Structures
Property 4.2. Closed-Form Controllability and Observability Grammians. In modal coordinates the diagonal entries of the controllability and
4.9 Time-Limited Grammians
Table 4.1.Hankel singular values.
Continuous time Discrete time 't= 0.7 s
Discrete time 't= 0.02 s
20.342 20.138 20.342
Mode 1
20.340 20.009 20.340
4.677 4.324 4.677
Mode 2
4.671 4.225 4.670
0.991 0.848 0.991
Mode 3
0.986 0.785 0.986
The table shows that for the sampling time ' t 0.7s the discrete-time Hankel singular values are smaller than the continuous-time values, especially for the third mode. In order to explain this, note that the natural frequencies are
1 0.771 rad/s,
Z Z2 2.160 rad/s, and Z3 3.121 rad/s. The sampling time must satisfy condition (3.52) for each mode. For the first mode S Z1 4.075, for the second mode S Z2 1.454, and for the third mode S Z3 1.007. The sampling time satisfies the condition (3.52). However, for this sampling time, one obtains
1 t 0.540,
Z' Z2' t 1.512, and Z3' t 2.185. It is shown in Fig. 4.5 that the discrete-time reduction of the Hankel singular values with respect to continuous- time Hankel singular values is significant, especially for the third mode.
This is changed for the sampling time ' t 0.02s. In this case one obtains
1 t 0.015,
Z' Z2' t 0.043, and Z3' t 0.062. One can see from Fig. 4.5 that for these values ofZi't the discrete-time Hankel singular values are almost equal to the continuous-time Hankel singular values.
Next, we verify the accuracy of the approximate relationship (4.83) between discrete- and continuous-time Hankel singular values. The accuracy is expressed with the coefficient , (4.84). The Hankel singular values were computed for different sampling times, and compared with the continuous-time Hankel singular values. Their ratio determines the coefficient . The plot of obtained for all three modes and the plot of the approximate coefficient from (4.84) are shown in Fig. 4.6.
The plot shows that the approximate curve and the actual curves are close, except for ki
ki ki
i t
Z' very close to S.
conveniently solved, and the properties of their solutions are not readily visible.
However, using the definitions from (4.3) we will derive the closed-form grammians over the finite interval T. Assume that a system is excited and its response measured within the time interval The grammians over this interval are defined as follows:
1 2
[ , ].
T t t
0 1 2 3
k1
k2
k3
approximate 1
0.8
ki 0.6
0.4 0.2
0
i t
Z'
Figure 4.6.The exact and approximate coefficients (k) coincide except for Zi't very close toS, which corresponds to the Nyquist frequency.
2 1
2 1
( ) exp( ) exp( ) ,
( ) exp( ) exp( ) .
t T T
c t
t T T
o t
W T A BB A d
W T A C C A d
W W W
W W W
³
³
(4.85)
For a stable matrix A these grammians are positive-definite.
First, we express the controllability and observability grammians over the interval (0, t) through the infinite-time controllability grammianWc.
Property 4.9(a). Grammians in Time Interval (0, t). The controllability grammian over the interval (0, t), and the observability grammian
over the interval (0, t), are obtained from the infinite-time controllability grammians and as follows:
(0, )
Wc t Wo(0, )t
Wc Wo
(4.86)
(0, ) ( ) ( ),
(0, ) ( ) ( ),
c c c T
o o T o
W t W S t W S t W t W S t W S t
where
( ) At.
S t e (4.87)
Proof. The controllability grammianWc Wc(0, ),f
0
A T AT
Wc
³
fe BB eW WdW can be decomposed for t f as follows:0t A T AT A T AT .
c t
W
³
e BB eW WdW³
fe BB eW WdW In the second integral, the variableW is replaced withT W t, yielding( ) ( )
0
0 ;
T T
T T
A T A A t T A t
t
At A T A A t At A tT
c
e BB e d e BB e d
e e BB e d e e W e
W W T T
T T
W T
T
f f
f
³ ³
³
therefore, combining the two latter equations one obtains (0, ) At A tT ,
c c c
W W t e W e
which, in turn, gives (4.86). The observability grammian is derived similarly. Note that equations (4.86) are the solutions of the Lyapunov differential equations (4.4). Indeed, from the first equation of (4.86) it follows that
. Note also that
(0, ) T
c c
W t SW S SW ST
c S dtd (eAt) AeAt AS;
T
T
T,
thus, Introducing the latter result and the first equation of (4.86) to (4.4) we obtain
(0, ) T T .
c c c
W t ASW S SW S A
T T T T T T T
c c c c c c
ASW S SW S A AW ASW S W A SW S A BB
or, after simplification,
0 AWcTW Ac T BB
which is, of course, a steady-state Lyapunov equation, fulfilled for stable systems.
