Small gain stability theory for matched basis function repetitive control

Acta Astronautica 95 (2014) 260–271

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Small gain stability theory for matched basis function repetitive control Yunde Shi a,1, Richard W. Longman b,n, Masaki Nagashima c a b c

Seagate Technology, 1200 Disc Drive, Shakopee, MN 55379, USA Department of Mechanical Engineering, MC4703, Columbia University, 500 West 120th Street, New York, NY 10027, USA Tula Technology, Inc., 2460 Zanker Road, San Jose, CA 95131, USA

a r t i c l e in f o

abstract

Article history: Received 11 April 2013 Received in revised form 2 August 2013 Accepted 29 September 2013 Available online 18 October 2013

Many spacecraft suffer from jitter produced by periodic vibration sources such as momentum wheels, reaction wheels, or control moment gyros. Vibration isolation mounts are needed for fine pointing equipment. Active control methods directly addressing frequencies of interest have the potential to completely cancel the influence of these disturbances. Typical repetitive control methods initially address all frequencies of a given period. Matched basis function repetitive control individually addresses each frequency, finding error components at these frequencies using the projection algorithm, and can converge to zero error, using only frequency response knowledge at addressed frequencies. This results in linear control laws but with periodic coefficients. Frequency domain raising produces a time invariant pole/zero model of the control law. A small gain stability theory is developed, that exhibits very strong stability robustness properties to model error. For convergence to zero tracking error it needs only knowledge of the phase response at addressed frequencies, and it must be known within an accuracy of 7 901. Controllers are then designed by pole-zero placement, bypassing the complexity of original periodic coefficient equations. Compared to the usual repetitive control approaches, the approach here eliminates the need for a robustifying zero phase low pass filter, eliminates the need for interpolation in data, and handles multiple unrelated frequencies easily and naturally. & 2013 Published by Elsevier Ltd. on behalf of IAA.

Keywords: Repetitive control Small gain stability theory Matched basis function repetitive control Frequency domain raising Spacecraft vibration isolation Spacecraft jitter mitigation

1. Introduction Many spacecraft suffer from jitter, vibrations produced by internal moving parts. These include slight imbalance in cryogenic pumps, in momentum wheels used to stiffen the attitude dynamics, three reaction wheels often used as actuators in the attitude control system, or four control moment gyros (CMGs) used for the same purpose. Jitter adversely affects fine pointing instruments on board. A common approach for active vibration isolation of such

n

Corresponding author. Tel.: þ 1 212 854 2959; fax: þ1 212 854 3304. E-mail addresses: [email protected] (Y. Shi), [email protected] (R.W. Longman), [email protected] (M. Nagashima). 1 Research conducted while a doctoral student, Columbia University, USA.

equipment is the filtered x-LMS or Multiple Error LMS, Refs. [1–3]. The approach requires a sensor giving a disturbance correlated signal and employs a real time adaptation of a finite impulse response filter that aims to model the needed transfer function from actuator to controlled location. Refs. [2,3] perform experiments on a spacecraft testbed using various different algorithms for jitter control. The testbed employs a 6 degree-of-freedom Stewart platform for vibration isolation, and geophone sensors. This same platform design has been used on orbit. This paper presents a new control approach for active jitter control. Refs. [4,5] perform experiments testing different control algorithms on a spacecraft testbed. Ref. [5] tests the method developed here on a fully floated spacecraft testbed employing a functioning attitude control system using CMG actuators and star tracker feedback. These experiments relate to the use of laser communication

0094-5765/$ - see front matter & 2013 Published by Elsevier Ltd. on behalf of IAA. http://dx.doi.org/10.1016/j.actaastro.2013.09.016

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between satellites which offers substantial advantages in bandwidth and power consumption for transmission of data over long distances. The control algorithms correct for jitter in the control of pan and tilt of the laser steering mirrors. This reference serves as a companion paper giving spacecraft hardware experiments with the algorithm developed here. Repetitive control (RC), sometimes called repetitive learning control, is a relatively new field that usually aims to eliminate the influence of periodic disturbances on a feedback control system, or it can aim for zero tracking error performing a periodic command, or both. See Refs. [6–10], and for the author's preferred design approaches see Refs. [11–13]. These methods assume one has a method of staying synchronized with the disturbance signal, for example, in Ref. [14], an index pulse is produced each rotation of the disturbance source. This is in fact easy to do for reaction wheels or CMGs based on the phase of the voltage supplied. Unlike the filtered x-LMS or Multiple Error LMS approaches, RC initially addresses all frequencies of the same period, i.e., the fundamental of the given period, DC or constant signals, and all harmonics up to Nyquist frequency. Because one usually cannot have a good model of the system all the way to Nyquist frequency, one normally needs to use a zero-phase cutoff filter to cut off the learning at high frequency when model error is too large for convergence. This is done with the Q filter of Ref. [10] or the enhanced filters discussed in Refs. [12,13]. RC methods normally only address one period, so in spacecraft applications they can apply to disturbance environments such as a cryogenic pump or a momentum wheel. Refs. [15–20] present repetitive control methods that address multiple unrelated periods, as would be needed to handle disturbances from imbalance in three reaction wheels or four CMGs. Robustness to model error deteriorates as more periods are included (Ref. [20]). These methods address all harmonics of each period, and one can introduce a zero-phase low-pass filter to cut off the learning above some frequency when model error becomes too large for convergence. Various iterative learning control and repetitive control approaches make use of the concept of basis functions, as in Refs. [21–24]. The most useful basis functions are simple sine and cosine functions of the frequencies of interest. Ref. [25] through [30] present matched basis function repetitive control (MBFRC), which uses the projection algorithm commonly applied in adaptive control (Ref. [31]) to obtain the components of the error on sines and cosines of the frequencies of interest, and applies sine and cosine modifications to the system input that include adjustment for the amplitude and phase change going through the system in order to have the output error be canceled. These adjustments define the matched basis functions, matching input sinusoids to the feedback control system to their resulting control system output sinusoids. As in other forms of RC, an integration is included to create convergence to zero error at the addressed frequencies in spite of substantial model error. Ref. [27] reports experimental tests on the same platform as used in Ref. [2]. Multiple period RC and MBFRC each have their own potential advantages. Multiple period RC simultaneously addresses all frequencies of the periods considered until

