HINFSTRUCT benchmarking

I. T HE TEST BENCH We used the COMPle ib benchmark library [2] to compare the performance of • HINFSTRUCT in the 2010b version of Robust Control Toolbox [3] • HIFOO version 2.0 with HANSO 1.01 [1] The COMPle ib benchmark library is ideally suited for performing our tests as it was already used by various authors, and is easily accessible. The acronyms AC1 – TL and so on in the left column of Tables I – VI allow to identify the model in [2]. Repetition of the same acronym means that the same open-loop model has been used to synthesize controllers of different orders. Column 2 of the tables labeled P/K/dimx gives the plant order, the controller order and the number of decision variables. Columns 3 and 4 give the cpu times that were achieved on a PC with Windows XP 32, Intel Core 2, 2Ghz and 4Gb memory for HIFOO and HINFSTRUCT, respectively. Columns 5 and 6 display the lowest value kTw→z (K(x)) k∞ achieved by HIFOO and HINFSTRUCT, respectively. Note no comparison is attempted to the value of the global minimum. The global minimum is not even known, except for a few of the toy test cases. Two types of tests have been performed which we refer to as TEST I and TEST II. They correspond to the two major modes of operation of HIFOO. In TEST I, HIFOO performs a gradient sampling phase to refine solutions while it is disabled in TEST II. We refer the readers to [1] for more details. Also to enhance HIFOO running times, we use H∞ computation from the SLICOT library. Our routine HINFSTRUCT has only one mode of operation and needs not to be set. For a fair evaluation, we use 3 restarts for each run which is the implemented default in HIFOO. For HINFSTRUCT, this amounts to adding 2 restarts to the default by passing an option structure: opts=hinfstructOptions(’RandomStart’,2).

Results for TEST I are displayed in tables I-III while those of TEST II are given in tables IV-VI Note for TEST II HIFOO simplifies to a more conventional BFGS minimization technique which disregards objective nonsmoothness. By omitting the final gradient sampling phase better running times are expected at the risk of worse objective values. Referring to the tables, an ’NaN’ in the cpu time column associated with an ’Inf’ in the H∞ -norm column indicates failure to find a stabilizing controller. II. R ESULTS AND COMMENTS Statistics in terms of execution times and achieved objective values for both techniques are summarized in figures 1

to 4. Note these plots do not account for failure occurrences (HIFOO failed in 7 runs while HINFSTRUCT failed 3 times). The left plot in figure 1 shows the horizontal bar diagram of H∞ -norm ratios: log10 (H∞ -norm HIFOO/H∞ -norm HINFSTRUCT) . The right diagram is a magnification of the central region for H∞ -norm ratios. The right diagram of figure 2 displays cpu time ratios: log10 (cpu time HIFOO/cpu time HINFSTRUCT) . The right diagram in figure 2 shows cpu time ratios for test cases with equal H∞ -norm objectives within 3% deviation. Figures 3 and 4 gives similar plots for TEST II, that is, when HIFOO does not run the gradient sampling phase. Note a right-half plane bar indicates advantage of HINFSTRUCT over HIFOO and conversely for left-half plane bars. A bar of unit length materializes improvement by a factor 10, a bar of length 2 a factor of 100, etc. We note that differences are more pronounced for problems with larger number of decision variables. As demonstrated in TEST II, short-circuiting the gradient sampling phase for HIFOO results in better execution times with some deterioration of the objective. Finally, we have noticed that running the test sets TEST I and TEST II several times does not alter much the stats which justifies our evaluation on a large test set. III. C ONCLUSION We have compared two nonsmooth algorithms for H∞ synthesis of fixed-order controllers. Both techniques are run in default mode as a non-expert user would do. This means we did not try to tweak algorithm constants and tolerances to improve speed and/or accuracy. Extensive numerical testing demonstrates that HINFSTRUCT is an attractive practical tool for solving difficult synthesis problems.

HIFOO left − HINFSTRUCT right

HIFOO left − HINFSTRUCT right

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TEST I - gradient sampling in HIFOO left: log10 of H∞ -norm ratios of H∞ -norm ratios - magnification of central region

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Fig. 1. right: log10

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HIFOO left − HINFSTRUCT right

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TEST II - without gradient sampling in HIFOO left: log10 of H∞ -norm ratios of H∞ -norm ratios - magnification of central region

HIFOO left − HINFSTRUCT right

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HIFOO left − HINFSTRUCT right

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TEST II - without gradient sampling in HIFOO left: log10 of cpu time ratios of cpu time ratios for problems with less than 3% deviation in objective

R EFERENCES [1] J. V. Burke, D. Henrion, A. S. Lewis, and M. L. Overton. HIFOO - a matlab package for fixed-order controller design and H∞ optimization. In 5th IFAC Symposium on Robust Control Design, Toulouse, France, July 2006. [2] F. Leibfritz. COMPLe IB, COnstraint Matrix-optimization Problem LIbrary - a collection of test examples for nonlinear semidefinite programs, control system design and related problems. Technical report, Universität Trier, 2003. [3] MATLAB. Robust Control Toolbox. The MathWorks Inc., Natick, MA, 2010.

