چرخه کنترل یادگیری تکراری شبه downsampled برای ردیابی عملکرد بالا

In this paper, a multirate cyclic pseudo-downsampled iterative learning control (ILC) scheme is proposed. The scheme has the ability to produce a good learning transient for trajectories with high frequency components with/without initial state errors. The proposed scheme downsamples the feedback error and input signals every mm samples to arrive at slower rate. Then, the downsampled slow rate signals are applied to an ILC algorithm, whose output is then interpolated and applied to an actuator. The main feature of the proposed scheme is that, for two successive iterations, the signal is downsampled with the same mm but the downsampling points are time shifted along the time axis. This shifting process makes the ILC scheme cyclic along the iteration axis with a period of mm cycles. Experimental results show significant improvement in tracking accuracy. Additional advantages are that the proposed scheme does not need a filter design and also reduces the computation and memory size substantially.

Currently, tracking accuracy requirements in many areas have come down to the nano- or micro-meter level. Due to modeling uncertainties and disturbances, feedback control design alone is certainly not enough. Iterative learning control (ILC), which was motivated by the growth of robots performing the same task repeatedly in the mid-eighties (Arimoto, Kawamura, & Miyazaki, 1984; Middleton, Goodwin, & Longman, 1989), becomes a simple and efficient solution to either improve tracking accuracy or remove the noise/disturbance. Though different from feedback control, ILC provides a feedforward control to the system. ILC improves the tracking performance by updating the input to the system based on the tracking error in previous iterations and, therefore, is suitable for most industrial systems that are repetitive in nature. However, a limitation of ILC is that the learning transient, or the decay of tracking error along the iteration axis, is often not monotonic. In the original work of Arimoto et al. (1984), the convergence of ILC is proven in the sense of the λλ-norm. The definition of the λλ-norm for a function f:[0,T]→Rnf:[0,T]→Rn is given by ∥f∥λ≜maxt∈[0,T]e-λt∥f∥∞∥f∥λ≜maxt∈[0,T]e-λt∥f∥∞ with ∥f∥∞≜max1⩽i⩽n|fi(t)|∥f∥∞≜max1⩽i⩽n|fi(t)| and λλ as a positive constant (Arimoto et al., 1984). From this definition, it is clear that for a large λλ, the errors at the end of the operation, where tt is often large, are much less weighted than those errors at the beginning of the operation. Then, for long trajectories, the tracking error at the end of the operation might rise to an unacceptable value in the sense of the ∞∞-norm while the λλ-norm is still a small value. For this reason, a huge overshoot of error might be observed; this phenomenon is referred to as a bad learning transient (Lee & Bien, 1997; Longman, 2000). There have been many efforts to generate a good learning transient(Cai, Freeman, Lewin, & Rogers, 2008; Chang, Longman, & Phan, 1992; Chen & Moore, 2001; Hakvoort, Aarts, van Dijk, & Jonker, 2008; Lee & Bien, 1997; Moore et al., 2002 and Moore et al., 2005; Sadegh, Hu, & James, 2002; Tomizuka, 1987; Tomizuka, Tsao, & Chew, 1989; Wang, 2000; Wang & Ye, 2005; Zhang, Wang, & Ye, 2005; Zhang, Wang, Ye, Wang, & Zhou, 2008). One simple way is to introduce a low-pass filter to cut off high frequency components that can cause the bad learning transient. However, ILC with such a low-pass filter does not have the ability to suppress those error components beyond the filter's cutoff frequency, and zero tracking error cannot be achieved. Therefore, this method introduces a trade-off between tracking accuracy and learning behavior. Another natural way is to tune the learning gain on the iteration axis (Wirkander & Longman, 1999) or on the time axis (Lee & Bien, 1997). The limitation of these learning gain tuning methods is that they require much knowledge of the system, and a very small learning gain can also yield a bad learning transient (Chang et al., 1992). Other methods include the bisection method (Chang et al., 1992) and a scheme with a reduced sampling rate in the first step to deal with initial state error (Hillenbrand & Pandit, 2000). The difficulty in the former is that it is difficult to choose the number of steps to meet a desired error tolerance restriction (Chang et al., 1992), while the latter only focuses on the initial state error. Consider that in the ∞∞-norm sense, an exponential convergence condition for a P-type ILC is derived (Moore, 2001). However, the