Text Illustrations in PPT
Chapter 3:
THEORETICAL FOUNDATION AND
BACKGROUND MATERIAL: COMPLEX VARIABLES,
DIFFERENTIAL EQUATIONS, LAPLACE TRANSFORM
Automatic Control Systems, 9th
Edition
Farid Golnaraghi, Simon Fraser University
Benjamin C. Kuo, University of Illinois
ISBN: 978-0-470-04896-2
Heating system block diagram (simplified).Actual temp. (output)
measured by sensor in the thermostat. Simple electronic circuit
(comparator) compares temps. Generates error voltage that acts as as
switch to open the gas valve to turn out the furnace. Opening windows
and doors etc. in room causes heat loss (disturbance). Process of sensing
output and comparing with input to generate error signal called
Feedback.
fig_03_01
(a) Open loop, dc-motor, speed control system (b) Block diagram; input
voltage to the motor, output of (non linear) power amp. Representation
issue for NL blocks.
fig_03_02
Common elements in block diagram of most control systems
(Compare to Figs.3-1, 3-2
. Comparators (electronic circuit measures error)
. BLOCKS representing transfer functions
. Reference sensor
. Output sensor
. Actuator
. Controller
. Plant
. INPUT or reference signals
. OUTPUT signals
. Disturbance signal
. Feedback loops
fig_03_03
fig_03_04
fig_03_05
X(s) = G(s) U(s)
fig_03_06
X(s) = G_1(s) G_2(s) U(s)
fig_03_07
G(s) = G_1(s) + G_2(s)
M(s) = Y(s) =
R(s)
M(s) = Y(s) =
R(s)
G(s)
1 + G(s) H(s)
G(s)
1 - G(s) H(s)
fig_03_08
negative feedback
positive feedback
ẍ(t) + 2ζωn ẋ(t) + ωn2 (t)x(t) = ωn2 u(t)
2ζωn2 sX(s)
fig_03_09
fig_03_10
fig_03_11
fig_03_12
fig_03_13
fig_03_14
ωn2
fig_03_15
Moving a branch point
H_1(s) G_2(s)
fig_03_16
Moving a comparator from RHS of G_2(s) to its LHS
fig_03_17
Block diagram reduction
first move branch pt. at Y_1 to left of G_2
fig_03_18a
1 + G_1G_2H_1
fig_03_18b
Block diagram of system undergoing disturbance
fig_03_19
We need to determine effects of D(s) on the system
Ytotal = YR |D=0 + YD |R=0
Y (s)
G1 (s)G2 (s)
=
R(s)
1 + G1 (s)G2 (s)H1 (s)
Y (s)
−G2 (s)
=
R(s)
1 + G1 (s)G2 (s)H1 (s)
So disturbance interferes with controller signal and adversely affects
system performance.
fig_03_20
Block diagram for D(s) = 0
fig_03_21
Block diagram for R(s)=0
fig_03_22
fig_03_23
Y (s) = [I + G(s)H(s)]−1 G(s)R(s)
Signal Flow Graphs
fig_03_24
Basic properties of SFG
. Only for linear systems
. Equations for SFG algebraic equations
. Nodes are used to represent variables
. Signals travel along branches only in direction of arrows
. Branch from node y_1 to y_2 shows dependence of node y_2 on node y_1
Other SFG terms
Input node (source) - A node that only has outgoing branches
Output node (sink) - A node that only has incoming branches
fig_03_25
fig_03_26
Can make y_2 an output node.
fig_03_27
Erroneous operation for SFG of Fig. 3-26(a)
Cannot make y_2 an input node; equation for
y_2 different from original SFG.
fig_03_28
4 loops in the SFG of Fig.3-25(d)
Node y_1 a summing point; also a transmitting point.
fig_03_29
fig_03_30
Simplification of SFG
fig_03_31
fig_03_32
SFG for feedback system of Fig.3-8
SFG Terms
Forward Path: a path that starts at an input node and ends at an output
node and along which no no node is traversed more than once.
Path Gain: The product of the branch gains encountered in traversing
a path.
Loop: A path that starts and terminates at the same node and along
which no other node is encountered more than once.
Forward-Path Gain: Path gain of a forward path.
Loop Gain: Path gain of a loop.
fig_03_33
What is the gain y7/y1 ?
fig_03_35
fig_03_36
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table_03_01