Continuous-time Systems Research Articles

Persistency and stability of a class of nonlinear forced positive discrete-time systems with delays

Persistence, excitability and stability properties are considered for a class of nonlinear, forced, positive discrete-time systems with delays. As will be illustrated, these equations arise in a number of biological and ecological contexts. Novel sufficient conditions for persistence, excitability and stability are presented. Further, similarities and differences between the delayed equations considered presently and their corresponding undelayed versions are explored, and some striking differences are noted. It is shown that recent results for a corresponding class of positive, nonlinear delay-differential (continuous-time) systems do not carry over to the discrete-time setting. Detailed discussion of three examples from population dynamics is provided.

Stable predictive control of continuous stirred-tank reactors using deep learning

In this work, a Lyapunov-based predictive control method utilizing deep learning techniques is proposed for driving continuous-time nonlinear processes towards the desired equilibrium point. Initially, a neural ordinary differential equation model is developed to learn the continuous-time dynamics from discrete-time sampling data. Subsequently, the theoretical conditions that the control Lyapunov function (CLF) developed on the learned continuous-time system is feasible to act on the actual system through a zero-order-holder are analyzed. Building upon the theoretical analysis and the constructed continuous-time dynamics, a deep neural network control Lyapunov function (DNN-CLF) controller is developed. Importantly, the DNN-CLF is independent of the affine system formulation, effectively addressing the challenge of designing the CLF. Benefiting from the stability guarantee provided by the established DNN-CLF, it is integrated into the model predictive control (MPC) framework to develop Lyapunov-based MPC to further improve the control performance. To ensure safety, considering error propagation and associated risks with enlarging the feasible region of the CLF, tubes are constructed and integrated into the MPC scheme. Finally, a numerical simulation of a chemical process is conducted to validate the efficiency of the proposed method.

Infinite-time robust optimal output tracking of continuous-time linear systems using undiscounted reinforcement learning

This research paper focuses on addressing the challenge of infinite-time linear quadratic tracking control (LQT) for linear systems with parametric uncertainty. Traditional solutions to the LQT problem often involve using a discount factor to prevent the cost function from growing unbounded over time. However, this approach can introduce instability in the closed-loop system. To overcome this issue, this paper proposes an alternative approach using an undiscounted cost function that ensures the asymptotic stability of the uncertain closed-loop system. To design a control scheme without requiring precise knowledge of the system dynamics, reinforcement learning (RL) algorithms are employed. However, for systems with uncertain parameters that may lead to instability, the convergence of RL algorithms to a stabilising solution is not guaranteed. To address this limitation, a robust optimal control structure is developed using on-policy and off-policy reinforcement learning algorithms, resulting in a model-free controller. The effectiveness of the proposed robust optimal controller is validated through comparative simulations on an uncertain model of a DC–DC buck converter connected to a constant power load. These simulations demonstrate the advantages and benefits of the robust optimal controller in handling parametric uncertainty and ensuring stability in the control system.

Study of Chaotic-based Audio Encryption Algorithms: A Review

Nowadays, multimedia communication has become very widespread and this requires it to be protected from attackers and transmitted securely for reliability. Encryption and decryption techniques are useful in providing effective security for speech signals to ensure that these signals are transmitted with secure data and prevent third parties or the public from reading private messages. Due to the rapid improvement in digital communications over the recent period up to the present, the security of voice data transmitted over various networks has been classified as a favored field of study in earlier years. The contributions to audio encryption are discussed in this review. This Comprehensive review mainly focuses on presenting several kinds of methods for audio encryption and decryption the analysis of these methods with their advantages and disadvantages have been investigated thoroughly. It will be classified into encryption based on traditional methods and encryption based on advanced chaotic systems. They are divided into two types, continuous-time system, and discrete-time system, and also classified based on the synchronization method and the implementation method. In the fields of information and communications security, system designers face many challenges in both cost, performance, and architecture design, Field Programmable gate arrays (FPGAs) provide an excellent balance between computational power and processing flexibility. In addition, encryption methods will be classified based on Chaos-based Pseudo Random Bit Generator, Fractional-order systems, and hybrid chaotic generator systems, which is an advantageous point for this review compared with previous ones. Audio algorithms are presented, discussed, and compared, highlighting important advantages and disadvantages. Audio signals have a large volume and a strong correlation between data samples. Therefore, if traditional cryptography systems are used to encrypt such huge data, they gain significant overhead. Standard symmetric encryption systems also have a small key-space, which makes them vulnerable to attacks. On the other hand, encryption by asymmetric algorithms is not ideal due to low processing speed and complexity. Therefore, great importance has been given to using chaotic theory to encode audio files. Therefore, when proposing an appropriate encryption method to ensure a high degree of security, the key space, which is the critical part of every encryption system, and the key sensitivity must be taken into account. The key sensitivity is related to the initial values and control variables of the chaotic system chosen as the audio encryption algorithm. In addition, the proposed algorithm should eliminate the problems of periodic windows, such as limited chaotic range and non-uniform distribution, and the quality of the recovered audio signal remains good, which confirms the convenience, reliability, and high security.

