Closed-form minimax time-delay filters for underdamped systems

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1 OPTIMAL CONTROL APPLICATIONS AND METHODS Otim. Control Al. Meth. 27; 28: Published online 4 December 26 in Wiley InterScience ( Closed-form minimax time-delay filters for underdamed systems Tarunraj Singh 1, *,y and Marco Muenchhof 2 1 Deartment of Mechanical and Aerosace Engineering, SUNY at Buffalo, Buffalo, NY 1426, U.S.A. 2 Institute for Automatic Control, TU Darmstadt, Darmstadt, Germany SUMMARY This aer derives closed-form solutions for the arameters of a time-delay filter designed to be robust to uncertainties in frequencies to be cancelled. It is shown that the sloe of the magnitude lot of the two time-delay filter is zero at the nominal frequency indicating that it is a local maximum. This information is used for deriving the solution of the arameters of the time-delay filter in closed form. Three time-delay filters are also designed which force a zero of the filter to be located at the nominal frequency of the system. Uniform and non-uniform distributions of the enalty over the uncertain regions are ermitted in this formulation. The alicability of the roosed technique for the control of multi-mode systems is also illustrated. Coyright # 26 John Wiley & Sons, Ltd. Received 1 Setember 24; Revised 19 October 26; Acceted 25 October 26 KEY WORDS: time-delay filter; minimax; vibration control 1. INTRODUCTION Vibration attenuation by shaing inut to underdamed systems has been addressed by numerous researchers [1 6] besides others. There has been an increased interest in the develoment of techniques to desensitize the controllers to uncertainties in the system model. The requirements of high recision in alications such as hard disk drives [7], telescoes [8, 9], etc. mandate that the controllers satisfy the ointing requirements in the resence of uncertainties in the model. Singer and Seering [2] roosed a technique to design a sequence of imulses with the objective of forcing the variation of the sensitivity of the residual energy of the system with resect to modelled daming or frequency to zero. The resulting controllers *Corresondence to: Tarunraj Singh, Deartment of Mechanical and Aerosace Engineering, SUNY at Buffalo, Buffalo, NY 1426, U.S.A. y tsingh@eng.buffalo.edu Contract/grant sonsor: von Humboldt Stiftung Coyright # 26 John Wiley & Sons, Ltd.

2 158 T. SINGH AND M. MUENCHHOF were called inut shaers. Singh and Vadali [5] illustrated that robustness to modelling errors can be achieved by cascading multile time-delay filters designed to cancel the nominal oles of the system. Singhose et al. [1] roosed a technique which they referred to as multi-hum extra-insensitive (EI) inut shaer where they determine the amlitudes of the imulses so as to maximize the uncertain domain where the residual vibration is below a secified threshold. Singh [11] roosed an otimization roblem where the maximum magnitude of the residual energy in an uncertain domain is minimized. The difference between the EI inut shaer and the minimax time-delay filter is that the EI shaer requires the residual energy curve to be zero at two frequencies which flank the nominal frequency of the considered mode. However, this constraint is not required in the minimax formulation and minimax design results in a larger uncertainty domain for any secified residual energy threshold, which corresonds to increased robustness. Singhose et al. [12] also illustrate the benefit of using the inut shaing technique (time-delay filter) comared to conventional Finite Imulse Resonse (FIR) and Infinite Imulse Resonse (IIR) filters. In this aer, the minimax roblem is first addressed for a single mode system. In numerous alications, there exists one dominant mode which is the main contributor to the residual energy of the manoeuvring structure. Thus, there is a motivation to derive the otimal time-delay filter which minimizes the maximum magnitude of the transfer function of the time-delay filter, in closed form. A simle technique for handling multile modes is also roosed. The resulting filter is designed by addressing the roblem as a series of single mode roblems. Section 2 focuses on the develoment of closed-form solutions for the arameters of minimax time-delay filters. The develoment is then modified to ermit differential weighting of the limiting and nominal frequencies. In Section 3, a simle aroach is roosed to ermit using the solution of the undamed systems to system with daming. The aroach to design minimax filters for multi-mode systems is described in Section 4. The aer concludes with some remarks in the final section. 2. OPTIMAL MINIMAX FILTERS FOR UNDAMPED SYSTEMS 2.1. Two time-delay filter A time-delay filter to modify the reference inut to a system to attenuate residual vibrations is shown in Figure 1. The transfer function of the time-delay filter is given by the equation A þ A 1 e st 1 þ A 2 e st 2 ð1þ which is used to refilter inuts to a system characterized by undamed modes. The frequency of the mode to be cancelled is uncertain, but the region of uncertainty is known. Assume that the nominal frequency is o and the uncertain frequency o lies in the range o l 4o42o o l ð2þ imlying that the uncertainty is symmetric about the nominal frequency. Figure 2 illustrates schematically the objective of the otimization roblem where the maximum magnitude of the cost function over the domain of uncertainty o l to 2o o l is to be minimized. To minimize the maximum magnitude of the magnitude lot of the time-delay filter in the region of uncertainty

