Ramesh S. Guttalu

Mechanics and Control

Two dozen papers promote the use of the advanced mechanics method in control theory, emphasizing the control of nonlinear mechanical systems subject to uncertainty. They cover control methodology, applications to aerospace systems, and the control of mechanical systems. Reproduced from typescript. A

On an application of dynamical systems theory to determine all the zeros of a vector function

Abstract A continuous-time dynamical system is constructed and analyzed in this paper to help locate all the zeros of nonlinear vector functions. Lyapunov stability technique is used to show that the zeros of the vector function become asymptotically stable equilibria for this dynamical system. A strict Lyapunov function is constructed to indicate local stability and to estimate the domains of attraction of all equilibria. Manifolds on which the Jacobian of the vector field is singular play a significant role in characterizing the global behavior of the system. Examples are provided to illustrate the extent of the theory. New computational techniques to determine all the zeros of a vector function can be developed based on the dynamical aspects of the theory given in this paper.

A Computational Method for Finding All the Roots of a Vector Function

Based on dynamical systems theory, a computational method is proposed to locate all the roots of a nonlinear vector function. The computational approach utilizes the cell-mapping method. This method relies on discretization of the state space and is a convenient and powerful numerical tool for analyzing the global behavior of nonlinear systems. Our study shows that it is efficient and effective for determining roots because it minimizes and simplifies computations of system trajectories. Since the roots are asymptotically stable equilibrium points of the autonomous dynamical system, it also provides the domains of attraction associated with each root. Other numerical techniques based on iterative and homotopic methods can make use of these domains to choose appropriate initial guesses. Singular manifolds play an important role in limiting the extent of these domains of attraction. Both a theoretical basis and a computational algorithm for locating the singular manifolds are also provided. They make use of similar state-space discretization frameworks. Examples are given to illustrate the computational approaches. It is demonstrated that for one of the examples (a mechanical system), the method yields many more solutions than those previously reported.

On the role of singularities in Branin's method from dynamic and continuation perspectives

A global analysis of Branins method (originally due to Davidenko) for finding all the real zeros of a vector function is carried out. The analysis is based on a global study of this method perceived of as a dynamical system. Since Branins algorithm is closely related to homotopy methods, this paper sheds some light on the global performance of these methods when employed for locating all the zeros of a vector function. Following the dynamical system approach, the performance of Branins algorithm is related to the existence of extraneous singularities as well as to the relative spatial distribution of the zeros of the vector function and singular manifolds. Branins conjectures regarding the types and the role of extraneous singularities are examined and counterexamples are provided to disprove them. We conclude that the performance of Branins method for locating all the zeros of a vector function is questionable even in the absence of extraneous singularities.

Analysis of Dynamical Systems by Truncated Point Mappings and Cell Mapping

Recent results obtained by the authors on the utility of truncated point mappings applicable to the analysis of multidimensional, multiparameter, periodic nonlinear systems are presented here. Based on multinomial truncation, an explicit analytical expression is determined for the point mapping in terms of the states and parameters of the system to any order of approximation. By combining this approach with analytical techniques, such as the perturbation method employed here, we obtain a powerful tool for finding periodic solutions and for analyzing their stability. A new approach for analyzing nonlinear systems which combines the techniques of truncated point mapping and cell mapping methods is also described here.

Stability study of a periodic system by a period-to-period mapping

Abstract Stability characteristics of period dynamical systems are investigated by the Poincare map ( point mapping ) analysis approach. The approach is based on a method for obtaining an analytical expression for the period-to-period mapping description of the dynamics of the system and its dependence on system parameters. Stability and bifurcation conditions are expressed analytically and functional relations between various system parameters are determined. The approach is applied to investigate the parametric stability of a double pendulum. Excellent agreement with direct numerical results, assumed to be the “exact solution” for the purpose of this study, was obtained. Analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems.

Stability of controllers with on-line computations

The dynamical systems theory developed by Zufiria [1], Zufiria and Guttalu [2, 3], and Guttalu and Zufiria [4] is applied to the stability analysis of control systems in which the feedback control law requires in real time the solution of a set of nonlinear algebraic equations. Since a small sampling period is assumed, the stability and performance of the controlled process can be studied with a continuous-time formulation. A singularly perturbed system is used to model both the dynamics of the system being controlled and a numerical iterative algorithm required to compute the control law. An updating control procedure has been proposed based on the iterative nature of the control algorithm. The results obtained by Zufiria [1] regarding the behavior of a dynamical system that models the numerical algorithms lead to a considerable simplification in the analysis. For the case of a control problem involving inverse kinematics, the numerical algorithm that solves for inverse kinematics can be considered as an observer (or an estimator) of the state-space variables. The study provides an estimate of the required speed of computations to preserve the stability of the controller.

Analysis of Nonlinear Systems by Truncated Point Mappings

This paper summarizes results obtained by the authors regarding the utility of truncated point mappings which have been recently published in a series of papers. The method described here is applicable to the analysis of multidimensional, multiparameter, periodic nonlinear systems by means of truncated point mappings. Based on multinomial truncation, an explicit analytical expression is determined for the point mapping in terms of the states and parameters of the system to any order of approximation. By combining this approach with analytical techniques, such as the perturbation method employed here, the authors obtain a powerful tool for finding periodic solutions and for analyzing their stability. The versatility of truncated point mapping method is demonstrated by applying it to study the limit cycles of van der Pol and coupled van der Pol oscillators, the periodic solutions of the forced Duffings equation and for a parametric analysis of periodic solutions of Mathieus equation.

Analytical stability analysis of periodic systems by Poincaré mappings with application to rotorcraft dynamics

A point mapping analysis is employed to investigate the stability of periodic systems. The method is applied to simplified rotorcraft models. The proposed approach is based on a procedure to obtain an analytical expression for the period-to-period mapping description of systems dynamics, and its dependence on systems parameters. Analytical stability and bifurcation conditions are then determined and expressed as functional relations between important system parameters. The method is applied to investigate the parametric stability of flapping motion of a rotor and the ground resonance problem encountered in rotorcraft dynamics. It is shown that the proposed approach provides very accurate results when compared with direct numerical results which are assumed to be an “exact solution” for the purpose of this study. It is also demonstrated that the point mapping method yields more accurate results than the widely used classical perturbation analysis. The ability to perform analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems. Stability analysis of higher dimensional systems, such as the ground resonance problems, by perturbation methods is not straightforward, and is usually very cumbersome.

Global Behavior of Branin’s Method

An analysis of Branin’s method for finding all the real zeros of a vector function is carried out. It is based on a global study of this method perceived of as a dynamical system. Since Branin’s algorithm is closely related to homotopy methods, this paper sheds some light on the global performance of these methods when employed for locating all the zeros of a vector function. We provide two examples for which the global behavior of Branin’s algorithm is carried out using the cell-to-cell mapping computational technique.