تقریب سریع نمونه برداشتن/سریع نگه داشتن اصلاح شده برای تجزیه و تحلیل سیستم نمونه گیری شده-داده
|کد مقاله||سال انتشار||مقاله انگلیسی||ترجمه فارسی||تعداد کلمات|
|27981||2008||11 صفحه PDF||سفارش دهید||6399 کلمه|
Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : European Journal of Control , Volume 14, Issue 4, 2008, Pages 286–296
This paper deals with theH∞ norm and frequency response gain analysis of sampled-data systems and provides a new approach, which we call modified fast-sample/fast-hold approximation. The new approximation approach discretizes the continuous-time generalized plant in a “γ-independent fashion” and leads to a discrete-time generalized plant with a similar structure to what is obtained by the conventional fast-sample/fast-hold approximation approach. Unlike the conventional approach, however, the modified fast-sample/fast-hold approximation approach can give both the upper and lower bounds of the H∞-norm and/or the frequency response gain for sampled-data systems. Furthermore, the gap between the upper and lower bounds can be bounded from above directly from the fast-sample/fast-hold parameter N and is independent of the controller. These features are quite useful when it is applied to control system design, and this study indeed has very close relation with the control system design via noncausal periodically time-varying scaling, a novel notion introduced recently.
In the analysis and design of sampled-data systems, it is essential to deal with the intersample behavior of continuous-time signals as it is, and sophisticated techniques have been well established to that end, e.g., the lifting technique [3, 19–21], the FR-operator technique [2, 7], the approach with jump systems [12, 18] and the parametric transfer function approach . These techniques can be regarded as providing methods for manipulating infinite-dimensional operators in the definitions of the H1-norm and/or frequency response gain of sampled-data systems  and then reducing the inherently infinite-dimensional analysis and design problems to finite-dimensional ones in an exact fashion. An independent method for dealing with sampleddata systems, called fast-sample/fast-hold (FSFH) approximation [15, 23], is also well known on the other hand, which reduces the infinite-dimensional problems to finite-dimensional ones \in an asymptotically exact fashion" in the sense that the approximation error is ensured to converge to zero as the approximation parameter tends to 1. The FSFH approximation, however, does not providea systematic way for determining some guaranteed upper and lower bounds of the H1-norm and/or frequency response gain of sampled-data systems for fixed N. This makes it difficult for the FSFH approximation approach to be applied to strict analysis and particularly control system design with guaranteed performance, essentially because we cannot determine when N is large enough to be able to ensure the associated analysis and design results exact and strict. Motivated by our recent study on noncausal periodically time-varyingscaling of sampled-data systems [6, 10], we provide yet another method for the analysis of the H1-norm and/or frequency response gain of sampled-data systems in this paper. Regarding the H1 analysis, the new method has a feature that the associated discretization of the continuoustime generalized plant is carried out in a simple \-independent fashion" unlike in the exact methods [3, 11, 13, 14] that require so-called \-dependent discretization." Furthermore, the resulting discretized generalized plant has a structure that is quite similar to what is obtained by the conventional FSFH approximation . While our new method is quite similar to the conventional FSFH approximation approach in these respects, ours has a distinctive advantage in that guaranteed upper and lower bounds of the H1-norm and frequency responsegain of sampled-data systems can be determined with it. Furthermore, a noteworthy fact regarding this advantage is that the gap between the upper and lower bounds can be bounded from above once we fix the continuous-time generalized plant and the approximation parameter N. This in particular implies that the gap is independent of the discrete-time controller. Hence, it follows that (i) we could judge whether or not the given approximation parameter N is large enough for the desired accuracy not only in analysis but also in design and (ii) the new method opens possibilities for sampled-data control system design with guaranteed robust stability or performance under plant uncertainties with some structures, unlike the conventional FSFH approximation. Taking all these advantages into consideration, we refer to our new method as modified fast-sample/fast-hold approximation. This paper is organized as follows. We first review the lifted representation of sampled-data systems in Section 2, which is used throughout the paper. Then, in Section 3, we employ an operator of \fast-lifting", denoted by LN and first introduced in , which plays a central role in this paper. Applying this operator, we then derive our main results about what we call modified FSFH approximation. Some remarks about the relationship with this study and noncausal periodically time-varying scaling [6, 10], as well as a relevant study giving another set of upper and lower bounds , are also given. Then, a numerical example is studied in Section 4 to verify the effectiveness of modified FSFH approximation. Finally, Section 5 concludes the paper.
نتیجه گیری انگلیسی
We applied to the sampled-data system analysis the \fast-lifting" idea, which was introduced in  in the context of what we call noncausal linear periodically time-varying (LPTV) scaling, and derived yet another method, called modified fast-sample/ fast-hold (FSFH) approximation, for the frequency response gain and H1-norm analysis. While the new method involves the discretization of the continuoustime generalized plant leading to a similar structure to what we have with the conventional FSFH approximation [15, 23], it has a distinctive advantage over the conventional one in that it can give both the upper and lower bounds of the target values. Moreover, the gap between these bounds can be computed in advance in the sense that it depends on the approximation parameter N as well as the continuous-time generalized plant but not on the controller, and these features are quite promising for its use in the context of controller design. In numerical studies, however, this paper confined itself to the problem of H1-norm computation, for simplicity, and demonstrated that our modified FSFH approximation approach does lead to faster as well as more accurate computations than the conventional FSFH approximation. Regarding the use of the modified FSFH approximation in the controller design, a fundamental use such as the H1 design is straightforward, and thus the technique developed in this paper can readily be applied to robust stabilization against uncertainties. In such a case, a possible advantage over the exact methods [3,11,13,14] includes the fact that the associated computations are more straightforward and intuitive. A simple application of H1 control, however, generally leads to conservativeness when we are faced with, e.g., structured/ unstructured static uncertainties, and robust stabilization against such uncertainties is a much involved problem in the sampled-data setting. It is important to remark that the idea of modified FSFH approximation can play a significant role with regard to such difficulties, too; roughly speaking, modified FSFH approximation would allow us to apply some sort of scaling approach on the resulting discretized system in such a way that some rigorous arguments can be developed about robust stabilization. Indeed, the discretization method derived in this paper is closely related to the noncausal LPTV scaling technique [6, 10] (see Remark 1), which is proved to be a promising technique for robust controller design under structured/unstructured static uncertainties. In this sense, the present paper can be said to lay a basis for a novel direction for robust stability of sampled-data systems. Developing further links will be a continued research topic in our future studies. Finally, we remark that the direct feedthrough matrix D11 from w to z is assumed to be zero in this paper. Even though this assumption has made the arguments much easier, we can also accommodate nonzero D11 without sacrificing the important features of the proposed method including the convergence property, and are eventually led to an only slightly modified discretized generalized plant even when D11 6¼ 0. Describing the details , however, takes further spaces, and thus will be reported independently.