Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
non-convex with respect to the lumped masses. This theory is supported by the fact that both the single SQP run and the MC SQP run resulted in the same values for the di’s; with different mass distributions. Referring back to the normalized parameter sensitivities in Table 2.3 it is apparent that changing the cross-sectional diameters has a greater impact on the cost than changing the lumped mass values. Therefore, the cost value is dominated by the di solution, which seems to be a global optimum and is found by all three algorithms. Table 2.6: PT optimization: MC SQP optimizations. J Tailoring, x∗ # [µm] d1[m] d2[m] m1[kg] m2[kg] 1 100.53 0.03 0.03 1.9410 0.0 2 102.88 0.03 0.03 0.0 0.0 3 102.88 0.03 0.03 0.0 0.0 4 100.53 0.03 0.03 1.9344 0.0 5 102.88 0.03 0.03 0.0 0.0 6 102.88 0.03 0.03 0.0 0.0 7 102.88 0.03 0.03 0.0 0.0 8 102.88 0.03 0.03 0.0 0.0 9 100.53 0.03 0.03 1.9311 0.0 10 100.53 0.03 0.03 1.9347 0.0 All three optimization algorithms result in a much improved PT-tailored design. The combination of SA and SQP converges on what appears to be a globally optimal design in around ten minutes, while running SQP optimizations at random initial guesses took almost twice as long (18 minutes) and 447 function evaluations. Since the solution space is non-convex due, in part, to the symmetry in the design masses, the SA algorithm provides a good starting point for the SQP optimization and improves the likelihood of locating the best design quickly. 2.5.2 Performance Tailored Design It is clear from comparing the cost values, RMS OPD, that the PT design is a large improvement over the nominal design in terms of performance. However, since the cost metric is RMS, an average energy over all frequencies, comparing these numbers 64
does not provide information on why the PT design performs so much better. In this section, the nominal and SQP PT designs are compared more closely. First, consider the output PSDs of each design obtained from the transfer functions using Equation 2.11. The PSDs are presented in the lower plot of Figure 2-7(a), and show a significant difference in the behavior of the two systems. The backbones of the systems are quite different and all of the modes in the PT design are shifted lower in frequency. The upper plot in the figure is the normalized cumulative variance, ¯σ 2 z,c . This quantity is the running total of energy in the system and is obtained by integrating the PSD over discrete frequency bins [50]. The cumulative variance presented here is normalized by the total system energy so that the two systems can be compared on a single plot: where f0 ∈ [fmin ...fmax] and¯σ 2 z,c ¯σ 2 z,c (f0) = 2 σ2 � +f0 [Szz (f)] df (2.46) z fmin � 1.0. The normalized cumulative variance plot is especially interesting because it indicates how the energy in the system accumulates as a function of frequency. It is possible to pick out the modes that are most critical to the performance by visually correlating the largest steps in the cumulative variance curve to the peaks in the output PSDs directly below it. The cumulative variance curve of the PT design shows that the energy is distributed somewhat evenly over a few critical modes instead of concentrated in a single mode, as in the nominal design. The critical modes and the distribution of energy among them for the two designs are shown graphically in a bar chart in Figure 2-7(b) and listed in full in Table 2-7(c). A critical mode is defined as one that accounts for at least 1% of the total output energy. The nominal and PT design data is presented side-by-side in the table with the modal frequencies in the first column, followed by the percent contribution of the mode to the total energy, and then the output variance and RMS attributed to the mode. The bar chart shows that the third mode, or second bending mode, is responsible for most of the energy in the nominal design (≈ 80%), while the PT design 65
- Page 13 and 14: Nomenclature Abbreviations ACS atti
- Page 15: dk optimization search direction f
- Page 18 and 19: 1.1 Space-Based Interferometry NASA
- Page 20 and 21: unfettered by the Earth’s atmosph
- Page 22 and 23: the SCI, both the size and flexibil
- Page 24 and 25: maybethatitbecomescertain that the
- Page 26 and 27: Table 1.1: Effect of simulation res
- Page 28 and 29: Table 1.2: Effect of simulation res
- Page 30 and 31: has been found that structural desi
- Page 32 and 33: precision telescope structure for m
