Global Robust Stability Research Articles

Uncertain impulsive differential systems of fractional order: almost periodic solutions

ABSTRACTIn this paper, we investigate the existence and stability of almost periodic solutions of impulsive fractional-order differential systems with uncertain parameters. The impulses are realised at fixed moments of time. For the first time, we determine the impact of the uncertainties on the qualitative behaviour of such systems. The main criteria for the existence of almost periodic solutions are proved by employing the fractional Lyapunov method. The global perfect robust uniform-asymptotic stability of such solutions is also considered. We apply our results to uncertain impulsive neural network systems of fractional order.

Robust Global Adaptive Exponential Stabilization of Discrete-Time Systems With Application to Freeway Traffic Control

This paper is devoted to the development of adaptive control schemes for uncertain discrete-time systems, which guarantee robust global exponential convergence to the desired equilibrium point of the system. The proposed control scheme consists of a nominal feedback law, which achieves robust global exponential stability properties when the vector of the parameters is known, in conjunction with a nonlinear dead-beat observer. The proposed adaptive control scheme depends on certain parameter observability assumptions. The obtained results are applicable to highly nonlinear uncertain discrete-time systems with unknown constant parameters. The successful applicability of the obtained results to real control problems is demonstrated by the rigorous application of the proposed adaptive control scheme to uncertain freeway models. A provided example demonstrates the efficiency of the approach.

Cooperative Global Robust Output Regulation for Nonlinear Output Feedback Multiagent Systems Under Directed Switching Networks

In this paper, we study the cooperative global robust output regulation problem for nonlinear multiagent systems in output feedback form with any relative degree subject to directed switching networks. Through the augmentation by a distributed internal model and a distributed observer of the given system, we first convert the original problem into the global robust stabilization problem of the so-called extended augmented system, which is a multi-input-coupled switched system. Then, we establish a few technical lemmas to lay the foundation for constructing a switching control law and the corresponding multiple Lyapunov functions for the switched system. Then, we present the solvability conditions of our problem via the multiple Lyapunov functions and average dwell time method. The main result will be applied to the leader-following synchronization problem for a group of hyperchaotic Lorenz multiagent systems with a directed switching network.

Dynamic Analysis of Hybrid Impulsive Delayed Neural Networks With Uncertainties.

Neural networks (NNs) have emerged as a powerful illustrative diagram for the brain. Unveiling the mechanism of neural-dynamic evolution is one of the crucial steps toward understanding how the brain works and evolves. Inspired by the universal existence of impulses in many real systems, this paper formulates a type of hybrid NNs (HNNs) with impulses, time delays, and interval uncertainties, and studies its global dynamic evolution by a robust interval analysis. The HNNs incorporate both continuous-time implementation and impulsive jump in mutual activations, where time delays and interval uncertainties are represented simultaneously. By constructing a Banach contraction mapping, the existence and uniqueness of the equilibrium of the HNN model are proved and analyzed in detail. Based on nonsmooth Lyapunov functions and delayed impulsive differential equations, new criteria are derived for ensuring the global robust exponential stability of the HNNs. Convergence analysis together with illustrative examples show the effectiveness of the theoretical results.

Robust adaptive passification and global stabilisation for uncertain switched nonlinear systems

In this paper, we focus on the problems of feedback passification and global stabilisation for a class of uncertain switched non-linear systems with parameter uncertainties and structural uncertainties. A state-dependent switching law and adaptive feedback controllers are designed to render the resulting closed-loop systems passive. The corresponding passivity-based adaptive stabilisation problems for these systems are solved. Finally, a numerical example is presented to illustrate the effectiveness of our method.

Event-Triggered Global Robust Output Regulation for a Class of Nonlinear Systems

This note studies the global robust output regulation problem for a class of nonlinear systems by an output-based event-triggered control law. First, we convert the problem into the event-triggered global robust stabilization problem of a well-defined augmented system based on the internal model approach. Then, we design an output-based event-triggered control law together with an output-based event-triggered mechanism to solve the stabilization problem, which in turn leads to the solution to the original problem by an output-based event-triggered control law. Finally, we illustrate our approach by an example.