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one cuts out high frequencies with a cutoff filter. MBFRC on the other hand, introduces a separate RC controller for each frequency to be addressed, requiring many controllers for many harmonics. If there are many harmonics that need to be addressed the former approach has an advantage. In exchange for the added complexity of one controller for each frequency, the problem of robustness to high frequency model error is alleviated when using MBFRC. One expects potential improvement in the waterbed effect using MBFRC by allowing high frequencies, above the desired cutoff, to absorb some of the required amplification. An advantage of MBFRC is that multiple unrelated frequencies are addressed in a simple manner, without the complexity needed in Refs. [16] through [20]. Yet another advantage relates to interpolation. The usual RC approaches require interpolation when the period of the addressed frequency is not an integer number of time steps (Ref. [32]). And the interpolation deteriorates at high frequencies. In MBFRC, the basis functions at each frequency allow one to “interpolate” with the actual frequency function of interest. MBFRC projects the error onto sinusoids and then applies the matched sinusoids to the system. This results in linear equations but with periodic coefficients. Refs. [25,26,28] use Floquet theory, or time domain raising, to study stability when the frequencies of interest have periods that are integer multiples of the sampling time interval. Under the same assumption, Ref. [29] developed stability analysis using the frequency raising technique. A very interesting result of this approach is that the controller involving linear equations involving periodic coefficients related to the basis functions, is seen to have a linear time invariant equivalent model. The purpose of this paper is to use the time invariant repetitive controller representation to develop very general and simple small gain stability robustness results. The result is obtained by using the departure angle condition from the theory of root locus plots. This approach was used previously to study the simplest form of RC in Ref. [33]. It represents a strong robustness result guaranteeing convergence to zero tracking error for all sufficiently small gain, provided the phase information about your system response for each addressed frequency is accurate to within 7 901. The result is independent of the system behavior at any other frequency. An additional bonus for the design method presented here is that it no longer requires that the frequencies being addressed have periods that are integer multiples of the sample time interval, or use interpolation. In the next sections, first the MBFRC algorithm is presented, then the equivalent time invariant control laws for each frequency. Then the case of addressing one frequency only is treated to determine the departure angles from poles on the unit circle, which is then generalized to apply to any number of addressed frequencies. Numerical examples are presented. 2. The matched basis function repetitive control algorithm This section summarizes the MBFRC algorithm. Usually the RC controller adjusts the command to a feedback control system, although it could simply be adjusting the input to

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some given system. Consider a single-input, single-output system xðk þ1Þ ¼ AxðkÞ þ BuðkÞ yf b ðkÞ ¼ CxðkÞ

ð1Þ

whose transfer function is given by GðzÞ, and the output yf b ðkÞ does not include the effects of any periodic disturbance. The output including the periodic disturbance is given by yðkÞ ¼ yf b ðkÞ þ wðkÞ. Wherever the disturbance enters in the feedback control system, there is an equivalent disturbance wðkÞ that can be added to the output, and this is done here. The desired output is yd ðkÞ and we seek to converge to whatever input uðkÞ is needed such that yðkÞ converges to this function. The associated tracking error is denoted eðkÞ ¼ yd ðkÞ  yðkÞ. The disturbance is the sum of sinusoids at N þ 1 frequencies ωn where n ¼ 1; 2; 3; :::; N, and n ¼ 0 is used for DC or zero frequency. The desired output yd ðkÞ may be a constant, or can be a sum of sinusoids at these frequencies. Of course, one can also consider commands yd ðkÞ and disturbances in wðkÞ at frequencies not being addressed in this set of N þ 1, and study the behavior of the resulting design for such situations. Since the final design produced is a time invariant linear systems, such a study is accomplished by simply finding the transfer function from disturbance to error and creating appropriate Bode plots. The frequency response of the system is such that an input uðkÞ ¼ cos ðωn kTÞ ¼ cos ðϕn kÞ where T is the sample time interval of the digital control, results in a steady state output given by yf b ðkÞ ¼ r n cos ðϕn k þ τn Þ. The radian frequencies addressed, ωn , have an upper limit of Nyquist frequency π=T, and the normalized frequencies ϕn then range from 0 to π. To find frequency response of a z-transfer function, one substitutes z ¼ expðiωn TÞ ¼ expðiϕn Þ, i.e., a point on the unit circle whose angle with the positive real axis direction represents this normalized frequency between zero and Nyquist. The matched input and output basis functions are then given as follows. The terms input and output refer to the input and output of the feedback control system. For frequency n one needs two output basis functions given in matrix H n ðkÞ, and the matched input basis function are then given in F n ðkÞ h i H n ðkÞ ¼ cos ðϕn kÞ sin ðϕn kÞ F n ðkÞ ¼ ½ð1=r n Þ cos ðϕn k  τn Þ

ð1=r n Þ sin ðϕn k  τn Þ

ð2Þ

thus, the first input basis function applied to the feedback control system produces the first output basis function after reaching steady state response. The input basis function has modified the cosine input magnitude and phase by the amount needed to produce the pure cosine output. Similarly for the second input basis function. The projection algorithm finds the components of the output error on the output basis functions (Ref. 31) βn ðk þ 1Þ ¼ ½I  aHTn ðk þ1ÞH n ðk þ1Þ βn ðkÞ þaH Tn ðk þ 1Þeðk þ1Þ ¼ Aβn ðkÞβn ðkÞ þBβn ðkÞeðk þ 1Þ n ¼ 1; 2; 3; :::; N

ð3Þ

Fig. 1. Block diagram of MBFRC.

the algorithm converges for 0 oa o2. To handle DC, note that ϕ0 ¼ 0 produces H 0 ¼ 1; F ¼ 1=r 0 which are now scalars instead of matrices. Repetitive control uses the discrete form of an integrator to force convergence to zero error. In the present context this becomes the discrete form of an integral of the output basis function components αn ðk þ 1Þ ¼ αn ðkÞ þ Λn βn ðkÞ. As with integral control that cannot tolerate a constant error without sending signals to infinity as time progresses, this summation cannot tolerate a constant amplitude sinusoidal error signal at the associated frequency, and accumulates corrective action until the error goes to zero or the system goes unstable. Then the command input is formed from the linear combination of the input basis functions. N

uðkÞ ¼ ∑ F n ðkÞαn ðkÞ n¼0

ð4Þ

For future reference, the gain Λn ¼ Φλn is split into an overall gain Φ and a separate gain λn to use for each frequency. The algorithm is summarized in Fig. 1. 3. The structure of the time invariant equivalent of MBFRC Ref. [29] used frequency raising as a method to study the stability of MBFRC. To do so it had to make the assumption that the period of the periodic function of interest is an integer N number of sample times T. This meant that ϕn must be an integer multiple of θn ¼ 2π=N. Under these conditions, Mathematica was used to manipulate the algebra relating error eðkÞ to the command uðkÞ for any addressed period, according to the equations summarized in Fig. 1. The somewhat surprising result is that this is a time invariant relationship, in spite of the periodic coefficients present in Eqs. (2)–(4). The result is an expression for the MBFRC controllers for each

