Reduction of multiple subsystem [compatibility mode]

Reduction of Multiple
Subsystems

INTRODUCTION

Complicated system/multiple subsystem are represented by the interconnection
of many subsystems.
Multiple subsystem are represented in two ways: as block diagrams and as
signal-flow graphs.

Block diagrams are usually used for frequency-domain analysis and design.

Signal-flow graphs for state space analysis and design.

Techniques to reduce multiple subsystem to a single transfer function:-
1) Block diagram algebra –to reduce block diagrams
2) Manson’s rule - to reduce signal-flow graphs.

BLOCK DIAGRAMS

Figure 1
Components of a block diagram for
a linear, time-invariant system

A. CASCADE FORM

Figure 2
a. Cascaded
subsystems;
b. equivalent transfer
function

Intermediate signal values are shown at the output of each system.
Each signal is derived from the product of the input times the transfer function.

The equivalent transfer function, Ge(s), shown in Figure 1(b), is the output
Laplace transform divided by the input Laplace transform from Figure 1(a), or
Ge ( s )  G3 ( s )G2 ( s )G1 ( s ) [1]

which is the product of the subsystems’ transfer functions.

Eq.[1] was derived under the assumption that interconnected subsystems do not
load adjacent subsystems.

Figure 3
Loading in cascaded
systems

B. PARALLEL FORM

Figure 4
a. Parallel
subsystems;
b. equivalent
transfer
function

Parallel subsystems have a common input and an output formed by the
algebraic sum of the outputs from all of the subsystems.

The equivalent transfer function, Ge(s), is the output transform divided by the
input transform from Figure 4(a) or

Ge ( s )  G1 ( s )  G2 ( s )  G3 ( s ) [2]

which is the algebraic sum of the subsystems’ transfer function; it appears in
Figure 5(b).

C. FEEDBACK FORM

Figure 5
a. Feedback control
system;
b. simplified model;
c. equivalent transfer
function

From Figure 5(b),

E ( s)  R( s)  C ( s) H ( s) [3]

But since C(s)=E(s)G(s),
C ( s) [4]
E ( s) 
G( s)
Substituting Eq. [4] into Eq.[3] and solving for the transfer function,
Ge(s)=C(s)/R(s), we obtain the equivalent, or closed-loop, transfer function
shown in Figure 5(c),

G( s)
Ge ( s)  [5]
1  G( s) H ( s)

The product, G(s)H(s), in Eq.[5] is called the open-loop transfer function, or loop
gain.

MOVING BLOCKS TO CREATE FAMILIAR FORMS

Figure 6
Block diagram
algebra for summing
junctions—
equivalent forms for moving a
block
a. to the left past a
summing junction;
b. to the right past a
summing junction

Figure 7
Block diagram algebra for
pickoff points—
equivalent forms for
moving a block
a. to the left past a pickoff
point;
b. to the right past a
pickoff point

Block diagram reduction via familiar forms
Example 1: Reduce the block diagram shown in Figure 8 to a single transfer
function.

Figure 8
Block diagram
for Example 1

Figure 9
Steps in solving
Example 1:
a. collapse summing
junctions;
b. form equivalent
cascaded system
in the forward path
and equivalent
parallel system in the
feedback path;
c. form equivalent
feedback system and
multiply by cascaded
G1(s)

Block diagram reduction by moving blocks
Example 2: Reduce the system shown in Figure 10 to a single transfer function.

Figure 10
Block diagram for
Example 2

Figure 11
Steps in the
block diagram
reduction for
Example 2

Example 3: Find the equivalent transfer function, T(s)=C(s)/R(s), for the system
shown in Figure 12.

Figure 12
Block diagram for
Example 3

Reduction of multiple subsystem [compatibility mode]

function, T(s)=C(s)/R(s). Use the block diagram reduction.

Figure 13

function, T(s)=C(s)/R(s). Use the block diagram reduction.