TABLE I TEST I - A IRCRAFT AND HELICOPTER MODELS model AC1 AC1 AC2 AC2 AC3 AC3 AC4 AC4 AC5 AC5 AC6 AC6 AC6 AC7 AC7 AC7 AC8 AC8 AC8 AC9 AC9 AC9 AC10 AC10 AC10 AC11 AC11 AC12 AC12 AC13 AC13 AC13 AC14 AC14 AC14 AC15 AC15 AC16 AC16 AC17 AC17 AC18 AC18 AC18 HE1 HE1 HE2 HE2 HE3 HE3 HE3 HE4 HE4 HE4 HE5 HE5 HE5 HE6 HE6 HE6 HE7 HE7 HE7 DIS1 DIS1 DIS1 DIS2 DIS2 DIS3 DIS3 DIS4 DIS4 DIS5 DIS5 JE1 JE1 JE1 JE2 JE2 JE2 JE3 JE3 JE3

order P/K/ dim x 5/0/9 5/2/25 5/0/9 5/2/25 5/0/8 5/2/24 4/0/2 4/2/12 4/0/4 4/2/16 7/0/8 7/2/24 7/4/48 9/0/2 9/3/20 9/5/42 9/0/5 9/3/32 9/5/60 10/0/20 10/3/56 10/5/90 55/0/4 55/3/25 55/10/144 5/0/8 5/2/24 4/0/12 4/2/30 28/0/12 28/2/30 28/7/110 40/0/12 40/2/30 40/10/182 4/0/6 4/2/20 4/0/8 4/2/24 4/0/2 4/2/12 10/0/4 10/2/16 10/5/49 4/0/2 4/2/12 4/0/4 4/2/16 8/0/24 8/2/48 8/4/80 8/0/24 8/2/48 8/4/80 8/0/8 8/2/24 8/4/48 20/0/24 20/2/48 20/8/168 20/0/24 20/2/48 20/8/168 8/0/16 8/2/36 8/4/64 3/0/4 3/2/16 6/0/16 6/2/36 6/0/24 6/2/48 4/0/4 4/2/16 30/0/15 30/2/35 30/8/143 21/0/9 21/2/25 21/8/121 24/0/18 24/2/40 24/8/154

cpu HIFOO 5.83 14.61 0.70 0.77 0.94 19.11 0.75 8.63 0.25 2.75 4.22 60.83 106.23 0.56 15.27 12.03 2.50 59.14 148.14 15.34 45.14 256.94 NaN NaN NaN 0.72 40.47 4.16 7.89 365.36 375.75 1139.55 272.97 90.64 61.64 2.72 7.72 1.84 3.34 0.05 0.13 0.94 41.42 107.44 0.17 36.44 0.55 26.44 88.20 60.48 197.13 22.58 7.13 48.56 5.17 33.17 109.08 60.64 257.44 402.70 111.41 208.03 384.34 5.17 8.63 15.89 0.80 6.92 20.09 11.25 2.97 10.34 1.80 32.64 227.50 381.67 1013.05 183.77 300.14 944.58 82.02 248.44 852.33

cpu HINFSTRUCT 5.72 7.06 1.05 0.70 1.41 3.45 1.08 1.42 1.13 1.53 0.88 3.52 6.89 0.63 3.77 5.55 0.80 5.53 4.83 3.47 4.73 11.81 18.19 66.70 294.50 1.98 1.56 1.34 5.41 11.36 14.53 28.08 9.05 6.53 8.28 0.94 1.56 0.89 1.11 0.31 0.28 2.59 2.34 5.66 0.69 1.98 0.70 2.36 2.77 2.98 8.92 2.98 2.31 4.81 0.80 2.08 7.03 8.09 38.41 78.89 18.73 46.80 85.97 1.23 1.20 2.84 0.45 1.06 2.50 3.17 1.25 1.69 1.34 2.50 38.39 46.30 103.64 12.95 37.52 88.23 11.25 29.28 32.94

