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This uncert ainty description, however, is not very suited for describing model deviations, where the number of poles in the right half plane might vary, since this means that em(w) must assume the value oo Note, that la and lm are frequency dependent scalars and, hence, that they are functions of w(rather than ju ) Often, however, ea()and em(w) will be represented by normal trans functions, where mainly the amplitude will be of significance 2.3 Nominal Stability A control sy stem is internally st able, if an excit ation by a bounded signal anywhere in the system can not stimulate an unbounded signal somewhere. In Figure 2.2 a controlled process is shown with three inputs(o, u, d) and three outputs(e, u, y to the overall system r)()()x()as G(s)m(s) u s Figure 2.2: Controlled s ystem with inp uts to the analysis of internal stability Analyzing internal st ability does not necessarily imply testing a 3 x 3 matrix, since several of the signals contain the same information from a st ability point of view. The signals d and r, for inst ance, have the same influence on the output u with respect to stability. Choosing r and u as inputs and y and u as output s, the transfer matrix below can be determined y(s r(s (2.6) From(2.6)it can be seen, that the controlled system is internally st able, only if none of the four elements in the matrix have poles in the right half plane. In a similar fashion, if both G(s and K(s)are stable, it suffices to analyze the characteristic equation 1+GK(S=0 Note that if G (s) has an unstable pole, it does not suffice to apply a compensator which exactly cancels the unstable pole in order to achieve internal st ability. This can be seen fromg(2.6), since a bounded input ul imply an unbounded output 2. 4 Robust stability. In the sequel, conditions for the controlled sy stem to be robustly stable will be studied. This means that the system is stable for all mo dels of the process cont ained in g. It is assumed that all mo dels in g has the same number of poles n in the right half plane Under these conditions, the controlled sy stem is stable, only if the Nyquist curve for GKgw) encomp asses the Ny quist point (-1, 0) exactly n times counter-clockwise / $ % ( " ) 22 ) 22! " 6 7 + ! $28% ) $28% 9 6 / 9 $28% : / /