Figure 14

ANALYSIS AND DESIGN OF FEEDBACK SYSTEMS

Figure 15
Second-order
feedback control
system

The transfer function K/s(s+a), can model the amplifiers, motor, load and gears.
From Eq.[5], the closed-loop transfer function, T(s), for this system is
K
T ( s)  2 [6]
s  as  K
where K models the amplifier gain, that is, the ratio of the output voltage to the
input voltage. As K varies, the poles move through three ranges of operation of
a second-order system: overdamped, critically damped, and underdamped.

For K between 0 and a2/4, the poles of the system are real and are located at

a a 2  4K [7]
s1, 2  
2 2
As K increases, the poles move along the real axis, and the system remains
overdamped until K=a2/4. At that gain, or amplification, both poles are real and
equal, and the system is critically damped.

For gain above a2/4, the system is underdamped, with complex poles located at

a 4K  a 2 [8]
s1, 2   j
2 2
As K increases, the real part remains constant and the imaginary part increases.
Thus, the peak time decreases and the percent overshoot increases, while the
settling time remains constant.

Example 4: For the system shown in Figure 14, find the peak time, percent
overshoot, and settling time.

Figure 16
Feedback system for
Example 4

The closed-loop transfer function Substituting Eq.[10] into Eq.[11],
found from Eq.[6] is
  0.5 [12]
25
T (s)  2 [9]
Using the values of  and  n
s  5s  25

From Eq.[9], Tp   0.726 s [13]
n 1   2

n  25  5 [10]
%OS  e  / 1 2
100  16.303 [14]

2 n  5 [11]
Ts 
4
 1 .6 s [15]
 n

Example 5: Design the value of gain, K, for the feedback control system of
Figure 15 so that the system will respond with a 10% overshoot.

Figure 15
Feedback system for
Example 5

The closed-loop transfer function of From Eq.[17] and Eq.[18],
the system is
5
K [16]   [19]
T (s)  2 2 K
s  5s  K
A 10% overshoot implies that
From Eq.[16],
 / 1 2
%OS  e 100  10%
2 n  5 [17]
  0.591 [20]

n  K [18] Substituting   0.591 Eq. [19]

K  17.9 [21]

Example 6: For a unity feedback control system with a forward-path transfer
function G(s)=16/s(s+a), design the value of a to yield a closed-loop step
response that has 5% overshoot.

SIGNAL-FLOW GRAPHS
A signal-flow graph consists only of branches, which is represent systems, and
nodes, which represent signals.

A system is represented by a line with an arrow showing the direction of signal
flow through the system.

Adjacent to the line we write the transfer function.

A signal is a node with the signal’s name written adjacent to the node.

Figure 17
Signal-flow graph components:
a. system;
b. signal;
c. interconnection of systems and signals

Each signal is the sum of signals flowing into it.

From Figure 16c, example for the signal,

V ( s )  R1 ( s )G1 ( s )  R2 ( s )G2 ( s )  R3 ( s )G3 ( s ) [1]

C2 ( s )  V ( s )G5 ( s )  R1 ( s )G1 ( s )G5 ( s )  R2 ( s )G2 ( s )G5 ( s )  R3 ( s )G3 ( s )G5 ( s )
[2]

C3 ( s )  V ( s )G6 ( s )   R1 ( s )G1 ( s )G6 ( s )  R2 ( s )G2 ( s )G6 ( s )  R3 ( s )G3 ( s )G6 ( s )
[3]

Notice that in summing negative signals we associate the negative sign with the
system and not with a summing junction, as in the case of block diagrams.

Converting common block diagrams to signal-flow graphs

Example 1: Convert the cascaded, parallel, and feedback forms of the block
diagrams in Figure 2a, 4a and 5b, respectively, into signal-flow graphs.

Figure 18
Building signal-flow
graphs: Figure 2(a)
a. cascaded system
nodes (from Figure 2(a));
b. cascaded system
signal-flow graph;

c. parallel system
nodes (from Figure 4(a));
d. parallel system
signal-flow graph;

Figure 4(a)

e. Feedback system nodes
(Figure 5(b))
f. feedback system
signal-flow graph

Figure 5(b)

Converting a block diagram to a signal-flow graph

Example 2: Convert the block diagram of Figure 10 to a signal-flow graph.

