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Aspects of nonlinear system stability. Christensen, Gustav Strom

Abstract

This thesis treats system stabilty from three separate points of view. 1. State Space Analysis 2. Complex Frequency Plane Analysis 3. Time Domain Analysis Asymptotic stability is considered in state space. Using state space and the gradient method an expression is derived for the total time derivative of the Liapunov function. This expression is a special case of the general Zubov equation, however, it does not lend itself to an explicit, exact solution except in special cases. Global asymptotic stability and bounded input -bounded output stability is considered in the complex frequency plane. Here a method developed by Sandberg has been applied to some systems the linear part of which has poles on the imaginary axis. The solution of an example of this type via the Sandberg method and the Popov method shows that the two methods give essentially the same result for the example considered. Bounded input - bounded output stability is considered in the time domain using two separate methods. One, a method developed by Barrett using Volterra series has been extended to cover cases with a nonlinearity of 2nd and 4th degree. Two, a method depending on the contraction mapping principle is developed and applied to several types of systems. It is shown that this method generates the Volterra series found by Barrett's method, and thus we can actually determine a region where, the solution of a given differential equation can be represented in the form of a Volterra series.

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