Stability Of Nonlinear Systems Research Articles

On Stabilization of Nonlinear Continuously-Discrete Dynamic Systems with a Constant Discretization Step

On Stabilization of Nonlinear Continuously-Discrete Dynamic Systems with a Constant Discretization Step

Time/Event‐Triggered Exponential Stabilization for Stochastic Systems: Enhancing Nonlinearity Tolerance

ABSTRACTThis article seeks the enhancement of nonlinearity tolerance in time/event‐triggered stabilization for stochastic systems. In the related results, the controller functions are required to obey global Lipschitz condition for the suppression of sampling/execution error, and the drift and/or diffusion coefficients of the stochastic systems are restricted to polynomial or, even, linear growth. These limitations deserve overcoming for enlarged applications. To this end, a distinct framework of time/event‐triggered controls is established for stochastic systems, targeted at exponential stabilization. Specifically, inclusive Lyapunov‐type feasibility conditions are proposed by capturing the effect of sampling/execution error and distinguishing the role of system nonlinearities. Particularly, different from the related results, the evolution of sampling/execution error is subtly exploited via Lyapunov functions to reveal the dynamic interaction between the sampling/execution errors and system state. Then, time‐triggered exponential stabilization via sampled‐data controller is achieved not only in the moment sense but also in the almost sure sense. Accordingly, Lyapunov function based analysis is performed for the composite dynamics of system state and sampling error, confronted with the coupling of discontinuous and stochastic features. To further reduce execution, periodic event‐triggered control is exploited to achieve exponential stabilization for stochastic nonlinear systems, by virtue of the relation with the dynamic evolution under sampled‐data control. Through typical examples, we demonstrate the potential of our framework in handling the cases with the controller functions violating global Lipschitz condition and with the system nonlinearities beyond polynomial growth.

Robust output feedback finite-time stabilization of nonlinear singular large-scale continuous-time systems with delays

Robust output feedback finite-time stabilization of nonlinear singular large-scale continuous-time systems with delays

Trade Off Analysis Between Fixed‐Time Stabilization and Energy Consumption of Nonlinear Stochastic Systems

ABSTRACTThe trade off analysis between the fixed‐time stabilization in probability and energy consumption of nonlinear stochastic system is studied in this paper. By constructing a switching controller and using inequality techniques, sufficient conditions for fixed‐time stabilization in probability in the Lyapunov sense are given, and the upper bounds of the settling time function and energy consumption are estimated. Then, by analyzing the relationship between control parameters, control time and energy consumption, the existence of trade off between control time and energy consumption is proposed, and the corresponding optimal parameter values are given. Finally, a numerical example is used to verify the validity of the theoretical results.

Stabilization of nonlinear safety-critical systems by relaxed converse Lyapunov-barrier approach and its applications in robotic systems

Combining safety objectives with stability objectives is crucial for safety-critical systems. Existing studies generally unified these two objectives by constructing Lyapunov-type barrier functions. However, insufficient analysis of key set relationships within the system may render the proposed safety and stability conditions conservative, and these studies also did not provide how to use such conditions to design safety-stability control strategies. This paper proposed a feasible and constructive design to achieve stabilization of safety-critical systems by a relaxed converse Lyapunov-barrier approach. By analyzing the relationships between a series of sets associated with the safety-critical system, the stability and safety conditions can be appropriately relaxed. Then, with the help of relaxed converse control Lyapunov-barrier functions (RCCLBFs), a theoretical result was obtained for the stability of affine nonlinear systems with safety constraints. Subsequently, a constructive method was developed for a second-order strict-feedback system to transform the process of solving RCCLBFs into a Lyapunov-like stabilization problem. Finally, the proposed safety-stability control method is exerted on a robotic system and demonstrated by simulations.

Suppression Limit Cycles in 2x2 Nonlinear Systems with Memory Type Nonlinearities

For several decades, the importance and weight-age of prediction of nonlinear self-sustained oscillations or Limit Cycles (LC) and their quenching by signal stabilization have been discussed, which is confined to Single Input and Single Output (SISO) systems. However, for the last five to six decades, the analysis of 2x2 Multi Input and Multi Output (MIMO) Nonlinear Systems gained importance in which a lot of literature available. In recent days’ people have started discussing suppression of LC which limits the performance of most of the physical systems in the world. It is a formidable task to suppress the limit cycles for 2x2 systems with memory type nonlinearity in particular. Backlash is one of the nonlinearities commonly occurring in physical systems that limit the performance of speed and position control in robotics, automation industry and other occasions like Load Frequency Control (LFC) in multi area power systems. The feasibility of suppression of such nonlinear self-oscillations has been explored in case of the memory type nonlinearities. Backlash is a common memory type nonlinearity which is an inherent Characteristic of a Governor, used for usual load frequency control of an inter-connected power system and elsewhere. Suppression LC using pole placement technique through arbitrary selection and optimal selection of feedback Gain Matrix K with complete state controllability condition and Riccati Equation respectively and is done through state feedback. The Governing equation is d/dt [X(t)] =(A-BK) X: which facilitates the determination of feedback gain matrix K for close loop Poles / Eigen values placement where the limit cycles are suppressed/eliminated in the general multi variable systems. The complexity involved in implicit non-memory type or memory type nonlinearities, it is extremely difficult to formulate the problem for 2x2 systems. Under this circumstance, the harmonic linearization/harmonic balance reduces the complexity considerably. Still the analytical expressions are so complex which loses the insight into the problem particularly for memory type nonlinearity in 2x2 system and the method is made further simpler assuming a 2x2 system exhibits the LC predominately at a single frequency. Hence in the proposed work an alternative attempt has been made to develop a graphical method for the prediction of Limit Cycling Oscillations in 2x2 memory type Nonlinear systems which not only reduces the complexity of formulations but also facilitates clear insight into the problem and its solution. The present techniques are well illustrated with an example and validated / substantiated by digital simulation (developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software. The present work has the brighter future scope of: Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional nonlinear systems through an exhaustive analysis. Analytical/Mathematical methods may also be developed for signal stabilization using both deterministic and random signals based on Dual Input Describing function (DIDF) and Random Input Describing Function (RIDF) respectively. The phenomena of Synchronization and De-synchronization can be observed/identified analytically using Incremental Input Describing Function (IDF), which can also be validated by digital simulations.

