Frequency Limited Grammians

In this section we interpret the controllability and observability grammians in frequency domain. In order to do this, note that from the Parseval theorem the time integrals (4.3), for the time span (0, <»), can be substituted with the following integrals in the frequency domain:

where

is the Fourier transform of eAt, and H* is a complex-conjugate transpose of H.

The above grammians are defined over the entire frequency range. The frequency band (-<», <») can be narrowed to (-», a>) where »<<» by defining the grammians in the latter band as

We show the following property:

Property 4.10(a). Grammians in Frequency Interval (0, to) are obtained from the following equations:

S(ffl) = jln ((H-1)*H) = jln ((A + jml)(A - jrnl)">) (4.101) and Wc is the controllability grammian obtained from the Lyapunov equation (4.5). Proof. Note first that

This can be shown by replacing A with Has in (4.98). Next, introduce BBt = -AWc -WcAtto (4.99), obtaining

Wc (ffl) =— I H(v)BB H (v) dv =--I H(v)(AWc + WcAT)H (v) dv.

Introducing (4.102) to the above equation we obtain

= Wc — r H>)dv + — _T H(V)dvWc = WcS>) + S(ffl)Wc, J-® J-®

2^ J-® 2^ J-® since S(®) in (4.101) is obtained as

The observability grammian is determined similarly. 0

Define the frequency band Q = [®j, ®2], such that <» > m2 >a>1 >0. It is easy to see that the grammians for the band Q are obtained as

W0 (Q) = Wo (®2) - Wo (©!). For this band the following property holds:

Property 4.11(a). Grammians in Frequency Interval (ah, (o2) are obtained from the following equations:

Wc (Q) = WcS *(fi) + S (Q)Wc, Wo (Q) = S *(n) Wo + WoS (Q), where

S (Q) = S (ffl2) - S (fflj). Proof. Introduce (4.100) to (4.104) to obtain (4.105). Next, we show the following property:

Property 4.12. Matrices A and S(g>) Commute.

Proof. Note first that

AH (®) = H (®) A, which we prove through the simple manipulations

AH(®) = A( jrnl - A)-' = [(jrnl - A)A"1]"1 = [(j^A^ -1XT1 = [ A'1 (jrnl - A)]"1 = (jrnl - Af1 A = H (a) A.

Equation (4.107) follows directly from (4.108) and the definition (4.103) of S(m). 0

Using the above property we derive an alternative formulation for the frequency-limited grammians.

Property 4.10(b). Grammians in Frequency Interval (0, ©) are determined from the following equations:

where

AWc (m) + Wc (a) At + Qc (m) = 0, ATWo (m) + Wo (m) A + Qo (m) = 0,

Qc (®) = 5 (®) BBT + BBTS * (®), Qo (») = S *(a)CTC + CTCS (®).

Proof. Use (4.100) and apply the commutative Property 4.12 to obtain

AWc (m) + Wc (m) At = AWcS * + ASWc + WcAT S" + SWcAT, S(WcAt + AWc) + (WcAt + AWc)S* = -SBBt -BBTS* = -Qc(®), i.e., the first of (4.109) is satisfied. The second (4.109) is proved similarly. 0

Next we determine the grammians over the interval Q = [®j, co2 ].

Property 4.11(b). Grammians in Frequency Interval Q=(©1, co2) are obtained from the following equations:

where

Example 4.10. Analyze a simple system as in Example 4.9, and obtain the frequency-limited grammians for Q = [0,®], where m is varying from 0 to 20 rad/s.

We obtain the grammians in modal coordinates from (4.111), and their plots for the three modes are shown in Fig. 4.8. The plots show that for m>mi, i=1,2,3, (mi is the ith natural frequency, a>1 = 2.55rad/s, m2 = 7.74 rad/s, and m3 = 12.38 rad/s), the grammians achieve constant value.

-

?-c

?—<

*

f

mode 3

8 10 12 frequency, rad/s

8 10 12 frequency, rad/s

Figure 4.8. Hankel singular values versus frequency for the system modes.

0 0

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