3 CLOSED-FORM MINIMAX TIME-DELAY FILTERS 159 Figure 1. Time-delay filter. max Figure 2. Sensitivity curve. (Figure 2), we require that the magnitude of the time-delay filter at the boundary be equal to that at the nominal frequency. Assuming further that T 1 ¼ and T 2 ¼ 2 ð3þ o o the magnitude of the transfer function of the time-delay filter can be shown to be FðoÞ ¼A 2 þ A2 1 þ A2 2 þ 2A A 1 cos o o þ 2A A 2 cos 2o o þ 2A 1 A 2 cos o o The location of the maximum of FðoÞ can be determined from the equation dfðoþ do ¼ 2A A 1 sin o 2 2A A 2 sin 2o 2A 1 A 2 sin o ¼ ð5þ o o o o o o It can be seen that o ¼ o satisfies Equation (5). To determine the arameters of the minimax time-delay filter, the magnitude of the time-delay filter at the boundary and the nominal frequency are equated, resulting in the equation o l A þ A 1 cos þ A 2 cos 2o 2 l þ A 1 sin o l þ A 2 sin 2o 2 l o o o o ¼ðA þ A 1 cosðþþa 2 cosð2þþ 2 ð4þ ð6þ

4 16 T. SINGH AND M. MUENCHHOF For undamed systems, we can assume that A 2 is equal to A : We also require the constraint A þ A 1 þ A 2 ¼ 1 ð7þ to be satisfied to enable assing DC signals without amlification or attenuation. Solving for A ; we have 2ð1 þ cosððo l =o ÞÞÞ A ¼ ð8þ 5 þ 4 cosððo l =o ÞÞ cosðð2o l =o ÞÞ Figure 3 illustrates the variation of the gains of the time-delay filter as a function of the normalized uncertain interval. It can be seen that the amlitudes are always ositive which is due to the fact that the delay time of the filter has been selected to be an integral multile of half the eriod of the frequency to be cancelled. In the limit when o l =o equals unity, the otimal solution boils down to the robust time-delay filter designed to lace two zeros at the estimated location of the oles of the system. Calculating the magnitude of the time-delay filter at o ¼ 2o o l using the solution for A given by Equation (8), we can show that it is equal to that at the nominal frequency o : Thus, the magnitude of the time-delay filter at the two limits of the uncertain frequency range and the nominal frequency are the same. Figure 4 illustrates the variation of the magnitude of the transfer function of the time-delay filter which is referred to as the sensitivity curve, for different uncertain regions. It is clear from the figure that as the uncertain region decreases, the maximum magnitude of the sensitivity curve becomes smaller. Since, the magnitude of the sensitivity lot at the nominal frequency is ða þ A 1 cosðþþa 2 cosð2þþ ¼ ða A 1 þ A Þ¼ð 1þ4A Þ¼ 1 þ cosðo l=o Þ ð9þ 3 cosðo l =o Þ A =A 2 A 1 Time-Delay Filter Gains Uncertain Region ω l /ω Figure 3. Time-delay filter arameters.

