Page 103 - Mechatronics with Experiments

CLOSED LOOP CONTROL 89
In most engineering systems, the transfer function has more poles than zeros. There-
fore, as we map the C contour with G(s),
1
The half circle arc (as s goes to infinity) will map to zero magnitude for deg(d(s)) >
deg(n(s)) or a finite value is deg(d(s)) = deg(n(s)) (Figure 2.33d).

The mapping of the lower half of the imaginary axis will be symmetric to the mapping
of the upper half of the imaginary axis (Figure 2.33d).
Hence, the Nyquist plot can be determined only from the mapping of the positive jw axis.
If there are poles or zeros on the imaginary axis, the contour C should either include
1
them or exclude them from being inside the C contour. Either approach would give the
1
same final conclusion regarding closed loop stability. It is customary to exclude the poles
and zeros on the imaginary axis from the C contour’s inside.
1
Relative Stability The relative stability of a CLS is quantified by the use of the
distance of the Nyquist plot from the (−1, 0) point. That is “how far is the closed loop
system from the stability boundary.”
There are two quantities defined for relative stability (Figure 2.34): the gain margin
(GM) and the phase margin (PM). The gain margin is the inverse of the magnitude of the
◦
loop transfer function when the phase is 180 . It indicates how much the loop gain can be
increased before the system reaches the stability boundary. The phase margin is the phase
◦
angle difference between the loop transfer function and −180 when the magnitude of the
loop transfer function is 1. The PM indicates the amount of phase lag that can be introduced
into the loop transfer function before it reaches the stability boundary. The measurement
of GM and PM on Bode and Nyquist plots is shown in Figure 2.34.

2.10 THE ROOT LOCUS METHOD

The root locus method is a graphical method for plotting the roots of an algebraic equation
as one or more parameters vary. It is used primarily in studying the effect of variations in
one parameter of a control system on the locations of closed loop system poles. In the next
section, we will use the root locus method in order to understand the characteristics of PID
type closed loop controllers. The basic mathematical functionality is to find and plot the
roots of an algebraic equation for various values of a parameter. The solution of algebraic
equations can be easily done by numerical means using a digital computer. Solving it for
various values of one or more parameters is nothing more than implementing the numerical
procedure in an iteration loop, that is FOR or DO loop in a high level programming language.
This is certainly a tool which became more and more effective with the availablity of CAD
tools for control system design. A control engineer must, however, always keep in mind
that the basic principle about computers is garbage in – garbage out. Therefore, it is very
important that a designer should be able to quickly verify in general terms a computer
calculation with hand calculations or analysis. The graphical hand sketching rules of the
root locus method provides such a tool. Understanding the graphical root locus method not
only provides a way of quickly checking the results of a computer calculations, but also
develops very valuable insights into the control system design.
Let us consider an algebraic equation, that is a polynomial of degree n,

n
a s + a s n−1 + ..... + a n−1 s + a = 0 (2.93)
n
1
0
which has n roots {s , s , ...., s }. If the value of any of a changes, there will be a new set
i
2
1
n
of n roots {s , s , ...., s }. If a particular parameter a varies from one minimum value to
2
1
i
n