Definition Root locus

for system, closed-loop transfer function given by

t
(
s
)
=



y
(
s
)


x
(
s
)



=



g
(
s
)


1
+
g
(
s
)
h
(
s
)





{\displaystyle t(s)={\frac {y(s)}{x(s)}}={\frac {g(s)}{1+g(s)h(s)}}}

thus, closed-loop poles of closed-loop transfer function roots of characteristic equation



1
+
g
(
s
)
h
(
s
)
=
0


{\displaystyle 1+g(s)h(s)=0}

. roots of equation may found wherever



g
(
s
)
h
(
s
)
=
−
1


{\displaystyle g(s)h(s)=-1}

.

in systems without pure delay, product



g
(
s
)
h
(
s
)


{\displaystyle g(s)h(s)}

rational polynomial function , may expressed as

g
(
s
)
h
(
s
)
=
k



(
s
+

z

1


)
(
s
+

z

2


)
⋯
(
s
+

z

m


)


(
s
+

p

1


)
(
s
+

p

2


)
⋯
(
s
+

p

n


)





{\displaystyle g(s)h(s)=k{\frac {(s+z_{1})(s+z_{2})\cdots (s+z_{m})}{(s+p_{1})(s+p_{2})\cdots (s+p_{n})}}}

where



−

z

i




{\displaystyle -z_{i}}





m


{\displaystyle m}

zeros,



−

p

i




{\displaystyle -p_{i}}





n


{\displaystyle n}

poles, ,



k


{\displaystyle k}

scalar gain. typically, root locus diagram indicate transfer function s pole locations varying values of parameter



k


{\displaystyle k}

. root locus plot points in s-plane



g
(
s
)
h
(
s
)
=
−
1


{\displaystyle g(s)h(s)=-1}

value of



k


{\displaystyle k}

.

the factoring of



k


{\displaystyle k}

, use of simple monomials means evaluation of rational polynomial can done vector techniques add or subtract angles , multiply or divide magnitudes. vector formulation arises fact each monomial term



(
s
−
a
)


{\displaystyle (s-a)}

in factored



g
(
s
)
h
(
s
)


{\displaystyle g(s)h(s)}

represents vector



a


{\displaystyle a}





s


{\displaystyle s}

in s-plane. polynomial can evaluated considering magnitudes , angles of each of these vectors.

according vector mathematics, angle of result of rational polynomial sum of angles in numerator minus sum of angles in denominator. test whether point in s-plane on root locus, angles open loop poles , zeros need considered. known angle condition.

similarly, magnitude of result of rational polynomial product of magnitudes in numerator divided product of magnitudes in denominator. turns out calculation of magnitude not needed determine if point in s-plane part of root locus because



k


{\displaystyle k}

varies , can take arbitrary real value. each point of root locus value of



k


{\displaystyle k}

can calculated. known magnitude condition.

a graphical method uses special protractor called spirule once used determine angles , draw root loci.

the root locus gives location of closed loop poles gain



k


{\displaystyle k}

varied. value of



k


{\displaystyle k}

not affect location of zeros. open-loop zeros same closed-loop zeros.

angle condition

a point



s


{\displaystyle s}

of complex s-plane satisfies angle condition if

∠
(
g
(
s
)
h
(
s
)
)
=
π


{\displaystyle \angle (g(s)h(s))=\pi }

which same saying that

∑

i
=
1


m


∠
(
s
+

z

i


)
−

∑

i
=
1


n


∠
(
s
+

p

i


)
=
π


{\displaystyle \sum _{i=1}^{m}\angle (s+z_{i})-\sum _{i=1}^{n}\angle (s+p_{i})=\pi }

that is, sum of angles open-loop zeros point



s


{\displaystyle s}

minus angles open-loop poles point



s


{\displaystyle s}

has equal



π


{\displaystyle \pi }

, or 180 degrees.

magnitude condition

a value of



k


{\displaystyle k}

satisfies magnitude condition given



s


{\displaystyle s}

point of root locus if

|

g
(
s
)
h
(
s
)

|

=
1


{\displaystyle |g(s)h(s)|=1}

which same saying that

k




|

s
+

z

1



|


|

s
+

z

2



|

⋯

|

s
+

z

m



|




|

s
+

p

1



|


|

s
+

p

2



|

⋯

|

s
+

p

n



|




=
1


{\displaystyle k{\frac {|s+z_{1}||s+z_{2}|\cdots |s+z_{m}|}{|s+p_{1}||s+p_{2}|\cdots |s+p_{n}|}}=1}

.




^ kuo 1967, p. 331.
^ kuo 1967, p. 332.
^ evans, walter r. (1965), spirule instructions, whittier, ca: spirule company