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frequency that is simply a well chosen pole/zero configuration that is given below. And now that we have such linear time invariant representations in terms of pole and zero locations, one can study the stability of MBFRC which is the subject of this paper. An important additional benefit of the pole/zero design is that it can be used for any frequency of interest and is not restricted to an integer multiple of θn . This fact directly addresses the interpolation problem that is present in other forms of RC, in effect it is doing the interpolation using the basis functions which is precisely what is needed for a perfect interpolation. The basic structure of the new version of MBFRC is given in Fig. 2. Each of the T n ðzÞ is the pole/zero compensator needed for frequency ϕn , and these will be discussed in detail in the next section. A new aspect has been introduced into the block diagram, the feedforward path that adds Y d ðzÞ to UðzÞ, see Ref. [13] for discussion of use of this loop in other forms of RC. Consider the closed loop behavior of this system. Define the repetitive controller as UðzÞ ¼ ΦRðzÞEðzÞ where N

RðzÞ ¼ ∑ λn T n ðzÞ n¼0

ð5Þ

First consider the case when the feedforward loop is not present. Then block diagram algebra establishes that the transfer functions from command and output disturbance to output and to error are given by YðzÞ ¼ T CL ðzÞY d ðzÞ þT S ðzÞWðzÞ EðzÞ ¼ T S ðzÞ½Y d ðzÞ  WðzÞ

ð6Þ

where the closed loop command to output transfer function T CL ðzÞ and the repetitive control sensitivity transfer function from output disturbance to error are ΦRðzÞGðzÞ 1 þ ΦRðzÞGðzÞ 1 T S ðzÞ ¼ 1 þ ΦRðzÞGðzÞ T CL ðzÞ ¼

ð7Þ

Now consider the case when the feedforward loop is present YðzÞ ¼ T FF ðzÞY d ðzÞ þ T S ðzÞWðzÞ EðzÞ ¼ T S ðzÞð1  GðzÞÞY d ðzÞ T S ðzÞWðzÞ T FF ðzÞ ¼ ½1 þ ΦRðzÞGðzÞ=½1 þΦRðzÞGðzÞ

ð8Þ

To interpret this result, consider that the feedback control system consists of a controller CðzÞ, a plant PðzÞ, and unity feedback. Also consider that the actual disturbance is WðzÞ which enters the feedback control system in the output disturbance location in the existing feedback control loop so that the equivalent output disturbance is WðzÞ ¼ SðzÞWðzÞ, where SðzÞ is the sensitivity transfer function of the feedback

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control system. It is given by SðzÞ ¼ 1=½1 þ PðzÞCðzÞ, and the feedback control system closed loop transfer function is GðzÞ ¼ PðzÞCðzÞ=½1 þ PðzÞCðzÞ. Therefore SðzÞ ¼ 1 GðzÞ. We conclude that EðzÞ ¼ T S ðzÞSðzÞ½Y d ðzÞ WðzÞ

ð9Þ

The difference between the transfer function from command to error with the feedforward signal and without the feedforward signal is of no significance if the command is composed only of frequencies being addressed. But if one wishes to give commands that are unrelated to the frequencies addressed, and are using the RC for the purpose of eliminating periodic disturbances, then one should use the feedforward signal to get the benefit of the feedback control system performance for these frequencies. 4. The pole/zero repetitive controllers for each frequency The repetitive control systems designed by MBFRC have poles on the unit circle at the frequencies for which we seek zero tracking error. This creates a discrete version of an integral, and just as integral control with a pole on the unit circle at z ¼ þ 1 will not tolerate a constant error, the poles on the unit circle at nonzero frequencies will not tolerate a steady state error at these frequencies. MBFRC only places poles at the addressed frequencies, while the usual RC designs that address all frequencies of a given period have poles evenly spaced around the unit circle, for the fundamental and all harmonics and DC. For stability, we need all poles on the unit circle to move toward the inside of the unit circle when the gain is increased from zero. We will prove that the MBFRC design method has the property that all such poles depart radially inward. We comment that if one wanted to design a controller by placing poles and zeros, with the aim of making every pole on the unit circle depart inward, it is not obvious how to accomplish this, especially with many frequencies being addressed. By taking the seemingly unlikely circuitous path of going to the projection algorithm and the matched basis functions in MBFRC, and then using frequency raising, followed by complex computations using Mathematica, we are able to design pole/zero configurations that do precisely what we want – depart radially inward (Ref. [29]). 4.1. The RC transfer functions for each frequency The repetitive controller for the nth addressed frequency has the following transfer function T n ðzÞ ¼ ðan =r n Þ 

½ cos ðϕn  τn Þz2  2 cos ðτn Þz þ cos ðϕn þ τn Þz ½z2 þ ðan  2Þ cos ðϕn Þz þ ð1  an Þ½z2  2 cos ðϕn Þz þ 1

n ¼ 1; 2; 3; :::; N

ð10Þ

This transfer function simplifies for the case of DC, n ¼ 0, by setting ϕ0 ¼ τ0 ¼ 0, and then noting that a factor of ½z 12 cancels from the numerator and denominator to result in the following: Fig. 2. Time invariant equivalent block diagram for MBFRC with feedforward signal.

T 0 ðzÞ ¼

ða0 =r 0 Þz ½z2 þ ða0  2Þz þ ð1  a0 Þ

ð11Þ

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Fig. 4. The value of an when the poles enter the real axis vs. the addressed frequency ϕn .

Fig. 3. Root locus for pole pn1;n 2 locations for different addressed frequencies ϕn .

