Continuous Time Discrete State Research Articles

Exponential Filtering for Uncertain Markovian Jump Time-Delay Systems With Nonlinear Disturbances

In this paper, we study the robust exponential filter design problem for a class of uncertain time-delay systems with both Markovian jumping parameters and nonlinear disturbances. The jumping parameters considered here are generated from a continuous-time discrete-state homogeneous Markov process, and the parameter uncertainties appearing in the state and output equations are real, time dependent, and norm bounded. The time-delay and the nonlinear disturbances are assumed to be unknown. The purpose of the problem under investigation is to design a linear, delay-free, uncertainty-independent state estimator such that, for all admissible uncertainties as well as nonlinear disturbances, the dynamics of the estimation error is stochastically exponentially stable in the mean square, independent of the time delay. We address both the filtering analysis and synthesis issues, and show that the problem of exponential filtering for the class of uncertain time-delay jump systems with nonlinear disturbances can be solved in terms of the solutions to a set of linear (quadratic) matrix inequalities. A numerical example is exploited to demonstrate the usefulness of the developed theory.

Cumulative Damage Survival Models for the Randomized Complete Block Design

AbstractA continuous time discrete state cumulative damage process {X(t), t ≥ 0} is considered, based on a non‐homogeneous Poisson hit‐count process and discrete distribution of damage per hit, which can be negative binomial, Neyman type A, Polya‐Aeppli or Lagrangian Poisson. Intensity functions considered for the Poisson process comprise a flexible three‐parameter family. The survival function is S(t) = P(X(t) ≤ L) where L is fixed. Individual variation is accounted for within the construction for the initial damage distribution {P(X(0) = x) | x = 0, 1, …,}. This distribution has an essential cut‐off before x = L and the distribution of L – X(0) may be considered a tolerance distribution. A multivariate extension appropriate for the randomized complete block design is developed by constructing dependence in the initial damage distributions. Our multivariate model is applied (via maximum likelihood) to litter‐matched tumorigenesis data for rats. The litter effect accounts for 5.9 percent of the variance of the individual effect. Cumulative damage hazard functions are compared to nonparametric hazard functions and to hazard functions obtained from the PVF‐Weibull frailty model. The cumulative damage model has greater dimensionality for interpretation compared to other models, owing principally to the intensity function part of the model.

On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters

In this paper, we investigate the stochastic stabilization problem for a class of bilinear continuous time-delay uncertain systems with Markovian jumping parameters. Specifically, the stochastic bilinear jump system under study involves unknown state time-delay, parameter uncertainties, and unknown nonlinear deterministic disturbances. The jumping parameters considered here form a continuous-time discrete-state homogeneous Markov process. The whole system may be regarded as a stochastic bilinear hybrid system that includes both time-evolving and event-driven mechanisms. Our attention is focused on the design of a robust state-feedback controller such that, for all admissible uncertainties as well as nonlinear disturbances, the closed-loop system is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are established to guarantee the existence of desired robust controllers, which are given in terms of the solutions to a set of either linear matrix inequalities (LMIs), or coupled quadratic matrix inequalities. The developed theory is illustrated by numerical simulation.

Stability analysis and controller design for a class of uncertain systems with Markovian jumping parameters

This paper investigates the problem of robust control for a class of systems with both Markovian jumping parameters and parametric uncertainty. The jumping parameters considered here form a continuous-time discrete-state homogeneous Markov process, and the parameter uncertainty appearing in the state equation is real, time-dependent, and normbounded. A method is proposed for the design of robust state-feedback controllers. Both the cases of finite and infinite horizon are addressed. We show that the problem of robust control for linear systems with Markovian jumping parameters can be solved in terms of the solutions to a set of either coupled Riccati differential equations, or algebraic Riccati equations.

Robust Control for Linear Systems with Markovian Jumping Parameters

This paper investigates the problem of robust control for a class of systems with both Markovian jumping parameters and parametric uncertainty. The jumping parameters considered here form a continuous-time discrete-state homogeneous Markov process, and the parameter uncertainty is real time-varying norm-bounded. A methodology is proposed for the design of robust state feedback controllers. We show that the problem of robust control for linear systems with Markovian jumping parameters can be solved in terms of the solutions to a set of either int ercoupled Riccati differential equations, or algebraic Riccati equations.

