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We address an iterative learning control (ILC) method for overcoming initial value problem caused by local convergence methods. Introducing a feedback recursive form of tracking errors into iterative learning law, this algorithm can avoid a crude linear approximation to nonlinear plants to reach global convergence property. The algorithm’s structure is entirely illustrated. Under assumptions, it is guaranteed that tracking errors of the closed-loop system converge to zero. Besides, we discuss the roles of parameters in iterative learning law for algorithm realization, and a nonlinear case study is presented to demonstrate the effectiveness and tracking performance of the proposed algorithm.

Iterative learning control (ILC) is a methodology for reducing errors from trial to trial for systems that operate repetitively. The objective of iterative learning control is to overcome the imperfect knowledge of system structure to improve tracking performance as few trials as possible. Since ILC issue is originally proposed by Arimoto et al. [

The research of iterative learning control has been focusing on nonlinear systems. ILC algorithms based on optimization theory are effective methods to improve tracking performance of nonlinear systems. Xu and Tan [

This paper develops an iterative learning control method with global convergence property for nonlinear systems. We disclose the relationship between optimization and general ILC nonlinear scheme. Based on this relationship, iterative learning control issue can be reasonably transformed into seeking root for an equation. Defining a new learning gain related with two contiguous iterations, we address iterative learning law with recursive form of errors, and the proposed algorithm avoids a crude linear approximation to nonlinear systems to overcome initial value issue. Giving algorithm’s structure entirely, we theoretically guarantee global convergence property of this algorithm. Afterwards, the importance of learning gain and parameter is discussed. A nonlinear case study is given to demonstrate the effectiveness and tracking performance of the proposed algorithm.

The remainder of this paper is organized as follows. In Section

The class of stable continuous-time nonlinear systems is considered:

The tracking error at the

The control input

Then, sampled nonlinear system (

Aiming at studying the effective global scheme for nonlinear ILC issue, we solve nonlinear equation (

Besides, we define

For

Input any initial

Compute a

Let

Choose

set

If tracking error

In this section, we turn our attention to establishing a convergence theorem for this ILC algorithm.

The following conditions for nonlinear system are assumed.

(i) The level set of control inputs

(ii) System function

In fact, these assumptions are reasonable for actual control process. Imply that the error generated by algorithm is bounded; that is, there is a constant

Under Assumption

Suppose that Assumption

From (

Note that it is necessary and important to discuss the convergence of infinite series including

Consider system (

From (

The proposed ILC method could enjoy the advantage of global convergence and solve the problem of the initial value close enough to the unknown desired input.

It is shown in the previous subsection that this ILC method could reach convergence under Assumption

This indicates that

Subsequently, the choice of learning gain is considered.

In this section, simulation example is presented to demonstrate the effectiveness of the proposed ILC method. The following system is the same as the first example in [

The reference output signal is

Set the initial condition of state

The performance comparison of the proposed ILC method and Newton method based ILC [

Simulation results of both the proposed ILC method and Newton method based ILC at the initial value

However, due to the unknown desired input, it is hard to get an ideal initial value close enough to the desired input without any additional requirement. As usual, the performances of algorithms vary with different initial inputs. If initial value of input is far from the desired one, a very large tracking error may be generated, and even divergence could happen. Therefore, an ILC method with global convergence property becomes a feasible alternative scheme for overcoming this problem.

To clarify the importance of initial control input, take the case

Convergence of the proposed ILC method at the initial value

The above results indicate that the proposed ILC method can converge even in the presence of different initial values. Although convergence speed of the proposed ILC method slows down with the system’s nonlinearity getting severe, it overcomes the limitation of divergence for local convergence algorithms when initial values are far from the unknown desired input.

This paper has proposed an iterative learning control method to avoid initial value issue caused by local convergence algorithms. The chief idea behind this algorithm is to construct iterative learning law with feedback recursive formula of tracking errors. Aiming to satisfy the global convergence performance, a proper learning gain is the key part in this formula, and the definition of learning gain is given. The problem of initial value close to unknown desired input is removed by this improvement in theoretical circumstance. Afterwards, under our assumptions, convergence proof of this algorithm is derived, and the proposed algorithm ensures that tracking errors of closed-loop system uniformly converge to zero. Besides, we discuss the role of learning gain for algorithm realization. Finally, a nonlinear case study is given to demonstrate the effectiveness of this ILC algorithm. In the future, interesting questions could include the possibility of uncertainty for plant and robustness issue.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (61074020).