Examine the poles and zeros of this case. There is one zero located at the origin. The two poles are located at þ1 and ð1  a0 Þ. Recall that the gain a0 must be in the open interval from zero to two, so the second pole starts at þ1 and goes to 1 as a0 increases from zero to two. When a0 ¼ 1, the pole is at the origin on top of the zero. 4.2. The pole locations for n ¼ 1; 2; 3; :::; N The integrator poles on the unit circle at the addressed frequency ϕn are given by ½z2  2 cos ðϕn Þz þ 1 ¼ ðz  P n Þðz  P n Þ

P n ¼ eiϕn

P n ¼ e  iϕn ð12Þ

The other two poles are pn1;n2 ¼ ð1=2Þðan  2Þ cos ϕn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ð1=2Þ ðan  2Þ2 cos 2 ϕn  4ð1 an Þ

ð13Þ

Note that when an ¼ 0 these poles are at P n and P n . When an ¼ 1 the roots are real and are at 0 and cos ϕn . When ϕn corresponds to 901 or π=2, i.e., for half Nyquist frequency, there is a repeated pole at the origin. When an ¼ 2 the poles are at 7 1. Fig. 3 presents the root locus for these two roots for different values of the addressed frequency ϕn as an goes from zero to two. Fig. 4 gives the values of an at which the roots enter the real axis for different addressed frequencies ϕn . This happens when the square root in Eq. (13) becomes zero. Note that when 1 r an r 2 the roots are always real, and otherwise they are complex when ϕn satisfies  pffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1  an 2 1  an cos  1 o ϕn oπ  cos  1 ð14Þ 2  an 2 an Fig. 5 gives the location of the entry to the real axis as a function of the addressed frequency ϕn . This is the value of the first term on the right of Eq. (13) when the value of an makes the square root zero. This an is given by 2 sin ϕn ð1  sin ϕn Þ= cos 2 ϕn which takes on the value þ1 when cos 2 ϕn ¼ 0. The resulting arrival position on

Fig. 5. Location of arrival at the real axis as a function of the addressed frequency ϕn .

the real axis is given by ð1  sin ϕn Þ= cos ϕn which has the value zero when the denominator is zero. Note that the arrival location to the real axis is approximately a linear function of the addressed frequency. 4.3. The zero locations for n ¼ 1; 2; 3; :::; N The poles discussed above are functions of the addressed frequency ϕn and the projection algorithm gain an . The zeros are independent of this gain, but are functions of the phase change through the system τn . One zero is always located at the origin. The two remaining zeros are located at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos τn 7 cos 2 τn  cos ðϕn  τn Þ cos ðϕn þ τn Þ zn1;n 2 ¼ cos ðϕn  τn Þ ð15Þ Consider the term inside the square root. Use the formula for cos ðx 7 yÞ, replace 1  cos 2 ϕn by sin 2 ϕn , and cos 2 τn þ sin 2 τn by unity to find that this term is equal to sin 2 ϕn . Then the zeros are zn1;n 2 ¼

cos τn 7 sin ϕn cos ðϕn τn Þ

ð16Þ

Note that these zeros are independent of an , and are always real. Figs. 6 and 7 show the zn1 and zn2 locations as a function

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5.1. The characteristic polynomial The characteristic polynomial from Fig. 2 for the case of addressing one frequency ϕn can be written in the form needed for a root locus plot for gain Φ as Φλn T n ðzÞGðzÞ ¼  1

ð17Þ

First we treat the special case of DC, and obtain the needed result. Using Eq. (11) in (17) Φðλ0 a0 =r 0 ÞzGðzÞ ¼ 1 ½z 1½z  ð1 a0 Þ Fig. 6. The location of the zero z1 as a function of ϕn  τn for different values of τn .

ð18Þ

We assume that GðzÞ is asymptotically stable, and for a0 in the required range for the projection algorithm, the root 1  a0 is inside the unit circle. The root locus condition that the locus exists on the real axis to the left of an odd number of real zeros plus real poles, indicates that the one root on the real axis, at z ¼ 1, departs radially inward along the real axis, giving the desired result for DC. 5.2. The root locus departure angle condition Now consider any nonzero frequency ϕn . Writing Eq. (17) in detail gives   an ½z  zn1 ½z  zn2 zGðzÞ Φ ¼ 1 ðλn cos ðϕn τn ÞÞ rn ½z  pn1 ½z pn2 ½z  P n ½z P n  ð19Þ

Fig. 7. The location of the zero z2 as a function of ϕn  τn for different values of τn .

of ϕn  τn for different values of the phase change τn through the system for the addressed frequency. Because the coefficient of z2 goes to zero when ϕn  τn reaches 901 or π=2, the root zn1 goes to plus infinity as the angle approaches 901 from below, and once it passes 901 and the coefficient is now negative, the roots start coming in from minus infinity. The other root zn2 stays finite, staying between  1 and approximately 1.5.

5. Small gain stability theory for a single addressed frequency In this section we develop a small gain stability result for MBFRC for the case when it addresses only one frequency, which can be DC. We use the departure angle condition of root locus plots to show that the integrator poles on the unit circle depart radially inward, becoming stable, as the overall gain Φ increases from zero, thus guaranteeing that for sufficiently small gain the MBFRC system is asymptotically stable. In the next section we then show that the MBFRC with as many frequencies as desired has the same property that all poles on the unit circle depart radially inward. This property is not influenced by what and how many frequencies are being addressed. Note that unlike the usual RC that applies to one period, where the frequencies all are fundamental and harmonics, the frequencies that MBRFC can address can be totally independent of each other.

  an ρ eiθnz1 ρnz2 eiθnz2 ρnz3 eiθnz3 r n eiτnw ðλn cos ðϕn  τn ÞÞ nz1 Φ ¼ 1 rn ρnp1 eiθnz1 ρnp2 eiθnz2 ρnP eiθnP ρ eiθnP nP

ð20Þ Any z satisfying Eq. (19) is on the root locus for some gain Φ. In Eq. (20) the complex number represented by each factor in Eq. (19) is written in polar form. Note that the phase response of GðzÞ is indicated here by τnw . For later use this indicates the actual phase response of the physical system, and τnm will be used to indicate the phase response of the model we use to design the MBFRC which might be incorrect. For purposes of this section we assume that they are equal. We assume that Φ is positive, but because cos ðϕn  τn Þ could be either positive or negative, we consider the possibility that one might want λn to be negative. Then when the product is positive, replace  1 by expiðπ 72πℓÞ where ℓ could be any integer. In the case the product is negative, replace 1 by expið 7 2πℓÞ and introduce absolute values on λn cos ðϕn τn Þ. Equating the angle of the complex number on the left of the equality to that on the right produces the root locus formula for the problem of interest ½ðθnz1 þ θnz2 Þ þ θz3 þ τnw   ½ðθnp1 þθnp2 Þ þ θnP þ θnP  ¼ ½ðθnz1 þθnz2 Þ þ ϕn þ τnw  ½ðθnp1 þ θnp2 Þ þθd þ π=2 ( π 7 ℓ2π f or λn cos ðϕn  τnm Þ 4 0 ¼ 7ℓ2π f or λn cos ðϕn τnm Þ o0

