Case studies of filtering techniques in multirate iterative learning control

Control Engineering Practice 26 (2014) 116–124

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Case studies of filtering techniques in multirate iterative learning control Bin Zhang a,n, Yongqiang Ye b, Keliang Zhou c, Danwei Wang d a

Department of Electrical Engineering, University of South Carolina, Columbia SC, USA College of Automation Engineering, Nanhang University, Nanjing, Jiangsu, China c Department of Electrical and Computer Engineering, University of Canterbury, New Zealand d School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore b

art ic l e i nf o

a b s t r a c t

Article history: Received 2 May 2013 Accepted 17 January 2014 Available online 11 February 2014

Iterative learning control (ILC) is a simple and efficient solution to improve tracking accuracy for systems that execute repetitively the same tracing operation. For engineering applications of ILC, the main concern is the monotonic decay of tracking errors, in the sense of infinity norm or peak error, along the trials. Low cost in implementation and robustness in performance are also critical factors. To achieve these important but sometimes contradicting goals, several multirate ILC schemes have been developed, in which different data sampling rates are used for feedback online loop and feedforward ILC offline loop. That is, multirate ILC uses a different (often lower) rate from the sampling rate of a feedback system to update input. Before the input signal is applied to the system for the next trial, it is upsampled to reach the original sampling rate. Since downsampling will cause distortion of frequency spectra, anti-aliasing and anti-imaging filters and signal extension are used together with downsampling and upsampling operations. In this paper, these technologies are integrated with three different multirate ILC schemes, pseudo-downsampled ILC, two-mode ILC, and cyclic pseudo-downsampled ILC, to achieve better performance. A series of experimental results on an industrial robot are presented to demonstrate the efficiency of multirate ILC schemes and compare the performance. The results demonstrate that multirate ILC schemes are able to achieve not only monotonic learning transient, but also much better tracking accuracy than conventional one-step-ahead ILC schemes. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Iterative learning control Multirate processing Anti-aliasing Anti-imaging

1. Introduction Iterative Learning Control (ILC) is an approach to find appropriate control input for systems that execute the same tracking task repeatedly. It aims to force the output of these systems to follow a trajectory yd(t) defined over a finite time duration T. With technology development, tracking accuracy requirements have come to nano- or micro-meter level. Feedback control alone is often not enough to achieve accuracy at this level due to modeling uncertainties and various disturbances. ILC provides a simple and effective feedforward channel to significantly improve the tracking accuracy with low cost. The basic idea of ILC is to update the input through the recorded tracking error in a previous trial, or iteration. ILC is a batch processing process. After the execution of one trial, the input and error signals are recorded in the memory. Before the start of

n

Corresponding author. E-mail addresses: [email protected] (B. Zhang), [email protected] (Y. Ye), [email protected] (K. Zhou), [email protected] (D. Wang). 0967-0661/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2014.01.018

the next iteration, feedforward ILC controller offline updates the input signal. When the next iteration starts, the calculated input signal is applied to the system. The features of batch processing and off-line calculation enable ILC to employ techniques that cannot be used in real time, such as non-causal filtering. The ILC update law has the general recursive form as uj þ 1 ¼ Hðuj ; ej Þ

ð1Þ

where H is the ILC input update function; tracking error is ej ¼ yd  yj with yd and yj being the desired trajectory and actual trajectory of the jth iteration, respectively. The objective is to make ej converge to zero as iterations go to infinity. To describe ej in a trial with a finite time duration or a finite number of sampling points, a certain norm J ej J is used. Therefore, ILC aims to achieve limj-1 J ej J -0. Note that ILC is a two-dimensional problem. On the one hand, the system performs the finite-time tracking command on the time axis. On the other hand, ILC adjusts the input to the system on the iteration axis. Time and iteration index are two independent variables (Elci, Longman, Phan, Juang, & Ugoletti, 2002).

