CONTROL SYSTEMS THEORY

Reduction of Multiple Subsystems

CHAPTER 3STB 35103

Objectives To reduce a block diagram of multiple

subsystems to a signal block representing the transfer function from input to output

Introduction Before this we only worked with individual

subsystems represented by a block with its input and output.

Complex systems are represented by the interconnection of many subsystems.

In order to analyze our system, we want to represent multiple subsystems as a single transfer function.

Block diagram A subsystems is represented as a block

with an input and output and a transfer function.

Many systems are composed of multiple subsystems. So, we need to add a few more schematic elements to the block diagram. Summing junction Pickoff points

Block diagram

Block diagram Summing junction

Output signal, C(s), is the algebraic sum of the input signals, R1(s), R2(s) and R3(s).

Pickoff point Distributes the input signals, R(s),

undiminished, to several output points.

Block diagram There are three topologies that can be

used to reduce a complicated system to a single block.

Cascade form Parallel form Feedback form

Block diagram Cascade form

a. cascaded subsystem b. equivalent transfer function

Equivalent transfer function is the output divided by the input.

Block diagram Parallel form

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

Block diagram Feedback form

It is the same as the closed loop system that we learn in Chapter 1.

a. closed loop system b. closed loop, G(s)H(s) is open loop transfer function

Block diagram Moving blocks to create familiar forms

Cascade, parallel and feedback topologies are not always apparent in a block diagram.

You will learn block moves that can be made in order to establish familiar forms when they almost exist. I.e. move blocks left and right past summing junctions and pickoff points.

Block diagramBlock diagramalgebra for summingjunctions—equivalent forms for moving a block

a. to the left past asumming junction;b. to the right past a summing junction

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

Block diagramBlock diagram reduction via familiar formsExample:Reduce the block diagram to a single

transfer function.

Block diagramSolution:Steps in solving

Example 5.1:a. collapse summingjunctions;b. form equivalentcascaded systemin the forward pathand equivalentparallel system in thefeedback path;c. form equivalentfeedback system andmultiply by cascadedG1(s)

Block diagramBlock diagram reduction by moving blocksExample:Reduce the system shown to a single transfer function.

Block diagramSolution:First, move G2(s) to the left past the pickoff point to

create parallel subsystems, and reduce the feedback system consisting of G3(s) and H3(s).

Block diagramSecond, reduce the parallel pair consisting of 1/g2(s) and unity and push G1(s) to the right past the summing junction, creating parallel subsystems in the feedback.

Block diagramThird, collapse the summing junctions, add the two feedback elements together, and combined the last two cascaded blocks.

Block diagramFourth, use the feedback formula to obtain figure below

Finally multiply the two cascaded blocks and obtain the final result.

Block diagramExercise:Find the equivalent transfer function, T(s)=C(s)/R(s)

Solution Combine the parallel blocks in the forward path. Then, push 1/s to

the left past the pickoff point. Combine the parallel feedback paths and get 2s. Apply the

feedback formula and simplify

Block diagram reduction rules Summary

Block diagram reduction rules

Signal-Flow graphs Alternative method to block diagrams. Consists of

(a) Branches Represents systems

(b) Nodes Represents signals

Signal-Flow graphs Interconnection of systems and signals

Example V(s)=R1(s)G1(s)-R2(s)G2(s)+R3(s)G3(s)

Signal-Flow graphs Cascaded system

Block diagram

Signal flow

Signal-Flow graphs Parallel system

Block diagram

Signal flow

Signal-Flow graphs Feedback system

Block diagram

Signal flow

SFG Question Given the following block diagram, draw a

signal-flow graph

Solution

Mason’s rule What?

A technique for reducing signal-flow graphs to single transfer function that relate the output of system to its input.

We must understand some components before using Mason’s rule Loop gain Forward-path gain Nontouching loops Nontouching-loop gain

Mason’s rule

Loop gain Product of branch gains found by going through a path

that starts at a node and ends at the same node, following the direction of the signal flow, without passing through any other node more than once.

G2(s)H1(s) G4(s)H2(s) G4(s)G5(s)H3(s) G4(s)G6(s)H3(s)

Mason’s rule

Forward-path gain Product of gains found by going through a path from the

input node of the signal-flow graph in the direction of signal flow.

G1(s)G2(s)G3(s)G4(s)G5(s)G7(s) G1(s)G2(s)G3(s)G4(s)G6(s)G7(s)

Mason’s rule

Nontouching loops Loops that do not have any nodes in common.

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)

Mason’s rule

Nontouching-loop gain Product of gains form nontouching loops taken

two, three, four, or more at a time. [G2(s)H1(s)][G4(s)H2(s)] [G2(s)H1(s)][G4(s)G5(s)H3(s)] [G2(s)H1(s)][G4(s)G6(s)H3(s)]

Mason’s rule The transfer function, C(s)/R(s), of a system

represented by a signal-flow graph is

k

kkT

sR

sCsG

)(

)()(

gainpath -forwardkth the

path forward ofnumber

kT

k

Mason’s rule

timeaat four taken gains loop gnontouchin

timeaat ee taken thrgains loop gnontouchin

timeaat taken twogains loop gnontouchin

gains loop -1

path. forwardth h the that toucgains loop those

from geliminatinby formed

kk

Example Draw the SFG representation

Solution SFG

Solution

Mason’s ruleQuestion Using Mason’s rule, find the transfer

function of the following SFG

Solution

Exercise 1 Apply Mason’s rule to obtain a single

transfer function

Exercise 21. Reduce to a single transfer function (BDR)2. Draw the SFG representation3. Apply Mason’s rule to obtain the transfer function