Similarly, we can show that the observability grammian from (4.86) satisfies the second equation of (4.4).
Denote the time interval T [ ,t to f] where tf !to.
Property 4.9(b). Grammians in Time Interval (to, tf). For the following holds:
o 0 t !
(4.88)
( ) ( ) ( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) ( ) ( ) ( ),
T T
c o c o f c f c o c
T T
o o o o f o f o o o
W T S t W S t S t W S t W t W t W T S t W S t S t W S t W t W t
f
f
where
(4.89) ( ) ( ) ( ),
( ) ( ) ( ).
c c T
o T o
W t S t W S t W t S t W S t Proof. To prove the first part we begin with the definition
( ) f T ,
o
t A T A
c t
W T
³
e BB eW WdW and the introduction of the new variable T W to, obtaining( ) ( )
0
0
( )
( )
f o T
o o
T T
f o T
o o o
t t A t T A t
c
t t
,
At A T A A t At o
c f o
W T e BB e d
e e BB e d e e W t t e
T T
T T
T T
³
³
A tT
which proves (4.88). The second part we prove by introducing (4.86) into (4.88),
( ) ( )
( ) Ato( A tf to A tT f to ) A tTf Ato A tTo Atf A tf.
c c c c c
W T e W e W e e e W e e W e
The time-limited grammians in modal coordinates have a simpler form, since the controllability grammian in modal coordinates is diagonally dominant and the matrixA is block-diagonal. The grammian block that corresponds to the ith mode has the form w t t Ici o( , )f 2, and the matrixA block is
i i i .
i i i
] Z Z Z ] Z
ª º
« »
¬ ¼
Introducing it into (4.88) we obtain
(4.90)
2 ( )
( ) 2 i i ot (1 i i tf to ).
ci ci
w T #w e] Z e] Z For the most practical case of to 0 we find
(4.91) (0, ) (1 2 i it).
ci ci
w t #w e] Z
The latter equations show that for a stable system the modal grammians of limited time are positive definite, and that they grow exponentially, with the time constant
i 1/ 2 i
T ] Zi.
The observability grammian in modal coordinates for structures is diagonally dominant, and the matrixA is block-diagonal. The grammian block that corresponds to the ith mode has the form . Similar to the controllability grammian we get
( , ) 2 oi o f
w t t I
(4.92)
2 ( )
( ) 2 i i ot (1 i i tf to ).
oi oi
w T #w e] Z e] Z For the most practical case of to 0 we obtain
(4.93) (0, ) (1 2 i it).
oi oi
w t #w e] Z
The latter equations show that for a stable system the modal grammians of limited time are positive definite, and that they grow exponentially, with the time constant
i 1/ 2 i
T ] Zi.
Define the Hankel singular values over the interval T ( , )t to f as follows:
( ) 1/ 2 ( ) ( ) .
i T i W T W Tc o
J O (4.94)
Introducing (4.90) and (4.92) to (4.94) we obtain
(4.95)
2 ( )
( ) 2 i i ot (1 i i tf to )
i T ie ] Z e ] Z
J #J
or
(4.96) (0, ) (1 2 i it).
i t i e ] Z
J #J
Example 4.9. Analyze a simple system with k1=10, k2=50, k3=50, k4=10, m1=m2=m3=1, with proportional damping matrix, D=0.005K+0.1M. The input is applied to the third mass and the output is the velocity of this mass. Calculate the time limited Hankel singular values for T [0, ]t , and for t is varying from 0 to 25 s using the exact equations (4.88), (4.94), and the approximate equation (4.96).
The plots of the Hankel singular values for the system three modes are shown in Fig. 4.7. The plots show close approximation for the first two modes, and not-so- close for the third mode and for a short time span (t1 s).
00
5 10 15 20 25
mode 2
mode3 mode 1 1.4
1.2 1
Hankel singular value
0.8 0.6 0.4 0.2
time, s
Figure 4.7. Hankel singular values versus time for the system modes: Exact (solid line) and approximate (dashed line).