H∞ -norm HIFOO 0.00 0.00 0.11 0.11 3.66 3.00 0.94 0.56 678.62 668.38 4.11 3.62 3.93 0.07 0.04 0.04 2.01 1.62 1.62 1.00 1.00 1.01 Inf Inf Inf 3.53 2.82 0.32 0.32 163.54 156.30 156.28 101.73 100.00 100.00 15.24 14.88 16.65 14.84 6.61 6.61 27.23 7.26 7.02 0.15 0.09 4.15 2.45 0.83 0.81 0.80 22.84 22.84 22.84 8.90 2.16 1.85 192.36 18.24 4.76 196.14 28.23 32.62 4.16 4.16 4.16 1.05 0.95 1.09 1.05 0.74 0.74 1035.53 746.42 23.41 17.56 19.37 183.35 96.12 80.82 5.54 2.90 2.89

H∞ -norm HINFSTRUCT 0.00 0.00 0.11 0.11 3.63 2.97 0.94 0.56 664.72 665.10 4.11 3.54 3.45 0.07 0.04 0.04 2.01 1.63 1.62 1.00 1.00 1.00 13.24 21.80 7.70 2.82 2.81 0.32 0.02 163.30 156.40 156.33 101.75 100.00 100.00 15.17 14.88 14.86 14.86 6.61 6.61 10.70 7.66 6.10 0.15 0.09 3.92 2.42 0.81 0.80 0.80 22.84 22.84 22.84 13.07 2.16 1.79 192.46 15.55 2.69 192.52 25.64 2.87 4.16 4.16 4.17 1.05 0.95 1.06 1.05 0.74 0.73 1035.53 667.41 10.17 4.05 3.94 183.50 68.94 49.59 5.10 2.90 2.89

TABLE II TEST I CONTINUED - MISCELLANEOUS model REA1 REA1 REA2 REA2 REA3 REA3 REA3 WEC1 WEC1 WEC1 WEC2 WEC2 WEC2 TG1 TG1 TG1 AGS AGS AGS PAS PAS TMD TMD TMD CM1 CM1 CM1 ROC1 ROC1 ROC1 ROC2 ROC2 ROC2 ROC3 ROC3 ROC3 ROC4 ROC4 ROC4 ROC5 ROC5 ROC5 IH IH IH TF1 TF1 TF1 TF2 TF2 TF2 TF3 TF3 TF3 HF2D10 HF2D10 HF2D11 HF2D11 HF2D12 HF2D12 HF2D13 HF2D13 CSE1 CSE1 CSE1 CSE2 CSE2 CSE2 NN1 NN1 NN2 NN2 NN3 NN3 NN4 NN4 NN5 NN5 NN5 NN6 NN6 NN6 NN7 NN7 NN7 NN8 NN8 NN9 NN9 NN11 NN11 NN11 NN12 NN12 NN13 NN13 NN14 NN14 NN15 NN15 NN16 NN16 NN16 NN17 NN17

order P/K/ dim x 4/0/6 4/2/20 4/0/4 4/2/16 12/0/3 12/2/15 12/5/48 10/0/12 10/2/30 10/4/56 10/0/12 10/2/30 10/4/56 10/0/4 10/2/16 10/4/36 12/0/4 12/2/16 12/4/36 5/0/3 5/2/15 6/0/8 6/2/24 6/4/48 20/0/2 20/2/12 20/7/72 9/1/9 9/3/25 9/5/49 10/1/12 10/3/30 10/5/56 11/1/25 11/3/49 11/5/81 9/1/9 9/3/25 9/5/49 7/1/24 7/3/48 7/5/80 21/0/110 21/2/156 21/8/342 7/0/8 7/2/24 7/4/48 7/0/6 7/2/20 7/4/42 7/0/6 7/2/20 7/4/42 5/0/6 5/2/20 5/0/6 5/2/20 5/0/8 5/2/24 5/0/8 5/2/24 20/0/20 20/2/48 20/8/180 60/0/60 60/2/128 60/10/480 3/0/2 3/2/12 2/0/1 2/2/9 4/0/1 4/2/9 4/0/6 4/2/20 7/0/2 7/2/12 7/4/30 9/0/4 9/2/18 9/4/40 9/0/4 9/3/28 9/5/54 3/0/4 3/3/25 5/0/6 5/3/30 16/0/15 16/2/35 16/6/99 6/0/4 6/2/16 6/0/4 6/2/16 6/0/4 6/2/16 3/0/4 3/2/16 8/0/16 8/2/36 8/4/64 3/0/2 3/2/12

