2 lec Eng

10

A ...

00

Diagonal form (The State Space)

dX t AX t Bu t dt

y t CX t Du t

0 ...	0	0				1
2 ...	0	0				1
... ... ...		...		B
					...
0 ...	n 1	0			1
0 ...	0					1
		n

C C1 C2

... Cn 1

Cn

D C0

Cascade form

It can be derived from the transfer function written as a product of simple factored terms, which could be presented by a series of cascaded blocks in a block diagram

Let m < n

W s

b0 s

m

b1s

m1

... bm

b0

m

s i

n

1

a0 sn a1sn 1

... an

s i

s j

a0

i 1

j m1

b0

m

i

i

n

1

1

s

i

s j

a0 i 1

j m1

Cascade form (The Block Diagram)

…

…

…

…

Typical Dynamic Links

The concept of a typical dynamic link

Typical dynamic links are defined by the root types of transfer

function polynomials

Roots

Real

complexconjugate

Typical dynamic links defined by the transfer function denominator

Name of

The Transfer

The Transfer

The Roots Type

Function

Dynamic Link

Function of the Link

Parameters

Stable

1

1

Im

T – the time

T

constant,

aperiodic

Ts 1

Re

sec

link

Unstable

1

1

Im

T – the time

T

aperiodic

constant,

Ts 1

Re

sec

link

1

0

Im

Re

Integrator

s

Typical dynamic links defined by the transfer function denominator

Name of

The Transfer

The Transfer

The Roots Type

Function

Dynamic Link

Function of the Link

Parameters

Stable

1

Im

1 2

1,2

T

j

T

T – the time

oscillatory

2

2

2 Ts 1

constant, sec

T

s

Re

– damping

link

coefficient

Im

T – the time

Unstable

1

1

2

1,2

T

j

T

constant, sec

oscillatory

2

2

2 Ts 1

Re

– damping

link

T

s

coefficient

1,2

Im

Conservative

1

j

1

T – the time

oscillatory

T

T 2 s2 1

Re

constant, sec

link

Typical dynamic links defined by the transfer function numerator

Name of

The Transfer

The Transfer

The Roots Type

Function

Dynamic Link

Function of the Link

Parameters

The 1st order

1

Im

T – the time

minimum-phase

Ts 1

T

constant,

differentiating

Re

sec

link

The 1

st

order

1

Im

T – the time

non-minimum-

Ts 1

T

constant,

phase

differentiating

Re

sec

Ideal

0

Im

differentiating

s

link

Re

Typical dynamic links defined by the transfer function numerator

	Name of	The Transfer		The Transfer
			The Roots Type	Function
	Dynamic Link	Function of the Link
				Parameters

The 2nd order

Im

1,2

j

1 2

T – the time

minimum-phase

2

2

2 Ts 1

T

T

constant, sec

differentiating T

s

Re

– damping

link

coefficient

The 2nd order

Im

T – the time

1,2

j

1 2

non-minimum- T

2

2

2 Ts 1

T

constant, sec

s

T

– damping

phase

Re

differentiating

coefficient

Singular

1,2 j

1

Im

T

2

s

2

1

T

T – the time

differentiating

constant, sec

link

Re

Description of a dynamic system as a set of typical dynamic elements (links)

The 1st order minimum-phase differentiating link

W s	s 1			s 1
	2s2 s 1			2s 1 s 1

Stable	Unstable
aperiodic link	aperiodic link