398 16 Stabi1it;v and Feedback Systems b > And f i < Tlie pole of thc system without feedback now lies ix~the Iefl, half of tlie s-plane For I< < it ~ n o w to s the right and at K = a,/&it, reaches thc right half of the plane Figure 16.8 shows the corresponding root locus Figurc 16.8: Root lrrcus of Figure lli.6 for n < (1, K < (1 The infiuenrc of the P-cirruit and the sign appearing at the ~ ~ i ~ x ~node ~ a ~ i ( ~ in the feedback- path on the system as a tvliote cari be mnmarised as follows: Positive feedback dcstabilises a byst em * Negative feedback stabilises a syhtcin ~ ~ n ~ o r t u ~ afor t c~l ~ i ~, ~ h ~ r -systems o r ~ e rthe r e l a ~ ~ i o nare ~ ~riot i i ~so ~ simple We cari show this with a, second-order plant depicted in Fignre 16.9 The tmrisfer fmiction in the forward path b lilfs) = - , U E I R (16.30) s2+u has two poles at s = k&Z and is therefore unstable for all U Using a P-circuit for fc,rdback the overall Iransfer funcriou is obtairred Figure 16.10 shows the rook locus for a < For K = all of the poles of the system lie on the real axis one in tlie right half-plane, For K < nothing chijnges, c%l; the poles just move further away For K > the poies move along the ~ n ~ a ~axis ~ ~and i a for ~ yEr' > ( - a ) / h they form a cornplcx c ~ ~ ~ jpole ~ ~pair a t on ~ the d iniaginaiy axis Clearly it is not pcnsible here troniove both poles into the Icft half of tlie s-planc using P-circuit feedbadc This can be confirmed by a Ilurwita, k s t The denominator polynomial s2 a K h in (16.31) is not a Hurwita polyriomial, as the coefficient of the linear term in Y iu cqual to zero The system c m be snccessfully stwbilisrd with proportional-differcrllial fwdtfaclc as in Figure 16.11 This Leads to an overall transfer fnnctitm for the system + +