Approximating Sampled-Data Systems with Applications to Digital Redesign Yutaka Yamamoto 1 Brian D. O. Anderson 2 Masaaki Nagahara 3 Abstract Despite the existence of methods for a direct optimal de- sign of sampled-data control systems, it is often desired to approximate sampled-data systems with discrete-time ones. This occurs frequently in the context of digital redesign, in which one retains intuition in the continuous-time con- text. This paper investigates and gives conditions under which such a method works. Under some conditions, we guarantee that fast-sampling approximation works well for H ∞ sampled-data design, and give an estimate for such an upsampling factor. As an application, we propose a new method for obtaining an FIR controller (possibly with first- order approximation). A comparison is made with an exist- ing method. 1 Introduction Modern sampled-data control theory enables us to design directly a digital controller that makes an analog perfor- mance optimal. It is also known that for finite-dimensional plants this design problem, for example, H ∞ control prob- lem having the mixed nature of both discrete-time and continuous-time, can be reduced to an equivalent discrete- time, finite-dimensional problem [3, 6, 7]. In spite of these results, however, there are still several rea- sons that lead us to approximate the design problem in a variety of ways by a more conventional approximate de- sign problem, and then obtain a digital controller. One rea- son is that we may be able to use a design software of our choice, and base our design intuition on such softwares we are familiar with. Once we have obtained an approximant, it is not difficult to invoke such a more conventional de- sign package. Yet another advantage is that we may wish to rely on some design intuition we have developed, and this is in many cases easier to maintain by resorting to approx- imations, rather than using the direct optimization methods mentioned above. So-called digital redesign is regarded as such a case: we first obtain a continuous-time controller, and then attempt to discretize it, while attempting to main- tain the desirable continuous-time performance. In such attempts, we of course wish to capture the 1 yy@i.kyoto-u.ac.jp, Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto Uni- versity, Kyoto 606-8501, JAPAN. Author to whom all correspondence should be addressed. 2 Brian.Anderson@anu.edu.au, Research School of Informa- tion Sciences and Engineering, Australian National University, Canbera ACT 0200, AUSTRALIA. 3 nagahara@acs.i.kyoto-u.ac.jp, the same address as Yu- taka Yamamoto. continuous-time performance, rather than the mere sample- point behavior. This requires approximation of the gen- eralized plant in the sampled-data setting, and desirably a convergence analysis. Keller and Anderson [1, 8] have proposed to use a fast-sampling/fast-hold approximation with its approximating performance taken to be the closed- loop behavior. Related design problems have been studied by Madievski and Anderson [10] for H ∞ control and by Bamieh et al. [4] for L 1 control problems. Later it was proven by Yamamoto et al. [14] that such fast-sampling ap- proximations converge uniformly in the frequency domain to the limiting sampled-data system. The following impor- tant issues still remain open, however: 1. How do we guarantee that a designed controller K (based on plant approximation/discretization) con- verges to the “right” controller in the limit? Since we do not know in advance which controller will be part of the closed-loop before the design, a conver- gence theorem with a fixed K would not be enough. We need some kind of uniformity in the controllers K in the convergence process to guarantee this. (While Bamieh et al. [4] successfully gave an explicit conver- gence bound for the L 1 control design problem, there is an important difference here. Although sampling is continuous with respect to the L ∞ norm where it is well defined, it is not continuous with respect to the L 2 norm. This makes the estimate of the convergence rate here in the H ∞ context more delicate, and quite different from that studied in [4].) 2. How fast should fast-sampling periods be? While there exist some empirical studies on estimates and indications in some special cases, there does not exist a precise estimate that guarantees quality of approxi- mation for a particular rate, to the best of the authors’ knowledge. This paper studies these problems. We first invoke the so- called FR-operator T by Araki and others [2]. Its principal submatrix T (M,M ) of size M tells us how many aliased components we need to obtain a desired degree of preci- sion. We then obtain an estimate on the fast-sampling rate using T (M,M ), based on the approximation estimate of sinusoids via piecewise step functions arising from sample and hold actions. We also apply the results to the digital redesign problem, and show that the FIR (finite-impulse response, i.e., those having finitely many Markov parameters) approximation problem of a stable controller for sampled-data systems can