struc- ture of both HTM poles and eigenvalues. 3.5 LPTV SYSTEM ANALYSIS USING HTMS Figure 3.17: Elementary feedback loop. Im{s} Im{G} contour at infinity Re{s} Re{G} Nyquist locus Figure 3.18: Nyquist stability is based on the study of Nyquist loci. A Nyquist locus is the graph traced by G(s) along (a) a contour that encircles the entire right-hand plane . (b) If G(s) contains no poles in the encircled area, the solid line indicates that the closed loop H(s) has no unstable poles as it does not encircle - 1 + j0. On the other hand, the dashed line, that does encircle - 1 + j0 a single time, denotes the presence of an unstable pole in the closed-loop behavior. 3.5.2.1 Recapitulating stability analysis of SISO LTI systems We begin our treatment with a brief review of traditional frequency-domain stability theory for single-input single-output (SISO) LTI systems [Fran94, Mac89]. Here, the starting point is the work of Nyquist8. Given the feedback loop in Fig. 3.17, with (3.105) Nyquist studied the stability of the closed-loop system H(s) in terms of the character- istics of the open-loop gain G(s). Nyquist stability analysis is based on the study of a Nyquist locus. As illustrated in Fig. 3.18, a Nyquist locus is the graph traced by G(s) as S traces a Nyquist contour, i.e. a contour that encircles the entire right-hand plane. If lims->o G(s) exists, then this 8 As a historical note aside: Nyquist's stability criterion essentially is an application of the argument principle which in turn builds on Cauchy's work on complex functions [Arf85]. Hence, the mathematical apparatus needed for the stability analysis of linear systems was already available by 1830. 3.5.2 Stability analysis graph is entirely determined by the values G(jw) for -00<00<00. Closing the loop using a contour at infinity is only relevant for theoretical purposes. The link between the properties of the closed-loop transfer function H(s) and the prop- erties of the open-loop gain G(s) is established by Nyquist's theorem [Bode45, Fran94, Mac89]: Theorem 