Discrete-time Case Research Articles

Uniform exponential contraction for viscous Hamilton–Jacobi equations

The well known phenomenon of exponential contraction for solutions to the viscous Hamilton-Jacobi equation in the space-periodic setting is based on the Markov mechanism. However, the corresponding Lyapunov exponent $\lambda(\nu)$ characterizing the exponential rate of contraction depends on the viscosity $\nu$. The Markov mechanism provides only a lower bound for $\lambda(\nu)$ which vanishes in the limit $\nu \to 0$. At the same time, in the inviscid case $\nu=0$ one also has exponential contraction based on a completely different dynamical mechanism. This mechanism is based on hyperbolicity of action-minimizing orbits for the related Lagrangian variational problem. In this paper we consider the discrete time case (kicked forcing), and establish a uniform lower bound for $\lambda(\nu)$ which is valid for all $\nu\geq 0$. The proof is based on a nontrivial interplay between the dynamical and Markov mechanisms for exponential contraction. We combine PDE methods with the ideas from the Weak KAM theory.

Discrete-time Flatness and Linearization along Trajectories

The paper studies the relation between a nonlinear time-varying flat discrete-time system and the corresponding linear time-varying systems which are obtained by a linearization along trajectories. It is motivated by the continuous-time case, where it is well-known that the linearization of flat systems along trajectories results in linear time-varying systems which are again flat. Since flatness implies controllability, this constitutes an important verifiable necessary condition for flatness. In the present contribution, it is shown that this is also true in the discrete-time case: We prove that the linearized system is again flat, and that a possible flat output is given by the linearization of a flat output of the nonlinear system. Analogously, the map that describes the parameterization of the system variables of the linear system by this flat output coincides with the linearization of the corresponding map of the nonlinear system. The results are illustrated by two examples.

Interval Observer for Switching Discrete-Time Linear Systems with Unknown Switching Sequences

The present paper deals with interval observers in the discrete-time case for switching linear systems. When the discrete state is known, several interval observers have been proposed. In the present paper, such an observer is proposed in the case there is no assumption nor knowledge on the switching parameter, and when the switching system considered is affected by bounded perturbations.

Tracking MPC Tuning in Continuous Time: A First-Order Approximation of Economic MPC

Economic MPC (EMPC) optimizes closed-loop performance by directly minimizing a given objective function, as opposed to Tracking MPC (TMPC) which instead penalizes deviations from a precalculated optimal reference. The main difference between the two approaches can be observed during transients, as the former always acts optimally, while the latter is only optimal when the reference is accurately tracked. Unfortunately, stability for EMPC is in general difficult to prove, as opposed to TMPC which builds on a rich theory. Additionally, many efficient algorithms are available for TMPC, while solving the EMPC problem can be much harder. In prior works 1, 2, a family of discrete-time TMPC schemes that provide approximate economic optimality has been developed in order to partially overcome these issues. In this paper, we aim at extending such a family of TMPC schemes to the continuous time case. Similarly to the discrete-time case, also in continuous-time we obtain a first-order approximation of the EMPC control law. We demonstrate the theory with a numerical example that confirms the first-order approximation and we show that our continuous-time formulation can be made equivalent to the discrete-time one.

Conley index theory without index pairs. I: The point-set level theory

We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion of index pairs. We introduce, instead, the notions of compactifiable subsets and index neighbourhoods and formulate and prove basic results in Conley index theory using these notions. We treat both the discrete time case and the continuous time case.

Jarzyski's Equality and Crooks' Fluctuation Theorem for General Markov Chains with Application to Decision-Making Systems.

We define common thermodynamic concepts purely within the framework of general Markov chains and derive Jarzynski's equality and Crooks' fluctuation theorem in this setup. In particular, we regard the discrete-time case, which leads to an asymmetry in the definition of work that appears in the usual formulation of Crooks' fluctuation theorem. We show how this asymmetry can be avoided with an additional condition regarding the energy protocol. The general formulation in terms of Markov chains allows transferring the results to other application areas outside of physics. Here, we discuss how this framework can be applied in the context of decision-making. This involves the definition of the relevant quantities, the assumptions that need to be made for the different fluctuation theorems to hold, as well as the consideration of discrete trajectories instead of the continuous trajectories, which are relevant in physics.

Finite-time input-to-state stability of nonlinear systems: the discrete-time case

This paper extends the finite-time input-to-state stability (FTISS) issue to the discrete-time dynamical systems. Based on the Lyapunov method, sufficient conditions are derived to check the FTISS property of the discrete-time nonlinear systems with inputs. In addition, the generalised class- functions and the settling-time function capturing the finite settling-time behaviour of the dynamical system are constructed explicitly and well studied. In the end of the paper, a numerical example is provided to illustrate the feasibility/effectiveness of the obtained results.

