Discrete-time Systems Research Articles

Affine formation control of discrete-time second-order multi-agent systems with a fixed communication period

Affine formation control of discrete-time second-order multi-agent systems with a fixed communication period

Two-Layer Reinforcement Learning for Output Consensus of Multiagent Systems Under Switching Topology.

In this article, the data-based output consensus of discrete-time multiagent systems under switching topology (ST) is studied via reinforcement learning. Due to the existence of ST, the kernel matrix of value function is switching-varying, which cannot be applied to existing algorithms. To overcome the inapplicability of varying kernel matrix, a two-layer reinforcement learning algorithm is proposed in this article. To further implement the proposed algorithm, a data-based distributed control policy is presented, which is applicable to both fixed topology and ST. Besides, the proposed method does not need assumptions on the eigenvalues of leader's dynamic matrix, it avoids the assumptions in the previous method. Subsequently, the convergence of algorithm is analyzed. Finally, three simulation examples are provided to verify the proposed algorithm.

Data-driven integral sliding mode predictive control with optimal disturbance observer

In this paper, a novel data-driven integral sliding mode predictive control algorithm based on an optimal disturbance observer (DDISMPC-ODO) is proposed for a class of nonlinear discrete-time systems (NDTS) subject to external disturbances. The designed optimal disturbance observer realizes the precise observation of the lumped disturbance, thus ameliorating the accuracy of the controller and weakening problems with chattering. In this work, a robust pseudo-partial derivative (PPD) estimation algorithm is introduced, which not only improves the system performance, but also facilitates theoretical proof of parameter estimation and tracking accuracy. The convergence of the PPD estimation error and disturbance observation error is proved. It is also proved that the accuracy of the disturbance observation error can converge to O(T3) and then the magnitude of the sliding variable and the tracking error are also reduced to O(T3) respectively. Finally, the effectiveness of the proposed method is demonstrated by a simulation example and an experiment.

Dynamics and control of two-dimensional discrete-time biological model incorporating weak Allee's effect.

Incorporating a weak Allee effect in a two-dimensional biological model in ℜ2, the study delves into the application of bifurcation theory, including center manifold and Ljapunov-Schmidt reduction, normal form theory, and universal unfolding, to analyze nonlinear stability issues across various engineering domains. The focus lies on the qualitative dynamics of a discrete-time system describing the interaction between prey and predator. Unlike its continuous counterpart, the discrete-time model exhibits heightened chaotic behavior. By exploring a biological Mmdel with linear functional prey response, the research elucidates the local asymptotic properties of equilibria. Additionally, employing bifurcation theory and the center manifold theorem, the analysis reveals that, for all α1 (i.e., intrinsic growth rate of prey), ð1˙ (i.e., parameter that scales the terms yn), and m (i.e., Allee effect constant), the model exhibits boundary fixed points A1 and A2, along with the unique positive fixed point A∗, given that the all parameters are positive. Additionally, stability theory is employed to explore the local dynamic characteristics, along with topological classifications, for the fixed points A1, A2, and A∗, considering the impact of the weak Allee effect on prey dynamics. A flip bifurcation is identified for the boundary fixed point A2, and a Neimark-Sacker bifurcation is observed in a small parameter neighborhood around the unique positive fixed point A∗=(mð1˙-1,α1-1-α1mð1˙-1). Furthermore, it implements two chaos control strategies, namely, state feedback and a hybrid approach. The effectiveness of these methods is demonstrated through numerical simulations, providing concrete illustrations of the theoretical findings. The model incorporates essential elements of population dynamics, considering interactions such as predation, competition, and environmental factors, along with a weak Allee effect influencing the prey population.

Data-driven discrete fractional chaotic systems, new numerical schemes and deep learning.

Parameter estimation is important in data-driven fractional chaotic systems. Less work has been reported due to challenges in discretization of fractional calculus operators. In this paper, several numerical schemes are newly derived for delay fractional difference equations of Caputo and Riemann-Liouville types. Then, loss functions are constructed and unknown parameters of the discrete fractional chaotic system are estimated by a neural network method. Parameter estimation results demonstrate high accuracy compared with real values. Robust analysis is provided under different noise levels. It can be concluded that this paper provides an efficient deep learning method based on fractional discrete-time systems.

