Second Method Of Lyapunov Research Articles

Control Lyapunov function optimal sliding mode controllers for attitude tracking of spacecraft

The attitude tracking control problem for a rigid spacecraft using two optimal sliding mode control laws is addressed. Integral sliding mode (ISM) control is applied to combine the first-order sliding mode with optimal control and is applied to quaternion-based spacecraft attitude tracking maneuvres with external disturbances and an uncertainty inertia matrix. For the optimal control part the control Lyapunov function (CLF) approach is used to solve the infinite-time nonlinear optimal control problem, whereas the Lyapunov optimizing control (LOC) method is applied to solve the finite-time nonlinear optimal control problem. The second method of Lyapunov is used to show that tracking is achieved globally. An example of multiaxial attitude tracking maneuvres is presented and simulation results are included to demonstrate and verify the usefulness of the proposed controllers.

Online Identification of PMSM Parameters: Parameter Identifiability and Estimator Comparative Study

In this paper, a model-reference-based online identification method is proposed to estimate permanent-magnet synchronous machine (PMSM) parameters during transients and in steady state. It is shown that all parameters are not identifiable in steady state and a selection has to be made according to the user's objectives. Then, large signal convergence of the estimated parameters is analyzed using the second method of Lyapunov and the singular perturbations theory. It is illustrated that this method may be applied with a decoupling control technique that improves convergence dynamics and overall system stability. This method is compared with an extended Kalman filter (EKF)-based online identification approach, and it is shown that, in spite of its implementation complexity with respect to the proposed method, EKF does not give better results than the proposed method. It is also shown that the use of a simple PMSM model makes estimated parameters sensitive to those supposed to be known whatever the estimator is (both the proposed method and EKF). The simulation results as well as the experimental ones, implemented on a nonsalient pole PMSM, illustrate the validity of the analytic approach and confirm the same conclusions.

Analysis and synthesis of discrete-time systems

This paper presents an introduction to the analysis and synthesis of sampled-data (discrete-time) systems, i.e. systems in which some or all signals can change values only at discrete values of time. The description of these systems is presented using state-space concepts. Following an introduction to linear discrete-time systems including systems where two or more samplers operate at different frequencies, here is a brief introduction to non-linear systems including the study of stability using the Second Method of Lyapunov. The latter part of the paper describes pulse-width modulated discrete systems. The final section considers the synthesis of systems designed to reach equilibrium states in the minimum number of sampling periods. The concepts discussed in the paper are illustrated with a large number of examples.

Optimal Sliding Mode Controllers for Attitude Stabilization of Flexible Spacecraft

The robust optimal attitude control problem for a flexible spacecraft is considered. Two optimal sliding mode control laws that ensure the exponential convergence of the attitude control system are developed. Integral sliding mode control (ISMC) is applied to combine the first‐order sliding mode with optimal control and is used to control quaternion‐based spacecraft attitude manoeuvres with external disturbances and an uncertainty inertia matrix. For the optimal control part the state‐dependent Riccati equation (SDRE) and optimal Lyapunov techniques are employed to solve the infinite‐time nonlinear optimal control problem. The second method of Lyapunov is used to guarantee the stability of the attitude control system under the action of the proposed control laws. An example of multiaxial attitude manoeuvres is presented and simulation results are included to verify the usefulness of the developed controllers.

Quasi-Continuous Higher Order Sliding-Mode Controllers for Spacecraft-Attitude-Tracking Maneuvers

This paper studies higher order sliding-mode-control laws to deal with some spacecraft-attitude-tracking problems. Quasi-continuous second- and third-order sliding controllers and differentiators are applied to quaternion-based spacecraft-attitude-tracking maneuvers. A class of linear sliding manifolds is selected as a function of angular velocities and quaternion errors. The second method of Lyapunov is used to show that tracking is achieved globally. An example of multiaxial attitude-tracking maneuvers is presented, and simulation results are included to verify and compare the practical usefulness of the various controllers.

On the Stability of a Friction Compensation Strategy for Flexible-Joint Manipulators

A new control strategy for flexible-joint manipulators with joint friction is proposed. The proposed controller includes two main components: a friction compensating torque, and a composite controller torque to compensate for the friction in the joints and the flexibility of the joints, respectively. The novel approach that is used to compensate for the friction in the joints includes a nonlinear one-state LuGre model and a PD control. The second method of Lyapunov is utilized to find sufficient conditions for the global asymptotic stability of the friction compensating torque and it is shown that the tracking error is bounded as long as appropriate PD gains are chosen. The performance of the proposed controller is experimentally verified using a setup of a two-rigid-link flexible-joint manipulator. Experimental results are presented for different flexibilities of the joints and also for different desired trajectories. These results demonstrate the high performance of the controller under different situations while the tracking error remains very small.

Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.

Optimal Sliding Mode Controllers for Spacecraft Attitude Manoeuvres

This paper studies optimal sliding mode controller designs using integral sliding mode control (ISM) to deal with some spacecraft attitude control problems. Integral sliding mode control combining the first order sliding mode and optimal control is applied to quaternion-based attitude control for rest-to-rest manoeuvres with external disturbances. For the optimal control part, the state dependent Riccati equation (SDRE) approach and control Lyapunov function (CLF) are used to solve the infinite-time nonlinear optimal problem. The second method of Lyapunov is used to show that asymptotic stability is achieved. An example of rest-to-rest attitude manoeuvres is presented and simulation results are included to verify and compare the usefulness of the various controllers.

Optimal sliding-mode guidance with terminal velocity constraint for fixed-interval propulsive maneuvers

An optimal strategy based on minimum control effort with terminal position and velocity constraints is developed for an exoatmospheric interceptor in order to generate effective intercept for fixed-interval propulsive maneuvers. It is then integrated with sliding-mode control theory to derive a robust optimal guidance law. In addition, this guidance law is generalized for intercepting an arbitrarily time-varying target maneuver. The new guidance method's robustness against disturbances and good miss distance performance are proved by the second method of Lyapunov and simulation results. The presented guidance law is simple to implement in practical applications and requires less acceleration command in comparison to optimal guidance law.

Mixed control of uncertain jumping time-delay systems

Abstract In this paper, we investigate a class of linear continuous-time systems with Markovian jump parameters. An integral part of the system dynamics is a delayed state with time-varying and bounded delays. The jumping parameters are modeled as a continuous-time, discrete-state Markov process. Employing norm-bounded parametric uncertainties and utilizing the second-method of Lyapunov, we examine the problem of designing a mixed H 2 / H ∞ controller which minimizes a quadratic H 2 performance measure while satisfying a prescribed H ∞ -norm bound on the closed-loop system. It is established that sufficient conditions for the existence of the mixed H 2 / H ∞ controller and the associated performance upper bound could be cast in the form of linear matrix inequalities.

Self-Organizing Radial Basis Function Network for Real-Time Approximation of Continuous-Time Dynamical Systems

Real-time approximators for continuous-time dynamical systems with many inputs are presented. These approximators employ a novel self-organizing radial basis function (RBF) network, which varies its structure dynamically to keep the prescribed approximation accuracy. The RBFs can be added or removed online in order to achieve the appropriate network complexity for the real-time approximation of the dynamical systems and to maintain the overall computational efficiency. The performance of this variable structure RBF network approximator with both Gaussian RBF (GRBF) and raised-cosine RBF (RCRBF) is analyzed. The compact support of RCRBF enables faster training and easier output evaluation of the network than that of the network with GRBF. The proposed real-time self-organizing RBF network approximator is then employed to approximate both linear and nonlinear dynamical systems to illustrate the effectiveness of our proposed approximation scheme, especially for higher order dynamical systems. The uniform ultimate boundedness of the approximation error is proved using the second method of Lyapunov.

Design of Observers for Takagi-Sugeno Systems with Immeasurable Premise Variables: an ℒ2 Approach

A new observer design method is proposed for Takagi-Sugeno systems with immeasurable premise variables. Since the state estimation error can be written as a perturbed system, then the proposed method is based on the ℒ2 techniques to minimize the effect of the perturbations on the state estimation error. The convergence conditions of the observer are established by using the second method of Lyapunov and a quadratic function. These conditions are expressed in terms of Linear Matrix Inequalities (LMI). Finally, the performances of the proposed observer are improved by eigenvalues clustering in LMI region.

Recursive Motion Recovery Based on Dynamic Vision

The motion recovery for a class of movements in the space by using stereo vision is considered by observing at least three points in this paper. The considered motion equation can cover a wide class of practical movements in the space. The observability of this class of movement is clarified. The estimations of the motion parameters which are all time-varying are developed in the proposed algorithm based on the second method of Lyapunov. The assumptions about the perspective system are reasonable, and the convergence conditions are intuitive and have apparently physical interpretations. The proposed recursive algorithm requires minor a priori knowledge about the system and can alleviate the noises in the image data. Furthermore, the proposed algorithm is modified to deal with the occlusion phenomenon. Simulation results show the proposed algorithm is effective even in the presence of measurement noises.