ð21Þ

The ℓ can be any integer. The first expression on the left applies to any z on the locus. Our interest is the departure angle from the pole at z ¼ eiϕn so that all angles are computed for z arbitrarily close to this value. Note that

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the departure angle of interest is θnP , which for clarity we denote by θd . Of course, the other departure angle of interest θnP , can be obtained immediately by the complex conjugate nature of the roots, once θd has been found. For the z of interest, θnP ¼ π=2 and θz3 ¼ ϕn , as indicated on the second line of Eq. (21). We seek to prove that the departure angle is radially inward. The complex number z ¼ eiϕn ¼ P is a phasor that is radially outward for this pole on the unit circle. Therefore the desired departure angle is θd ¼ ϕn þ π modulo 2π. In the next three subsections, we will show that ( ϕn  τnm þπ f or λn cos ðϕn  τnm Þ 4 0 ðθnz1 þ θnz2 Þ ¼ f or λn cos ðϕn  τnm Þ o 0 ϕn  τnm

ðθnp1 þ θnp2 Þ ¼ ϕn þ π=2 8 an

wn1 wn2 ¼ ½4 cos 2 ðϕn  τn Þ1=2 ¼ 2Δ cos ðϕn τn Þ

sin ðθnz1 þθnz2 Þ ¼ sin θnz1 cos θnz2 þ cos θnz1 sin θnz2 cos ðθnz1 þ θnz2 Þ ¼ cos θnz1 cos θnz2  sin θnz1 sin θnz2 ð28Þ From the complex numbers for each angle

ð23Þ

cos θnz1 ¼ Δ½ 1  sin ðϕn  τn Þ=wn1

ð24Þ

5.3. Departure angle contribution from zn1;n2

ð27Þ

Since these are magnitudes, we must use the positive square root as indicated. Our objective is to compute both of the following:

sin θnz1 ¼ ½Δ cos ðϕn  τn Þ=wn1

Therefore, when the phase change through the system in the model used to design the MBFRC matches that in the real world, so that ðτnm  τnw Þ ¼ 0, the departure angle is the desired radially inward direction from the poles on the unit circle. And when there is error in the phase information, this error is the amount of deviation from radially inward departure.

sin θnz2 ¼ ½Δ cos ðϕn τn Þ=wn2 cos θnz2 ¼ Δ½ 1  sin ðϕn  τn Þ=wn2

ð29Þ

one computes that sin ðθnz1 þ θnz2 Þ ¼  Δ sin ðϕn τn Þ cos ðθnz1 þ θnz2 Þ ¼  Δ cos ðϕn  τn Þ

ð30Þ

If Δ is negative, it is evident that θnz1 þ θnz2 ¼ ϕn  τn , and otherwise one adds 1801 to the right hand side. This establishes Eq. (22). 5.4. The special case of cos ðϕn  τn Þ ¼ 0

The objective of this section is to establish Eq. (22). We examine the departure from the pole at P n , so that the angles θnz1;nz2 represent the angles the complex number P n  zn1;n 2 makes with the positive real axis. Hence they can be written as   θnz1;nz2 ¼ ∡ P n zn1;n 2   cos ϕn cos ðϕn  τn Þ  cos τn 7 sin ϕn þ i sin ϕn ¼∡ cos ðϕn  τn Þ ð25Þ We adopt the convention that the angle for the first subscript, n1, refers to the upper sign, and the second subscript to the lower sign when there are sign choices in the equation. Examine the numerator in the real part of the complex number. Using the trigonometric identity to write cos ðϕn τn Þ in terms of trigonometric functions of each angle, factoring out sin ϕn , and recognizing sin ðϕn τn Þ in the result, produces the numerator in the form sin ϕn ½ 81  sin ðϕn  τn Þ. Note that both the real part and the imaginary part contain the factor sin ϕn which is always positive (DC has been treated separately), so this factor can be eliminated without changing the angle. Since cos ðϕn  τn Þ can be both positive and negative, let Δ ¼ sgn½ cos ðϕ n  τn Þ. Then multiply both real and imaginary parts by cos ðϕn  τn Þ and factor out Δ to obtain θnz1;nz2 ¼ ∡Δ½ 81  sin ðϕn  τn Þ þ i cos ðϕn τn Þ

wn1;n2 ¼ ½2 7 2 sin ðϕn  τn Þ1=2

ð22Þ

Substituting these into the second version of Eq. (21) produces the following result ðϕn  θd Þ  ðτnm  τnw Þ ¼ π 7 ℓ2π

then

ð26Þ

For later use, let wn1 and wn2 be the magnitudes of the complex numbers for each of the angles respectively,

For this special case, the compensator becomes T n ðzÞ ¼ ðan =r n Þ 

½  2 cos ðτn Þ½z  cos ðϕn þ τn Þ=ð2 cos ðτn ÞÞz ½z2 þ ðan  2Þ cos ðϕn Þz þ ð1  an Þ½z2  2 cos ðϕn Þz þ 1

ð31Þ One of the zeros has disappeared. The angle for the remaining zero is h i ∡ eiϕn  cos ðϕn þ τn Þ=2 cos ðτn Þ   cos ϕn cos τn þ sin ϕn sin τn þ i sin ϕn ð32Þ ¼∡ 2 cos ðτn Þ The condition cos ðϕn τn Þ ¼ 0 means that ϕn ¼ τn þ π=2 7ℓ0 π for some integer value of ℓ0 . If, ϕn ¼ τn þ π=2 then cos ϕn ¼  sin τn and sin ϕn ¼ cos τn . The signs are reversed if ϕn ¼ τn π=2. Other values of ℓ0 repeat these two possibilities. In both cases the real part is zero in Eq. (32), and hence contribution for this zero,  the angle  denoted θnz12 , is ∡ i sin ϕn , or 901. Note that the gain for this root locus, Φλn ½  2 cos ðτn Þ, could be either positive or negative. Since there is only one instead of two extra zeros in this case, angle condition Eq. (21) becomes ½θnz12 þθz3 þ τnw   ½ðθnp1 þ θnp1 Þ þ θnP þθnP  ¼ ½π=2 þ ϕn þ τnw   ½ðϕn þ π=2Þ þ θd þ π=2 ( o 0 7 l2π f or  λn cos ðτnw Þ 4 0 ¼ π 7l2π f or  λn cos ðτnw Þ o 0

