Lecture 8 Notes
The Routh Stability Criterion
Suppose we have a transfer function
Y psq b0 sm ` b1 S m1 ` . `
bm Rpsq sn ` a1 sn1 ` .
` an
We can always factor as
m
T psq
i1 ps
pi q
The closedloop system is stable if
Rppi q 0, @ i
NB: It might turn out
Lecture 5 Notes
Dynamic Response:
Usually, we nd the response of a system using Laplace techniques. Often, do within Matlab.
Example: DC Motor.
Suppose:
J 0.01 kgm2 ; b 0.001 Nmsec
Kt Ke 1 nM/A 1 V/
(rad/sec) Ra 10, L 1 H
Then
psq
s3 ` 10.1s2 ` 101s
Lecture 3 Notes
Modeling principles:
1. Identify the states of the system:
positions
velocitie
s
inductor
currents
capacitor
voltages etc
2. Use physics to nd dx1 cfw_dt, dx2 cfw_dt,.
3. Organize as:
dt
f px, uq
y gpx, uq
where
xstate
vector u
control
Lecture 4 Notes
Block Diagram Manipulations:
G1
G1G2
G1
+
=
G2
G1 + G2
G1
+

G2
=
G
1+G1G2
The gain of a single loop feedback
system (with sign 1 in the loop)
is the
forward gain divided by the sum of
1 plus the loop gain.
r
e
r
=

1
G2

z
G2
e
G2
z
Lecture 7 Notes
Eects of Zeros on Step Response
Weve looked at the response of a secondorder system:
Gpsq
2
s2 ` 2(nn s+n
What if we had a zero in the numerator? How would that change the response? Consider:
ps `
1qn s2 `
2( n s+n
The step response is t
Lecture 6 Notes
Time Domain Specications:
Many control systems are dominated by a second order pair of poles. So look at time
response (to step input) of
Hpsq
2
s2 ` 2nn s+n
Typical response:
1.4
1.2
1
0
.
9
0.5
0.1
0
0
2
4
14
6
r
8
10
12
Time, t (sec)
M
Lecture 10 Notes
PID Control
A common way to design a control system is to use PID control.
PID = proportionalintegralderivative
Will consider each in turn, using an example transfer function
Gpsq
s2 ` a1 s ` a2
Proportional (P) control
In proportional
Lecture 11 Notes
The Root Locus Method
Often, it is useful to nd how the closedloop poles of a system change as a single
parameter is varied. To do this, we use the root locus method.
Root  root of s polynomial equation
Locus  Set of points (plural  l
Lecture 9 Notes
Unity Feedback Control With Noise
Consider a typical unity feedback control system
d,
r
e
disturbance controller plant
+
+
e

+

+
v
sensor noise
e1 is the error perceived by the control system; e is the actual error. The important
trans
Lecture 2 Notes
Reasons for using automatic control:
Reduce workload
Perform tasks people cant
Reduce the eects of disturbances
Reduce the eects of plant variations
Stabilize an unstable system
Improve the performance of a system (time response)
Im