Germán Andrés Ramos Fuentes¹, Josep Olm Miras², Ramón Costa Castelló³

¹ Master in Industrial Automation, Universidad Nacional de Colombia. Assistant Professor, Universidad Nacional de Colombia, Colombia. garamosf@unal.edu.co

²Ph.D. of Science (Physics), Universitat Politécnica de Catalunya. Dr. Assistant Professor, Universitat Politecnica de Catalunya, Spain. josep.olm@upc.edu

³ Ph.D. in Computer Science, Universitat Politecnica de Catalunya. Professor, Universitat Politecnica de Catalunya, Spain. ramon.costa@upc.edu

ABSTRACT

Digital repetitive control is a strategy that allows tracking/ rejecting periodic signals. Repetitive controllers are designed assuming that the exogenous signal period is constant and known, its main drawback being the dramatic loss of performance when signal frequency varies. This paper reviews the most relevant proposals advanced for overcoming this problem.

Keywords: repetitive control, varying frequency, high order repetitive control, varying sampling period.

Received: May 3th 2010

Accepted: July 1th 2011

Introduction

Repetitive control (RC) (Costa-Castelló et al., 2005; Tomizuka, 2008; Wang et al., 2009) is an internal model principle (IMP)- based technique (Francis and Wonham, 1976). The IMP establishes that in order to track/reject a certain signal the generator of such signal has to be included in the control loop. RC uses the IMP for tracking/rejecting periodic signals, assuming that these signals’ frequency remains constant and known. This yields the selection of a number of samples per period which are structurally incorporated into the control algorithm.

However, it has been shown that even slight changes in signal frequency yield dramatic performance decay (Steinbuch, 2002). This paper reviews the most relevant proposals for overcoming this problem.

Digital repetitive control

Digital implementation of RC implies external signal discretisation. Hence, a continuous T_p-periodic signal sampled with period T_s becomes a discrete N -periodic signal (N= T_p / T_s). The internal model of these types of signals came from Tomisuka (Tomisuka et al., 1989)⁴:

[1]

Other types of internal models have been proposed depending on the specific type of signal to be dealt with (Escobar et al., 2007; Griñó and Costa-Castelló, 2005).

The main feature of (1) is that it has N poles uniformly distributed on the complex plane' s unit circle. Hence, the controller provides infinite gain at ω =2kπ/N frequencies, with k = 0, 1, 2, …,N-1. This high gain compromises system stability and robustness. Two additional filters are included to address this fact: Lz), used to assure stability and Q(z) aiming at providing robustness regarding unmodelled dynamics. Although ther are several structures, the most usual one is that shown in Figure 1. Here the RC is added to a controller, C(z) , specifically designed to stabilise the plant P(z). Therefore (1) becomes:

[2]

The RC structure was originally proposed in continuous time (Hara et al., 1988; Inoue et al., 1981) while the first digital implementation was reported in Nakano and Hara (1986).

Stability

Stability analysis of a closed-loop system (Figure 1) is important and this is why one follows the scheme proposed in Inoue (Inoue et al., 1981). The closed-loop system sensitivity function is:

[3]

With S_o(z) = 1/(1+C(z)P(z)) and T_o(z) = C(z)P(z)/(1+C(z)P(z)) being, respectively, the system' s sensitivity function and complementary sensitivity function without RC, and S_M(z) being the modifying sensitivity function, which shows the effect of the RC on s y s tem sensitivity:

[4]

Hence, the system shown in Figure 1 would be stable if:

1. S_o(z) is stable, which arises from inner loop stability;

2. (1 - z^-NQ(z)) is stable, which may be fulfilled by defining Q(z) as a FIR filter;

3. ⁄Q(z)⁄< 1 ; and

4. ⁄1 - T_o(z)L(z)⁄ <1

The design of L(z) is based on phase cancellation of T_o(z) (Inoue, 1990). For minimum phase systems, one takes L(z) = k_Υ T_o^-1(z), where 0 < kΥ < ‚2 ; and k_Υ' are selected through trade-off between robustness and performance (Hillerström and Lee, 1997).

Applications

RC has been widely used in different areas, such as hard disk and CD drives (Chew and Tomizuka, 1990), robotics (Kasac et al., 2008; Sun et al., 2007), electro-hydraulic actuators (Kim and Tsao, 2000), vibration suppression (Daley et al., 2006; Hattori et al.,000), heat exchangers (álvarez et al., 2007), friction compensation (Huang et al., 1998), electrical injection molding machines (Nan et al., 2006), piezoelectric actuators (Choi et al., 2002), linear motor machine control (Chen and Hsieh, 2007), electronic rectifiers (Zhou and Wang, 2003), PWM inverters (Li et al., 2006; Wang et al., 2007b) and active filters (Costa-Castelló et al., 2009).

Performance

Performance in varying frequency conditions

Frequency is not exactly known or varies in most practical applications. Figure 2 shows internal model gain for signal period variations at harmonics k=1, 2 and 3. It can be noticed that performance decays dramatically the further away it is from the nominal period, and the situation becomes even worse for higher harmonics.

Performance faced with non-harmonic errors

Due to the waterbed effect (Songchon and Longman, 2001), RC controller sensitivity function shows amplification of frequencies between consecutive harmonics. This degrades system behavior and must be taken into account.

Organisation of the approaches

Several approaches have been proposed for resolving the performance issue when working with time-varying frequency signals. They can be grouped into two main sets: high order repetitive control (HORC) which uses several delay blocks to improve robustness and adaptive RC in which one or more controller parameters are adjusted on-line so as to maintain performance. The adjusted parameters are either sampling period (T_s) or the number of samples per period (N).

High order repetitive control

The internal model architecture used in HORC is shown in Figure 3, its transfer function being:

[5]

where , and usually . The modifying sensitivity function with HORC is:

[6]

The stability conditions are coincident with already established ones, just replacing condition iii) by |W(z)Q(z) |< ‚ 1.

Weighting selection methods

Because of the excluding relationship between robustness regarding frequency variation and sensitivity to non-harmonic frequencies, selecting weighting w_i is posed as an optimisation problem.

HORC is used in Inoue (Inoue, 1990) to improve harmonic frequency behaviour. The cost function to be minimised is the mean value of the square of S_MW(z), , which implies that all weighting must be equal and have a w_i=1/m value.

The goal in Chang (Chang et al., 1995) was to reduce the error spectrum, considering both harmonic and non-harmonic components,i.e. .

Steinbuch (Steinbuch, 2002) aimed at reducing sensitivity to signal period variations. The first m derivatives of W(z) regarding signal period were zeroed. This allowed keeping a high gain for (5) in a wider neighbourhood around the harmonic frequencies. This problem exhibits an analytic solution.

Chang’s results (Chang et al., 1995) have been generalised in Steinbuch (Steinbuch et al., 2007) through the solution of being a weighting function establishing the interval in which the norm of S_MW (ω) is to be minimised. Steinbuch (Steinbuch, 2002) and Chang' s (Chang et al., 1995) results corresponded to specific cases of this formulation.

Differently from the preceding approaches, the constraint was removed in Pipeleers (Pipeleers et al., 2008). This reduces harmonic frequencies gains which leads to obtaining better performance results for non-harmonic frequencies. Steinbuch (Steinbuch, 2002; Steinbuch et al., 2007) and Chang’s (Chang et al., 1995) results corresponded to specific cases regarding this formulation.

Figure 4 gives a comparison of the above proposals for m=3. Pipeleers' design (Pipeleers et al., 2008) for 20% uncertainty during the period had better results than Steinbuch (Steinbuch, 2002) for robust and non-harmonic performance. Notice that the gain at harmonic frequencies was not zero and it was lower for a wider frequency interval around the harmonics too. It may also be observed that, besides improving inter-harmonic behaviour, the approaches in Inoue (Inoue, 1990) and Chang (Chang et al., 1995) did not exhibit a robust design regarding period changes. The weights obtained with the aforementioned approaches are summarised in Table 1.

