Stability Analysis Of Nonlinear Systems Research Articles

Contraction theory based adaptive synchronization of chaotic systems

Contraction theory based stability analysis exploits the incremental behavior of trajectories of a system with respect to each other. Application of contraction theory provides an alternative way for stability analysis of nonlinear systems. This paper considers the design of a control law for synchronization of certain class of chaotic systems based on backstepping technique. The controller is selected so as to make the error dynamics between the two systems contracting. Synchronization problem with and without uncertainty in system parameters is discussed and necessary stability proofs are worked out using contraction theory. Suitable adaptation laws for unknown parameters are proposed based on the contraction principle. The numerical simulations verify the synchronization of the chaotic systems. Also parameter estimates converge to their true values with the proposed adaptation laws.

On the stability analysis of nonlinear systems using polynomial Lyapunov functions

In the stability study of nonlinear systems, not to found feasible solution for the LMI problem associated with a quadratic Lyapunov function shows that it doesn’t exist positive definite quadratic Lyapunov function that proves stability of the system, but doesn’t show that the system isn’t stable. So, we must search for other Lyapunov functions. That's why, in the present paper, the construction of polynomial Lyapunov candidate functions is investigated and sufficient conditions for global asymptotic stability of analytical nonlinear systems are proposed to allow computational implementation. The main keys of this work are the description of the nonlinear studied systems by polynomial state equations, the use of an efficient mathematical tool: the Kronecker product; and the non-redundant state formulation. These notations allow algebraic manipulations and make easy the extension of the stability analysis associated to quadratic or homogeneous Laypunov functions towards more general functions. The advantage of the proposed approach is that the derived conditions proving the stability of the studied systems can be presented as linear matrix inequalities (LMIs) feasibility tests and the obtained results can show in some cases how the polynomial Lyapunov functions leads to less conservative results than those obtained via quadratic (QLFs) or monomial Laypunov functions. This contribution to the stability analysis of high order nonlinear continuous systems can be extended to the stability, robust analysis and control of other classes of systems.

FUZZY LYAPUNOV METHOD FOR STABILITY CONDITIONS OF NONLINEAR SYSTEMS

This paper proposes a fuzzy Lyapunov method for stability analysis of nonlinear systems represented by Tagagi-Sugeno (T-S) fuzzy model. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Based on fuzzy Lyapunov functions, some stability conditions are derived to ensure nonlinear systems are asymptotic stable. By using parallel distributed compensation (PDC) scheme, we design a nonlinear fuzzy controller for the nonlinear system. This control problem will be reformulated into linear matrix inequalities (LMI) problem.

An approximate stability analysis of nonlinear systems described by Universal Learning Networks

Stability is one of the most important subjects in control systems. As for the stability of nonlinear dynamical systems, Lyapunov's direct method and linearized stability analysis method have been widely used. But, it is generally recognized that finding an appropriate Lyapunov function is fairly difficult especially for the nonlinear dynamical systems, and also it is not so easy for the linearized stability analysis to find the locally asymptotically stable region. Therefore, it is crucial and highly motivated to develop a new stability analysis method, which is easy to use and can easily study the locally asymptotically stable region at least approximately, if not exactly. On the other hand, as for the calculation of the higher order derivative, Universal Learning Networks (ULNs) are equipped with a systematic mechanism that calculates their first and second order derivatives exactly. So, in this paper, an approximate stability analysis method based on @h approximation is proposed in order to overcome the above problems and its application to a nonlinear dynamical control system is discussed. The proposed method studies the stability of the original trajectory by investigating whether the perturbed trajectory can approach the original trajectory or not. The above investigation is carried out approximately by using the higher order derivatives of ULNs. In summarizing the proposed method, firstly, the absolute values of the first order derivatives of any nodes of the trajectory with respect to any initial disturbances are calculated by using ULNs. If they approach zero at time infinity, then the trajectory is locally asymptotically stable. This is an alternative linearized stability analysis method for nonlinear trajectories without calculating Jacobians directly. In the method, the stability analysis of time-varying systems with multi-branches having any sample delays is possible, because the systems are modeled by ULNs. Secondly, the locally asymptotically stable region, where asymptotical stability is secured approximately, is obtained by finding the area where the first order terms of Taylor expansion are dominant compared to the second order terms with @h approximation assuming that the higher order terms more than the third order are negligibly small in the area. Simulations of an inverted pendulum balancing system are carried out. From the results of the simulations, it is clarified that the stability of the inverted pendulum control system is easily analyzed by the proposed method in terms of studying the locally asymptotically stable region.

Switching Model Construction and Stability Analysis for Nonlinear Systems

This paper presents switching model construction and stability analysis for a class of nonlinear systems. A switching fuzzy model newly developed in this paper is employed to represent the dynamics of a nonlinear system. A key feature of the switching fuzzy model construction is to find the so-called minimum distance sector by solving a nonlinear optimization problem. Next, we discuss the stability of a switching fuzzy model. To take advantage of the switching fuzzy model, we introduce a piecewise Lyapunov function that mirrors its structure. We show that the piecewise Lyapunov function approach provides less conservative results for the typical quadratic Lyapunov function approach. Illustrative examples demonstrate the utility of the switching model construction and the stability analysis.