ð33Þ

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where θnz12 ¼ π=2 from above and ðθnp1 þ θnp1 Þ ¼ ðϕn þ π=2Þ (see Eq. (23)). Then ( f or  λn cos ðτnw Þ 40 ðτnw  ϕn Þ  π=2 8l2π θ d  ϕn ¼ ðτnw  ϕn Þ  π=2 8l2π  π f or  λn cos ðτnw Þ o 0 ð34Þ Pick sgnðλn Þ ¼ þ sgnðτnw Þ when τnw  ϕn ¼ þ π=2, and sgnðλn Þ ¼  sgnðτnw Þ when τnw  ϕn ¼ π=2. In either case θd ϕn ¼  π þl2π. Therefore, even if cos ðϕn  τn Þ ¼ 0, the departure angle is still radially inward provided one picks the sign of λn according to sgnðλn Þ ¼ sgnð cos ðτnw ÞÞsgnðϕn  τnw Þ

ð35Þ

If τnw  ϕn ¼ þπ=2 and cos ðτn Þ 4 0, then λn 4 0, and the same is true with  π=2 and o 0. And with þπ=2 and o0, or  π=2 and 4 0 one uses λn o 0. 5.5. Departure angle contribution from pn1;n2 , real root case This section establishes Eq. (23) for the case that the poles pn1;n2 are real so that pffiffiffiffi pn1;n 2 ¼  ð1=2Þðan  2Þ cos ϕn 7 ð1=2Þ Γ Γ ¼ ðan  2Þ2 cos 2 ϕn  4ð1  an Þ

ð36Þ

h pffiffiffiffii P n pn1;p2 ¼ ð1=2Þan cos ϕn 8 ð1=2Þ Γ þ i sin ϕn

ð37Þ

θnp1 ¼ ∡ðP n  pn1 Þ;

ð38Þ

θnp2 ¼ ∡ðP n pn2 Þ;

Instead of wn1 and wn2 , denote the magnitudes of the complex number in Eq. (37) by r n1 and r n2 which are given by n pffiffiffiffio1=2 r n1;n2 ¼ ð1=2Þ a2n cos 2 ϕn þ Γ þ 4 sin 2 ϕn 8 2an cos ϕn Γ ð39Þ ðða2n =2Þ 

The first three terms in the curly brackets equal an Þ cos 2 ϕn þ an including the (1/2) factor. Then r n1 r n2 ¼ an sin ϕn , making use of the fact that the product is a magnitude and that sin ϕn is always positive. Then we write Eq. (28) for the argument ðθnp1 þ θnp2 Þ and in place of Eq. (26) we have pffiffiffiffi  sin θnp1 ¼ sin ϕn =r n1 cos θnp1 ¼ ð1=2Þ an cos ϕn  Γ =r n1 sin θnp2 ¼ sin ϕn =r n2

h pffiffiffiffii cos θnz2 ¼ ð1=2Þ an cos ϕn þ Γ =r n2

ð40Þ producing the following result that establishes Eq. (23) sin ðθnp1 þ θnp2 Þ ¼ cos ϕn cos ðθnp1 þθnp2 Þ ¼  sin ϕn

ð41Þ

5.6. Departure angle contribution from pn1;n 2 , complex root case When the square rootpin (36) is the root of a pffiffiffiffi ffiffiffiffi Eq. negative number, replace Γ ¼ i Ω where Ω ¼  Γ, and then Eq. (37) becomes h pffiffiffiffii P n pn1;p2 ¼ ð1=2Þan cos ϕn þ i sin ϕn 8 ð1=2Þ Ω ð42Þ

267

The magnitudes of these two complex numbers are given by n pffiffiffiffio1=2 sn1;n2 ¼ ð1=4Þa2n cos 2 ϕn þ sin 2 ϕn þ Ω=4 8 2 sin ϕn Ω ð43Þ The first three terms are equal to ð2 an Þ sin 2 ϕn , and the product of the magnitudes matches the product obtained in the real root case. In the ðθnp1 þ θnp2 Þ version of Eq. (28) we substitute pffiffiffiffi sin θnp1 ¼ ½ sin ϕn  Ω=2=sn1 cos θnp1 ¼ ð1=2Þan cos ϕn =sn1 pffiffiffiffi sin θnp2 ¼ ½ sin ϕn þ Ω=2=sn2 cos θnz2 ¼ ð1=2Þan cos ϕn =sn2

ð44Þ

and after simplifying, obtain Eq. (41), which establishes the desired result. 5.7. Single addressed frequency result The previous subsections have proved the following asymptotic stability theorem. Theorem 1. Given a MBFRC system as in Fig. 2: (I) With or without the feedforward loop. (II) Which addresses a single frequency ϕn . This frequency can be DC. (III) The system transfer function GðzÞ is known to be asymptotically stable. (IV) The phase response τn is known at the addressed frequency, where Gðeiϕn Þ ¼ r n eiτn (with r n 4 0 by definition). (V) And sgnðλn Þ is chosen according to Eq. (35) as needed. (VI) Then the MBFRC system has the following properties: (1) For any an A ð0; 2Þ and any r n 4 0, the system is asymptotically stable for all sufficiently small gains Φ 40. (2) If the disturbance wðkÞ and the desired output yd ðkÞ are either zero or signals of the addressed frequency, then the output will converge to the desired output as k-1 producing zero tracking error for all sufficiently small gains Φ4 0. Note that the only knowledge needed about the system is that it is linear, time invariant, and asymptotically stable, and one needs the phase change through the system at the addressed frequency. No other knowledge is needed and hence the asymptotic stability result is robust to inaccuracy to all other system properties. The robustness to error in the phase information is given by Eq. (24). Theorem 2. Asymptotic stability and convergence to zero tracking error according to Theorem 1, is also achieved for all phase discrepancies between the phase used to design the MBFRC law, and the real world phase at the addressed frequency, that satisfy 901 o ðτnm  τnw Þ o901

ð45Þ

We comment that this condition should easily be satisfied in applications.

