Abstract:
Research on the frequency domain stability analysis of non-linear systems has not received significant attention in recent times, when compared to its time domain or linear counterparts, due to the complex nature of the mathematics involved in analysing such systems. Despite these apparent obstacles, this thesis aims to derive necessary frequency domain stability conditions for a wide class of single input-single output non-linear systems that can be expressed in terms of polynomial non-linearities. Necessary frequency domain stability conditions for a wide class of single input-single output non-linear systems, which can be expressed in terms of polynomial non-linearities, are derived for open-loop and closedloop control systems. With regards to the open-loop system, only if the linear characteristic polynomial of the open-loop Generalized Frequency Response Function (GFRF) is stable can the system be input-output stable. This is derived from the sufficient condition that the GFRF be bounded in order to guarantee stability. These conditions are explicitly stated and verified for the general open-loop system, as well as, the special cases of the open-loop system; including pure input, pure output and pure cross-product systems. On the other hand, for the closed-loop system, the system can be internally stable and input-output stable only if the linear characteristic polynomial of the closed-loop GFRF is stable. This necessary condition is derived from the sufficient condition for -stability, once a relationship between open-loop GFRFs and closed-loop GFRFs is established. These conditions are explicitly stated and verified for two closed-loop stability problems, namely the tracking and disturbance stability problems. The input-output stability established does not take into account non-zero initial conditions and, given that stability of non-linear systems is dependent on initial conditions, Input-to-State Stability (ISS) is chosen as a stronger notion of stability for non-linear systems. ISS is formulated in terms of input-output stability, under suitable reachability and observability conditions which are evaluated using a minimal state-space realization constructed using phase variables. These time domain conditions are then restated in the frequency domain to obtain sufficient frequency domain conditions for the time domain property of ISS.