Robust asymptotic stability of interval fractional-order nonlinear systems with time-delay

This paper studies the global asymptotic stability of a class of interval fractional-order (FO) nonlinear systems with time-delay. First, a new lemma for the Caputo fractional derivative is presented. It extends the FO Lyapunov direct method allowing the stability analysis and synthesis of FO nonlinear systems with time-delay. Second, by employing FO Razumikhin theorem, a new delay-independent stability criterion, in the form of linear matrix inequality is established for ensuring that a system is globally asymptotically stable. It is shown that the new criterion is simple, easy to use and valid for the FO or integer-order interval neural networks with time-delay. Finally, the feasibility and effectiveness of the proposed scheme are tested with a numerical example.     

A neutral system approach to stability of singular time-delay systems

This paper is concerned with the problem of delay-dependent stability for a class of singular time-delay systems. By representing the singular system as a neutral form, using an augmented Lyapunov–Krasovskii functional and the Wirtinger-based integral inequality method, we obtain a new stability criterion in terms of a linear matrix inequality (LMI). The criterion is applicable for the stability test of both singular time-delay systems and neutral systems with constant time delays. Illustrative examples show the effectiveness and merits of the method.     

New finite-time stability for fractional-order time-varying time-delay linear systems: A Lyapunov approach

The primary goal of this paper is to examine the finite-time stability and finite-time contractive stability of the linear systems in fractional domain with time-varying delays. We develop some sufficient criteria for finite-time contractive stability and finite-time stability utilizing fractional-order Lyapunov-Razumikhin technique. To validate the proposed conditions, two different types of dynamical systems are taken into account, one is general time-delay fractional-order system and another one is fractional-order linear time-varying time-delay system, furthermore the efficacy of the stability conditions is demonstrated numerically.     

Stochastic finite-time stabilization for discrete-time positive Markov jump time-delay systems

In this paper, the problems of stochastic finite-time stability and stabilization of discrete-time positive Markov jump systems are investigated. To deal with time-varying delays and switching transition probability (STP), stochastic finite-time stability conditions are established under mode-dependent average dwell time (MDADT) switching signal by developing a stochastic copositive Lyapunov-Krasovskii functional approach. Then a dual-mode dependent output feedback controller is designed, thus stochastic finite-time stabilization is achieved based on linear programming technique. Finally, two examples are given to show the effectiveness of our results.     

On the excitability of a class of positive continuous time-delay systems

This article is on the excitability of positive linear time-invariant systems subject to internal point delays. It is proved that excitability independent of delay is guaranteed if an auxiliary delay-free system is excitable. Necessary and sufficient conditions for excitability and transparency independent of the delay size are formulated in terms of the parameterization of the dynamics and control matrices. Some particular results are also given for the properties being dependent on the size of the point delay and for any possible finite values of the delay. The same formulation is given in parallel in terms of strict positivity of a matrix of an associate system obtained from the influence graph of the original system. The excitability and transparency properties are both testable through simple algebraic tests involving a moderate computational effort that is directly related to the system's order.     

Event-triggered controller design for positive T-S fuzzy systems with random time-delay

This paper is concerned with the design of event-triggered controller for positive Takagi-Sugeno (T-S) fuzzy systems with a random time-delay. The random time-delay is described as a Markov process. A controller switched at different event-triggered instant is proposed. By constructing a new event-triggered instant-dependent linear co-positive Lyapunov function, the design criteria of event-triggered controller is derived to ensure the positivity and stability of the closed-loop system. These criteria can be solved by linear programming (LP) technique. A positive lower bound on the inter-execution time is ensured, which means that there is Zeno-free phenomenon. Finally, the simulation has demonstrated the effectiveness and merit of the proposed results.     

一类非线性不确定多时滞中立型系统的鲁棒自适应控制（英文）

针对一类不确定多时滞中立型非线性系统，在其非线性不确定项的范数有界，但其上界未知的情况下，论证了自适应鲁棒控制器存在的条件，并给出了能适应未知参数变化且使得最终闭环系统一致最终有界的鲁棒控制律的设计方法。最后，具体算例的仿真结果说明了此法的有效性。     

Comparison principle for positive time-delay systems: An extension and its application

This paper proposes an extended comparison principle for continuous-time linear positive time-delay systems. Unlike the existing comparison principle, which uses a constant initial function for the comparison system, instead, we propose a time-varying initial function in order to derive a more general solution comparison. Based on this extended comparison principle, we develop a novel computational method, which exploits more effectively the information of the initial value function, to derive tighter exponential estimates for the state vector of positive time-delay systems. The effectiveness of our developed method is illustrated through two numerical examples.     

Finite-time stability of positive switched time-delay systems based on linear time-varying copositive Lyapunov functional

In this paper, we deal with the finite-time stability of positive switched linear time-delay systems. By constructing a class of linear time-varying copositive Lyapunov functionals, we present new explicit criteria in terms of solvable linear inequalities for the finite-time stability of positive switched linear time-delay systems under arbitrary switching and average dwell-time switching. As an important application, we apply the method to finite-time stability of linear time-varying systems with time delay.     

Memory-based controller design for neutral time-delay systems with input saturations: A novel delay-dependent polytopic approach

In this paper, a new memory-based control problem is addressed for neutral systems with time-varying delay, input saturations and energy bounded disturbances. Attention is focused on the design of a memory-based state feedback controller such that the closed-loop system achieves the desirable performance indices including the boundedness of the state trajectories, the H_∞ disturbance rejection/attenuation level as well as the asymptotic stability. By using the combination of a novel delay-dependent polytopic approach, augmented Lyapunov–Krasovskii functionals and some integral inequalities, delay-dependent sufficient conditions are first proposed in terms of linear matrix inequalities. Then, three convex optimization problems are formulated whose aims are to, respectively, maximize the disturbance tolerance level, minimize the disturbance attenuation level and maximize the initial condition set. Finally, simulation examples demonstrate the effectiveness and benefits of the obtained results.     