cpu HIFOO 0.23 0.38 0.50 0.73 0.27 0.23 0.89 66.23 7.98 2.05 38.55 13.31 164.45 7.45 17.20 28.80 2.31 55.78 5.08 0.09 1.36 4.13 45.53 Inf 21.98 4.66 31.06 8.94 63.06 121.39 13.92 64.03 140.14 105.03 165.80 313.75 7.14 77.39 119.63 44.14 22.58 131.80 523.86 634.73 812.34 9.92 68.22 112.19 0.03 0.05 0.05 10.58 59.27 88.08 11.09 40.03 10.02 54.45 1.30 3.59 0.17 0.33 0.31 1.19 6.30 2.20 4.38 58.38 0.33 1.64 0.17 1.02 NaN NaN 2.63 4.38 0.88 26.89 66.66 20.33 56.69 110.75 3.69 73.92 137.31 0.58 2.56 5.81 66.55 0.19 1.28 67.67 7.42 40.23 5.52 33.61 3.19 35.81 0.06 0.45 32.17 24.41 14.45 0.53 4.59

cpu HINFSTRUCT 0.83 0.50 1.16 0.63 0.48 0.86 1.42 2.42 1.14 2.42 2.70 1.13 2.22 1.59 2.08 2.75 1.52 2.25 4.55 2.11 3.05 1.19 2.89 5.23 1.83 3.22 6.50 1.30 2.63 3.64 2.95 5.50 13.19 10.09 31.00 42.97 3.23 3.31 7.67 11.98 12.67 31.31 53.45 66.38 164.72 3.27 7.67 5.25 0.19 0.23 0.27 1.56 4.20 5.66 0.86 1.42 0.73 1.09 0.31 0.52 0.34 0.36 2.00 2.83 6.25 14.13 16.52 61.59 NaN 1.22 0.53 0.66 NaN 2.70 1.17 1.58 0.53 1.53 2.69 1.45 6.44 25.23 0.83 7.50 12.16 0.67 1.13 1.39 4.88 1.20 4.84 31.50 NaN 7.75 1.69 2.20 1.78 4.80 0.42 0.80 2.81 1.75 4.11 0.44 1.36

H∞ -norm HIFOO 0.90 0.87 1.15 1.13 74.25 74.25 74.25 4.98 3.64 3.64 4.25 3.60 3.62 12.85 3.47 3.47 8.17 8.96 8.17 42.85 0.01 2.56 2.38 Inf 0.82 0.82 0.82 1.24 1.20 1.15 0.05 0.05 0.05 174060.81 28790.58 35549.48 302.21 259.30 250.00 0.02 0.00 0.01 1.54 0.65 1.38 0.38 0.26 0.25 5200.00 5200.00 5200.00 0.41 0.31 0.27 80010.36 79663.41 77237.84 76637.78 1037666.22 1037666.22 101548.53 101548.53 0.02 0.02 0.02 0.02 0.02 0.02 14.13 13.51 2.22 1.76 Inf Inf 1.36 1.29 266.54 238.47 239.20 5602.55 335.12 169.90 74.08 42.60 41.01 2.90 2.36 28.84 14.14 0.11 0.10 0.04 17.08 11.83 14.06 11.62 17.52 9.71 0.10 0.10 0.96 0.96 0.96 11.22 4.95

H∞ -norm HINFSTRUCT 0.87 0.86 1.15 1.13 74.25 74.25 74.25 4.05 3.64 3.64 4.25 3.60 3.60 12.85 3.47 3.47 8.17 8.17 8.17 0.00 0.00 2.52 2.19 2.14 0.82 0.82 0.82 1.24 1.17 1.19 0.05 0.04 0.04 469.56 183.69 201.65 302.21 155.36 155.30 0.00 0.00 0.00 2.33 1.38 0.53 0.38 0.25 0.25 5200.00 5200.00 5200.00 0.42 0.26 0.25 80776.07 79668.40 77259.65 76480.80 1037666.22 1037666.22 101548.53 101548.53 0.02 0.02 0.02 0.02 0.02 0.02 Inf 13.13 2.22 1.76 Inf 465.67 1.37 1.29 266.54 241.37 238.68 5604.12 181.60 140.65 74.08 37.30 37.28 2.89 2.36 28.80 13.65 0.09 0.05 0.01 Inf 7.39 14.07 10.47 17.48 9.46 0.10 0.10 0.96 0.96 0.96 11.22 2.76

