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Register a new userStability Results on Synchronized Queues in Discrete-Time for Arbitrary Dimension
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Added by
Richard Schoeffauer
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In a batch of synchronized queues, customers can only be serviced all at once or not at all, implying that service remains idle if at least one queue is empty. We propose that a batch of $n$ synchronized queues in a discrete-time setting is quasi-stable for $n in {2,3}$ and unstable for $n geq 4$. A correspondence between such systems and a random-walk-like discrete-time Markov chain (DTMC), which operates on a quotient space of the original state-space, is derived. Using this relation, we prove the proposition by showing that the DTMC is transient for $n geq 4$ and null-recurrent (hence quasi-stability) for $n in {2,3}$ via evaluating infinite power sums over skewed binomial coefficients. Ignoring the special structure of the quotient space, the proposition can be interpreted as a result of Polyas theorem on random walks, since the dimension of said space is $d-1$.
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This paper deals with the stability analysis problem of discrete-time switched linear systems with ranged dwell time. A novel concept called L-switching-cycle is proposed, which contains sequences of multiple activation cycles satisfying the prescribed ranged dwell time constraint. Based on L-switching-cycle, two sufficient conditions are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. It is noted that two conditions are equivalent in stability analysis with the same $L$-switching-cycle. These two sufficient conditions can be viewed as generalizations of the clock-dependent Lyapunov and multiple Lyapunov function methods, respectively. Furthermore, it has been proven that the proposed L-switching-cycle can eventually achieve the nonconservativeness in stability analysis as long as a sufficiently long L-switching-cycle is adopted. A numerical example is provided to illustrate our theoretical results.
Differentiation is an important task in control, observation and fault detection. Levants differentiator is unique, since it is able to estimate exactly and robustly the derivatives of a signal with a bounded high-order derivative. However, the convergence time, although finite, grows unboundedly with the norm of the initial differentiation error, making it uncertain when the estimated derivative is exact. In this paper we propose an extension of Levants differentiator so that the worst case convergence time can be arbitrarily assigned independently of the initial condition, i.e. the estimation converges in emph{Fixed-Time}. We propose also a family of continuous differentiators and provide a unified Lyapunov framework for analysis and design.
Many processes, such as discrete event systems in engineering or population dynamics in biology, evolve in discrete space and continuous time. We consider the problem of optimal decision making in such discrete state and action space systems under partial observability. This places our work at the intersection of optimal filtering and optimal control. At the current state of research, a mathematical description for simultaneous decision making and filtering in continuous time with finite state and action spaces is still missing. In this paper, we give a mathematical description of a continuous-time partial observable Markov decision process (POMDP). By leveraging optimal filtering theory we derive a Hamilton-Jacobi-Bellman (HJB) type equation that characterizes the optimal solution. Using techniques from deep learning we approximately solve the resulting partial integro-differential equation. We present (i) an approach solving the decision problem offline by learning an approximation of the value function and (ii) an online algorithm which provides a solution in belief space using deep reinforcement learning. We show the applicability on a set of toy examples which pave the way for future methods providing solutions for high dimensional problems.
Constructing differentiation algorithms with a fixed-time convergence and a predefined Upper Bound on their Settling Time (textit{UBST}), i.e., predefined-time differentiators, is attracting attention for solving estimation and control problems under time constraints. However, existing methods are limited to signals having an $n$-th Lipschitz derivative. Here, we introduce a general methodology to design $n$-th order predefined-time differentiators for a broader class of signals: for signals, whose $(n+1)$-th derivative is bounded by a function with bounded logarithmic derivative, i.e., whose $(n+1)$-th derivative grows at most exponentially. Our approach is based on a class of time-varying gains known as Time-Base Generators (textit{TBG}). The only assumption to construct the differentiator is that the class of signals to be differentiated $n$-times have a $(n+1)$-th derivative bounded by a known function with a known bound for its $(n+1)$-th logarithmic derivative. We show how our methodology achieves an textit{UBST} equal to the predefined time, better transient responses with smaller error peaks than autonomous predefined-time differentiators, and a textit{TBG} gain that is bounded at the settling time instant.
We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others.
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