Abstract

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.

1. 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 [2–8].

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 [18–21]. 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.

2. Problem Formulation

Consider the following discrete-time system:where denotes the th state variable at the th time instant of the th iteration, , is the state vector with random initial condition in each iteration , is the system input, is an unknown iteration-varying bounded parameter, is a known nonlinear regressor function, and , is the unknown time-varying input gain function. The case that , , can be considered similarly by redefining the input profile.

Define the desired trajectory as , , and assume that is bounded and generated by the following reference model:where the function is continuous with respect to its arguments.

Then, the tracking error at the th iteration is . From (1) and (2), it follows that

The control target is to find a sequence of system input so that the system state of (1) can converge to the desired trajectory asymptotically along the iteration axis.

We shall make some assumptions first.

Assumption 1 (see [16]). The iteration-varying parameter satisfies where , , are known constant parameters and the initial parameters are unknown functions that are linearly independent. In other words, satisfies HOIM with order .

Assumption 2. The nonlinear function satisfies the linear growth condition; that is, , where and are positive constants.

The following lemma will be used in deriving the learning convergence of the proposed control scheme.

Lemma 3 (the Key Technical Lemma [22]). If where , , and are real scalar sequence and is a real vector sequence, and the following two conditions hold: (1)uniform boundedness conditions and for all ;(2)linear boundedness condition where and , then we have and is bounded. denotes the Euclid norm.

Define , where and with . Then we can rewrite (4) as where Repeating (8), we obtain Let be the last row of the matrix ; it renders to Notice that . Owing to the boundedness of , is also bounded; that is, there exists such that for and . As such, the last equation of system (1) can be written in a more compact form:where and .

Remark 4. From system (12), the estimation of the iteration-varying parameter is transformed to that of the iteration-invariant parameter . It implies that the parametric updating law and the control law can be, more conveniently, designed in the iteration domain. This is the main reason why the iteration-varying parameters satisfying HOIM can be addressed along the proposed way.

3. 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 aswhere , , and are the estimates of and at the th iteration, respectively, , and is a projection operator defined as [23] where is the lower bound of the unknown control gain . By using the projection operator function, the possible singularity in (13) can be avoid. In fact, if the initial condition is chosen as .

Observing (13), we have Then, by the definition of state tracking error , where , , , , , and .

The parametric updating laws for and are directly given as follows: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].

4. 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.

Theorem 6. For nonlinear discrete-time system (1), under Assumptions 1 and 2, control law (13) and parametric updating laws (17) ensure that (1)the parametric estimation , is always bounded for all iterations,(2)the tracking errors , , , will converge to zero asymptotically as .

Proof. The whole proof is divided into two parts. Part 1 derives the boundedness of , and Part 2 addresses the asymptotical convergence of .
Part 1 (the boundedness of ). 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 (19), applying the learning laws (17) yields For the second part of the right hand side of (19), similar procedure leads to Now, substituting (20) and (21) into (19) renders to implying that or equivalently Considering the boundedness of , , , and , it follows that is bounded, and therefore is bounded, , .
Part 2 (the asymptotical convergence of ). Our idea is to first prove the asymptotical convergence of , and then the asymptotical convergence of , , can be obtained immediately by the canonical form of the system.
On the one hand, we have from (22) Since is bounded and , it is clear to see that On the other hand, noticing the learning laws (17) and once again, we have yielding Combining (26) and (28) leads to Since includes as the last entry, the following relationship is directly obtained from By the definition of , (30) renders to Further, observing , it is clear that As a result, Since the right hand side of (33) satisfies (31), Now, noticing the error dyanmics of , that is, (16), the relationships (27) and (35) render to In order to prove the asymptotical convergence of via Lemma 3, namely, the Key Technical Lemma, it suffices to prove where and are certain finite constants. This will be addressed in the following.
By the definition of ,First evaluate the upper bound of . From the expression of the proposed controller (13), where and , and . The relationship is adopted in deriving (38).
Second, we evaluate the upper bound of . By the definition of , it follows that Notice that . Then, by Assumption 2, namely, the linear growth condition for the nonlinear regressor , we haveCombining (39) and (40) gives where is an upper bound of , .
Combining (38) and (41) yields where and .
Now, the remaining is to find the relationship between and the quantity . Observing the state error dynamics (3), implying As such, we obtain Hence, by substituting (45) into (42), we have where and .
At last, according to (43), the asymptotical convergence of , , guarantees the asymptotical convergence of , , . The proof is complete.