Page 107 - Mechatronics with Experiments

CLOSED LOOP CONTROL 93
root locus is the collection of points on the s-plane that makes the phase angle of the
◦
transfer function equal to 180 , since
1
K =− ; (2.102)
G(s)
Since, K is a positive real number, the G(s) complex function will have negative real
values at the collection of s-points that makes up the root locus. In other words, the
◦
phase angle of G(s) at all points that are part of the root locus is 180 .
1 | | j180 ◦
1
G(s) =− = | | ⋅ e (2.103)
K | |
| K |
3. If there are n poles, and m zeros, m of the poles end up at the m zero locations
as parameter K goes to infinity. The remaining n − m poles go to infinity along
asymptotes. If n − m = 1, then the one extra pole goes to infinity along the negative
real axis. If n − m ≥ 2, then they go to infinity along the asymptotes defined by the
asymptote center and angles as follows:
∑ ∑
p − z i
i
= (2.104)
n − m
180 + l ⋅ 360
= ; l = 0, 1, 2, … , n − m − 1. (2.105)
l
n − m
where p ’s are the pole locations z ’s are the zero locations of the G(s),
i i
Π m (s − z )
i
i=1
G(s) = n (2.106)
Π (s − p )
i=1 i
The quick hand sketches of the root locus allow the designer to quickly check the computer
analysis results for correctness and provide valuable insight in controller design. Note
that when the excessive number of poles than zeros is n − m = 0, 1, 2, 3, 4, the angles of
asymptotes are none, 180, {90, 270}, {60, 180, 300}, {45, 135, 225, 315}, respectively.

2.11 CORRELATION BETWEEN TIME DOMAIN AND
FREQUENCY DOMAIN INFORMATION

Three major groups of events which are not under our control and affect the system
performance are:

1. variations in the process parameters and dynamics,
2. disturbances,
3. sensor noise.

A desired performance specification for any CLS includes specifications regarding
1. stability,
2. response quality (transient and steady state),
3. robustness of stability and response quality despite real-world imperfections, that is
variations in the process dynamics, disturbances, and sensor noise.
The stability of CLS requires that all of the CLS poles be in the left half of the
s-plane (LHP). If a certain degree of relative stability is required, then we can further