5 CLOSED-FORM MINIMAX TIME-DELAY FILTERS < ω < < ω < < ω < 1.2 Magnitude ω/ω Figure 4. Sensitivity curve. the uncertain region can be solved given a ermissible maximum magnitude of the sensitivity curve. For a given magnitude M; the lower bound of the uncertain frequency range is o l ¼ o 3M 1 cos 1 ð1þ M þ 1 or the width of the uncertainty is given by the equation 2ðo o l Þ¼2o 1 1 3M 1 cos 1 ð11þ M þ 1 It is also clear from Equation (9) that A lies in the range 1 4 4A ) 4M41 ð12þ since the magnitude of the sensitivity curve should not be greater than 1 at the nominal frequency. The location of zeros of the minimax filter can be solved for by substituting the closed-form solutions for the arameters of the time-delay filter into Equation (1). The equations resulting by equating the real and imaginary arts of Equation (1) to zero lead to A þ A 1 cosðotþþa 2 cosð2otþ ¼ ð13þ A 1 sinðotþþa 2 sinð2otþ ¼ ð14þ where T ¼ =o ; which have to be solved to determine the locations of the zeros of the timedelay filter. Equation (14) can be rewritten as ð1 2A Þ sinðotþþa sinð2otþ ¼sinðoTÞð1 2A þ 2A cosðotþþ ¼ ð15þ

6 162 T. SINGH AND M. MUENCHHOF Table I. Minimax solutions. Maximum magnitude Zero location Zerolocation 2A 1 2A j 1 2A 1 T cos 1 2A þ 2o Uncertain range ð 1 þ 4A Þ 1 T cos 1 :65o51:4.288 :7271j 1:2729j :75o51: :7919j 1:281j :85o51:2.51 :8598j 1:142j j Table II. Analytical and numerical minimax solutions. Uncertain range Numerical otimal cost Closed-form cost % Change :65o51: :63E 7 :75o51: :98E 9 :85o51: :29E 6 which can be solved resulting in the equation cosðotþ ¼ 2A 1 2A ð16þ which satisfies Equation (13) as well. Thus, the zeros of the time delay filter ðs ¼ ojþ are given by o ¼ 1 2A 1 T cos 1 þ 2n 2A T ¼o 2A 1 cos 1 þ 2no ð17þ 2A where n is an integer. Table I lists the location of the zeros and the maximum magnitude of the sensitivity lot as a function of the uncertain range. The roosed aroach is comared to a numerical minimax otimization [11] aroach for three uncertain intervals and the results are tabulated in Table II. The minimax roblem corresonds to the determination of the arameters of the time-delay filter to minimize the maximum magnitude of the function FðoÞ (Equation (4)). It can be seen that the difference between the numerical and the roosed aroach is negligible Non-uniform weighting To ermit differential weighting of the frequencies in the uncertain region, the revious develoment can be modified to include a enalty at the boundary of the uncertain region. The weight at the nominal frequency is always assumed to be unity. Including the enalty in the formulation results in Equation (6) being modified to o l W A þ A 1 cos þ A 2 cos 2o 2 l þ A 1 sin o l þ A 2 sin 2o 2 l o o o o ¼ðA þ A 1 cosðþþa 2 cosð2þþ 2 ð18þ

7 CLOSED-FORM MINIMAX TIME-DELAY FILTERS 163 For instance, if a gaussian enalty is assumed then W ¼ ex ðo l o Þ 2 2s 2 ð19þ where s is the variance of the gaussian distribution. Assuming that A 2 ¼ A and therefore A 1 ¼ 1 2A ; we can solve Equation (18), resulting in the quadratic equation A 2 o l 6ðW 1Þ 8 1þ cos 2 1 cos 2o l o o o l þ A 4ðW 1Þþ4 1þ cos þðw 1Þ ¼ ð2þ o which can be used to solve for A : Figure 5 illustrates the otimal solutions as a function of different enalties on the boundary and nominal magnitude of the sensitivity curves. It can be seen that as s decreases, the magnitude of the transfer function of the filter at the nominal frequency also decreases Three time-delay filter It can be seen from the two time-delay filter that the magnitude of the time-delay filter transfer function at the nominal frequency is non-zero. To force the magnitude at the nominal frequency to zero while minimizing the maximum magnitude of the transfer function over the uncertain domain, a three time-delay filter is roosed as shown in Figure 6. As in the develoment for the two time-delay filter, the time delays are assumed to be T 1 ¼ ; T 2 ¼ 2 and T 3 ¼ 3 ð21þ o o o σ =.1 σ =.2 σ = 1.5 Magnitude ω/ω Figure 5. Robust minimax control: varying enalty.