Robust control of a family of uncertain nonminimum‐phase systems via continuous‐time and sampled‐data output feedback

SummaryThe problem of global robust stabilization is studied by both continuous‐time and sampled‐data output feedback for a family of nonminimum‐phase nonlinear systems with uncertainty. The uncertain nonlinear system considered in this paper has an interconnect structure consisting of a driving system and a possibly unstable zero dynamics with uncertainty, ie, the uncertain driven system. Under a linear growth condition on the uncertain zero dynamics and a Lipschitz condition on the driving system, we show that it is possible to globally robustly stabilize the family of uncertain nonminimum‐phase systems by a single continuous‐time or a sampled‐data output feedback controller. The sampled‐data output feedback controller is designed by using the emulated versions of a continuous‐time observer and a state feedback controller, ie, by holding the input/output signals constant over each sampling interval. The design of either continuous‐time or sampled‐data output compensator uses only the information of the nominal system of the uncertain controlled plant. In the case of sampled‐data control, global robust stability of the hybrid closed‐loop system with uncertainty is established by means of a feedback domination method together with the robustness of the nominal closed‐loop system if the sampling time is small enough.

Robust stability analysis of impulsive complex-valued neural networks with time delays and parameter uncertainties

The present study considers the robust stability for impulsive complex-valued neural networks (CVNNs) with discrete time delays. By applying the homeomorphic mapping theorem and some inequalities in a complex domain, some sufficient conditions are obtained to prove the existence and uniqueness of the equilibrium for the CVNNs. By constructing appropriate Lyapunov-Krasovskii functionals and employing the complex-valued matrix inequality skills, the study finds the conditions to guarantee its global robust stability. A numerical simulation illustrates the correctness of the proposed theoretical results.

Global robust stability and stabilization of Boolean network with disturbances

Based on semi-tensor product of matrices, global robust stability and stabilization of Boolean network (BN) with disturbances are investigated. Firstly, the definition of global robust stability for BN with disturbances is presented. By using the algebraic state space representation of disturbed BN, some necessary and sufficient criteria are obtained to ensure the global robust stability with respect to (w.r.t.) a fixed point or w.r.t. a limit cycle. In contrast, if a given disturbed BN is not globally robust stable w.r.t. a fixed point or w.r.t. a limit cycle, system can achieve stability by a matrix transformation technique. However, it is a difficult task to find such a suitable matrix transformation. In this paper, a pinning state feedback control design is proposed to find a suitable transformation. The obtained results are well illustrated by numerical example.

Global exponential stabilisation of acyclic traffic networks

ABSTRACTThis work is devoted to the construction of explicit feedback control laws for the robust global exponential stabilisation of general uncertain discrete-time acyclic networks. We consider discrete-time uncertain network models, which satisfy very weak assumptions. The construction of the controllers and the rigorous proof of the robust global exponential stability for the closed-loop system are based on recently proposed vector-Lyapunov function criteria, as well as the fact that the network is acyclic. It is shown, in this study, that the latter requirement is necessary for the existence of a robust global exponential stabiliser of the desired uncongested equilibrium point of the network. Our main focus is on traffic networks and all assumptions are related to features appearing in traffic models. An illustrative example demonstrates the applicability of the obtained results to realistic traffic flow networks.

Further Improvement on Delay-Dependent Global Robust Exponential Stability for Delayed Cellular Neural Networks with Time-Varying Delays

This paper is concerned with global robust exponential stability for a class of delayed cellular neural networks with time-varying delays. Some new sufficient conditions are presented for the uniqueness of equilibrium point and the global stability of cellular neural networks with time varying delay by constructing Lyapunov functional and using linear matrix inequality and the integral inequality approach. Numerical examples are illustrated to show the effectiveness of the proposed method. From the simulation results, significant improvement over the recent results can be observed.

Lur\u2019e-Postnikov Lyapunov function approach to global robust Mittag-Leffler stability of fractional-order neural networks

In this paper, the global robust Mittag-Leffler stability analysis is preformed for fractional-order neural networks (FNNs) with parameter uncertainties. A new inequality with respect to the Caputo derivative of integer-order integral function with the variable upper limit is developed. By means of the properties of Brouwer degree and the matrix inequality analysis technique, the proof of the existence and uniqueness of equilibrium point is given. By using integer-order integral with the variable upper limit, Lur’e-Postnikov type Lyapunov functional candidate is constructed to address the global robust Mittag-Leffler stability condition in terms of linear matrix inequalities (LMIs). Finally, two examples are provided to illustrate the validity of the theoretical results.