Finite-time fractional-order adaptive intelligent backstepping sliding mode control of uncertain fractional-order chaotic systems

This paper precedes chaos control of fractional-order chaotic systems in presence of uncertainty and external disturbances. Based on some basic properties on fractional calculus and the stability theorems, we present a hybrid adaptive intelligent backstepping-sliding mode controller (FAIBSMC) for the finite-time control of such systems. The FAIBSMC is proposed based on the concept of active control technique. The asymptotic stability of the controller is shown based on Lyapunov theorem and the finite time reaching to the sliding surfaces is also proved. Illustrative and comparative examples and simulation results are given to confirm the effectiveness of the proposed procedure, which consent well with the analytical results.     

New fractional-order integral inequalities: Application to fractional-order systems with time-varying delay

In this paper, the problem of delay-dependent stability analysis of fractional-order systems with time-varying delay is investigated. First, a class of novel fractional-order integral inequalities for quadratic functions by constructing appropriate auxiliary functions is proposed, which has been proven to be useful in analyzing fractional-order systems with time-varying delay. Based on these proposed inequalities, the Lyapunov–Krasovskii functions are designed to deal with the time-varying delay terms, reducing the conservatism of the stability criteria. Furthermore, delay-dependent criteria are derived to achieve asymptotic stability of fractional-order systems with time-varying delay. Finally, two examples are provided to illustrate the effectiveness and feasibility of the proposed stability criteria.     

Sampled-data control for time-delay systems

The sampled-data systems are hybrid ones involving continuous time and discrete time signals, which makes the traditional analysis and synthesis methodologies of time-delay systems unable to be directly used in the cases of hybrid systems with time-delay. The primary disadvantages of current design techniques of sampled-data control are their inabilities to deal effectively with time-delay and the model uncertainty. In this paper, we generalized the analysis methodology of time-delay systems to that of the hybrid systems with time-delay and uncertainty, which developed a design procedure of sampled-data control for time-delay systems. Asymptotic stability of the time-delay hybrid systems was developed. The time-delay dependent robust sampled-data control for the time-varying delay of an uncertain linear system was then discussed. The results were described as linear matrix inequalities, which can be solved using newly released LMITool.     

Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay

In this paper, a sufficient condition is established for the controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay in Banach spaces. The approach used is analytic semigroups and fractional powers of closed operators and nonlinear alternative of Leray–Schauder type for multivalued maps due to D. O'Regan.     

Containment control of heterogeneous fractional-order multi-agent systems

Fractional-order calculus has been studied deeply because many networked systems can only be described with fractional-order dynamics in complex environments. When different agents of networked systems show diverse individual features, fractional-order dynamics with heterogeneous characters will be used to illustrate the multi-agent systems (MAS). Based on the distinguishing behaviors of agents, a compounded fractional-order multi-agent systems(FOMAS) is presented with diverse dynamical equations. Suppose multiple leader agents existing in FOMAS, containment consensus control of FOMAS with directed weighted topologies is studied. By applying frequency domain analysis theory of the fractional-order operator, an upper bound of delays is obtained to ensure containment controls of heterogenous FOMAS with communication delays. The consensus results of delayed fractional-order dynamics in this paper can be expanded to the integer-order models. Finally, the results are verified by simulation examples.     

Iterative learning control for fractional-order multi-agent systems

In this paper, we apply iterative learning control to both linear and nonlinear fractional-order multi-agent systems to solve consensus tacking problem. Both fixed and iteration-varying communicating graphs are addressed in this paper. For linear systems, a PD^α-type update law with initial state learning mechanism is introduced by virtue of the memory property of fractional-order derivative. For nonlinear systems, a D^α-type update law with forgetting factor and initial state learning is designed. Sufficient conditions for both linear and nonlinear systems are established to guarantee all agents achieving the asymptotic output consensus. Simulation examples are provided to verify the proposed schemes.     

Chaotic synchronization between different fractional-order chaotic systems

Based on the idea of tracking control and stability theory of fractional-order systems, a novel synchronization approach for fractional order chaotic systems is proposed. We prove that the synchronization between drive system and response system with different fractional order q can be achieved, and the synchronization between different fractional-order chaotic systems with different fractional order q can be achieved. Two examples are used to illustrate the effectiveness of the proposed synchronization method. Numerical simulations coincide with the theoretical analysis.     

On the robust stability of commensurate fractional-order systems

Based on a recent generalised version of the Mikhailov stability criterion, this paper presents a Kharitonov–like test for a class of linear fractional–order systems described by transfer functions whose coefficients are subject to interval uncertainties. To this purpose, first the transfer function is associated with an integer-order complex polynomial function of the generalised frequency (i.e. the current coordinate along the boundary radii of the instability sector) whose coefficients are uncertain. Then the geometrical form of the value set of this characteristic polynomial is determined from the direct examination of its monomial terms. To show how the test operates, it is finally applied to two fractional–order transfer functions whose coefficients belong to given intervals.     

Lag projective synchronization of fractional-order delayed chaotic systems

This paper considers the lag projective synchronization of fractional-order delayed chaotic systems. The lag projective synchronization is achieved through the use of comparison principle of linear fractional equation at the presence of time delay. Some sufficient conditions are obtained via a suitable controller. The results show that the slave system can synchronize the past state of the driver up to a scaling factor. Finally, two different structural fractional order delayed chaotic systems are considered in order to examine the effectiveness of the lag projective synchronization. Feasibility of the proposed method is validated through numerical simulations.