Discrete-time Case Research Articles

Equivalent conditions for the synchronization of identical linear systems over arbitrary interconnections

We propose necessary and sufficient conditions for the synchronization of N identical single-input–single-output (SISO) systems, connected through a directed graph without imposing any assumption on the graph interconnection. We consider both the continuous-time and the discrete-time case, and we provide conditions that are equivalent to the uniform global exponential stability, with guaranteed convergence rate, of the closed and unbounded attractor that corresponds to the synchronization set.

Multivariate regularly varying insurance and financial risks in multidimensional risk models

AbstractMultivariate regular variation is a key concept that has been applied in finance, insurance, and risk management. This paper proposes a new dependence assumption via a framework of multivariate regular variation. Under the condition that financial and insurance risks satisfy our assumption, we conduct asymptotic analyses for multidimensional ruin probabilities in the discrete-time and continuous-time cases. Also, we present a two-dimensional numerical example satisfying our assumption, through which we show the accuracy of the asymptotic result for the discrete-time multidimensional insurance risk model.

Enhancing rational approximation of wideband resonant MIMO systems with frequency-domain data

Vector Fitting (VF) is a successful methodology for both time and frequency domain estimation of dynamic systems. These algorithms have gained popularity as a system identification tool due to their ability to provide a practical procedure for accurately approximating the measured frequency response behavior of complex and highly resonant dynamic systems using rational models. Instrumental Variable (IV) techniques can also be added to the VF algorithm to enhance the optimality of the solutions in the presence of measurement noise. In this context, this paper presents a multiple-input multiple-output (MIMO) generalization of instrumental variable (IV-)VF. The proposed algorithm can be applied to accurately estimating models formed by either continuous- or discrete-time rational basis functions (RBFs). We show that such a generalization preserves the main implementation features observed in Standard VF, but with improved optimality property for its solutions. Additionally, in order to alleviate convergence problems that may arise from the fixed high-frequency normalization, we also present in this paper how to incorporate in the proposed IV-VF method the relaxed formulation found in standard VF for both the continuous and discrete-time cases. The resulting MIMO IV-VF algorithm can be easily understood by VF users, and its applicability is discussed by using an estimation case study with two power system pieces of equipment (an 88 kV Transformer and a 550 kV Instrument Transformer) for which actual frequency domain measurements were collected in the field. It is shown that the proposed algorithm significantly improves the modeling accuracy when compared to standard VF algorithms.

On irreducibility and positivity improvement of norm continuous quantum Markov semigroups

We show that for a norm continuous quantum Markov semigroup irreducibility and positivity improvement for any time [Formula: see text] are equivalent under some general conditions that hold, in particular, in the finite-dimensional case. This result is applied to quantum Markov semigroups of weak coupling limit type. We also discuss the discrete-time case.

Stability robust for fractional generalized multi-dimensional state–space models

In this work, we consider a new class of generalized fractional linear multidimensional state–space systems described by the Roesser model. We discuss a novel technique for analysing robust stability, focusing specifically on the stability of the closed-loop system in terms of the H 2 and H ∞ norms. Both discrete-time and continuous-time cases are addressed across various regions of the complex plane. An extension of the bounded real lemma is proposed, dealing with both continuous and discrete cases. This lemma is used to provide sufficient conditions, in the form of linear matrix inequalities, to ensure stability margin for the perturbed system. Motivating examples are presented to demonstrate the effectiveness of our main results.

A geometrical approach for consensus security

Guaranteeing security against eavesdroppers while allowing agents to reach an agreement on some shared variables is an essential feature to foster the adoption of distributed protocols. In the literature, a wide variety of methodologies have been developed in the discrete-time case, and most of them rely on encryption and undirected graphs, often exhibiting quantization or accuracy issues. In this paper, we develop a continuous-time secure consensus algorithm for agents that communicate via a broadcast-only protocol over directed and strongly connected graphs. In detail, the legitimate agents interact by exchanging an extended state, which can be converted into the hidden consensus state via a pre-deployed shared vector, thus preventing eavesdropping. In this paper, we take a geometrical perspective, and we implement our encryption via coordinate shifts. This choice guarantees, on one side, the existence of an uncountable infinity of possible vectors and, on the other side, provides a control-theoretical viewpoint on encryption that naturally blends with a dynamical system. We complement our framework by providing an additional layer of protection against honest but curious legitimate nodes, preventing them from gaining insights into the initial condition of their peers.

