Complex Scaling Approach for Stability Analysis and Stabilization of Multiple Time-Delayed Systems

Abstract

Based on complex scaling, the paper copes with stability analysis and stabilization of linear time-invariant continuous-time systems with multiple time delays in state, control input, and measured output, under state and/or output feedback. More specifically, the paper establishes stability criteria for exponential/asymptotical stability with Hurwitz complex scaling being applied to the related characteristic polynomials. The stability conditions are necessary and sufficient, delay-dependent, and independent of feedback structures and open-loop poles distribution. The criteria can be implemented graphically with locus plotting or numerically without it; moreover, no prior frequency sweeping is involved, and the contour and locus encirclement orientations can be self-defined. Exploiting the complex scaling approach and embracing its technical merits, it is considered to design static state feedback control for robustly stabilizing time-delayed systems. A small-gain interpretation for the suggested stabilization is also elaborated. Examples are included to illustrate the main results.

Stability analysis and stabilisation in linear continuous-time periodic systems by complex scaling

“…To surmount various algebraic and structural obstacles in stability analysis and stabilisation (Zhou, Qian, & Lu, 2017), the complex scaling technique is suggested and brings us with fruitful results in complicated feedback systems, say fractionalorder (Zhou, 2017), sampled-data (Zhou & Qian, 2017b), timedelayed (Zhou, Gao, & Lu, 2018) and even nonlinear systems (Zhou, 2018). It must be stressed that although the same mathematical tool is adopted in the studies, different types of return difference relations with essentially different properties must be dealt with.…”