Reduction of Multiple Subsystems

Reduction of Multiple Subsystems

Ref: Control System Engineering

Norman Nise : Chapter 5

RM 1511 – AUTOMATIC CONTROL

Multiple subsystems - 2

Chapter objectives :

� How to reduce a block diagram of multiple subsystems to a

single block representing the transfer function from input

to output

� How to analyze and design transient response for a system

consisting of multiple subsystems

� How to represent in state space a system consisting of

multiple subsystems



1. Block Diagrams for Dynamic Systems

Block diagram � an interconnection of blocks representing

basic mathematical operations in such a way that the overall

diagram is equivalent to the system’s mathematical model.

In such a diagram, the lines interconnecting the blocks

represent the variables describing the system behaviour.

Kx f

A block diagram representing f = Kx



Summer � addition and subtraction of variables x1

x2

x3

+

+

-

y

A summer representing y = x1 + x2 - x3

Pickoff point � input signal distribution to several output point



Constant � has no input, and its output never changes

cy

Gain � multiplication of a single by a constant (exp. spring)

Integrator � integration with respect to time

u y y& y

∫ dt∫ dt

The block diagram for an integrator



Rules for altering diagram structure

Transfer functions which are generally the ratio of two polynomials are

often denoted by F(s), G(s) or H(s). When the transfer function is a constant,

then that block reduces to a gain block.

Series combination



Parallel combination

Example 1

Evaluate the transfer functions Y(s)/U(s) and Z(s)/U(s) for the block diagram

below � give the results as rational functions of s



Equivalent diagrams for

the diagram shown in

Example 1



Moving a pick off point � a point where an incoming variable in the diagram

is directed into more than one block

1 Original diagram, 2 & 3 equivalent diagrams

(2)

(1)

(3)

Moving block to create familiar forms



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



Moving a summing junction

Ahead of a block After a block



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



Example 2 � Modify the bock diagram in (a) to remove the right summing

junction, leaving only the left summing junction

(a) Original diagram, (b), (c) & (d) equivalent diagrams



Reducing diagrams

for feedback systems

G(s) = Y(s)/V(s) � forward transfer

function

H(s) = Z(s)/Y(s) � feedback transfer

function

G(s)H(s) � open-loop transfer

function

T(s) = Y(s)/U(s) � closed-loop

transfer function

H(s) = 1 � unity feedback

system



Block diagram reduction via familiar form

Example 3 � reduce the block diagram shown below to a single transfer

function



Steps in solving Example 3:

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 4 � reduce the block diagram shown below to a single transfer

function



Steps in the

block diagram

reduction for

Example 4



Example 5 � find the equivalent transfer function T(s)=C(s)/R(s)



Example 6 � Find the closed-loop transfer function for the feedback

system below. Compare the locations of the poles of the open-loop and

closed-loop transfer function in s-plane.



Example 7 � Find the closed-loop transfer

function of the two-loop feedback system in

Fig 1. Also express the damping ratio and the

un-damped natural frequency of the closed-

loop system in terms of the gains a0 and a1.

Equivalent block diagrams

Figure 1



2. Analysis and Design of Feedback System

Immediate application of the principles of block diagram.

Example 9 � find the peak time, percent overshoot and settling time.

Example 10 � design the value gain K for the system below so that the

system will respond with a 10 % overshoot



3. Signal-Flow Graphs

Signal flow graphs are alternative to block diagram.

A signal flow graph consists only of branches, which represent systems, and

nodes, which represent signals.

Signal-flow graph components:

a. system;

b. signal;

c. interconnection of systems

and signals



a. cascaded system nodes; b. cascaded system signal-flow graph;

Converting common block diagrams to signal-flow graphs



c. parallel system nodes; d. parallel system signal-flow graph;



e. feedback system nodes; f. feedback system signal-flow graph;



Example 11 � Convert the

block diagram in Example

4 to signal-flow graph.

Signal-flow graph

development:

a. signal nodes;

b. signal-flow graph;

c. simplified signal-flow

graph



Example 12 � Convert the block diagram below to signal-flow graph

Reduction of Multiple Subsystems

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