IEEE TRANSACTIONS ON AUTOMATICCONTROL, VOL. zyxwvutsrqp 38, NO. 4, APRIL 1993 zyxwvut 581 Technical Notes and Correspondence On the Projection Algorithm and Delay of Peaking in Adaptive Control Bernard Delyon, Rauf Izmailov, and Anatoli Juditsky Abstract-Adaptive deadbeat control is considered for the determinis- tic linear plant without any persistent excitation assumption imposed. The upper bound on the rate of convergence is provided for the control algorithm that uses the projection identification algorithm. On the other hand, we show that convergence is delayed if the “regularized” identifi- cation algorithm is used. It is shown that arbitrarily large delays can precede an overshoot under certain choices of initial conditions. zyxwvutsr An estimate is also provided for the amplitude of the peak. zyxwvutsrqpo Problem Statement and zyxwvutsrq Main Result: Consider a scalar system described by the following deterministic autoregressive equation (DAR-model): N Y,,~ zyxwvutsrqponm = ary(t - i + 1) + u(t). (1) i= 1 Here, a;;..,a;Er are unknown scalar parameters, zyxwvutsrq yt is a scalar output, and U, is a scalar control. The control is designed to bring the output y, to zero. The deadbeat control N u(t) = - zyxwv C ai(t)y(t - i + 1) (2) i= 1 is used, where a,(t>;.*, a Jt) are the estimates of the parame- ters af;--, a$ available at time zyxwvutsrq C. The closed-loop system (11, (2) may be written in the form N Y,+~ = (a: - ai(t))y(t - i + 1). (3) i= 1 Let 9, = (y,;.., ytPN+ l)T and a, = (a,(t),-.., (here T denotes a transposition). To obtain the estimates a, of parame- ters we use the following “projection” algorithm (see [1]-[3]) at+, = a, + Y((PrT’PI)-l(P,Yt+l (4) where 0 < y < 2 is a scalar gain. Some care has to be taken since for a set of initial conditions of Lebesgue measure zero (4) may not be defined, To avoid problems we define ate+, = ato if Iq,,l = 0. Then Iqtl = 0 for all t 2 to. To complete the descrip- tion of the system, we define the vector of identification errors The asymptotic behavior of this system has been studied, for example, in 121. It has been shown that if the system is persis- tently excited, the sequence of parameters and outputs con- Manuscript received July 26, 1991; revised December 20, 1991. B. Delyon is with IRISA/INRIA, Campus de Beaulieu, 35042 Rennes, R. Izmailov is with the Institute for Information Transmissions Prob- IEEE Log Number 9203124. France. lems, 19 Ermolovoy Str. 101447 Moscow, USSR. verges asymptotically, i.e., 6, -+ 0 and (o, -+ 0 as t -+ 00 (see, for instance, [2, lemma 3.4.41). The exponential rate of convergence of parameter estimates to the true values has been also proved. We consider the situation of the complete absence of excitation in the input. The following theorem provides an asymptotic estimate for the convergence rate. Theorem I: For any initial conditions (So, Po) and t > N21SOl2 (5) Commentary: Theorem 1 states that superlinear asymptotic convergence takes place for the adaptive control algorithm (31, (4). In fact, estimate (5) also provides the bound for the duration of the transient period of the algorithm: lytl < Iqol as soon as t > N216012/(2y - 7’). An exponential bound has been proved in [5] for a more general object. However, for that situation, a condition of “stable invertibility” is required. The proof of Theorem 1 is based on the following inequality. Proposition 1: Let (xi), i = -N + 1, t be a sequence of posi- tive real numbers and N a positive integer; set mi = max(x,,xi-lr...,xj-N+,) for i = 0, t. Then In particular x1 x2 x3 x, - + - + - + .-. +- 2 t/Nmin mn mi mz mt-1 Prooj Inequality (7) is an immediate consequence of the inequality (6). We will prove the latter by induction on t. The proposition is obvious in the case t = 1. Let us consider t 2 2 and suppose that the proposition is true for t - 1. We will need those notations: min &‘Ik, “(x) = r/N<kst xk-l ‘k +-+-. XI x s --+ 1+ ... k- “0 m1 mk-2 mk-l for 1 I k I t. Consider the last term of the expression S, and let v be an integer such that mt-l = x,. Then we have Finally, we have only to prove that +AY) + X/Y 2 MX) for any positive real numbers x and y and any U such that t - N I v s t - 1. Using the inequality x,y 2 0 0018-9286/93$03.00 0 1993 IEEE