Asymptotic Stability Of Nonlinear Systems Research Articles

Self-Triggered Impulsive Control for Lyapunov Stability of Nonlinear Systems in Discrete Time.

This article studies Lyapunov stability for nonlinear systems based on discrete-time self-triggered impulsive control (STIC). With the help of comparison approach, some Lyapunov-based sufficient conditions guaranteeing nonZeno behavior and global asymptotic stability of nonlinear systems under STIC are derived. Different from the existing self-triggered mechanisms using implicit expressions to compute the next impulse instant, which may result in computational burden, we propose a novel discrete-time self-triggered mechanism (STM) by which the next impulse instant can be predicted directly by the information of the comparison system. Moreover, the resulting algorithm is easy to be implemented. It is shown that the designed STIC strategy can not only achieve a tradeoff between computational complexity, communication resource usage and control performance, but also has strong robustness and flexibility. Finally, an illustrative simulation is provided showing the effectiveness of the results.

Mode decomposed passivity-based speed control of DC drive with bidirectional Zeta-SEPIC DC-DC converter for light electric vehicles

Currently, light electric vehicles are rapidly developing in various kinds. To power these vehicles with batteries, the simplest electric drive system is a DC motor controlled by a DC-DC converter. This work utilizes a bidirectional Zeta-SEPIC DC-DC converter with an integrated DC motor. This implementation enables control of motor speed and torque in traction and regenerative braking modes. Additionally, it allows for the use of a lower voltage battery compared to the motor's rated voltage, reducing battery weight and increasing safety. In this work, a decomposition approach is applied. Two separate port-controlled Hamiltonian subsystems are obtained to adjust the motor angular velocity in the traction (Zeta) and braking (SEPIC) modes of the DC-DC converter. The Passivity-Based Control (PBC) method is used to synthesize the drive control subsystems in these modes. This method is based on the energy laws of processes in systems and provides asymptotic stability of nonlinear systems, in this case, two fourth-order subsystems for speed control. Two third-order current control subsystems synthesized by the PBC were used to limit the motor current at a given level. The synthesis resulted in sets of possible structures of control influence formers (CIFs) for all PBC subsystems using Zeta and SEPIC DC-DC converters. The study analyzed the operation of the obtained structures of the CIFs, selected the most effective ones, and determined the laws of adaptation of their parameters to the value of the motor angular velocity through computer simulation in Matlab/Simulink. The results of the simulation showed that the drive operated well in both static and dynamic modes.

Analysis and design of control systems via parameter‐based approach

Analysis and design of control systems via parameter‐based approach

Asymptotic stability for nonlinear time-varying systems on time scales

This paper is concerned with nonlinear time-varying systems on time scales, which can cover not only trivial continuous and discrete dynamical systems, but also some other nontrivial cases. Some less conservative sufficient conditions for asymptotic stability of nonlinear systems are derived. Compared with the traditional stability results, the negative definiteness of the time derivatives of Lyapunov functions is weakened, that is, the time derivatives of related Lyapunov functions in this paper are allowed to be indefinite. Two examples about trivial and nontrivial systems are included to illustrate the effectiveness of the results obtained.

Application of LaSalle’s Invariance Principle on Polynomial Differential Equations Using Quantifier Elimination

LaSalle’s invariance principle is a commonly used extension of Lyapunov’s second method to study asymptotic stability of nonlinear systems. If the system can be written in polynomial form, the examination can be automated using algebraic geometry and quantifier elimination. This article addresses this automated examination using a method relying on polynomial ideals and applies it on some example systems. In addition, some properties of these special ideals are derived that allow to reduce the computational effort significantly.

Practical implementable controller design with guaranteed asymptotic stability for nonlinear systems

Establishing the stability of model predictive control formulation is challenging. Using a combination of the terminal cost term and the terminal inequality constraint i.e. terminal set results in nominal asymptotic stability. Larger terminal set results in shorter minimum prediction horizon length required for feasibility from identical initial condition. Current work presents novel linear quadratic Gaussian regulator based approach for the terminal set characterization for the discrete time model predictive control scheme. Proposed approach provides large degrees of freedom in the form of additive matrices as tuning parameters for enlarging the terminal sets as against a single scalar for the literature approaches. Efficacy of the novel approach is demonstrated using a benchmark four tank system. It was observed that the average computation time reduces to significantly when compared to the approaches from the literature for identical initial conditions. This makes the controller design suitable for practical implementation.

