Canonical Transformation

25
President University Erwin Sitompul SMI 5/1 Dr.-Ing. Erwin Sitompul President University Lecture 5 System Modeling and Identification http://zitompul.wordpress.com

description

Chapter 3. State Space Process Models. Canonical Transformation. Chapter 3. State Space Process Models. Canonical Transformation. As the result,. Chapter 3. State Space Process Models. Order Reduction. - PowerPoint PPT Presentation

Transcript of Canonical Transformation

Page 1: Canonical Transformation

President University Erwin Sitompul SMI 5/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 5System Modeling and Identification

http://zitompul.wordpress.com

Page 2: Canonical Transformation

President University Erwin Sitompul SMI 5/2

Canonical Transformation0 19 30 1

( ) 1 0 0 ( ) 0 ( ),0 1 0 0

t t u t

x x

( ) 0 2 1 ( ).y t t x

1 5 2 3 3 2

Chapter 3 State Space Process Models

0 19 301 0 00 1 0

A

1

25,5

1

e

( ) ii 0A I e19 30

1 00 1

i

ii

i

0e 2

9,3

1

e 3

4.2

1

e

Page 3: Canonical Transformation

President University Erwin Sitompul SMI 5/3

Chapter 3 State Space Process Models

Canonical Transformation 1 2 3T e e e

25 9 45 3 2

1 1 1

1

0.0179 0.0893 0.10710.1250 0.3750 1.250.1429 0.2857 2.1429

T

1

5 0 00 3 00 0 2

A T AT

1

0.01790.12500.1429

B T B

9 7 5 C CT

As the result, 5 0 0 0.0179

( ) ( ) ( )0 3 0 0.12500 0 2 0.1429

t t u t

x x

( ) 9 7 5 ( )y t t x

Page 4: Canonical Transformation

President University Erwin Sitompul SMI 5/4

A mathematical model which is constructed using physical or chemical laws may have a high order due to the number of interacting components inside the model.

In many cases, a model with lower order is wanted because it is easier to handle.

Thus, it is wished that the order of a model can be made low. But, it is also wished that the model still can be interpreted physically.

The method in doing so is called “Order Reduction”.

Chapter 3 State Space Process Models

Order Reduction

Page 5: Canonical Transformation

President University Erwin Sitompul SMI 5/5

Chapter 3 State Space Process Models

Order ReductionStarting point:

A state space of order n x Ax Buy C x

We define:xs : part of x that contains states

considered to be significant, such as measured variables or other critical variables.

xr : the remaining part of x which are not the member of xs.After the definition process, the states are to be

reconstructed as:s

r

xx x

Page 6: Canonical Transformation

President University Erwin Sitompul SMI 5/6

Chapter 3 State Space Process Models

Order ReductionThe reconstruction of state vector yields an implication

on matrices A, B, and C. If the ith state is switched with the jth state,

xi xj

ith column of C jth column of C

ith row of B jth row of B

ith column & ith row of A

jth column & jth

row A

Page 7: Canonical Transformation

President University Erwin Sitompul SMI 5/7

Chapter 3 State Space Process Models

Order ReductionResult:

A new state space of order p, as an approximation of the original state space of order n

ss x Ax Bu

sy C x

Question:How to find the ideal order p of the new state space?

• What is p equal to?

Page 8: Canonical Transformation

President University Erwin Sitompul SMI 5/8

Chapter 3 State Space Process Models

Order Reduction

x T z

The order reduction is performed using the Canonical Transformation:

• Modal Coordinate• Matrix of Eigenvectors of A

x Ax Bu

T z AT z Bu1 1 z T AT z T Bu

Λ *B

y C x

y CT z

*C[nn] [nm] [rn]

Page 9: Canonical Transformation

President University Erwin Sitompul SMI 5/9

Chapter 3 State Space Process Models

Order Reduction

* z Λz B u*y C z

As the results of the Canonical Transformation, in time and frequency domain we can write:

*1( ) ( ) ( )s s s Z I Λ B U*( ) ( )s sY C Z

* *1( ) ( ) ( )s s s Y C I Λ B U

1

11

1 2

1

0 0

0 0( )

