Circuit theory approach to stability and passivity analysis of nonlinear
dynamical systems
Abstract
In this paper, we address the problem of global asymptotic stability and
strong passivity analysis of nonlinear and nonautonomous systems
controlled by second-order vector differential equations. First, we
construct this system or the differential equation from a nonlinear time
varying network of the circuit theory. Our system and with its real
energy function generalize and improve upon some well-known studies in
the literature. This system and its special forms have ample
applications in many scientific investigations. We realized that most of
the first- and second-order ordinary differential equations can be
represented by LRC circuits. So, the energy (Lyapunov) functions of the
systems can be constructed directly without much trial and error. By
this way, the application of Lyapunov’s direct method may become a
standard technique for physical systems. We illuminate this idea with
many applications and improvements. We also compare the Lyapunov
stability theory with Hamiltonian and Lagrangian systems in the sense of
conservative and dissipative systems. Then, we provide new explicit
stability and passivity results with minimum criteria.