This paper deals with the problem of the estimation of regions of asymptotic stability for continuous, autonomous, nonlinear systems. The first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields. In the second part certain topological considerations are first developed and the "trajectory reversing method" is then presented together with a theorem on which it is based. In the final part, several examples of application are reported, showing the efficiency of the proposed technique for low-order (second and third) systems.

On the estimation of asymptotic stability regions: State of the art and new proposals

The paper deals with the problem of the estimation of regions of asymptotic stability for continuous, autonomous, nonlinear systems. After an outline of the main approaches available in the literature, the "trajectory reversing method" is presented as a. powerful numerical technique for low order systems. Then, an analytical procedure based on the same topological approach is developed, and a comparison is made with the classical Zubov method, noting the possibility of overcoming some of the classical method's drawbacks (e.g., its nonuniform convergence). Several examples of applications of the "trajectory reversing method" both in the numerical and analytical formulation are reported.

New techniques for constructing asymptotic stability regions for nonlinear systems

We present a method of modeling the basin of attraction as a three-dimensional function describing a two-dimensional manifold on which the dynamics of the system evolves from experimental time series data. Our method is based on the density of the data set and uses numerical optimization and data modeling tools. We also show how to obtain analytic curves that describe both the contours and the boundary of the basin. Our method is applied to the problem of regaining balance after perturbation from quiet vertical stance using data of an elite athlete. Our method goes beyond the statistical description of the experimental data, providing a function that describes the shape of the basin of attraction. To test its robustness, our method has also been applied to two different data sets of a second subject and no significant differences were found between the contours of the calculated basin of attraction for the different data sets. The proposed method has many uses in a wide variety of areas, not just human balance for which there are many applications in medicine, rehabilitation, and sport.

Modeling the basin of attraction as a two-dimensional manifold from experimental data: applications to balance in humans.

The results of a numerical investigation of a number of techniques for stability analysis of non-linear systems are presented. The methods compared are the Ingwerson and Zubov procedures for construction of Liapounov functions, a number of methods for obtaining optimum Liapounov functions, tracking functions, the alternating extreme radius path, and the method of Luus and Lapidus.

Comparison of numerical methods in stability analysis

A numerical algorithm for the construction of Liapunov functions is described. This algorithm is based upon the method of Szego and is applicable to a particular class of systems. The algorithm is described for second-order systems although extension to systems of higher order would appear to be possible.

https://ieeexplore.ieee.org/iel5/9/24105/01099085.pdf

Numerical application of Szego's method for constructing Liapunov functions

The Zubov and Ingwerson procedures for constructing Lyapunov functions depend upon an initial choice of some function of the state variables. A direct search procedure is described for obtaining the optimum function, and hence the Lyapunov function of an arbitrary order which gives the best estimate of the region of asymptotic stability. The method is illustrated by an example.

Optimisation of the Zubov and Ingwerson methods for constructing Lyapunov functions

A computing algorithm based on the tracking-function method is presented for obtaining regions of practical stability and ultimate boundedness for second-order autonomous differential equations. It is shown that the algorithm obviates certain problems of the tracking-function method as originally formulated. The Van der Pol equation is analysed as an example.

Computer application of the tracking-function approach to practical stability

A simple, but in some cases more powerful, modification of the tracking-function method of practical stability analysis is described.

J.R. Hewit

Papers

Optimisation of the Zubov and Ingwerson methods for constructing Lyapunov functions

Computer application of the tracking-function approach to practical stability

Practical stability in polar co-ordinates