Razumikhin Theorem Research Articles

Hybrid control design for Mittag-Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects

This article discusses the global Mittag-Leffler projective synchronization (GM-LPS) on fractional-order quaternion-valued neural networks (FOQVNNs) including mixed delays and impulses. The effects of various types of delay factors and impulses on Mittag-Leffler projective synchronization performance are simultaneously explored. Based on Banach contractive mapping principle, the existence result of solution to the considered systems is derived Applying the Lyapunov direct method, inequality analysis technique and generalized fractional Razumikhin theorem, the algebraic criteria for ensuring global Mittag-Leffler projective synchronization by designing the hybrid controllers are proposed. The result of algebraic inequality greatly reduces the computational complexity to some extent. Combining the different orders of Caputo derivative and MATLAB toolbox, the simulations are given to further validate the obtained results.

Non-Fragile Fault-Tolerant Control Design for Fractional-Order Nonlinear Systems With Distributed Delays and Fractional Parametric Uncertainties

This work discuss the stabilization issue for a class of fractional-order nonlinear systems together with time delay, parametric uncertainties and actuator faults. Precisely, the considered system comprises of two delays namely distributed delay and time-varying delay. Moreover, the occurrence of the actuator faults and fractional parametric uncertainties may induce poor performance of the systems. To overcome these issue, a non-fragile fault-tolerant controller is designed which makes the system asymptotically stable with the specified mixed <inline-formula> <tex-math notation="LaTeX">$H_{\infty} $ </tex-math></inline-formula> and passive performance index. A fractional Razumikhin theorem is applied to handle the distributed delay term in the stabilization analysis. With the aid of suitable Lyapunov-Krasovskii functional, the sufficient conditions are established in terms of linear matrix inequalities together with Razumikhin stability theorem for getting the required results. By virtue of this, the controller gain matrix is obtained by solving the obtained LMIs and the graphical results are simulated using FOMCON toolbox. Later, the potency of the developed results are validated by virtue of three numerical examples including a rocket motor chamber.

Global decentralized control for uncertain large-scale feedforward nonlinear time-delay systems via output feedback

Abstract In this paper, the problem of global decentralized output feedback control is addressed for a class of large-scale nonlinear systems with zero dynamics and unknown time-varying delay. System disturbance nonlinearities are subject to feedforward growth restrictions with unknown growth rate. In the spirit of dynamic scaling change technique, a novel pair of time-varying-gain observer and controller is proposed. Compared with the existing results, the controller proposed is capable of handling the difficulties caused by the zero-dynamics, the uncertainties and the unknown time-varying delay. With the help of the Razumikhin theorem, it is shown that the closed-loop states asymptotically converge to zero. Finally, the effectiveness of the proposed control scheme is illustrated by a numerical example.

Further results on asymptotic and finite-time stability analysis of fractional-order time-delayed genetic regulatory networks

This research work aims at investigating the fractional-order genetic regulatory networks (FOGRNs) with feedback regulation delays. First, we have established the existence and uniqueness of considered systems based on the Banach fixed point theorem. Second, based on a Lyapunov method and fractional-order Razumikhin theorem, some novel sufficient criteria are obtained to ensure the global asymptotic stability for addressing fractional-order systems. To this end, a suitable controller that includes discontinuous terms is designed to guarantee the system states tends to zero asymptotically in a finite time when the system states do not admit stable performance. Moreover, the upper bound of the settling-time for finite-time stability is also estimated accurately. Finally, two numerical examples are presented to verify the correctness of the obtained stability results.

Robust finite-time consensus subject to unknown communication time delays based on delay-dependent criteria

In this paper, robust finite-time consensus of a group of nonlinear multi-agent systems in the presence of communication time delays is considered. In particular, appropriate delay-dependent strategies which are less conservative are suggested. Sufficient conditions for finite-time consensus in the presence of deterministic and stochastic disturbances are presented. The communication delays don’t need to be time invariant, uniform, symmetric, or even known. The only required condition is that all delays satisfy a known upper bound. The consensus algorithm is appropriate for agents with partial access to neighbor agents’ signals. The Lyapunov–Razumikhin theorem for finite-time convergence is used to prove the results. Simulation results on a group of mobile robot manipulators as the agents of the system are presented.