Fuzzy Control for Nonlinear Systems via Neural-Network-Based Approach

The stabilization problem is considered in this study for a nonlinear system. It is shown that the stability analysis of nonlinear systems can be reduced into linear matrix inequality (LMI) problems. First, the neural-network (NN) model is employed to approximate a nonlinear system via the backpropagation algorithm. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. In terms of Lyapunov's direct method, a sufficient condition is provided to guarantee the stability of nonlinear systems. Based on this criterion, a model-based fuzzy controller is then designed to stabilize the nonlinear system and the H∞ control performance is achieved at the same time. Finally, two examples with numerical simulations are given to illustrate the control methodology.

A new approach to exponential stability analysis of nonlinear systems

An effective method for analyzing the stability of nonlinear systems is developed. After introducing a novel concept named the point-wise generalized Dahlquist constant for any mapping and presenting its useful properties, we show that the point-wise generalized Dahlquist constant is sufficient for characterizing the exponential stability of nonlinear systems.

STRICT LYAPUNOV FUNCTION AND CHETAEV FUNCTION FOR STABILITY/INSTABILITY ANALYSIS OF THE PENDULUM

Stability analysis of nonlinear systems can be carried out by the Lyapunov's first method (linearization technique). However, by utilizing this approach most insight about the physical system may be lost. Alternative approaches are available to exploit system characteristics such as those associated to energy concepts for stability and instability analysis. In this paper, with reference to a simple nonlinear system —the pendulum—, we proposed first, a simple strict Lyapunov function motivated by energy consideration to study asymptotic stability of some equilibria, and second a Chetaev function to analyze instability of the remaining ones. A remarkable point is that both functions have a common structure based on the system's energy, and differs only in a single scalar parameter.

AUTOMATIC GENERATION OF LYAPUNOV FUNCTIONS USING GENETIC PROGRAMMING

The selection of a suitable Lyapunov function is a critical step in the stability analysis of nonlinear systems. The appropriate choice can lead to the possibility of guaranteeing increased regions of attraction in the vicinity of a desired stable equilibrium point. Typically, its selection is limited to well-known functions and on the intuition of the researcher, which can lead to suboptimal functions. This paper describes the preliminary results obtained using a novel method to identify Lyapunov functions using genetic programming.

Modelling, control, and stability analysis of non-linear systems using generalized fuzzy neural networks

This paper presents an adaptive fuzzy neural controller (AFNC) suitable for modelling and control of MIMO non-linear dynamic systems. The proposed AFNC has the following salient features: (1) fuzzy neural control rules can be generated or deleted dynamically and automatically; (2) uncertain MIMO non-linear systems can be adaptively modelled on line; (3) adaptation and learning speed is fast; (4) expert knowledge can be easily incorporated into the system; (5) the structure and parameters of the AFNC can be self-adaptive in the presence of uncertainties to maintain a high control performance; and (6) the asymptotical stability of the system is established using the Lyapunov approach. Simulation studies on a two-link robot manipulator show that the performance of the proposed controller is better than that of some existing fuzzy/neural methods.

A WAVELET-GALERKIN PROCEDURE TO INVESTIGATE TIME-PERIODIC SYSTEMS: TRANSIENT VIBRATION AND STABILITY ANALYSIS

A wavelet-Galerkin procedure is introduced in order to obtain transient and periodic solutions of multi-degree-of-freedom (d.o.f.s) dynamical systems with time-periodic coefficients. Numerical comparisons, achieved with a Runge–Kutta method, emphasize that the wavelet-based procedure is reliable even in the case of problems involving both smooth or non-smooth parametric excitations and a relatively large number of degrees of freedom. The procedure is then applied to study the vibrations of some theoretical parametrically excited systems. Since problems of stability analysis of non-linear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, the method is shown to be effective for the computation of the Floquet exponents that characterize stable/unstable parameters areas and consequently allows estimators for stability/instability levels to be provided. Stability diagrams of some theoretical examples including a single-degree-of-freedom Mathieu oscillator and a two-degree-of-freedom parametrically excited system, illustrate the relevance of the method. Finally, future studies are outlined for the extension of the wavelet method to the non-linear case.

Relaxed persistency of excitation for uniform asymptotic stability

The persistency of excitation property is crucial for the stability analysis of parameter identification algorithms and adaptive control loops. We propose a relaxed definition of persistency of excitation and we use it to establish uniform global asymptotic stability (UGAS) for a large class of nonlinear, time-varying systems. Our relaxed definition is conceptually equivalent to the definition introduced in Loria et al. (1999), but is easier to verify. It is useful for stability analysis of nonlinear systems that arise, for example, in nonlinear adaptive control, stabilization of nonholonomic systems, and tracking control. Our proof of UGAS relies on some integral characterizations of UGAS established in Teel et al. (2000). These characterizations also streamline the proof of UGAS in the presence of (uniform) classical persistency of excitation.

