lt in a great waste of efforts. Fortunately, the structure inherent to HTMs makes simplification possible. As men- tioned in section 3.3.4, both the poles and eigenvalues of a HTM G(s) exhibit a repet- itive structure. If p is a pole of G(s), SO is p jwo. Furthermore, the set {2k(s)} of 3.5.2 Stability analysis 21(jw) 22(jw) Figure 3.19: The two characteristic loci for the MIMO system defined by equation (3.110). eigenvalues of G(s) is periodic in a sense that (3.112) Both observations taken together imply that it is sufficient to trace S over the frequency interval -wo/2,000/2] in drawing the characteristic loci of G(s) and in determining its poles. Due to the repetitive nature of the poles and eigenvalues of G(s) in the direction of the imaginary axis, no new information will be revealed as we trace S beyond this interval. This brings us to the Nyquist criterion for LPTV systems [Were91a, Moll00]9: Theorem (Nyquist LPTV): A closed-loop LPTV system that is represented by the HTM H(s) = (I+G(s))- G(s) contains no unstable poles if the characteristic loci of the open-loop gain G(s), taken together and drawn for S = jw for - wo/2 < w< wo/2, encircle -1 - + j0 as many times anti-clockwise as G(s) has poles located in the right- hand plane with an imaginary part that lies in the interval [-wo/2,000/2]. Although satisfying from a theoretical point of view, questions rise as to the practical applicability of this Nyquist criterion for LPTV systems. For a particular value S = jw, G(s) is still an oo-dimensional matrix. Hence, it has an infinite number of eigenvalues. Fortunately, for many practical systems, most of the eigenvalues have a magnitude that is well below 1. Since their locus cannot help to encircle - 1 + j0, they are irrelevant as far as stability is concerned. 9 As presented here, the criterion slightly deviates from the one presented in [Were91a, Moll00]. Contrary to the procedure suggested there, the periodic structure of the set of eigenvalues {Ak(s)} renders it unneces- sary to close the curve by tracin