Unconditional error analysis of the linearized transformed [formula omitted] virtual element method for nonlinear coupled time-fractional Schrödinger equations

This paper constructs a linearized transformed L1 virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical s-fractional differential system derived from a smoothing transformation of variables t=s1/α, 0<α<1. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in L2-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.

Lyapunov-based robust model predictive tracking controller design for discrete-time linear systems with ellipsoidal terminal constraint

In this article, a robust model predictive control (MPC) method is introduced based on Lyapunov-theory to control the discrete-time uncertain linear systems due to external disturbances. The structure of the proposed controller consists of two parts designed based on the offline/online schemes. In the offline-scheme, an H ∞ -based robust tracking controller is provided to satisfy the robust and tracking performance. Based on the H ∞ performance, a new linear matrix inequality problem is developed to calculate the offline-controller gains. By considering the Lyapunov-function of the offline-controller as the terminal cost of the MPC and the designed offline-controller, the online optimisation problem (OOP) is structured. This means, the MPC is designed based on the stabilised system to satisfy the physical limitations of the overall controller and the system states. Moreover, to improve the feasibility problem of the OOP, an ellipsoidal terminal constraint is added to the proposed MPC problem. Therefore, compared with the existing min–max MPC and tube MPC, the advantages of the overall controller are feasibility improvement and reducing the computational complexity with less conservatism while the robustness against parametric uncertainties and disturbances is achieved. Two simulation examples are employed to show the superiorities of the proposed robust MPC.

An approach to mismatched disturbance rejection control for uncontrollable systems

This study focuses on the problem of optimal mismatched disturbance rejection control for uncontrollable linear discrete-time systems. In contrast to previous studies, by introducing a quadratic performance index such that the regulated state can track a reference trajectory and minimize the effects of disturbances, mismatched disturbance rejection control is transformed into a linear quadratic tracking problem. The necessary and sufficient conditions for the solvability of this problem over a finite horizon and a disturbance rejection controller are derived by solving a forward–backward difference equation. In the case of an infinite horizon, a sufficient condition for the stabilization of the system is obtained under the detectable condition. Additionally, in combination with the generalized extended state observer, a controller design method is proposed, and the stability analysis of the system under this controller is presented. This paper details our novel approach to disturbance rejection. Finally, four examples are provided to demonstrate the effectiveness of the proposed method.

Model-free aperiodic tracking for discrete-time systems using hierarchical reinforcement learning

In the operation of practical systems, it is often a challenging task to deal with limited knowledge of the system dynamics, therefore, it is needed to obtain a framework for the simultaneous learning and control of dynamic systems, able to cope with uncertain dynamics. This paper proposes a model-free aperiodic tracking control method based on MAXQ hierarchical reinforcement learning for unknown dynamics in discrete-time systems. Firstly, a MAXQ hierarchical framework is developed for aperiodic tracking control that decomposes the task into pre-control and accurate control subtasks. The former achieves sub-optimal tracking which prompts the implementation of accurate control. The latter implements aperiodic tracking based on the pre-control result to reduce resource occupation. The structure of the MAXQ framework simplifies the complexity of the tracking task, and the incorporation of pre-control accelerates the learning process in the formal control stage. Secondly, considering the disparities between the control inputs and the control update instants, pre-control is decomposed to learn the triggering strategy and control policy, respectively. Meanwhile, the control policy learns optimal control inputs by minimizing the cost function. Similarly, the triggering strategy and control policy are also separately learned in accurate control. In contrast to traditional aperiodic triggering mechanisms, in accurate control stage, the triggering strategy is developed based on the cumulative error of the pre-control result, thus avoiding the influence of accidental factors and the cumulative effects of errors, enhancing system robustness. Thirdly, the proposed tracking control is derived by using only the input, output, and reference signal data from the system, without relying on system dynamics. It is applicable to both linear and nonlinear systems, demonstrating strong generalizability. Finally, simulation examples are provided to validate the effectiveness and superiority of the proposed method.

