This document discusses describing function analysis of nonlinear systems. It defines various components in a control system including the reference variable, error signal, control variable, manipulated input, controlled variable, measured output, feedback signal, disturbance, measurement noise, actuator, process, controller, transmitter, and filter. It states that in many cases, a system's nonlinearity can be fully characterized by its static characteristics, and its dynamics can be neglected. It provides examples of nonlinearities including saturated actuators, relay control, gear backlash, hysteresis in magnetic materials, and dead zones in electromechanical systems.

1. Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
r: reference variable
e: error signal
u: control variable
m: manipulated input
y: controlled variable
ym: measure of the output
z: feedback signal
d: disturbance
n: measurement noise
A
actuator
P
process
C
controller
T
trasmitter
+
+
+r(t) e(t) u(t) m(t)
d(t)
y(t)
ym(t)
_
F
filter
z(t)
Controller
+
n(t)
+
Plant

2. Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
In many cases the system presents a nonlinear phenomenon which is fully
characterised by its static characteristics, i.e., its dynamics can be neglected
Saturated actuators
Relay control
Gears backlash
Hysteresis in magnetic
materials
Dead zone in electro-
mechanical systems
u
m
+M
-M

3. Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
In many cases the system presents a nonlinear phenomenon which is fully
characterised by its static characteristics, i.e., its dynamics can be neglected
Saturated actuators
Relay control
Gears backlash
Hysteresis in magnetic
materials
Dead zone in electro-
mechanical systems
e
e
+U
-U
+E
-E

4. Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
In many cases the system presents a nonlinear phenomenon which is fully
characterised by its static characteristics, i.e., its dynamics can be neglected
Saturated actuators
Relay control
Gears backlash
Hysteresis in magnetic
materials
Dead zone in electro-
mechanical systems
u (driving gear)
+U
-U
m (driven gear)

5. Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
In many cases the system presents a nonlinear phenomenon which is fully
characterised by its static characteristics, i.e., its dynamics can be neglected
Saturated actuators
Relay control
Gears backlash
Hysteresis in magnetic
materials
Dead zone in electro-
mechanical systems
u
m
+M0
-M0

6. Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
In many cases the system presents a nonlinear phenomenon which is fully
characterised by its static characteristics, i.e., its dynamics can be neglected
Saturated actuators
Relay control
Gears backlash
Hysteresis in magnetic
materials
Dead zone in electro-
mechanical systems
u
+U
-U
m