Reviewing Stability Criteria for Positive Homogeneous Systems and Adding One for Discrete-Time Cases with Degree Less Than One

Reviewing Stability Criteria for Positive Homogeneous Systems and Adding One for Discrete-Time Cases with Degree Less Than One

(Q, S, R)-Dissipativity Analysis of Large-Scale Networked Systems

This brief investigates the (Q,S,R)-dissipativity of large-scale networked systems in the discrete-time case. The networked system consists of a large number of subsystems, each of which can have different dynamic characteristics, and the connections among subsystems are arbitrary. When the number of subsystems is large, the traditional analysis method based on lumped description may have computational difficulties. To solve this problem, some computationally attractive conditions are derived, which essentially depend on the block-diagonal structure of the system parameter matrices and the sparseness of the subsystem connection matrix. Numerical simulations illustrate that compared with the existing results, the conditions suggested in this brief have higher computational efficiency in the dissipative analysis of large-scale networked systems.

Budget deficit and money holding when consumers live forever in an endogenous growth model

<p class="MsoNormal" style="margin-top: 12pt; text-align: justify;"><span lang="EN-US" style="font-family: arial, helvetica, sans-serif;">In this paper I will show that budget deficit (or fiscal deficit) is necessary to achieve full employment under constant prices or inflation, using a model of endogenous growth in which consumers hold money for the reason of liquidity and live forever. Budget deficit need not be offset by future budget surpluses. I consider the continuous time case by taking the limit of the discrete time case when the time interval approaches zero. A continuous time dynamic model seems to be more general than a discrete time model. When the actual budget deficit is greater (smaller) than the value which is necessary and sufficient for full employment under constant prices, an inflation (a recession) occurs. The main argument of this paper is that a growing economy requires the continuation of budget deficit, and that we should not think of paying off the resulting government debt with taxes.</span></p>

Generalized Lyapunov stability theory of continuous-time and discrete-time nonlinear distributed-order systems and its application to boundedness and attractiveness for networks models

The Lyapunov direct method is a powerful tool for analyzing the stability of nonlinear dynamical systems. However, most existing research has primarily focused on asymptotic stability, neglecting the crucial aspect of boundedness. In light of this, this paper aims to address the issues concerning the generalized Lyapunov stability theory of distributed-order systems and its application to ensure uniform boundedness and global attractiveness for distributed-order Hopfield neural networks (DOHNNs) with time-varying external input. To achieve this, several essential inequalities that form the foundation of our main work are proposed. Subsequently, two stability criteria in the form of generalized distributed-order Lyapunov theorem are derived by employing Laplace transform techniques. Meanwhile, building upon the proposed inequalities and Lyapunov theorem, sufficient conditions that guarantee the uniform boundedness and global attractiveness of DOHNNs with time-varying external input are established. By the way, all our proposed conclusions are presented in both continuous-time and discrete-time cases to present the distinction and relationship between these two cases. Finally, various simulation examples are designed meticulously to validate the correctness of our proposed theorems and the effectiveness to real-world applicability of our approach.

A design method of distributed algorithms via discrete-time blended dynamics theorem

We develop a discrete-time version of the blended dynamics theorem for the use of designing distributed computation algorithms. The blended dynamics theorem enables to predict the behavior of heterogeneous multi-agent systems. Therefore, once we get a blended dynamics for a particular computational task, design idea of node dynamics for individual heterogeneous agents can easily occur. In the continuous-time case, prediction by blended dynamics was enabled by high coupling gain among neighboring agents. In the discrete-time case, we propose an equivalent action, which we call multi-step coupling in this paper. Compared to the continuous-time case, the blended dynamics can have more variety depending on the coupling matrix. This benefit is demonstrated with three applications; distributed estimation of network size, distributed computation of the PageRank, and distributed computation of the degree sequence of a graph, which correspond to the coupling by doubly-stochastic, column-stochastic, and row-stochastic matrices, respectively.

