Stability Analysis and Neuro-control of Nonlinear Systems: a Dynamic Pole Motion Approach

Abstract

In a linear time-invariant system, the parameters are constant thereby poles are static. However, in a linear time-varying system since the parameters are a function of time, therefore, the poles are not static rather dynamic. Similarly, the parameters of a nonlinear system are a function of system states, and that makes nonlinear system poles dynamic in the complex plane. The location of nonlinear system poles are a function of system states explicitly and time implicitly. Performance characteristics of a dynamic system, e.g., stability conditions and the quality of response depend on the location of dynamic poles in the complex plane.

In this thesis, a dynamic pole motion in the complex g-plane based approach is established to enhance the performance characteristics of a nonlinear dynamic system. g-plane is a three-dimensional complex plane.

The stability approach, initiated by Sahu et al. (2013), was an exertion of the dynamic Routh's stability criterion by constructing a dynamic Routh's array to examine the absolute stability of a nonlinear system in time domain. This thesis extends the work to investigate the relative stability as well as stability in the frequency domain with the introduction of the dynamic Nyquist and Bode plots. A dynamic Nyquist criterion together with the concept of the dynamic pole motion is developed. The locations of the dynamic poles are executed by drawing a dynamic root locus from the dynamic characteristic equation of a nonlinear system.

The quality of the response of a nonlinear dynamic system is enhanced by using a dynamic pole motion based neuro-controller, introduced by Song et al. (2011). In this thesis, we give a more comprehensive descriptions of the neuro-controller design techniques and illustrate the neuro-controller design approach with the help of several nonlinear dynamic system examples. The controller parameters are a function of the error, and continually relocate the dynamic poles in the complex g-plane to assure a higher bandwidth and lower damping for larger errors and lower bandwidth and larger damping for smaller errors. Finally, the theoretical concepts are further corroborated by simulation results.