TABLE III TEST I CONTINUED - MISCELLANEOUS model UWV UWV UWV EB1 EB1 EB1 EB2 EB2 EB2 EB3 EB3 EB3 AC9 AC13 AC14 HE6 HE7 JE2 JE3 REA3 WEC1 IH CSE2 JE2 JE3 IH BDT2 HE6 HE7 AC10 AC13 AC14 CM1 CM1 CM1 CM2 CM2 CM2 CM3 CM3 CM3 CM4 CM4 CM4 TL TL

order P/K/ dim x 8/0/4 8/2/16 8/4/36 10/0/1 10/2/9 10/4/25 10/0/1 10/2/9 10/4/25 10/0/1 10/2/9 10/4/25 10/8/156 28/10/182 40/10/182 20/8/168 20/10/224 21/10/169 24/10/208 12/10/143 10/9/156 21/10/420 60/10/480 21/15/324 24/20/598 21/15/650 82/30/1156 20/15/399 20/15/399 55/20/484 28/20/552 40/20/552 20/0/2 20/3/20 20/8/90 60/0/2 60/4/30 60/11/156 120/0/2 120/5/42 120/14/240 240/0/2 240/5/42 240/17/342 256/0/4 256/2/16

cpu HIFOO 1.42 5.97 13.44 0.81 0.39 1.97 0.11 4.94 8.02 0.08 4.22 15.70 217.52 1150.19 66.52 368.48 494.78 929.61 697.00 0.73 359.14 972.77 45.38 1171.14 1228.94 1309.23 11782.78 499.55 3046.06 NaN 1554.92 9149.83 5.97 30.84 29.72 384.41 228.27 250.19 261.81 3428.73 1471.11 9453.56 13141.66 12623.02 NaN 31365.89

cpu HINFSTRUCT 0.80 1.78 2.59 0.81 0.73 1.11 0.69 1.09 1.77 1.05 1.08 2.73 24.56 70.80 10.75 80.39 104.88 101.70 40.83 2.45 5.23 233.38 46.52 200.17 201.64 981.55 4407.03 215.25 206.45 821.70 136.05 38.44 2.03 4.55 12.02 10.11 41.64 97.00 66.05 117.48 298.75 304.44 786.16 1165.11 13419.77 8903.50

H∞ -norm HIFOO 0.00 0.00 0.00 3.12 3.10 3.10 2.02 1.78 1.78 2.06 1.87 1.81 1.01 156.31 100.00 9.76 3.10 107.46 2.90 74.25 3.64 1.35 0.02 111.84 2.90 1.39 0.47 15.26 2.84 Inf 156.30 100.00 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 Inf 13060.11

TABLE IV TEST II - A IRCRAFT AND HELICOPTER MODELS H∞ -norm HINFSTRUCT 0.00 0.00 0.00 3.12 3.10 3.10 2.02 1.79 1.78 2.06 1.87 1.82 1.00 156.32 100.00 2.82 2.76 53.59 2.89 74.25 3.64 0.70 0.02 62.18 2.89 0.92 0.47 2.50 2.62 12.16 156.43 100.00 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 12781.47 29257.41

model AC1 AC1 AC2 AC2 AC3 AC3 AC4 AC4 AC5 AC5 AC6 AC6 AC6 AC7 AC7 AC7 AC8 AC8 AC8 AC9 AC9 AC9 AC10 AC10 AC10 AC11 AC11 AC12 AC12 AC13 AC13 AC13 AC14 AC14 AC14 AC15 AC15 AC16 AC16 AC17 AC17 AC18 AC18 AC18 HE1 HE1 HE2 HE2 HE3 HE3 HE3 HE4 HE4 HE4 HE5 HE5 HE5 HE6 HE6 HE6 HE7 HE7 HE7 DIS1 DIS1 DIS1 DIS2 DIS2 DIS3 DIS3 DIS4 DIS4 DIS5 DIS5 JE1 JE1 JE1 JE2 JE2 JE2 JE3 JE3 JE3

order P/K/ dim x 5/0/9 5/2/25 5/0/9 5/2/25 5/0/8 5/2/24 4/0/2 4/2/12 4/0/4 4/2/16 7/0/8 7/2/24 7/4/48 9/0/2 9/3/20 9/5/42 9/0/5 9/3/32 9/5/60 10/0/20 10/3/56 10/5/90 55/0/4 55/3/25 55/10/144 5/0/8 5/2/24 4/0/12 4/2/30 28/0/12 28/2/30 28/7/110 40/0/12 40/2/30 40/10/182 4/0/6 4/2/20 4/0/8 4/2/24 4/0/2 4/2/12 10/0/4 10/2/16 10/5/49 4/0/2 4/2/12 4/0/4 4/2/16 8/0/24 8/2/48 8/4/80 8/0/24 8/2/48 8/4/80 8/0/8 8/2/24 8/4/48 20/0/24 20/2/48 20/8/168 20/0/24 20/2/48 20/8/168 8/0/16 8/2/36 8/4/64 3/0/4 3/2/16 6/0/16 6/2/36 6/0/24 6/2/48 4/0/4 4/2/16 30/0/15 30/2/35 30/8/143 21/0/9 21/2/25 21/8/121 24/0/18 24/2/40 24/8/154

