Semi-definite Derivative Research Articles

On the motion stabilization of a three-link robotic manipulator with incomplete measurement

Abstract. This paper considers a mathematical model of a manipulator which consists of a vertical column, two links, connected to it in series, and a gripper with a load. The column resting on a fixed base can rotate around its vertical axis. The links are connected by cylindrical hinges allowing them to rotate in the same vertical plane. The column and the links are modeled as rigid bodies with the links having unequal principal moments of inertia. The position of the manipulator in space is determined by three rotation angles of the column and the links. The manipulator can have several types of steady-state program movements. When gravitational torques are compensated by control torques applied in the cylindrical hinges, the manipulator has a program equilibrium position. The manipulator can also have a program motion when the column rotates at a given constant angular velocity, and the links have given relative equilibrium positions in their plane. The stabilization problem of manipulator motion is investigated by means of control torques with feedback when only the rotation angles of the column and links are measured. The problem posed is solved in the form of a nonlinear proportional-integral controller taking into account the cylindrical phase space of the manipulator's mathematical model. The solution includes construction of a Lyapunov functional with a semi-definite derivative and application of the corresponding theorems on the asymptotic stability of non-autonomous functional differential retarded-type equations. The obtained conditions for the program motion stabilization are robust with respect to the mass-inertial parameters of the manipulator. The numerical simulation results demonstrate global attraction to its given position in cylindrical phase space.

On the stability of a nonlinear nonautonomous scalar equation with variable delay

The stability problem of a scalar functional differential equation is a classical one. It has been most fully studied for linear equations. Modern research on modeling biological, infectious and other processes leads to the need to determine the qualitative properties of the solutions for more general equations. In this paper we study the stability and the global limit behavior of solutions to a nonlinear one-dimensional (scalar) equation with variable delay with unbounded and bounded right-hand sides. In particular, our research is reduced to a problem on the stability of a non-stationary solution of a nonlinear scalar Lotka-Volterra-type equation, on the stabilization and control of a non-stationary process described by such an equation. The problem posed is considered depending on the delay behavior: is it a bounded differentiable function or a continuous and bounded one. The study is based on the application of the Lyapunov-Krasovsky functionals method as well as the corresponding theorems on the stability of non-autonomous functional differential equations of retarded type with finite delay. Sufficient conditions are derived for uniform asymptotic stability of the zero solution, including global stability, for every continuous initial function. Using the theorem proven by one of the co-authors on the limiting behavior of solutions to a non-autonomous functional differential equation based on the Lyapunov functional with a semidefinite derivative, the properties of the solutions’ attraction to the set of equilibrium states of the equation under study are obtained. In addition, illustrative examples are provided.

A Discussion on the Existence of Smooth Lyapunov Functions for Continuous Stable Systems

Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability.

Invariance-Like Results for Nonautonomous Switched Systems

This paper generalizes the Lasalle-Yoshizawa Theorem to switched nonsmooth systems. Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A candidate common Lyapunov function that has a negative semidefinite derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle-Yoshizawa results for the switched system. Results for regular and non-regular candidate Lyapunov functions are presented using an appropriate generalization of the time derivative. The developed generalization is motivated by adaptive control of switched systems where the derivative of the candidate Lyapunov function is typically negative semidefinite.

On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

Existence and construction of a function with sign semidefinite derivative for dynamic systems

The problem of finding a function with sign semidefinite derivative along the solutions to the system of ordinary differential equations is considered. It is shown that a sufficient existence condition is the existence of trajectories that leave a neighborhood of the equilibrium point both in forward or backward time. Two approaches to the construction of the desired function in the autonomous and nonautonomous cases are proposed. The results are illustrated by the example of a nonlinear autonomous system (Artstein's circle).

Can we prove stability by using a positive definite function with non sign-definite derivative?

Novel criteria for global asymptotic stability of nonlinear uncertain finite-dimensional systems are presented. The results are obtained by a combination of the discretization approach and the ideas contained in the proof of the original Matrosov's result. The results can be used for the proof of global asymptotic stability by using continuously differentiable, positive definite functions which do not have a negative semi-definite derivative. Illustrating examples are provided.

Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem

The classical Matrosov theorem concludes uniform asymptotic stability of time-varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi-definite derivative along solutions) and another auxiliary function with derivative that is strictly nonzero where the derivative of the Lyapunov function is zero (Mastrosov in J Appl Math Mech 26:1337–1353, 1962). Recently, several generalizations of the classical Matrosov theorem have been reported in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005). None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) which is a very useful analysis and controller design tool for nonlinear systems. Inspired by generalized Matrosov conditions in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005), we provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function-based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a simple Euler-Lagrange system controlled by an adaptive controller and use this result to determine an ISS controller.

A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems

A new infinitesimal sufficient condition is given for uniform global asymptotic stability (UGAS) for time-varying nonlinear systems. It is used to show that a certain relaxed persistency of excitation condition, called uniform /spl delta/-persistency of excitation (U/spl delta/-PE), is sufficient for uniform global asymptotic stability in certain situations. U/spl delta/-PE of the right-hand side of a time-varying differential equation is also shown to be necessary under a uniform Lipschitz condition. The infinitesimal sufficient condition for UGAS involves the inner products of the flow field with the gradients of a finite number of possibly sign-indefinite, locally Lipschitz Lyapunov-like functions. These inner products are supposed to be bounded by functions that have a certain nested, or triangular, negative semidefinite structure. This idea is reminiscent of a previous idea of Matrosov who supplemented a Lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the Lyapunov function is zero. For this reason, we call the main result a nested Matrosov theorem. The utility of our results on stability analysis is illustrated through the well-known case-study of the nonholonomic integrator.

Input-to-state stability of PD-controlled robotic systems

A sufficient condition for input-to-state stability is presented for systems admitting a Lyapunov function with negative semi-definite derivative. An application to PD-controlled robotic manipulators is also described.

Asymptotic stability of nonautonomous systems by Liapunov's direct method

This paper discusses the asymptotic stability of nonautonomous systems by means of the direct method of Liapunov. We assume existence of a positive time-invariant Liapunov function with negative semi-definite derivative. We focus on the extra conditions needed in order to guarantee asymptotic stability and present a theorem which offers a new criterion for asymptotic stability. The theorem is illustrated first by recovering Barbashin's principle for autonomous and periodic systems and second by studying the nonlinear pendulum with time-varying damping for which we derive a new asymptotic stability result.

A New Criterion for Asymptotic Stability of Nonautonomous Differential Equations: An Illustrative Example

This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. We consider a positive time-invariant Liapunov function with negative semi-definite derivative. We focus on the extra conditions needed in order to guarantee asymptotic stability and propose a new criterion. The theorem is illustrated with a simple example: we discuss the harmonic oscillator with time-variant damping for which we derive a new asymptotic stability result.

Stability of nonautonomous systems by Liapunov's direct method

This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. The existence of a positive time-invariant Liapunov function with negative semi-definite derivative is assumed. The paper focuses on the extra c

Necessary and sufficient conditions for the existence of liapunov function of Lur'e form with $$\dot V \leqslant 0$$

This paper is concerned with the problem of absolute stability for a control system with several executive elements. Necessary and sufficient conditions are obtained for the existence of Liapunov function of Lur'e form with negative semi-definite derivative (i.e.\(\dot V \leqslant 0\)).