Concept Of Lyapunov Exponents Research Articles

Admittance-Controlled Teleoperation of a Pneumatic Actuator: Implementation and Stability Analysis

Implementation, experimental evaluation and stability analysis of an admittance-controlled teleoperated pneumatic system is presented. A master manipulator navigates a pneumatic slave actuation, which interacts with a human arm as an environment. Considering the external force in the position control loop in the admittance control, enables the slave to handle the external force independent of the master. The proposed control system is evaluated experimentally using the admittance models with different settings. Stability of the control system is analyzed using the concept of Lyapunov exponents. Parametric stability analysis is conducted to show the effect of changing system parameters on stability.

Nonlinear Phenomena in Cournot Duopoly Model

The economic world is very dynamic, and most phenomena appearing in this world are mutually interconnected. These connections may result in the emergence of nonlinear relationships among economic agents. Research discussions about different markets’ structures cannot be considered as finished yet. Even such a well-known concept as oligopoly can be described with different models applying diverse assumptions and using various values of parameters; for example, the Cournot duopoly game, Bertrand duopoly game or Stackelberg duopoly game can be and are used. These models usually assume linear functions and make analyses of the behavior of the two companies. The aim of this paper is to consider a nonlinear inverse demand function in the Cournot duopoly model. Supposing there is a sufficiently large proportion among the costs of the two companies, we can possibly detect nonlinear phenomena such as bifurcation of limit values of production or deterministic chaos. To prove a sensitive dependence on the initial condition, which accompanies deterministic chaos, the concept of Lyapunov exponents is used. We also point out the fact that even though some particular values of parameters are irrelevant for the above-mentioned nonlinear phenomena, it is worth being aware of their existence.

Lyapunov exponents-based stability analysis and integrated control of rollover mitigation and yaw stabilisation of ground vehicles

Yaw stabilisation and rollover prevention are two key factors in safety of vehicles. Designing a controller that can address both of the safety concerns and evaluating the performance of such a control system using a proper stability analysis tool is of interest. The complexities in dynamics of vehicles, such as tyre dynamics, make the stability analysis of such systems more challenging. In addition, maintaining both of the above objectives, the yaw motion control and rollover mitigation, is contradictory when a vehicle experiences high lateral accelerations. In this research, a T-S fuzzy controller is proposed which prioritises the concern that must be addressed at each time instant based on the state of the vehicle. The controller compromises between the two contradictory objectives of tracking a desired value for the yaw rate and maintaining the rollover index within a limit to ensure wheel lift-off does not occur. A novel method using the concept of Lyapunov exponents is employed to perform system and structural stability analysis. Results of the stability analysis together with simulations demonstrate the advantages of the proposed controller.

Hybrid algorithm for rotor angle security assessment in power systems

Transient rotor angle stability assessment and oscillatory rotor angle stability assessment subsequent to a contingency are integral components of dynamic security assessment (DSA) in power systems. This study proposes a hybrid algorithm to determine whether the post-fault power system is secure due to both transient rotor angle stability and oscillatory rotor angle stability subsequent to a set of known contingencies. The hybrid algorithm first uses a new security measure developed based on the concept of Lyapunov exponents (LEs) to determine the transient security of the post-fault power system. Later, the transient secure power swing curves are analysed using an improved Prony algorithm which extracts the dominant oscillatory modes and estimates their damping ratios. The damping ratio is a security measure about the oscillatory security of the post-fault power system subsequent to the contingency. The suitability of the proposed hybrid algorithm for DSA in power systems is illustrated using different contingencies of a 16-generator 68-bus test system and a 50-generator 470-bus test system. The accuracy of the stability conclusions and the acceptable computational burden indicate that the proposed hybrid algorithm is suitable for real-time security assessment with respect to both transient rotor angle stability and oscillatory rotor angle stability under multiple contingencies of the power system.