cpu HIFOO 4.47 6.78 0.53 0.61 0.78 5.61 0.75 1.92 0.20 0.89 8.95 13.08 15.14 0.47 9.83 14.98 1.47 15.00 14.33 12.84 17.38 41.45 NaN NaN 50.33 2.44 6.19 5.89 5.52 70.13 91.47 137.13 13.02 18.59 31.52 2.25 4.73 1.84 2.30 0.08 0.09 1.44 2.17 1.23 0.23 6.30 0.41 6.20 8.94 7.88 19.75 9.73 9.59 19.52 1.30 11.20 14.11 12.80 19.73 12.45 10.44 19.64 25.98 2.75 4.78 13.33 1.14 3.67 5.75 9.52 3.64 4.33 1.50 7.09 101.81 132.72 199.38 64.56 115.94 170.20 48.95 69.28 107.30

cpu HINFSTRUCT 3.88 5.42 0.97 0.53 1.25 2.50 0.98 1.44 0.97 1.34 1.39 4.56 6.95 0.48 3.61 5.47 0.77 6.66 6.11 3.31 7.05 12.58 18.19 110.53 299.89 1.41 1.73 1.44 3.75 10.75 13.58 26.45 8.81 6.77 9.97 0.83 1.59 0.89 1.17 0.30 0.27 1.94 1.88 5.66 0.64 3.28 0.83 2.09 2.34 3.09 7.88 2.88 2.50 5.17 0.70 2.44 7.42 6.80 45.73 79.89 18.69 43.20 83.22 1.08 1.17 3.25 0.61 1.19 2.39 2.25 1.23 2.05 1.39 2.86 37.67 49.63 94.69 11.42 32.34 86.06 13.81 16.27 25.63

H∞ -norm HIFOO 0.00 0.00 0.11 0.11 4.75 3.25 0.94 0.56 693.85 667.26 4.11 3.81 3.48 0.07 0.04 0.04 2.01 1.63 1.62 1.00 1.01 1.00 Inf Inf 182.61 2.87 2.82 0.32 0.32 163.73 156.89 156.84 102.09 100.00 100.00 15.48 15.07 14.87 14.86 6.61 6.61 27.24 38.77 15.96 0.15 0.09 4.37 2.44 0.84 0.83 0.82 22.84 22.84 22.84 8.90 2.16 1.96 192.36 190.16 7.41 192.41 25.72 5.88 4.17 4.16 4.17 1.03 0.95 1.10 1.05 0.74 0.74 1035.53 689.28 23.48 22.46 31.72 183.78 145.25 80.84 5.13 2.90 2.99

H∞ -norm HINFSTRUCT 0.00 0.00 0.11 0.11 3.63 2.98 0.94 0.56 665.22 665.10 4.11 3.53 3.45 0.07 0.04 0.04 2.01 1.63 1.62 1.00 1.00 1.00 13.24 9.03 7.88 2.82 2.81 0.32 0.02 163.30 156.32 156.33 101.83 100.00 100.00 15.17 14.87 14.86 14.86 6.61 6.61 10.70 7.66 6.13 0.15 0.07 3.91 2.42 0.81 0.80 0.80 22.84 22.84 22.84 13.07 2.16 1.80 192.37 15.50 2.59 192.60 25.62 2.97 4.16 4.16 4.16 1.02 0.95 1.07 1.05 0.74 0.73 1035.53 667.16 10.17 4.05 3.94 183.45 76.92 52.82 5.10 2.90 2.89

TABLE V TEST II CONTINUED - MISCELLANEOUS model REA1 REA1 REA2 REA2 REA3 REA3 REA3 WEC1 WEC1 WEC1 WEC2 WEC2 WEC2 TG1 TG1 TG1 AGS AGS AGS PAS PAS TMD TMD TMD CM1 CM1 CM1 ROC1 ROC1 ROC1 ROC2 ROC2 ROC2 ROC3 ROC3 ROC3 ROC4 ROC4 ROC4 ROC5 ROC5 ROC5 IH IH IH TF1 TF1 TF1 TF2 TF2 TF2 TF3 TF3 TF3 HF2D10 HF2D10 HF2D11 HF2D11 HF2D12 HF2D12 HF2D13 HF2D13 CSE1 CSE1 CSE1 CSE2 CSE2 CSE2 NN1 NN1 NN2 NN2 NN3 NN3 NN4 NN4 NN5 NN5 NN5 NN6 NN6 NN6 NN7 NN7 NN7 NN8 NN8 NN9 NN9 NN11 NN11 NN11 NN12 NN12 NN13 NN13 NN14 NN14 NN15 NN15 NN16 NN16 NN16 NN17 NN17

