Problem Of Global Stabilization Research Articles

Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay

Abstract In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2\pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2\pi )$-norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.

Global Prescribed-Time Stabilization for a Class of Uncertain Feedforward Nonlinear Systems

The problem of global prescribed-time stabilization (PTS) is addressed in this brief for a class of uncertain feedforward nonlinear systems. To achieve this task, a new time-scale transformation (TST) is first introduced, which has the good property that it changes the original global PTS into the global asymptotic stabilization of the transformed one but does not changes the system structures. Then, under the new framework, both state and output feedback controllers are developed to achieve the requiring performances with the help of the non-recursive technique. At last, simulation results of a nonlinear liquid level control resonant circuit (NLLCRC) system are provided to show the validity of the given approach.

Predictor-Based Global Sampled-Data Output Feedback Stabilization for Nonlinear Uncertain Systems Subject to Delayed Output

In this article, we address the problem of global sampled-data output feedback stabilization for a class of nonlinear uncertain systems with delayed output using the continuous-discrete method. Thanks to the prediction technique and feedback domination method, a novel coupled design method of predictor-based continuous-discrete observer and linear controller is proposed when only delayed sampled-data output is accessible. The proposed predictor-based observer can effectively estimate the unknown state by compensating the influences of sampling and output delay. The main advantage of the proposed control method is that the full state information and accurate model nonlinearities do not need to be known any more. The global exponential stability of the overall hybrid control system can be ensured when there hold some sufficient conditions with respect to the maximum allowable sampling period and output delay.

A Linear Time-Varying Inequality Approach for Prescribed Time Stability and Stabilization.

This article studies the problem of finite-time, fixed-time, and prescribed-time stability analysis and stabilization. First, a linear time-varying (LTV) inequality-based approach is introduced for prescribed-time stability analysis. Then, it is shown that the existing nonlinear Lyapunov inequalities-based finite- and fixed-time stability criteria can be recast into the unified framework of the LTV inequality-based approach for prescribed-time stability. Finally, the unified LTV inequality-based approach is used to solve the global prescribed-time stabilization problem of the attitude control system of a rigid spacecraft with disturbance, and a bounded nonlinear time-varying controller is proposed via back stepping. Numerical simulations are presented to show the effectiveness of the proposed methods.

New Results on Fixed-Time Stabilization of Switched Uncertain Nonlinear Systems in p-Normal Form

In this article, the global fixed-time stabilization problem is addressed for a class of switched uncertain nonlinear systems in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -normal form. Different from the existing results, the bounds of the unknown system parameters, including dead-zone parameters and control coefficients, are not required to be known. By combining the adding a power integrator technique with the common Lyapunov function method, a novel adaptive controller is proposed and a dynamic controller parameter is introduced to cope with the unknown system parameters. Besides, an effective common regulate rule is designed based on the improved fixed-time stability framework. Furthermore, it is shown that the controller parameter can be regulated online by the switching mechanism to compensate the unknown system parameters and the system states can converge to zero in fixed time under arbitrary switchings. The effectiveness of the proposed method is verified by a simulation example.

A new design method to global asymptotic stabilization of strict-feedforward stochastic nonlinear time delay systems

The stabilization problems of stochastic nonlinear systems remain challenging because many new obstacles will be encountered and need to be cautiously handled. This paper discusses the global asymptotic stabilization problem of stochastic nonlinear systems which possess strict-feedforward structure and multiple time delays. In the first place, we are trying to transform the considered system into an equivalent one by constructing the novel parameter-dependent state feedback controller and the suitable coordinate transformation. And then, through using the stochastic time delay system stability criterion and introducing the proper Lyapunov–Krasovskii functional, the global asymptotic stability of stochastic closed-loop system is proved. Finally, simulation results show the effectiveness of the parameter-dependent state feedback controller.

Generalized stability criteria of switched linear systems with time-varying delays

In this paper, the global exponential stability problem of the switched linear system with time-varying delays is considered. We establish a new stability criterion for the switched linear time-delay system under the Φ-dependent average dwell time technique. By adopting different grouping methods for subsystems, the corresponding stability criteria based on average dwell time and mode-dependent dwell time techniques can be unified to the new results. Finally, an example is given to illustrate the effectiveness of the obtained results.

On robust approximate feedback linearisation with an event-triggered controller

ABSTRACT In this paper, we consider a problem of global stabilisation of approximate feedback linearised systems with perturbed nonlinearity by an event-triggered control. We categorise the perturbed nonlinearity into three cases and obtain a characterisation function for each case. Then we propose an event-triggered controller with a gain-scaling factor along with an event-triggering condition which is developed based on the characterisation function. Utilising the system analysis results, we suggest a control parameter selection rule depending on the priority on either system output regulation speed or size of the interexecution times. Via practical examples, we demonstrate the validity of our control approach.

