Discrete Time Instants Research Articles

Deep neural network based adaptive learning for switched systems

In this paper, we present a deep neural network based adaptive learning (DNN-AL) approach for switched systems. Currently, deep neural network based methods are actively developed for learning governing equations in unknown dynamic systems. However, their efficiency can degenerate for switching systems, where structural changes exist at discrete time instants. In this new DNN-AL strategy, observed datasets are adaptively decomposed into subsets, such that no structural change within each subset. During the adaptive procedures, DNNs are hierarchically constructed, and unknown switching time instants are gradually identified. Especially, network parameters at previous iteration steps are reused to initialize networks for the later iteration steps, which provides efficient training procedures for the DNNs. For the DNNs obtained through our DNN-AL, the bounds of the prediction error are established. Numerical studies are conducted to demonstrate the efficiency of DNN-AL.

Guest Editorial: Learning, optimisation and control of cyber‐physical systems

Guest Editorial: Learning, optimisation and control of cyber‐physical systems

Multi-Period Portfolio Management and a Simple Method for Calculating the Realized Return with Transaction Costs

In this work the authors provide a detailed analysis of multi-period portfolio transactions with transaction costs under a fixed and finite investment horizon. It is assumed that changes in asset prices occur only at discrete instants of time. The authors propose a simple method for calculating the realized return of multi-period portfolio transactions with transaction costs. This method is based on the so-called temporary decomposition of the transaction and on the detailed analysis of the history of all opened positions.

Predefined‐time distributed event‐triggered algorithms for resource allocation

The resource allocation problem in a distributed multi-agent system is considered in this study. First, the authors develop a predefined-time distributed algorithm and analyse its convergence analysis using the Lyapunov stability theory, in which the local constraint is ensured by a differential projection operator. Thus, a predefined time is obtained by a time-varying time-based generator. Second, to reduce the communication consumption between agents, the authors develop a static as well as a dynamic-based event-triggered control scheme, where the information broadcast only occurs at some discrete time instants. Moreover, the three proposed algorithms converge precisely to the global optimal solution. Besides, the Zeno behaviour is excluded in the above static and dynamic event-triggered mechanisms. Finally, the authors test the proposed algorithms' efficiency based on the provided numerical examples.

Cooperative Control of UAVs Over an Unreliable Communication Network

This article proposes a method for the cooperative control of two unmanned aerial vehicles (UAVs) to guarantee collision avoidance. The UAVs are locally controlled and are able to change their trajectories at any time. They are connected over an unreliable communication network that may induce packet losses and transmission delays. The network has no coordinator and necessary information for collision avoidance can solely be communicated. In order to reduce the communication effort, information are only transmitted at discrete time instants when they are necessary. Depending on the current situation, one UAV is responsible for avoiding the collision by adjusting its trajectory. To this aim, a control unit is introduced for the UAVs that uses the idea of event-based control and combines methods from communication technology and control theory. The control unit has to execute four tasks: estimation of the current network properties, prediction of the movement of the neighboring object, invocation of communication in an event-based fashion, and planning of the collision avoiding trajectory. It is proved that the control aims are satisfied even in the presence of time delays and packet losses induced by the communication network. A simulation study with two quadrotors shows the suitability of the method.

Adaptive Fuzzy Control for a Hybrid Spacecraft System With Spatial Motion and Communication Constraints

This article proposes an adaptive fuzzy control approach with an event-triggered mechanism and spatial motion constraint for a hybrid spacecraft system. The spacecraft system is composed of a rigid body and a slender flexible panel, with coupled dynamics captured by three ordinary differential equations and two partial differential equations. The overall control objective lies in utilizing an event-triggered control input to regulate the angular velocities of the rigid body and stabilize the vibrations of the flexible panel under unknown input disturbances and prescribed spatial motion performance. We collectively address the posture regulation and disturbance rejection purposes by introducing a barrier Lyapunov function and a fuzzy logic system. The event-triggered solution only updates the control signals at some discrete-time instants, and hence the communication burden is reduced significantly. The potential effectiveness and thrifty efficiency of the developed control strategy are theoretically demonstrated and numerically verified.