Synchronizations control of fractional-order multidimension-valued memristive neural networks with delays

In this paper, systems of the fractional-order multidimension-valued memristive neural networks (FMVMNNs) with leakage delay and time-varying delays are studied in continuous-time and discrete-time cases. Comparative studies on synchronizations control between discrete-time FMVMNNs and the continuous-time system are investigated directly rather than through the decomposition method. By employing Lyapunov theory and fractional derivative inequalities, sufficient criteria about the quasi-synchronization and global ultimate Mittag-Leffler lag quasi-synchronization are obtained in multidimension-valued linear matrix inequality form, while succinct conclusions of asymptotical synchronization and global Mittag-Leffler synchronization are derived in vector form. The obtained criteria have many advantages in a less computation, lower conservatism, more generality and higher flexibility. Finally, four examples are given to demonstrate the effectiveness and availability of the derived conclusions.

Reference regulation of constrained uncertain discrete-time systems via robust predictive integral control

Reference regulation of uncertain systems is investigated in the discrete-time case. Hence, including the uncertainties and constraints, through a prediction-based controller, the set-point following will be robustly fulfilled in a class of uncertain systems. A novel idea is addressed to achieve this goal optimally. Thus the control synthesis is deliberately transformed into an optimisation issue subjected to some matrix inequality. In the real-time implementation, the controller’s parameters would be obtained when a minimisation problem is solved. The applicability of the method is evaluated in some examples. The superiority and robustness of the suggested procedure are remarkably validated by comparing it with a similar one.

Kernel-Based Impulse Response Identification With Side-Information on Steady-State Gain

In this paper, we consider the problem of system identification when side-information is available on the steady-state gain (SSG) of the system. We formulate a general nonparametric identification method as an infinite-dimensional constrained convex program over the reproducing kernel Hilbert space (RKHS) of stable impulse responses. The objective function of this optimization problem is the empirical loss regularized with the norm of RKHS, and the constraint is considered for enforcing the integration of the SSG side-information. The proposed formulation addresses both the discrete-time and continuous-time cases. We show that this program has a unique solution obtained by solving an equivalent finite-dimensional convex optimization. This solution has a closed-form when the empirical loss and regularization functions are quadratic and exact side-information is considered. We perform extensive numerical comparisons to verify the efficiency of the proposed identification methodology.

Stability Analysis of Second-Order Linear PDEs on Time Scales

This paper presents the stability analysis of a class of second-order linear partial differential equations (PDEs) on time scales with diffusion operator and first-order partial derivative. According to the time scale theory, the Lyapunov functional method, and some inequality techniques, sufficient conditions for exponential stability are strictly obtained, and the results are generalized for that where both the discrete-time and continuous-time cases are considered jointly. In addition, the theoretical results are applied to exponential synchronization of reaction-diffusion neural networks (RDNNs). Simulation examples are given to verify the feasibility of our results.

On mixed censored [formula omitted]-shock models

In this paper we introduce the mixed censored δ-shock model which combines the censored δ-shock model and the classical extreme shock model. Under the mixed censored δ-shock model, the system fails whenever no shock occurs within a δ-length time period from the last shock, or the magnitude of the shock is larger than another critical threshold γ>0. For the discrete-time case of occurrence of shocks, by assuming the dependence between intershock times and the corresponding magnitudes of shocks we derive the probability generating function (pgf) of the lifetime of the system, and a matrix-based expression is obtained for the exact distribution of the system’s lifetime when the distribution of intershock times and the magnitudes of shocks have a discrete bivariate phase-type distribution. Similar results are obtained by assuming the independence between intershock times and the corresponding magnitudes of shocks, and by proving that the distribution of the shifted lifetime at δ, is the convolution of a discrete compound geometric distribution and a discrete compound Bernoulli distribution, we get several results concerning the distribution of system’s lifetime, like as, simple and efficient recursions for evaluating the survival function and the probability mass function (pmf), the mean and the variance of system’s lifetime, as well as discrete Lundberg-type upper bounds for the reliability function. For the continuous-time case of occurrence of shocks we obtain an exact formula for the reliability function and the Laplace–Stieltjes transform of system’s lifetime by assuming the dependence between intershock times and the corresponding magnitudes of shocks, and under the independence setup, we obtain the reliability function when the intershock times have the uniform distribution, and we give an asymptotic result under the Poisson process for the arrival of shocks. Similar to the discrete-time case, it is shown that the distribution of the shifted lifetime at δ, is the convolution of a compound geometric distribution and a compound Bernoulli distribution and using this we obtain two-sided Lundberg-type bounds for the survival function. Finally, some numerical examples to illustrate our results, are also given.