B. Zhang et al. / Control Engineering Practice 26 (2014) 116–124

Generally speaking, there are two configurations of the ILC scheme. The first one is parallel configuration (Bristow, Tharayil, & Alleyne, 2006; de Roover & Bosgra, 2000), in which ILC adjusts the commands to the plant directly. The second one is serial configuration (Longman, 2000), in which ILC adjusts the commands given to the existing closed-loop feedback control system. It has been proved that these two configurations are mathematically equivalent (Solcz & Longman, 1992). Since many commercial products have feedback controllers and it is not desirable to open the feedback control loops, the second configuration is relatively easier for practical implementations. When an iterative learning controller is designed, an important issue that needs to be taken into account is the monotonic convergence of tracking error along the iteration axis (Chang, Longman, & Phan, 1992; Longman, 2000; Wang, 2000). As mentioned earlier, a proper norm is used to describe the tracking error in a trail such that the convergence is considered in the sense of this selected norm. It is well-known that ILC often shows bad transient behavior. That is, the tracking error goes down in the initial iterations but goes up again, usually to a very huge value, before it finally converges to zero. The reason is that many previous analyses are either in the α-norm or λ-norm. The λ-norm for a function f(t) is ‖f ‖λ 9 supt A ½0;T e  λt maxjf ðtÞj with λ being a positive scalar that usually needs to be sufficiently large. The α-norm is defined as J f ðÞ J α ¼ supk A N f ðkÞαk with 0 o α o 1. In the sense of these two norms, the error near the terminal phase of the operations is much less weighted than those at the beginning phase of the operations. Due to this decreasing weighting factor, a huge overshoot of error can appear and indicate a bad convergence performance in the sense of the 1-norm, given by ‖f ‖1 ¼ supk A N f ðkÞ, even with the presence of a mathematical convergence analysis with α-norm or λ-norm. To overcome this bad transient behavior, the convergence should be investigated in the 1-norm and many approaches have been developed (Frueh & Phan, 2000; Jang, Chio, & Ahn, 1995; Kuc, Lee, & Nam, 1992; LeeGlauser, Juang, & Longman, 1996; Park & Bien, 2002; Ye, Wang, Zhang, & Wang, 2009; Zhang, Wang, & Ye, 2009; Zhang, Wang, & Ye, 2010). The explanation in the frequency domain is that bad transient behavior is due to high frequency error components violating the condition of monotonic decay or the error signal contains a component beyond the ILC system's learnable bandwidth (Zhang, Wang, & Ye, 2005). A widely used method to achieve good learning behavior is to introduce a low-pass filter (Chen & Moore, 2001; Zhang, Wang, & Ye, 2009). However, ILC with such a filter will no longer be able to achieve zero tracking since it cuts off high frequency components. If desired performance requires elimination of error components in high frequencies, this method results in poor tracking accuracy. It is desirable, therefore, to develop ILC to guarantee both transient behavior and high tracking accuracy in the form of infinite norm. In Moore, Chen, and Bahl (2005), Moore et al. derived an exponential convergence condition for P-type ILC and they used time-varying gain to make the condition hold. For most systems in use, the limitation is that the feedback controller is encapsulated and the condition from Moore et al. (2005) often cannot be satisfied. Redesign feedback controller to satisfy the condition is inconvenient (Moore, Chen, & Bahl, 2002). Alternatively, a simple solution to make the condition in Moore et al. (2005) hold is to reduce the sampling rate. Since it is not easy to change sampling frequency for most physical systems, multirate ILC schemes are developed in which the sampled data are processed at different rates. This will bring the design of ILC into the multirate signal processing domain (Zhang, Wang, Wang, Ye, & Zhou, 2008; Zhang, Wang, Ye, Wang, & Zhou, 2007, 2008; Zhang, Wang, Ye, Zhou, & Wang, 2010). Some other approaches using the

117

multirate concept include optimal ILC for multirate physical systems and multirate model inverse (Oomen, Wijdeven, & Bosgra, 2009; Shiraishi & Fujimoto, 2010). Another advantage of multirate ILC is that it is able to deal with initial state error properly. The original definition of ILC problem requires the same initial state of each iteration (Longman, 2000). This makes analysis simple and makes zero-error tracking possible. However, this assumption may not hold for real systems because the same initial state sometimes cannot be guaranteed in practice. A research on continuous D-type ILC shows that initial state error can make the learning process unstable (Lee & Bien, 1991). Some methods are proposed to achieve good learning behavior with the presence of initial state error (Chen, Wen, Gong, & Sun, 1999; Chen, Wen, Xun, & Sun, 1996; Hillenbrand & Pandit, 2000; Sun & Wang, 2002; Wang, 2000). It is worth noting that multirate control itself is not new and its capabilities and limitations have been well-studied (Moore, Bhattacharyya, & Dahleh, 1993). One limitation is the degradation in the intersample behavior. This is also true for multirate ILC and we will design novel ILC schemes to overcome this limitation. Based on the work in Moore et al. (2005), several multirate ILC schemes are developed and successfully applied to an industrial robot system (Zhang, Wang, Wang, et al., 2008; Zhang et al., 2007; Zhang, Wang, Ye, Zhou, & Wang, 2009; Zhang, Wang, Ye, et al., 2008; Zhang, Wang, Ye, Zhou, et al., 2010). In these schemes, the downsampling and upsampling cause distortion in signal frequency spectra and deteriorate the learning performance. To solve this problem, some considerations in data processing including signal extension, anti-aliasing, and anti-imaging are investigated in this paper and integrated with these schemes. The remainder of the paper is organized as follows: Section 2 discusses the idea of multirate ILC with its necessary signal processing techniques. Section 3 enhances several multirate ILC schemes by applying the techniques in Section 2. Each of them is followed by another one having better tracking performance. Section 4 presents a series of experimental results of the proposed multirate ILC schemes and their performances are compared, which is followed by concluding remarks in Section 5.