A linearized BDF2 virtual element method for the unsteady Brinkman–Forchheimer equations with variable time step

This paper is devoted to a lowest order virtual element method for the unsteady incompressible Brinkman–Forchheimer equations on polygonal mesh, while a linearized variable-time-step second order backward differentiation formula is adopted in time. We utilize discrete orthogonal convolution and discrete complementary convolution kernels to obtain error estimates for the solutions of time-discrete system. By virtue of the temporal–spatial error splitting approach, and under the current mildest adjacent time-step ratio condition: 0<rk≔τk/τk−1≤rmax≈4.8645, we have established the L∞ boundedness of the fully discrete velocity solution and its L2-norm optimal error estimate. Numerical experiments are conducted on various polygonal meshes, validating the accuracy of the theoretical analysis.

Snapback Repellers, Computational Chaos and Extreme Multistability in Discrete-Time Memristor Murali–Lakshmanan–Chua Circuit

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves.

A compact meta-learned neuro-fuzzy technique for noise-robust nonlinear control

Neuro-fuzzy systems show promise for adaptive control but can become complex due to the need to learn many parameters. This paper presents a resilient nonlinear controller that combines a simplified neuro-fuzzy system (Simp_NFS) and simplified neural network (Simp_NN) with only two meta-learnable parameters. This architecture enables fast and stable adaptation in uncertain nonlinear discrete-time systems. Simp_NFS utilizes interpretable hyperplane-based rules without antecedent parameters, simplifying the learning process to consequent weights. Simp_NN reduces complexity by replacing hidden-output weights with their mean. The hybrid auto-adaptive controller (HAC) combines the advantages of Simp_NFS and Simp_NN, significantly reducing the number of adaptive parameters compared to standard neuro-fuzzy methods for real-time control with limited resources. Simp_NFS provides structural adaptivity to handle uncertainties, while Simp_NN ensures reliable disturbance attenuation. The stability of HAC is proven using Lyapunov analysis. Extensive testing on challenging single-input single-output (SISO) and multi-input multi-output systems (MIMO) demonstrates that HAC improves performance by up to 82.55% compared to existing techniques. Key innovations include an ultra-compact meta-learned architecture, transparent online evolution of hyperplane clusters, and enhanced modeling capability for nonlinear uncertain systems. This interpretable neuro-fuzzy approach could enhance autonomy and safety by maintaining model transparency. The implementation of HAC is publicly available on GitHub at https://github.com/m-ferdaus/HAC.

Formation control of high-order linear discrete-time multiple unmanned aerial vehicles systems in noisy environments

Formation control of high-order linear discrete-time multiple unmanned aerial vehicles (multi-UAVs) systems with multiplicative noises is studied in this article. Because of the existence of multiplicative noises, classical methods for discrete-time system control problems should be improved. Here, we develop a new discrete-time formation protocol for multi-UAVs systems. The stochastic Lyapunov stability theory and the positive definiteness theory of a generalized algebraic Riccati equation are then used to derive sufficient conditions for formation feasibility.

Novel generalized policy iteration for efficient evolving control of nonlinear systems

In this article, we construct a novel generalized policy iteration framework to address optimal regulation problems for discrete-time nonlinear systems in a more efficient way. Relevant properties are investigated for the framework, including monotonicity and convergence of the iterative value function sequence as well as the admissibility of the iterative control policy. Additionally, an innovative approach is developed to seek an initial admissible control policy for the framework with an adjustable searching speed. Based on these, an evolving control algorithm is presented with stability guarantee. This algorithm employs iterative control policies for system control during the computation of the optimal control policy, as opposed to waiting for the generation of the optimal control policy before implementing control. Eventually, two simulation experiments are conducted with real-world physical backgrounds, in order to illustrate the performance of the proposed strategy.