Bilateral teleoperation of a pneumatic actuator: experiment and stability analysis

This paper presents design, implementation and stability analysis of a bilateral teleoperated pneumatic actuation system whereby a low-cost pneumatic actuator is navigated by an operator using a commercially-available haptic device. The actuator is subject to an external force, the value of which is scaled and rendered on the haptic device to provide the operator with a feeling of the interaction at the remote site. Sliding mode control scheme is implemented for positioning the pneumatic actuator. The performance of the system is experimentally evaluated through several experiments including interaction with springs having different stiffnesses and an arbitrary resistive/assistive force applied by a human at the actuator side. Stability of the entire system is theoretically proven using the concept of Lyapunov exponents that quantitatively measures convergence/divergence of initially infinitely close solution trajectories.

A novel method to identify boundaries of basins of attraction in a dynamical system using Lyapunov exponents and Monte Carlo techniques

Stability analysis is a critical part of the study of a dynamical system that includes: finding the attractors of the system, determining their stability or instability, and identifying basins of attraction (BoAs) of the attractors. There are some classical methods for finding parts of BoAs, like Lyapunov-based and Non-Lyapunov-based methods; however, due to the limitations of the current methods, the identification of a complex boundary of a BoA is not feasible in addition to the high computational load of these methods. The concept of Lyapunov exponents is a powerful tool for performing stability analysis of highly nonlinear systems. In this paper, a new and intelligent approach for identifying the boundary of the entire BoA using Lyapunov exponents, Monte Carlo techniques, and support vector machine learning algorithm is developed. The proposed approach can identify the boundary of BoAs due to the intelligent sampling brought by Monte Carlo technique, which also dramatically reduces the computational load. Three illustrative examples are presented to demonstrate the effectiveness of the proposed approach.

On Constrained and Energy Efficient Balance Control of a Standing Biped: Experimentation and Stability Analysis

Balancing control is important for biped standing. In spite of large efforts, it is very difficult to design balancing control strategies satisfying three requirements simultaneously: maintaining postural stability, improving energy efficiency, and satisfying the constraints between the biped feet and the ground. To implement such a control, inclusion of the actuators' dynamics is necessary, which complicates the overall system, obstructs the control design, and makes stability analysis more difficult. In this paper, a constrained balancing control meeting all three requirements is designed for a standing bipedal robot. The dynamics of the selected actuators has been considered for developing the motion equations of the overall control system, which has usually been neglected in simulations. In addition, stability analysis of such a complex biped control system has been provided using the concept of Lyapunov exponents (LEs), which shows the significance of actuators' dynamics on the stability region. The paper contributes to balancing standing biped in both the theoretical and the practical sense.

Power system transient stability analysis via the concept of Lyapunov Exponents

Transient stability of an electrical power system refers to the ability of the system to settle at the stable equilibrium point in the post-fault system subsequent to a specific fault scenario. This stability problem can be studied either as a system stability or a structural stability problem. In this paper, the concept of Lyapunov Exponents (LEs) is used to analyze the transient stability of IEEE 3-generator 9-bus and IEEE 16 generator 69 bus benchmark systems. A spectrum of LEs is calculated from the mathematical models and the Largest Lyapunov Exponent (LLE) being negative implies the exponential stability of the post-fault power system. The system stability regions for specific fault scenarios are determined using the invariance property of the LEs from the initial conditions. Furthermore, the structural stability regions in terms of parameters of the pre-fault system dynamic equations are also determined. This study demonstrates that the concept of LEs can be used as a novel method to determine if the post-fault power system is within the stability region of attraction of the new stable equilibrium point, and hence to determine the stability region. It is also proposed that the largest average exponential rate calculated over a short time window is a sound approach for early prediction of the stability.