Control of a Discrete Dynamical System with Noise

For a discrete dynamical control system with noise, we consider the problem of retaining a phase point in a given family of sets at discrete instants of time. We analyze the case where the control vectogram is a polyhedron defined by a system of linear inequalities. We prove some properties of specific polyhedra satisfying the linearity condition, which allows us to obtain a condition of retention in the explicit form. Necessary and sufficient conditions for the possibility of retention are given. The results obtained are illustrated by an example.

Machine learning approach to the Floquet-Lindbladian problem.

Similar to its classical version, quantum Markovian evolution can be either time-discrete or time-continuous. Discrete quantum Markovian evolution is usually modeled with completely positive trace-preserving maps, while time-continuous evolution is often specified with superoperators referred to as "Lindbladians." Here, we address the following question: Being given a quantum map, can we find a Lindbladian that generates an evolution identical-when monitored at discrete instances of time-to the one induced by the map? It was demonstrated that the problem of getting the answer to this question can be reduced to an NP-complete (in the dimension N of the Hilbert space, the evolution takes place in) problem. We approach this question from a different perspective by considering a variety of machine learning (ML) methods and trying to estimate their potential ability to give the correct answer. Complimentarily, we use the performance of different ML methods as a tool to validate a hypothesis that the answer to the question is encoded in spectral properties of the so-called Choi matrix, which can be constructed from the given quantum map. As a test bed, we use two single-qubit models for which the answer can be obtained using the reduction procedure. The outcome of our experiment is that, for a given map, the property of being generated by a time-independent Lindbladian is encoded both in the eigenvalues and the eigenstates of the corresponding Choi matrix.

Settling Time Estimation in Synchronization of Impulsive Networks With Switching Topologies

This article addresses the problem of synchronization of impulsive networks with switching topologies. A new synchronization framework is established with an emphasis on settling time estimation. The impulsive networks consist of physical nodes and cyber modules. For physical nodes, states are changed impulsively at discrete time instants due to some switching phenomena or unexpected sudden noises. For cyber modules, two cases of switching scenarios are considered for information exchange patterns during specific time intervals. In the first case, cyber modules lose all the communication links with others, resulting in disconnected topologies. Then, a distributed controller is proposed for nodes without intrinsic nonlinear dynamics. A distinguished feature of this controller is its capability to estimate a bound for settling time, beyond which the synchronization with respect to a virtual target is guaranteed. In the second case, cyber modules lose some communication links but build other new ones with the help of a smart communication center to form connected topologies. A distributed controller is further designed for nodes in the presence of intrinsic nonlinear dynamics. Accordingly, a sufficient condition is derived to achieve synchronization with an estimated settling time bound. For both cases, the estimated bounds are able to reveal the relationship between the impulsive strength and the synchronization performance. Finally, numerical examples including a case study on a modified IEEE 34 bus test feeder are provided to demonstrate the effectiveness of the proposed controllers.

An Ellipsoidal Predictor–Corrector State Estimation Scheme for Linear Continuous-Time Systems With Bounded Parameters and Bounded Measurement Errors

For linear time-invariant dynamic systems with exactly known coefficients of their system matrices for which measurements with bounded errors are available at discrete time instants, an optimal polygonal state estimation scheme was recently published. This scheme allows for tightly enclosing all possible state trajectories in presence of uncertain, but bounded, system inputs which may be varying arbitrarily within in their bounds. Moreover, this approach is also capable of accounting for uncertainty related to the measurement time instants. However, the drawback of this polygonal technique is its rapidly increasing complexity for larger system dimensions. For that reason, the polygonal state enclosures are replaced by a computationally less expensive, but nearly optimal, ellipsoidal enclosure technique in this paper. Numerical simulations for representative benchmark examples focusing both on applications with precisely known and uncertain parameters conclude this contribution.