2. Downsampled learning The downsampled learning can deal with both single input single output (SISO) and multiple-input multiple-output (MIMO) systems. For simplicity, consider a discrete-time linear single input single output (SISO) system ( xf ;j ðk þ 1Þ ¼ Af xf ;j ðkÞ þ Bf uf ;j ðkÞ þ wf ;j ðkÞ ð2Þ yf ;j ðkÞ ¼ C f xf ;j ðkÞ þ vf ;j ðkÞ with a one-step-ahead learning law in serial configuration of ILC: ( uf ;j ðkÞ ¼ yd ðkÞ þ uL;f ;j ðkÞ ð3Þ uL;f ;j þ 1 ðkÞ ¼ uL;f ;j ðkÞ þ Γ ef ;j ðk þ 1Þ where k A ½0; p  1, p is the number of total sampling points of a given trajectory to be followed, state xf ;j is a n dimensional vector, input uf ;j and output yf ;j are both scalars, subscript j is the iteration index, f denotes the feedback system sampling rate, and wf ;j and vf ;j are the repeated state disturbances and output disturbances, respectively. The error is ef ;j ðkÞ ¼ yd ðkÞ  yf ;j ðkÞ with yd as the desired trajectory. Γ is the learning gain. It is worth noting that for other ILC schemes, the downsampled learning scheme is also applicable. With the assumption of same initial state for all trials, we have ef ;j þ 1 ¼ Qef ;j

ð4Þ

118

B. Zhang et al. / Control Engineering Practice 26 (2014) 116–124

where ef ;j ¼ ½ef ;j ð1Þ; ef ;j ð2Þ; …; ef ;j ðpÞT 2 0 1  Γ C f Bf 6  ΓC A B 1  Γ C f Bf 6 f f f Q ¼6 6 ⋮ ⋮ 4  Γ C f Apf  1 Bf  Γ C f Apf  2 Bf

and

us(0)

⋯ ⋯

0 0

⋱

⋮

⋯

1  Γ C f Bf

3 7 7 7 7 5

us(2)

us(1)

uf(0)

uf(1)

uf(2)

uf(3)

uf(4)

uf(5)

uf(6)

uf(7)

uf(8)

uf(9)

k=0

k=1

k=2

k=3

k=4

k=5

k=6

k=7

k=8

k=9

For monotonic decay of the 1-norm of tracking error along the trial axis, the following condition must be satisfied:

K=0

K=1

K= 2

K= 3

‖ef ;j þ 1 ‖1 r J Q J 1 J ef ;j J 1

us(0)

us(1)

us(2)

us(3)

ð5Þ

or in other words,

Fig. 2. Illustration of downsampling.

J Q J 1 r1

ð6Þ

The condition for monotonic decay of J ef ;j J 1 can be derived from (6) as (Moore, 2001) p1

jC f Bf jZ ∑ jC f Aif Bf j

ð7Þ

i¼1

if ð1  Γ C f Bf Þ 4 0 and j1  Γ C f Bf jo 1, which indicate 1=C f Bf o Γ o 2=C f Bf for non-minimum phase systems. Similarly, condition (6) for monotonic decay of J ef ;j J 1 becomes (Moore, 2001) p1   C f Bf  o 2  ∑ jC f Ai Bf j f

Γ

time index

i¼1

ð8Þ

if ð1  Γ C f Bf Þ o0 and j1  Γ C f Bf j o 1, which indicate 0 o Γ o 1=C f Bf for non-minimum phase systems. Consider non-minimum phase systems, condition (7) is only related to system Markov parameters and if it does not hold, monotonic decay of tracking error cannot be guaranteed. For Condition (8), gain Γ can be adjusted to make it hold. However, Γ needs to be in a very small range to make (8) and 1=C f Bf o Γ o 2=C f Bf hold at the same time. Note that there is a hidden freedom – sampling rate – that can be used to make these two conditions easier to satisfy. For a continuous-time system Ac, its zero order holds equivalent with a sampling period of T is Hillenbrand and Pandit (2000): A ¼ eAc T . If Ac is stable, all eigenvalues of Ac are located in the left half plane. Then, all eigenvalues of eAc T are inside the unite circle (Hillenbrand & Pandit, 2000). If the sampling rate is reduced and the sampling period T-1, then limT-1 A-0. This makes condition (7) easier to satisfy. As for condition (8), with a reduced sampling rate, a large learning gain Γ can be used to make the condition easier to satisfy. This also helps to achieve a fast convergence speed. With this in mind, suppose the system with a sampling period T (feedback rate) cannot make the monotonic decay condition (7) or (8) hold, the sampling period can be increased to mT (ILC rate). The multirate ILC can be illustrated in Fig. 1, in which ½s;j and ½f ;j denote signal at ILC rate and feedback rate from the jth trial, respectively. Theoretically, m can be any real number larger than 1 in order to reduce the sampling rate. If a rational value is used, say m ¼ F 1 =F 2 , the signal is upsampled by a factor of F1 and followed

