70 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 1, JANUARY 1999 [10] R. K. Miller and A. N. Michel, “On existence of periodic motions in nonlinear control systems with periodic inputs,” SIAM J. Contr. Opt., vol. 18, no. 5, pp. 585–598, 1980. [11] A. N. Michel, “Stability: The common thread in the evolution of feedback control,” IEEE Contr. Syst., vol. 16, no. 3, pp. 50–59, June 1996. [12] I. W. Sandberg, “Approximately-finite memory and input–output maps,” IEEE Trans. Circuits Syst., vol. 39, no. 7, pp. 549–556, July. 1992. [13] , “Uniform approximation and the circle criterion,” IEEE Trans. Automat. Contr., vol. 38, pp. 1450–1458, Oct. 1993. [14] K. K. Clarke and P. M. Hess, Communication Circuits: Analysis and Design. Reading, MA: Addison-Wesley, 1971. [15] A. Ushida and L. O. Chua, “Frequency-domain analysis of nonlinear circuits driven by multi-tone signals,” IEEE Trans. Circuits Syst., vol. CAS-31, pp. 766–779, Sept. 1984. [16] J. K. Hale, Ordinary Differential Equations. New York: Wiley- Interscience, 1954. [17] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. New York: Springer-Verlag, 1975. [18] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992. [19] L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations. New York: Van Nostrand, 1971. On the Parameter Convergence Properties of a Combined VS/Adaptive Control Scheme During Sliding Motion Giorgio Bartolini and Antonella Ferrara Abstract—The effects of the use of a variable structure control com- ponent within an adaptive control scheme, previously proposed by the authors, are investigated in this paper. The discontinuous control action does not allow the explicit identification of the coefficients of the filters, but only the identification of the so-called equivalent control coinciding with the ideal control. By coupling the variable structure control strategy with a continuous parameter adjustment mechanism it is instead possible to obtain an equivalent control which, during the sliding motion, turns out to be equal to a suitably constructed prediction error. As long as the parameter adaptation is driven by both the tracking error (which vanishes in finite time) and the prediction error, then the controller parameters prove to be exponentially convergent to the ideal values, provided that the reference signal is sufficiently persistently exciting. Index Terms—Adaptive control, parameter convergence, sliding modes, variable structure control. I. INTRODUCTION In recent years the attention of many researchers has focused on the possibility of combining variable structure control (VSC) and model reference adaptive control (MRAC) in order to improve the robustness features and the performances of the resulting control schemes. As far as the control of linear time-invariant (LTI) single- input/single-output (SISO) plants is concerned, many proposals of combined control strategies have appeared in the literature (see, for instance, [8], [9], [16], [19], [20], and [23]). In [20], a VSC approach is followed to generate a component of the control signal ensuring Manuscript received June 5, 1994; revised January 11, 1995. G. Bartolini is with the Istituto di Elettrotecnica, University of Cagliari, 09123 Cagliari, Italy. A. Ferrara is with the Department of Communication, Computer and System Sciences, University of Genova, 16145 Genova, Italy (e-mail: fer- rara@dist.unige.it). Publisher Item Identifier S 0018-9286(99)00585-1. the zeroing of the tracking error also in the presence of unmodeled dynamics. In [16], the suggested control law is characterized by an adaptive term plus a term depending on a continuous approximation of a sign function, the error signal driving these two terms being the tracking error. The effect of the VSC component asymptotically vanishes, so that its main purpose consists of the improvement of the transient response of the controlled plant. The use of a discontinuous component in the derivative of a suitable prediction error is instead presented in [19], leading to a combined VSC/indirect adaptive control scheme with relevant robustness properties when bounded unmeasurable disturbances act on the plant. Finally, in [8] and [9], the analysis of the stability and asymptotic zeroing of the tracking error of an adaptive control scheme with discontinuous adjustment of the parameters is presented. The scheme exploits the possibility of obtaining, as output of an high-gain filter, the so-called equivalent control, and is capable of determining, through a series of filtering operations, a control signal to be applied to the plant coinciding with the ideal (known parameter) control. More recently, in [10], the same authors have also analyzed the possibility of using, within an adaptive control framework, a VSC control strategy of binary type, but no explicit identification of the controller parameters is performed, while the problem of parameter identification in a VSC context is dealt with in [23] for systems with accessible state. The aim of the present paper is that of studying the effects of the use of a VSC component within a particular adaptive control scheme, previously proposed by the authors [1], [2]. Such a scheme differs from classical proposals since it enables the relaxation of some of the relevant assumptions on which MRAC schemes are usually based [12], [14], [15]. In particular, the complexity of the filters constituting the controller is independent of the relative degree of the plant to control. If the relative degree is greater than one, then the sign of the high-frequency gain of the plant can be unknown (in the case of relative degree equal to one, it is instead required to have some a priori knowledge of an upper bound of the leading coefficient of the plant numerator to make the previous assumption true). The applicability of the scheme is extended to a class of nonminimum phase systems (those whose zeros can be robustly located in the complex l.h.p. by means of a relative-degree-one parallel compensator). The considered scheme can be viewed as belonging to a class of adaptive schemes with parallel feedforward filters which has recently obtained the attention of many researchers (see, for instance, [11]). Note that, for the sake of simplicity, in this paper a first-order parallel compensator is adopted, even if the problem of the choice of the parameters of a parallel compensator of higher order has been satisfactorily addressed in recent works [7], [17]. The control approach is based on a two-step procedure: a pole assignment control problem is indirectly solved by directly solving a model tracking auxiliary problem. Indeed, as a by-product of the solution to the auxiliary problem, a subset of the parameters of the feedback part of the controller are identified. These latter are used, according to a suitably established relationship, as parameters of a prefilter, thus attaining the major control aim. Yet, the adaptive control scheme in question requires the convergence of the parameters of the controller to the ideal value to actually guarantee that the controlled output of the plant presents the desired behavior. For this reason, the application of VSC techniques within the mentioned adaptive scheme, although appearing suitable for robustness issues, cannot be effectively actuated. More precisely, in this paper, it 0018–9286/99$10.00  1999 IEEE