Discrete-time Dynamical Systems Research Articles

Synchronous stability of the fractional-order discrete-time dynamical network system model with impulsive couplings

In this paper, the synchronization of a class of fractional-order discrete-time dynamical networks is concerned. The new model of fractional-order discrete-time dynamical network system with impulsive couplings among nodes is constructed. Based on the multiple Lyapunov function method, the global synchronous stability and the local synchronous stability of the system are studied and several synchronous criteria are obtained. In the end, an example with numerical simulations is presented to verify the effectiveness of our obtained findings.

On the Existence of the Stabilizing Solution of a Class of Periodic Stochastic Riccati Equations

This paper is devoted to the characterization of existence and uniqueness conditions for the stabilizing solution of a large class of Riccati equations arising in stochastic dynamic games. As an application of the obtained results, we consider in a second step the problem of a zero-sum two players linear quadratic difference game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such a control problem, a crucial role is played by the stabilizing solution of the considered class of Riccati difference equations.

Optimal Secure Control With Linear Temporal Logic Constraints

Prior work on automatic control synthesis for cyber-physical systems under logical constraints has primarily focused on environmental disturbances or modeling uncertainties, however, the impact of deliberate and malicious attacks has been less studied. In this paper, we consider a discrete-time dynamical system with a linear temporal logic (LTL) constraint in the presence of an adversary, which is modeled as a stochastic game. We assume that the adversary observes the control policy before choosing an attack strategy. We investigate two problems. In the first problem, we synthesize a robust control policy for the stochastic game that maximizes the probability of satisfying the LTL constraint. A value iteration based algorithm is proposed to compute the optimal control policy. In the second problem, we focus on a subclass of LTL constraints, which consist of an arbitrary LTL formula and an invariant constraint. We then investigate the problem of computing a control policy that minimizes the expected number of invariant constraint violations while maximizing the probability of satisfying the arbitrary LTL constraint. We characterize the optimality condition for the desired control policy. A policy iteration based algorithm is proposed to compute the control policy. We illustrate the proposed approaches using two numerical case studies.

Cruise Flight Performance Optimization for Minimizing Green Direct Operating Cost

To cope with the environmental impact of aviation and pollution problems in the future, airlines need to assess environmental impacts and offer countermeasures in advance. In order to measure the influence of environment on the airlines’ operational costs, this paper establishes an aircraft green direct operating cost (GDOC) model to quantify adverse environmental effects, such as air pollution and greenhouse effects, into the direct operating cost (DOC). Furthermore, fuel consumption, flight time, and distance in the cruising stage account for about 80% of the entire flight mission, and optimizing cruise flight performance can contribute greatly to reduce GDOC. Therefore, this paper sets up an optimal control model to minimize GDOC, establishes a discrete time dynamic system for optimizing the cruise altitude and speed profiles, and searches the optimal results by using dynamic programming. Besides, as meteorological conditions affect aircraft aerodynamics, fuel flow rate, contrail formation, and so on, this paper analyzes meteorological uncertainty by using historic meteorological data. Finally, a route is selected as an example, and the rationality of the optimal results is proven by comparing GDOC with DOC. The results and discussion of the numerical test also show that environmental effects on aircraft operation can be reduced significantly by adopting GDOC as the optimization objective, especially the contrail cost, and the step-climb cruise mode can further reduce GDOC effectively.

\u03b1-Rank: Multi-Agent Evaluation by Evolution

We introduce α-Rank, a principled evolutionary dynamics methodology, for the evaluation and ranking of agents in large-scale multi-agent interactions, grounded in a novel dynamical game-theoretic solution concept called Markov-Conley chains (MCCs). The approach leverages continuous-time and discrete-time evolutionary dynamical systems applied to empirical games, and scales tractably in the number of agents, in the type of interactions (beyond dyadic), and the type of empirical games (symmetric and asymmetric). Current models are fundamentally limited in one or more of these dimensions, and are not guaranteed to converge to the desired game-theoretic solution concept (typically the Nash equilibrium). α-Rank automatically provides a ranking over the set of agents under evaluation and provides insights into their strengths, weaknesses, and long-term dynamics in terms of basins of attraction and sink components. This is a direct consequence of the correspondence we establish to the dynamical MCC solution concept when the underlying evolutionary model’s ranking-intensity parameter, α, is chosen to be large, which exactly forms the basis of α-Rank. In contrast to the Nash equilibrium, which is a static solution concept based solely on fixed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley’s Fundamental Theorem of Dynamical Systems, and the core ingredients of dynamical systems: fixed points, recurrent sets, periodic orbits, and limit cycles. Our α-Rank method runs in polynomial time with respect to the total number of pure strategy profiles, whereas computing a Nash equilibrium for a general-sum game is known to be intractable. We introduce mathematical proofs that not only provide an overarching and unifying perspective of existing continuous- and discrete-time evolutionary evaluation models, but also reveal the formal underpinnings of the α-Rank methodology. We illustrate the method in canonical games and empirically validate it in several domains, including AlphaGo, AlphaZero, MuJoCo Soccer, and Poker.

