044501-7 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) 60 -mode 1 10 -mode 2 50 mode 3 -mode 4 40 10 10 27 80 10 10 10 10 frequency (Hz) damping factor(kg/s) FIG.8.Open loop Bode plot comparing damping laws for yaw.The damp- FIG.7.Settling times for the four x modes of a quadruple suspension with ing law,low-pass filter,and mechanical plant are combined.The solid (blue) pure velocity damping of variable strength at stage 1.The lowest mode (1) curves represent the differentiator law,with a suitable low-pass filter.The dominates yielding a shortest settling time of 18s(to 2%).with damping dashed (green)line shows the truncated differentiator with the pole at 3.5 Hz. strength 55 kg/s. The dotted(red)curve represents the interrupted differentiator.Finally the dashed-dotted(cyan)curve shows an example with resonant zeros and poles. The gains are adjusted to match at 10 Hz.Other filter parameters are given in the text. and,apart from the special case with the masses and stage lengths set equal,numerical methods were more convenient. In the equal-mass,equal-length case the minimum O for the same attenuation at 10 Hz,provides a slightly more favor- lower mode is 2.As the mass or length ratios deviate from able balance of phase margin and damping.The transfer func- unity,keeping the same total mass and length,the minimum tion of the resulting loop,with the pole at 3.5 Hz,is added to O and settling time tend to increase.This trend continues as Figure 8. more stages are added,conserving total length.The quadruple A filter with approximately f amplitude response has suspensions have transfer functions that are octic in s requir- /4 phase lead so at a given gain the velocity component is ing numerical methods. reduced by a factor cos(-/4)=1/v2.However due to the To estimate the shortest settling time,a model of the four lower gradient,the gain is increased at the lowest mode by a x DOFs was built with velocity damping applied at stage 1. factor /L:a little over 2 in this case,so there is net ben- The results are shown in Figure 7.As we are interested in efit.The f law can be approximated by inserting an extra the behavior over many 1/e-fold decay periods the slowest- pole and zero to reduce the slope of the differentiator (hence decaying mode dominates.The shortest 1/e decay constant is this filter is called the interrupted differentiator).Performance 4.5s. can be optimized by adjusting the frequencies of these new In the multi-mode system the most important attributes elements.The result,with zeros at 0.04 and 2 Hz and poles at of the controller are:(i)the gain and phase at the lowest- 0.75 and 4.5 Hz,is added to Figure 8. frequency mode,at frequency fi-these determine the settling The final control law considered has a resonant pair of time,(ii)the phase margin at the highest crossing of unity gain zeros and a resonant pair of poles replacing the real zero and immediately above the highest mode,at frequency fi.and(iii) pole of the simplest filter.The idea was to introduce addi- the noise attenuation at 10 Hz.To avoid excessive ringing and tional parameters,namely the two Os,but no performance ad- risk of instability,we aim for ~30 phase margin. vantage was found in increasing either O beyond 0.5,so this The MATLAB model corresponding to a particular in- filter has just two useful parameters.A version with zeros at stance of a suspension from the production series was em- 0.2 Hz and poles at 1.5 Hz,all with =0.5,is added to the ployed.We present results for yaw in detail. comparison graph. In the simplest filter a zero is placed no higher than The low-pass filters already shown are of one generic 0.04 Hz,and a pole is placed at 40 Hz or higher.Note that type but differ in detail to optimize phase margin and atten- real poles and zeros are indicated unless stated explicitly.A uation in each case.The steeply falling response of the sus- low-pass filter (see below)is added and the result is shown in pension implies that strongest attenuation is needed at 10 Hz Figure 8. and immediately above.The key to achieving better perfor- Damping requires a significant feedback component in mance than any of the standard (e.g.,elliptical)filter topolo- the velocity quadrature.We look for control laws with a shal- gies is found in two aspects of the steeply falling frequency lower response slope,trading-off phase lead at fi for increased response of the mechanical system.Even below 10 Hz there gain,and a larger useful feedback component.Modal damp- is a downward slope in the magnitude of the transfer function ing provides a solution matched to the suspension,but we from force applied to the top stage to its displacement.The search for methods that may be more tolerant of variation in same is true of the transfer function from force applied to the the plant. top stage to mirror motion.The former behavior permits Placing a pole to truncate the rising response of the dif- the insertion of complex poles of moderately high into the ferentiator,and modifying the low-pass filter to provide the loop while the latter allows complex zeros to be added. Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/rights-and-permissions.Download to IP:183.195.251.6 On:Fri.22 Apr 2016 00:5549044501-7 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) FIG. 7. Settling times for the four x modes of a quadruple suspension with pure velocity damping of variable strength at stage 1. The lowest mode (1) dominates yielding a shortest settling time of 18 s (to 2%), with damping strength 55 kg/s. and, apart from the special case with the masses and stage lengths set equal, numerical methods were more convenient. In the equal-mass, equal-length case the minimum Q for the lower mode is 2. As the mass or length ratios deviate from unity, keeping the same total mass and length, the minimum Q and settling time tend to increase. This trend continues as more stages are added, conserving total length. The quadruple suspensions have transfer functions that are octic in s requir￾ing numerical methods. To estimate the shortest settling time, a model of the four x DOFs was built with velocity damping applied at stage 1. The results are shown in Figure 7. As we are interested in the behavior over many 1/e-fold decay periods the slowest￾decaying mode dominates. The shortest 1/e decay constant is 4.5 s. In the multi-mode system the most important attributes of the controller are: (i) the gain and phase at the lowest￾frequency mode, at frequency fL—these determine the settling time, (ii) the phase margin at the highest crossing of unity gain immediately above the highest mode, at frequency fH, and (iii) the noise attenuation at 10 Hz. To avoid excessive ringing and risk of instability, we aim for ∼30◦ phase margin. The MATLABR model corresponding to a particular in￾stance of a suspension from the production series was em￾ployed. We present results for yaw in detail. In the simplest filter a zero is placed no higher than 0.04 Hz, and a pole is placed at 40 Hz or higher. Note that real poles and zeros are indicated unless stated explicitly. A low-pass filter (see below) is added and the result is shown in Figure 8. Damping requires a significant feedback component in the velocity quadrature. We look for control laws with a shal￾lower response slope, trading-off phase lead at fL for increased gain, and a larger useful feedback component. Modal damp￾ing provides a solution matched to the suspension, but we search for methods that may be more tolerant of variation in the plant. Placing a pole to truncate the rising response of the dif￾ferentiator, and modifying the low-pass filter to provide the FIG. 8. Open loop Bode plot comparing damping laws for yaw. The damp￾ing law, low-pass filter, and mechanical plant are combined. The solid (blue) curves represent the differentiator law, with a suitable low-pass filter. The dashed (green) line shows the truncated differentiator with the pole at 3.5 Hz. The dotted (red) curve represents the interrupted differentiator. Finally the dashed-dotted (cyan) curve shows an example with resonant zeros and poles. The gains are adjusted to match at 10 Hz. Other filter parameters are given in the text. same attenuation at 10 Hz, provides a slightly more favor￾able balance of phase margin and damping. The transfer func￾tion of the resulting loop, with the pole at 3.5 Hz, is added to Figure 8. A filter with approximately √ f amplitude response has π/4 phase lead so at a given gain the velocity component is reduced by a factor cos(−π/4) = 1/ √2. However due to the lower gradient, the gain is increased at the lowest mode by a factor √ fH/ fL: a little over 2 in this case, so there is net ben￾efit. The √ f law can be approximated by inserting an extra pole and zero to reduce the slope of the differentiator (hence this filter is called the interrupted differentiator). Performance can be optimized by adjusting the frequencies of these new elements. The result, with zeros at 0.04 and 2 Hz and poles at 0.75 and 4.5 Hz, is added to Figure 8. The final control law considered has a resonant pair of zeros and a resonant pair of poles replacing the real zero and pole of the simplest filter. The idea was to introduce addi￾tional parameters, namely the two Qs, but no performance ad￾vantage was found in increasing either Q beyond 0.5, so this filter has just two useful parameters. A version with zeros at 0.2 Hz and poles at 1.5 Hz, all with Q = 0.5, is added to the comparison graph. The low-pass filters already shown are of one generic type but differ in detail to optimize phase margin and atten￾uation in each case. The steeply falling response of the sus￾pension implies that strongest attenuation is needed at 10 Hz and immediately above. The key to achieving better perfor￾mance than any of the standard (e.g., elliptical) filter topolo￾gies is found in two aspects of the steeply falling frequency response of the mechanical system. Even below 10 Hz there is a downward slope in the magnitude of the transfer function from force applied to the top stage to its displacement. The same is true of the transfer function from force applied to the top stage to mirror motion. The former behavior permits the insertion of complex poles of moderately high Q into the loop while the latter allows complex zeros to be added. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:55:49