Stability Of Nonlinear Systems Research Articles

Adaptive anti‐windup output feedback control for nonlinear systems and its stability analysis

SummaryIn the practical application of adaptive control methods, a windup phenomenon for the systems with input saturation may occur because there exists an input restriction such as saturation of actuator in most actual systems due to physical reasons. In this paper, to alleviate the influence of the windup phenomenon by input saturation, we propose an output feedback exponentially passive (OFEP)‐based adaptive model recovery anti‐windup (AMRAW) control strategy for the stable nonlinear systems with input saturation. Moreover, in order to realize an augmented OFEP system for non‐OFEP nonlinear systems that do not satisfy the OFEP properties, a parallel feedforward compensator (PFC) is introduced, and a radial basis function (RBF) neural network (NN)‐based adaptive feedforward input is designed to attain accurate tracking performance for systems with the PFC. Furthermore, the detailed stability analysis of the proposed AMRAW control system is provided, and the performance of the proposed method is validated through appropriate numerical simulations.

Parameter estimation of biased harmonic signal

This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9, 10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].

Uniform asymptotic and input to state stability by indefinite Lyapunov functions

In this work, we study uniform, uniform asymptotic, and input-to-state stability conditions for nonlinear time-varying systems. We introduce an easily verifiable condition for uniform attractivity by utilizing an indefinite sign upper bound for the derivative of the Lyapunov function. With this bounding structure, we propose novel conditions that enable us to test uniform stability, uniform asymptotic stability, and ISS, easily. As a result, the constraints on the coefficients of the bound that identify uniformity for stability and attractivity, and many of the available conditions have been relaxed. The results are also used for the perturbation problem of uniformly stable and uniformly asymptotically stable linear time-varying systems. Consequently, we demonstrate that uniform asymptotic stability of nonlinear time-varying systems can be robust for perturbations, but with special time-varying coefficients.

Frequency criterion for stability «as a whole» of modulation-type pulse voltage converters according to Lyapunov

Background. The use of methods for researching and designing stable nonlinear dynamic systems by constructing the frequency characteristics of the loop gain of the negative feedback of a pulsed voltage converter through time characteristics obtained by numerical methods for solving nonlinear differential equations that describe electromagnetic processes in the converter takes a lot of time and has low accuracy in calculating optimal modes, when you need to calculate hundreds of options.
 Aim. Therefore, it seems necessary to further develop the theory of stability of modulation-type pulse voltage converters using frequency stability criteria, Lyapunov functions, amplitude and phase frequency characteristics of the converter loop gain.
 Methods. Stability in the «small» is solved using linearized models, which are constructed using the so-called first Lyapunov method - the «first approximation» method. The second method for studying the stability of nonlinear dynamic systems is the harmonic linearization method, when the nonlinear element is replaced by a harmonically linearized link with a transmission coefficient only for the first harmonic.
 Results. The Lyapunov function for a pulse voltage converter with nonlinear relay-type elements was found. Conclusion. The results obtained in the work make it possible to simplify the design of Lyapunov-stable pulse voltage converters through the use of the method of amplitude-frequency characteristics used for stationary linear systems.

Stabilization of time-delay nonlinear systems using Takagi-Sugeno fuzzy models

We propose a controller design method for time-delay nonlinear systems with delays affecting both the states and the inputs, represented by Takagi-Sugeno fuzzy models with nonlinear consequents. To handle the nonlinearities in the consequents we assume that they are slope-bounded. Linear matrix inequality conditions are formulated to design the controller. The obtained results are compared to state-of-the-art approaches and illustrated on two examples.

Decomposition of Equations of Nonlinear Affine Control Systems and its Application to the Synthesis of Regulators

The article deals with the decomposition of nonlinear differential equations based on the group-theoretic approach. At the beginning, the decomposition of differential equations of linear systems using a transition matrix of state is presented, and then, based on the theory of continuous groups (Lie groups), the process of decomposition of differential equations of nonlinear systems is shown. The decomposition approach is based on the isomorphism theorem of the space of vector fields and Lie derivatives, which allows us to consider vector fields as differential operators of smooth functions. A formula is derived about the adjoin representation of a Lie group in its Lie algebra, which actually determines the finding of a vector field that characterizes the interaction of two or more vector fields. The Lie algebra of derivatives makes it possible to determine the infinitesimal action of the Lie group, i.e. the linearization of this action is carried out (transformation of the points of the trajectory space of the original system in a small neighborhood). Decomposition allows, as in the linear case, to separate the finding of an action (only locally) of a group of transformations from the transformed points themselves. For linear systems, this separation is global. It is also shown that the decomposition of linear equations is a particular case of the decomposition of nonlinear equations. An algorithm of the method of model predictive control with Gramian weighting using this decomposition is presented. A practical example of decomposition and application of the model predictive control for stabilization of a nonstationary nonlinear system is considered.