Local and Global Stabilization of Switched Linear Systems With Actuator Saturation

Utilizing the min-composite Lyapunov function and the min-projection switching strategy, we solve the local and global stabilization problems for switched linear systems by saturated, not necessarily stabilizing individually, linear feedback laws. Sufficient conditions are derived in the form of matrix inequalities under which the closed-loop system is locally asymptotically stable with an estimate of the domain of attraction or is globally asymptotically stable. The derived conditions are shown to be less conservative than the existing ones and, when used to guide the design of feedback laws, enable local stabilization with a larger estimate of the domain of attraction or global stabilization for a larger class of systems. Numerical examples illustrate the theoretical conclusions.

Memoryless parameter-dependent control strategy of stochastic strict-feedback time delay systems

For a class of stochastic strict-feedback nonlinear systems subject to different time delay states, this paper mainly concerns the problem of global asymptotic stabilization. Two new control strategies that the memoryless parameter-dependent state feedback control and the memoryless parameter-dependent output feedback control are taken into consideration, respectively. By skillfully constructing the Lyapunov-Krasovskii (L-K) functional, taking the proper determined parameter and employing the stochastic nonlinear time delay system (SNTDS) stability theory, the global asymptotic stability of the stochastic closed-loop system can be achieved. The proposed output feedback control scheme is finally utilized for the control design of the one-link manipulator system and two-stage chemical reactor system, which can verify the availability of the control approach.

Global Practical Stabilization of the Double Integrator System In the Presence of Output Saturation and a Bounded Disturbance

In this paper we consider the problem of global practical stabilization of the double integrator system in the presence of output saturation and a bounded disturbance additive to the output. Because of the disturbance in the output, not all system states can be driven into an arbitrarily small set, as in the usual practical stabilization problem, the output can only be driven into a bounded set whose size is dependent on the bound of the disturbance. It is shown that there exist linear feedback laws that globally practically stabilize the double integrator system in the presence of output saturation and the bounded output disturbance, and we propose linear low gain based dynamic output feedback laws to solve the global practical stabilization problem. Simulation results illustrate the effectiveness of the proposed control design.

A New Finite-Time Stabilizing Design for a Class of High-Order Uncertain Nonlinear Systems and Its Application in Maglev Systems

In this article, the problem of global fixed-time stabilization for a class of high-order uncertain nonlinear systems has been investigated. Quite different from traditional methods, a novel finite-time control scheme is presented for the first time based on a serial of exponential functions and fractional power integration with nested sign functions, which can guarantee that the convergent time of the states of the closed-loop systems is finite and independent of any initial conditions by the simple choice of design parameters. The remarkable contribution of this article lies in the fact that it provides an alternative to manipulate the possibility of initial states being far from the origin. As a practical application, the finite-time stabilizing design of maglev systems is provided to demonstrate the effectiveness and the superiority of the proposed strategy.

Impulsive control for stationary oscillation of nonlinear delay systems and applications

&lt;abstract&gt;&lt;p&gt;In this paper, the problem of existence-uniqueness and global exponential stability of periodic solution (i.e., stationary oscillation) for a class of nonlinear delay systems with impulses was studied. Some new sufficient conditions ensuring the existence of stationary oscillation for the addressed equations were derived by using the inequality technique that has been reported in recent publications. Our proposed method, which is different with the existing results in the literature, shows that nonlinear delay systems may admit a stationary oscillation using proper impulsive control strategies even if it was originally unstable or divergent. As an application, we considered the single species logarithmic population model and established a new criterion to guarantee the existence of positive stationary oscillation. Some numerical examples and their computer simulations were also given at the end of this paper to show the effectiveness of our development control method.&lt;/p&gt;&lt;/abstract&gt;

Global exponential stability conditions for quaternion-valued neural networks with leakage, transmission and distribution delays

&lt;abstract&gt;&lt;p&gt;This paper studies the global exponential stability problem of quaternion-valued neural networks (QVNNs) with leakage, transmission, and distribution delays. To address this issue, a direct method based on system solutions is proposed to ensure the global exponential stability of the considered network models. In addition, this method does not need to construct any Lyapunov-Krasovskii functional, which greatly reduces the amount of computation. Finally, a numerical example is given to demonstrate the effectiveness of the proposed results.&lt;/p&gt;&lt;/abstract&gt;

Direct quaternion method-based stability criteria for quaternion-valued Takagi-Sugeno fuzzy BAM delayed neural networks using quaternion-valued Wirtinger-based integral inequality

