Chapter 4 Block Diagrams of Control Systems
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Transcript of Chapter 4 Block Diagrams of Control Systems
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EEE 352 Automatic Control Systems
Chapter 4: Block Diagrams of Control Systems
Prof. Dr. Ahmet Uçar
© Dr. Ahmet Uçar EEE 352 Chapter 4 1
Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 2
A block diagram of a system is a pictorial representation of the functions
performed by each component and of the flow of signals. Such a diagram depicts
the interrelationshipsthat exist among the various components.
Differing from a purely abstract mathematical representation, a block diagram
has the advantage of indicating more realistically the signal flows of the actual
system.
The nonlinear and linear systems canbe represented by block diagrams.
The nonlinear systems
F ( x, u)y (t )
Output
signal
u(t )
Input
signal
State initial condition;
x(t =0) 0
The linear systems
Transfer
function
G(s)
R(s)
Input
signal
Y (s)
Output
signal
State initial condition;
x(t =0) 0
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Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 3
Drawing a Block Diagram of linear systems:
The transfer functions of the components are usually entered in the
corresponding blocks, which are connected by arrows to indicate the direction of
the flow of signals.
Transfer
function
G(s)
R(s)
Input
signal
Y (s)
Output
signal
Branch point
G( s) R( s) Y ( s)
H ( s)
E (s)Gc( s)
Summing point
Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 4
Drawing a Block Diagram of linear systems with the Laplace transforms.
•Write the equations that describe the dynamic behavior of each component.
•Take the Laplace transforms of these equations, assuming zero initial
conditions.
•Represent each Laplace-transformed equation individually in block form.
•Assemble the elements into a complete block diagram.
ei e0 R
C I
Example 4.1 a): Draw block diagram of the following RC circuit with the
Laplace transforms?
Solution 4.1 a): Let define the error between input and output signal as E (s)
)1()()()( 0 s E s E s E i
E i( s)
E 0( s)
E ( s)
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Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 5
Example 4.1: a) Write current I(s)
)2()(1
)( s E R
s I
)3()0()(1
1),0(
1
00
00
e s I Cs
E
sdt e Idt
C e
Write output E 0(s)
ei e0 R
C I E (s) I(s)
R
1
Finally, assemble the elements into a complete
block diagram.
E 0(s)I(s)
Cs
1
e0(0)0
E i( s) E 0( s) E ( s) I ( s)
R
1
Cs
1
e0(0)0
Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 6
Drawing a Block Diagram of linear systems with time domain:
•Write the equations that describe the dynamic behavior of each component.
•Represent each equation individually'inblock form.
•Assemble the elements into a complete block diagram.
ei e0, e0(0)0 R
C I
Example 4.1 b): Draw block diagram of the following RC circuit without Laplace
transform ?
Solution 4.1 b): Let define the error between input and output signal as e(t )
)1()()()( 0 t et et e i
ei(t )
e0(t )
e(t )
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Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 7
Example 4.1 b): Write current i (t )
)2()(1
)( t e R
t i
)3()0()(1
)( 00 t edt t iC t e
Write output e0(t )
ei e0 R
C I e(s) i (t )
R
1
Finally, assemble the elements into a complete block diagram.
e0(t )i(t )
e0(0)0
ei(t ) e0(t )e(t ) i(t )
R
1
e0(0)0
C
1
Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 8
Block diagram of continuous time control system represented in state space
r mnm
r nnn
R D RC t Dut Cxt y
R B R At But Axt x
,)()()(
,)()()(
s1 B
D
A
C U ( s) Y ( s) X ( s)
In time domain:
In Laplace domain:
dt B
D
A
C u(t ) )(t x x(t ) y(t )
x0(0)0
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Block Diagrams of Control SystemsRules of Block Diagram Algebra
© Dr. Ahmet Uçar EEE 352 Chapter 4 9
Original Block Diagram Equivalent Block Diagram
1.Combining blocks in
cascade
2. Moving a summing
point behind a block
A complicated block diagram involving many feedback loops can be simplified by
a step-by-step rearrangement. Simplification of the block diagram by
rearrangements considerably reduces the labour needed for subsequent
mathematical analysis.The following rules block diagram algebra can be used.
