Chapter 4: System Modeling with Block Diagrams

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Chapter 4 System Modeling with Block Diagrams
2
Outline

• Block diagrams what why
• Sample block diagrams of known systems
• Block diagram construction with example
• Aggregation of blocks
• System analysis using block diagrams
• Reduction of complex block diagrams
• Matlab

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Introduction

• Pictorial representation of a system
• Components in the system
• Signal flow
• Used for understanding a system
• Used for analyzing a system

u(k1) a1u(k) a2u(k-1) be(k1)
y(k1) cy(k) du(k)
Controller
Target System
r(k)
e(k)
u(k)
y(k)

-
e(k) y(k) r(k)
Transducer
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Description

• Signals - expressed in z-domain
• Blocks - labeled by transfer function

U(z)
U(z)
Y(z)
G(z)
where, Y(z) G(z)U(z)
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Description contd

• Summation point
• Branching point or take-off point

R(z)
E(z) R(z) W(z)


W(z)
Y(z)
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Block diagrams of a few systems
Feedback Control System
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Block diagrams of a few systems contd
Supervisory Control System
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Supervisory Control Example Profit Maximization
Supervisory Logic
Revenue r (number of Completed transactions)
Controller design Reference value selection
Cost c (number of responses longer than
the response time constraint W)
Profit model
Profit Revenue - Cost
System identification
Users
Reference value
MaxUsers
Completed Transactions
Feedback Controller
Response Time
Server Log
Administrator
Server
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The Big Picture
Start
System Modeling
Controller Design
Block diagram construction for Target System
Controller Evaluation-using block diagrams
Transfer function formulation and validation
Objective achieved?
Y
Stop
Model Ok?
Y
N
N
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Block diagram Construction
Example M/M/1/K Feedback control System

• Work flow of an M/M/1/K system

l
m

K

• Functional block for M/M/1/K system

M/M/1/K
U(z)
Y(z)
G(z)
Response time
Buffer size
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Example contd

• Modeling disturbance and noise

Noise N(z)
Disturbance D(z)
M/M/1/K
T(z)
G(z)
U(z)
V(z)
Y(z)




Measured Response time
Actual Response time
Buffer size

• Reference signal and controller

Controller
K(z)
R(z)
U(z)
Buffer size
Desired Response time
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Example contd
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Example contdComputation of System Transfer
function

• G(z) for M/M/1/K
• From system identification,
• y(k1) 0.49y(k) 0.033u(k)
• Taking z-transform,
• z?k0 to 8 y(k1)z-(k1) ?k0 to80.49y(k)z-k
  ?k0 to 8 0.033u(k)z-k
• zY(z) 0.49 Y(z) 0.033 U(z)
• Y(z)/U(z) G(z) 0.033 / (z - 0.49)

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Example contdComputation of Controller Transfer
function

• K(z) for M/M/1/K Controller
• Lets assume,
• u(k1) 0.35u(k) 0.01u(k-1) 0.75e(k)
• Taking z-transform,
• z?k0 to 8 u(k1)z-(k1) ?k0 to80.35u(k)z-k
  
• z-1 ?k0 to 8 0.01u(k-1)z-(k-1)
• ?k0 to 8 0.75e(k)z-k
• zU(z) 0.35 U(z) z-1 0.01U(z) 0.75 E(z)
• U(z)/E(z) K(z) 0.75z / (z2 - 0.35z - 1)

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Aggregation of Blocks
E(z)
U(z)
Y(z)
G(z)
K(z)
E(z)
Y(z)
K(z)G(z)
Feedforward transfer function
Y(z) U(z)
FFF(z) Y(z) / E(z)
U(z) E(z)
G(z)K(z)
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Aggregation of Blocks contd
E(z)
U(z)
Y(z)
G(z)
K(z)
H(z)
W(z)
E(z)
W(z)
K(z)G(z)H(z)
Loop transfer function
W(z) Y(z) U(z)
FLP(z) W(z) / E(z)
Y(z) U(z) E(z)
H(z)G(z)K(z)
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Transfer function of System Control Analysis
Controller
Target System
K(z)
G(z)
R(z)
Y(z)
U(z)
E(z) Error
W(z)
H(z)
R(z)
Y(z)
K(z)G(z)
1 K(z)G(z)H(z)
Reference Feedback transfer function

• E(z) R(z) W(z)
• R(z) E(z)K(z)G(z)H(z)
• R(z) E(z) 1 K(z)G(z)H(z) (1)
• Y(z) E(z)K(z)G(z) (2)
• Using (1) and (2),
• FR(z) Y(z) / R(z)

FFF
K(z)G(z)

1 FLP
1K(z)G(z)H(z)
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Analysis of Error, Disturbance and Noise
D(z)
N(z)
Controller
Target System


K(z)
G(z)


R(z)
Y(z)
U(z)
T(z)
E(z) Error
W(z)
H(z)
-G(z)H(z)

• FDE E(z) / D(z)

1 K(z)G(z)H(z)
1

• FN T(z) / N(z)

1 K(z)G(z)H(z)
-H(z)

• FNE E(z) / N(z)

1 K(z)G(z)H(z)
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• Analysis of M/M/1/K Feedback Control System

Noise N(z)
Workload Variations
Controller
M/M/1/K


K(z)
G(z)


R(z)
Y(z)
U(z)
T(z)
E(z) Error
V(z) System input
Actual Response time
Desired Response time
Measured Response time
Buffer size
K(z) 0.75z / (z2 - 0.35z - 1) G(z) 0.033 / (z
- 0.49) H(z) 1
FR(z), FRE(z), FD(z), FDE(z), FN(z), FNE(z) ?
Matlab exercise
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Block Diagram Reduction/Aggregation
Example Cascaded Control
Measured System Utilization
D(z)
Desired System Utilization
Desired Response time
Measured Response time
users
K1(z)
K2(z)
G2(z)
G1(z)

R(z)

Y(z)


-
-
H2(z)
H1(z)
K2(z)G2(z)
K1(z)
G1(z)
R(z)
Y(z)

-
1K2(z)G2(z)H2(z)
H1(z)
K1(z)G1(z)K2(z)G2(z)
1K2(z)G2(z)H2(z)
R(z)
Y(z)
1 K1(z)G1(z)K2(z)G2(z)H1(z)
1K2(z)G2(z)H2(z)
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Its not always simple
H1(z)
K1(z)
-
K2(z)
G(z)
Y(z)
R(z)


-
H2(z)
???
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Reduction Rules
X
X
P2
P1P2
Y
Y
P1
X
X
Y
P1/- P2
Y
P1

/-
P2
z
KI
z-1

KP

PI Controller
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More rules
Y
P1P2
Y
P1
X
X
1/P2


/-
/-
P2
X
Z
P
X
P

Z

/-
/-
Y
1/P
Y
X
P
X
P
Y
Y
P
Y
Y
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Block Diagram Reduction - Example
C
D
Y
X
A
B

-
E
F
D
C/A

Y
X
A
B

-
E
F
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Example contd
D/B
Y
X
AB
1C/A

-
E
F
D/B
Y
X
AB
1C/A

-
E
F
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Example contd
Y
X
AB
1D/B
1C/A

-
E
F
Y
X
AB
1D/B
1C/A
1ABE
F
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Example contd
Y
X
AB(1D/B)
1C/A
1ABE
F
(AC)(BD)
1ABE
Y
X
(AC)(BD)F
1
1ABE