Figure 10

Figure 19
Signal-flow graph
development:
a. signal nodes;
b. signal-flow graph;
c. simplified signal-flow
graph

Example 3: Convert the block diagram of Figure 12 to a signal-flow graph.

Figure 12

MASON’S RULE

Mason’s signal – a technique for reducing signal-flow graphs to single transfer
functions that relate the output of a system to its input.

Definitions

[1]Loop gain
The product of branch gains found by traversing a path that starts at a node and
ends at the same node, following the direction of signal flow, without passing
through an other node more than once.

There are four loop gains (Refer Figure 20):

(1) G2 ( s) H1 ( s) [4a]

(2) G4 ( s) H 2 ( s) [4b]

(3) G4 ( s )G5 ( s ) H 3 ( s ) [4c]

(4) G ( s )G6 ( s ) H 3 ( s ) [4d]
4

Figure 20
Signal-flow graph
for demonstrating
Mason’s rule

[2]Forward-path gain
The product of gains found by traversing a path from the input node to the
output node of the signal-flow graph in the direction of signal flow.

These are two forward-path gains:

(1) G1 ( s )G2 ( s )G3 ( s )G4 ( s )G5 ( s )G7 ( s ) [5a]

(2) G1 ( s )G2 ( s )G3 ( s )G4 ( s )G6 ( s )G7 ( s ) [5b]

[3]Nontouching loops
Loops that do not have any nodes in common. In Figure 20, loop G2(s)H1(s)
does not touch loops G4(s)H2(s), G4(s)G5(s)H3(s), and G4(s)G6(s)H3(s).

[4]Nontouching-loop gain
The product of loop gains from nontouching loops taken two, three, four, or
more at a time. In Figure 20 the product of loop gain G2(s)H1(s) and loop gain
G4(s)H2(s) is a nontouching-loop gain taken two at a time. In summary, all three
of the nontouching-loop gains taken two at a time are

(1) G2 ( s) H1 ( s)G4 ( s) H 2 ( s) [6a]

(2) G2 ( s) H1 ( s)G4 ( s)G5 ( s) H 3 ( s) [6b]

(3) G2 ( s) H1 ( s)G4 ( s)G6 ( s) H 3 ( s) [6c]

Mason’s Rule

The transfer function, C(s)/R(s), of a system represented by a signal-flow graphs
is
C ( s )  k Tk  k
G (s)   [7]
R( s) 

where

k Number of forward paths
Tk  The kth forward-path gain

 1-  loop gain +  nontouching-loop gains taken two at a
time -  nontouching-loop gains taken three at a time + 
nontouching-loop gains taken four at a time - …

 k   -  gain term in  that touch the kth forward path. In
loop
other words,  k is formed by eliminating from  those loop
gains that touch the kth forward path.

Transfer function via Mason’s rule

Example 1: Find the transfer function, C(s)/R(s), for the signal-flow graph in
Figure 21.

Figure 21
Signal-flow graph

Step 1: Identify the forward-path gains. In this case, there is only one.

G1 ( s )G2 ( s )G3 ( s )G4 ( s )G5 ( s ) [8]

Step 2: Identify the loop gains. There are four, as follow:

(1) G2 ( s ) H1 ( s ) [9.1]

(2) G4 ( s ) H 2 ( s ) [9.2]

(3) G7 ( s ) H 4 ( s ) [9.3]

(4) G2 ( s )G3 ( s )G4 ( s )G5 ( s )G6 ( s )G7 ( s )G8 ( s ) [9.4]

Step 3: Identify the nontouching loops taken two at a time.