Positive semi-definite Lyapunov function-based adaptive control for nonlinear discrete-time systems with application to chaotic Duffing oscillator

Positive semi-definite Lyapunov function-based adaptive control for nonlinear discrete-time systems with application to chaotic Duffing oscillator

Learning-based robust output tracking control for unknown discrete-time nonlinear systems with dynamic uncertainty

Although numerous approximate dynamic programming (ADP) methods have been proposed and applied to robust control problems, research on the use of ADP methods for robust tracking control in nonlinear systems remains limited, especially for discrete time (DT) nonlinear systems. In this paper, we propose an efficient robust learning-based output tracking control method, referred to as r-LTC, for a class of unknown DT nonlinear systems with dynamic uncertainty. The objectives are achieved by: 1) using a system identification technique based on network-type coefficients Auto-Regressive with eXogenous variables (ARX) model to identify completely unknown systems; 2) reformulating the identified model into a state space model in terms of deviations from the reference trajectory, such that the robust tracking problem can be converted into a robust regulation problem; 3) transforming the robust regulation problem into an equivalent optimal regulation problem for a nominal system with a modified utility function; 4) alternately implementing the performance evaluation and control policy update through two neural networks (NNs) to numerically solve DT Hamilton-Jacobi-Bellman (HJB) equation, such that the approximated optimal feedback control inputs for nominal system can be obtained, allowing for realization of all unknown bounded uncertainties. The proposed method does not require a state observer and any steady state knowledge, only the measurable system input and output data are utilized. The convergence of NNs’ approximate weights and stability of the closed-loop uncertain systems are rigorously proven using the Lyapunov method. Lastly, a case study is performed using real data collected from the Shenzhen Metro Line 12 tunneling project to examine the performance of the proposed r-LTC algorithm in terms of the set value tracking accuracy, control system stability, and robustness against different types of disturbances.

GA based order abatement technique for linear dynamic systems for continuous time system and discrete time system

GA based order abatement technique for linear dynamic systems for continuous time system and discrete time system

A novel control approach for discrete-time 2D Roesser systems under input saturation and intrinsic nonlinearity: A region of stability investigation

This paper presents the control system design for the nonlinear discrete-time Roesser systems under saturated control input. A deficiency in the existing literature has been addressed in this study for the control of two-dimensional (2D) systems. A generalized boundary condition analysis is presented in the form of tractable invariant sets for the derivation of region of stability to design a suitable state feedback control. A mechanism for the evaluation of controller parameters is presented by considering the generalized structure of 2D Lyapunov function. The work has been extended when external disturbances are present in the 2D systems, and a robust design condition by accounting the perturbations as well as the input saturation has been established. In comparison to the existing works, a control system design for 2D Roesser systems under input saturation and intrinsic nonlinearity by application of a generalized Lyapunov function has been revealed. Furthermore, a tractable 2D region of stability for the local control of the 2D systems is achieved, rather than the conventional global controllers. Furthermore, a method has also been studied for obtainment of the controller gain matrix through convex optimization regarding the complete structure of the 2D Lyapunov function. Simulation results are provided for the stabilization of an unstable nonlinear 2D system under input saturation.

Robust Invariance Conditions of Uncertain Linear Discrete Time Systems Based on Semidefinite Programming Duality

This article proposes a novel robust invariance condition for uncertain linear discrete-time systems with state and control constraints, utilizing a method of semidefinite programming duality. The approach involves approximating the robust invariant set for these systems by tackling the dual problem associated with semidefinite programming. Central to this method is the formulation of a dual programming through the application of adjoint mapping. From the standpoint of semidefinite programming dual optimization, the paper presents a novel linear matrix inequality (LMI) conditions pertinent to robust positive invariance. Illustrative examples are incorporated to elucidate the findings.

Structure-preserving model reduction based on multiorder Arnoldi for discrete-time second-order systems

This paper investigates the new structure-preserving model reduction methods based on multiorder Arnoldi for discrete-time second-order systems by using discrete Laguerre polynomials. The expansion coefficients of discrete-time second-order system and its equivalent first-order system satisfy the implicit recursive expressions in space spanned by discrete Laguerre polynomials, and then multiorder Arnoldi algorithm is applied to the implicit recursive expressions to produce the column-orthogonal matrices, where the orthogonal projection matrix for the equivalent first-order system is constructed by partitioning the resulting column-orthogonal matrix appropriately. The reduced-order systems can preserve the special structures of the original systems as well as match a desired number of expansion coefficients of the original outputs. The error estimate of output variables is also given for discrete-time second-order system. Two benchmark examples are simulated to illustrate the effectiveness of the proposed methods.