Operator-based adaptive robust control for uncertain nonlinear systems by combining coprime factorization and fuzzy control method1

In this paper, the robust stability of nonlinear system with unknown perturbation is considered combining operator-based right coprime factorization and fuzzy control method from the input-output view of point. In detail, fuzzy logic system is firstly combined with operator-based right coprime factorization method to study the uncertain nonlinear system. By using the operator-based fuzzy controller, the unknown perturbation is formulated, and a sufficient condition of guaranteeing robust stability is given by systematic calculation, which reduces difficulties in designing controller and calculating inverse of Bezout identity. Implications of the results related to former results are briefly compared and discussed. Finally, a simulation example is shown to confirm effectiveness of the proposed design scheme of this paper.

Finite-time stabilization of nonlinear time-varying systems involving impulsive action

This article investigates the finite-time stabilization of time-varying impulsive systems by time-varying feedback control. Moreover, the stabilizing and destabilizing effects of impulses on the stability of the system are considered, where the stabilizing impulses are used to regulate the unstable continuous dynamics. In addition, it can be found that the local and global characteristics of the finite-time stability may be affected by the stabilizing impulses. By establishing the relationship among the time-varying gain functions, the system structure and impulsive time sequences, some sufficient conditions for finite-time stability of time-varying systems involving impulsive action are proposed. Furthermore, based on the results mentioned above, a new class of finite-time controllers is designed for time-varying affine systems with external noises. The effectiveness of the proposed results is illustrated by three examples.

Input-to-state stability of stochastic nonlinear system with delayed impulses.

Stochastic input-to-state stability (SISS) of the stochastic nonlinear system has received extensive research. This paper aimed to investigate SISS of the stochastic nonlinear system with delayed impulses. First, when all subsystems were stable, using the average impulsive interval method and Lyapunov approach, some theoretical conditions ensuring SISS of the considered system were established. The SISS characteristic of the argumented system with both stable and unstable subsystems was also discussed, then the stochastic nonlinear system with multiple delayed impulse jumps was considered and SISS property was explored. Additionally, it should be noted that the Lyapunov rate coefficient considered in this paper is positively time-varying. Finally, several numerical examples confirmed validity of theoretical results.

Domain of Stability Characterization for Power Systems: A Novel Individual Invariance Method

In this paper, we approach the problem of stability in nonlinear systems through a new perspective that views them as a combination of individual artificial systems carefully chosen to simplify the complex structure of nonlinear systems. This is achieved by recasting nonlinear vector fields as an algebraic sum of individual vector fields for which artificial systems with known invariant sets or at least in forms that allow for tractable approximation of their invariant sets. This attempt to restructure nonlinear systems stands out in comparison to other previous attempts like Lure' systems or network based models as a purely mathematical structuring technique that transcends the physical constraints and dependencies within dynamical models and allows the user to creatively construct artificial systems with the sole focus on the overall stability. The theoretical foundation is provided for a theorem about individual invariance to relate the invariant sets of individual artificial systems to the invariant set of their original system in a way that significantly simplifies the task of approximating regions of attraction. Several examples are used to demonstrate this theorem and we also evaluate the use this theorem for the challenging power system stability problems in both AC and DC grids. The proposed method is successfully applied to the IEEE 39- bus New England system, and a DC converter with constant power load giving accurate and realistic estimations of the critical clearing time and stability regions in comparison to state of the art approaches.

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

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

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.

Almost Sure Stability of Dual Switching Continuous-time Nonlinear System With Deterministic and Stochastic Subsystems

Almost Sure Stability of Dual Switching Continuous-time Nonlinear System With Deterministic and Stochastic Subsystems