Performance analysis of Time Warp with homogeneous processors and exponential task times

The behavior of n interacting processors synchronized by the "Time Warp" protocol is analyzed using a discrete state continuous time Markov chain model. The performance and dynamics of the processes are analyzed under the following assumptions: exponential task times and times-tamp increments on messages, each event message generates one new message that is sent to a randomly selected process, negligible rollback, state saving, and communication delay, unbounded message buffers, and homogeneous processors that are never idle. We determine the fraction of processed events that commit, speedup, rollback probability, expected length of rollback, the probability mass function for the number of uncommitted processed events, and the probability distribution function for the virtual time of a process. The analysis is approximate, so the results have been validated through performance measurements of a Time Warp testbed (PHOLD workload model) executing on a shared memory multiprocessor.

Estimating a parametric trend component in a continuous-time jump-type process

We consider stochastic processes with continuous time parameter and discrete state space processing an intensity process. We assume that the intensity process depends on a parameter β, the maximum likelihood (m.l.) estimator β of which enjoys the usual asymptotic properties. Now a trend is defined by a factor multiplied to the intensity which may depend on a parameter α. We present two different types of trend functions (polynominal and reciprocal functions) under which the asymptotic properties of β are inherited by the m.l. estimator ( α, β ) of (α,β). These trend functions, in particular, can be consistently estimated. Examples where the theory presented applies are Markov processes of jump-type, Markov branching processes with immigration and linear OM- (or learning-) processes.

Aspects of non-stationarity

This paper presents descriptive continuous time discrete state non-stationary Markov models that are useful exploratory tools in a first stage of longitudinal data analysis. Because the estimation schemes do not require a lot of arbitrary parametric structure they are useful in investigating data and isolating facts in areas in social science where the relevant theory is not well enough developed to suggest precise functional forms for estimating equations. The paper treats two problems that are especially difficult in non-parametric estimation of longitudinal models: non-stationarity and time aggregation. General non-parametric procedures are presented that enable analysts to assess whether or not data arise from certain special classes of inhomogeneous Markov chains that are of general interest. Explicit models of consumer purchase behaviour, labor market search and occupational mobility are presented.

Chernoff bounds for discriminating between two markov processes

We study a statistical hypothesis testing problem, where a sample function of a Markov process with one of two sets of known parameters is observed over a finite time interval. When a log likelihood ratio test is used to discriminate between the two sets of parameters, we give bounds on the probability of choosing an incorrect hypothesis, and on the total probability of error, for both discrete and continuous time parameter, and discrete and continuous state space. The asymptotic behavior of the bounds is examined as the observation interval becomes infinite.

Numerical methods for stochastic processes

Two numerical methods aimed at discrete state continuous time stochastic processes are discussed. The first (inversion) method is applicable to processes determined by the equation for their probability transform. For instance, many Markov processes are determined by a partial differential equation. The intractability of the presently available analytic methods is overcome by requiring only a numerical solution to the transform equation. Applying the inverse Fourier transform to this solution then yields an approximation to the distribution of a process at an epoch. Useful error bounds are presented. The second (recursion) method, which can be used in conjunction with the first, is of much greater generality, but less efficient. By recursively applying the Chapman- Kolmogorov equation to some initial distribution, the probability distribution for any sequence of epochs can be computed, hence giving a complete description of a process through time. This method can be applied to absorbing processes, and it is shown that the order of the approximation is O( h).

A Stochastic Model for Change in Group Structure

Abstract This paper develops and tests a continuous time discrete state Markov Chain as a model for change in group structure. Movement of triads among different states is focused upon. It is shown formally that a triad cannot move to a state where all the pair relationships are mutual without incurring intransitivity. An empirical test on the change in sociometric structures provides support for the model.

A stochastic model for change in group structure

This paper develops and tests a continuous time discrete state Markov Chain as a model for change in group structure. Movement of triads among different states is focused upon. It is shown formally that a triad cannot move to a state where all the pair relationships are mutual without incurring intransitivity. An empirical test on the change in sociometric structures provides support for the model.