Journal of Control Science and Engineering

Volume 2019, Article ID 8023730, 10 pages

https://doi.org/10.1155/2019/8023730

## Robust Conditions for Iterative Learning Control in State Feedback and Output Injection Paradigm

Department of Electronics Engineering, College of Technological Studies, PAAET, Kuwait

Correspondence should be addressed to Muhammad A. Alsubaie; wk.ude.teaap@eiabusla.am

Received 7 October 2018; Accepted 2 January 2019; Published 20 January 2019

Academic Editor: Petko Petkov

Copyright © 2019 Muhammad A. Alsubaie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A robust Iterative Learning Control (ILC) design that uses state feedback and output injection for linear time-invariant systems is reintroduced. ILC is a control tool that is used to overcome periodic disturbances in repetitive systems acting on the system input. The design basically depends on the small gain theorem, which suggests isolating a modeled disturbance system and finding the overall transfer function around the delay model. This assures disturbance accommodation if stability conditions are achieved. The reported design has a lack in terms of the uncertainty issue. This study considered the robustness issue by investigating and setting conditions to improve the system performance in the ILC design against a system’s unmodeled dynamics. The simulation results obtained for two different systems showed an improvement in the stability margin in the case of system perturbation.

#### 1. Introduction

Repetitive systems repeatedly perform a predefined task, , with a fixed time duration, , with high precision [1]. Iterative Learning Control (ILC) is a control technique inspired by human learning methods that use repetition to improve performance. ILC was developed in the mid s [2], where the first introduced control law required system stability within the trial time duration as a condition for error convergence from trial to trial. The impact of ILC on industry can be found in many systems such as those used in pick-and-place operations. One example is a robot arm used to move cans, one at a time, from one moving belt to another without content spillage for an infinite number of trials. Other applications of the ILC theory can be found in chemical patch processes and automated manufacturing plants.

In repetitive systems, ILC uses information provided from previous executions/trials to update the current control input with the purpose of enhancing the performance and accommodating periodic disturbances from trial-to-trial. Thus, a system performs a predefined task, records the input, measures the output, and uses the error signal () as a forcing term to update the new control signal. After each trial, the system has to reset to its initial position to start the next one. All of the calculations take place in the stoppage time, the time required for the system to reset between trials. One common approach to ILC takes the form , where is the input, is the learning gain, is the error signal, and index denotes the trial. For more details on ILC theory and applications, refer to [3, 4].

Repetitive control (RC) [5] is another technique used to accommodate periodic disturbances and enhance the performance of repetitive systems without the need for reset between trials; the reference to follow is continuous with , where is the number of samples.

It has been found that any periodic signal can be generated by an autonomous system containing a delay model along the forward path with a positive feedback loop [6]. An accommodation of this type of signal can be achieved using the internal model principle by duplicating the system inside a feedback loop. The work introduced in [7, 8] gives the appropriate selection of the required controller, RC or ILC, to accommodate the periodic signal depending on the location of the internal model of the disturbance. The designed framework explicitly incorporates the current error feedback, while in [9] a modified framework was introduced that incorporated the current error feedback as well as feeding forward the previous error with experimental verification. Basically, the idea in both frames was that a solution for one controller, RC or ILC, was the solution for both.

The novelty of this paper lies with setting new robust conditions for different cases in the ILC design within the framework proposed in [9]. The simulation results obtained showed the reliability of the new design that included unmodeled system dynamics. The results obtained showed the advantage of the new robust conditions over the proposed designs in terms of system perturbation and modeling mismatch.

The following section briefly discusses the ILC design in the general case under the framework proposed in [9]. The design robustness and performance against the unmodeled dynamics of the proposed ILC designs are presented in Section 3. Section 4 considers two examples to illustrate the design advantage, where the first is a -order model of the -axis of a gantry robot and the second is a Nonminimum Phase Plant (NMP). The simulation results obtained are also considered. The conclusion and future work are discussed in Section 5.

#### 2. State Feedback and Output Injection ILC Background

Let the system to be considered in this paper have outputs, inputs, and states. The discrete linear time-invariant system’s overall transfer function in the state space form is given as , where the matrices , and have appropriate dimensions. If the system output is defined as , with as the input, the output equation is .

Considering a single trial with a finite time duration having samples, the model of the system dynamics can be expressed as follows:

where . After each trial, the system resets to its initial position. Thus, there is no loss of generality to set .

Many ILC designs rely on expressing the model in trial notation only, rather than using both the time and trial, because the time is fixed for each trial. These designs include the norm optimal ILC [12] and predictive norm optimal ILC [13]. We here introduce the following supervectors: These allow the dynamics for each trial to be written as follows: with where the matrix elements are the Markov parameters. In the same manner, the reference can be defined with the following vector form: Then, the ILC objective is to generate a new input signal for each trial such that the system output follows the reference trajectory with high precision. Many ILC designs can be found in the literature, where one basic choice is to select an input with the form , where the error vector of trial is and is the learning gain [4, 14].

A periodic signal with appropriate boundary conditions can be generated by an autonomous system consisting of a positive feedback control loop with a pure time delay in the forward path. Thus, a periodic signal of length samples in discrete-time can be modeled as follows: ( is the time instant of trial )with matrix of size given asand the row vector as

Then, the control problem can be defined to find a robust controller (where denotes the discrete-time delay operator) for the robust periodic control problem as follows.

Given an transfer-function matrix with an input vector consisting of both plant and disturbance inputs, , the output signal is defined as in (3), and reference signal . It is necessary to design such that the overall closed loop system is asymptotically stable; the tracking error, , tends to zero along the trial domain; and the previous two conditions are robust.

The solution considered in [7, 8] uses the internal model principle [15] and small gain theorem to set stability conditions to design both feedback and observer gains using the Linear Quadratic Regulator (LQR) in the current error feedback case, where the periodic disturbances act on the system input (ILC). The study [9] considered a more general case that incorporated both the current error and past error in the designed framework.

The work presented in [9] considered two design schemes for ILC. The first used state feedback, and the second used output injection. Each case had two different stability conditions, which depended on using either the current error feedback or past error feedforward. The following subsection briefly explains the design steps in [9] and the stability condition for each case.

##### 2.1. State Feedback-Based ILC

For a single channel, consider the system in (6), which also introduces the following vector:and

for a more general design. A multi-input multioutput case is considered by defining to be a diagonal matrix consisting of along its diagonal.

The same is true for , and , where each diagonal block is repeated times (acting on the system input). Thus, when considering the periodic problem proposed in Figure 1, the transfer function of is given as follows: