Analysis of equilibrium points for stability of nonlinear dynamical systems using DPMA

In this paper, an innovative approach named the dynamic pole motion approach (DPMA) has been explained for analyzing equilibrium points for the stability of nonlinear dynamical systems. The stability of a nonlinear dynamical system can be analyzed in g-space using DPMA without linearization. In a linear system, an equilibrium point is uniquely located at the origin; however, in nonlinear systems, it may have several isolated equilibrium points. DPMA describes the motion of dynamic poles in g-space, where ‘g’ is represented in a three-dimensional g-space; g(t)=σ(t)+jω(t) (where σ,ω ϵ R). For a stable system, the characteristic equation's roots (poles) of the dynamical system must lie on the left-hand side of the g-space. This implies that the net feedback in the system is negative. One or more poles in the g-space's right-hand side will provide positive net feedback, causing instability. Stability of L.T.I. systems is shown in the s-plane. However, the ‘g’ operator and g-space can be used for both linear and nonlinear systems. The stability and classification of equilibrium points for nonlinear systems are also explained by using DPMA. There is less mathematics by using DPMA, which simplifies the analysis of equilibrium points for the stability of the nonlinear dynamical systems with constraints.