with dowsampled by a factor of F2. The analysis remains the same. In this paper, m is selected as an integer to simplify signal processing. This downsampling process of the error signal can be illustrated in Fig. 2, in which m¼ 3 is used as an example. The sampling index at the feedback rate is indicated by k, k ¼ 0; 1; …; p  1, while those sampling points marked with solid points are denoted by downsampling points, and the sampling index at the ILC rate is indicated by K, K ¼ 0; 1; …; ps  1. With a sampling period of mT, the matrices in system (2) m1 become As ¼ Am Bf þ ⋯ þ Af Bf þ Bf Þ, Cs ¼Cf, and the f , Bs ¼ ðAf number of sampling points of the trajectory becomes ps. Here, subscript s indicates the ILC rate and ps is given by   p 1 þ1 ð9Þ ps ¼ int m where intðÞ gets the integer part of a number. If trajectory length p is not a multiple of m, some sampling points are added by repeating e(p). The error and input signals are downsampled and denoted as es;j and us;j , respectively. Since ILC algorithm is carried out on signals es;j and us;j , the monotonic tracking error decay conditions (7) and (8) for nonminimum phase systems become 8 ps  1 1 2 > > > jC s Bs jZ ∑ jC s Ais Bs j if oΓ o > < C s Bs C s Bs i¼1 ð10Þ > 2 ps  1 1 > i > > : jC s Bs jo Γ  ∑ jC s As Bs j if 0 o Γ o C s Bs i¼1 When a signal is downsampled, the Nyquist criterion must still be satisfied with respect to the new sampling period to avoid aliasing. Downsampling by m causes aliasing of any frequencies in the original signal above jωj 4 π =m and this makes the information of the signal indistinguishable from its image. To prevent this aliasing, the signal is usually filtered by a low-pass anti-aliasing filter, denoted as F a;a , with appropriate cutoff frequency to reduce the bandwidth of the signal. Since a practical low-pass filter does not have perfect cutoff frequency, the cutoff frequency should be set below the theoretical cutoff frequency π =m. With this consideration, the downsampled error signal can be written as ( e~ f ;j ¼ F a;a ðef ;j Þ ð11Þ F down : es;j ðK þ 1Þ ¼ e~ f ;j ðmK þ 1Þ: Then, the ILC input update is F ilc : ΔuL;s;j ðKÞ ¼ Γ es;j ðK þ 1Þ

ð12Þ

Before the input update signal ΔuL;s;j is used to adjust the ILC input for the next trial, it is recovered to a signal with sampling period of T at feedback rate by upsampling, which is a process of increasing the sampling rate of a signal. When a signal is upsampled, the first step is to use a zero-order-hold to hold the values between every two samples. That is Fig. 1. Multirate ILC scheme.

F zoh : Δu~ L;f ;j ðmK þ iÞ ¼ uL;s;j ðKÞ;

i ¼ 0; …; m  1:

ð13Þ

B. Zhang et al. / Control Engineering Practice 26 (2014) 116–124

119

The upsampled signal needs to be filtered with a low-pass antiimaging filter with a theoretical cutoff frequency of π =m, denoted as F a;i , to prevent the distortion of frequency spectra.

ΔuL;f ;j ¼ F a;i ðΔu~ L;f ;j Þ

ð14Þ

The filtering operation is influenced by the initial condition. As Longman (2000) pointed out, the filter response can be considered steady state after roughly one settling time. For this reason when anti-aliasing and anti-imaging filters are implemented, the signal is extended to sufficiently remove the influence of the initial condition. After the filtering operation, the extended signal is truncated to recover its original signal length. A thorough study of different signal extension approaches (Plotnik & Longman, 1999; Zhang, Wang, Wang, et al., 2008) shows that the signal extended by repeating the end points of the signal has the best performance. The signal extension with this approach can be expressed, if n data points are extended on both ends, as 8 > < eext ð1 : nÞ ¼ eð1Þ F ext : eext ð1 : p þ 2nÞ ¼ eext ðn þ 1 : n þ pÞ ¼ eð1 : pÞ ð15Þ > : e ðn þ p þ1 : n þ p þnÞ ¼ eðpÞ ext and signal truncation can be expressed as F tru : etru ð1 : pÞ ¼ eðn þ1 : n þ pÞ:

ð16Þ

Then, the signal flow in the multirate ILC is illustrated in Fig. 3. For description simplicity, the preprocessing and post-processing of data in Fig. 3 and next section are denoted, unless otherwise defined, as F preproc : es;j ¼ F down F tru F a;a F ext ðef ;j Þ F postproc : ΔuL;f ;j ¼ F tru F a;i F ext F zoh ðΔuL;s;j Þ;

ð17Þ

respectively.