0 0

s

s

s n

s

I Λ

1

2

0 00 0

0 0 n

Λ

Page 10: Canonical Transformation

President University Erwin Sitompul SMI 5/10

Chapter 3 State Space Process Models

Order ReductionExamining the output equations in frequency domain,

the relationship between the jth input Uj(s) and the ith output Yi(s) can be formulated as:

* *1( ) ( ) ( )s s s Y C I Λ B U

1

11

2

1

0 0

0 0

0 0

s

s

s n

* *

1

1( ) ( )

n

ik kji jk k

c bY s U ss

*1

1* *1

*

( ) ( ) ( ) j

i ji in

nj

bY s s U sc c

b

I Λ

Page 11: Canonical Transformation

President University Erwin Sitompul SMI 5/11

Chapter 3 State Space Process Models

The Procedure of Order ReductionStep 1:

The finding of dominant eigenvalues

The step response of the state space model is given as:* *

1

1 1( )n

ik kjik k

c bY ss s

* *

1

1( ) 1k

nt

i ik kjk k

y t c b e

• approaches –1 for stable λ,

can be normed• measures the influence of λk in the

connection between uj(t) and yi(t)

Page 12: Canonical Transformation

President University Erwin Sitompul SMI 5/12

Chapter 3 State Space Process Models

The Procedure of Order ReductionWe now define a Dominance Measure,

* *ik kj

ikjk

c bD

The value of Dikj will be small for large λk (a λ with large value, away from imaginary axis, is not a dominant λ).

The value of Dikj will be zero for cik* = 0

(the kth proper motion of the ith output is not observable).

The value of Dikj will be zero for bkj* = 0

(the kth proper motion of the jth input is not controllable).

Page 13: Canonical Transformation

President University Erwin Sitompul SMI 5/13

Chapter 3 State Space Process Models

The Procedure of Order ReductionThe Dominance Measures of each eigenvalue are

further analyzed to yield two measures, the maximum and the sum:

1 1100 max max

r m

k ikji jM D

1 1

100r m

k ikji j

S D

Maximum of Dominance Measure

Sum of Dominance Measure

Page 14: Canonical Transformation

President University Erwin Sitompul SMI 5/14

Chapter 3 State Space Process Models

The Procedure of Order Reduction The dominant eigenvalues are chosen according to the

following criteria:1. All unstable eigenvalues (λ>0) are dominant.2. The stable eigenvalues with the largest Mk are

dominant.Generally, for normalized system with gain equals to 1,

if Mk>20, then λk is dominantif Mk <2, then λk is not dominantif 2<Mk<20, consider Sk

Result of Step 1:Out of n eigenvalues, p dominant eigenvalues are chosen: λ1, λ2, …, λp

Page 15: Canonical Transformation

President University Erwin Sitompul SMI 5/15

Chapter 3 State Space Process Models

The Procedure of Order ReductionStep 2:

The calculation of A, B, and C~ ~ ~

The state vector z is now reorganized as follows:1

1

p

p

n

z

zz

z

z d

n

zz

Where:zd : states of z with dominant

eigenvalues.zn : the remaining states of z, which are the

states with not dominant eigenvalues.

Page 16: Canonical Transformation

President University Erwin Sitompul SMI 5/16

Chapter 3 State Space Process Models

The Procedure of Order Reduction

* z Λz B u*

d 1d d*

nn n 2

z Λ z Buz Λ z B

00 1

d

p

Λ 0

0

1

n

p

n

Λ 0

0

The canonical form of the state space, including the dominance consideration, is:

d* *1 2

n

zy C C z

*y C z

• What is B1*, B2

* ?

• What is C1*, C2

*?

Page 17: Canonical Transformation

President University Erwin Sitompul SMI 5/17

Chapter 3 State Space Process Models

The Procedure of Order Reduction

x T z

The transformation matrix T is also reorganized, considering dominant states (eigenvalues):

11 12s d

21 22r n

x T T zx T T z

• Chosen freely according to a certain technical criteria defined by model maker

• Chosen according to a mathematical criteria the Dominance Measures

The matrix multiplication yields:11 12s d n x T z T z 11s d x T z

• Omitted, because not significant/dominant

111 sdz T x

Page 18: Canonical Transformation

President University Erwin Sitompul SMI 5/18

Chapter 3 State Space Process Models

The Procedure of Order Reduction*

d 1d d*

nn n 2

z Λ z Buz Λ z B

00

We go back now to the state equations:

