Stability Of Nonlinear Systems Research Articles

Adaptive Prescribed-Time Stabilization of Uncertain Nonlinear Systems With Unknown Control Directions

Adaptive Prescribed-Time Stabilization of Uncertain Nonlinear Systems With Unknown Control Directions

Strong Prescribed-Time Stabilization of Uncertain Nonlinear Systems by Periodic Delayed Feedback

Strong Prescribed-Time Stabilization of Uncertain Nonlinear Systems by Periodic Delayed Feedback

Advanced stability assessment for control systems: Gelfand’s method with iterated shanks transformation

ABSTRACT In this paper, we present an efficient method for evaluating the stability of both linear time-invariant (LTI) systems and continuous-time nonlinear systems. Our approach centres around the modulus of the spectral radius derived from a specific matrix transformation of the original system state matrix in the case of LTI systems, or the Jacobian of the nonlinear system. We enhance computational speed by employing the iterated Shanks transformation on Gelfand’s formula, eliminating the need to rely on the system’s characteristic equation. Notably, our stability criterion avoids the computation of all eigenvalues of the system matrix. Illustrative examples, including a fluidised bed reactor, a continuous stirred tank reactor and a hypothetical 20th-order dynamical system, are provided. Additionally, we conduct a CPU-time analysis, revealing that our method demonstrates several orders of magnitude faster computational speed compared to the classical Routh-Hurwitz stability test for systems ranging from two to 80 dimensions.

Hybrid stabilization of nonlinear systems based on a fully actuated system approach

This article studies the stabilization problem for a category of nonlinear systems. It introduces a novel hybrid control strategy specifically for nonlinear high-order fully actuated (HOFA) systems. The stabilization problem for these nonlinear HOFA systems is transformed into a stabilization problem for impulsive switched systems. This issue is addressed by applying impulsive control in discrete-time and switching control based on the HOFA system approach in continuous-time. Using switched Lyapunov functions, we obtain the criteria for the exponential and asymptotic stabilization of these systems. The effectiveness of this newly presented hybrid control strategy is exemplified by simulations of a robotic manipulator system and a reduced model of the lac operon.

Asymptotic adaptive output-feedback stabilisation of nonlinear systems with unknown growth rate and sensor uncertainty

For a class of nonlinear systems with sensor uncertainty, the problem of adaptive output feedback stabilisation is investigated in this paper. It is assumed that only the output is available for feedback control, which suffers from sensor uncertainty. At the same time, the nonlinear function in the system model has an unknown growth rate. To solve this problem, a set of state observers is designed to estimate the unknown system states based on the uncertain output, and an adaptive output-feedback control is designed based on this. By using the corollary of Barbalet's Lemma, it is shown that the equilibrium of the closed-loop system is asymptotically stable. Finally, simulation results verify the effectiveness of the control scheme

Disturbance rejections of polynomial fuzzy systems under equivalent-input-disturbance estimator approach

This paper proposes an integrated robust stabilization and anti-disturbance control scheme for nonlinear systems using a polynomial fuzzy model approach. To estimate unknown disturbances, an equivalent-input-disturbance (EID) estimator is employed. The proposed approach incorporates a novel fuzzy model assisted by EID estimator into the sum-of-squares-based approach, utilizing a polynomial fuzzy model-based observer to estimate the disturbance effect. A suitable fuzzy rule-based control law is developed by utilizing the parallel distributed compensation approach and the output of the EID estimator. To ensure stability, fuzzy membership functions are converted into sum-of-square polynomials using a polynomial curve fitting approach, allowing their exact shape information to be used in the stability condition. The addressed system is transformed into an augmented system by incorporating the system, observer, and filter states, simplifying the analysis. Gain matrices for the controller and observer are obtained using Lyapunov stability theory and sum-of-squares methods to confirm asymptotic stabilization of the fuzzy system. Two numerical examples are presented to demonstrate the effectiveness of the proposed control design method.

The Second-Order Sliding Mode Control Algorithm for Fixed-Time Stability of Nonlinear Systems

The Second-Order Sliding Mode Control Algorithm for Fixed-Time Stability of Nonlinear Systems

Feedback Stabilization of Quasi-One-Sided Lipschitz Nonlinear Discrete-Time Systems with Reduced-Order Observer

The feedback stabilization problem for nonlinear discrete-time systems with a reduced-order observer is investigated, in which the nonlinear terms of the systems satisfy the quasi-one-sided Lipschitz condition. First, a discrete-time reduced-order observer for nonlinear systems is designed. Then, a feedback controller with a reduced-order observer is designed for realizing the stabilization of nonlinear discrete-time systems. We prove that the design of a feedback controller and reduced-order observer of systems can be carried out independently in the case of discrete-time with nonlinear terms, which largely reduces the computational complexity of the observer and controller. The introduction of the quasi-one-sided Lipschitz condition simultaneously enhances the robustness and stability of nonlinear control systems. Finally, the feasibility and effectiveness of the proposed design approach is verified by a numerical simulation.