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6. Small gain stability theorem for arbitrary number of addressed frequencies

7. Numerical examples

This section generalizes the results to the case when the MBFRC addresses an arbitrary number of frequencies N and also DC if desired. The characteristic polynomial becomes

by

Consider applying MBFRC to a third order system given

Φ½λ0 T 0 ðzÞ þ λ1 T 1 ðzÞ þ⋯ þλN T N ðzÞGðzÞ ¼  1

ð46Þ

The system GðzÞ is assumed asymptotically stable so it has all poles inside the unit circle. Each T n ðzÞ has four poles, two are inside the unit circle, and two are on the unit circle at P n ¼ eiϕn and P n ¼ e  iϕn . When the left hand side is put over a common denominator, the resulting set of poles is composed of all of the poles of each term. When the gain Φ ¼ 0, all of the roots of the MBFRC system are at these poles. To study the departure angle from any chosen pole on the unit circle P n , one picks a z arbitrarily close to this pole but not at the pole. Then the denominator of T n ðzÞ contains the factor z  P n which must approach zero as Φ approaches zero. All other terms T m ðzÞ, m an will have denominators bounded away from zero. We can rewrite Eq. (46) as Φ½λ0 T 0 ðzÞ þ ⋯ þ λn  1 T n  1 ðzÞGðzÞ þ Φλn T n ðzÞGðzÞ þΦ½λn þ 1 T n þ 1 ðzÞ þ ⋯ þ λN T N ðzÞGðzÞ ¼ 1

ð47Þ

 GðsÞ ¼

  b ω2u 2 2 s þb s þ2ζωu s þ ωu

ð49Þ

where b ¼ 44, ζ ¼ 0:5, and ωu corresponds to 29:5Hz. This system is fed by a zero order hold sampling at 100 Hz. The MBFRC law uses an ¼ 0:7 and λn ¼ 1 for all addressed frequencies which include 10 frequencies, 0 Hz up to 9 Hz in increments of 1 Hz. The overall gain is Φ ¼ 0:01. Fig. 8 shows the sensitivity transfer function T S ðzÞ for the system in Fig. 2 without the feedforward signal, giving the error as a function of frequency according to Eq. (6). We see that it does produce zero error at each of the 10 addressed frequencies. The waterbed effect (Bode integral theorem) produces significant amplification of errors at frequencies between those addressed. The feedforward signal is introduced in Fig. 9, which shows the frequency response of the new sensitivity transfer function T S ðzÞSðzÞ ^ according to Eq. (9). For disturbances WðzÞ the output disturbance location in the existing feedback control loop, there is no influence of the feedforward signal. Using the feedforward signal is only important if one may want to

As Φ approaches zero, the first and third terms on the left of this equation are Φ times bounded functions, and hence they approach zero. As zero is approached only the middle term is able to match the -1 on the right hand side, since it is formed as a product of Φ going to zero times something with z  P n in the denominator, which is also going to zero. Hence, to study the departure angle from P n one only needs to examine the equation Φλn T n ðzÞGðzÞ ¼  1

ð48Þ

Therefore, Theorem 1 derived for the case of addressing only one frequency, also applies to each frequency independently when addressing multiple frequencies. Theorem 3. The MBFRC of Fig. 2 having an arbitrary number of addressed frequencies, including DC if desired, and satisfying the conditions of Theorem 1 for each frequency independently, has the following properties: (1) The MBFRC system is asymptotically stable for all sufficiently small Φ 40, for correctly chosen sgnðλn Þ, for any r n 40, and for any choice of an A ð0; 2Þ, for all n, (2) If the disturbance wðkÞ and the desired output yd ðkÞ are zero or signals composed of linear combinations of the addressed frequencies, then the tracking error will approach zero as k-1 for all sufficiently small Φ4 0. (3) The above two properties are maintained in the presence of modeling errors of the phase change through the system at the addressed frequencies, satisfying Eq. (45) at all n. Convergence is independent of the frequency response at any other frequency, and independent of the magnitude response at each addressed frequency.

Fig. 8. Sensitivity transfer function magnitude response without the feedforward signal.

Fig. 9. Sensitivity transfer function magnitude response with feedforward signal.

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the controllers are added together in Eq. (46). Note that introducing DC into the control law modifies the zero location that was on the positive real axis. Without this happening, the departure angle from the DC pole at þ 1 would be in the wrong direction.

8. Conclusions

Fig. 10. Error vs. time using MBFRC that is turned on at 4 s.

Fig. 11. Root locus plot for MBFRC addressing two frequencies.

Fig. 12. Root locus plot including DC.

apply commands that are not restricted to linear combinations of the addressed frequency, in which case we want the sensitivity transfer function SðzÞ exhibiting the performance of the feedback controller response to commands to be present as in Eq. (9). Fig. 10 shows the performance of the MBFRC when the command is set to zero, and the disturbance wðkÞ is a linear combination of cosines of the 10 addressed frequencies above, with amplitudes 0.7. 0.5, 0.3, 0.4, 0.2, 0.1, 0.1, 0.2, 0.1, 0.15 for the frequencies in increasing order. When the MBFRC controller is turned on at 4 s the tracking error decreases to a small value relatively quickly. Figs. 11,12 give examples of the root locus plots. Fig. 11 considers two addressed frequencies at 30% and 48% Nyquist frequency. The phases of the system frequency response for these two frequencies are  126:38 ̂ and 186:82 ̂ respectively. Fig. 12 shows the change when DC is included. The corresponding gains λn are 0.7 for DC, and 1 and 0.7 for the other frequencies. All projection gains an are 0.7. We observe the radially inward departures from the poles on the unit circle. The poles are all ones that we know from the denominators of the compensators and the system GðzÞ. The zeros however are altered when

