An Iterative Learning Control Design Method for Nonlinear Discrete-Time Systems with Unknown Iteration-Varying Parameters and Control Direction

An iterative learning control (ILC) scheme is designed for a class of nonlinear discrete-time dynamical systems with unknown iteration-varying parameters and control direction.The iteration-varying parameters are described by a high-order internal model (HOIM) such that the unknown parameters in the current iteration are a linear combination of the counterparts in the previous certain iterations. Under the framework of ILC, the learning convergence condition is derived through rigorous analysis. It is shown that the adaptive ILC law can achieve perfect tracking of system state in presence of iteration-varying parameters and unknown control direction. The effectiveness of the proposed control scheme is verified by simulations.


Introduction
Iterative learning control (ILC) is an effective control method in improving the transient response and tracking performance of controlled system when the control task is performed repeatedly in a finite time interval [1].The main idea of ILC is to modify the control input profile by using the deviation of the system output and the desired trajectory so that the track performance can be improved continuously along the iteration axis.Recently, more and more attentions have been put towards ILC design under more general problem settings as well as application of the well-established ILC schemes to industrial and engineering processes [2][3][4][5][6][7][8].
Traditional framework of ILC design needs the strict repeatability of processes, which however is hard to be met in practice.As a result, ILC design with iteration-varying factors is a problem of considerable importance in both theory and practical applications [9].For example, the iteration-varying initial state [10,11], reference [12,13], and disturbances [14,15] have been frequently encountered.In practice, along the iterative axis, these factors can be described by high-order internal models (HOIMs) [16]; that is, the iteration-varying factors in the current iteration are linear combinations of the counterparts in the previous certain iterations [17].It is worth noticing that although HOIM information has been considered to expedite the learning convergence of ILC in [9,16,17], there have been no works addressing ILC design of nonlinear discrete-time systems with iteration-varying HOIM-type uncertainties.
The main contribution of the paper lies in the fact that HOIM-based ILC scheme is proposed for a class of nonlinear discrete-time systems with unknown control direction [18][19][20][21].The learning convergence condition is derived through rigorous analysis.It is shown that the proposed adaptive ILC law can achieve perfect tracking of system state in presence of iteration-varying parameters and unknown control direction.The paper is organized as follows.In Section 2, the problem formulation is given.In Section 3, an adaptive ILC scheme is proposed to achieve perfect tracking. of system output.In Section 4, the learning convergence of the proposed control scheme is addressed rigorously.In Section 5, the effectiveness of the proposed control scheme is verified by simulations.Section 6 concludes the work.
where the function  is continuous with respect to its arguments.
We shall make some assumptions first.

Controller Design
In this section, by making full use of the HOIM information of the parametric uncertainties   (), an ILC controller is designed for the considered nonlinear discrete-time system (1).Notice that the dynamics of  , in (1) has been reformulated as (12), where the parametric uncertainties () and () are iteration-invariant.The control law is given as where φ, (),  = 1, 2, . . ., , and b () are the estimates of   () and () at the th iteration, respectively, φ () ≜ [φ 1, (), . . ., φ, ()]  , and proj(⋅) is a projection operator defined as [23] proj where  min is the lower bound of the unknown control gain ().By using the projection operator function, the possible singularity in ( 13) can be avoid.
Then, by the definition of state tracking error  , (), where φ () ≜ [φ where  is a positive constant.
Remark 5.The ILC (13) with parameter updating laws ( 17) is an adaptive scheme, which is an extension of typical adaptive controller and repetitive control [24].Moreover, this ILC borrows the idea of the HOIM-based ILC in [9,16,17,25].

Convergence Analysis
In this section, the learning convergence of the proposed ILC scheme, that is, control law ( 13) and parametric updating laws (17), will be analyzed in a rigorous way.
Proof.The whole proof is divided into two parts.Part 1 derives the boundedness of φ (), and Part 2 addresses the asymptotical convergence of  , ().
Part 1 (the boundedness of φ ()).Define the composite energy function at the th iteration as whose difference in two consecutive iterations is For the first part of the right hand side of ( In order to prove the asymptotical convergence of  , via Lemma 3, namely, the Key Technical Lemma, it suffices to prove where  1 and  2 are certain finite constants.This will be addressed in the following. where  1 ≜  −1 min  and  2 ≜  −1 min max ∈[0,] ‖φ  ()‖, and  ≜ max ∈[0,] ‖x  ()‖.The relationship |proj(  ())| ≥  min is adopted in deriving (38).
Remark 7. The learning convergence of the proposed ILC scheme, that is, control law (13) and parametric updating laws (17), is proved rigorously for any random bounded initial states.In other words, the perfect tracking can be achieved for any random bounded initial conditions.The main reason is that the desired states at  = 1, 2, . . .,  of system (2) are directly utilized to regulate control input (13) and the effect of the state at  = 0 can be ignored.In order to achieve perfect tracking, traditional ILC schemes restrict the initial states to be identical or convergent [9][10][11].Hence, the efficiency in dealing with any random initial conditions is another contribution of our paper.
In addition, the random initial condition of the system state, x  (0), is shown in Figure 1.
The tracking performance is shown in Figures 2 and 3.More clearly, Figure 2 gives the maximum tracking error of  2, (),  ∈ [1,200], along the iteration axis.It can be seen that the tracking error is decreased significantly in 10 iterations and becomes invisible after 50 iterations.For illustration, the state profile of the system in the 70th iteration and its desired trajectory are given simultaneously in Figure 3.All these simulation results verify the effectiveness of the proposed ILC scheme.

Conclusions
In this paper, an iterative learning control scheme is presented for a class of nonlinear discrete-time systems with unknown iteration-varying parameters and unknown control direction, where the unknown iteration-varying parameters are assumed to satisfy a structure of high-order internal model (HOIM).By making full use of the information embedded in the HOIM, two efficient parametric updating laws are proposed to learn the system uncertainties.The learning convergence of the proposed control scheme is ensured through rigorous analysis.Our next research phase is to exploit the ILC design for systems with HOIM-type uncertainties but without linear growth conditions, as well as its applications.

Figure 1 :
Figure 1: The random initial condition of the system state, x  (0), versus the iteration number.
Our idea is to first prove the asymptotical convergence of  , (), and then the asymptotical convergence of  , (),  ≤  − 1, can be obtained immediately by the canonical form of the system.