Stability Of Nonlinear Systems Research Articles

Predefined-time stabilisation of stochastic low-order nonlinear systems with asymmetric output constraint

This paper researches the predefined-time stabilisation problem for a class of stochastic low-order nonlinear systems with asymmetric output constraint whose nonlinearities satisfy high-order and low-order growth conditions. By introducing a novel nonlinear transformation and utilising the adding a power integrator technique, a continuous state feedback controller is designed. The proposed control scheme can guarantee that the output constraint will not be violated almost surely and the trivial solution of the closed-loop system is stochastic predefined-time stable. Finally, two simulation examples are provided to demonstrate the effectiveness of this control method.

Feedback Stabilization of Chain-Like MDOF Nonlinear Structural Systems Under Purely Parametric Gaussian White Noises

A feedback control method is proposed to asymptotically stabilize the chain-like multi-degree-of-freedom (MDOF) nonlinear structural system under purely parametric Gaussian white noises. First, the motion equation of a chain-like nonlinear structural system with [Formula: see text] coupled elements and an undetermined control law is derived. It is reduced to a one-dimensional partially averaged Itô stochastic differential equation of the controlled Hamiltonian by applying the stochastic averaging method, which bypasses the challenge of calculating multiple Lyapunov exponents. Next, based on the dynamical programming principle, the dynamical programming equation of the averaged system with an undetermined cost function is established and addressed to obtain the optimal control law. Then, the Lyapunov exponent of the completely averaged Itô equation is derived. For the given requirement of stabilizing the system, the Lyapunov exponent is entirely fixed and used to analyze the stability of the originally chain-like MDOF nonlinear structural system. A controlled chain-like 8-DOF nonlinear structural system is taken as an example to illustrate the application and effectiveness of the proposed procedure and the effect of control forces on stabilizing the system.

Fast finite‐time stabilization of stochastic nonlinear systems with output constraints and an application to single‐link robot

AbstractThis article studies the fast finite‐time stabilization problem for stochastic nonlinear systems with asymmetric output constraints. To address the issue of slow convergence encountered in previous finite‐time control designs of stochastic nonlinear constrained systems, this paper extends the fast finite‐time stabilization mechanism to such systems. First, a novel asymmetric barrier Lyapunov function is employed to tackle the output constraint. Then, by adding a power integrator technique and the barrier Lyapunov function, a state‐feedback controller for stochastic nonlinear systems is explicitly designed using the backstepping method. The designed controller achieves fast finite‐time stability in probability of the stochastic nonlinear system while ensuring no violation of the output constraints. Finally, two examples are presented to demonstrate the effectiveness of the controller scheme.

Fractional-Order Modeling and Stochastic Dynamics Analysis of a Nonlinear Rubbing Overhung Rotor System

To study the nonlinear dynamic behavior and system stability of a rubbing overhung rotor with viscoelastic and memory-effect damping and random uncertain parameters, this paper introduces a fractional-order modeling and stochastic dynamic analysis method for the nonlinear overhung rotor system with frictional impact faults. Firstly, the dynamic equations of the overhung rotor considering friction effect and fractional damping effect are established based on the transfer matrix method and fractional order derivative. Then, the time-domain response of the fractional-order dynamic equations is solved by combining the Runge–Kutta method and the continuous fractional expansion, and the steady-state response characteristics of different fractional damping are analyzed in the deterministic case. Finally, to analyze the response of the system under the effect of stochastic parameters, the sparse grid-based PCE metamodel of the system response is developed. Statistical moments, probability distributions, and sensitivity indices of the response of stochastic systems are revealed. The results of this paper provide a theoretical basis for efficient and accurate prediction of the stochastic response of nonlinear rubbing overhung rotor systems.

