Grammians - Controllability and Observability of a Second-Order Modal Model a Second-Order Mo

Structures

Property 4.2. Closed-Form Controllability and Observability Grammians. In modal coordinates the diagonal entries of the controllability and

4.6 Controllability and Observability of a Second-Order Modal Model a Second-Order Modal Model

4.6.1 Grammians

The grammians and the balanced models are defined exclusively in the state-space representation, and they do not exist in the second-order form. This is a certain disadvantage since the second-order structural equations are popular forms of structural modeling. We will show, however, that for flexible structures one can find a second-order model which is almost balanced, and for which Hankel singular values can be approximately determined without using a state-space representation.

First, we determine the grammians for the second-order modal model.

Property 4.6. Controllability and Observability Grammians of the Second-Order Modal Model. The controllability and observability grammians of the second-order modal model are given as

(w_c) (w_o)

(4.57)

1 1

0.25 diag( ),

c T

o mT

w B

w C

= :

m m m

B C

where diag(B B_{m m}^T) and diag(C C^T_{m m}) denote the diagonal part of B B and_{m m}^T respectively,

T ,

C Cm m B is given by (2.23),_m C_m ª¬C_mq:¹ C_mvº¼, and are defined as in (2.24) and(2.25). Therefore, the ith diagonal entries of and are

C ,mq C_mv wc w_o

2 2

4 ,

mi ci

i i mi oi

i i

w b

w c

] Z

(4.58)

and b_mi is the ith row of B , and _m c_mi is the ith column of C_m.

Proof. In order to show this we introduce a state-space representation by defining the following state vector:

m .

m m

x q q ª: º « »

¬ ¼

The following state-space representation is associated with the above vector 0 1

0 , ,

2 _m ^mq ^mv

A B C C

: ª º

ª º

C .

ª º

« »

«: =:» ¬ : ¼

¬ ¼ ¬ ¼

By inspection, for this representation, the grammians are diagonally dominant, in the form

0 0

, ,

0 0

c o

w w

W W

w w

ª º ª

#« » #«

¬ ¼ ¬

º»

where w_c and w_o are the diagonally dominant matrices, w_c#diag(wci) and Introducing the last two equations to the Lyapunov equations (4.5)

we obtain (4.58).

diag( ).

wo # w_oi

Having the grammians for the second-order models, the Hankel singular values are determined approximately from (4.58) as

2 2, 1,..., . 4

mi mi

i ci oi

i i

b c

w w i n

J ^# ] Z ^(4.59)

4.6.2 Approximately Balanced Structure in Modal Coordinates Second-order modal models are not unique, since they are obtained using natural modes that are arbitrarily scaled. Hence we have a freedom to choose the scaling factor. By a proper choice of the scaling factors we introduce a model that is almost balanced, i.e., its controllability and observability grammians are approximately equal and diagonally dominant. The second-order almost-balanced model is obtained by scaling the modal displacement (q_m) as follows:

(4.60)

1 ,

ab m

q R q that is,

m a ,

q Rq _b (4.61)

and q_ab is the almost-balanced displacement.

The transformation R is obtained as follows. Denote b_mi ₂ and c_mi ₂ as the input and output gains, then

= diag ( ),_i 1,..., ,

R r i n (4.62)

and the ith entry r_i is defined as a square root of the gain ratio

2 2 mi .

i mi

r b

c (4.63)

Using (2.38) one obtains

2 2

2 ^qi ,

mi vi

c c

c 2 (4.64)

while

2 2

2 ^T and 2 .

qi qi qi vi vi vi

c c c c c c^T (4.65)

In the above equations b_mi is theith row of B_m, and are the ith columns of , , and respectively. c_qi,c c_vi, _mi

mq mv

C C C_m,

Introducing (4.60) and (4.62)–(4.65) to the modal equation (2.19) we obtain the almost-balanced second-order modal model

(4.66)

2 2 ,

ab ab ab ab

abq ab abv ab

q q q B

y C q C q

=: :

where

ab 1 m

B R B (4.67)

and

, R,

abq mq abv mv

C C R C C (4.68)

while the output matrix C_ab is defined as

1 .

ab abq abv

C ª¬C : C º¼ (4.69)

This has the following property:

2 1 2

2 2 .

ab abq abv

C C : C ²₂ (4.70)

