Pergamon 0005-1098(95)OOl22-0 Aulomafica, Vol. 31, No. 12, pp. 1893-1895, 1995 Copyright @ 1995 Elsevier Science Ltd Printed in Great Britain. All rights resewed ooo5-1098/95 $9.50 + 0.00 Technical Communique zyxwvutsrqpon A Comment on ‘A Time-varying Sliding Surface for Fast and Robust Tracking Control of Second-order Uncertain Systems’* ANDRZEJ BARTOSZEWICZt Key Words-Variable structure control; sliding surfaces. Abstract-In a recent paper, Seung-Bok Choi, Dong-Won Park and Suhada Jayasuriya (Automatica, 30,899-904, 1994) have presented a stepwise time-varying sliding surface for variable structure control of a class of second-order uncertain systems. As the surface does not truly guarantee insensitivity of the systems to parameter variations and external disturbances, in this comment we propose a continuously time-varying surface that allows faster tracking and really guarantees robust bebaviour of the systems. 1. Introduction Seung-Bok Choi et al. (1994) considered a second-order nonlinear uncertain dynamic system. They introduced a time-varying sliding surface that is adaptable to arbitrary initial conditions of the system. At the beginning, the surface passes through the representative point specified by any given or measured initial errors, and then the surface is moved step by step to its predetermined, desired position. Seung-Bok Choi et al. have shown that their variable structure control scheme with the proposed surface guarantees tracking time essentially shorter than conven- tional VSC with time-invariant sliding surfaces. However, the proposed scheme does not guarantee insensitivity to parameter variations or external disturbances, and therefore it does not allow one to enforce a desired dynamic behaviour of the system. In the proposed scheme the sliding surface is repeatedly, instantaneously moved: shifted or/and rotated. As a consequence, after each movement of the surface, the system representative point is no longer on the surface, and the system is, at least for some time, not insensitive to disturbances and parameter uncertainties. In other words, after each instantaneous movement of the sliding surface, the system is in the phase of reaching of a new surface-thus it is not robust. Whether or not it actually reaches the new surface before the surface is moved again depends on the relation between dwelling time Ar, constants AL and Afs (all three of them set arbityarily), current disturbance and parameter variations. In this comment we propose to substitute the stepwise time-varying sliding surface (line) with a continuously time-varying one. Then we introduce a modified variable structure control law that satisfies sufficient conditions for the existence of a sliding mode on our surface. Actually, to achieve the sliding mode, we add to the control signal of Seung-Bok Choi et a/.: a proportional term for rotated sliding *Received 10 October 1994; revised 22 March 1995; received in final form 26 July 1995. This paper was not presented at any IFAC meeting. This paper was recom- mended for publication in revised form by Peter Dorato. Corresponding author Andrzej Bartbszewicz. Tel. (48 42) 312554: Fax (4842) 312551: E-mail ~ndpb~rt@lodzl.p.lodz.pl. ’ ’ t Institute of Automatic Control I-13, Technical University of Lodz, 18/22 Stefanowskiego St.. 90-924 Lodz, Poland. lines and a constant term for shifted ones. By this means, the representative point of the considered system is forced to stay always on our sliding line. Therefore the system is truly insensitive to parameter variations and external disturbances; hence its dynamic behaviour can be precisely predetermined. We also avoid setting the constants AT, Ah, A& arbitrarily, and, if required, we can further decrease the tracking time of the system. Finally, we point out a minor ambiguity in the illustrative example presented by Seung-Bok Choi et al. 2. Continuously time-varying sliding surface Consider a second-order nonlinear uncertain dynamic system described by i,(t) =X2(t), J&(t) = 2 f;(x,(th x20), t) + 5 wtlgi(~Io), X20), 0 (1) r=l i=, + W,(t), x20), OuW + 4th with given initial conditions xl@,,), x2(to), where Gai(t)gi(x,(t), x2(t), I) and d(t) represent plant uncertainty and external disturbance respectively. Let us denote the tracking error by e(r) = [e,(t) e2WlT = h(t) - xdlW 120) - h2WlT and define the sliding line by zyxwvutsrqponmlkjihgfedcbaZYXWVU a(eW, t) = c(t)e,O) + e2(t) + a(t), (2) where c(t) = At + B and a(t) = Ct + D, with A, B, C and D constant. Taking into account the function V(c) = :02(e(t), t), (3) which is positive definite with respect to a, and provided that uncertainties and disturbances are bounded, zyxwvutsrqponmlkjihgfed (“i)min 5 hi 5 (ai)maxt (4) YI 5 d(t) 5 ~2, (5) the control law u(t) = [-(k + ,$ Ikfdxt 1)l) W ta) - ,$ ftx9 t, - ,$ zyxwvutsrqponml Ei(X9 t, - cx2 + cxd2 + id2 - Ae, - C zyxwvutsrqponmlkjihgfed I/ b(x, t), (6) where Zitx~ t, = zigi(x, t). di = l[(ai)rnin + (ai)msxl7 &qx, t) = a,g& t), lij = (aJmax- iii, k>max(l~,l, Ir21) (7) 1893