Quenching and Suppression of Limit Cycles in 3x3 Nonlinear Systems

For several decades, the importance and weight-age of prediction of nonlinear self-sustained oscillations or Limit Cycles (LC) and their quenching by signal stabilization have been discussed which is confined to Single Input and Single Output (SISO) system. However, for the last five to six decades, the analysis of 2x2 Multi Input and Multi Output (MIMO) Nonlinear Systems gained importance in which a lot of literature available. In recent days few literatures are available which addresses the exhibition of LC and their quenching/suppression in 3x3 MIMO Nonlinear systems. Poor performances in many cases like Load Frequency Control (LFC) in multi area power system, speed and position control in robotics, automation industry and other occasions have been observed which draws attention of Researchers. The complexity involved, in implicit nonmemory type and memory type nonlinearities, it is extremely difficult to formulate the problem in particular for 3x3 systems. Under this circumstance, the harmonic linearization/ harmonic balance reduces the complexity considerably. Still the analytical expressions are so complex which loses the insight into the problem particularly for memory type nonlinearity in 3x3 system. Hence in the present work a novel graphical method has been developed for prediction of limit cycling oscillations in a 3x3 nonlinear system. The quenching of such LC using signal stabilization technique using deterministic (Sinusoidal) and random (Gaussian) signals has been explored. Suppression LC using pole placement technique through arbitrary selection and optimal selection of feedback Gain Matrix K with complete state controllability condition and Riccati Equation respectively. The method is made further simpler assuming a 3x3 system exhibits the LC predominantly at a single frequency, which facilitates clear insight into the problem and its solution. The proposed techniques are well illustrated with example and validated/substantiated by digital simulation (a developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software. The Signal stabilization with Random (Gaussian) Signals and Suppression LC with optimal selection of state feedback matrix K using Riccati Equation for 3x3 nonlinear systems have never been discussed elsewhere and hence it claims originality and novelty. The present work has the brighter future scope of: i. Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional nonlinear systems through an exhaustive analysis. ii. Analytical/Mathematical method may also be developed for signal stabilization using both deterministic and random signals based on Dual Input Describing function (DIDF) and Random Input Describing Function (RIDF) respectively. iii. The phenomena of Synchronization and De-synchronization can be observed/identified analytically using Incremental Input Describing Function (IDF), which can also be validated by digital simulations.

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 input-to-state stability and its converse Lyapunov theorem

This paper establishes input-to-state stability (ISS) for fractional-order nonlinear systems, providing a stability constraint framework for fractional-order systems affected by external inputs. The general fractional comparison principle aids in constructing Lyapunov functions, offering criteria for input-to-state stability of fractional-order nonlinear systems. The concept of weak robustness and the converse Lyapunov theorem for fractional-order nonlinear systems further enhance the completeness of input-to-state stability in these systems. By designing systems to maintain fractional-order input-to-state stability (FOISS), the system can remain robust against the effects of bounded external inputs. Serving as a foundational ”toolbox,” this work provides insights into the stability analysis of fractional-order nonlinear systems, benefiting researchers and practitioners in the field of control theory.

Global Stability of Fractional Nonlinear Differential Systems With State-Dependent Delayed Impulses.

The fractional-order (FO) nonlinear differential system with state-dependent (SD) delayed impulses (DI) is considered in this brief. The considered impulses are related to the delayed state of the system and the delays are SD. A novel lemma for the monotonicity of the solution of Caputo's FO derivative equation is given. By means of linear matrix inequality (LMI) and several comparative arguments, criteria of uniform stability, uniform asymptotical stability, and Mittag-Leffler stability are obtained. Compared with other works on integer-order (IO) impulsive delayed systems with SD delays or fixed delays, how to impose constraints on parameters and impulses is explored, without imposing the boundedness on the state delays. Two examples are implemented to examine the practicality and sharpness of our theoretical analysis.

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

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.

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.

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