Lyapunov exponents for systems with unbounded coefficients†

In this article we propose a generalization of the concept of Lyapunov exponents for discrete linear systems which may be used in the case of unbounded coefficients. We show some simplest properties of this generalization and apply it to define a generalization of regular system. Finally, we discuss the problem of stability by linear approximation. † This article was originally published with an error. This version has been corrected. Please see Corrigendum (http://dx.doi.org/10.1080/14689367.2012.756700)

Stability Analysis of Bipedal Robots Using the Concept of Lyapunov Exponents

The dynamics and stability of passive bipedal robot have an important impact on the mass distribution, leg length, and the angle of inclination. Lyapunov’s second method is difficult to be used in highly nonlinear multibody systems, due to the lack of constructive methods for deriving Lyapunov fuction. The dynamics equation is established by Kane method, the relationship between the mass, length of leg, angle of inclination, and stability of passive bipedal robot by the largest Lyapunov exponent. And the Lyapunov exponents of continuous dynamical systems are estimated by numerical methods, which are simple and easy to be applied to the system stability simulation analysis, provide the design basis for passive bipedal robot prototype, and improve design efficiency.

Stability analysis via the concept of Lyapunov exponents: a case study in optimal controlled biped standing

Balancing control is important for biped standing. In spite of large efforts, it is very difficult to design balancing control strategies satisfying three requirements simultaneously: maintaining postural stability, improving energy efficiency and satisfying the constraints between the biped feet and the ground. In this article, a proportional-derivative (PD) controller is proposed for a standing biped, which is simplified as a two-link inverted pendulum with one additional rigid foot-link. The genetic algorithm (GA) is used to search for the control gain meeting all three requirements. The stability analysis of such a deterministic biped control system is carried out using the concept of Lyapunov exponents (LEs), based on which, the system stability, where the disturbance comes from the initial states, and the structural stability, where the disturbance comes from the PD gains, are examined quantitively in terms of stability region. This article contributes to the biped balancing control, more significantly, the method shown in the studied case of biped provides a general framework of systematic stability analysis for certain deterministic nonlinear dynamical systems.

LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

On stability analysis via Lyapunov exponents calculated based on radial basis function networks

The concept of Lyapunov exponents is a powerful tool for analysing the stability of nonlinear dynamic systems, especially when the mathematical models of the systems are available. For real world systems, such models are often unknown. Estimating Lyapunov exponents using a time series has the advantage in that no mathematical model is required. However the time-series-based methods are believed to be reliable only for estimating positive exponents. Furthermore, when nonlinear mapping is applied for deriving the neighbourhood-to-neighbourhood matrices, the loads of mathematical deduction and programming increase significantly, which makes it unfeasible to nonlinear systems with high dimensions. In contrary, the model-based methods are constructive and reliable for calculating both positive and non-positive exponents. The use of the system Jacobians is the key to the advantage of the model-based methods. In this article, a novel approach is proposed, where the system Jacobians are derived based on system approximation using the radial basis function network. The proposed method inherits the advantage of the model-based methods, yet no mathematical model is required. Two case studies are presented to demonstrate the efficacy of the proposed method. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either difficult to model due to complexities or unknown.

Switching contact task control in hydraulic actuators: Stability analysis and experimental evaluation

AbstractA switching contact task control for hydraulic actuators is proposed. The controller is built upon three individually designed control laws for three phases of motion: (1) position regulation in free space, (2) impact suppression and stable transition from free to constrained motion and (3) force regulation in sustained‐contact motion. The position and force control schemes are capable of asymptotic set‐point regulation in the presence of actuator friction and without the complexity of sliding mode or adaptive control techniques. The intermediate impact control scheme is included for the first time to dampen the undesirable impacts and dissipate the impact energy that could potentially drive the whole system unstable. The solution concept and the stability of the complete switching control system are analyzed rigorously using the Filippov's solution concept and the concept of Lyapunov exponents. Both computer simulations and experiments are carried out to demonstrate the efficacy of the designed switching control law. Copyright © 2009 John Wiley & Sons, Ltd.

Nearly optimal neural network stabilization of bipedal standing using genetic algorithm

In this work, stability control of bipedal standing is investigated. The biped is simplified as an inverted pendulum with a foot-link. The controller consists of a general regression neural network (GRNN) feedback control, which stabilizes the inverted pendulum in a region around the upright position, and a PID feedback control, which keeps the pendulum at the upright position. The GRNN controller is also designed to minimize an energy-related cost function while satisfying the constraints between the foot-link and the ground. The optimization has been carried out using the genetic algorithm (GA) and the GRNN is directly trained during optimization iteration process to provide the closed loop feedback optimal controller. The stability of the controlled system is analyzed using the concept of Lyapunov exponents, and a stability region is determined. Simulation results show that the controller can keep the inverted pendulum at the upright position while nearly minimizing an energy-related cost function and keeping the foot-link stationary on the ground. The work contributes to bipedal balancing control, which is important to the development of bipedal robots.