condition in (Moore, 2001) is often difficult to satisfy. To design a feedback controller to ensure that the condition holds is inconvenient, time-consuming, and induces a high cost. Alternatively, a simple and effective solution is to reduce the sampling rate to force the condition in Moore (2001) to hold. Based on this idea, a pseudo-downsampled ILC (Zhang et al., 2008) is proposed. In this scheme, the downsampled signals are used in learning, which results in loss of information for those in-between sampling points. A two-mode ILC (Zhang, Wang, Ye, Wang, & Zhou, 2007) is proposed to compensate for this loss. In the two-mode ILC, a conventional ILC with the system sampling rate is used in the low frequency range, while a pseudo-downsampled ILC is applied to high frequency components beyond the learnable bandwidth. Although two-mode ILC can compensate for the lost information in the low frequency range, the lost information in the high frequency range cannot be compensated. Therefore, in theory, these two schemes cannot achieve zero tracking error. In this paper, a new multirate cyclic pseudo-downsampled ILC is proposed to track trajectories with high frequency components. In this scheme, the feedback control system has a sampling rate with a period of TT (sampling period of the feedback system), which is referred to as the feedback sampling rate hereinafter. ILC has a sampling rate with a period of mTmT, which is a downsampled slower rate and is referred to as the ILC sampling rate hereinafter. The ratio mm between the two sampling periods is referred to as the sampling ratio . Since all the signals are sampled at the feedback sampling rate while ILC merely uses the downsampled signals (realized by software), this downsampling process is termed as pseudo-downsampling. With this downsampling, ILC updating is carried out at every mm sampling points and these sampling points are referred to as downsampling points . For the next iteration, the downsampling points shift forward by a time interval of TT. Because of this time shift, downsampling is a cyclic process with a period of mm cycles on the iteration axis and therefore, the input to every sampling point at the feedback sampling rate is updated once every mm cycles. Due to this cyclic input update based on the pseudo-downsampled signals, this ILC scheme is referred to as the cyclic pseudo-downsampled ILC. The benefits of this scheme include the tracking of trajectories with high frequency components, the ability to deal with initial state error, elimination of the need for a filter design, improvement of the tracking accuracy, and the reduction of computation and memory size. Experimental results are presented to verify the proposed method. The paper is organized as follows. In Section 2, the idea of downsampled learning is briefly introduced, which is followed by design and implementation of the proposed cyclic pseudo-downsampled ILC in Section 3. A series of experiments are presented in Section 4 and concluding remarks are given in Section 5.

Motivated by the objectives of tracking trajectories with high frequency components and dealing with initial state errors, a multirate cyclic pseudo-downsampled ILC is proposed. The proposed scheme downsamples the feedback sampling rate to a slower ILC sampling rate with a ratio mm for the feed-forward ILC input. Based on downsampled error signals, the ILC updates and interpolates the input signal. Over iterations, the scheme downsamples the signals with the same ratio mm and with a shift. This way, the input update is a cyclic process on the iteration axis, and the input at any sampling point is updated once every mm cycles. The main advantage of the proposed method is that it can achieve zero tracking error even with the presence of high frequency components. The proposed method also possesses the ability to deal with the initial state error. Experimental results on an industrial robot show that the proposed method can greatly improve tracking accuracy. Although the proposed method can properly suppress error components on the entire frequency band, there are some limitations. The first is that the learning is carried out every mm sampling points in an iteration, which may result in a slow convergence rate, although the experiments do not show this tendency. To avoid this, it is desirable to select the sampling ratio mm to be as small as possible. Second, the conditions given in Theorem 1 are sufficient conditions and are conservative. It is desirable to find necessary and sufficient conditions to further improve the learning performance.