Adaptive Nonlinear Synchronization Control of Twin-Gyro Precession

The slewing motion of a truss arm driven by a V-gimbaled control-moment gyro is studied. The V-gimbaled control-moment gyro consists of a pair of gyros that must precess synchronously. For open-loop slewing motion control, the controller design problem is simplified into finding a feedback controller to steer the two gyros to synchronously track a specific command. To improve the synchronization performance, the integral of synchronization error is introduced into the design as an additional state variable. Based on the second method of Lyapunov, an adaptive nonlinear feedback controller is designed. For more accurate but complicated closed-loop slewing motion control, the feedback linearization technique is utilized to partially linearize the nonlinear nominal model, where two specific output functions are chosen to satisfy the system tracking and synchronization requirements. The system tracking dynamics are bounded by properly determining system indices and command signals. For the partially linearized system, the backstepping tuning function design approach is employed to design an adaptive nonlinear controller. The dynamic order of the adaptive controller is reduced to its minimum. The performance of the proposed controllers is verified by simulation.

MOTION RECOVERY BY USING DYNAMIC VISION

The recovery of motion for a class of movements in the space by using the perspective observation of one point is considered in this paper. The motion equation can cover a wide class of practical movements in the space. The estimations of the position and motion parameters which are all time-varying are simultaneously developed in the proposed algorithm. The formulated problem can be converted into the observation of a dynamical system with nonlinearities. The proposed observer is based on the second method of Lyapunov. First, the parameters relating to the rotation of the motion are identified, where only one camera is needed. Then the position of the moving object is identified, where the stereo vision is necessary. In the third step, the parameters relating to the straight movement are identified. The assumptions about the perspective system are reasonable, and the convergence conditions are intuitive and have apparently physical interpretations. The proposed method requires minor a priori knowledge about the system and can cope with a much more general class of perspective systems. Furthermore, the algorithm is modified to deal with the occlusion phenomenon.

Robust control of multilink flexible manipulators

This paper deals with some robustness aspects of a model based controller used for trajectory tracking in multi-link flexible manipulators. It is known in literature that the finite element formulation over-estimates the natural frequencies of the original system. We show that over-estimation of natural frequencies may lead to unstable closed-loop response for flexible manipulators using a model based inversion control algorithm. A robust controller design based on the second method of Lyapunov using simple quantitative bounds on the model uncertainties is illustrated for use during the trajectory tracking phase in multi-link manipulator control. In order to actively suppress the link vibrations excited during the trajectory-tracking phase, a second controller based on end-point sensing and the rigid Jacobian of the manipulator is used. The performance of the two-stage controller is illustrated with the help of numerical simulations of a flexible elbow manipulator.

FINITE ELEMENT MODELLING OF LAMINATED PIEZO-ELASTIC STRUCTURES: LYAPUNOV STABILITY ANALYSIS

In this study, the dynamic stability of finite element modelling of laminated piezo-elastic structures is investigated. The criteria for stability are established based on the second method of Lyapunov, which considers the energy of the system. The results show that the equations of motion are asymptotically stable. However, energy dissipation through the piezoelectric effect continued at zero feedback gain. This implied that the structure was controlled unconditionally. Subsequently, a control strategy that satisfies the condition of no piezoelectric effects when the gain set to zero is developed.

Adaptive control for mobile robot using wavelet networks

This work improves recent results concerning the adaptive control of mobile robots via neural and wavelet networks, in the sense that the stability proof, based on the second method of Lyapunov, encompasses (1) unmodeled dynamics and disturbances in the robot model; (2) adaptation of all parameters in the wavelet networks; and (3) a flexible procedure for automatically adjusting the wavelet architecture. Prior knowledge of dynamic of the mobile robot and network training is not necessary because the controller learns the dynamics online. The wavelet network's parameters and structure are also adapted online. Simulation results are presented by using parameters of the Magellan mobile robot from IS Robotics, Inc.

Stability of a Trivial Solution of a Mixed Problem for a Parabolic-Type, Distributed-Parameter System

The Lyapunov stability condition is established for a parabolic-type system with delay. N. N. Krasovskii's idea [1] of using functionals instead of functions for the second method of Lyapunov is borrowed here.

Robust Suboptimal Control of Hybrid Stochastic Systems With Time-Delay

A class of hybrid in state control system which modelled as a finite family of differential equations with time-delay is considered. Each model of this family describes the individual mode of the system. The discontinuous transitions between these modes are described by the homogeneous Markov chain. The purpose of the presented paper is to obtain a constant (nonswitching) state feedback control law, which guarantees asymptotic stability in the mean square independently on transition probabilities of the Markov chain and with arbitrary time-delay. The approach to the solution is based on the second method of Lyapunov with using the Lyapunov-Krasovskii functional. The control law is obtained through solving linear matrix equations and linear matrix inequalities.