Leader-follower non-fragile consensus of delayed fractional-order nonlinear multi-agent systems

This paper addresses the leader-follower non-fragile consensus of nonlinear fractional-order (FO) multi-agent systems (FOMAS) with state time delay. The structured uncertainties occurring in both the plant and the controller are considered for the first time. Using the linear matrix inequality approach and the FO Razumikhin theorem, a delay- and order-dependent protocol is obtained to guarantee the leader-follower non-fragile consensus of the FOMAS with uncertain parameters. New sufficient conditions for the leader-follower non-fragile consensus of FO linear multi-agent systems are presented. The feasibility and effectiveness of the protocol proposed is verified with three numerical examples. Compared with the existing schemes, the approach reveals good robustness and can be extended to different kinds of consensus problems.

Guaranteed Cost Leaderless Consensus Protocol Design for Fractional-Order Uncertain Multi-Agent Systems with State and Input Delays

This paper addresses the guaranteed cost leaderless consensus of delayed fractional-order (FO) multi-agent systems (FOMASs) with nonlinearities and uncertainties. A guaranteed cost function for FOMAS is proposed to simultaneously consider consensus performance and energy consumption. By employing the linear matrix inequality approach and the FO Razumikhin theorem, a delay-dependent and order-dependent consensus protocol is formulated for FOMASs with input delay. The proposed protocol not only guarantees the robust stability of the closed-loop system error but also ensures that the performance degradation caused by the system uncertainty is lesser than that obtained with other approaches. Two numerical examples are provided in order to verify the effectiveness and accuracy of the proposed protocol.

Global regulation by output feedback for feedforward systems with time delays

In this paper, an output feedback controller with logarithmic function is constructed to address the objective of global regulation for a class of feedforward nonlinear systems with time delays. Firstly, discrete delays and distributed delays are allowed in the input and output as well as states of the considered system. Secondly, compared to the conventional research contributions of state feedback regulation, not only the condition that the output signal is needed to design the control strategy become relaxed, but also the information about the discrete delays and distributed delays that need to be known is less. Finally, different from the existing research results, based on the time-varying gain technology and Lyapunov–Razumikhin theorem, the output feedback global regulation controller constructed here is more lucid in structure and more perfect in function. Two examples are presented to demonstrate the effectiveness of global regulation about the proposed approach for the considered system.

Asymptotic stabilization of nonlinear systems with long input delay via memoryless feedback: A linearization method

This paper considers the problem of how to control nonlinear systems with a large delay in the input. A linearization method is presented to achieve local asymptotic stabilization (LAS) via memoryless state and/or output feedback, for multiple-input/multiple-output (MIMO) nonlinear systems with a large input delay. The development of the linearization technique is based on the Razumikhin theorem, a forward completeness lemma of the FDE (functional differential equation) on a closed and bounded interval, and the previous work on truncated predictor feedback control of linear systems with input delay.

Multiple asymptotic stability of fractional-order quaternion-valued neural networks with time-varying delays

In this paper, the multiple asymptotic stability is investigated for fractional-order quaternion-valued neural networks (FQVNNs) with time-varying delays. The activation function is a nonmonotonic piecewise nonlinear activation function. By applying the Hamilton rules, the FQVNNs are transformed into real-valued systems. Then, according to the Brouwer’s fixed point theorem, three new conditions are proposed to ensure that there exist 34n equilibrium points. Moreover, by virtue of fractional-order Razumikhin theorem and Lyapunov function, a new condition is derived to guarantee the FQVNNs have 24n locally asymptotic stable equilibrium points. For the first time, the multiple asymptotic stability of delayed FQVNNs is investigated. Contrast to multistability analysis of integer-order quaternion-valued neural networks, this paper present different conclusions. Finally, two numerical simulations demonstrate the validity of the results.