3. Multirate ILC 3.1. Pseudo-downsampled ILC In the pseudo-downsampled ILC, the input updated is carried out on the downsampled signals directly. The pseudo-downsampled ILC is illustrated in Fig. 4. With ps being trajectory length with respect to ILC rate, the ILC update law in this scheme is given as follows (Zhang, Wang, Ye, et al., 2008): 8 uf ;j ðkÞ ¼ yf ;d ðkÞ þ uL;f ;j ðkÞ > > > > > uL;f ;j ðkÞ ¼ uL;f ;j  1 ðkÞ þ ΔuL;f ;j  1 ðkÞ > <   ΔuL;f ;j  1 ¼ F postproc ΔuL;s;j  1 ð18Þ > > > Δ u ðKÞ ¼ Γ e ðK þ 1Þ; L;s;j  1 s;j  1 > > > :e ¼F ðe Þ s;j  1

preproc

f ;j  1

Fig. 4. Pseudo-downsampled ILC.

Fig. 5. The two-mode ILC scheme.

3.2. Two-mode ILC In the two-mode ILC scheme, the error signal is decomposed with reference to low and high frequency bands. Then, two different learning laws are used for the decomposed error signals. On the low frequency, the input updated is carried out with the original sampling frequency while on the high frequency, the input update is carried out with the downsampled frequency. The scheme is shown in Fig. 5 in which F is a low pass filter used to decompose error signals. The update law is summarized as follows (Zhang et al., 2007): 8 uf ;j ðkÞ ¼ yf ;d ðkÞ þ uL;f ;j ðkÞ; > > > > > l h > u > > L;f ;j ðkÞ ¼ uL;f ;j  1 ðkÞ þ ΔuL;f ;j  1 ðkÞ þ ΔuL;f ;j  1 ðkÞ; > > > > > ΔuhL;f ;j  1 ¼ F postproc ðΔuhL;s;j  1 Þ; > > > > > < ΔuhL;s;j  1 ðKÞ ¼ Γ h ehs;j  1 ðK þ 1Þ; ehs;j  1 ¼ F preproc ðehf;j  1 Þ; > > > > > > Δul ðkÞ ¼ Γ l el > > L;f ;j f ;j  1 ðk þ 1Þ; > > > h > > e ¼ ef ;j  1  elf ;j  1 ; > > > f ;j  1 > > : elf ;j  1 ¼ Fðef ;j  1 Þ;

ð19Þ

where Γl and Γh are the learning gains on the low and high frequency bands, respectively.

3.3. Cyclic pseudo-downsampled ILC The cyclic pseudo-downsampled ILC aims to compensate for the information loss caused by downsampling. For this reason, the downsampling point and input update are shifted along an iteration axis. The scheme is illustrated in Fig. 6. The ILC update law can be written as (Zhang, Wang, Ye, Zhou, et al., 2010) 8 uf ;j ðkÞ ¼ yf ;d ðkÞ þ uL;f ;j ðkÞ; > > > > > ðkÞ ¼ uL;f ;j  1 ðkÞ þ ΔuL;f ;j  1 ðkÞ; u > < L;f ;j ΔuL;f ;j  1 ¼ F tru F a;i F ext F shift_zoh ðΔuL;s;j  1 Þ; > > > > ΔuL;s;j  1 ðKÞ ¼ Γ es;j  1 ðK þ 1Þ; > > :e ¼F F F F ðe Þ s;j  1

shift_down tru a;a ext

ð20Þ

f ;j  1

in which the downsampling process is ( F shift_down : Fig. 3. The signal flow in the multirate ILC.

es;j ð1Þ ¼ ef ;j ð1Þ es;j ðis Þ ¼ ef ;j ðr þ mðis  2Þ þ 1Þ;

is ¼ 2; …; ps ;

ð21Þ

Markov parameters

B. Zhang et al. / Control Engineering Practice 26 (2014) 116–124

Fig. 6. Downsampling shift over iteration axis.

