2 lec Eng

Block diagram transformations

The Transfer function does not contain any information about the structure

u(t)

Input signal

			dX t		AX t Bu t
System	y(t)		dt
	Output signal		y t CX t Du t
…

X(t) – the State vector

~	t T X t – another State vector for this system
X

where T is the transformation matrix from one vector to

another			~		~ ~
		dX t				~
choosing a new State vector					AX	t Bu t
			dt
we get an equivalent State					~ ~	~
space representation		y t CX t Du t
	~					~
	A TAT 1					B TB
	~		CT 1			~
	C					D D

Conclusion

For the same system represented by the

transfer function, we can construct a

variety of different equivalent State space

representations as well as Block diagrams

Obtaining methods of Block Diagrams and State Equations from Transfer functions

•The form with the feedbacks from the output

•Canonical form (with the feedbacks from the state variables)

•Diagonal form (Parallel realization)

•Cascade form (Series realization)

The form with the feedbacks from the output (The Block Diagram)

It can be derived directly from the expanded polynomial form of

the transfer function W s b0 sm b1sm1 ... bm1s bm

sn a1sn 1 ... an 1s an

Let m = n

…

The form with the feedbacks from the output (The State Space)

dX t AX t Bu t

dt

y t CX t Du t

	a1
		a2
		a3
	A
	...
	a		n 1
		a
				n

	1	0	0	0	...	0
	0	1	0	0	...	0
	0	0	1	0	...	0
	... ... ...			...	...	...
	0	0	0	0	...	1
	0	0	0	0	0	0

	b1	a1b0
	b	a	b
	2		2	0
	b3	a3b0
B
		...
b		a		b
	n 1	a	n	1 0
	b		b
	n		n	0
C 1		0		0	0 ... 0

D b0

Canonical form (The Block Diagram)

It can be derived directly from the expanded polynomial form of

the transfer function			b0 sm b1sm1 ... bm1s bm
	W s		sn a sn 1	... a		s a
					n 1		n
		1

Let m = n

…

000

A

...

0an

Canonical form (The State Space)

						dX t		AX t Bu t
							dt
						y t CX t Du t
														0
1		0			0			...	0		0
													0
0		1			0			...	0		0
														0
0		0			1			...	0		0		B
...		...			... ... ...						...		...
														0
0		0			0			...	0		1
a		a			a				a		a			1
	n 1		n	2			n 3	...		2
											1

C bn b0an

bn 1 b0an 1

... b1 b0a1

D b0

Diagonal form

It can be derived from the transfer function written as a sum of partial fraction expansion terms, which would appear as parallel blocks on a block diagram

Let m = n

n

W s

b0 s

n

b1s

n 1

... bn

b0

s i

n

Ci

i 1

C0

a

sn

a sn 1

... a

n

s

a

i 1

0

1

n

0

s i

i

i 1

where i , i – the polynomial roots of the numerator and denominator respectively

Diagonal form (The Block Diagram)

…