Wiley signals and systems e book TLFe BO 413

398

16. Stabi1it;v and Feedback Systems

b > 0 And f i < 0. Tlie pole of thc system without feedback now lies ix~the Iefl,
half of tlie s-plane. For I< < 0 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 < 0.
For K = 0 all of the poles of the system lie on the real axis. one in tlie right
half-plane, For K < 0 nothing chijnges, c%l; the poles just move further away. For
K > 0 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

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