&lt;abstract&gt;&lt;p&gt;This paper investigates the global asymptotic stability problem for a class of quaternion-valued Takagi-Sugeno fuzzy BAM neural networks with time-varying delays. By applying Takagi-Sugeno fuzzy models, we first consider a general form of quaternion-valued Takagi-Sugeno fuzzy BAM neural networks with time-varying delays. Then, we apply the Cauchy-Schwarz algorithm and homeomorphism principle to obtain sufficient conditions for the existence and uniqueness of the equilibrium point. By utilizing suitable Lyapunov-Krasovskii functionals and newly developed quaternion-valued Wirtinger-based integral inequality, some sufficient criteria are obtained to guarantee the global asymptotic stability of the considered networks. Further, the results of this paper are presented in the form of quaternion-valued linear matrix inequalities, which can be solved using the MATLAB YALMIP toolbox. Two numerical examples are presented with their simulations to demonstrate the validity of the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

System decomposition method-based global stability criteria for T-S fuzzy Clifford-valued delayed neural networks with impulses and leakage term

&lt;abstract&gt;&lt;p&gt;This paper investigates the global asymptotic stability problem for a class of Takagi-Sugeno fuzzy Clifford-valued delayed neural networks with impulsive effects and leakage delays using the system decomposition method. By applying Takagi-Sugeno fuzzy theory, we first consider a general form of Takagi-Sugeno fuzzy Clifford-valued delayed neural networks. Then, we decompose the considered $ n $-dimensional Clifford-valued systems into $ 2^mn $-dimensional real-valued systems in order to avoid the inconvenience caused by the non-commutativity of the multiplication of Clifford numbers. By using Lyapunov-Krasovskii functionals and integral inequalities, we derive new sufficient criteria to guarantee the global asymptotic stability for the considered neural networks. Further, the results of this paper are presented in terms of real-valued linear matrix inequalities, which can be directly solved using the MATLAB LMI toolbox. Finally, a numerical example is provided with their simulations to demonstrate the validity of the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Global Small Solutions to a Special $$2\frac{1}{2}$$-D Compressible Viscous Non-resistive MHD System

This paper solves the global well-posedness and stability problem on a special \(2\frac{1}{2}\)-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in \(L^p\)-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced dissipation and stabilizing effect resulting from the background magnetic field, a phenomenon observed in physical experiments. In addition, we obtain the solution’s optimal decay rate when the initial data is further assumed to be in a Besov space of negative index.

Stability Analysis of Impulsive Switched Nonlinear Systems With Double State-Dependent Delays

This brief addresses the problem of global asymptotic stability for impulsive switched nonlinear systems (ISNSs) with double state-dependent delays (DSDDs), where the state-dependent delays exist in both subsystem functions and impulsive functions. The innovations of our results are that: 1) Due to the ubiquitous asynchronous phenomenon between the switches and impulses in many practical systems, we investigate the asynchronous phenomenon for ISNSs with DSDDs. 2) Meanwhile, the stabilizing and destabilizing effects of state-dependent delays impulses are fully considered. 3) By utilizing multiple Lyapunov functions, the switching strategy and the impulsive strategy, based on the asynchronous phenomenon, some stability criteria on global asymptotic stability are presented for ISNSs with DSDDs. As a result, the existing results are improved and can be regarded as a special case of the stability criteria in this brief. A simulation example is provided to verify the effectiveness of the developed methods.

Stability and exponential decay of the magnetic Bénard system with horizontal dissipation

This article is concerned with the global stability problem for the two-dimensional magnetic Bénard system. The focus here involves only horizontal dissipation, horizontal magnetic diffusion, and horizontal thermal diffusivity. The lack of vertical dissipation makes it difficult to study the stability and long-time behavior of the magnetic Bénard system. To handle these, we use a new method to decompose the velocity u, magnetic field b, and temperature Θ into the horizontal average (u‾,b‾,Θ‾) and the oscillation part (u˜,b˜,Θ˜). Then we establish the global stability by using several anisotropic inequalities when the initial data are small in H2(Ω). Moreover, we mathematically show that the H1-norm of the oscillation part eventually decays exponentially to zero. Furthermore, the solution (u,b,Θ) approaches the horizontal average (u‾,b‾,Θ‾) asymptotically.

Exponential stability of neural networks with a time-varying delay via a cubic function negative-determination lemma

The problem of global exponential stability of neural networks with a time-varying delay is studied in this article. Firstly, to fully utilize the cross-term relationships among state variables, an improved augmented delay-product-type Lyapunov-Krasovskii functional, including an extra double integral state, is established for the stability analysis. Accordingly, this augmented LKF derivative is a higher-order function of the time-varying delay. Then, three state vectors are considered to reduce the order of the function to cubic. So, to obtain the feasible negative-definiteness condition of this LKF derivative of non-convexity, a negative-determination lemma for cubic functions is employed to handle this problem. As a result, a novel stability criterion is obtained. Two well-known numerical examples illustrate the effectiveness of the criterion.