New delay‐dependent finite‐time stabilization of Takagi–Sugeno fuzzy approaches for delayed nonlinear systems

AbstractIn this paper, the finite‐time stability and stabilization of nonlinear systems with delays is studied, via a Takagi–Sugeno approach. By using a novel Lyapunov–Krasovskii functional and introducing some fuzzy free‐weighting matrices, sufficient conditions are derived, for bounded and differentiable time‐varying delays in terms of an upper bound of the delay derivatives. Then, we achieve closed‐loop stabilization in finite time through an efficient parallel distributed compensation design. The sufficient conditions are formulated as linear matrix inequalities to achieve the desired performance. Finally, the proposed methodology is applied to various case studies, highlighting its significance.

Finite-time and fixed-time stability for nonlinear systems via aperiodically intermittent control

Finite-time and fixed-time stability for nonlinear systems via aperiodically intermittent control

ℒ2-gain performance of nonlinear sampled-data systems with gain-scheduled control

This paper presents new sufficient conditions for stabilisation of sampled-data nonlinear systems, recast as quasi-linear parameter-varying models. The adopted control strategy considers gain-scheduled state-feedback controllers subject to induced input delay. The L 2 -gain performance is investigated in the sense of Lyapunov. After the proposition of a Lyapunov function and with the application of Wirtinger's inequality, the derived conditions are rewritten in terms of linear matrix inequalities. Aiming at the enlargement of the aperiodic sampling time and at the minimisation of the L 2 -gain, the effectiveness of the developed approach is assessed through numerical simulations.

Improving extensions for the global uniform asymptotic stability of continuous-time nonlinear systems

This paper considers the classical problems of global asymptotic stability (GAS) of nonlinear autonomous (time-invariant) systems and global uniform asymptotic stability (GUAS) of nonlinear non-autonomous (time-varying) systems. By imposing new conditions on the Lyapunov function and its derivative, the state convergence to the origin from any initial condition with bounded norm in the classical setting is extended to the initial conditions with unbounded norms in the present work. The proposed new notions of stability are called extended global asymptotic stability (EGAS) and extended global uniform asymptotic stability (EGUAS) in this paper. The proposed Lyapunov-based criteria yield the global asymptotic stability of the origin and fixed-time/predefined-time attractiveness of any selected neighbourhood of the origin. The result is that the dynamical behaviour of the state trajectories from the convergence time point of view is improved considerably. The relationship between the smoothness and input-to-state stability (ISS) properties is also studied from a quantitative point of view. Under some new sufficient conditions, it is shown that dynamical systems with smaller Lipschitz constants represent smaller ultimate bounds, and as a result, are more robust. The desirable dynamical behaviour and robustness property of the proposed stability notion makes it comparable with the finite/fixed/predefined/prescribed (exact) stable systems that are naturally non-Lipschitz at the origin. Numerical results illustrate the effectiveness of the proposed theoretical results.

Adaptive Output Formation Tracking for Nonlinear Multiagent Systems With Double Semi-Markovian Switching Topologies and Time-Varying Actuator Faults.

This article investigates adaptive output formation tracking control of nonlinear multiagent systems with time-varying actuator faults and unknown nonidentical control directions under double semi-Markovian switching topologies. Considering the dynamic changes of communication connections in uncertain environments, a double semi-Markov process is first introduced into the leader-follower structure to describe the random switching of communication topologies. Then, a novel adaptive distributed fault-tolerant output formation tracking control framework is established using the backstepping and Nussbaum gain technique to address matched/mismatched uncertainties and disturbances, time-varying actuator faults, and unknown nonidentical control directions. In this control framework, the independent variable of the Nussbaum function is designed as a non-negative function that monotonically increases with respect to time, thereby overcoming the presence of the absolute value of its derivative in the integration process. Based on the distributed structure, an adaptive fault-tolerant controller is further proposed to achieve the asymptotic output formation tracking in mean-square sense. The stability of the closed-loop nonlinear multiagent systems is analysed through the contradiction argument and Lyapunov theorem. The simulation example verifies the effectiveness of the proposed control strategy.

