Vector Lyapunov Function Approach Research Articles

Finite-Time Stabilization for Continuous Triangular Systems via Vector Lyapunov Function Approach

This article provides finite-time stabilizing control laws for the continuous triangular systems, in which the nominal dynamics is the chain of power integrators with the power of a positive odd rational number. For a class of continuous lower-triangular systems, the linear controllers are designed. By invoking a vector Lyapunov function theory, there is no need to use the usual back-stepping technique and a quite number of inequality computations are avoided. For a class of continuous upper-triangular systems, the nested-saturation controllers are constructed. In this case, the use of a vector Lyapunov function theory also helps to reduce computational burden, and allows us to summarize a same group of parameter conditions guaranteeing both the reduction of saturated terms and the finite-time stability of the reduced system.

Tracking Control with Robotic Systems for a Moving Target: A Vector Lyapunov Function Approach

Abstract In this paper, we develop a controller for tracking a moving target using a manipulator equipped with a vision-sensor at the end-effector. The proposed control system is composed of a target velocity observer, a position-based visual servo (PBVS) and a proportional-derivative (PD+) tracking controller. This control scheme is analyzed as an interconnected system where input-to-state stability is proved for PBVS and PD+ subsystems, and the overall stability is proved using vector Lyapunov functions. The analysis technique further aids the determination of control parameters which defines the response of the whole system. A simplified notion of a manipulator is used to develop the theory and the application is demonstrated through simulation results in the context of on-orbit servicing where the objective is to track a non-cooperative tumbling satellite with a free-floating space manipulator.

Vector Lyapunov Function based Input-to-State Stability of Stochastic Impulsive Switched Time-Delay Systems

In this paper, the input-to-state stability is studied for stochastic impulsive switched time-delay systems. Using the vector Lyapunov function, average dwell time, and the properties of $M$ -matrix, different types of sufficient conditions are established. Both the case that the continuous dynamics is stable and the case that the discrete dynamics is stable are addressed, and the stability conditions are obtained. In the obtained stability conditions, different components of the vector Lyapunov function are allowed to be coupled; the information in consecutive impulsive switching intervals is also allowed to be coupled. Therefore, the magnification on the corresponding coupling items is avoided and the obtained results are more general and less conservative than the existing results. Furthermore, we investigate the relationships among the vector Lyapunov function approach, the approach based on the comparison principle and the scalar Lyapunov function approach. According to the vector Lyapunov function, the comparison system is constructed and the scalar-Lyapunov-function-based stability conditions are established. Finally, the applicability of our results is illustrated through two examples from neural systems and the synchronization problem of chaos-based secure communication systems.

Nonlinear impulsive systems: 2D stability analysis approach

This paper contributes to the stability analysis for nonlinear impulsive dynamical systems based on a vector Lyapunov function and its divergence operator. The new method relies on a 2D time domain representation. Different types of stability notions for a class of nonlinear impulsive systems are studied using a vector Lyapunov function approach. The results are applied to analyze the stability of a class of Lipschitz nonlinear impulsive systems based on Linear Matrix Inequalities. Some numerical examples illustrate the feasibility of the proposed approach.

Stability of a Class of Nonlinear Repetitive Processes with Application to Iterative Learning Control

The paper considers a discrete repetitive process described by the state-space model with static nonlinearity satisfying quadratic constraints (in particular, the SISO case with the sector bounded nonlinearity). Based on the vector Lyapunov function approach developed by the authors, sufficient conditions of pass profile exponential stability are obtained in the form of linear matrix inequalities. These conditions are then applied to the design of an iterative learning control law for a simple nonlinear mechanical system.

Stability of Infinite Dimensional Interconnected Systems with Impulsive and Stochastic Disturbances

Some research on the stability with mode constraint for a class of infinite dimensional look-ahead interconnected systems with impulsive and stochastic disturbances is studied by using the vector Lyapunov function approach. Intuitively, the stability with mode constraint is the property of damping disturbance propagation. Firstly, we derive a set of sufficient conditions to assure the stability with mode constraint for a class of general infinite dimensional look-ahead interconnected systems with impulsive and stochastic disturbances. The obtained conditions are less conservative than the existing ones. Secondly, the controller for a class of look-ahead vehicle following systems with the above uncertainties is constructed by the sliding mode control method. Based on the obtained new stability conditions, the domain of the control parameters of the systems is proposed. Finally, a numerical example with simulations is given to show the effectiveness and correctness of the obtained results.

Control of a Parametrically Excited Crane: A Vector Lyapunov Approach

In this brief, a controller is proposed in order to avoid the parametric resonance effect and to attenuate the load oscillations in a parametrically excited crane, ensuring precise load transfer during the load movement despite model uncertainties and un-modeled dynamic actuators. The nonlinear controller proposed in this brief is motivated by the Twisting algorithm and the design uses the vector Lyapunov function approach ensuring the ultimate bounded stability of the overall closed-loop system. The experiments conducted over a laboratory platform resemble quite well the simulations, confirming the obtained results.

