independent sequence of random variables with known mean and variance. We consider the cases where the state is known outright and where the state is estimated via an observer.

Stochastic time scales: Quadratic Lyapunov functions and probabilistic regions of stability

We present a version of Lyapunov theory for stochastically generated time scales. In the case of quadratic Lyapunov functions for the LTI case, our results improve the requirement that spec(A) ⊂ Hmin. Our approach also allows us to consider a special class of LTV problems where the dependence on time is only through the graininess.

On the stability of mu-varying dynamic equations on stochastically generated time scales

In their 2001 paper, Potzsche, Siegmund and Wirth gave necessary and sufficient conditions for an LTI system on a time scale to have exponentially stable solutions based on pole placement. We find simple conditions for the stability of mu-varying scalar dynamic equations on time scales which are stochastically generated. As a special case, we examine the region in the complex plane which will guarantee the exponential stability of solutions of LTI systems. Via a decay analysis, we show how the tendency of the solution to grow or decay at each time step is determined by the pole placement within the region of exponential stability1.

Is Deterministic Real Time Control Always Necessary? A Time Scales Perspective

Practitioners of feedback control design often must spend a great deal of time and effort dealing with the complexities of deterministic, or “real time” computing. In this paper, we argue that if certain conditions are met, stable feedback control is possible under non-deterministic conditions. In particular, certain classes of linear systems may be uniformly exponentially stabilized by placing the closed-loop poles within an “osculating circle” if the statistics of the controller’s sampling times are known.Copyright

The Poisson process and associated probability distributions on time scales

Duals of probability distributions on continuous (R) domains exist on discrete (Z) domains. The Poisson distribution on R, for example, manifests itself as a binomial distribution on Z. Time scales are a domain generalization in which R and Z are special cases. We formulate a generalized Poisson process on an arbitrary time scale and show that the conventional Poisson distribution on R and binomial distribution on Z are special cases. The waiting times of the generalized Poisson process are used to derive the Erlang distribution on a time scale and, in particular, the exponential distribution on a time scale. The memoryless property of the exponential distribution on R is well known. We find conditions on the time scale which preserve the memorylessness property in the generalized case.

The emergence of chaos on non-uniformly timed systems

This paper is motivated by the problems posed in control design when actuators, sensors, and/or computational nodes connect via unreliable or unpredictable communications channels. In these cases, non-uniformities are introduced into the underlying time domain of the system. Our central question is whether chaos can emerge in a system as a result of changing only the time domain. We answer the question in the positive by producing an example. Additionally, we find fractal structure in the emergence of chaos in this example.

Numeric and Analytic Investigations of Mean-Square Exponential Stability for Stochastically Timed Systems

This paper is motivated by the problems posed in feedback control design when actuators, sensors, and/or computational nodes connect via unreliable or unpredictable communications channels. In these cases, there is a degree of stochastic uncertainty to the timing of the system’s discretizing elements, such as digital-to-analog converters. Several theorems related to the stability of non-uniformly sampled discrete dynamical systems have recently been proposed; here we examine through numeric investigation the characteristics of systems which are mean square exponentially stable (MSES). In particular we present a method to compute the range of mean and variance that a nonuniformly discretized feedback control system may tolerate while remaining MSES. Several examples are presented.Copyright

Experimental Investigation of a Time Scales-Based Stability Criterion Over Finite Time Horizons

Feedback control systems that employ large area networks or other unpredictable or unreliable communications protocols between sensors, actuators, and controllers may experience nonuniform sampling characteristics. Previous work by Poulsen, et. al. gives a criterion for exponential stability of non-uniformly discretized feedback control systems, assuming sample periods drawn from a known statistical distribution. However, the given stability theorem assumes an infinite time horizon. This work therefore examines the exponential stability criterion experimentally, over a finite time horizon, on a 2nd-order servo mechanism. This paper is the first to experimentally investigate the validity of this time scales stability criterion.Copyright