(Nyquist): A closed-loop system H(s) = G(s)/ +G(s)) contains no un- stable poles if the Nyquist locus of the open-loop gain G(s) encircles - 1 + j0 as many times anti-clockwise as G(s) has poles located in the right-hand plane. For example, if G(s) contains no poles in the right-hand plane, the Nyquist locus indi- cated by the solid line in Fig. 3.18(b) proves that the closed-loop transfer function (s) is stable as it does not encircle -1 - + j0. The dashed line, encircling - 1 + j0 a single time, indicates the presence of a single unstable closed-loop pole. Note that by tracing curves other than the one in Fig. 3.18(a), similar results are ob- tained that provide more detailed information on the position of the closed-loop poles. For example, plotting the graph of G(s) for S varying along a o-shifted frequency axis, us to determine whether the closed-loop system contains poles with a real part that is greater than 0. This is useful as, often, designers are not interested in shear stability. They would also like the closed-loop system to settle sufficiently fast, i.e. all poles must have a real part that is smaller than a certain negative real 0. This implies that the closed loop should not contain any poles located in the plane to the right of the o-shifted frequency axis. For a more detailed discussion, see [Mac89]. 3.5.2.2 Recapitulating stability analysis of MIMO LTI systems Since HTMs can be considered as the limit of a sequence of finite-dimensional, multi- input multi-output (MIMO) LTI systems, the question rises whether Nyquist can be generalized to handle this kind of systems. A generalization like this is discussed in [Deso80, Mac89]. For MIMO LTI systems, the open-loop gain in Fig. 3.17 is no longer scalar- but matrix- valued, i.e. G1,N(s) (3.106) GN,1(s) GN,N(s) For the moment, G(s) E CNxN is considered an arbitrary matrix-valued transfer func- tion (no structural constraints). As for LTI transfer functions, if im||G(s)||= = , (3.107) then p E C is said to be a pole of G(s). 3.5 LPTV SYSTEM ANALYSIS U