Almost sure convergence of ILC for networked linear systems with random link failures

The iterative learning control (ILC) problem for networked linear system with random link failure is addressed in this paper. The link failures arise both from the controller to the plant and from the plant to the controller. The random link failure is modeled by an independent Bernoulli random variable. Two P-type update laws are designed and critically analyzed. The almost sure convergence property is established according to this case for the first time. Illustrative simulations show the effectiveness of the proposed algorithms.     

Almost sure exponential convergence to consensus of random gossip algorithms

In this paper, we provide a proof of almost sure exponential convergence to consensus for a general class of ergodic edge selection processes. The proof is based on the multiplicative ergodic theorem of Oseledec and also applies to continuous time gossip algorithms. An example of exponential convergence in a non ergodic case is also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Almost sure stability for uncertain differential equation with jumps

Uncertain differential equation with jumps, as a crucial tool to deal with a discontinuous uncertain system, is a type of differential equation driven by both canonical Liu process and uncertain renewal process. So far, a concept of stability in measure for an uncertain differential equation with jumps has been proposed. As a supplement, this paper proposes a concept of almost sure stability for an uncertain differential equation with jumps. A sufficient condition is derived for an uncertain differential equation with jumps being stable almost surely. As a corollary, a sufficient condition is also given for a linear uncertain differential equation with jumps being stable almost surely.     

Almost sure convergence of stochastic approximation algorithms with non-additive noise

Stochastic approximation algorithms with non-additive noise are discussed. In studying strong convergence of such algorithms, traditionally one assumes that the iterates return to a bounded or compact set infinitely often, or that the function under consideration grows with certain rate. The usual projection algorithms require that the bounded projection region is known beforehand. It is desirable to weaken these ‘boundedness’ conditions. By introducing randomly varying truncations, Chen and Zhu (1986) achieved this for stochastic approximation algorithms with additive noise. Here, we extend their result to a more general setting.     

Almost sure convergence of iterative learning control for stochastic systems

This paper proposes an iterative learning control (ILC) algorithm with the purpose of controling the output of a linear stochastic system presented in state space form to track a desired realizable trajectory. It is proved that the algorithm converges to the optimal one a.s. under the condition that the product input-output coupling matrices are full-column rank in addition to some assumptions on noises. No other knowledge about system matrices and covariance matrices is required.     

Random solution of a stochastic integral equation :almost sure and mean square convergence of successive approximations

A non-linear random integral equation of the Volterra type of the form was considered by Tsokos (1960), where ω?Ω, the underlying set of the probability measure space (Ω A, P). He was concerned with the existence of a unique random solution to this equation, where a random solution is defined to be a second-order stochastic process, x(t; ω), which satisfies the equation almost surely. The present paper shows that a sequence of successive approximations, x_n)tω) converges to the unique random solution at each t≥0 with probability one and in mean square under the conditions of Tsokos' theorem. The rate of convergence of the successive approximations and a bound on the mean square error of approximation are also given.     

Almost sure parameter estimation and convergence rates for hidden Markov models

A continuous-time version of Kronecker's Lemma is established and used to give rates of convergence for parameter estimates in hidden Markov models.     

Almost sure convergence of least squares estimates for regular multivariate ARX systems

We establish almost sure convergence of least squares estimates for general multivariate ARX(p, s) systems, with stochastic input signal. Results of strong consistency and speed of convergence are obtained with a regularity assumption on the AR part of the system.     

Almost sure convergence of extremum seeking algorithm using stochastic perturbation

This paper presents an extremum seeking (ES) algorithm where the perturbation signal is a martingale difference sequence (m.d.s.) with a vanishing variance. The measurement noise at the plant output is modeled by a superposition of deterministic component, and a non-stationary colored noise signal. The optimizing set point of the uncertain reference-to-output equilibrium map is estimated by a stochastic approximation (SA)-type algorithm. The algorithm has a vanishing gain sequence dependent on the set point estimates. By utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point, in spite of the presence of a measurement noise. This result is derived without requiring boundedness or any prior condition on the set point estimates.     

Almost sure convergence of classification procedures using Hermite series density estimates

A multidimensional classification procedure is examined derived from the multiple Hermite series estimate of probability density functions. Conditions for the almost sure convergence of the integrated square error for the estimate are presented and the rate of the convergence is studied. The probability of misclassification, conditioned on a learning sequence of length n, is shown to converge to the Bayes risk almost surely as rapidly as $\text{O}\text{(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2+\delta </mtext>}}\text{)}$, δ positive.     

Almost sure rate of convergence of the parameter estimates instochastic approximation algorithm

This paper presents novel results on the almost sure convergence of the parameter estimates in stochastic approximation algorithm. It is proved that this rate has the same order as the best one established for the least squares algorithm. Although we consider the case of self-tuning controllers, the presented results can easily be extended to some other adaptive processes     

Almost sure convergence rates for system identification using binary,quantized, and regular sensors

This paper presents almost sure convergence rates for system identification under binary, quantized, and regular sensors. To accommodate practical model complexity constraints, the system under consideration is represented by a modeled part together with an unknown-but-bounded unmodeled dynamics. Under uncorrelated noise sequences, identification errors with different sensor types are studied and tight error bounds are obtained without information or constraints on noise moment conditions. The results are then extended to correlated noise sequences whose remote past and distant future are asymptotically independent. In both cases, almost sure error bounds of the laws of iterated logarithms type are derived.     

Adaptive ILC for a class of discrete-time systems with iteration-varying trajectory and random initial condition

In this work we present a discrete-time adaptive iterative learning control (AILC) scheme to deal with systems with time-varying parametric uncertainties. Using the analogy between the discrete-time axis and the iterative learning axis, the new adaptive ILC can incorporate a Recursive Least Squares (RLS) algorithm, hence the learning gain can be tuned iteratively along the learning axis and pointwisely along the time axis. When the initial states are random and the reference trajectory is iteration-varying, the new AILC can achieve the pointwise convergence over a finite time interval asymptotically along the iterative learning axis.     

具有脉冲效应的切换随机系统的几乎必然稳定性

针对一类在切换时刻具有脉冲行为的Markov切换非线性随机系统,首先,应用切换的Lyapunov函数方法研究系统的稳定性,给出系统几乎必然稳定的充分条件,该条件不依赖于系统的矩稳定性;然后,进一步对线性系统的稳定化问题进行分析与设计,对随机子系统的控制结构同时出现在方程的位移部分与扩散部分,给出相应的状态反馈增益矩阵的求解方法;最后,数值算例说明了所设计方法的有效性.     

Almost sure exponential stability of stochastic fluid networks with nonlinear control

Due to the objective of controlling fluid flows, fluid networks are high-order nonlinear stochastic systems in essential. Taking the nonlinearities in pressure drop of fan branch, measurement error of inertia coefficients, and random factors into consideration, this paper presents a stochastic fluid networks model described by Itô stochastic differential equations. As drift term of the stochastic fluid networks does not satisfy the linear growth condition, the almost sure exponential stability cannot get from the moment exponential stability. We directly investigate the almost sure exponential stability of the tracking and regulation problems for fluid networks under decentralised and adaptive control. It shows that the fluid flows in all branches can reach their reference values if we set controller only in links of the networks. We illustrate the results with two examples.     

The asymptotic self-similar behavior for the quasilinear heat equation with nonlinear boundary condition

In this paper the quasilinear heat equation with the nonlinear boundary condition is studied. The blow-up rate and existence of a self-similar solution are obtained. It is proved that the rescaled function

v(y,t)=(T−t)^{1/(2p+α−2)}u((T−t)^{(p−1)/(2p+α−2)}y,t),