0.1 0.05 0 −0.05 20

40

60

80

100

120

140

160

180

200

sampling points

ð22Þ where   8 j > > ; rem > < m r¼ > > > : m;

0.15

0

and the zero-order-hold in the upsampling is 8 > < ΔuL;f ;j ði1 Þ ¼ ΔuL;s;j ð0Þ; F shift_zoh : ΔuL;f ;j ðr þ mði3  1Þ þ i2 Þ ¼ ΔuL;s;j ði3 Þ; > : i ¼ 0; 1; …; r  1; i ¼ 0; …; m  1; i ¼ 1; …; p  1 1 2 3 s



 j a 0; m   j if rem ¼ 0; remðÞ gets residual m

if rem

Markov parameters

120

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 0

5

10

15

ð23Þ

20

25

30

35

40

samplign points Fig. 7. Markov parameters of different sampling rates.

and

  pr ps ¼ int 1 þ þ1 ; m

ð24Þ

intðÞ gets integer

In this law, F shift_down and F shift_zoh shift the downsampling points along the iteration axis to make sure that the input at every sampling point at the feedback rate can be updated.

4. Experiments Experiments are carried out on an industrial SCARA robot, SEIKO TT3000. One joint moving in the horizontal plane is used to test our proposed method. The nominal model of the closed-loop joint is given as follows: Gp ðsÞ ¼

948 s2 þ 42s þ948

ð25Þ

The desired trajectory is given as follows: 51

yd ðtÞ ¼ ∑ an j1  cos ωn tj þ 0:15½1  cos ð16π tÞ þ 0:05½1  cos ð20π tÞ n¼1

ð26Þ . where t A ½0; 10 s, ωn are ½0:1π ; 2π ; 4π ; …; 100π , and an ¼ 80e Here, frequency components at 8 Hz and 10 Hz are introduced. The system has a sampling frequency of 100 Hz. Since the conventional one-step-ahead ILC scheme with learning gain 0.5 has learnable bandwidth of 3 Hz, it cannot deal with these two frequency components.  ωn t

4.1. Multirate ILC design 4.1.1. Learning gain A high learning gain, although it can generate a fast convergence speed, may degrade the tracking performance in steady state response in the sense that random noise going through the learning law will be amplified (Longman, 2000). Hence, a low value learning gain is suggested. The learning gain Γ should make 0 o Γ o1=C s Bs hold. For system (25), when the sampling ratio changes from 1 to 10, 1=C s Bs for these different sampling ratios are larger than 1. Therefore, Γ is selected conservatively as 0.5. For two-mode ILC, both Γl and Γh are also selected as 0.5. 4.1.2. Sampling ratio m With T ¼0.01 s, the discretized system has Markov parameters as shown in the upper subfigure of Fig. 7. To make it clear, only the

first 200 parameters are shown. It is clear that (10) is not satisfied. Increase the sampling period and when it becomes 0.05 s, Markov parameters are shown in the lower subfigure of Fig. 7. Under this sampling period of 0.05 s, (10) is satisfied by the first Markov parameter being 0.5548 while the sum of all remaining Markov parameters' absolute value is 0.5523. Note that the number of sampling points is reduced by a ratio of 5. Therefore, m is chosen as 5. With Γ ¼ 0:5 and m¼ 5, all the stability conditions for the three multirate ILC schemes are satisfied (Zhang et al., 2007; Zhang, Wang, Ye, et al., 2008; Zhang et al., 2009). 4.1.3. Filter design As mentioned earlier, anti-aliasing and anti-imaging low-pass filters need to be applied to prevent the distortion of frequency spectra. Suppose the signal is downsampled by a ratio m, the bandwidth should be band limited by π =m. With Nyquist frequency 50 Hz and m ¼5, the cutoff frequency fc is given as 10 Hz. A comparison study shows that a zero-phase finite-duration impulse response (FIR) window filter is a good choice (Zhang, Wang, Wang, et al., 2008). Window FIR filter has the advantage of easy implementation and low computation burden. For a filter with a given cutoff frequency, its impulse response sequence can be easily obtained from its frequency response. The generated impulse response sequence is not implementable because it is infinite. To solve this problem, a Hamming window is employed to truncate the infinite impulse response. After being truncated by this window, the generated impulse response sequence becomes the weighting factor of each sampling point. The filter in this case has a length of 2q þ 1 and has the form of q

q

i¼1

i¼1

Q ðzÞ ¼ ∑ αi z  i þ a0 þ ∑ αi zi

4.2. Experimental results Some experimental results are presented to demonstrate the efficiency of multirate ILC schemes. Since other multirate ILC schemes in literature (Oomen et al., 2009; Shiraishi & Fujimoto, 2010) are not designed for similar systems, it is hard to conduct a fair comparison. Therefore, the results are compared against that of a conventional one-step-ahead ILC with a lowpass filter (the cutoff frequency of the filter is given as 3 Hz as this is the learnable