Asymptotic stabilization for nonlinear systems with time‐varying delays

In this paper, asymptotic stabilization problem for a class of nonlinear systems with unknown time-varying delays is addressed. With the introduced static gain functions and new Lyapunov–Krasovskii functionals, a novel output feedback control strategy is developed. A memoryless output feedback controller which does not need precise knowledge of time delays is designed. The asymptotic stabilization and the boundedness of the closed-loop system are proved via Lyapunov analysis. Finally, numerical simulation validates the effectiveness of this method for the nonlinear time-delay system.

K‐filter‐based adaptive output feedback control for high‐order nonlinear systems subject to actuator and sensor attacks

AbstractIn this article, the security control problem of a class of high‐order nonlinear cyber‐physical systems is investigated where the actuator and sensor are suffered from deceptive cyber‐attacks. By assuming that only the contaminated system output is available for feedback control, a set of K‐filters are applied to develop a novel observer to reveal the state information, and based on which an adaptive control law is designed. It is shown that asymptotic stability of nonlinear systems and boundedness of all the closed‐loop signals can be guaranteed using backstepping method even in presence of adversarial attacks. Simulations results illustrate the effectiveness of the proposed algorithms.

Asymptotic stabilization of nonlinear systems with convex-polytope input constraints by continuous input

In this paper, we propose an asymptotic stabilization controller for a nonlinear system with a convex-polytope input constraint. Moreover, we show that the controller has inverse optimality in the domain of attraction. The controller input is continuous in the set in which the asymptotic stabilization is guaranteed. We consider the asymmetric constraint for zero-input, and this bias is adjusted by using a non-uniform scaling on each vertex point in the neighborhood of the origin. Therefore, the input cost of the inverse optimal performance functional is approximated by a quadratic function near the origin.

Improved Controller Design with Two Gain-scaling Factors for Global Asymptotic Stabilization of Nonlinear Systems via Matrix Inequality Approach

Improved Controller Design with Two Gain-scaling Factors for Global Asymptotic Stabilization of Nonlinear Systems via Matrix Inequality Approach

Asymptotic stabilization of nonlinear systems with long input delay via memoryless feedback: A linearization method

This paper considers the problem of how to control nonlinear systems with a large delay in the input. A linearization method is presented to achieve local asymptotic stabilization (LAS) via memoryless state and/or output feedback, for multiple-input/multiple-output (MIMO) nonlinear systems with a large input delay. The development of the linearization technique is based on the Razumikhin theorem, a forward completeness lemma of the FDE (functional differential equation) on a closed and bounded interval, and the previous work on truncated predictor feedback control of linear systems with input delay.

Adaptive NN control for nonlinear systems with uncertainty based on dynamic surface control

The problem of adaptive neural network (NN) control and the dynamic surface control (DSC) method for a series of nonlinear systems with uncertainty is discussed in this paper. Unknown smooth functions can be approximated by radial basis function-neural networks (RBF-NNs) with arbitrary accuracy in nonlinear systems. The DSC scheme is introduced for nonlinear systems with uncertainty to overcome the “explosion of complexity” compared with the conventional backstepping approach. Meanwhile, the global asymptotic stability of nonlinear systems is manifested via the Lyapunov stability theory, which indicates the uniform boundedness of all closed-loop signals is ensured and dynamic surface errors are arbitrarily small in the compact set by selecting proper parameters. Finally, the effectiveness of the proposed control technique is validated by two simulation examples.

Semiglobal asymptotic stabilization of nonlinear systems with triangular zero dynamics by linear feedback

This paper studies the problem of semiglobal asymptotic stabilization (SGAS) by state feedback for a class of nonlinear systems beyond the normal form. The zero dynamics is of a triangular structure and assumed to satisfy minimum-phase and ISS conditions. By taking advantage of the triangular structure, we develop a semiglobal backstepping approach for the construction of a set of parameterized Lyapunov functions, associated sublevel sets and nested high-gain linear state feedback laws, step-by-step. To prevent the shrinking of domains of attraction and overcome the complexity of the semiglobal design based on the rational Lyapunov functions (Teel and Praly, 1995), we introduce the notion of sublevel set and a set of parameterized Lyapunov functions that are well-defined in the entire space, and show how they can be constructed recursively, which are instrumental in solving the SGAS problem by partial linear state feedback.