Continuous-time MISO fractional system identification using higher-order-statistics

Continuous-time MISO fractional system identification using higher-order-statistics

Advanced stability assessment for control systems: Gelfand’s method with iterated Shanks transformation

ABSTRACT In this paper, we present an efficient method for evaluating the stability of both linear time-invariant (LTI) systems and continuous-time nonlinear systems. Our approach centres around the modulus of the spectral radius derived from a specific matrix transformation of the original system state matrix in the case of LTI systems, or the Jacobian of the nonlinear system. We enhance computational speed by employing the iterated Shanks transformation on Gelfand’s formula, eliminating the need to rely on the system’s characteristic equation. Notably, our stability criterion avoids the computation of all eigenvalues of the system matrix. Illustrative examples, including a fluidised bed reactor, a continuous stirred tank reactor and a hypothetical 20th-order dynamical system, are provided. Additionally, we conduct a CPU-time analysis, revealing that our method demonstrates several orders of magnitude faster computational speed compared to the classical Routh-Hurwitz stability test for systems ranging from two to 80 dimensions.

On the effect of clock offsets and quantization on learning-based adversarial games

In this work, we consider systems whose components suffer from clock offsets and quantization and study the effect of those on a reinforcement learning (RL) algorithm. Specifically, we consider an off-policy iterative RL algorithm for continuous-time systems, which uses input and state data to approximate the Nash-equilibrium of a zero-sum game. However, the data used by this algorithm are not consistent with one another, in that each of them originates from a slightly different time instant of the past, hence putting the convergence of the algorithm in question. We prove that, given that these timing inconsistencies remain below a certain threshold, the iterative off-policy RL algorithm will still converge epsilon-closely to the desired Nash policy. However, this result is conditional to a certain Lipschitz continuity and differentiability condition on the input-state data collected, which is indispensable in the presence of clock offsets. A similar result is also derived when quantization of the measured state is considered. Finally, unlike prior work, we provide a sufficiently rich data condition for the execution of the iterative RL algorithm, which can be verified a priori across all iteration indices. Simulations are performed, which verify and clarify theoretical findings.

Consistent spectral approximation of Koopman operators using resolvent compactification

Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L 2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.

Practical exponential stability and applications of switched homogeneous nonlinear systems with external inputs and impulses

The practical exponential stability of switched homogeneous nonlinear systems (SHNS) with external inputs and impulses is studied in this paper, which includes two types of continuous-time systems and discrete-time systems. Specifically, we add a kind of nonnegative impulses which is consistent with the switching behaviours for the systems. We first study switched homogeneous positive nonlinear systems (SHPNS) case, then we extend the SHPNS to a time-varying SHNS case. The primary difficulty of this research is the estimation of Lyapunov function value at switching-impulse points under the coupling effects of synchronous switches and state jumps. By employing the max-separable Lyapunov function approach, we present new sufficient stability criteria and design new switching-impulse signals, and successfully establish the new relation between the average dwell time parameter and impulse strength. Then, we apply the obtained results to the time-varying vehicle formation model with impulses. At last, four examples are given to demonstrate the feasibility of our results, which includes the specific algorithm verification process.

Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra

Quantum simulation is capable of simulating certain dynamical systems in continuous time—Schrödinger’s equations being the most direct and well known—more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger’s equations via a method called Schrödingerisation (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.

Integral sliding mode-based event-triggered optimal fault tolerant tracking control of continuous-time nonlinear systems

In this paper, the integral sliding mode-based event-triggered optimal fault tolerant tracking control of continuous-time nonlinear systems is investigated via adaptive dynamic programming. The developed control scheme consists of two parts, i.e., integral sliding mode control and event-triggered optimal tracking control. For the first part, an integral sliding mode controller is designed to eliminate the affect of actuator fault and the dynamics of nominal nonlinear systems is obtained. For the second part, a novel quadratic cost function with respect to the tracking error and its dynamics is developed such that the feedforward control law or the discount factor is not required, which reduces the complexity of the control method and guarantees the tracking performance. Moreover, a critic-only structure is established to obtain the solution of tracking Hamilton–Jacobi–Bellman equation. It should be noted that the optimal tracking control law is updated only at triggering moments in order to preserve computing and communication resources. Finally, the effectiveness of the present approach is demonstrated through simulation examples of a robotic arm system and a Van der Pol circuit system.

Distributed Minimum-Energy Containment Control of Continuous-Time Multi-Agent Systems by Inverse Optimal Control

Dear Editor, This letter focuses on the distributed optimal containment control of continuous-time multi-agent systems (CTMASs) with respect to the minimum-energy performance index over fixed topology. To achieve this, we firstly investigate the optimal containment control problem using the inverse optimal control method, where all states of followers asymptotically converge to the convex hull spanned by the leaders while some quadratic performance indexes get minimized. A sufficient condition for existence of the distributed optimal containment control protocol is derived. By introducing the parametric algebraic Riccati equation (PARE), it is strictly proved that the global performance index can be used to approximate the standard minimum-energy performance index as the parameters tends to infinity. In consequence, the standard minimum-energy cooperative containment control can be solved by local steady state feedback protocols. Finally, numerical examples are given to demonstrate the effectiveness of theoretical results.

