SING HTMS For the feedback system in Fig. 3.17, the closed-loop transfer matrix H(s) is found to equal H(s)=(I+G(s)) - G(s) (3.108) with I E CNxN the N-dimensional unity matrix. In analyzing the stability or settling behavior of the closed-loop system, i.e. the absence of poles of H(s) in a particular region of the complex plane, we compute the eigenvalues Nk(s) of the open-loop gain G(s), i.e. EUk(s)EC:G(s)Uk(s)=Nk(s)Uk(s (3.109) for k = 1, , N. The graphs of the Nk(s) as S traces a Nyquist contour are called the characteristic loci of G(s). The properties of the characteristic loci of G(s) are linked to the properties of the closed-loop system H(s) by the following theorem [Deso80, Mac89]: Theorem (Generalized Nyquist): A closed-loop MIMO system H(s) = G(s) contains no unstable poles if the characteristic loci of the open-loop gain G(s), taken together, encircle + j0 as many times anti-clockwise as G(s) has poles located in the right-hand plane. Example (MIMO stability): Consider the system with open-loop gain (3.110) The eigenvalues of this system are (3.111) Fig. 3.19 plots the characteristic loci for both (s) and 22 (s). Since G(s) has no un- stable poles and since the joint characteristic loci encircle - 1 + j0 a single time, it is concluded that the closed-loop system H(s) determined by (3.108) will be unstable. Note also the importance of drawing both characteristic loci. If only one of them is drawn, the overall locus is not closed and the generalized Nyquist criterion has there- fore no meaning. 3.5.2.3 Stability analysis of LPTV systems The correspondence between the HTM representation of an LPTV system and the transfer matrix of a MIMO LTI system suggests to apply the results above to establish a frequency-domain Nyquist criterion that can handle LPTV system stability. How- ever, the o-dimensional structure of the HTMs causes trouble. It implies both an infi- nite number of poles and eigenvalues. Straightforward application of the generalized Nyquist criterion for MIMO LTI systems would therefore resu