On the Module of Almost Periodic Output for Sampled-Data Feedback Control Systems
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On the Module of Almost Periodic Output for Sampled-Data Feedback Control Systems Rong Yuan
Received: 9 February 2007 / Accepted: 23 March 2007 / Published online: 18 July 2007 © Springer Science + Business Media B.V. 2007
Abstract The present paper is concerned with the module containment of almost periodic output for sampled-data feedback control system. Some known (periodic, quasi-periodic) results would be expanded. A special case of sampled-data feedback control systems is detailed. Keywords Sampled-data feedback control system · Hybrid dynamical system · Plant output · Controller · Almost periodicity · Module containment Mathematics Subject Classification (2000) 93C23 · 93B52 · 34K14
1 Introduction Many mathematical models in practice, e.g., automatic control, power systems, etc., could be written as the sampled-data feedback control systems of the form ⎧ p ⎪ N
⎪ ⎪ , + bj ()u(n − j ), n ≤ t < n + 1, x(t) ˙ = F t, x(t), x([t − j ]) ⎪ 0 ⎪ ⎨ j =0 (1.1) p N ⎪
⎪ ⎪ ⎪ u(n + 1) = cj ()u(n − j ) + dj ()x(n − j ) + g(n, ). ⎪ ⎩ j =0
j =0
The sampled-data feedback control system is a class of hybrid dynamical systems which are in emphasis recently (see [1, 5–9, 14, 19, 20] and references therewith). The sampled-data feedback control system consists of an interconnection of a nonlinear plant (described by a system of first order differential equations ) and a linear digital controller (described by
This work was partially supported by the National Natural Science Foundation of China and RFDP. R. Yuan () Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China e-mail: [email protected]
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a system of higher order linear difference equations). The interface between the plant and the controller is an A/D converter, and the interface between the controller and the plant is a D/A converter. The qualitative behavior of sampled-data feedback control systems has been under continual investigation for many years, with an emphasis on linear systems (see [5, 8, 9]). The problem on stability of digital controller and plant output reduces to the study of stability of steady position for sampled-data feedback control system so that the obtained results in [14, 20] can be applied. Periodic digital controllers were discussed in [5]. Quasi-periodic and almost periodic digital controllers were studied in [19]. The time almost periodic dependence in (1.1) reflects the effects of certain “seasonal” variations which are roughly but not exactly periodic. Models with such time dependence are characterized more appropriately by quasi-periodic or almost periodic equations rather than by periodic ones. It is shown in [19] that we can choose a suitable initial position and change the coefficients such that the output of the plant is periodic, quasi-periodic, and almost periodic in time for periodic, quasi-periodic, and almost periodic digital controllers, respectively. The main purpose of this paper is to investigate the module containment of almost periodic output to the samp
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