Cooperative Containment Control for Multiagent Systems With Reduced-Order Protocols.

This article addresses the problem of containment control for continuous-time multiagent systems. A containment error is first given to show the coordination between the outputs of leaders and followers. Then, an observer is designed based on the neighbor observable convex hull state. Under the assumption that the designed reduced-order observer is subject to external disturbances, a reduced-order protocol is designed to realize the containment coordination. In order to ensure the designed control protocol can achieve the effect of the main theories, a corresponding Sylvester equation is given with a novel approach which proves that the Sylvester equation is solvable. Finally, a numerical example is given to verify the validity of the main results.

Quantized cooperative output regulation of continuous-time multi-agent systems over switching graph

Quantized cooperative output regulation of continuous-time multi-agent systems over switching graph

Bayesian inference and optimisation of stochastic dynamical networks

Network inference has been extensively studied in several fields, such as systems biology and social sciences. Learning network topology and internal dynamics is essential to understand mechanisms of complex systems. In particular, sparse topologies and stable dynamics are fundamental features of many real-world continuous-time (CT) networks. Given that usually only a partial set of nodes are able to observe, we consider linear CT systems to depict networks since they can model unmeasured nodes via transfer functions. Additionally, measurements tend to be noisy and with low and varying sampling frequencies. This paper applies dynamical structure functions (DSFs) derived from linear stochastic differential equations (SDEs) to describe networks of measured nodes. A numerical sampling method, preconditioned Crank–Nicolson (pCN), is used to refine coarse-grained trajectories to improve inference accuracy. The proposed method can handle sparsely sampled data and unmeasurable nodes. Monte Carlo simulations indicate that the proposed method outperforms state-of-the-art methods with various network topologies. The developed method can be applied under a wide range of contexts, such as gene regulatory networks, social networks and communication systems.

Transfer matrices with positive coefficients of descriptor linear systems

Transfer matrices with positive coefficients of descriptor positive linear continuous-time systems are addressed. Two methods of checking of the positivity of descriptor linear systems are proposed. It is shown that if the positive descriptor system is asymptotically stable then all coefficients of its transfer matrix are positive.

Robust Recursive Least-Squares Fixed-Point Smoother and Filter using Covariance Information in Linear Continuous-Time Stochastic Systems with Uncertainties

This study develops robust recursive least-squares (RLS) fixed-point smoothing and filtering algorithms for signals in linear continuous-time stochastic systems with uncertainties. The algorithms use covariance information, such as the cross-covariance function of the signal with the observed value and the autocovariance function of the degraded signal. A finite Fourier cosine series expansion approximates these functions. Additive white Gaussian noise is present in the observation of the degraded signal. A numerical simulation compares the estimation accuracy of the proposed robust RLS filter with the robust RLS Wiener filter, showing similar mean square values (MSVs) of the filtering errors. The MSVs of the proposed robust RLS fixed-point smoother are also compared to those of the proposed robust RLS filter.

H∞ fault detection and control of Takagi–Sugeno continuous-time conic-type nonlinear systems

In this study, control and fault detection are examined for Takagi–Sugeno (T–S) fuzzy continuous-time systems with conic-type nonlinearity. In a conic-type dynamic system, Takagi–Sugeno fuzzy models are applied. A conic-type non-linearity must lie in an n-dimensional hyper-sphere with an uncertain centre to satisfy the restrictive condition. A combined error system of estimation and control error is obtained by combining the observations and the controller. Our main goal is to design an observer and controller for fault detection that minimizes the impact of unknown inputs and faults on the system in terms of the H ∞ performance. The Lyapunov function is used to construct the residual signal using an observer and a controller. Using Linear Matrix Inequalities (LMIs), we can express the performance conditions for H ∞ . The scheme's effectiveness is demonstrated with numerical examples.

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.

Privacy-preserving bipartite output consensus for continuous-time heterogeneous multi-agent systems via event-triggered impulsive control

This paper studies the problem of differentially private bipartite output consensus in continuous-time heterogeneous multi-agent systems (MASs) characterized by antagonistic interactions. A novel hybrid privacy-preserving event-triggered impulsive consensus protocol is introduced to protect the agent’s initial information from disclosure, which involves a discrete-time information transmission based on an event-triggering mechanism. Using stochastic Lyapunov method, sufficient conditions have been obtained to achieve mean square bipartite output consensus with a guaranteed level of privacy. Furthermore, the differential privacy of competitive agent pairs is exclusively secured by the proposed control scheme by injecting Laplace noise. The protocol also effectively prevents Zeno behavior by imposing a lower bound for impulsive intervals under all event-triggered conditions. A simulation example is provided to validate the effectiveness of the theoretical result.