Discretized fast–slow systems near pitchfork singularities

ABSTRACTMotivated by the normal form of a fast–slow ordinary differential equation exhibiting a pitchfork singularity we consider the discrete-time dynamical system that is obtained by an application of the explicit Euler method. Tracking trajectories in the vicinity of the singularity we show, how the slow manifold extends beyond the singular point and give an estimate on the contraction rate of a transition mapping. The proof relies on the blow-up method suitably adapted to the discrete setting where precise estimates for a cubic map in the central rescaling chart make a key technical contribution.

A new Neuro-Fuzzy Inference System with Dynamic Neurons (NFIS-DN) for system identification and time series forecasting

A new Neuro-Fuzzy Inference System with Dynamic Neurons or NFIS-DN is presented here for discrete time dynamic system identification and time series forecasting problems. The proposed dynamic system based neuron, referred to as Dynamic Neuron (DN) is realized by a discrete-time nonlinear state-space model. The DN is designed such way, that the output considers only the effect of finite past instances, enabling the system with finite memory. The NFIS-DN model has five layers, and DNs are employed only in the layers handling crisp values. The antecedent and the consequent parameters of NFIS-DN are updated using a self-regulated backpropagation through time learning algorithm. The performance evaluation of NFIS-DN has been carried-out using benchmark problems in the areas of nonlinear system identification and time series forecasting. The results are compared with the state-of-the-art method on the neural fuzzy networks. The obtained results clearly suggest that the NFIS-DN performs significantly better while using a smaller or similar number of fuzzy rules. Finally the practical application of the NFIS-DN has been demonstrated using two real-world problems.

Stochastic sensitivity analysis of chaotic attractors in 2D non-invertible maps

Abstract A response of chaotic discrete-time dynamical systems on parametric random disturbances is considered. For two-dimensional stochastic systems with non-invertible maps, a dispersion of random states near chaotic attractors is studied. To analyse the dispersion near the border of the chaotic attractor, we elaborate an asymptotic approach based on the stochastic sensitivity technique. In this analysis, critical curves defining parts of this border are used. An application of this theory to the study of the dispersion of random states near borders of chaotic attractors is given on the example of the Sprott model. Constructive abilities of the elaborated approach for the analysis of noise-induced escape from the basin of the chaotic attractor are demonstrated.

Bayesian filter for nonlinear systems with randomly delayed and lost measurements

In this note, a modified-likelihood Bayesian filter is proposed for a class of discrete-time nonlinear dynamical systems whose outputs are transmitted to the estimator with random delay and dropout. The likelihood function of the filter is computed by marginalizing out the delay variable to extract accurate information from the delayed measurements. Moreover, the measurement update stage of the filtering algorithm is omitted when no new measurement is received. Simulation results are presented to verify the superior performance of the introduced filter compared to some existing ones in the literature.

Delayed‐impulsive control for difference systems with actuator saturation and its synchronisation application

This study focuses on the problem of time-delayed impulsive control with actuator saturation for discrete-time dynamical systems. By establishing a delayed-impulsive difference inequality, combining with convex analysis and inequality techniques, some sufficient conditions are obtained to ensure exponential stability for discrete-time dynamical systems via time-delayed impulsive controller with actuator saturation. And those results are also used in the problem of synchronisation of the discrete-time complex networks. The designed controller admits the existence of some transmission delays in impulsive feedback law, and the control input variables are required to stay within an availability zone. Several numerical simulations are also given to demonstrate the effectiveness of the proposed results.

