ElmoMC SimplIQ Digital Servo Drives-Bell Getting Started - Manual (Page 82)

The SimplIQ for Steppers Getting Started & Tuning and Commissioning Guide
MAN-BELGS (Ver. 1.1)
82
Appendix B: A Short Course in
Linear Control
This section goes over some theoretical matters that are the essence of the
Conductor.
It brings continuous time theory although the SimplIQ is a sampled, discrete
system. This simplifies the theoretical presentation while maintaining physical
understanding.
B.1 Linear Systems and Transfer Functions
A system is something that has inputs and outputs. The inputs (deliberate inputs
and disturbances) cause the system to respond. The outputs are what can be
observed from the response of the system to the inputs. For example, consider a
system consisting of an analog servo amplifier, a motor, and a shaft encoder. The
input to the system is the ±10 V command voltage of the servo amplifier. The
output of the system is the reading of the shaft encoder. A schematic
representation of a system, named S, whose input is
u and output ) ( u S y
=
, is
shown in Figure 74 figure below.
System S(.)
Input u
Output y
Figure 74: Block diagram of a system S with input u and output ) ( u S y
=
In the above block diagram, the letter S hides very complicated electronics and
physical phenomena. Writing a true model of the system is too complex. For the
purpose of controller design, we are content with a simple, approximate model.
The performance we want to obtain dictates the level of model complexity. At
first approximation, the example system is LTI (Linear and Time Invariant). LTI
models (not necessarily for truly LTI systems) have a great advantage – a well-
established theory of dealing with them. LTI systems have the following useful
properties:
1. The superposition principle works. Mathematically, if the system’s output is
1
y for input
1
u and
2
y for input
2
u , then the system’s output for the sum of
inputs,
2 1
u u
+
, is the sum of outputs,
2 1
y y
+
. A simple consequence of this is
that if the system’s input is amplified by a fixed value, so is its output.
2. They always behave the same way. If ) ( t y is the output for the input ) ( t u then
) ( T t y
−
is the output to the delayed input ) ( T t u
−
, for any time shift T .
To summarize, if the signals
1
u and
2
u are applied to a system S and yield the
outputs ) (
1
u S and ) (
2
u S , and for any constant a , time shift T and input u
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
T t u S T t y t u S t y
u S a u a S
u S u S u u S
− = − ⇒ =
⋅ = ⋅
+ = +
2 1 2 1
(1)