Guaranteed cost control of fractional‐order linear uncertain systems with time‐varying delay

AbstractThis article is concerned with the guaranteed cost control problem for a class of fractional‐order (FO) uncertain delayed linear systems with norm‐bounded time‐varying parametric uncertainty. Based on whether the states are available or not, guaranteed‐cost state‐based feedback stabilization and guaranteed‐cost observer‐based feedback stabilization for such system are addressed, respectively. By employing the linear matrix inequality (LMI) approach and FO Razumikhin theorem, two delay‐independent guaranteed cost controller design methods and their guaranteed costs are first formulated, respectively, in which sufficient conditions of the state feedback and observer gains are presented in form of LMI. Two numerical examples are presented to demonstrate the effectiveness of the theoretical formulation.

A Razumikhin approach to stability and synchronization criteria for fractional order time delayed gene regulatory networks

<abstract> This manuscript is concerned with the stability and synchronization for fractional-order delayed gene regulatory networks (FODGRNs) via Razumikhin approach. First of all, the existence of FODGRNs are established by using homeomorphism theory, 2-norm based on the algebraic method and Cauchy Schwartz inequality. The uniqueness of this work among the existing stability results are, the global Mittag-Leffler stability of FODGRNs is explored based on the fractional-order Lyapunov Razumikhin approach. In the meanwhile, two different controllers such as linear feedback and adaptive feedback control, are designed respectively. With the assistance of fractional Razumikhin theorem and our designed controllers, we have established the global Mittag-Leffler synchronization and adaptive synchronization for addressing master-slave systems. Finally, three numerical cases are given to justify the applicability of our stability and synchronization results. </abstract>

Delay-Dependent and Order-Dependent Guaranteed Cost Control for Uncertain Fractional-Order Delayed Linear Systems

This paper addresses the guaranteed cost control problem of a class of uncertain fractional-order (FO) delayed linear systems with norm-bounded time-varying parametric uncertainty. The study is focused on the design of state feedback controllers with delay such that the resulting closed-loop system is asymptotically stable and an adequate level of performance is also guaranteed. Stemming from the linear matrix inequality (LMI) approach and the FO Razumikhin theorem, a delay- and order-dependent design method is proposed with guaranteed closed-loop stability and cost for admissible uncertainties. Examples illustrate the effectiveness of the proposed method.

Input Delay Tolerance of Nonlinear Systems Under Smooth Feedback: A Semiglobal Control Framework

This article studies the problem of when a globally asymptotically stabilizable nonlinear system by smooth feedback is tolerable with respect to input delay. We illustrate, by means of theoretic and practical examples, some fundamental limitations including: 1) A globally exponentially stabilizable (GES) nonlinear system may not guarantee global asymptotic stabilizability (GAS) of the nonlinear system even with an arbitrarily small input delay; 2) GAS cannot even ensure semiglobal asymptotic stabilizability of the nonlinear system with a small input delay. To overcome these obstacles, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">semiglobal input delay tolerance</i> (SGIDT) and present a semiglobal control framework for the asymptotic analysis/synthesis of input delay tolerance of multi-input–multi-output (MIMO) nonlinear systems under smooth feedback. With the aid of Razumikhin theorem and the converse Lyapunov theorem on global asymptotic local exponential stability (GALES), it is proved that in the case of state feedback, GALES does ensure the SGIDT of a MIMO nonlinear system. In the case of output feedback, it is further proved that GALES and uniform observability imply the SGIDT of the nonlinear system.

Control of a quadrotor with network induced time delay

A backstepping controller augmented with a state predictor is proposed to control a quadrotor over a network subjected to both state and input time delay. The state predictor predicts the future values of the states by taking the measured delayed states as input. A backstepping control law is further designed based on these predicted states. It is shown with the aid of the Lyapunov–Razumikhin theorem that the error dynamics of the predictor is asymptotically stable. The cascade of state predictor and backstepping controller makes the tracking error dynamics of the quadrotor asymptotically stable. Simulation results are presented to validate the proposed approach.