Stability analysis for stochastic nonlinear systems with state-dependent switching and time-delay

The stability of stochastic nonlinear systems with time-delay and state-dependent switching is studied in this paper. Firstly, system maximal solution is constructed and further extended as the global solution under certain mild conditions. Secondly, several Krasovskii-type stability and Razumikhin-type theorems are proposed by uniformly asymptotically stable functions (UASFs). These stability criteria overcome some stringent constraints associated with Lyapunov functional/functions. Finally, the theoretical findings are validated through two simulation examples.

Multi-mechanical frequency locking mechanism and robustness analysis of MDOF coupled resonator

Mechanical frequency locking can overcome the dependence of peak frequency on the driving amplitude and improve the frequency stability in nonlinear resonant systems. Most research focuses on the single mechanical frequency locking phenomenon, but few studies focus on multi-mechanical frequency locking phenomenon. In this study, the multi-mechanical frequency locking mechanism and robustness analysis of the peak frequency is proposed through multi-mode coupled vibration. Firstly, a multi-degree-of-freedom (MDOF) coupled resonator consisting of a low-frequency resonant element and multiple high-frequency resonant elements is designed, and the vibration energy of the resonant system can be transferred from the low-frequency resonant element driven by a harmonic excitation to the high-frequency resonant elements via the 1:3 internal resonance. Through perturbation and bifurcation analysis, the nonlinear dynamic behaviors of MDOF coupled resonator are analyzed in detail. It is found that the peak frequency of the low-frequency resonant element can remain stable in multiple drive intervals, leading to multi-mechanical frequency locking phenomenon. Then, a 3-DOF magnetic coupled resonant system is experimentally designed to demonstrate the physical conditions and influencing factors of the double mechanical frequency locking phenomenon. Typically, introducing external disturbances, the frequency locking is experimentally demonstrated to improve the robustness of peak frequency. The experimental results show that when the mechanical frequency locking phenomenon occurs, the influence of small external noise disturbance on the peak frequency can be ignored, which greatly improves the frequency stability of nonlinear resonant systems.

Unified Sufficient Conditions for Predefined-Time Stability of Non-Linear Systems and Its Standard Controller Design

This paper presents a unified Lyapunov-based predefined-time stability theorem that includes three sufficient conditions. The standard theoretical analysis method for achieving predefined-time stability of non-linear systems using this theorem is provided within the framework of Lyapunov theory. The developed Lyapunov-based theorem facilitates the establishment of equivalence between the existing Lyapunov theorems concerning predefined-time stability. Furthermore, when the presented sufficient conditions are relaxed, the predefined-time stability conclusion for non-linear systems degenerates into a finite-time one. Consequently, a standard non-singular sliding mode control framework based on the unified Lyapunov-based theorem is developed for a Lagrangian system to ensure its predefined-time stability. Exemplary numerical simulation results are subsequently given, in order to illustrate the convergence behavior of the system states and confirm that the controlled systems are predefined-time stable.

Stability and Stabilizability Criteria of Nonlinear Interval Systems

Stability and Stabilizability Criteria of Nonlinear Interval Systems

Simultaneous robust model predictive control and state estimation for nonlinear systems with state‐ and input‐dependent uncertainties

AbstractThe convergence and stability of uncertain nonlinear systems is a challenging problem in the nonlinear control area. Besides, in many practical cases, all states are not measurable and are affected by measurement noise. Based on this motivation, the first objective of this paper is to design a novel output feedback robust model predictive control approach for nonlinear systems with state‐ and input‐dependent uncertainties and measurement noise. This approach combines state estimation and robust model predictive control (MPC) into one min–max optimization and by solving the optimization, these two tasks are performed simultaneously. The studied nonlinear system comprises a linear part, a nonlinear part, and a function that denotes the state‐ and input‐dependent uncertainties. Therefore, the other objective is to reduce the computational complexity; thus, the system's nonlinear term and the aforementioned uncertainties are converted into additional disturbances. Subsequently, the optimization problem becomes a quadratic form, which leads to global convergence with the appropriate selection of objective function weights. Besides, this paper explores the convergence of the closed‐loop system states and the sufficient synthesis conditions to guarantee input‐to‐state stability. The implementation on a numerical example and a CSTR process demonstrate the applicability and reliability of the proposed approach.

Output Feedback Stabilization of High-Order Nonlinear Time-Delay Systems With Low-Order and High-Order Nonlinearities

In this paper, the output feedback stabilization of a class of high-order nonlinear time-delay systems with more general low-order and high-order nonlinearities is investigated. By constructing the new Lyapunov-Krasovskii functional and reduced-order observer, based on homogeneous domination theory together with the adding a power integrator method, an output feedback controller is developed to guarantee the equilibrium of the closed system globally uniformly asymptotically stable.