Boolean network analysis through the joint use of linear algebra and algebraic geometry

Among the various phenomena that can be modeled by Boolean networks, i.e., discrete-time dynamical systems with binary state variables, gene regulatory interactions are especially well known. Therefore, the analysis of Boolean networks is critical, e.g., to identify genetic pathways and to predict the effects of mutations on the cell functionality. Two methodologies (i.e., the semi-tensor product and the Gröbner bases over finite fields) have recently been proposed to tackle the problem of determining cycles and attractors (with the corresponding basin of attraction) for such systems. Here, it is shown that, by suitably coupling methodologies taken from these two fields (i.e., linear algebra and algebraic geometry), it is not only possible to determine cycles and attractors, but also to find closed-form solutions of the Boolean network. Such a goal is pursued by finding an immersion that recasts the Boolean dynamics in a linear form and by computing the closed-form solution of the latter system. The effectiveness of this technique is demonstrated by fully computing the solutions of the Boolean network modeling the differentiation of the Th-lymphocyte, a type of white blood cells involved in the human adaptive immune system.

Identification of nonlinear block-oriented systems with backlash and saturation

Abstract A new approach to modeling and identification of discrete-time nonlinear dynamic systems with input backlash and output saturation nonlinearities is presented. The proposed three-block cascade mathematical model results from successive applications of the key-term separation principle. This provides special nonlinear model description that is linear in parameters. An iterative technique with internal variable estimation is proposed for estimation of all the model parameters based on measured input/output data and minimizing the least-squares criterion. Illustrative example of cascade system identification with backlash and saturation is included.

The Weighted Nearest Neighbor Estimate for Hammerstein System Identification

This paper concerns the nonparametric identification problem for a class of nonlinear discrete-time dynamical systems that is characterized by its cascade structure. This is a Hammerstein system being a series connection of a nonlinear memoryless element followed by a linear dynamic system. The input–output training data generated from the system are dependent and they do not reveal the strong mixing property. The nonlinear part of the system is recovered with the weighted $k$ -nearest neighbor regression estimate. The a priori information is nonparametric, both the nonlinear characteristic and the impulse response of the linear part are completely unknown and can be of any form. Local and global properties of the estimate are examined. Whatever the probability density of the input signal, the estimate converges at every continuity point of the characteristic as well as in the global sense. We derive the formulas for asymptotic bias and the variance and evaluate the corresponding rate of convergence. The convergence rate is independent of the shape of the input density and is proved to be optimal. These results allow us to find a set of optimal nonnegative weights that further improve the accuracy of our estimation algorithm. Our findings are supported by simulation experiments.

Evaluating the Evolution of Distribution Networks under Different Regulatory Frameworks with Multi-Agent Modelling

In the context of increasing decentralised electricity generation, this paper evaluates the effect of different regulatory frameworks on the evolution of distribution networks. This problem is addressed by means of agent based modelling in which the interactions between the agents of a distribution network and an environment are described. The consumers and the distribution system operator are the agents, which act in an environment that is composed by a set of rules. For a given environment, we can simulate the evolution of the distribution network by computing the actions of the agents at every time step of a discrete time dynamical system. We assume the electricity consumers are rational agents that may deploy distributed energy installations. The deployment of such installations may alter the remuneration mechanism of the distribution system operator. By modelling this mechanism, we may compute the evolution of the electricity distribution tariff in response to the deployment of distributed generation.

Neimark–Sacker Bifurcation with $$\mathbb {Z}_n$$ Z n -Symmetry and a Neural Application

Effects of $$\mathbb {Z}_n$$ -symmetry, ( $$n\ge 2$$ ), on normal form of Neimark–Sacker bifurcation in discrete time dynamical systems are investigated. As an application, we consider three dimensional discrete Hopfield neural network with $$\mathbb {Z}_2$$ -symmetry. We drive analytical conditions for stability and bifurcations of the trivial fixed point of the system and compute analytically the normal form coefficients for the codimension 1 and codimension 2 bifurcation points including pitchfork, period-doubling, Neimark–Sacker, $$\mathbb {Z}_2$$ -symmetric Neimark–Sacker and resonance 1:4. By using numerical continuation in numerical software matcontm, we compute bifurcation curves of trivial fixed point and cycle with period 4 under variation of one and two parameters, and all codimension 1 and codimension 2 bifurcations supported by matcontm, on the corresponding curves are computed.