Stability analysis for switched stochastic delayed systems under asynchronous switching: A relaxed switching signal

SummaryThis article deals with the stochastic input‐to‐state stability (SISS) for a class of switched stochastic systems under asynchronous switching. Asynchronous switching refers to that the switching of the candidate controllers does not coincide with the switching of system modes. In addition, the time delays appear in the state. Unlike the existing works, we not only allow the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be increasing during the matched time interval, but also in the unmatched time interval. Meanwhile, by using the switching signals which prevail over the traditional average dwell time scheme and the Razumikhin theorem, the SISS property is obtained. Especially our results improve some existing ones regarding asynchronous switching in the literature. Two examples are used to demonstrate the applicability of the results.

Non‐fragile control for a class of fractional‐order uncertain linear systems with time‐delay

This study addresses the non-fragile controller design of a class of fractional-order (FO) linear delayed systems assumed to have structured uncertainties in both the plant and the controller. By using the linear matrix inequality method, singular value decomposition and FO Razumikhin theorem, the delay-dependent and order-dependent robust non-fragile state feedback controller and non-fragile observer-based robust state feedback controller are synthesised, respectively, based on whether the states are available or not. Three examples are presented to verify the effectiveness of the proposed schemes.

Tracking control and dynamic regulation of time-varying delay nonlinear systems with actuator saturation via multi-dimensional Taylor networks

For desirable tracking control and transient optimization of time-varying nonlinear delay systems with actuator saturation, a controller with a fixed structure, i.e., multi-dimensional Taylor networks (MTN) controller, is proposed here. Based on the extension of the Lie derivative, a series of assumptions for most nonlinear time-varying delays systems are identified. Based on the filter variable and Lyapunov–Razumikhin theorem, it is proven that the MTN controller can track the reference satisfactorily, which conforms to the assumption. Furthermore, the dynamic regulation to improve system transient response using the MTN controller is realized through polynomial approximation. Parameters of the controller are determined by means of the appropriate optimization method, and its general design is explored and detailed. As verified by the results from two numerical examples, the proposed approach is feasible and effective. Depending on the conclusion of the interconnected control, the application of MTN controller in MIMO time-varying delays nonlinear system with actuator saturation is discussed.

Global regulation for feedforward systems with both discrete delays and distributed delays

In this work, a global regulation controller with a time-varying gain is constructed for a class of feedforward nonlinear time-delay systems, and the time-varying function with the logarithmic form is for the first time proposed to copy with the unknown parameter. The uncertain nonlinearities are assumed to be bounded by unknown constants multiplied by states, and both discrete delays and distributed delays are allowed in both states and input of the involved systems. Based on the state transformation method and Lyapunov–Razumikhin theorem, a global regulation controller with the simple form is elaborately constructed for the given systems. Contrary to most existing control designs for feedforward nonlinear systems, the celebrated forwarding design and saturation design are not used here. Via two examples, we demonstrate that a class of the nonlinear time-delay systems which cannot be dealt by the existing methods are successfully regulated by our state feedback controllers.

Global synchronization of fractional-order quaternion-valued neural networks with leakage and discrete delays

In this paper, without transforming the quaternion-valued neural networks into two equivalent complex-valued systems or four equivalent real-valued systems, a novel class of fractional-order quaternion-valued neural networks with leakage and discrete delays is investigated. To this end, two novel inequalities are established with the aid of properties of quaternion and Caputo fractional derivative. By exploiting Lyapunov method, our established inequalities, fractional-order Razumikhin theorem and some analysis techniques, some criteria ensuring the global asymptotical synchronization and the global Mittag-Leffler synchronization of the considered networks are obtained through designing suitable controllers. Finally, two numerical examples are utilized to show the derived theoretical results.