**Sampled Tracking for Delayed Systems Using Two-Time-Scale Sampled-data Controllers**

**Haoping WANG ^{1}, Christian VASSEUR^{2}, Vladan KONCAR^{3}, Afzal CHAMROO^{4}, Nicolai CHRISTOV^{5} **

^{1,2,5}LAGIS CNRS FRE 3303, Université Lille 1 Sciences et Technologies, Bât. P2,

59655 Villeneuve d’Ascq, France,

{haoping.wang, christian.vasseur, nicolai.christov}@univ-lille1.fr

^{3}GEMTEX, ENSAIT,

9 Rue de l’Ermitage, BP 30329, 59056 Roubaix, France,

vladan.koncar@ensait.fr

^{4}LAII, Université Poitiers,

40 av. du Recteur Pineau, 86022 Poitiers, France,

afzal.chamroo@univ-poitiers.fr

**Abstract**: This article deals with the trajectory tracking of linear plants with sampled and delayed outputs. A class of sampled-data controllers with two time-scales is proposed which allows sampled tracking with a delay twice as much as that of the plant output. Numerical simulation results are presented to demonstrate the effectiveness of the proposed approach.

**Keywords**: Sampled tracking, sampled-data controllers, two time-scale systems, optimal control, Luenberger discrete-time observer.

**Full text **

**CITE THIS PAPER AS**:

Haoping WANG, Christian VASSEUR, Vladan KONCAR, Afzal CHAMROO, Nicolai CHRISTOV**, ****Sampled Tracking for Delayed Systems Using Two-Time-Scale Sampled-data Controllers, ***Studies in Informatics and Control*, ISSN 1220-1766, vol. 19 (4), pp. 339-346, 2010.

**1. Introduction**

During the last two decades, two-time-scale sampled-data control of continuous-time plants has been studied in several papers dealing with both its theory and applications [1]-[5]. For instance, a sampled-data control scheme which control actions are scheduled at two different sampling rates (slow and fast) is suggested in [1]. This sampled-data control is in composite form and is computed as the sum of the slow and fast control signals. A two-time-scale digital controller is derived by using the singular perturbation theory and is applied for motor position servoing [2]. Recently, a robust two-time-scale control based on the singular perturbation method and time delay control is proposed for a pneumatic vibration isolator [4]. A decentralised two-time-scale motions control of linear time-invariant plants with unstable decentralised fixed modes (UDFM) is designed in [5]. The method used generalizes the sampled-data hold function introduced by Kabamba [6] to eliminate UDFM and to decouple the discrete-time plant model into independent input-output channels.

An alternative approach for two-time-scale controller design is proposed in [3] using the theory of the sampled-data systems with piecewise functioning (SPF) [3]. The developed controller, however, needs full information for the plant state which limits its practical application.

Figure 1. System to be controlled

In this paper we deal with the case when the only available plant information is delivered from the plant output via a digital sensor introducing a delay corresponding to the time needed to process the information (see Figure 1).

Further on we consider that , where is the sampling period. Such systems are frequently encountered in industry when digital technology is used for measurement (e.g. camera) and/or control [8]. Their dynamics can be described as

(1c)

where , are constant matrices and * represents sampling operation with constant period .

To ensure efficient tracking control of the considered systems, we propose a new type of SPF based two-time-scale sampled-data controllers (SDC) which use sampled and delayed plant output measurements.

The paper is organized as follows. In Section 2 the SPF theory and the existing full state information SDC are briefly presented. In Section 3 we develop SDC which use delayed state measurements and in Section 4 we generalize these controllers for the case when only delayed plant output measurements are available. The performance analysis of the new class of SDC by numerical simulation is presented in Section 5. Finally, in Section 6 some concluding remarks are given.

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