Delay-dependent stability of discrete-time interconnected systems

This paper studies the delay-dependent stability problem of discrete-time interconnected systems with time-varying delays. By using vector Lyapunov function approach and linear matrix inequalities (LMIs), new stability conditions are derived. These results proposed in this paper are all at subsystems level. After comparing with the existing results, it is shown that these conditions are less conservative. A numerical example is presented to demonstrate the effectiveness of the results.

Design of finite‐time stabilizing controllers for nonlinear dynamical systems

AbstractFinite‐time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite‐time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non‐Lipschitzian dynamics. Sufficient conditions for finite‐time stability have been developed in the literature using Hölder continuous Lyapunov functions. In this paper, we extend the finite‐time stability theory to revisit time‐invariant dynamical systems and to address time‐varying dynamical systems. Specifically, we develop a Lyapunov‐based stability and control design framework for finite‐time stability as well as finite‐time tracking for time‐varying nonlinear dynamical systems. Furthermore, we use the vector Lyapunov function approach to study finite‐time stabilization of compact sets for large‐scale dynamical systems. Copyright © 2008 John Wiley & Sons, Ltd.

Stability analysis of uncertain fuzzy large-scale system

Abstract In this paper the stability problem of continuous and discrete time fuzzy large-scale systems with uncertainties is studied. Each of these systems consists of J interconnected subsystems which are represented by Takagi–Sugeno models. The stability conditions are derived by using vector Lyapunov function approach and expressed in terms of linear matrix inequalities. Two examples are given to demonstrate the correctness of the results.

Stability analysis of fuzzy multivariable systems: Vector Lyapunov function approach

A design methodology for the stabilisation of a class of nonlinear systems which can be identified in the form of a fuzzy dynamic model is presented. The identified model is a fuzzy aggregation of a set of local linear models. The stability condition of the fuzzy model is devised by using the vector Lyapunov function approach and some important issues of stability analysis are discussed. A parameter optimisation algorithm is proposed to construct a set of local linear control laws for the fuzzy system. The performance of the fuzzy model based control technique is illustrated by a simulation example.

Control Design of Nonlinear Systems Using Fuzzy Model and Genetic Algorithm

A control design methodology for stabilization of a class of nonlinear systems is presented. The systems are identified in the form of a fuzzy model and its structure is a fuzzy aggregation of a set of local linear models. The stability condition of the fuzzy dynamic model is determined by using the vector Lyapunov function approach. A parameter optimization algorithm is proposed to construct a set of local linear control laws for the fuzzy systems by implementing the modified Genetic Algorithm. The performance of the fuzzy model based control technique is illustrated by a simulation example.

A vector energy function approach for security analysis of AC/DC systems

The authors examine dynamic behavior in system models that reflect reasonably detailed third-order high-voltage DC (HVDC) dynamics along with AC system models that include reactive flows, and frequency and voltage-dependent load models. A vector Lyapunov function approach is used to define a system-wide energy function that can be used for general security analysis. They describe the derivation of individual component Lyapunov functions for simplified models of HVDC links connected to infinitely strong AC systems, along with a standard AC only system Lyapunov function. A novel method of obtaining weighting coefficients to sum these components for the overall system energy function is proposed. Use of the new energy function for transient stability and security analysis was illustrated in a test system. >

A constructive methodology for estimating the stability regions of interconnected nonlinear systems

A constructive methodology for estimating the stability regions of interconnected nonlinear systems is presented. One appealing figure of the methodology is that it significantly reduces the undesirable conservativeness of the Lyapunov function approach. This constructive methodology is iterative in character and yields a sequence of estimated stability regions which is shown to be a strictly monotonic increasing sequence and yet each of the iterations lies inside the entire stability region. The methodology applies to both the scalar Lyapunov function approach and the vector Lyapunov function approach and has a sound theoretical basis. Examples are provided to illustrate the constructive methodology.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stability analysis of nonlinearly interconnected systems—Application of M-functions

In the large number of papers on the stability analysis of interconnected systems which have been published these ten years, actual stability criterions have been most frequently derived on the basis of M-matrices. However, the application of M-matrices in both I-O stability analysis and vector lyapunov function approach for composite systems resticts the connections of subsystems to those satisfying linear estimations such as

Stability analysis of composite systems

The stability of composite systems is investigated in terms of their subsystems and their interconnecting structure. Whereas previous investigators utilized vector Lyapunov functions in their approach, scalar functions consisting of weighted sums of scalar Lyapunov functions of subsystems of the composite systems are employed presently. It is shown that the scalar Lyapunov function approach may in general yield stability results which are less conservative than those obtained by the vector Lyapunov function approach. Both methods are applied to specific examples considered previously. These examples demonstrate the improvement of the present results.