Page 105 - Mechatronics with Experiments

CLOSED LOOP CONTROL 91

r(t) + y(t)
K G(s)
–
FIGURE 2.35: Basic root locus problem.

where the poles of the closed loop system are given by the roots of the denominator,
Δ (s) = 1 + KG(s) (2.95)
cls
The standard root locus analysis problem involves the sketch of the locus of the roots of
this equation as K varies from zero to infinity. From the above general discussion, it is clear
that the root locus method is not limited to that type of problem. It can address any problem
which involves finding roots of an algebraic equation as any one or more parameters vary.
Consider another example as shown in Figure 2.36, where a is a parameter. We would like
to study the locations of closed loop system poles as the parameter a varies from zero to
infinity.
y(s) 1 1
= = (2.96)
r(s) 1 + s(s + a) s + as + 1
2
The characteristic equation is
2
Δ (s) = s + as + 1 = 0 (2.97)
cls
which can be expressed in standard root locus formulation form suitable for graphical
sketching as
s
1 + a ⋅ = 0 (2.98)
2
s + 1
The graphical root locus method rules are developed for sketching the roots of a polyno-
mial equation as one parameter varies (Figure 2.37). The polynomial equation is always
expressed in the form of
numerator
1 + (parameter) ⋅ = 0 (2.99)
denominator
Therefore, the locus of roots can be studied as a function of any parameter in the closed loop
system, not just the gain of the loop transfer function. Let us assume that we are interested
in studying the locus of the roots of the folowing equation as parameter b varies from zero
to infinity,
2
3
s + 6s + bs + 8 = 0 (2.100)
This problem can be expressed in a form suitable for the application of the root locus
sketching rules as follows,
s
1 + b ⋅ = 0 (2.101)
3
2
s + 6s + 8
MATLAB ® provides the rlocus(...) function for root locus. The rlocus() function
is overloaded and can accept different parameters. In principle, it takes the loop transfer

r y
+ 1
– s (s + a)

FIGURE 2.36: An example:
closed loop transfer function
poles as parameter a varies.