G1( s) G2( s)Y ( s) R( s)
G1( s) R( s)
G2( s) Y ( s) X ( s)
R( s)G1( s)
Y ( s)
B( s)
R( s) Y ( s)
G1( s)
G1( s)
B( s)
Block Diagrams of Control SystemsRules of Block Diagram Algebra
© Dr. Ahmet Uçar EEE 352 Chapter 4 10
Original Block Diagram Equivalent Block Diagram
3. Moving a branch point
ahead of a block R( s)G1( s)
Y ( s)
Y ( s)
R( s) Y ( s)
G1( s)
G1( s)
Y ( s)
4. Moving a branch
point behind a block R( s)
G1( s) Y ( s)
R( s)
R( s) Y ( s)G1( s)
R( s)
)(
1
1 sG
R( s)G1( s)
Y ( s)
B( s)
R( s)G1( s)
B( s)
)(
1
1 sG
Y ( s)
5. Moving a summing point
ahead of a block
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Block Diagrams of Control SystemsRules of Block Diagram Algebra
© Dr. Ahmet Uçar EEE 352 Chapter 4 11
Original Block Diagram Equivalent Block Diagram
6. Eliminating a feedback
loop Y ( s)G( s)
R( s) E ( s)+
H ( s)
Y ( s) R( s)
)()(1
)(
s H sG
sG
Example 4.2: Consider the system shown in the following Figure.
Simplify this diagram?
G1 R
H 2
G2 G3Y
H 1
Block Diagrams of Control Systems
Rules of Block Diagram Algebra
© Dr. Ahmet Uçar EEE 352 Chapter 4 12
Example 4.2:(cont.)
G1 R
H 2
G2 G3Y
H 1
G1 R
H 2/G1
G2 G3Y
H 1
By moving the summing point of the negative feedback loop containing H2 outside the
positivefeedback loop containing H1:
Eliminating the positive feedback loop:
R
H 2/G1
G3Y
121
21
1 H GG
GG
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Block Diagrams of Control Systems
Rules of Block Diagram Algebra
© Dr. Ahmet Uçar EEE 352 Chapter 4 13
Example 4.2 : (cont.)
The elimination of the loop containing H2/G1:
R Y
232121
321
1 H GG H GG
GGG
R
H 2/G1
G3Y
121
21
1 H GG
GG
Finally, eliminating the feedback loop results overall system transfer function:
R Y
321232121
321
1 GGG H GG H GG
GGG
Remark. The numerator of the closed-loop transfer function Y (s)/R(s) is the product of
the transfer functions of the feedforward path: G1(s)G2(s)G3(s)
The denominator of Y (s)/R(s) is equal to
1 - (product of the transfer functions around each loop)
Block Diagrams of Control Systems
Rules of Block Diagram Algebra
© Dr. Ahmet Uçar EEE 352 Chapter 4 14
Example 4.2 :(cont.)
The system transfer function: R Y
321232121
321
1 GGG H GG H GG
GGG
Remark.
The numerator of the closed-loop transfer function Y (s)/R(s) is the product of
the transfer functions of the feedforward path: G1(s)G2(s)G3(s)
The denominator of Y (s)/R(s) is equal to
1 -
(product of the transfer functions around each loop)
G1 R
H 2
G2 G3Y
H 1
321232121
321232121
1
)(1
GGG H GG H GG
GGG H GG H GG
The positive feedback loop yields a negative term in the denominator.
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Block Diagrams of Control Systems in MATLAB
© Dr. Ahmet Uçar EEE 352 Chapter 4 15
Example 4.3: Find the transfer function of the following system with given blocks
transfer functions.