From Eq.[9] and Figure 21, loop 1 does not touch loop 2, loop 1 does
not touch loop 3 and loop 2 does not touch loop 3. Notice that loops 1,
2, and 3 all touch loop 4. Thus, the combinations of nontouching loops
taken two at a time are as follows:

G2 ( s ) H1 ( s )G4 ( s ) H 2 ( s ) [10.1]
Loop 1 and loop 2:

Loop 1 and loop 3: G2 ( s ) H1 ( s )G7 ( s ) H 4 ( s ) [10.2]

Loop 2 and loop 3:
G4 ( s ) H 2 ( s )G7 ( s ) H 4 ( s ) [10.3]

Finally, the nontouching loops taken three at a time as follows:

Loop 1, 2 and 3: G2 ( s ) H 1 ( s )G4 ( s ) H 2 ( s )G7 ( s ) H 4 ( s ) [11]

Step 4: From Eq.[7] and its definitions, we form  and  k . Hence,
  1  [G2 ( s ) H 1 ( s )  G4 ( s ) H 2 ( s )  G7 ( s ) H 4 ( s ) [12]
 G2 ( s )G3 ( s )G4 ( s )G5 ( s )G6 ( s )G7 ( s )G8 ( s )]
 [G2 ( s ) H 1 ( s )G4 ( s ) H 2 ( s )  G2 ( s ) H 1 ( s )G7 ( s ) H 4 ( s )
 G4 ( s ) H 2 ( s )G7 ( s ) H 4 ( s )]

 [G2 ( s ) H 1 ( s )G4 ( s ) H 2 ( s )G7 ( s ) H 4 ( s )]

We form  k by eliminating from  the loop gains that touch the kth forward
path:

1  1  G7 ( s) H 4 ( s) [13]

Step 5: Expressions [8], [12] and [13] are now substituted into Eq.[7], yielding
the transfer function:

T11 [G1 ( s)G2 ( s)G3 ( s)G4 ( s)G5 ( s)][1  G7 H 4 ( s)] [14]
G( s)  
 

Since there is only one forward path, G(s) consists of only one term, rather than
a sum of terms, each coming from a forward path.

Example 2: Use Mason’s rule to find the transfer function of the signal-flow
diagram shown in Figure 22.

Figure 22
Signal-flow graph

Example 3: Use Mason’s rule to find the transfer function of the signal-flow
diagram shown in Figure 23.

Figure 23

SIGNAL-FLOW GRAPHS OF STATE EQUATIONS

Consider the following state and output equations:

x1  2 x1  5 x2  3x3  2r
 [15a]
x2  6 x1  2 x2  2 x3  5r [15b]


x3  x1  3x2  4 x3  7r
 [15c]
y  4 x1  6 x2  9 x3 [15d]

Step 1: Identify three nodes to be the three state variables, x1, x2 and x3; also
identify three nodes, placed to the left of each respective state variable, to be
derivatives of the state variables. Also identify a node as the input, r, and another
node as the output, y.

Step 2: Next interconnect the state variables and their derivatives with the
defining integration, 1/s.

Step 3: Then using Eqs.[15], feed to each node the indicated signals.

Example:

Eq.[15a] - x1 receives 2 x1  5 x2  3x3  2r (Figure 24c)


Eq.[15b] - x2 receives 6 x1  2 x2  2 x3  5r (Figure 24d)


Eq.[15c] - x3 receives x1  3x2  4 x3  7r (Figure 24e)


Eq.[15d] – the output,  4 x1  6 x2  9 x3 (Figure 24f)

Figure 24f – the final phase-variable representation, where the
state variables are the outputs of the integrators.

Figure 24
Stages of
development of a
signal-flow graph
for the system of
Eqs.15:
a. place nodes;
b. interconnect
state variables and
derivatives;
c. form dx1/dt ;
d. form dx2/dt
(figure continues)

Figure 24
(continued)
e. form dx3 /dt;
f. form output

Example 1: Draw a signal-flow graph for the following state and output
equations:

 2 1 0 0 
x  0

  3 1  x  0 r
  
 3  4  5
  1 
 
y  0 1 0 x

Reduction of multiple subsystem [compatibility mode]

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