8 164 T. SINGH AND M. MUENCHHOF Figure 6. Time-delay filter. Further, assuming that A ¼ A 3 and A 1 ¼ A 2 ð22þ and with the requirement that A þ A 1 þ A 2 þ A 3 ¼ 1 ð23þ it can be shown that the magnitude of the transfer function of the time-delay filter FðoÞ ¼8 cos 3 o A 2 o þð4a 8A 2 Þ o cos2 þ 1 o 2 8A2 cos o þ 1 o 2 4A þ 8A 2 ð24þ is at o ¼ o ; the nominal frequency. The location of the extremum of the sensitivity curve can be determined from the equation dfðoþ do ¼ 48A 2 o 2o cos2 o þð16a 32A 2 o Þ cos þ 1 16A 2 o sin o o Equating the sin term to zero, we have o ¼ o ð26þ which corresonds to the minimum at the nominal frequency. The other solution is determined by solving the quadratic equation in the cos term, resulting in o ¼ o 4A 1 cos 1 þ 2no ; 4A n ¼ 1; 2; 3... ð27þ which corresonds to the minimum and o ¼ o cos 1 4A þ 1 þ 2no ; 12A n ¼ 1; 2; 3... ð28þ which corresonds to the maximum. The magnitude of the sensitivity curve at the maximum is F o ¼ o cos 1 4A þ 1 ¼ 512A3 192A2 þ 24A 1 12A 54A ð29þ Equating the magnitude of the transfer function of the time-delay filter at the lower limiting frequency to the maximum, we have 8 cos 3 o l A 2 o þð4a 8A 2 Þ cos2 ¼ 512A3 192A2 þ 24A 1 54A o l o þ 1 2 8A2 cos o l o ¼ þ 1 2 4A þ 8A 2 ð25þ ð3þ

9 CLOSED-FORM MINIMAX TIME-DELAY FILTERS 165 we can solve the cubic equation for A ; resulting in the solution 1 A ¼ 3 cosðo l =o Þ 5 ; A 1 ¼ 4ð1 þ 3 cosðo l =o ÞÞ 1 A ¼ 4ð1 þ 3 cosðo l =o ÞÞ and ð31þ The second and the third solutions, which are identical, force the boundary to be a maximum resulting in a subotimal minimax filter. The first solution results in two maxima which lie within the uncertain interval resulting in the otimal minimax solution. Figure 7 illustrates the variation of the gains of the time-delay filter as a function of the normalized uncertain interval. It can be seen that the amlitudes are always ositive as in the two time-delay case. It should be noted that at o l =o ¼ 1; the solution corresonds to the filter where three identical single time-delay filters, which are designed to cancel the nominal frequency of the system, are cascaded. Figure 8 illustrates the variation of the sensitivity curve for different uncertain regions. Comared to Figure 4, it can be seen that the maximum magnitude of the three time-delay filter is significantly smaller than the two time-delay filter. Figure 9 illustrates the variation of the uncertain region as a function of ermissible maximum magnitude of the two and three time-delay filter. It is clear that as the maximum ermissible magnitude of the time-delay filter, M; is increased, the uncertain region monotonically increases. However, the rate of increase of the uncertain region of the three time-delay filter is large comared to the two time-delay filter in the vicinity of zero ermissible magnitude. This imlies that for small ermissible magnitude, the uncertain region of the three time-delay filter is large A =A 3 A 1 =A 2.4 Time-Delay Filter Gains Uncertain Region ω l /ω Figure 7. Three time-delay filter arameters.