Design of fixed-time sliding mode control using variable exponents

This paper proposes the design of a robust fixed-time sliding mode controller for a second-order nonlinear system with matched and unmatched uncertainties. First, a fixed-time stable scalar dynamics with a variable exponent is introduced. We establish the global fixed-time stability results for the proposed dynamics using Lyapunov analysis. The maximum convergence time has been derived in terms of the design parameters, which is independent of the initial conditions. These results are then extended to solve the robust fixed-time stabilization for uncertain nonlinear second-order system. Specifically, we proceed with the sliding mode control design, where the sliding surface and the control law are motivated by the proposed scalar dynamics. Moreover, the proposed design is free from singularity and guarantees fixed-time robust stabilization. Simulation results with comparative analysis has been included. Further, experimental validation on a single link manipulator is provided to demonstrate the performance of the proposed approach.

Adaptive Fuzzy Predefined-Time Synchronization of Different Dimensional Delayed Chaotic Systems: A Scheme for Realizing Predefined Time Stability of Nonlinear Systems

Adaptive Fuzzy Predefined-Time Synchronization of Different Dimensional Delayed Chaotic Systems: A Scheme for Realizing Predefined Time Stability of Nonlinear Systems

Stability Analysis and Stabilization of General Conformable Polynomial Fuzzy Models with Time Delay

This paper introduces a sum-of-squares (S-O-S) approach to Stability Analysis and Stabilization (SAS) of nonlinear dynamical systems described by General Conformable Polynomial Fuzzy (GCPF) models with a time delay. First, we present GCPF models, which are more general compared to the widely recognized Takagi–Sugeno Fuzzy (T-SF) models. Then, we establish SAS conditions for these models using a Lyapunov–Krasovskii functional and the S-O-S approach, making the SAS conditions in this work less conservative than the Linear Matrix Inequalities (LMI)-based approach to the T-SF models. In addition, the SAS conditions are found by satisfying S-O-S conditions dependent on membership functions that are reliant on the polynomial fitting approximation algorithm. These S-O-S conditions can be solved numerically using the recently developed SOSTOOLS. To demonstrate the effectiveness and practicality of our approach, two numerical examples are provided to demonstrate the effectiveness and practicality of our approach.

Designing a switching law for Mittag-Leffler stability in nonlinear singular fractional-order systems and its applications in synchronization

In this work, the stability and synchronization issue of switched singular continuous-time fractional-order systems with nonlinear perturbation are examined. Using the fixed-point principle and S-procedure lemma, a sufficient condition for the existence and uniqueness of the solution to the switched singular fractional-order system is first stated. Next, using the Lyapunov functional method in combination with some techniques related to singular systems and fractional calculus, a switching rule for regularity, impulse-free, and Mittag-Leffler stability is developed based on the formation of a partition of the stability state regions in convex cones. For synchronizing switched fractional singular dynamical systems, we propose a state feedback controller that ensures regularity, impulse-free, and Mittag-Leffler stable in the error closed-loop system. Finally, the ease of use and computational convenience of our proposed methods are illustrated by two numerical examples and a practical example about DC motor controlling an inverted pendulum accompanied by simulation results.

Finite-time contractive stability for fractional-order nonlinear systems with delayed impulses: Applications to neural networks

This paper deals with finite-time stability (FTS) and finite-time contractive stability (FTCS) criteria of fractional-order nonlinear systems (FONSs), where the state of the systems consists of a finite number of delayed impulsive instants. To achieve these results, we have extended the theories on FTS and FTCS criteria from integer-order NSs to fractional-order systems. Employing the fractional-order Lyapunov stability theory, the sufficient conditions of the above-mentioned stability criteria are derived for a class of FONSs with state-dependent delayed impulses. Then, to ensure the applicability of the proposed results, we have analyzed the desired performances for several neural network (NN) models, such as non-autonomous NNs (NANNs), Cohen–Grossberg NNs (CGNNs), switched NNs (SNNs) and bidirectional associative memory NNs (BAMNNs). Finally, four representative numerical examples are illustrated to demonstrate the assurance of the obtained stability conditions of the respective state-dependent impulsive fractional-order NNs (FONNs).