order P/K/ dim x 4/0/6 4/2/20 4/0/4 4/2/16 12/0/3 12/2/15 12/5/48 10/0/12 10/2/30 10/4/56 10/0/12 10/2/30 10/4/56 10/0/4 10/2/16 10/4/36 12/0/4 12/2/16 12/4/36 5/0/3 5/2/15 6/0/8 6/2/24 6/4/48 20/0/2 20/2/12 20/7/72 9/1/9 9/3/25 9/5/49 10/1/12 10/3/30 10/5/56 11/1/25 11/3/49 11/5/81 9/1/9 9/3/25 9/5/49 7/1/24 7/3/48 7/5/80 21/0/110 21/2/156 21/8/342 7/0/8 7/2/24 7/4/48 7/0/6 7/2/20 7/4/42 7/0/6 7/2/20 7/4/42 5/0/6 5/2/20 5/0/6 5/2/20 5/0/8 5/2/24 5/0/8 5/2/24 20/0/20 20/2/48 20/8/180 60/0/60 60/2/128 60/10/480 3/0/2 3/2/12 2/0/1 2/2/9 4/0/1 4/2/9 4/0/6 4/2/20 7/0/2 7/2/12 7/4/30 9/0/4 9/2/18 9/4/40 9/0/4 9/3/28 9/5/54 3/0/4 3/3/25 5/0/6 5/3/30 16/0/15 16/2/35 16/6/99 6/0/4 6/2/16 6/0/4 6/2/16 6/0/4 6/2/16 3/0/4 3/2/16 8/0/16 8/2/36 8/4/64 3/0/2 3/2/12

cpu HIFOO 0.11 0.22 0.16 0.66 0.33 0.36 0.53 12.67 4.39 1.73 11.17 6.53 6.06 3.30 14.06 16.66 1.02 3.61 5.19 0.25 0.19 5.44 5.41 9.44 4.22 7.67 23.42 4.48 4.09 13.17 9.36 15.02 17.58 NaN 11.06 25.64 8.00 11.06 10.64 6.80 6.13 8.16 83.72 93.58 178.42 8.47 10.59 8.67 0.03 0.05 0.06 2.80 8.89 12.67 1.75 3.03 1.77 4.80 0.23 0.41 0.14 0.42 0.31 0.98 1.36 2.73 17.70 43.52 0.41 1.73 0.17 0.52 NaN 1.09 3.09 3.19 0.80 3.53 10.50 11.19 13.72 21.58 2.69 14.00 17.42 0.41 1.22 3.34 7.63 0.17 12.41 34.14 NaN 7.88 2.98 4.95 1.45 4.22 0.08 0.38 7.22 9.61 16.55 0.52 1.34

cpu HINFSTRUCT 1.00 0.45 1.13 0.48 0.48 1.05 1.42 2.30 0.95 2.50 2.56 1.02 2.81 1.00 1.81 3.38 0.78 1.70 3.17 NaN 5.55 1.61 2.61 6.38 2.00 3.27 6.11 1.13 2.34 3.72 2.92 3.78 13.83 14.05 31.38 46.73 1.64 3.17 7.64 11.72 14.06 32.73 54.13 67.50 163.89 1.44 5.13 5.44 0.11 0.23 0.25 2.66 6.11 5.94 0.95 1.19 0.77 1.34 0.36 0.42 0.28 0.53 1.88 2.44 5.83 12.59 15.56 54.61 NaN 1.28 0.52 0.66 NaN 2.92 1.30 1.47 0.55 1.64 2.81 1.30 7.83 27.27 0.89 7.42 9.64 0.66 1.20 1.77 4.25 1.16 13.53 33.52 NaN 7.33 1.19 2.56 1.91 4.80 0.59 0.75 2.22 2.77 6.88 0.38 0.98

H∞ -norm HIFOO 0.89 0.86 1.18 1.13 74.25 74.25 74.25 4.05 3.64 3.67 4.33 3.60 3.60 12.85 3.48 3.47 8.17 8.17 8.17 48.53 861165.74 2.52 2.40 2.28 0.82 0.82 0.82 1.24 1.24 1.21 0.05 0.04 0.04 Inf 53471.40 7853.62 302.21 253.23 269.12 0.00 0.00 0.01 1.62 1.66 1.36 0.38 0.25 0.25 5200.00 5200.00 5200.00 1.34 0.30 0.28 80175.86 79643.35 77245.79 76686.97 1037666.22 1037666.22 101548.53 101548.53 0.02 0.02 0.02 0.02 0.02 0.02 13.97 15.65 2.22 1.77 Inf 84.03 1.36 1.30 266.54 238.88 238.64 5603.46 263.06 242.22 74.08 42.36 49.56 2.90 2.38 28.89 14.14 0.11 0.05 0.04 Inf 37.48 16.55 18.23 18.26 21.69 0.11 0.10 0.96 0.96 0.96 11.22 5.20