Repetitive control aims to produce zero tracking error to periodic commands and to do so in the presence of periodic disturbances. Matched basis function repetitive control was initially based on using the projection algorithm from adaptive control to determine the components of the error on frequencies of interest. To get the repetitive control property of convergence to zero error, it introduced something equivalent to an integral at any frequency of interest. The resulting controller equations are linear with periodic coefficients. In a previous work the authors used frequency raising and the assumption that each addressed period was an integer number of time steps, and obtained a linear time invariant pole/zero transfer function that is equivalent to the periodic coefficient repetitive controller. This paper examines this pole/ zero design. For each frequency addressed, it uses two poles on the unit circle at this frequency which produce the error integral that demands convergence to zero tracking error at this frequency. There are two additional poles inside the unit circle, one zero at the origin, and two additional zeros. Thinking in terms of classical control system design, when there are many unrelated poles on the unit circle, corresponding to many addressed frequencies, it is very hard to find a compensator that will pull these roots on the unit circle stability boundary into the stable region inside the unit circle as the gain is turned up. This circuitous route of the projection algorithm, integration, periodic coefficient equations, and frequency raising succeeds in producing a simple set of pole/zero locations to do this, and they are uncoupled, one frequency at a time. The main result of this paper establishes that this design approach is guaranteed to produce asymptotic stability for all sufficiently small gains. The only information that one needs to know about is the phase change from command to response of the feedback control system being used (and knowledge that it is asymptotically stable). The result does not require any additional knowledge about the system behavior at any other frequency, does not require knowledge of the order, or number and locations of system zeros or poles. And the approach allows one to address an arbitrary number of frequencies that can be totally unrelated, e.g they need not be harmonics. Concerning the accuracy needed for this one piece of information required, the phase change through the system at frequencies of interest, it is shown that asymptotic stability and convergence to zero tracking error is obtained for all phases used in the design, provided they do not differ by more than 7 901 from the true phase. For each frequency one chooses to address, one expects to be able to determine the phase information to within this generous accuracy limit. Thus, the small gain stability and

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the convergence to zero error properties are extremely robust. A secondary result of this paper is to eliminate the assumption needed in stability analysis using either time domain raising or frequency domain raising, that the periods of the frequencies being addressed are all an integer number of time steps. Frequency raising showed us the pole zero design under this assumption. All of the small gain stability results here do not need this assumption. Hence, the approach developed here proves asymptotic stability for situations for which we do not have a proof for the repetitive controllers employing the projection algorithm. One can compare the MBFRC design to more standard repetitive control design methods and to the filtered x-LMS approach. Concerning the latter, MBFRC assumes that you can stay synchronized to the disturbance frequency using for example and index pulse for each revolution of a momentum wheel. And it uses integral action to converge to zero tracking error employing knowledge of the phase change through the system at addressed frequencies. Convergence to zero error is obtained by what might be considered a direct approach. Filteres x-LMS instead requires a disturbance correlated signal. And it adaptively produces a finite impulse response model of the system which produces the needed phase information. It can be analogous to indirect adaptive approaches, and the imperfect FIR model structure could compromise the zero error performance. Both approaches need to use one controller for each frequency to be addressed. We can make a series of comments to compare the approach developed here to other repetitive control design approaches as in Reference [13]. 8.1. The price paid The price paid to obtain all of the good robustness properties listed above, is mainly related to the requirement of sufficiently small gain. Nothing is saying how small the gain has to be to have the stability and robustness properties. Whereas the robustness property is independent of the system being controlled in nearly all aspects except the phase information at addressed frequencies, the gain limit is likely very much system dependent. Limited experience suggests that this gain limit may be influenced adversely when one needs to address many frequencies. If the limit is low, then one is forced to have slow convergence. 8.2. Frequency vs. period MBFRC uses one controller for each frequency addressed, while in typical RC all frequencies with the same period are addressed simultaneously, i.e. DC, the fundamental, and all harmonics up to Nyquist frequency. Depending on how many harmonics are of interest, handling all harmonics at once can be an advantage. 8.3. Cutoff filter One very likely has large error in any model at high frequencies, due to inability to measure response at very

high frequencies, and due to parasitic or residual modes, etc. Since usual RC methods address all harmonics up to Nyquist, it is necessary for stability robustness to high frequency model error, to introduce a zero phase low pass filter. The model independence of the small gain results here indicate that no such filter is needed in MBFRC. 8.4. Controller order The usual RC design methods require a controller order higher than the number of time steps in a period. In addition, the zero-phase cut-off filter can significantly increase the order of the controller. Unless the MBFRC is addressing very many harmonics, its order is likely to be significantly smaller than the RC design. 8.5. Interpolation Usual RC design methods require interpolation of error signals when the frequencies of interest do not have periods that are an integer number of time steps. Interpolation compromises performance. MBFRC uses sine and cosine functions of the frequency of interest to perform the interpolation, and these are precisely the functions that should be used. 8.6. Multiple unrelated periods Usual RC design methods need a special structure to address multiple periods (Refs. [16] through [19] or [13]). This adds complexity to the control law, and also has an adverse influence on robustness to model error (Ref. [20]). The MBFRC approach handles multiple unrelated frequencies as effortlessly as it does frequencies that are harmonics. References [1] B. Widrow, S.C. Stearns, Adaptive Signal Processing, Prentice-Hall, Englewood Cliffs, New Jersey, 1985. [2] S.G. Edwards, B.N. Agrawal, M.Q. Phan, R.W. Longman, Disturbance identification and rejection experiments on an ultra quiet platform, Adv. Astronaut. Sci. 103 (1999) 633–651. [3] H.-J. Chen, B.N. Agrawal, R.W. Longman, M.Q. Phan, S.G. Edwards, Rejection of multiple periodic disturbances using MELMS with disturbance identification, Adv. Astronaut. Sci. 108 (2002) 587–606. [4] E.S. Ahn, R.W. Longman, J.J. Kim, B.N. Agrawal, Evaluation of five control algorithms for addressing CMG induced jitter on a spacecraft testbed, Adv. Astronaut. Sci. (2013). (in press). [5] E.S. Ahn, R.W. Longman, J.J. Kim, B.N. Agrawal, Improving laser communications between formation flying satellites using repetitive control jitter mitigation,” in: Proceedings of the 7th International Workshop on Satellite Constellations and Formation Flying, Lisbon, Portugal, 2013. [6] T. Inoue, M. Nakano, S. Iwai, High accuracy control of a proton synchrotron magnet power supply, in: Proceedings of the 8th World Congress of IFAC, 1981, pp. 216–221. [7] R.H. Middleton, G.C. Goodwin, R.W. Longman, A method for improving the dynamic accuracy of a robot performing a repetitive task, Int. J. Rob. Res. 8 (1989) 67–74. (also); R.H. Middleton, G.C. Goodwin, R.W. Longman, University of Newcastle, Newcastle, Australia, Department of Electrical Engineering Technical Report EE8546, 1985. [8] S. Hara, Y. Yamamoto, Synthesis of repetitive control systems and its applications, in: Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, FL, 1985, pp. 326–327. [9] M. Nakano, S. Hara, Microprocessor-based repetitive control, Microprocess.-Based Contr. Syst. (1986) 279–296.

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