In order to improve the tracking performance of a non-minimum phase plant, a new method called the reference shift algorithm has been developed to overcome the problem of output lag encountered when using traditional feedback control combined with basic forms of iterative learning control. In the proposed algorithm a hybrid approach has been adopted in order to generate the next input signal. One learning loop addresses the system lag and another tackles the possibility of a large initial plant input commonly encountered when using basic iterative learning control algorithms. Simulations and experimental results have shown that there is a significant improvement in tracking performance when using this approach compared with that of other iterative learning control algorithms that have been implemented on the non-minimum phase experimental test facility.
Iterative learning control is a technique applicable to systems operating in a repetitive mode with the additional requirement that a specified output trajectory r(t)r(t) defined over a finite interval [0,T][0,T] is followed to a high precision. Examples of such systems are robot manipulators that are required to repeat a given task to high precision, chemical batch processes or, more generally, the class of tracking systems. Motivated by human learning, the basic idea of iterative learning control is to use information from previous executions of the task in order to improve performance from trial to trial in the sense that the tracking error is sequentially reduced.
The original work in this area can be traced back to Arimoto, Miyazaki, and Kawamura (1984). For example, suppose that the signal to be tracked over [0,T][0,T] on trial k>0k>0 then the basic P-type ILC algorithm consists of an update incorporating the error from the previous trial of the form
equation(1)
uk+1(t)=uk(t)+Lek(t),uk+1(t)=uk(t)+Lek(t),
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where L is the proportional scalar learning gain, uk(t)uk(t) is the control signal applied on trial k , and the tracking error on this trial is given by ek(t)=r(t)-yk(t)ek(t)=r(t)-yk(t) where yk(t)yk(t) denotes the corresponding plant trial output. Many other simple structure ILC algorithms have also been considered such as those termed the D-type, delay type (see, for example, Barton, Lewin, & Brown, 2000) and phase-lead algorithms (see, Freeman et al., 2005 and Wang and Longman, 1996). Moreover, ILC algorithms have moved beyond these relatively simple structure types and now encompass a wide range of (linear and nonlinear) plant models and control law structures (for examples of nonlinear applications see Hakvoort et al., 2007 and Heertjes and Tso, 2007). In terms of actual application, however, it is clear that the ILC law with the simplest structure which can meet the performance requirements should be used.
In some cases, the development of the algorithm has been accompanied by experimental studies but only a somewhat limited amount of work has been directed at ILC for plants which can be adequately modelled by a linear but non-minimum phase model. This is despite the fact that such models arise in many relevant applications areas and also there are theoretical results which show that obtaining high performance from ILC applied to such plants could well be problematic (Amann & Owens, 1994).
Given the statement above about the complexity of the algorithm for actual implementation, this paper focuses on applying the simple phase-lead law and attempting to overcome the difficulties that have previously been encountered when using it. A specially constructed experimental non-minimum phase linear system is used and the focus of the paper is on the practical issues involved with achieving a high level of tracking performance whilst applying an intuitively simple ILC algorithm. For this reason, analysis is restricted only to showing that the algorithm used satisfies a well known convergence criterion, whilst the major contribution of the paper lies in the experimental results and associated discussion which examine the effect of filtering and compare the present algorithm with a well established but far more complex model-based ILC law. Such information informs the controller design process. Justification for adopting the intuitively simple approach is provided by results that show that it achieves faster convergence and less fluctuation in tracking error than other approaches
In this paper, a method has been developed to improve tracking performance which incorporates shifting the reference signal from trial to trial and the use of ILC to update the control input. It has been found that the two learning loops perform an optimal plant identification in terms of a specific weighting between learning and stability. The corresponding plant inverse is then used in the ILC law. Frequency domain filters have been used to satisfy the monotonic convergence criteria in order to prevent the system from going unstable. Experimental results have shown that the reference shift algorithm achieves faster convergence and lower final error than those algorithms previously used to control the non-minimum phase test facility.
However, some aspects of the reference shift algorithm require further attention. For example, the calculations necessary between each trial are complex and may limit its use when applied to rapid industrial applications. Also, as illustrated in Fig. 6, the convergence performance is a function of the time shift which prompts the use of an adaptive filter to improve the convergence speed and maintain the final system stability. To implement this, the relationship between the required cut-off frequency and the reference shifted time must first be derived. In addition, the learning gain could also be varied adaptively in order to improve the learning performance and enable it to be even more applicable to engineering and industrial practice.