A flexible structure in modal coordinates is described by its natural modes, I_i, Similarly the almost-balanced modal representation is a modal representation with a unique scaling, and is described by the almost-balanced modes,

1, , . i !n

Iabi, i 1, , .! n The latter ones we obtain by rescaling the natural modes n

, 1, , ,

abi ri i i

I I ! (4.71)

with the scaling factor given by (4.63), andr_i

ab R,

) ) (4.72)

where )_ab [I_ab₁I_ab₂!I_abn], and ) is a modal matrix, as in (2.12). In order to show this, note that from (2.18) one obtains q )q_m or, equivalently

(4.73)

n i mi i

¦

But, from (4.61) it follows that q_mi r q_{i abi}; thus, (4.73) is now

(4.74)

1 1

n n

i i abi abi abi

i i

¦

r qI

¦

I q whereI_abi is a balanced mode as in (4.71).

Property 4.7. Grammians of the Almost-Balanced Model. In the almost- balanced model the controllability and observability grammians are approximately equal,

2 2

4 .

mi mi

cabi oabi i

i i

b c

w w J

# # ] Z # (4.75)

Proof. From (4.58) we have

4 4

abi abi

cabi oabi

i i i i

b c

w w ².

] Z ] Z ^(4.76)

However, from (4.49) and (4.52) it follows that

2 2

, ²

mi mi

abi mi abi mi

mi mi

c b

b b c c

b c . (4.77)

Introducing the above equation to (4.76) we obtain approximately equal grammians

as in (4.75).

Define b_abi ₂ and c_abi ₂ as the input and output gains of the ith almost- balanced mode, respectively, and we find that these gains are equal.

Property 4.8. Gains of the Almost-Balanced Model. In the second-order almost-balanced model, the input and output gains are equal,

2 .

abi abi

b c ₂ (4.78)

Proof. The transformationR as in (4.62) is introduced to (4.67) and (4.68) obtaining

2 2 2 2

1 mi

abi mi mi mi mi

i mi

b b c b b c

r b ²

and

2 2 2 2 2

2 mi

abi mi i mi mi mi abi

c c r c b b c b

c ²

»»

»¼

Example 4.7. Determine the almost-balanced model of a simple structure from Example 2.2.

We obtain the transformation matrixRfrom (4.62) and (4.63) as R=diag(0.6836, 0.7671, 0.8989). Next, we find the almost-balanced input and output matrices from (4.67), (4.68), and (4.69), knowing from Example 2.2 that

hence,

: (3.1210, 2.1598, 0.7708);

diag

1 2 3

0.4798

0.7705 ,

0.8198

0.1294 0.2617 0.3825 0 0 0

0 0 0 , 0.4040 0.5653 0.2948 ,

0 0 0 0.2242 0.4534 0.6625

ab ab

abq abv

B b

C C

ª º

« »

« » « »

« »

¬ ¼ ¬ ¼

ª º ª

« » «

: « » «

« » «

¬ ¼ ¬

The output matrix C_ab is obtained by putting together C_abq:¹ and , such that the first column of is followed by the first column of , followed by the second column of

Cabv abq 1

C : C_abv

abq 1

C : , followed by the second column of C_abv, etc., i.e.,

>

1 2 3

@

0.1294 0 0.2617 0 0.3825 0

0 0.4040 0 0.5653 0 0.2948 .

0 0.2242 0 0.4534 0 0.6625

ab ab ab ab

C c

ª º

« »

¬ ¼

c c

The almost-balanced mode matrix is obtained from (4.72), 0.4040 0.5653 0.2948 0.5038 0.2516 0.5313 . 0.2242 0.4534 0.6625

ª º

« »

) « »

« »

¬ ¼

Finally, it is easy to check that the input and output gains are equal,

12 1 2

2 2 2 2

3 2 3 2

0.4798, 0.7705, 0.8198.

ab ab

b c

Also, from (4.76) we obtain w_c₁ w_o₁ 1.1821, w_c₂ w_o₂ 6.3628, and which shows that the model is almost balanced, since the exact Hankel singular values for this system are

3 3 55.8920,

c o

w w

1 1.1794,

J J₂ 6.3736, and

3 56.4212.

4.7 Three Ways to Compute Hankel

Dalam dokumen Advanced Structural Dynamics and Active Control of Structures (Halaman 108-114)