ElmoMC SimplIQ Digital Servo Drives-Bell Getting Started - Manual (Page 91)

The SimplIQ for Steppers Getting Started & Tuning and Commissioning Guide
MAN-BELGS (Ver. 1.1)
91
The sensor, which measures the plant output, adds noise to the measurement.
The controller amplifies the measurement noise. A highly amplified
measurement noise may even saturate the plant input, which might destabilize
the system, or produce an unacceptable output.
Feedback should be carefully designed in order to avoid loss of stability due to
plant uncertainty and plant input saturation.
B.4.2 Open Loop, Gain Margin and Phase Margin,
Bandwidth and Stability
A feedback-controlled system should be carefully designed so that it will not
loose stability, will not oscillate too much and will have good performance in its
entire operating envelope. Stability, bandwidth and gain and phase margins are
the key parameters used to describe a feedback design. These parameters are
now explained, but first we define the open loop of a feedback system.
Open loop – The open loop of the system is the transfer function:
( ) ( ) ( ) ( )
s Sensor s C s P s L
⋅ ⋅ = (15)
where ) s ( P , ) s ( C and
( )
s Sensor are the transfer functions of the plant
controller
and sensor, respectively.
The design criterion to guarantee stability of an LTI feedback system is the
Nyquist stability criterion, which is a frequency domain criterion applied on the
open loop transfer function. A simplification of the Nyquist criterion, adequate
for most motion applications, simply requires that the Bode plot of
( )
ω
j L will
have a phase larger than ο
180
− at the frequency
0
ω
where
( )
1
0
= ω
j L . Clearly,
the Nyquist criterion guarantees that there exists no frequency such that
( )
1
− = ω
j L . A feedback design for which
( )
ω
j L is far from the ‘dangerous’
value -1 at all frequencies guarantees the following two very important closed
loop properties:
The plant output will not oscillate during operation.
If the plant transfer function (characteristic),
) s ( P , slightly changes during
operation, the changed open loop, ) j ( L
ω
, will also satisfy the Nyquist stability
criterion, which means that stability remains under slight model changes.
How much the open loop, ) j ( L
ω
, is far from the
1
− value is, therefore, a critical
design criterion, and is measured by the open loop gain and phase margins.
Gain margin - The gain margin of the transfer function ) ( s L is k , if k is the
smallest positive value such that the plant ) ( s P k
⋅
becomes unstable. For simple
plants,
k
is the smallest positive value for which there exists a frequency,
GM
ω
,
so that
( ) ( )
ο
180 20
− = ω − = ω ⋅
GM GM
j L arg , k j L log and
(16)