A novel iterative learning control (ILC) algorithm is proposed to produce output curves that pass close to the desired trajectory. The key advantage of the proposed algorithm is introducing forgetting factor, which is a function of the number of iterations. Due to the forgetting factor characteristic of ILC, the proposed scheme not only stabilizes the nonlinear system with uncertainties but also weakens interference on the tracking desired trajectory. Simulation examples are included to demonstrate feasibility and effectiveness of the proposed algorithm.

Control schemes for tracking problems can be divided into two steps: trajectory planning and tracking control. In these schemes, the trajectory planner attempts to generate a desired trajectory, which is set in advance. Then, the controller, which is designed to track the desired trajectory, focuses on the system dynamics to generate a sequence of inputs. To improve the accuracy in trajectory tracking, various control schemes such as feedback control [

The algorithm based on ILC is further improved by combining with existing feedback controller, such as PID [

The contribution of this paper is a combination of the high-order feedback iterative learning control and forgetting factor. Through a new ILC algorithm, the perfect tracking control performance of the robot manipulators can be achieved in the repetitive nonlinear time-varying systems [

The remainder of this paper is organized as follows. The iterative learning control problem is described in a general setting in Section

Now, let us consider the repetitive nonlinear time-varying systems with uncertainty and disturbance:

Functions

Initial error of the system at the

Let us make the definitions of norm to simplify formula:

Let

The high-order feedback iterative learning control algorithm with forgetting factor is applied to nonlinear time-varying systems. As

When

To make further improvement on the tracking precision of system, we will find an ideal input

Denote the output tracking error

Starting from an arbitrary continuous initial control input

We propose high-order feedback iterative learning controller as follows. At the

In the work, the forgetting factor

The high-order feedback ILC controller with forgetting factor is constructed as follows:

From (

Let us take norm on both sides of (

Inserting (

Assume that there is a positive real sequence

If

Inserting

When conditions

To demonstrate the effectiveness of the proposed ILC algorithm, we consider a two-degree-of-freedom planar manipulator with revolute joints and use the most basic nonlinear strong coupling example. This paper deals with nonlinear dynamical system (

Equation (

Let

Simulations are carried out on a planar directly driving two-joint robot. And the matrices for the robot arm in the state space are^{2}, ^{2}, and ^{2}.

The expected output trail is

We choose a high-order feedback iterative learning control algorithm with forgetting factor:

We choose a high-order feedback iterative learning control algorithm:

Given to Lemma

After checking the convergence condition in Lemma

The first algorithm is as follows:

The second algorithm is as follows:

Simulation results are shown in Figures

Control input trajectories.

Desired and actual position trajectories.

Max absolute values of the tracking error.

In Figure

Figure

A high-order feedback ILC algorithm with forgetting factor for a class of nonlinear systems with uncertain and nonrepetitive disturbance is introduced in this paper. The main contribution of this paper is adding a forgetting factor to the existing high-order feedback ILC method for nonlinear systems. A rigorous proof is given to show the effectiveness of the proposed algorithm and the asymptotic error convergence along the iteration axis. Simulation results demonstrate that the proposed algorithm has faster convergence speed, greatly reduced tracking error, and better performance in stability and robustness. The future work aims to apply the proposed algorithm to actual robot control and biochemical reactions process.

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 (Grant no. 61473248).