Parameter estimation for threshold Ornstein–Uhlenbeck processes from discrete observations

Assuming that a threshold Ornstein–Uhlenbeck process is observed at discrete time instants, we propose generalized moment estimators to estimate the parameters. Our theoretical basis is the celebrated ergodic theorem. With the sampling time step arbitrarily fixed, we prove the strong consistency and asymptotic normality of our estimators as the sample size tends to infinity.

Improving Energy Efficiency and QoS of LPWANs for IoT Using Q-Learning Based Data Routing

Recent proliferation of Internet of Things (IoT) demands large scale connectivity among smart IoT devices over a vast geographical area. However, limited radio range and lack of scalability of conventional wireless sensor networks do not allow a wide area connectivity among IoT devices. To address these challenges, Low-Power Wide-Area Networks (LPWANs) are emerging to provide long-range communication capability with low-power consumption of the end devices. Nevertheless, given the demand in delivering an increasingly large volume of data generated by IoT devices, the direct data transmission model is not suitable due to its poor network lifetime. Therefore, in this work, a multi-hop data routing method is proposed for LPWANs. Since multi-hop data transmission faces several challenges such as increased data latency, higher interference, and reduced data throughput (i.e., poor bandwidth utilization), we propose a reinforcement learning method to address those challenges. The proposed method updates the Q-matrix of the network at varying discrete time instants and selects relay devices in such a way that maximizes the cumulative reward value between selected device-gateway pairs. The applicability and effectiveness of the proposed method are illustrated over both simulated LPWAN testbed and real field data sets. The obtained results clearly demonstrate the improved network performance in terms of energy efficiency and QoS of the proposed method as compared to various existing methods.

Online Smoothing for Diffusion Processes Observed with Noise

We introduce a methodology for online estimation of smoothing expectations for a class of additive functionals, in the context of a rich family of diffusion processes (that may include jumps) – observed at discrete-time instances. We overcome the unavailability of the transition density of the underlying SDE by working on the augmented pathspace. The new method can be applied, for instance, to carry out online parameter inference for the designated class of models. Algorithms defined on the infinite-dimensional pathspace have been developed the last years mainly in the context of MCMC techniques. There, the main benefit is the achievement of mesh-free mixing times for the practical time-discretised algorithm used on a PC. Our own methodology sets up the framework for infinite-dimensional online filtering – an important positive practical consequence is the construct of estimates with variance that does not increase with decreasing mesh-size. Besides regularity conditions, our method is, in principle, applicable under the weak assumption – relatively to restrictive conditions often required in the MCMC or filtering literature of methods defined on pathspace – that the SDE covariance matrix is invertible.

Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-informed Deep Generative Models

We propose a new method for inferring the governing stochastic ordinary differential equations (SODEs) by observing particle ensembles at discrete and sparse time instants, i.e., multiple "snapshots". Particle coordinates at a single time instant, possibly noisy or truncated, are recorded in each snapshot but are unpaired across the snapshots. By training a physics-informed generative model that generates "fake" sample paths, we aim to fit the observed particle ensemble distributions with a curve in the probability measure space, which is induced from the inferred particle dynamics. We employ different metrics to quantify the differences between distributions, e.g., the sliced Wasserstein distances and the adversarial losses in generative adversarial networks (GANs). We refer to this method as generative "ensemble-regression" (GER), in analogy to the classic "point-regression", where we infer the dynamics by performing regression in the Euclidean space. We illustrate the GER by learning the drift and diffusion terms of particle ensembles governed by SODEs with Brownian motions and Levy processes up to 100 dimensions. We also discuss how to treat cases with noisy or truncated observations. Apart from systems consisting of independent particles, we also tackle nonlocal interacting particle systems with unknown interaction potential parameters by constructing a physics-informed loss function. Finally, we investigate scenarios of paired observations and discuss how to reduce the dimensionality in such cases by proving a convergence theorem that provides theoretical support.

