classical control revision[1]

7/31/2019 Classical Control Revision[1]

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1. Classical Control Revision

Classical control which was studied for relatively simple systems, consisted of these techniques:

Modelling Process

Representation by TransferFunction

Analysis and Design by:

(i) Direct Methods

(i) Frequency Response(i) Root Locus

You have already considered relatively simple systems, which could be described by linear differential

equations. In this course we will briefly revise and enlarge the material of the Part 3 Course, and include twonew topics:

Modern Control Theory

Digital Control Theory

Modern control theory does not replace Classical control. Rather, each technique has its use, A control

engineer must be familiar with a range of techniques to solve a wide range of problems that will be encountered.

Only with experience does a control engineer become accustomed to which tool should be use for which task.

Digital control theory, as the name suggests, is concerned with applying control theory (both Classical and

Modern) into a digital environment. Due to the massive computerisation of all industries this area of control is

becoming very important.

Both these forms of control theory are highly mathematical. Thus it is advisable that you not only revise the

material from the Part 3 course, but you also do some Mathematics revision, especially Matrix Theory.

1.1 Manipulations of Block Diagrams

Manipulation follows standard common-sense rules. Note that we assume one block does not load another - the

output of a block is unaffected by what it feeds into.

For example:

v

vs

sTT R C

o

i

( )

1

1 11 1 1where

v

vs

sT sT

o

i

( )

1

1

1

11 2

v

vs

sT sT

o

i

( )

1

1

1

11 2

Thus when considering block diagrams, it is assumed that the physical circuits will be similar to the buffered

type. Care must therefore be taken when designing control circuits, to ensure minimal loading, i.e. input/output

impedance matching. This is not a concern when considering digital circuits.

vi vo

R1

C1

v i vo

R1

C1

R2

C2

vi vo

R1

C1

R2

C2+1

Buffer


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Problem 1-1

G2

H1

Show that:

_

G1

HG

GH1

3

1

2

G2

G1

Reduces to:

R

H2

G2

_

C

G2

_

R C

Problem 1-2

G2

H1

Show that:

+

G G

G G H G H G G H

1 2

2 3 2 4 2 1 2 11

G1

Reduces to:

R

H2

G3+

C

G

G3

4

2

_

R C

+ +

G4

+

G G G G G

G G H G H G G H G G G G G

1 2 3 1 4

2 3 2 4 2 1 2 1 1 2 3 1 41

Which Simplifies to:

This method of block reduction, especially for multi-loop systems, can become very tedious. There are othermethods of tackling the problem: graph methods and Signal-flow Diagrams.


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1.2 Signal-Flow Diagrams

Signal-flow diagrams are an alternative way of representing systems graphically, i.e. instead of using block

diagrams.

For example a typical block diagram would be:

Equivalent signal-flow diagram (or signal-flow graph):

Advantages:1) Although the information is the same (it represents equations in graphical form), it is

sometimes easier to draw. Loading effects can sometimes be accommodated for.

1) Solutions, e.g.C

RG G 1 2 from above, are directly available byMasons Rules.

G1 G2

R CX1

CX1RG1 G2

Nodes

Branches


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1.2.1 Basic Rules

X1 G1

Note: Input to Nodes always added

XG

GHX2 1

1

-H

X2

X1

Moving Feedback Links

H

G1X2 G2

X3 X1 G1X2 G2

X3

X1 G1X2 G2

X3

Thus

G1H

G1G2H

CX1RG1 G2

Multiplication

CX1G1G2

X1 G1

G2

Addition

X G X G X G X4 1 1 2 2 3 3 G3

X3

X2

X4

X1 G1

FeedBack

XG

GHX2 1

1

H

X2

XG G

G G HX3

1 2

1 2

11


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Masons Rules

Example

1.2.2 Definitions

Loop and Loop Gain: A loop is a path which starts and ends at the same node, not passing through any nodemore than once. e.g. there are four loops in the example:

(i) -G2H1 Loop1

(i) -G4G5H2 Loop2

(i) -G1G2G3G4G5H3 Loop3

(i) - G1G2G6G5H3 Loop4

Forward Path Gain: A forward path gain goes from input to output. In the above example there are twoforward paths :

(i) G1G2G3G4G5 Path1(i) G1G2G6G5 Path2

Non-Touching Loops: Loops which have no nodes in common. In the example above there are two suchloops: Loop1 and Loop2.

Non-Touching Loop Gain: Product of loop gains from non-touching loops taken two, three,.. at a time. Inthe example above Loop1 and and Loop2.

4. Mason;s Rule:

C

R

Pk kk

Where:

Pkis the kth

forward path

(network determinant) is defined as:

1 1 2 3 1 2 1 3 2 3 1 2 3 1 2 3 4L L L L L L L L L L L L L L L L

Product of all non-touching loops taken two at a time +

Product of all non-touching loops taken three at a time +

Product of all non-touching loops taken four at a time +

kis co-factor of path k and is defined as:

k loop gains touching k forward pathth

i.e. eliminate those gains which touch kth forward path

C

-H1

R1 G1 G2 G3 G4 G5

-H2

-H3

G6


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In the example above:

P G G G G G

P G G G G

P G G G G G G G G G G G G G G G

L L L L L L

C

R

G G G G G G

G H G G H G G G H G G G G G G

k kk

1 1 2 3 4 5 1

2 1 2 6 5 2

1 2 3 4 5 1 2 6 5 1 2 5 3 4 6

1 2 3 4 1 2

1 2 5 3 4 6

2 1 4 5 2 1 2 5 3 3 4 6 2 4 5

1

1

1

1

H H1 2

Problem 1-3

For Problem 1-2, construct a Signal Flow graph and validate the answer you got.

classical control revision[1]

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