On the practical ψ^γ-exponential asymptotic stability of nonlinear systems of differential equations

On the practical ψ^γ-exponential asymptotic stability of nonlinear systems of differential equations

Prediction‐based sliding mode control of non‐linear systems with input delay using disturbance observer

In this study, stabilisation of input-delay non-linear systems in the presence of disturbance based on prediction variable is studied. For both disturbance estimation and delay compensation, an extended prediction variable as well as adaptive sliding mode controller are proposed in which the open-loop system is not required to be stable. In the proposed prediction variable one needs to estimate disturbance in the future. In order to solve this problem a new non-linear disturbance observer (NDOB) is proposed. Considering the NDOB in the closed-loop system, the stability analysis of error dynamic is performed. And the asymptotic stability of non-linear system with input delay is guaranteed for some class of disturbance. Simulation results finally validate the good performance and robustness of the proposed controller based on disturbance observer.

Uniform global asymptotic stability for nonlinear systems with delay and sampling

Nowadays, the time-delay and sampling of the control systems are considered as one of the most significant issues in the field of nonlinear control systems. It is undeniable that applying measurement delay with controls can cause the sampling of control laws with the delay in the behavior of nonlinear control systems. As a result, this paper introduces a new Lyapunov Krasovskii functional to prove the Uniformly Globally Asymptotic Stability (UGAS) for a class of continues nonlinear systems with the time delay and sampling. Hence, firstly, some assumptions are considered to simplify the complexity of the stability analysis of such systems. Secondly, the upper bounds of time-delays are obtained based on the suggested novel approach under a new theorem. Finally, an illustrative example is presented to show the effectiveness of our proposed method.

Global asymptotic stability for the SEIRS models with varying total population size

In this paper, SEIRS epidemiological model with disease caused death and varying total population size is discussed. Based on the geometric approach developed by Li and Muldowney, a new criterion to determine the global asymptotic stability for nonlinear system is proposed. By applying this new criterion, global asymptotic stability of the endemic equilibrium when it is unique is proved. The above global result shows that the basic reproduction number is a sharp threshold for SEIRS model which removes restrictions of rate of loss of immunity and rate of disease caused death in Li and Muldowney’s result.

Asymptotic Stabilization of Nonlinear Systems with Convex-Polytope Input Constraints and its Inverse Optimality

In this study, we propose an asymptotic stabilization controller for a nonlinear system with a convex-polytope input constraint. Moreover, we establish a condition that the controller has inverse optimality in the neighborhood of the origin. The controller input is continuous in the set where the asymptotic stabilization is guaranteed. We consider the asymmetric constraint for zero-input, and this bias is adjusted by using an un-uniform scaling on each vertex point, in the neighborhood of the origin. Therefore, the input cost in the inverse optimal performance functional is approximated by a quadratic form near the origin.

Stabilisation of non‐linear DISS systems with uncertainty via encoded feedback

The global asymptotic stability of non-linear systems with uncertainty through sampled encoded measurements is investigated. With the objective of achieving global asymptotic stability, some sufficient conditions are derived on sampling time and the quantity of uncertainty under differentiably ISS (DISS) condition, which can be relaxed under input-to-state stability (ISS) condition. Additionally, as a special case, the result when the non-linear system is fixed without uncertainty is improved compared with that in Liberzon and Hespanha, where ISS, a strictly stronger condition than DISS, is employed.

A common fixed point theorem for semigroups of nonlinear uniformly continuous mappings with an application to asymptotic stability of nonlinear systems

In this paper, we study the existence of a common fixed point for uniformly continuous one parameter semigroups of nonlinear self-mappings on a closed convex subset C of a real Banach space X with uniformly normal structure such that the semigroup has a bounded orbit. This result applies, in particular, to the study of an asymptotic stability criterion for a class of semigroup of nonlinear uniformly continuous infinite-dimensional systems.