Fast and Accurate Non-Negative Latent Factor Analysis of High-Dimensional and Sparse Matrices in Recommender Systems

A fast non-negative latent factor (FNLF) model for a high-dimensional and sparse (HiDS) matrix adopts a Single Latent Factor-dependent, Non-negative, Multiplicative and Momentum-incorporated Update (SLF-NM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> U) algorithm, which enables its fast convergence. It is crucial to achieve a rigorously theoretical proof regarding its fast convergence, which has not been provided in prior research. Aiming at addressing this critical issue, this work theoretically proves that with an appropriately chosen momentum coefficient, SLF-NM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> U enables the fast convergence of an FNLF model in both continuous and discrete time cases. Empirical analysis of HiDS matrices generated by representative industrial applications provides empirical evidences for the theoretical proof. Hence, this study represents an important milestone in the field of HiDS matrix analysis.

Consensus of Multi-Agent Systems With Asymmetric Input Saturation Over Directed Graph

In this brief, the consensus issue of continuous-time first-order multi-agent systems (MASs) subject to asymmetric input saturation under the directed graph containing a directed spanning tree is investigated. First, a fully distributed consensus protocol is proposed. We discuss the scenario where there are zeros in the partial saturation bounds. Based on the contradiction method and the graph theory, a necessary and sufficient criterion for guaranteeing the global consensus is given. After that, we extend the above findings to the discrete-time case, where an upper bound of the sample time is explicitly formulated. Finally, simulation examples verify the effectiveness of our proposed strategy.

An in-silico framework for modeling optimal control of neural systems.

Brain-machine interfaces have reached an unprecedented capacity to measure and drive activity in the brain, allowing restoration of impaired sensory, cognitive or motor function. Classical control theory is pushed to its limit when aiming to design control laws that are suitable for large-scale, complex neural systems. This work proposes a scalable, data-driven, unified approach to study brain-machine-environment interaction using established tools from dynamical systems, optimal control theory, and deep learning. To unify the methodology, we define the environment, neural system, and prosthesis in terms of differential equations with learnable parameters, which effectively reduce to recurrent neural networks in the discrete-time case. Drawing on tools from optimal control, we describe three ways to train the system: Direct optimization of an objective function, oracle-based learning, and reinforcement learning. These approaches are adapted to different assumptions about knowledge of system equations, linearity, differentiability, and observability. We apply the proposed framework to train an in-silico neural system to perform tasks in a linear and a nonlinear environment, namely particle stabilization and pole balancing. After training, this model is perturbed to simulate impairment of sensor and motor function. We show how a prosthetic controller can be trained to restore the behavior of the neural system under increasing levels of perturbation. We expect that the proposed framework will enable rapid and flexible synthesis of control algorithms for neural prostheses that reduce the need for in-vivo testing. We further highlight implications for sparse placement of prosthetic sensor and actuator components.

Faster quantum sampling of Markov chains in nonregular graphs with fewer qubits

Sampling from the stationary distribution is one of the fundamental tasks of Markov chain-based algorithms and has important applications in machine learning, combinatorial optimization and network science. For the quantum case, qsampling from Markov chains can be constructed as preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. In this paper, a new qsampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks and works without any limit compared with existing results. In detail, we build a qsampling algorithm that not only accelerates non-regular graphs but also keeps the speed-up of existing quantum algorithms for regular graphs. In non-regular graphs, the invocation of the quantum fast-forward algorithm accelerates existing state-of-the-art qsampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms we reduce log n, where n is the number of graph vertices. In regular graphs, our result matches other quantum algorithms, and our reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared to the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm is the first algorithm to achieve complete quadratic acceleration (without log factor) over the classical case without any limit. To enlarge success probability amplitude amplification is introduced. We construct a new reflection on stationary state with fewer ancilla qubits and think it may have independent application.