B. Zhang et al. / Control Engineering Practice 26 (2014) 116–124

0.1

0.1

Conventional ILC

Pseudo−downsampling without Fa,a and Fa,i 0.08 0.06

0.04

0.08 0.06

Pseudo−downsampling

0

50

100

150

200

300

250

Error (deg) at 300th iteration

cycle index Error: Pseudo−downsampled ILC (black) vs ILC without Fa,a and Fa,i (blue)

0.2

RMS error (deg)

RMS error (deg)

RMS error comparison

121

Pseudo−downsampling without Fa,a and Fa,i

Pseudo−downsampling

0.04 Two−mode

0.02

0.01

0.1 0

Cyclic pseudo−downsampling 0.005

−0.1 −0.2

0 0

Pseudo−downsampling 4 6

2

50

100

150

10

8

200

250

300

cycle index Fig. 9. Comparison of RMS error.

time (second) 0 Pseudo−downsampling without Fa,a and Fa,i −20

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bandwidth Zhang et al. (2005) of a one-step-ahead ILC with learning gain of 0.5.). The first experiment uses pseudo-downsampled ILC to demonstrate the necessity of the use of anti-aliasing and antiimaging filters. The second experiment is designed to show the capability of multirate ILC to improve the tracking accuracy by suppressing high frequency error components. The third one shows that multirate ILC schemes can deal with initial position offset properly. Experiment 1. Effect of anti-aliasing and anti-imaging filters. Previous results in Zhang et al. (2007), Zhang, Wang, Ye, et al. (2008), Zhang et al. (2009) and Zhang, Wang, Wang, et al. (2008) show the improvement of tracking accuracy via multirate ILC design. In those schemes, anti-aliasing filter F a;a and anti-imaging filter F a;i are not used for implementation simplicity. In this experiment, the pseudo-downsampled ILC schemes are implemented in two versions: one is without these two filters and another one is with them. The results are illustrated in Fig. 8. In this figure, the top subfigure shows root mean square (RMS) error. The RMS errors of pseudo-downsampled ILC without and with F a;a and F a;i are 0.06931 and 0.04651, respectively, after learning gets into steady state. The improvement of tracking accuracy is about 33%. The mid subfigure shows the tracking error at the 300th iteration. The peak error of pseudo-downsampled ILC without and with F a;a and F a;i is 0.17051 and 0.08751, respectively. The reduction of peak error is about 50%. The bottom subfigure shows the power spectra of the tracking error. Compared to error components from pseudo-downsampled ILC, it is clear that the introduction of antialiasing filter F a;a and anti-imaging filter F a;i partially suppresses the error components in [6, 13] Hz. The error components caused by downsampling and upsampling operations in [26, 33] Hz and [45, 50] Hz are completely suppressed. These results show that well-designed anti-aliasing and anti-imaging filters are critical to high performance tracking in multirate ILC. Experiment 2. Trajectory without initial state error. Fig. 9 shows the convergence of RMS of tracking error along the iteration axis. It is clear that all of them can achieve monotonic decay of tracking error. However, multirate ILC schemes show

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substantial improvement of tracking accuracy. The RMS error of conventional one-step-ahead ILC reaches a value of 0.12551 after learning gets into steady state. As a comparison, the RMS error of pseudo-downsampled ILC, two-mode ILC, and cyclic pseudodownsampled is 0.04651, 0.04051, and 0.00891, respectively. The tracking accuracy of pseudo-downsampled ILC and two-mode ILC is very comparable while the two-mode ILC shows slightly better performance. The reason is that two-mode ILC only slightly reduces the tracking error components below 3 Hz. Therefore, the improvement is limited. Cyclic pseudo-downsampled ILC enjoys the highest tracking accuracy because it handles all the error components missed by pseudo-downsampling ILC. Compared with conventional one-step-ahead ILC, the improvement of tracking accuracy of pseudo-downsampled ILC, two-mode ILC, and cyclic pseudo-downsampled ILC is about 63%, 68%, and 93%. The improvement of tracking accuracy is significant. The tracking error signals are shown in Fig. 10. The error signal of conventional one-step-ahead ILC contains rich low frequency error components. Although pseudo-downsampled ILC and two-mode ILC reduce the tracking error substantially, the tracking errors of them are still large. This is caused by the loss of error component in downsampling processing. The peak errors of the four ILC schemes