On stabilization of bipedal robots during disturbed standing using the concept of Lyapunov exponents

Design of balancing control and the stability analysis of a biped during disturbed standing are investigated. A PD-based switching state feedback control is used to stabilize the biped at the upright position while satisfying the constraints between the feet and the ground. The concept of Lyapunov exponents is used for the stability analysis, and a stability region is determined. Furthermore, the stability region is compared and agrees well with the one from the previous work that predicts the feasible movement during which balance of human standing can be maintained. This agreement shows the potential of the concept of Lyapunov exponents to be used as a measure of balancing of human standing. The work contributes to bipedal balancing control, which is important in the development of bipedal robots.

Calculation of Lyapunov Exponents using Nonstandard Finite Difference Discretization Scheme: A Case Study

Stability analysis of nonsmooth systems, using Lyapunov direct method, is a difficult task. An alternative way of studying the stability in such systems is to employ the concept of Lyapunov exponents. However, two difficulties in the calculation of Lyapunov exponents, i.e. numerical artifacts and low computational efficiency, often prohibit the application of this otherwise powerful method. In this paper, a case study of the stability analysis of a switching contact task control system using the concept of Lyapunov exponents is presented. The goal is to carry out a numerical investigation of the method of nonstandard finite difference discretization, for constructing discrete models of differential equations that describe the Lyapunov exponents for such a nonsmooth system. It is shown that, as compared with the standard fourth-order Runge-Kutta method, the nonstandard finite difference discretization scheme provides numerically stable results with a larger critical integration step-size and less computation time. Therefore, from numerical stability and computational efficiency viewpoints, the nonstandard finite difference discretization method is meritorious and should be given consideration when stability analysis of systems using Lyapunov exponents is of concern.

Pathological foliations and removable zero exponents

The ergodic theory of uniformly hyperbolic, or “Axiom A”, diffeomorphisms has been studied extensively, beginning with the pioneering work of Anosov, Sinai, Ruelle and Bowen ([An], [Si], [Ru], [Bo]). While uniformly hyperbolic systems enjoy strong mixing properties, they are not dense among C diffeomorphisms [Sm], [AS]. Using the concept of Lyapunov exponents, Pesin introduced a weaker form of hyperbolicity, which he termed nonuniform hyperbolicity. Nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones. Our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms. As a curious by-product of our construction, we prove that a pathological feature of central foliations – the complete failure of absolute continuity – can exist in a C-open set of volume-preserving diffeomorphisms. We recall the definition of a nonuniformly hyperbolic diffeomorphism. A real number λ is a Lyapunov exponent of the diffeomorphism g : M → M if there exists a nonzero vector v ∈ TM such that

Algebraic Approach to the Calculation of Lyapunov Exponents

The concept of Lyapunov exponents is presented. The possibility of formulating methods for algebraically determining the signs and magnitudes of Lyapunov exponents is investigated. Special considerations have been given for the determination of zero Lyapunov exponents. Proposed methods are applied to a number of Hamiltonian and non-Hamiltonian systems.

Instability And Chaos In The Elastica Type Problem Of Parametrically Excited Columns

The analysis of a parametrically excited simply supported column, made of non-linear elastic material, including the effects of the (elastica type) large deflections and the changes in its length, is presented. The equation of motion derived is in the form of the non-linear Mathieu equation ÿ + (ω 2 + a cos θ t) y + b cos θ ty 3 + cy 3 = 0, which was analytically investigated in a previous paper by the present authors and their colleagues [1]. The main results as obtained there are described. The influence of the various parameters involved and the initial conditions on the instability regions, as well as on the nature of the system's response, are investigated. The possibility of chaotic behaviour is examined within the concept of Lyapunov exponents.