Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

the output covariance matrix, Σz, is found by integrating Szz over frequency:
Sigmaz =
� ∞
−∞
Szz (ω) dω (2.12)
In general, Σz, is a matrix, and the diagonal elements are the performance variances,
σ2 , corresponding to the system outputs, zi.
zi
Alternatively, if the disturbance input is white noise then the output variance
is obtained more directly using the Lyapunov equation to obtain the modal state
covariance matrix, Σq:
AΣq +ΣqA T + BB T = 0 (2.13)
The performance variance for the i th output is obtained by pre and post-multiplying
the state covariance matrix by the appropriate rows of the state-space output matrix
C:
σ 2 zi = CiΣqC T i
(2.14)
The disturbance input for the SCI model is white noise, therefore Equation 2.14 is
used in the implementation to compute performance variance. However, a Lyapunov
analysis can only be conducted on a stable system. Recall from Figure 2-2 and
Table 2.2 that this model contains three rigid body modes with zero frequency. These
modes must be removed or stabilized in order to perform the necessary jitter analysis.
The translational rigid body modes are not observable in the output and, therefore,
can be removed from the model without consequences. The rotational rigid body
mode, on the other hand, does result in positive OPD. In practice, such motion
is controlled with an attitude control system (ACS). The ACS is modeled with a
rotational spring with stiffness, kACS =10, 000 Nm
rad
at the rotational node of the
combiner. The value of kACS is chosen to produce an ACS mode that has a frequency
well below the flexible modes. The effect of the ACS model on the transfer functions
from Tz to OPD is shown in Figure 2-3. Notice that the transfer function with ACS
does not have a negative slope at zero frequency. The addition of the ACS model does
affect the first few asymmetric bending modes, but only slightly. The new frequency
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