Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 7072032, 15 pages

https://doi.org/10.1155/2018/7072032

## Output Information Based Fault-Tolerant Iterative Learning Control for Dual-Rate Sampling Process with Disturbances and Output Delay

Correspondence should be addressed to Hongfeng Tao; moc.liamtoh@gnefgnohoat

Received 18 August 2017; Accepted 8 February 2018; Published 19 March 2018

Academic Editor: Andrzej Swierniak

Copyright © 2018 Hongfeng Tao 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

For a class of single-input single-output (SISO) dual-rate sampling processes with disturbances and output delay, this paper presents a robust fault-tolerant iterative learning control algorithm based on output information. Firstly, the dual-rate sampling process with output delay is transformed into discrete system in state-space model form with slow sampling rate without time delay by using lifting technology; then output information based fault-tolerant iterative learning control scheme is designed and the control process is turned into an equivalent two-dimensional (2D) repetitive process. Moreover, based on the repetitive process stability theory, the sufficient conditions for the stability of system and the design method of robust controller are given in terms of linear matrix inequalities (LMIs) technique. Finally, the flow control simulations of two flow tanks in series demonstrate the feasibility and effectiveness of the proposed method.

#### 1. Introduction

In industrial applications, many engineering plants operate in continuous time while the system inputs and outputs are sampled, yielding discrete-time signals. Moreover, due to the hardware limitations, process characteristics, and other reasons, sampling each variable with the same frequency is not necessary and realistic. Therefore, the measurable output and input information is usually sampled in different rates from different types of sensors, manual sampling, or laboratory analyses [1, 2]. These systems are often termed as multirate sampling process and dual-rate sampling process is a special case; sometimes the sampling periods of the slow rate sampled variables are integer multiples of the fast rate sampled ones [3]. For example, the control of the bottom and top composition products of a distillation column by acting on the reflux and vapor flow rates is a typical case, where the input control signals can be rapidly adjusted, while the infrequent and delayed composition measurements are only obtained by gas chromatography [4]. In some vehicle control systems, displacement and velocity are measured by using ultrasonic sensors; the two different groups of sensors are located at different locations of the vehicle and have different sampling periods [5]. During the last several decades, this corresponding control problem has attracted considerable attention, including the model identification [6–8] and control algorithms [9–11]. But the control problem of dual-rate sampling systems still has achieved relatively little research results compared with the single-rate sampling case. Moreover, to the best of our knowledge, there are few papers dealing with iterative learning control (ILC) problems for dual-rate sampling process with time delay and actuator fault.

The idea of ILC arose from Uchiyama in 1978 [12]; it represents a powerful approach for high performance tracking control of systems, which execute the same task over a finite duration repeatedly with a given desired trajectory and reset to the starting location once each execution is complete. Each execution of the task is known as a trial, or pass, and its duration is termed the trial length. ILC is currently mainly used to control single-rate sampling process. Compared with standard control scheme, the distinguishing feature of the ILC dynamic sequence of operations is to use the information from previous trials to update the control signal applied on the next one; the major advantage of ILC is the ability to improve system performance from trial to trial and include temporal information from previous trials that would be noncausal in standard systems. Over the past few decades, ILC has drawn significant research attention and increasingly been employed in many industrial processes, such as traffic system [13], networked stochastic system [14], robotic manipulator system [15], multiagent system [16], chemical pharmaceutical crystallization [17], and industrial injection molding batch processes [18].

The design of an ILC law starts, as always, with performance specifications where the novel feature for ILC is the reference trajectory or vector, which is assumed to be the same for all trials in most of the ILC literature. In the case of discrete dynamics, let , denote the output and input, respectively, on trial ; denotes the sampling number over the trial duration. Then if the error on trial is , where denotes the reference signal, and the basic ILC design problem is to construct a sequence of input function that forces the error sequence and input sequence to converge to zero and , respectively, or to within an acceptable tolerance, is termed learned control. It is a common approach to ILC design for discrete dynamics to use the lifting technique [19]. For the SISO systems with a nature extension to the multiple-input multiple-output (MIMO) systems, the input and output on any trial can be represented by supervectors formed by assembling the values at the sample instants into a column vector. Once the ILC law is applied, the propagation of the error can be represented by a linear difference equation and discrete linear systems theory can be utilized for trial-to-trial error convergence analysis and control law design.