Y ( s)G2
R( s) H 2
G3 G4
H 1
G1
H 3
10
1)(1
s
sGMATLAB Command
%%%%%%%%%%%%%%%%%%%%%
% Modelling blocks
ng1=[1]; dg1=[1 10];sysg1=tf(ng1,dg1)
ng2=[1]; dg2=[1 1];sysg2=tf(ng2,dg2)
ng3=[1 0 1]; dg3=[1 4 4];sysg3=tf(ng3,dg3)ng4=[1 1]; dg4=[1 6];sysg4=tf(ng4,dg4)
nh1=[1 1]; dh1=[1 2];sysh1=tf(nh1,dh1)
nh2=[2]; dh2=[1];sysh2=tf(nh2,dh2)
nh3=[1]; dh3=[1];sysh3=tf(nh3,dh3)
1
1)(2
s
sG
44
1
)( 2
2
3
s s
s
sG
6
1)(4
s
s sG
2
1)(1
s
s s H
2)(2 s H
1)(3 s H
Block Diagrams of Control Systems in MATLAB
© Dr. Ahmet Uçar EEE 352 Chapter 4 16
Example 4.3:
Y ( s)G2
R( s)
H 2/G4
G3 G4
H 1
G1
H 3
sys2
sys4 sys3
sys1
sys6
sys
sys5
MATLAB Command
% Block Diagram Reduction
sys1=sysh2/sysg4
sys2=series(sysg3,sysg4)
sys3=feedback(sys2,sysh1,+1)
712219631282517106620512
25664
)(
)(23456
2345
s s s s s s
s s s s s sys
s R
sY
sys4=series(sysg2,sys3)
sys5=feedback(sys4,sys1,-1)
sys6=series(sysg1,sys5)
sys=feedback(sys6,sysh3)
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Block Diagrams of Control Systems in MATLAB
© Dr. Ahmet Uçar EEE 352 Chapter 4 17
MATLAB Command
sysmr = minreal(sys)
Eliminates cancels pole-zero pairs in transfer functions or zero-pole-gain
models.
The output sysmr has minimal order and the same response characteristics
as the original model sys.
Ones many feedback loops simplified it is difficult to observe pole zero pairs toreduce system transfer function futher. However this can readly achived with
Matlab.
712219631282517106620512
25664
)(
)(23456
2345
s s s s s s
s s s s s sys
s R
sY
Example 4.14: (cont.)
Block Diagrams of Control Systems in MATLAB
© Dr. Ahmet Uçar EEE 352 Chapter 4 18
MATLAB Commandsysmr = minreal(sys)
712219631282517106620512
25664
)(
)(23456
2345
s s s s s s
s s s s s sys
s R
sY
33.597.12313775.7208.16
1667.025.025.025.008333.0
)(
)(2345
234
s s s s s
s s s s sysmr
s R
sY
Both transfer functions have the same response characteristics
Example 4.3:
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Block Diagrams of Control Systems in MATLAB
© Dr. Ahmet Uçar EEE 352 Chapter 4 19
Homework 4.1: Simplify the block diagram shown in Figure and obtain theclosed loop transfer function Y (s)/R(s).
Homework 4.2: Simplify the block diagram shown in Figure and obtain the closed
loop transfer function Y (s)/R(s).
R( s) Y ( s)
G1( s) G2( s) G3( s)
H 1( s)
H 2( s)
H 3( s)
R( s) Y ( s) E 1(s)G1( s) G2( s)
H 1( s) H 2( s)
H 3( s)
G3( s) G4( s)
E 2(s)
Block Diagrams of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 20
Drawing a Block Diagram of nolinear systems:
•Write the equations that describe the dynamic behavior of each component.
•Represent each equation individually'inblock form.
•Assemble the elements into a complete block diagram.
Example 4.4: Draw block diagram of the following nonlinear system for non
zero state initial conditions? The system output is y = x ?
x y
x x x x
036.0 2
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Block Diagram of Nonlinear Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 21
x y
x x x x
036.0 2
Solution 4.4: We start to draw the block diagram from the system output y
1 x(t ) y(t )
and considering the integration process as; x(t )
x0(0)0
)(t x
1 y(t ) x(t )
x0(0)0
)(t x
)(t x
0)0(0 x
0.6
3
( )2
Block Diagram of Nonlinear Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 22
Example 4.5: Draw block diagram of the following nonlinear system for non
zero state initial conditions? The system output is y = x 1 ?
1
21
2
12
21
)1(
x y
u x x x x
x x
m u
x1 xe= 022 )( x xc
3
111)( x x xk
Physical
System
f ( x )
Output
y s(t )
Input
u(t ) 0
State initial
condition ;
x(t =0) 0
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Block Diagram of Nonlinear Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 23
1
21
2
12
21
)1(
x y
u x x x x
x x
m u
x1 xe= 022 )( x xc
3
111)( x x xk
Solution 4.5: We start to draw the block diagram from the system output y andconsidering the integration process.
1 y(t ) x1(t )
x10(0)0
)(2 t x
)(2 t x
1
( )3
u(t )
x20(0)0
Block Diagram of Nonlinear Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 24
Homework 4.3: Draw block diagram of the following nonlinear system for non
zero state initial conditions? The system output is y = x .