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Experiment 3. Trajectory with bounded random initial position offset. Although ILC requires the same initial position for each trial, the initial position for real systems is often random values within a bound. In this experiment, these multirate ILC schemes will be investigated under such kind of initial position offset. The desired trajectory has the same shape but begins from 0.81. A random but bounded initial position offset is obtained by using the command 0:8  ð1 7 randÞ before the operation of each cycle, where rand generates a random value between [0,1]. Therefore, the bound of the initial position offset is ½  0:81; 0:81. Note that this initial position offset causes tracking error divergence of the conventional one-stepahead ILC with cutoff frequency of 3 Hz. To maintain monotonic decay tracking error for conventional one-step-ahead ILC, its cutoff frequency is adjusted to 2 Hz. Fig. 12 shows the RMS errors. Because of the random initial offset, the RMS curves stabilize in a range after learning gets into steady state. For conventional one-step-ahead ILC, RMS error is in a range of [0.1381, 0.1491]. The RMS errors of pseudo-downsampled ILC, twomode ILC, and cyclic pseudo-downsampled ILC show much improvement and stay in range of ½0:05271; 0:06711, ½0:04211; 0:07131, and ½0:02151; 0:05221, respectively. The average of RMS errors of these four ILC schemes in the last 50 iterations is 0.14461, 0.0611, 0.05661, 0.03941, respectively, after learning gets into steady state. In general, the cyclic pseudo-downsampled ILC has the best performance, two-mode ILC has a comparable but slightly better performance than pseudo-downsampled ILC. Compared to conventional one-stepahead ILC in terms of averaged RMS error, pseudo-downsampled ILC, two-mode ILC, and cyclic pseudo-downsampled ILC have about 58%, 61%, and 73% performance improvement, respectively. Fig. 13 illustrates the position errors at the 300th iteration. Similar to those results in the 1st experiment, the tracking error of

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are 0.31951, 0.08751, 0.08081, and 0.04771, respectively. Cyclic pseudo-downsampled ILC enjoys a very high level of tracking accuracy. This shows that cyclic pseudo-downsampled ILC is able to suppress high frequency error components. The results of power spectra of tracking error are illustrated in Fig. 11. The power spectra of tracking error from conventional one-step-ahead ILC show that main error components are below 13 Hz. Since the conventional one-step-ahead ILC has a cutoff frequency of 3 Hz, the error components in 3–13 Hz are left unlearned and result in a poor tracking performance. With pseudo-downsampled ILC and two-mode ILC, error components at 2–6 Hz are well suppressed and those in 7–10 Hz are partially suppressed. However, error components in [11, 13] Hz increase a little. The cyclic pseudo-downsampled ILC suppresses almost all error components and, therefore, has the best performance.

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conventional one-step-ahead ILC is large. Although the tracking errors of pseudo-downsampled ILC and two-mode ILC have much improvement, the tracking error is still large. Since these results come with random initial position offsets, the peak errors are compared after 0.1 s of the start of the trial. The peak errors are 0.23781 for conventional one-step-ahead ILC, 0.08951 for pseudodownsampled ILC, 0.08731 for two-mode ILC, and 0.02471 for cyclic pseudo-downsampled ILC. By comparison, it can be seen that the cyclic pseudo-downsampled ILC has nearly perfect tracking performance and the tracking error is close to zero. Note that, because of random initial position offset, the tracking error in the beginning of the trial is large as shown in Fig. 14. This large initial position offset does not influence the tracking performance as it is clear that tracking error for cyclic pseudo-downsampled ILC converges quickly. The power spectra of tracking error at the 300th iteration are illustrated in Fig. 15. For the cyclic pseudo-downsampled method, the error components are very small in the entire frequency range. The conventional one-step-ahead ILC has large error components on [3, 13] Hz while the pseudo-downsampled ILC and two-model ILC have large error components on [7, 14] Hz.

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results show that, with a proper design, these multirate ILC schemes can guarantee the good transient behavior and achieve a much better tracking performance. In general, pseudo-downsampled ILC is the simplest with very good tracking accuracy, two-mode ILC enhances pseudo-downsampled ILC a little, and cyclic pseudo-downsampled ILC has the best performance.

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References

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Note: Although cyclic pseudo-downsampled ILC has the best performance, it requires more conditions to guarantee good transient behavior. The pseudo-downsampled ILC is the simplest one with very good tracking performance. Two-mode ILC can further enhance the tracking performance, especially in the low frequency band below the learnable bandwidth. The convergence condition is the same with that of pseudo-downsampled ILC. Therefore, for real applications, pseudo-downsampled ILC should be the first choice. Two-mode ILC can be selected if more error components in low frequency band should be suppressed. Cyclic pseudo-downsampled ILC should be used when pseudo-downsampled ILC and two-mode ILC cannot reach the required tracking accuracy.

5. Conclusion In this paper, the ILC problem is considered in the multirate signal processing domain. Three different multirate ILC schemes are developed in which signal extension, anti-aliasing, and antiimaging are considered. The features and performances of these different multirate ILC schemes are studied and compared with a series of experimental results on an industrial robot system. The

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