*d 1d d z Λ z B u

1 *11 d 11 11 1s s

x T Λ T x T B u

A B

111 sdz T x

[pp] [pm]

Page 19: Canonical Transformation

President University Erwin Sitompul SMI 5/19

Chapter 3 State Space Process Models

The Procedure of Order Reduction

C

* 1111 sy C T x

Also, we go back to the output equations:d* *

1 2n

zy C C z

* *1 2d n y C z C z

• Omitted, because not significant/dominant

111 sdz T x

[rp]

Page 20: Canonical Transformation

President University Erwin Sitompul SMI 5/20

Chapter 3 State Space Process Models

The Procedure of Order ReductionResult of Step 2:

Out of a state space of order n, we will have a state space of order p:

s s x Ax Bu

sy C x

Page 21: Canonical Transformation

President University Erwin Sitompul SMI 5/21

Recollecting the last example, the original state space is given as:

Chapter 3 State Space Process Models

Example: Order Reduction

1 4 1 4 13 4( ) 0 3 0 ( ) 1 ( )

0 0 2 1t t u t

x x

( ) 1 0 0 ( )y t t x

After canonical transformation:11 0 0

( ) ( ) 1 ( )0 3 00.250 0 2

t t u t

x x

( ) 1 2 1 ( )y t t x

Simplify the 3rd order state space into a 2nd order state space using the Order Reduction, if the significant states are xs =[x1 x2]T.

Page 22: Canonical Transformation

President University Erwin Sitompul SMI 5/22

Chapter 3 State Space Process Models

Example: Order Reduction

1

1 2 0.250 1 00 0 0.25

T

1 2 10 1 00 0 4

T

1

1 0 00 3 00 0 2

Λ T AT

* 1

11

0.25

B T B

* 1 2 1 C CT

n=3 k=[1…n]m=1 j=[1…m]r=1 i=[1…r]

* *11 11

1111

c bD

* *12 21

1212

c bD

(1)(1)( 1)

1

(2)(1)( 3)

0.667

* *13 31

1313

c bD

(1)(0.25)( 2)

0.125

1 1100 max max

r m

k ikji jM D

1 111100M D 100

2 121100M D 66.7

3 131100M D 12.5

• Dominant• Dominant

Page 23: Canonical Transformation

President University Erwin Sitompul SMI 5/23

Chapter 3 State Space Process Models

Example: Order Reduction

1 1

2 2

3 3

1 2 10 1 00 0 4

x zx zx z

11T 111 d 11

A T Λ T

1 2 1 0 1 20 1 0 3 0 1

1 40 3

*11 1B T B

1 2 10 1 1 31

* 1111C C T

1 21 2

0 1

1 0

s s

1 4 30 3 1

x x u

s1 0y x

1 0 00 3 00 0 2

Λ

*

11

0.25

B

* 1 2 1C

*1B

*1C

Page 24: Canonical Transformation

President University Erwin Sitompul SMI 5/24

Chapter 3 State Space Process Models

Order Reduction Homework 5

A linear time-invariant system is given as below:3 1 0 1

( ) 1 3 0 ( ) 0 ( )3 5 6 4.5

t t u t

x x

0 1 0( ) ( )

0 0 1t t

y x

a) Calculate the eigenvalues and the eigenvectors of the system.b) A second order model is now wished to approximate the system.

The second and the third state are chosen to be the significant states. Perform the Order Reduction based on the chosen significant states. Regarding the Dominance Measure, which eigenvalues of the original model should be considered in the new reduced-order model?

c) Write the complete reduced-order model in state space form. Hint: This model must be a second order model.

Page 25: Canonical Transformation

President University Erwin Sitompul SMI 5/25

Chapter 3 State Space Process Models

Homework 5: Order ReductionRedo b) and c) of the previous homework problem if, for this time, only the first state is considered to be the significant state. This is equivalent to replacing the output equation with:

NEW

( ) 1 0 0 ( )t ty x

3 1 0 1( ) 1 3 0 ( ) 0 ( )

3 5 6 4,5t t u t

x x