Fully Distributed Event-Triggered Affine Formation Maneuver Control Over Directed Graphs

This paper studies event-triggered affine formation maneuver control problem for general linear multi-agent systems over directed graphs. Unlike most existing papers where a diagonal stabilizing matrix is required to achieve affine localizability and require continuous communication, this paper presents a novel adaptive dynamic event-triggered protocol, which is fully distributed, independent of any global information of the network topology, and under which the agents only need to sample and communicate with their neighbors at discrete time instants.

Collision-Free Formation Tracking of Multi-Agent Systems Under Communication Constraints

This letter deals with the problem of collision-free formation tracking of second-order Multi-Agent Systems (MAS) under communication constraints. It is assumed that an agent can only transmit its position to its neighbors at asynchronous and aperiodic discrete time instants. Velocity and acceleration of the agents are not available. Moreover, the dynamics of the leader can be controlled through an external input. Using continuous-discrete time observers, an output feedback formation controller is designed to drive the agents to a desired formation shape and to ensure that the formation tracks the leader trajectory under directed communication topology. A potential function based mechanism is incorporated into the proposed formation tracking algorithm to avoid collision between agents. The simulation results have shown the efficacy of the proposed algorithm.

Parameter Identification of Lorenz System with Incomplete Information: Case of One Known and Two Unknown Functions

In the present paper, which is the continuation of the previous one, the problem of parameter identification of the Lorenz system is solved in assumption that only one of three functions is known at discrete time instants on finite time initial time interval. Two other functions are assumed to be unknown. The regular methods of guess values determination of the unknown parameters are developed. They are based on the Lagrange multiplier and auxiliary parameters approaches. A novel method of initial value problem solution is proposed in which the abovementioned guess values are used for more accurate estimation of the system parameters. It is demonstrated that the proposed IVP method simultaneously solves three different tasks: the problem of function interpolation from its discrete values on the initial time interval; the problem of unknown functions reconstruction on the same time interval, and the problem of extrapolation of all functions on limited time interval. It is also shown that the proposed method reconstructs the Lorenz attractor from limited data volume and data including random components.

A Method of Finding Discrete Controls for Switched Dynamical Systems by Applying Continued Fractions

Control systems with a finite number of control settings are considered. It is assumed that any polysystem operates in continuous time and control switchings occur at some certain discrete time instants. A control goal is to transfer a polysystem from an initial state to a final state. Controllability of systems switched in discrete time is studied. Controls are constructed by using the theory of generalized multicomponent continued fractions and the congruences theory. Applications of the proposed control method to specific systems are discussed.

Optimal inspections and mission abort policies for multistate systems

At many instances, information on the state of a multistate system executing a mission is obtained via the costly inspections performed at discrete instants of time. Therefore, the corresponding cost-wise optimal mission abort policy that takes into account the state of a system should be designed accordingly. In this paper, we introduce an optimal mission abort policy that minimizes the expected costs due to inspections, mission failure and loss of a system. A system operates in a random environment modeled by a renewal process of shocks. With each shock, its state can deteriorate with certain probabilities that can eventually result in the total failure. The decision to abort or to continue operation depends on the number of experienced shocks and on the state of a system that is revealed only at inspections. The corresponding cost minimization problem is formulated and the necessary relationships for solving it are derived. Detailed numerical illustrations and discussions are presented.

Estimation of all parameters in the reflected Ornstein–Uhlenbeck process from discrete observations

Assuming that a reflected Ornstein–Uhlenbeck process is observed at discrete time instants, we propose generalized moment estimators to estimate all the drift and diffusion parameters via the celebrated ergodic theorem. With the sampling time step h>0 arbitrarily fixed, we prove the strong consistency and asymptotic normality of our estimators as the sampling size n tends to infinity. This provides a complete solution to an open problem left in Hu et al. (2015).