Construction of Piecewise Linear Lyapunov Functions and Its Fast Method

In this paper, the issue of constructing piecewise linear Lyapunov functions (PWLLF) for stability analysis of nonlinear systems satisfying generalized sector conditions is addressed. The existence condition of PWLLF is represented by a linear programming problem (LP in brief). If the optimal value of the LP is negative, then PWLLF is constructed by using the optimal solution of it. When the optimal value of the LP is nonnegative, the candidate of PWLLF is modified with additional freedom and a new LP is formulated corresponding to the new PWLLF candidiate. It is shown that the optimal value of the resulting new LP is always less than or equal to that of the old LP. The main purpose of this paper is to propose a fast method to solve LPs which appear repeatedly in constructing PWLLF. The method is based on a kind of sensitivity analysis approach of LP and utilizes the special structure of LPs that appears in constructing PWLLF.

Estimating the domain of attraction for power systems via a group of damping-reflected energy functions

The well-known Lyapunov direct method provides precise and rigorous theoretical backgrounds for stability analysis of nonlinear systems. However, incorporating damping effects into accurate estimates of the domain of attraction has proved difficult, which is pointed as a minor but crucial drawback. This paper presents a new method to derive a group of Lyapunov functions to reflect the damping effects by considering the first integral of motion obtained from the system governing equations when transfer conductances are neglected. A systematical approach of converting some part of damping loss into some appropriate system energy terms is developed. Examples show that the proposed method remarkably improves the estimation of the region of attraction compared with the conventional approach.

Some remarks on "System analysis via integral quadratic constraints"

The author analyses results obtained by Megretski and Rantzer (ibid., vol.42, p.819-30, 1997) on the stability analysis of nonlinear systems. He establishes the connections and relations between their results and the abstract quadratic criterion developed by Yakubovich (Automation Remote Control, p.5-30, 1967).

Stability analysis of fuzzy control systems

In this paper, we present a general review of the stability issue as related to fuzzy control systems. The concept of stability and the general criterion used for fuzzy control systems are discussed. The intuitive and nonlinear system stability analysis of fuzzy control systems are given. Popov's technique is proposed to test the stability of fuzzy systems.

A solution to the two-dimensional findpath problem

The ‘findpath problem’ a well-known problem in robotics, is the problem of finding a path for a moving solid among other solid obstacles. In this paper, a solution is proposed for the two-dimensional case where two point masses are required to move to designated areas or targets located in the horizontal plane while avoiding moving or stationary planar objects. The main tool used to solve the problem is the 'second or direct method of Liapunov', a powerful mathematical tool usually associated with the stability analysis of nonlinear systems. The theory developed from solving the two-dimensional findpath problem is then applied to the problem of cooperation between two planar robot arms. Computer simulations show the effectiveness of the proposed method

Energy function for power systems with transmission losses

Stability analysis of nonlinear systems draws heavily on energy functions. For power systems, energy functions have been developed for single machine systems and for the special case of constant power loads and lossless transmission lines, extended to multimachine systems. In this paper a circuit energy approach is reconciled with the traditional machine equations using the steady state solution for the transmission network. Physically based development of the energy function starts from the model analysed in a stationary reference frame. The monotonic convergence of the energy is demonstrated on a 3 machine system with transmission losses.

Vector Lyapunov Function Method: Theory and Application to Complex Industrial Systems

Subject of this lecture is Vector Lyapunov functions (VLF) method in nonlinear theory for large-scale systems and there industrial applications. Abstract control theory like axiomatic theory of system of processes with controls and perturbations: concepts, dynamic properties, discrete and continuous time: axiomatic VLF method. Algorithms for inference of theorems on dynamical properties for nonlinear systems. Software packages for theorem inference, for stability analysis of nonlinear systems. Application to industrial problems. Nonlinear dynamics of orbital telescopes and geostationary spacecraft. Nonlinear methods, results and software for dynamic investigations of fail-safe gyromoment attitude control systems of communication spacecrafts. Method oriented software DYNAMICS for CAD-systems.

MICROPROCESSOR BASED NONLINEAR EXCITATION CONTROLLER FOR SYNCHRONOUS MACHINES

ABSTRACT The synchronous machines in power systems have the excitation systems of field winding to improve the transient stability of power systems. The synchronous machines are nonlinear systems, however, the excitation systems have been designed based on linearized systems, because the stability analysis of nonlinear systems is difficult and the generalized control method of nonlinear systems has not been developed yet. This paper proposes a nonlinear excitation controller for synchronous machines in power systems. For the sake of simplification of the nonlinear state feedback control, this paper considers both the transient stability and the suppression of the rotor swings in transient conditions. The effect of the automatic voltage regulator (AVR) for the synchronous generator is not discussed here. This controller is designed based on Lyapunov's direct method that can directly judge the stability of nonlinear systems. The usefulness and validity of the proposed excitation controller are confirmed by ...