Adaptive repetitive control

Delay adaptation

The adjusted parameter is delay N in this strategy. There are two approaches: integer adaptation and fractional adaptation.

Integer delay. In varying frequency conditions (i.e. with variable Tp), the η = T_p/T_s ratio may take a non-integer value. Hence, the adaptation strategy consists of approximating N to the nearest integer to η, (Dotsch et al., 1995; Manayathara et al., 1996); a stability proof for systems with varying N is available in Hu (Hu, 1992).

Fractional delay. In this proposal, the integer part of η· keeps its original implementation , while the fractional part is implemented using an FIR filter to approximate it (Laakso et al., 1996). FIR filter parameters are obtained through on-line optimisation using frequency error as a cost function (Wang et al., 2007a; Yu and Hu, 2001). Figure 5 plots a comparison of the results obtained with three different techniques.

Varying sampling period

This section includes approaches where sampling period, T_s, is varied, aiming to keep the N=T_p/T_s,/ ratio constant.

An RC is applied to a peristaltic bomb in Hillerström (Hillerström, 1994) where disturbances are directly related to a mechanism' s angular position. Sampling period variations imply that system behaviour is not that of a linear time invariant (LTI) system; controller parameters therefore depend on sampling period.

This strategy’s effect on system stability is analysed in Ramos (Ramos et al., 2009) via a linear matrix inequality (LMI)-gridding approach (Apkarian and Adams, 1998; Sala, 2005).

Differently from the preceding one, Olm (Olm et al., 2011) presents a methodology for stability analysis based on robust control theory (Fujioka, 2008; Suh, 2008). Time varying components are treated as norm-bounded uncertainties. The obtained results yield conservatism reduction and wider stability intervals, compared to Ramos (Ramos et al., 2009).

A methodology for compensating for parametric changes due to sampling period variations has been presented by Olm (Olm et al., 2010). This is achieved by including a pre-compensator that cancels out the effect of the time-varying sampling and makes the closed-loop system invariant to sampling period changes.

Integer adaptation of delay with variable sampling period

N is obtained in Tsao (Tsao et al., 2000) via an integer approximation of the Tp/Ts ratio, as previously described, and Ts is readjusted so that external period Tp results in an integer multiple N of Ts . Although experimental results are provided, the study does not include a formal stability analysis.

Interpolators

Cao and Ledwich' s proposal (Cao and Ledwich, 2002) includes a system having two independent sampling periods; one is based on a loop with nominal sampling which is constant and includes the plant and a stabilising controller, while RC works with variable period using Tsao' s adaptation technique (Tsao et al., 2000). These parts are assembled with two interpolators which are located at RC control loop input and output. Each interpolator is in charge of reconstructing the signal from input samples and delivering it in the format of the corresponding output samples so that the two control actions can be added up. This structure is shown in Figure 6.

Spatial repetitive control

An RC for a rotatory system in which the disturbances directly depend on angular position is described in Chen and Yang (Chen and Yang, 2007). Instead of using non-uniform sampling (Hillerstra, 1994), the proposal is to use spatial transformation resulting in a nonlinear position invariant (NPI) system. It is then linearised by feedback adaptation and a standard RC is introduced. The use of a nonlinear observer in Yang and Chen (Yang and Chen, 2008) means that the need for state measures can be removed.

Conclusions

This paper has described techniques proposed for tackling the problem of digital RC performance loss when working in variable frequency conditions. HORC leads to a higher computational burden and memory requirements, while sampling period variation involves higher technological complexity.

One has to estimate on-line signal frequency in adaptation schemes while the estimator is not actually needed in HORC schemes. HORC allows LTI analysis, while sampling period variation results in a linear time varying (LTV) framework which entails increased theoretical complexity for stability proofs.

Acknowledgements

This work has been partially supported by the Spanish ME (project DPI2010-15110).

Footnotes

4 This internal model was obtained from the Z-transform of discrete Fourier series expansion of periodic signal

References