Model reference adaptive iterative learning control for linear systems

Abstract

SUMMARYIn this paper, we propose a model reference adaptive control (MRAC) strategy for continuous-time singleinput single-output (SISO) linear time-invariant (LTI) systems with unknown parameters, performing repetitive tasks. This is achieved through the introduction of a discrete-type parametric adaptation law in the 'iteration domain', which is directly obtained from the continuous-time parametric adaptation law used in standard MRAC schemes. In fact, at the first iteration, we apply a standard MRAC to the system under consideration, while for the subsequent iterations, the parameters are appropriately updated along the iteration-axis, in order to enhance the tracking performance from iteration to iteration. This approach is referred to as the model reference adaptive iterative learning control (MRAILC). In the case of systems with relative degree one, we obtain a pointwise convergence of the tracking error to zero, over the whole finite time interval, when the number of iterations tends to infinity. In the general case, i.e. systems with arbitrary relative degree, we show that the tracking error converges to a prescribed small domain around zero, over the whole finite time interval, when the number of iterations tends to infinity. It is worth noting that this approach allows: (1) to extend existing MRAC schemes, in a straightforward manner, to repetitive systems; (2) to avoid the use of the output time derivatives, which are generally required in traditional iterative learning control (ILC) strategies dealing with systems with high relative degree; (3) to handle systems with multiple tracking objectives (i.e. the desired trajectory can be iteration-varying). Finally, simulation results are carried out to support the theoretical development.

Repetitive Learning Control Design and Period Uncertainties

“…represents the two-point backward difference approximation for the time derivative of u * (t) while the delayed termû * (t − T) satisfies the saturation constraint | |û * (t − T) | | ≤ M u with M u taking into account the bound on |u * (t)|. In particular, the function W (t) -including the quadratic termsỹ 2 (t) and ( −1)Π 2 (t) -allows to show that the closed loop error system in suitable coordinates (enlightening the minimum-phase properties of system (1)) is made strictly passive (for t ≥ T) -by choosing sufficiently high k P and k D (when = 2) -with respect to the input (û * − u * ) and to the output y p = ( − 1)Π + (2 − )ỹ(t) so that conditions required by Meyer-Kalman-Yakubovich Lemma in [15] (see also [21]) are satisfied. The "adaptive" term…”

A self‐tuning iterative learning controller for time variant systems

“…Also, according to (9), the controller parameters F(i, j) and G(i, j) are bounded. Thus, by attention to this fact that our system is linear and operates over a finite-time interval 0 ≤ i ≤ M (for example 0 ≤ i ≤ 10) in all iterations, the boundedness of system parameters, desired output trajectory and controller parameters guarantee the boundedness of the state x(i, j) and input u(i, j) (for a similar case see [17]). Thus, r 1 (i, j) and r 2 (i, j) are bounded in all iterations for finite time interval 0 ≤ i ≤ M. Hence, according to (19) there are numerous values for the step size (i, j) that guarantees the convergence of the presented STILCS.…”

Rohrs' Example Revisited: On the Robustness of Adaptive Iterative Learning Control

“…Inspired by this question, in this paper, we use the model reference adaptive ILC (MRAILC) scheme of to demonstrate the lack of robustness of AILC to unmodeled dynamics. The MRAILC scheme of for systems with relative degree 1 is a direct extension of the basic MRAC scheme scrutinized in Rohrs' counterexample, and is therefore a good avenue to explore the convergence properties of AILC in the presence of unmodeled dynamics.…”