Stability Analysis and Control of Linear Periodic Delayed Systems using Chebyshev and Temporal Finite Element Methods

This chapter provides a brief literature review together with detailed descriptions of the authors’ work on the stability and control of systems represented by linear time-periodic delay-differential equations using the Chebyshev and temporal finite element analysis (TFEA) techniques. Here, the theory and examples assume that there is a single fixed discrete delay which is equal to the principal period. Two Chebyshev-based methods, Chebyshev polynomial expansion and collocation, are developed. After the computational techniques are explained in detail with illustrative examples, the TFEA and Chebyshev collocation techniques are both applied for comparison purposes to determine the stability boundaries of a single degree-of-freedom model of chatter vibrations in the milling process. Subsequently, it is shown how the Chebyshev polynomial expansion method is utilized for both optimal and delayed state feedback control of periodic delayed systems.