Given the finite trial length, trial-to-trial error convergence can occur even if the system is unstable since such a system can only produce a bounded output over a finite time duration. Literature reviews [20–22] confirm that an alternative approach to ILC design is to first apply a feedback control law to produce acceptable dynamics along the trial and then apply ILC to accomplish trial-to-trial error convergence of the resulting 2D system. A drawback of the two-step synthesis procedure is that it does not lead to an optimal combination of the feedback and feedforward actions. Based on an abstract model in a Banach space setting, repetitive process is a particular subclass of 2D system that operates over a subset of the upper right quadrant of the 2D plane and is characterized by a series of sweeps, or passes, through a set of dynamics defined over a finite duration known as the pass length. It is a nature setting for ILC analysis and design; the main advantage is that it gives a systematic way to simultaneously consider behaviour along the time axis and from trial to trial [23]. A detailed treatment of the dynamics of these processes, including their origins in the modeling of mining operations, can be found in [24].

Time delay is also frequently encountered in the transmission of material or information between different parts of a system, including biology, chemistry, economics, population dynamics, and engineering applications. Time delay is one of the main causes of instability and poor performance in process control systems [25, 26], and currently many ILC algorithms have been applied to time delay single-rate sampling process by treating them as batch processes in a finite time on every trial. For example, a robust 2D closed-loop ILC combined with the output feedback scheme has been applied to batch processes with state delay and time-varying uncertainties [27, 28]. Composite iterative learning feedback controllers combined with state and output information are designed in [29]; then the sufficient conditions for delay dependent stability are obtained. However, these proposed methods are just based on the single-rate sampling process model, so that they cannot be directly applied to dual-rate sampling processes. Furthermore, these methods only use consistent slow sampling data and do not fully utilize all sampling data with different sampling intervals to improve control performance.

Moreover, the involved industrial control systems under challenging environment are vulnerable to faults. A fault in a single component may have major efforts on the large system as a whole. Actuator faults will reduce the stability and performance of control systems and may even cause complete breakdown of these systems. Fault-tolerant control is a special action that ensures a fail-safe operation under real-time conditions if components in the control system fail or become faulty [30]. Fault-tolerant ILC design which is sensitive to faults is especially required in application for ILC scheme due to the repeated nature of the control actuator. The extensions to deal with faults for ILC systems receive increasing attentions [31, 32]. For example, a robust fault-tolerant iterative learning control design method is proposed and illustrated by an electric motor system in [31] and also [32] gives iterative learning fault-tolerant control for a class of linear differential time delay uncertain systems with actuator faults in finite frequency domains. The challenge of fault-tolerant control here is how to design a reliable ILC scheme against fault based on the inconsistent dual-rate sampling input and output information with time delay.

This paper develops new results for ILC design applied to time delay dual-rate sampling process with the following contributions:(i)The output information based ILC law design is extended to the fault-tolerant control problem for dual-rate sampling process with time delay and actuator faults.(ii)Monotonic trial-to-trial error convergence conditions for the controlled ILC dynamics are derived.(iii)Robust control issue for disturbance attenuation performance is solved.

This paper is organized as follows: Section 2 describes a dual-rate sampling process with output delay and disturbance in the ILC setting by the state-space model with actuator faults; then it is transformed into a discrete system model form in slow sampling rate without time delay by using the lifting technology. Section 3 formulates the output information based fault-tolerant ILC design problem in the repetitive process setting. Some repetitive process stability theories are given as background in Section 4. Then the sufficient conditions for the stability of the controlled dynamic and the design method of robust controller are analyzed and given in corresponding linear matrix inequalities form in Section 5. Section 6 verifies the effectiveness of the proposed method by the flow control simulations of two flow tanks in series. Finally, some conclusions are given in Section 7.

Throughout this paper, the null and identity matrices with the required dimensions are denoted by 0 and , respectively, and the notation (resp., ) is used to represent the negative definite (resp., positive definite) matrix The notation denotes the transpose of elements in the symmetric position in a matrix. The symbol denotes a block-diagonal matrix with diagonal blocks and . The symbol represents the largest integer which is less than or equal to .

#### 2. System Description

Consider a class of SISO linear continuous processes in Figure 1 with output delay and disturbance; system dynamics are described in the ILC setting by the following state-space model:where the symbol denotes the trial number, is the fixed and finite trial length, , , , and are the system state, input, output, and disturbance vectors, respectively, is the time delay constant, and , , , and are system matrices of appropriate dimensions. Without loss of generality, assume , , on each trial.