Novel Robust Estimation-Based Control of One-Sided Lipschitz Nonlinear Systems Subject to Output and Input Delays

This paper highlights the design of a controller established on estimated states for one-sided Lipschitz (OSL) nonlinear systems subject to output and input delays. The controller has been devised by involving Luenberger-like estimated states. The stability of the time-delayed nonlinear system is reckoned by assuming a Lyapunov functional for delayed dynamics and for which a delay-range dependent criterion is posed with a delay ranging between known upper and lower bounds. The time derivative of the functional is further exploited with linear matrix inequality (LMI) procedures, and employing Wirtinger’s inequality for the integral terms instead of the traditional and more conservative Jensen’s condition. Moreover, a sufficient and necessary solution is derived for the proposed design by involving the tedious decoupling technique to attain controller and observer gain simultaneously. The proposed methodology validates the observer error stability between observers and states asymptotically. The solution of matrix inequalities was obtained by employing cone-complementary linearization techniques to solve the tiresome constraints through simulation tools by convex optimization. Additionally, a novel scheme of an observer-based controller for the linear counterpart is also derived for one-sided Lipschitz nonlinear systems with multiple delays. Finally, the effectualness of the presented observer-based controller under input and output delays for one-sided Lipschitz nonlinear systems is validated by considering a numerical simulation of mobile systems in Cartesian coordinates.

Predefined-time stabilization of stochastic nonlinear systems with application to UAVs

The paper presents a new Lyapunov-type predefined-time stabilization control algorithm for stochastic high-order nonlinear systems with asymmetric output constraints. In contrast to stochastic finite-time and fixed-time stabilization, the average value of the settling-time function for stochastic predefined-time stabilization control is independent of both the initial value and the control factors. To mitigate the significant uncertainties arising from the asymmetric output constraint, a tan-type barrier Lyapunov function is formulated. Furthermore, by harnessing the previously mentioned barrier Lyapunov function and integrating the power integrator technique, a controller design strategy is formulated based on the backstepping method. The rigorous analysis in this study proves that the designed controller ensures both the attainment of predefined-time convergence of the system states to the origin in probability and the satisfaction of the output constraint. Finally, an example of a roll angle subsystem for quadrotor UAVs and a numerical illustration are presented to corroborate the theoretical analysis.

Reliable stabilization of discrete-time nonlinear singular systems with time-varying delays

The problem of delay-dependent reliable control for uncertain discrete-time nonlinear singular system with time-varying delays is investigated. A new set of delay-dependent sufficient conditions are established for the existence of a reliable controller which ensures the singular system to be regular, causal (absence of impulses) and stable for all admissible parameter uncertainties. First, the controller for singular system in the presence of known actuator fault matrix is designed. Further, the result is extended to study the robust stabilization of singular systems with unknown actuator fault matrix. The sensor failures of actuator are considered by a variable, which is varying in a given interval. The conditions for the existence of proposed controller design are established in the form of linear matrix inequality (LMI), which can be easily solved via standard numerical software. Finally, numerical examples with simulation results are provided to illustrate the usefulness of control design and results are compared with existing results to show the less conservatism of the proposed theory.

Stabilization of nonlinear stochastic systems with input and output delays via event‐triggered predictive control

AbstractThe stabilization of nonlinear stochastic systems with input and output delays is investigated in this article. First, an observer that includes input delay information is designed to estimate the state information after eliminating the effect of input delay. Second, a predictive scheme with delay information is utilized to predict the system state affected by output delay, and an event‐triggered predictive controller is established to eliminate the effects of input and output delays simultaneously. Subsequently, the convergence of the prediction error is analyzed by utilizing a recursive method, and a sufficient condition to ensure the global exponential stability is given by constructing a new closed‐loop system. Finally, numerical simulations of two nonlinear stochastic systems with different delay sizes are utilized to verify the impact of the event‐triggered predictive control in addressing the stabilization of nonlinear stochastic systems with time delays.

Semi-global exponential stabilization of stochastic nonlinear systems with distributed delay under discrete-time state observations and application to circuit systems

This paper is aimed at studying the semi-global exponential stabilization problem for stochastic nonlinear systems with distributed delay under discrete-time state observations. An extended definition of horizontal translation of the segment function along the investigated systems is given, related to the stochastic disturbance and Brownian motion. Basically, the right and upper Dini’s derivative of the constructed functional is presented. Combining this with the global exponential stability of the stochastic nonlinear systems with distributed delay under continuous-time feedback control, the Lyapunov method and Halanay inequality, the maximum observation interval and some stabilization criteria to make the investigated systems under discrete-time state observations reach semi-global exponential stabilization are provided. The results indicate that the semi-global exponential stabilization is dependent upon stochastic disturbance, initial condition range, and maximum observation interval. Finally, an application to the circuit systems is discussed and the involved sufficient conditions for the circuit systems to keep semi-global stabilization are also proposed. Besides, the numerical simulation results confirm the effectiveness of the theoretical 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.