10 166 T. SINGH AND M. MUENCHHOF < ω < < ω < < ω < Magnitude ω/ω Figure 8. Sensitivity curve Time delay Filter 3 Time delay Filter (Uncertain Region)/ω Maximum Magnitude (M) Figure 9. Uncertain range vs ermissible magnitude. 3. DAMPED SYSTEMS To design minimax filters for systems characterized by under-damed behaviour, a simle aroach is roosed which uses a transformation to reresent the damed system as an

11 CLOSED-FORM MINIMAX TIME-DELAY FILTERS 167 undamed system in the new sace. For an under-damed system given by the transfer function we define a transformation GðsÞ ¼ o 2 s 2 þ 2zos þ o 2 s ¼ zo Substituting Equation (33) into Equation (32), we have GðÞ ¼ 2 þ o 2 ð ð34þ Þ which reresents an undamed system with a natural frequency of o ffiffiffiffiffiffiffiffiffiffiffiffi : The closed-form solution derived earlier for undamed systems can now be used for this transformed system. The time-delay filter can then be transformed back into the original sace to arrive at the minimax filter for the damed system. For instance, the transfer function of a time-delay filter designed to cancel the oles of an undamed system (Equation (34)) is given by the equation FðÞ ¼1 þ ex o ffiffiffiffiffiffiffiffiffiffiffiffi ð35þ as shown by Singh and Vadali [5]. Transforming this filter into the original sace, we have FðsÞ ¼1 þ ex ðs þ zoþ o ffiffiffiffiffiffiffiffiffiffiffiffi ð36þ which can be rewritten as z z FðsÞ ¼ex ffiffiffiffiffiffiffiffiffiffiffiffi ex ffiffiffiffiffiffiffiffiffiffiffiffi þ ex s o ffiffiffiffiffiffiffiffiffiffiffiffi ð37þ The requirement that the final value of the outut of the time-delay filter subject to a unity ste inut, be unity, requires scaling of the gains of the time-delay filter, resulting in the solution FðsÞ ¼ exðz= ffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þþexð s=o ð ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffi ð38þ exðz= Þþ1 which is identical to the time-delay filter designed to cancel the damed oles as shown by Singh and Vadali [5]. Therefore, the closed-form solution for the gains of the two time-delay minimax filter for a damed system with uncertainty in frequency is given by the equations 2ð1 þ cosððo l =o ÞÞÞ A ¼ 5 þ 4 cosððo l =o ÞÞ cosðð2o l =o ÞÞ ex 2z ffiffiffiffiffiffiffiffiffiffiffiffi ð39þ o 2 ð32þ ð33þ 2ð1 þ cosððo l =o ÞÞÞ A 1 ¼ 1 2 ex 5 þ 4 cosððo l =o ÞÞ cosðð2o l =o ÞÞ z ffiffiffiffiffiffiffiffiffiffiffiffi ð4þ

12 168 T. SINGH AND M. MUENCHHOF and and the delay times are A 2 ¼ T 1 ¼ 2ð1 þ cosððo l =o ÞÞÞ 5 þ 4 cosððo l =o ÞÞ cosðð2o l =o ÞÞ o ffiffiffiffiffiffiffiffiffiffiffiffi and T 2 ¼ o 2 ffiffiffiffiffiffiffiffiffiffiffiffi ð41þ ð42þ and the arameters of the three time-delay minimax filter for a damed system are given by the equations 1 A ¼ 5 3 cosðo l =o Þ ex 3z ffiffiffiffiffiffiffiffiffiffiffiffi ð43þ A 1 ¼ ex 5 3 cosðo l =o Þ 2z ffiffiffiffiffiffiffiffiffiffiffiffi ð44þ A 2 ¼ ex 5 3 cosðo l =o Þ z ffiffiffiffiffiffiffiffiffiffiffiffi ð45þ A 3 ¼ cosðo l =o Þ ð46þ T 1 ¼ o ffiffiffiffiffiffiffiffiffiffiffiffi; T 2 ¼ o 2 ffiffiffiffiffiffiffiffiffiffiffiffi and T 3 ¼ o 3 ffiffiffiffiffiffiffiffiffiffiffiffi ð47þ The gains A ; A 1 ; A 2 and A 3 have to be normalized which is achieved by dividing each of the gains by A þ A 1 þ A 2 for the two time-delay filter case and by A þ A 1 þ A 2 þ A 3 for the three time-delay filter case. Equations (39) (42) and (43) (47) are the exact minimax solutions for an uncertain region which lies along the vertical line in the comlex lane assing through the nominal damed oles. This imlies that both the daming ratio and the natural frequency are varying. The otimal solution when uncertainty exists in the estimated natural frequency imlies that the uncertain region is along a straight line assing through the damed nominal ole and the origin. Figure 1 illustrates the difference between the numerical and the closed-form solution for a system with a nominal daming ratio of.1 and uncertainty in the natural frequency. The difference between the numerical and closed form increases with increasing uncertainty and it is clear that the difference is not large and occurs at the uer limit of the uncertain region.