New results on asymptotic stability of time-varying nonlinear systems with applications

Abstract. In this paper, we present a converse Lyapunov theorem for the new notion of global generalized practical uniform h-stability of nonlinear systems of differential equations. We derive some sufficient conditions which guarantee the global generalized practical uniform h-stability of time-varying perturbed systems. In addition, these results are used to study the practical h-stability of models of infectious diseases and vaccination. Mathematics Subject Classification (2010): 35B40, 37B55, 34D20, 93D15, 92D30. Keywords: Epidemic models, generalized practical uniform h-stability, Gronwall’s inequalities, Lyapunov functions.

Stabilization of fractional nonlinear systems with disturbances via sliding mode control

AbstractIn this article, the sliding mode control (SMC) of fractional nonlinear systems (FNSs) with disturbances is studied. First, a useful fractional power‐rate inequality for is extended to a more general form . Based on the generalized inequality, the method of a modified fractional integral SMC is completed, in which the parameter of the symbolic function is extended to and . This increases the degree of freedom of the sliding mode surface (SMS). Then, the SMC of FNSs is studied for the cases of known and unknown disturbances. In the case of known disturbance, it is proved by the generalized inequality and quadratic Lyapunov function method that the state of the FNSs can converge asymptotically to zero on the SMS. For the case of unknown disturbance, a fractional disturbance observer is used to estimate the disturbance by introducing auxiliary variables . The disturbance estimation error of the proposed FNSs is proved to be bounded. An equivalent global control law is obtained and asymptotic convergence of the FNSs under the action of the controller is proved. Finally, two numerical simulation examples are given to verify the feasibility of the method proposed in this article.

Decentralized control of nonlinear complex systems

Abstract In the paper, a novel approach to decentralized controller design for nonlinear systems is introduced. The proposed method is based on the relationship between the stability of complex nonlinear systems and the stability of their subsystems. The design procedure of the decentralized controller consists of three steps. In the first step, the stability of the complex nonlinear system is calculated. In the second step, stability conditions at the subsystem level are obtained such that guarantee the stability of the complex nonlinear system. Finally, in the third step, a controller design method is used to ensure that the subsystem stability conditions obtained in the second step are met. As an example, to better understanding the proposed method two simple nonlinear models are used to demonstrate the effectiveness of the proposed method.

Positive Effect of Switching on Input-to-State Stability of Nonlinear Systems

Positive Effect of Switching on Input-to-State Stability of Nonlinear Systems

Finite time stability of nonlinear impulsive stochastic system and its application to neural networks

In this paper, we employ the Lyapunov theory to generalize the finite time stability (FTS) results from general deterministic impulsive systems to impulsive stochastic time-varying systems, which overcomes inherent challenges. Sufficient conditions for the FTS of the system under stabilizing and destabilizing impulses are established by using the method of average dwell interval (ADT). For FTS of stabilizing impulses, we relax the constraint on the differential operator by allowing it to be indefinite rather than strictly negative or semi-negative definite. Furthermore, the theoretical results are applied to impulsive stochastic neural networks. Finally, two numerical examples are given to validate the reliability and practicability of the obtained results.

Robust stabilisation of uncertain nonlinear systems with matched and unmatched uncertainties

This paper considers the problem of robust stabilisation of uncertain nonlinear systems with both matched and unmatched uncertainties. A new robust controller is proposed to guarantee practical or asymptotic stability under non-vanishing or vanishing unmatched uncertainties. First, the system uncertainties are decomposed optimally into the matched and unmatched portions with an objective of the minimum norm for the unmatched part. Next, instead of a robust control design based on only the matched part of uncertainties commonly used in the previous works, a robust controller is synthesised based on both matched and mismatched portions of the uncertainties. It is shown that in uncertain systems whose state cannot simultaneously stay in a so-called null set and diverge to infinity, the proposed controller guarantees stability no matter how large the input-unrelated unmatched uncertainties are. Numerical examples illustrate that the proposed controller is more robust against uncertainties, and offers improved damping and stabilisation features.