Jun Guo Lu

Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order

This technical note firstly presents a sufficient and necessary condition for the robust asymptotical stability of fractional-order interval systems with the fractional order α satisfying 0 < α < 1. And then a sufficient condition for the robust asymptotical stabilization of such fractional-order interval systems is derived. All the results are obtained in terms of linear matrix inequalities. Finally, two illustrative examples are given to show that our results are effective for checking the robust stability and designing the robust stabilizing controller for fractional-order interval systems.

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This technical note presents necessary and sufficient conditions for the stability and stabilization of fractional-order interval systems. The results are obtained in terms of linear matrix inequalities. Two illustrative examples are given to show that our results are effective and less conservative for checking the robust stability and designing the stabilizing controller for fractional-order interval systems.

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In this paper, a stability theorem of nonlinear fractional-order differential equations is proven theoretically by using the Gronwall-Bellman lemma. According to this theorem, the linear state feedback controller is introduced for stabilizing a class of nonlinear fractional-order systems. And, a new criterion is derived for designing the controller gains for stabilization, in which control parameters can be selected via the pole placement technique of the linear fractional-order control theory. Finally, the theoretical results are further substantiated by simulation results of the fractional-order chaotic Lorenz system with desired design requirements.

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This brief presents a sufficient condition for the existence, uniqueness, and robust global exponential stability of the equilibrium solution for a class of interval reaction diffusion Hopfield neural networks with distributed delays and Dirichlet boundary conditions by constructing suitable Lyapunov functional and utilizing some inequality techniques. The result imposes constraint conditions on the boundary values of the network parameters. The result is also easy to verify and plays an important role in the design and application of globally exponentially stable neural circuits.

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Abstract This paper develops a general approach for constructing a response system to implement linear generalized synchronization (GS) with the drive continuous-time chaotic system. Some sufficient conditions of global asymptotic linear GS between the drive and response continuous-time chaotic systems are attained from rigorously modern control theory. Finally, we take Chua’s circuit as an example for illustration and verification.

Robust Stability and Stabilization of Fractional-Order Interval Systems: An LMI Approach

This paper considers the problems of robust stability and stabilization for a class of fractional-order linear time-invariant systems with convex polytopic uncertainties. The stability condition of the fractional-order linear time-invariant systems without uncertainties is extended by introducing a new matrix variable. The new extended stability condition is linear with respect to the new matrix variable and exhibits a kind of decoupling between the positive definite matrix and the system matrix. Based on the new extended stability condition, sufficient conditions for the above robust stability and stabilization problems are established in terms of linear matrix inequalities by using parameter-dependent positive definite matrices. Finally, numerical examples are provided to illustrate the proposed results.

Stability Analysis of a Class of Nonlinear Fractional-Order Systems

This brief investigates backstepping and adaptive-backstepping design for the control of a class of discrete-time chaotic systems with known or unknown parameters. The proposed method presents a systematic procedure for the control of a class of discrete-time chaotic systems. It can be used for the stabilization of discrete-time chaotic systems to a steady state as well as tracking of any desired trajectory. Moreover, dead-beat control and tracking, exact stabilization at a fixed point and tracking of any desired trajectory in finite time can be achieved. The chaotic Henon system with known or unknown parameters is taken as an example to illustrate the applicability and effectiveness of the backstepping design.

Robust Global Exponential Stability for Interval Reaction–Diffusion Hopfield Neural Networks With Distributed Delays

This paper investigates the robust stability and stabilization of fractional order linear systems with positive real uncertainty. Firstly, sufficient conditions for the asymptotical stability of such uncertain fractional order systems are presented. Secondly, the existence conditions and design methods of the state feedback controller, static output feedback controller and observer-based controller for asymptotically stabilizing such uncertain fractional order systems are derived. The results are obtained in terms of linear matrix inequalities. Finally, some numerical examples are given to validate the proposed theoretical results.

Linear generalized synchronization of continuous-time chaotic systems

Abstract This paper investigates the maximal perturbation bound problem for robust stabilizability of the fractional-order system with two-norm bounded perturbations or infinity-norm bounded perturbations. Firstly, a necessary condition and several sufficient conditions for robust stabilization are derived. Secondly, linear matrix inequality approaches for computing the maximal robust stabilizability perturbation bound of such perturbed fractional-order system with a linear state feedback controller, simultaneously obtaining the corresponding linear state feedback stabilizing controller are presented. With the help of the linear matrix inequality solvers, we can easily obtain the maximal robust stabilizability perturbation bound and the corresponding linear state feedback stabilizing controller. Finally, simulation examples are given to demonstrate the effectiveness of the proposed approaches.

Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties

This paper considers the robust stability bound problem of uncertain fractional-order systems. The system considered is subject either to a two-norm bounded uncertainty or to a infinity-norm bounded uncertainty. The robust stability bounds on the uncertainties are derived. The fact that these bounds are not exceeded guarantees that the asymptotical stability of the uncertain fractional-order systems is preserved when the nominal fractional-order systems are already asymptotically stable. Simulation examples are given to demonstrate the effectiveness of the proposed theoretical results.