Numerical dynamics of integrodifference equations: Basics and discretization errors in a C0-setting

Besides being interesting infinite-dimensional dynamical systems in discrete time, integrodifference equations successfully model growth and dispersal of populations with nonoverlapping generations, and are often illustrated by simulations. This paper points towards and initiates a mathematical foundation of such simulations using generic methods to numerically discretize (and solve) integral equations. We tackle basic properties of a flexible class of integrodifference equations, as well as of their collocation and degenerate kernel semi-discretizations on the state space of continuous functions over a compact domain. Moreover, various estimates for the global discretization error are provided. Numerical simulations illustrate and confirm our theoretical results.

Data-Driven Robust Control of Discrete-Time Uncertain Linear Systems via Off-Policy Reinforcement Learning.

This paper presents a model-free solution to the robust stabilization problem of discrete-time linear dynamical systems with bounded and mismatched uncertainty. An optimal controller design method is derived to solve the robust control problem, which results in solving an algebraic Riccati equation (ARE). It is shown that the optimal controller obtained by solving the ARE can robustly stabilize the uncertain system. To develop a model-free solution to the translated ARE, off-policy reinforcement learning (RL) is employed to solve the problem in hand without the requirement of system dynamics. In addition, the comparisons between on- and off-policy RL methods are presented regarding the robustness to probing noise and the dependence on system dynamics. Finally, a simulation example is carried out to validate the efficacy of the presented off-policy RL approach.

Adaptive Leader Following in Networks of Discrete-Time Dynamical Systems: Algorithms and Global Convergence Analysis

Most of the results on adaptive leader following in multiagent systems pertain to continuous-time agent dynamics. It is implicitly presumed that the information flow among agents is continuous and infinitely fast. Since in the discrete-time setting, there no need for this assumption, it is of interest to develop adaptive methods for leader-follower synchronization in networks of discrete-time dynamical systems. There main challenges are as follows: 1) in general, the sample values of the leader's trajectory arrive at each agent with different delays; 2) since the received leader's trajectory samples do not carry the imprint of a leader, and are immersed in the state values of respective neighbors, agents do not have explicit knowledge of the leader's samples; and 3) the uncertainty in the agent dynamics calls for an appropriate adaptive rule for generating interagent coupling gains. This paper considers unknown linear agent dynamics of general order and arbitrary leader's trajectory, on a directed graph topology. Two novel distributed algorithms are proposed for both cases of known and unknown control directions. It is proven that all agents synchronize to the leader's trajectory, the tracking error is an $l_2$ sequence, the input signals remain bounded, and the parameter estimates are convergent sequences.

Inhomogeneous percolation on multilayer networks

A generalization of high-order inhomogeneous bond percolation is studied on a multilayer finite fixed network (MFFN) and a multilayer infinite random network (MIRN). The inhomogeneous bond percolation means that edges in different layers are occupied or removed with distinct probabilities, independently and randomly. Firstly, an analytical approach to simplify the inhomogeneous bond percolation on MFFN is presented, by decomposing the layers into disjoint ones, fusing inhomogeneous bond percolation process and combining weighted layers together to form a monolayer network. Then employing generalised recursive approach, discrete-time dynamical system analysis and fixed-point iteration method, we derive the results for percolating probability and the critical occupation probability, demonstrate the percolation transition and critical phenomenon of inhomogeneous bond percolation on these two networks. It is found that, for inhomogeneous bond percolation on MFFN, the high fraction of intersection speeds up the percolating process; for inhomogeneous bond percolation on MIRN, the increasing mean degree will enlarge the supercritical region and enhance the intensity of percolating process.

A discrete-time dynamical system with four types of codimension-one bifurcations

Usually, several discrete-time difference equations are shown in introductory courses on dynamical systems theory, in order to illustrate the occurrence of the most common bifurcations, which are saddle-node, transcritical, pitchfork, and flip. For instance, transcritical and flip bifurcations are found in the well-known logistic map. Here, a first-order difference equation undergoing these four types of bifurcations is presented. The bifurcation diagram is analytically derived and the rationale behind the construction of this equation is explained. The main goal of this didactic work is to give tips on how to write difference equations exhibiting various types of bifurcations, which can be associated with real-world scenarios.