Iterative Learning Control and Recursive Identification

This abstract discusses our investigations relating Iterative Learning Control (ILC) for periodic systems on the one hand, and the class of Recursive Identification (RI), Gradient Descent (GD), Stochastic Approximation (SA) and adaptive filtering algorithms on the other. The benefit of such is the straightforward transfer of results in the latter context which is useful to study different design decisions made for the former. We discuss briefly the possible relevance of this observation for (i) design and analysis of suitable gain factors, (ii) working with constrained control signals, (iii) designing a model-free control strategy. For a survey of design, analysis and applications of ILC, see e.g. (1; 2). For an overview of practical and theoretical studies of RI and SA see (3), GD (4; 5), and adaptive filtering (6; 7). In this note we articulate the basic idea, and discuss further work which may be expected from this. 1 GRADIENT DESCENT AND ILC Given two timeseries (u(t))t and (y(t))t, respectively giving the (control) input to the system, and the desired output signal to which the system is steered to. The control problem we are interested in has the overall property that the desired output is repetitive, with a fixed periodsay n ∈ N. Now we let ut ∈ R denote the input signal during n consecutive instants, starting from u(t+ 1), or u = (u(tn+ 1), . . . , u(tn+ n)) T ∈ R, ∀t = 1, . . . , T. (1) and let uk = u(tn+ k) be the kth element of this vector. Equivalently, we define the output signal in period t as y ∈ R as y = (y(tn+ δ + 1), . . . , y(tn+ δ + n)) T ∈ R, ∀t = 1, . . . , T. (2) where δ ∈ N0, is a given delay of the system. and let y k = y(tn + k + δ + 1) denote the kth element of this vector. Finally, let y∗ ∈ R represent the reference output signal, which is assumed to be constant over the different periods in this paper. Let y∗ k denote the kth element of this vector. At first, we consider the noiseless case where the system to be controlled can be written as follows. Assume the system be represented as a matrix S =  s11 0 . . . 0 s21 s22 0 s31 s32 s33 .. . . . sn1 . . . snn  , (3) with {sij}i≤j denoting fixed scalars. then the inputoutput behavior of the system can be written as y , y t 1 .. y n  = S u t 1 .. un + b1 .. bn  = Su + b. (4) where b = (b1, . . . ,bn) ∈ R is a vector of fixed scalars, representing the intercept terms at the different steps in any interval. Note that in case sij = hi−j for scalars {hτ}τ=0, this would represent a Linear Time Invariant (LTI) system, and the following engineering representation is used:‘