Frequency response of sampled-data systems

Abstract

This paper introduces the concept of frequency response for sampled-data systems and explores some basic properties as well as its computational procedures. It is shown that 1) by making use of the lifting technique, the notion of frequency response can be naturally introduced to sampled-data systems in spite of their the-varying characteristics, 2) it represents a frequency domain steady-state behavior, and 3) it is also closely related to the original transfer function representation via an integral formula. It is shown that the computation of the frequency response can be reduced to a finite-dimensional eigenvalue problem, and some examples are presented to illustrate the results. I. JNTRODUCTIONHE importance of the notion of frequency response for T continuous-time, time-invariant systems needs no justification. It is used in various aspects of system performance evaluation and still is at the center of many design methods.This fact is only reinforced by the now-s&ndard H" control theory, and attempts have been made to generalize this design methodology to various new directions. In the setting of sampled-data systems, there are now quite a few investigations along this line-for example, [lo], 171, [171, [181, [31, f271, [26], and [29], to name just a few. The difference here from the classical theory lies in the emphasis upon the importance of built-in intersample behavior in the model, so that it is part of the design specifications. As a result, in this approach the sampled-data systems are viewed as hybrid systems, and their performance is evaluated in the continuous time.An important problem in this context of sampled-data systems is that of frequency domain analysis. In classical treatments (see, e.g., [25]) the frequency domain analysis of sampled-data systems has been carried out. The classical approach is via inffinite sum formulas for sampled signals and their transforms. The mixture of continuous-and discretetime systems introduces a time-varying periodic characteristic in sampled-data systems, and this has made the classical frequency domain treatment of sampled-data systems rather awkward. It should be noted that in the classical treatment the signals are always a.ccompanied with (either real or fictitious) samplers, while in the modern point of view the actual continuous-time response is analyzed. Frequency domain anal-.Publisher Item Identifier S 0018-9286 (96)00974-9. ysis in the setting of sampled-data systems has been revisited in recent years from the modern operator theoretic standpoint in [20] and [ 111, and robust stability condition in the frequency domain has been analyzed in [9]. The works of [32], [l], and [2] pursue the justification of the notion of frequency response as a steady-state response; the former uses so-called lifting, and the latter impulse modulation. Since the advent of the lifting technique [3], [4], [19], [29], [30], it has become possible to view sampled-data systems as time-invariant discrete-time systems with built-in intersample behavior. This time-invariance g...

Optimal Sampled-Data Control

“…For simplicity of treatments, assume D 11 in (11) to be zero; for the general case, see Yamamoto and Khargonekar (1996). Let G.z/ be the transfer operator G.z/ WD D C C.zI A/ 1 B.…”

Best Achievable Tracking Performance in Sampled-Data Systems via LTI Controllers

“…Then, the output response can be expressed in the frequency domain as where is the sensitivity function of a discretized system. Employing the parameterization (3), we obtain the error response as (5) and thus the tracking performance can be expressed in the frequency domain as (6) Frequency-domain lifting techniques were developed in, e.g., [1], [8], [18], [37], which have been the subject of numerous studies of sampled-data control problems (see also [38] for an extension to linear, periodically time-varying systems). In what follows we briefly describe this procedure, while referring to [1], [8] Consequently, the tracking performance can be quantified using the lifted version of ; specifically…”

Constrained robust sampled‐data control for nonlinear uncertain systems

“…Many existing works deal with linear sampled-data control problems such as H and H problems and servo problems using the frequency response approach [3,4], lifting technology [5], and¸ approach [6,7]. Also, results of the robustness and stability of the uncertain sampled-data systems are reported [7}9].…”