On the Global Stability of Nonlinear Hurwitz Control Systems

The problem of designing nonlinear control systems by the algebraic polynomial-matrix method using quasilinear models is considered. The quasilinear models of nonlinear plants are easily created on the basis of their nonlinear equations in the Cauchy form. To create these models only the differentiability of the plants nonlinearities is required. The solution to the design problem of the nonlinear Hurwitz control systems using the algebraic polynomial-matrix method is available if the quasilinear model of the plant is controllable. The design with application of this method consists of creating the quasilinear model of the nonlinear plant, generating several polynomials, composing and solving a system of linear algebraic equations. The theorem about the global stability of the equilibrium of the nonlinear systems, represented in the quasilinear model, is proved by the method of Lyapunov functions. The numerical examples of the design of nonlinear Hurwitz control systems are given. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article is concerned with the creation of stable nonlinear control systems with nonlinear elements since the linear systems can’t fulfill the quality requirements demanded from modern control systems. Additionally, the known design methods of nonlinear systems, such as the input-state feedback linearization, backstepping, passivity and others, require the transformation of the original nonlinear equations of the plant into some special form. These transformations are often very difficult to find and to execute. A new algebraic polynomial-matrix design method of the nonlinear control systems using the quasilinear models of nonlinear plants is proposed in the article. This method can be applied if the nonlinearities of the plant are differentiable, and the quasilinear model of the plant is controllable. The theorem about global stability of the equilibrium is proved for the systems designed with this approach. The quasilinear models are easily created on the basis of the original equations in the Cauchy form for a given nonlinear plant. The algebraic polynomial-matrix design method is very simple: the quasilinear model is created; several polynomials are calculated with the use of this model and the linear algebraic equations system is composed. The solution of this system allows us to write down the expression which defines the control law as a nonlinear function of the plant’s state variables. This system will be globally or locally stable if the conditions of the theorem or the corollaries, proved in this article, are satisfied. The numerical examples illustrate the application of the suggested approach to the design of nonlinear Hurwitz control systems for the nonlinear plants. The main advantages of this approach: the quasilinear models are created quite easily; the nonlinear control law is found as a solution to the system of linear equations. The results can be applied to the creation of nonlinear control systems of nonlinear plants in many industries: shipbuilding, aircraft construction, automobile construction, agriculture and many others.

Uniform Local and Global Asymptotic Stabilization of Nonlinear Periodic Discrete-Time Systems by State Feedback

Uniform Local and Global Asymptotic Stabilization of Nonlinear Periodic Discrete-Time Systems by State Feedback

Event-Triggered Robust Adaptive Dynamic Programming for Multiplayer Stackelberg-Nash Games of Uncertain Nonlinear Systems.

In this article, an event-triggered robust adaptive dynamic programming (ETRADP) algorithm is developed to solve a class of multiplayer Stackelberg-Nash games (MSNGs) for uncertain nonlinear continuous-time systems. Considering the different roles of players in the MSNG, the hierarchical decision-making process is described as the designed value functions for the leader and all followers, which assist to transform the robust control problem of the uncertain nonlinear system into an optimal regulation problem of the nominal system. Then, an online policy iteration algorithm is formulated to solve the derived coupled Hamilton-Jacobi equation. Meanwhile, an event-triggered mechanism is designed to alleviate computational and communication burdens. Moreover, critic neural networks (NNs) are constructed to obtain the event-triggered approximate optimal control polices for all players, which constitute the Stackelberg-Nash equilibrium of the MSNG. By using Lyapunov's direct method, the stability of the closed-loop uncertain nonlinear system is guaranteed under the ETRADP-based control scheme in the sense of uniform ultimate boundedness. Finally, a numerical simulation is provided to demonstrate the effectiveness of the present ETRADP-based control scheme.

A new design method to global asymptotic stabilization of strict-feedforward nonlinear systems with state and input delays

&lt;abstract&gt;&lt;p&gt;This paper studies the global asymptotic stabilization problem of strict-feedforward nonlinear systems with state and input delays. We will first transform the considered system into an equivalent system by constructing the novel parameter-dependent state feedback controller and introducing the appropriate coordinate transformation. After that, the global asymptotic stability of the closed system is proved by giving the proper Lyapunov-Krasovskii functional and using the stability criterion of time-delay system.&lt;/p&gt;&lt;/abstract&gt;

Adaptive event-triggered prescribed-time stabilization of uncertain nonlinear systems with asymmetric time-varying output constraint

Adaptive event-triggered prescribed-time stabilization of uncertain nonlinear systems with asymmetric time-varying output constraint

Stabilization and decay rate estimation of nonlinear flexible marine riser system with the rotational inertia under nonlinear boundary controls

Stabilization and decay rate estimation of nonlinear flexible marine riser system with the rotational inertia under nonlinear boundary controls

Stability Results for Nonlinear Fractional Differential Equations with Incommensurate Orders

This paper aims to establish some novel conditions for revealing the stability of nonlinear incommensurate fractional-order systems formulated using Caputo differential operator. This will be accomplished through introducing satisfactory proofs for these conditions with the help of using Lyapunov function method. All theoretical findings will be verified numerically by observing the wished stability is occurring on the systems at hand.