On the optional and orthogonal decompositions of a class of semimartingales

We consider a set \mathcal Q of probability measures, which are absolutely continuous with respect to the physical probability measure \mathbb{P} and at least one is equivalent to \mathbb{P} . We investigate necessary and sufficient conditions on \mathcal Q , under which any \mathcal Q -supermartingale can be decomposed into the sum of a local \mathcal Q -martingale and a decreasing process. We also provide an orthogonal decomposition of square integrable semimartingale as the orthogonal sum of a local \mathcal Q -martingale and a square integrable semimartingale. As one application, we state the orthogonal decomposition in an appropriate sense of the polar set of \mathcal Q . We generalise then the results of a previous article (2021), from finite probability space and discrete time case to general continuous time case.

Prioritized Safe Interval Path Planning for Multi-Agent Pathfinding With Continuous Time on 2D Roadmaps

We address a challenging multi-agent pathfinding (MAPF) problem for hundreds of agents moving on a 2D roadmap with continuous time. Despite its known potential for producing better solutions compared to typical grid and discrete-time cases, few approaches have been established to solve this problem due to the intractability of collision checks on a large scale. In this work, we propose Prioritized Safe-Interval Path Planning with Continuous-Time Conflicts (PSIPP/CTC) that extends a scalable prioritized planning algorithm to work on the 2D roadmap and continuous-time setup by alleviating intensive collision checks. Our approach involves a novel concept named Continuous-Time Conflict (CTC), which describes a pair among vertices and edges associated with continuous-time intervals within which collisions can happen between agents. We pre-compute CTCs using geometric neighbor-search and sweeping techniques and annotate roadmaps with the CTCs just once before planning starts. Doing so allows us to efficiently enumerate collision-free time intervals for all vertices and edges and find each agent's path with continuous time in prioritized planning. Extensive experimental evaluations demonstrate that PSIPP/CTC significantly outperforms existing methods in terms of planning success rate and runtime while maintaining an acceptable solution quality. As a proof of concept, we also confirmed the effectiveness of the proposed approach on a physics simulation with differential wheeled robots.

Stochastic representation under g-expectation and applications: The discrete time case

In this paper, we address the stochastic representation problem in discrete time under (non-linear) g-expectation. We establish existence and uniqueness of the solution, as well as a characterization of the solution. As an application, we investigate a new approach to the optimal stopping problem under g-expectation and the related pricing of American options under Knightian uncertainty. Our results are also applied to a (non-linear) Skorokhod-type obstacle problem.

Control of dead-time process: From the Smith predictor to general multi-input multi-output dead-time compensators

This review paper deals with the analysis, design, and tuning of dead-time compensators for stable and unstable multi-input multi-output (MIMO) processes with multiple time delays. It is well known that, even in the single-input single-output case, processes with significant dead times are difficult to control using standard feedback controllers. For MIMO systems, the study of processes with dead time is more involved, particularly when the process behavior exhibits different dead times in the different input-output relationships. Because of this, much research has been conducted in the last 50 years on this subject, with different approaches and proposals of controllers for covering a variety of objectives. Thus, this paper gives an overview of this important topic, focusing on the solutions derived from the Smith Predictor. First, a historical perspective of the different controllers proposed in the literature is presented. Then, the general solution of the problem is developed, paying particular attention to robustness and disturbance rejection properties, because of their importance and usefulness in industrial processes. All the development is done in the discrete-time case, which allows direct digital implementation. Two simulation case studies are presented to illustrate some of the ideas discussed in the paper, and an experimental case study is used to discuss aspects of practical implementation.

Stability analysis of quaternion-valued BAM neural networks fractional-order model with impulses and proportional delays

In this paper, a class of quaternion-valued BAM neural networks (QVBAMNNs) fractional-order model with impulses and proportional delays is proposed in discrete-time case. The QVBAMNNs fractional-order model is investigated directly rather than through real decomposition method or the plural one. By employing homeomorphic mapping theorem, Lyapunov stability theory and inequality technology, several criteria for the discrete-time QVBAMNNs fractional-order model are derived to guarantee the existence, uniqueness and global exponential stability (GES) of equilibrium point. The novelty of these results comes from the generality with the fractional-order systems and the integer-order ones. Finally, two examples are given to demonstrate the effectiveness and availability of the derived criteria.

Observer-based output feedback control for networked systems with dual-channel event-triggered sampling and quantization

This paper investigates the problem of observer-based output feedback control for linear networked systems with dual-channel event-triggered mechanisms and quantization. Both continuous-time and discrete-time event detection cases are discussed. In the continuous-time case, the stability of observer error dynamics and closed-loop system are analyzed respectively, and it is proved that Zeno behavior would not occur. In order to approach engineering practice, in the discrete-time case, two types of network attacks including denial-of-service (DoS) and fault data injection (FDI) attacks are considered, whose nature property is characterized by Bernoulli variables. By combining these factors and transmission delay, a novel augmented system model is proposed, and some sufficient conditions are derived based on Lyapunov functional approach and linear matrix inequalities (LMIs). Compared with the existing results, this framework is more comprehensive and practical, and the global uniform ultimate boundedness of closed-loop systems can be guaranteed. Finally, simulation examples are given to demonstrate the effectiveness of the proposed method.