13 CLOSED-FORM MINIMAX TIME-DELAY FILTERS Numerical Solution Closed Form Solution.3.25 Magnitude Figure 1. Filter comarison. 4. MULTI-MODE SYSTEMS The roosed aroach can be used to design robust filters for systems whose transfer function includes multile modes. Time-delay filters are designed for each uncertain frequency and they are subsequently convolved to arrive at a time-delay filter which is robust to all uncertain frequencies. To illustrate the roosed technique, assume that the two frequencies to be attenuated lie in the range :84o 1 41:2 and 3:64o 2 44:4 ð48þ and the nominal frequencies are selected to be at the midoint of the uncertain regions. The transfer functions of the minimax time-delay filters for each of the frequencies are F 1 ðsþ ¼:2625 þ :4749 exð sþþ:2625 exð 2sÞ ð49þ and F 2 ðsþ ¼:2531 þ :4938 ex 4 s þ :2531 ex 2 4 s ð5þ Figure 11 illustrates the magnitude lot of the minimax filters. The dashed line and the dash-dot lines are the magnitude lots of the filters F 1 ðsþ and F 2 ðsþ; resectively. The solid line is the magnitude lot of the final filter which is FðsÞ ¼F 1 ðsþf 2 ðsþ ð51þ It can be seen that the filter FðsÞ is robust to uncertainties in both the frequencies.

14 17 T. SINGH AND M. MUENCHHOF 1.9 Filter for ω = 1 Filter for ω = 4 Final Filter Magnitude Frequency Figure 11. Robust minimax control: multi-mode system. 5. PERFORMANCE ANALYSIS The roosed technique minimizes the maximum magnitude of the transfer function of a timedelay filter given knowledge of the uncertain domain of the under-damed mode. From a ractical viewoint, the residual energy of the manoeuvring structure is the metric of interest to control designers. It is therefore of interest to determine how the roosed technique for the design of refilters comares to the minimax filter designed to minimize the maximum residual energy over the uncertain domain, a technique develoed by Singh [11]. The residual energy, defined by F ¼ 1 2 yt M y þ 1 2 yt Ky ð52þ where M and K are the mass and stiffness matrices and y and y are the velocity and osition vectors of the system in consideration, is used to comare the erformance of the two filters. The two time-delay minimax filter which minimizes the maximum magnitude of the residual energy is GðsÞ ¼:2439 þ :4554 e 3:527s þ :37 e 2ð3:527Þs ð53þ and the minimax filter using the closed-form solution roosed in this work is GðsÞ ¼:2787 þ :4426 e s þ :2787 e 2s where the uncertain frequency o lies in the range :74o41:3 ð54þ ð55þ It can be seen that the time-delay of the filter which minimizes the residual energy is smaller comared to that of the closed-form solution. To comare the erformance of the minimax

15 CLOSED-FORM MINIMAX TIME-DELAY FILTERS 171 Square Root of Residual Energy Transfer Function Minimax Robust Time Delay EI Inut Shaer Residual Energy Minimax ω /ω Figure 12. Filter erformance: 2 time-delay filter. filters to the ZVD [1] inut shaer which has the same number of delays, the ZVD shaer is designed and is given by the transfer function GðsÞ ¼:25 þ :5e s þ :25 e 2s ð56þ Figure 12 illustrates the variation of the square root of the residual energy over the uncertain frequency for three manifestations of the two time-delay filter. It can be seen that the minimax filter, designed using the residual energy as the cost (solid line), erforms better than the roosed filter (dashed line). The two minimax filters are also comared to the ZVD filter (dashdot line) which is designed to lace multile zeros of the time-delay filter at the nominal location of the uncertain oles of the system and the EI inut shaer (dotted line). It is clear that the maximum magnitudes of the minimax filters are significantly smaller than the ZVD filter over the uncertain frequency domain. The roosed minimax filter is also marginally better than the EI inut shaer. The next filter which constrains the residual energy to be zero at the nominal frequency results in the filters GðsÞ ¼:1474 þ :3492 e 3:1415s þ :3526 e 2ð3:1415Þs þ :158 e 3ð3:1415Þs ð57þ for the residual energy cost and GðsÞ ¼:1479 þ :3521 e s þ :3521 e 2s þ :1479 e 3s for the cost roosed in this work. The transfer function for the ZVDD [1] is GðsÞ ¼:125 þ :375 e s þ :375 e 2s þ :125 e 3s which laces three sets of zeros of the time-delay filter at the nominal location of the oles of the system. ð58þ ð59þ