TABLE VI TEST II CONTINUED - MISCELLANEOUS H∞ -norm HINFSTRUCT 0.87 0.86 1.15 1.13 74.25 74.25 74.25 4.05 3.64 3.64 4.25 3.60 3.60 12.85 3.47 3.47 8.17 8.17 8.17 Inf 0.00 2.24 2.17 2.14 0.82 0.82 0.82 1.24 1.19 1.19 0.05 0.04 0.04 563.53 182.68 238.01 302.21 155.33 87.55 0.00 0.00 0.00 1.93 1.54 1.14 0.38 0.25 0.25 5200.00 5200.00 5200.00 0.42 0.25 0.26 80549.11 79742.50 77246.16 76480.80 1037666.22 1037666.22 101548.53 101548.53 0.02 0.02 0.02 0.02 0.02 0.02 Inf 13.13 2.22 1.76 Inf 20.48 1.36 1.29 266.54 238.60 238.68 5612.12 176.89 138.20 74.08 37.40 37.71 2.89 2.36 28.80 13.65 0.09 0.02 0.01 Inf 7.47 14.07 10.47 17.48 9.46 0.10 0.10 0.96 0.96 0.96 11.22 5.15

model UWV UWV UWV EB1 EB1 EB1 EB2 EB2 EB2 EB3 EB3 EB3 AC9 AC13 AC14 HE6 HE7 JE2 JE3 REA3 WEC1 IH CSE2 JE2 JE3 IH BDT2 HE6 HE7 AC10 AC13 AC14 CM1 CM1 CM1 CM2 CM2 CM2 CM3 CM3 CM3 CM4 CM4 CM4 TL TL

order P/K/ dim x 8/0/4 8/2/16 8/4/36 10/0/1 10/2/9 10/4/25 10/0/1 10/2/9 10/4/25 10/0/1 10/2/9 10/4/25 10/8/156 28/10/182 40/10/182 20/8/168 20/10/224 21/10/169 24/10/208 12/10/143 10/9/156 21/10/420 60/10/480 21/15/324 24/20/598 21/15/650 82/30/1156 20/15/399 20/15/399 55/20/484 28/20/552 40/20/552 20/0/2 20/3/20 20/8/90 60/0/2 60/4/30 60/11/156 120/0/2 120/5/42 120/14/240 240/0/2 240/5/42 240/17/342 256/0/4 256/2/16

cpu HIFOO 1.06 4.44 11.34 0.72 1.31 0.28 0.05 4.44 8.70 0.06 3.03 7.31 52.44 91.25 37.23 22.67 20.83 186.08 121.06 1.55 14.16 241.94 34.02 286.25 413.08 413.06 2166.39 22.25 38.06 190.92 248.53 87.09 5.69 18.45 28.77 54.64 55.14 224.39 267.13 851.19 1411.11 2027.39 5535.83 7664.95 NaN 27852.92

cpu HINFSTRUCT 0.81 1.69 2.92 0.92 0.78 0.75 0.69 1.27 1.75 1.05 1.02 1.72 21.91 39.73 8.97 80.28 102.13 108.78 39.27 2.73 5.66 399.05 60.98 208.67 208.41 611.16 4342.08 230.59 225.73 776.11 155.27 34.78 2.03 4.83 12.38 7.08 37.56 44.13 49.33 142.84 292.64 294.66 642.67 1559.41 15109.67 10270.98

H∞ -norm HIFOO 0.00 0.00 0.00 3.12 3.10 3.10 2.02 1.78 1.78 2.06 1.87 1.82 1.02 157.19 100.00 4.81 7.26 126.68 2.89 74.25 3.64 2.26 0.02 94.03 2.89 2.00 0.46 424.82 6.44 1068.77 157.19 100.00 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 Inf 13464.93

H∞ -norm HINFSTRUCT 0.00 0.00 0.00 3.12 3.10 3.10 2.02 1.79 1.78 2.06 1.91 1.82 1.00 156.32 100.00 2.93 2.83 55.78 2.89 74.25 3.64 1.11 0.02 60.65 2.89 1.27 0.47 2.40 2.62 8.27 156.43 100.00 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 12781.68 12781.46