Homework 4.4: Draw block diagram of the following nonlinear system for non
zero state initial conditions? The system output is y = x 1 .
u x x x x )1( 2
2
3
112
21
x x x x
x x
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MATLAB/Simulink
© Dr. Ahmet Uçar EEE 352 Chapter 4 25
Simulink provides a graphical editor, customizable block libraries, and solvers for
modeling and simulating dynamic systems. It is integrated with MATLAB, enabling
you to incorporate MATLAB algorithms into models and export simulation results
to MATLAB for further analysis.
The construction of a model is simplified with click-and-drag mouse operations.
Simulink includes a comprehensive block library of toolboxes to analyze and
design the linear and nonlinear systems.
Open Simulink Library Browser
Start MATLAB, and then in the MATLAB Command Window,
a) enter
» simulink or
b) by clicking the Simulink icon in the MATLAB toolbar or
Simulink Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 26
c ) by clicking the MATLAB Start button, then selecting Simulink > Library Browser
The Simulink Library Browser in Figure 2 opens.
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Simulink Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 27
Create a New Simulink Model from Simulink Library Browser1. From the Simulink Library Browser menu, select File> New> Model.
An empty model opens in the Simulink Editor;
2. Use Simulink library and build the system block diagram.
3. In the Simulink Editor, select File > Save.
4. In the Save As dialog box, enter a name for your model, and then click Save.
Simulink saves your model.
Simulink Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 28
Example 4.6: Obtain SIMULINK model the following RC circuit.
ei e0, e0(0)0 R
C I
The complete block diagram of the system is;
ei(t ) e0(t )e(t ) i(t )
R
1
e0(0)0
C
1
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Simulink Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 29
The block diagram of the circuit/system can be directly implemented in
SIMULINK, as shown in the following Figure:
Solution 4.6:Assume ei(t ) is a DC input.
ei(t ) e0(t )e(t ) i(t )
R
1
e0(0)0
C
1
yo
ei; e0
ei; e0
ei
-1
e0(0)
1
sxo
Integrator
1
Gain1
1/2
1/R
1
1/C
Simulink Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 30
Example 4.7: A Damped Pendulum System is depicted Figure shows it can
rotate freely around its fixed point P. It consists of a rod with angle
represented by from the vertical position. Using the equation of motion for
a damped pendulum given by
Obtain SIMULINK model the following nonlinear systems.
2122
21
sinml
T x
l
g x
ml
c x
x x
c
v
h
P
2sin
ml
T
l
g
ml
c c
here is the angle of the pendulum from the vertical (in radians),c= 0.15 is the velocity damping term (in 1/sec),
m= 1 is the mass of the pendulum (in kilograms),
l =2.5 is the length (in meters),
g= 9.81 is the acceleration due to gravity (in m/s2) and
T c is the force input (in N).
Changing state variable as leads to),(),( 21 x x
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Simulink Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 31
The SIMULINK model is given in the following Figure:
Solution 4.7:
2122
21
sinml
T x
l
g x
ml
c x
x x
c
x2
x2x1
x1
r
r
1
s
int x2
1
s
int x1
-K-c/ml1
K
c/ml
XY Graph
Tc
sin
Sin(x1)
Scope
K
1/ml
Simulink Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 32
Open an Existing Model
Open an existing Simulink model from the Simulink Library Browser.
1. From the Simulink Library Browser menu, select File > Open.
2. In the Open dialog box, select the model file that you want to open, and then
click Open.
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Block Diagram of Control Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 33
Homework 4.5: Model the following system in SIMULINK and depict the timeresponse of the system output y = x for;
a) u=0 and state initial condition; x0=(1,2).
b) u=1 and state initial condition; x0=(0,0).
u x x x x )1( 2
2
2
112
21
)1( x x x x
x x
Homework 4.6: Model the following system in SIMULINK and depict x1 versus
x2 for
a) state initial condition; x0=(0.2,0.2).
b) state initial condition; x0=(4,4).
Block Diagram of Nonlinear Systems
© Dr. Ahmet Uçar EEE 352 Chapter 4 34
2
3
112
21
x x x x
x x
Homework 4.7: Model the following system in SIMULINK.
a) depict the time response of the system output y = x 1 for state initial condition;
x0=(-1,1).
and depict x1 versus x2 for
b) state initial condition; x0=(-0.5,0).
c) state initial condition; x0=(0.5,0).
d) state initial condition; x0=(1.5,0).
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EEE 352 Automatic Control Systems
Chapter 4: Block Diagrams of Control Systems
Remarks and Questions?