- Page 34 and 35: attractive, and more conservative a
- Page 36 and 37: to solve the performance tailoring
- Page 39 and 40: Chapter 2 Performance Tailoring A c
- Page 41 and 42: ometer (SCI). In the following sect
- Page 43 and 44: The equations of motion of the unda
- Page 45 and 46: The frequency response functions fr
- Page 47 and 48: the output covariance matrix, Σz,
- Page 49 and 50: where the subscript indicates the i
- Page 51 and 52: 2.3.3 Design Variables The choice o
- Page 53 and 54: and then, by inspection, the inerti
- Page 55 and 56: algorithms begin at an initial gues
- Page 57 and 58: at least locally optimal, and the s
- Page 59 and 60: initial design variable state, x =
- Page 61 and 62: and the RMS OPD is computed using E
- Page 63: # Designs 25 20 15 10 5 Accepted, b
- Page 67 and 68: energy is distributed almost evenly
- Page 69 and 70: also symmetric as seen in the figur
- Page 71 and 72: Chapter 3 Robust Performance Tailor
- Page 73 and 74: through careful and experienced mod
- Page 75 and 76: described above. However, one can r
- Page 77 and 78: ic, σz(�x, �p), that is depend
- Page 79 and 80: Magnitude, OPD/F x [µm/N] Magnitud
- Page 81 and 82: % Energy 100 90 80 70 60 50 40 30 2
- Page 83 and 84: metric to the cost function. Note,
- Page 85 and 86: tion: ∂hi (z,�x, �pi) ∂�x
- Page 87 and 88: values are chosen from their statis
- Page 89 and 90: Table 3.3: Algorithm performance: a
- Page 91 and 92: Statistical Robustness The statisti
- Page 93 and 94: Performance [µm] 1400 1200 1000 80
- Page 95 and 96: (Figure 3-6(b)). The nominal perfor
- Page 97 and 98: Norm. Cum. Var. [µm 2 ] PSD [µm 2
- Page 99 and 100: energy by mode for easy comparison.
- Page 101 and 102: Y−coordinate [m] Y−coordinate [
- Page 103 and 104: RMS performance, [µm] 400 350 300
- Page 105 and 106: The requirement chosen here is some
- Page 107 and 108: Chapter 4 Dynamic Tuning Robust Per
- Page 109 and 110: on a physical truss. Since tailorin
- Page 111 and 112: Table 4.1: Tuning parameters for SC
- Page 113 and 114: m 2 [kg] J ∗ # # time y ∗ [kg]
does not provide information on why the PT design performs so much better. In this<br />
section, the nominal and SQP PT designs are compared more closely.<br />
First, consider the output PSDs of each design obtained from the transfer functions<br />
using Equation 2.11. The PSDs are presented in the lower plot of Figure 2-7(a), and<br />
show a significant difference in the behavior of the two systems. The backbones of<br />
the systems are quite different and all of the modes in the PT design are shifted lower<br />
in frequency. The upper plot in the figure is the normalized cumulative variance,<br />
¯σ 2 z,c . This quantity is the running total of energy in the system and is obtained<br />
by integrating the PSD over discrete frequency bins [50]. The cumulative variance<br />
presented here is normalized by the total system energy so that the two systems can<br />
be compared on a single plot:<br />
where f0 ∈ [fmin ...fmax] and¯σ 2 z,c<br />
¯σ 2 z,c (f0) = 2<br />
σ2 � +f0<br />
[Szz (f)] df (2.46)<br />
z fmin<br />
� 1.0. The normalized cumulative variance plot is<br />
especially interesting because it indicates how the energy in the system accumulates<br />
as a function of frequency. It is possible to pick out the modes that are most critical<br />
to the performance by visually correlating the largest steps in the cumulative variance<br />
curve to the peaks in the output PSDs directly below it. The cumulative variance<br />
curve of the PT design shows that the energy is distributed somewhat evenly over a<br />
few critical modes instead of concentrated in a single mode, as in the nominal design.<br />
The critical modes and the distribution of energy among them for the two designs<br />
are shown graphically in a bar chart in Figure 2-7(b) and listed in full in Table 2-7(c).<br />
A critical mode is defined as one that accounts for at least 1% of the total output<br />
energy. The nominal and PT design data is presented side-by-side in the table <strong>with</strong><br />
the modal frequencies in the first column, followed by the percent contribution of<br />
the mode to the total energy, and then the output variance and RMS attributed<br />
to the mode. The bar chart shows that the third mode, or second bending mode, is<br />
responsible for most of the energy in the nominal design (≈ 80%), while the PT design<br />
65
Change language