16 172 T. SINGH AND M. MUENCHHOF.12.1 Transfer Function Minimax ZVDD Filter Residual Energy Minimax Square Root of Residual Energy ω/ω Figure 13. Filter erformance: 3 time-delay filter. Figure 13 illustrates that the difference between the two minimax filters is not very significant with the roosed refilter roviding a erformance which is better than the filter designed to minimize the maximum residual energy, over a large segment of the uncertain range. As in the two time-delay filter, the minimax filters outerform the ZVDD (dash-dot line). The benefit of the roosed technique is that the closed-form solution ermits its use in real-time filter design which would be of interest for the design of adative filters. 6. CONCLUSIONS The contribution of this aer is the develoment of a closed-form solution for the arameters of a time-delay filter which minimizes the maximum magnitude of the transfer function of the time-delay filter. A simle technique to design minimax time-delay filters for underdamed systems is roosed. The minimax two time-delay filter results in non-zero magnitude at the nominal frequency. This magnitude can be reduced by enalizing the magnitude at the nominal frequency comared to the limiting frequencies in the uncertain domain. Closed-form solutions for non-uniform weighting are also derived. Closed-form solutions for the three time-delay filter are also derived which force the magnitude at the nominal frequency to zero. For systems with non-symmetric uncertain regions around the nominal frequency, the roosed aroach can be used by selecting the mean frequency of the uncertain region as the nominal frequency in the design rocess. The roosed technique can also be used for multi-mode systems by concatenating the minimax filters designed for each uncertain frequency.

17 CLOSED-FORM MINIMAX TIME-DELAY FILTERS 173 ACKNOWLEDGEMENTS This work was comleted during the first author s sabbatical stay at the Technical University of Darmstadt under the sonsorshi of the von Humboldt Stiftung. The first author gratefully acknowledges their suort. REFERENCES 1. Smith OJM. Posicast control of damed oscillatory systems. Proceedings of the IRE 1957; 45: Singer NC, Seering WP. Preshaing command inuts to reduce system vibrations. Journal of Dynamic Systems, Measurement and Control (ASME) 199; 112: Swigert CJ. Shaed torques techniques. Journal of Guidance and Control 198; 3: Farrenkof RL. Otimal oen-loo maneuver rofiles for flexible sacecraft. Journal of Guidance and Control 1979; 2: Singh T, Vadali SR. Robust time-delay control of multimode systems. International Journal of Control 1995; 62(6): Junkins JL, Rahman ZH, Bang H. Near-minimum-time maneuvers of distributed arameter systems: analytical and exerimental results. Journal of Guidance, Control and Dynamics 1991; 14(2): Al Mamun A, Ge SS. Precision control of hard disk drives. IEEE Control System Magazine 25; Ye X, Yajima N, Ai G, Hu N. Attitude control and high recision ointing control system of a large balloon Borne solar telescoe. Advances in Sace Research 2; 26(9): Nelan E, Meixner M, Valenti J, McCullough P. JWST small angle maneuver ointing requirements. Technical Memorandum, STSCI-JWST-TM-23-15A, Sace Telescoe Science Institute, Setember Singhose W, Porter L, Singer N. Vibration reduction using multi-hum extra-insensitive inut shaers. American Control Conference, vol. 5, Seattle, WA, 1995; Singh T. Minimax design of robust controller for flexible systems. Journal of Guidance, Control and Dynamics 22; 25(5): Singhose W, Singer N, Seering W. Comarison of command shaing methods for reducing residual vibration. Proceedings of the 1995 Euroean Control Conference, vol. 2, Rome, Italy, Setember 1995;