Global and Robust Stability in a General Price and Assortment Competition Model

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the regular extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models such as the MNL model or any of its variants. We show that a special equilibrium in this model has global robust stability. This means that, from any starting point, the market converges to this equilibrium when firms use a particular response mapping to dynamically adjust their own prices in response to their competitors' prices. The mapping involves each firm optimizing its own prices over a limited subset of possible prices and requires each firm to only know the demand function and cost structure for its own products (but not for other firms' products).

Global asymptotic and robust stability of inertial neural networks with proportional delays

This paper is concerned with the global asymptotic stability and global robust stability of inertial neural networks with proportional delays. First, by using linear matrix inequality and constructing appropriate Lyapunov functional, several sufficient conditions are obtained for the global asymptotic stability of inertial neural networks with proportional delays. Furthermore, the problem of global robust stability of the network under the assumptions that the network parameters are uncertain and bounded is studied. Finally, some simulation examples are presented to demonstrate the effectiveness of the proposed method.

Nonlinear Feedback Control for General Acyclic Traffic Networks

This work is devoted to the construction of explicit feedback control laws for the robust global exponential stabilization of general uncertain discrete-time acyclic traffic networks. We consider discrete-time uncertain network models which satisfy very weak assumptions. The construction of the controllers is based on recently proposed vector-Lyapunov function criteria, as well as the fact that the network is acyclic. The latter requirement is necessary for the existence of a robust, global, exponential stabilizer of the desired uncongested equilibrium point of the network. An illustrative example demonstrates the applicability of the obtained results to realistic traffic flow networks.

Global Robust Output Regulation for a Class of Nonlinear Systems via Event-Triggered Control

In this paper, we study the event-triggered global robust output regulation problem for a class of nonlinear systems in output feedback form with unity relative degree. We first convert the problem into the event-triggered global robust stabilization problem of a well defined augmented system by the internal model approach. Then, we solve the stabilization problem by an output-based event-triggered control law together with an output-based event-triggered mechanism. Finally, the theoretical result is applied to the Lorenz systems.

A Nonsmooth Hybrid Invariance Principle Applied to Robust Event-Triggered Design

We first propose a nonsmooth hybrid invariance principle with relaxed conditions stemming from the fact that flowing solutions evolve only in the tangent cone, and complete jumping solutions cannot jump outside the jump and flow sets. We then show an application consisting in the design of event-triggered rules to stabilize a class of uncertain linear control systems. The event-triggering rule depends only on local information, that is it uses only the output signals available to the controller. The approach proposed combines hybrid and sampled-data tools. The proposed design conditions are formulated in terms of linear matrix inequalities (LMIs) ensuring global robust asymptotic stability of the closed-loop system. A tunable parameter is also available to guarantee an adjustable dwell-time property of the solutions. The effectiveness of the approach is evaluated through an example borrowed from the literature.

Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties

This paper addresses the problem of robust stability for quaternion-valued neural networks (QVNNs) with leakage delay, discrete delay and parameter uncertainties. Based on Homeomorphic mapping theorem and Lyapunov theorem, via modulus inequality technique of quaternions, some sufficient conditions on the existence, uniqueness, and global robust stability of the equilibrium point are derived for the delayed QVNNs with parameter uncertainties. Furthermore, as direct applications of these results, several sufficient conditions are obtained for checking the global robust stability of QVNNs without leakage delay as well as complex-valued neural networks (CVNNs) with both leakage and discrete delays. Finally, two numerical examples are provided to substantiate the effectiveness of the proposed results.

A Tool for the Global Stabilization of Stochastic Nonlinear Systems

This technical note investigates the robust global stabilization problem for nonlinear systems subject to dynamic and stochastic disturbance, from the perspective of stochastic input-to-state stability. A new stochastic gain assignment method is developed. This method can be combined with the nonlinear small-gain theorem to design partial-state feedback controllers for stochastic nonlinear systems. A numerical example is given to illustrate the effectiveness of the proposed method.

Global robust stabilization of nonlinear cascaded systems with integral ISS dynamic uncertainties

The authors consider a global robust asymptotic stabilization problem (GRS) for cascaded systems having dynamic uncertainties that are not necessarily input-to-state stable (ISS). Specifically, a recursive Lyapunov design approach is developed by induction on the system relative degree, providing a smoothly globally stabilizing controller. It is applicable to nonlinear cascaded systems having multiple distinct iISS dynamic uncertainties. The proposed design is constructive and leads to an iISS-Lyapunov characterization in a superposition form for the closed-loop system.