Deterministic homogenization under optimal moment assumptions for fast–slow systems. Part 2

We consider deterministic homogenization for discrete-time fast-slow systems of the form $$ X_{k+1} = X_k + n^{-1}a_n(X_k,Y_k) + n^{-1/2}b_n(X_k,Y_k)\;, \quad Y_{k+1} = T_nY_k\;$$ and give conditions under which the dynamics of the slow equations converge weakly to an It\^o diffusion $X$ as $n\to\infty$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. This extends the results of [Kelly-Melbourne, J. Funct. Anal. 272 (2017) 4063--4102] from the continuous-time case to the discrete-time case. Moreover, our methods (c\`adl\`ag $p$-variation rough paths) work under optimal moment assumptions. Combined with parallel developments on martingale approximations for families of nonuniformly expanding maps in Part 1 by Korepanov, Kosloff & Melbourne, we obtain optimal homogenization results when $T_n$ is such a family of maps.

The Infinite-Dimensional Standard and Strict Bounded Real Lemmas in Continuous Time: The Storage Function Approach

The bounded real lemma (BRL) is a classical result in systems theory, which provides a linear matrix inequality criterium for dissipativity, via the Kalman-Yakubovich-Popov (KYP) inequality. The BRL has many applications, among others in H^infty control. Extensions to infinite dimensional systems, although already present in the work of Yakubovich, have only been studied systematically in the last few decades. In this context various notions of stability, observability and controllability exist, and depending on the hypothesis one may have to allow the KYP-inequality to have unbounded solutions which forces one to consider the KYP-inequality in a spatial form. In the present paper we consider the BRL for continuous time, infinite dimensional, linear well-posed systems. Via an adaptation of Willems’ storage function approach we present a unified way to address both the standard and strict forms of the BRL. We avoid making use of the Cayley transform and work only in continuous time. While for the standard bounded real lemma, we obtain analogous results as there exist for the discrete time case, when treating the strict case additional conditions are required, at least at this stage. This might be caused by the fact that the Cayley transform does not preserve exponential stability, an important property in the strict case, when transferring a continuous-time system to a discrete-time system.

Stability analysis and stabilization of linear symmetric matrix-valued continuous, discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.

Event-Triggered Asynchronous Filter of Positive Switched Systems with State Saturation

This paper investigates the event-triggered asynchronous filter of positive switched systems with state saturation using linear programming and multiple Lyapunov functions. First, a filter is constructed for continuous-time positive switched systems. Under the asynchronous switching law, an error system is proposed with respect to positive switched systems and their filters, where the state saturation term is described in a polytopic form by virtue of the saturation property. A novel event-triggering condition is addressed based on a 1-norm inequality. Under the event-triggering condition, the error system is transformed into an interval system with lower and upper bounds. By using multiple Lyapunov functions and linear programming, the positivity and stability of the error system are achieved by considering the corresponding properties of the lower and upper bounds, respectively. Then, the event-triggered l1-gain filter and nonfragile filter are also proposed for the systems with disturbances. Moreover, the presented filter framework is extended to the discrete-time case. Finally, two examples are given to verify the effectiveness of the proposed filters.

Generalizing the Letov formula for the discrete-time case

Generalizing the Letov formula for the discrete-time case

A Mean-Field Approximation of SIR Epidemics on an Erdös-Rényi Network Model.

The stochastic nature of epidemic dynamics on a network makes their direct study very challenging. One avenue to reduce the complexity is a mean-field approximation (or mean-field equation) of the dynamics; however, the classic mean-field equation has been shown to perform sub-optimally in many applications. Here, we adapt a recently developed mean-field equation for SIR epidemics on a network in continuous time to the discrete time case. With this new discrete mean-field approximation, this proof-of-concept study shows that, given the density of the network, there is a strong correspondence between the epidemics on an Erdös-Rényi network and a system of discrete equations. Through this connection, we developed a parameter fitting procedure that allowed us to use synthetic daily SIR data to approximate the underlying SIR epidemic parameters on the network. This procedure has improved accuracy in the estimation of the network epidemic parameters as the network density increases, and is extremely cheap computationally.

Global Exponential Stability of Both Continuous-Time and Discrete-Time Switched Positive Time-Varying Delay Systems with Interval Uncertainties and All Unstable Subsystems

The global stability problem for a class of linear switched positive time-varying delay systems (LSPTDSs) with interval uncertainties by means of a fast average dwell time (FADT) switching is analyzed in this paper. A distinctive feature of this research is that all subsystems are considered to be unstable. Both the continuous-time and the discrete-time cases of LSPTDSs with interval uncertainties and all unstable subsystems (AUSs) are investigated. By constructing a time-scheduled multiple copositive Lyapunov-Krasovskii functional (MCLKF), novel sufficient conditions are derived within the framework of the FADT switching to guarantee such systems in the case of continuous-time to be globally uniformly exponentially stable. Based on the above approach, the corresponding result is extended to the discrete-time LSPTDSs including both interval uncertainties and AUSs. In addition, new stability criteria in an exponential sense are formulated for the studied systems without interval uncertainties. The efficiency and validity of the theoretical results are shown through simulation examples.