Mathematical Problems in Engineering

Volume 2016, Article ID 1316046, 8 pages

http://dx.doi.org/10.1155/2016/1316046

## Disturbance Rejection for Fractional-Order Time-Delay Systems

School of Electric Power, South China University of Technology, Guangzhou, Guangdong 510641, China

Received 8 April 2016; Accepted 10 May 2016

Academic Editor: Riccardo Caponetto

Copyright © 2016 Hai-Peng Jiang and Yong-Qiang Liu. 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

This paper presents an equivalent-input-disturbance (EID-) based disturbance rejection method for fractional-order time-delay systems. First, a modified state observer is applied to reconstruct the state of the fractional-order time-delay plant. Then, a disturbance estimator is designed to actively compensate for the disturbances. Under such a construction of the system, by constructing a novel monochromatic Lyapunov function and using direct Lyapunov approach, the stability analysis and controller design algorithm are derived in terms of linear matrix inequality (LMI) technique. Finally, simulation results demonstrate the validity of the proposed method.

#### 1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order [1]. It was found that the description of some dynamic systems in interdisciplinary fields is more accurate when the fractional derivative is used [2, 3]. In the meantime, time-delays are inherent in many dynamic systems [4]. The delay terms may degrade the achievable control performance or even cause instability. In recent years, considerable attention has been paid to fractional-order time-delay systems [5, 6].

The question of stability is of main interest in the control theory. For integer-order systems, the second method of Lyapunov provides a way to analyze the stability of a system without explicitly solving the differential equations. On the base of this method, Li et al. proposed Lyapunov direct theorem for fractional system [7]. Based on the same idea, Baleanu et al. extended the theorem to fractional systems with delay and propose a fractional Lyapunov-Krasovskii stable theorem [8]. The idea of these fractional Lyapunov theorems is derived by constructing a positive definite function and calculating the fractional derivative of this function. However, by now there is not an effectively approach to deal with it, especially the fractional system with delay.

On the other hand, to enhance the control performance, the disturbance rejection for time-delay systems has been addressed in the control theory and engineering. Balochian et al. proposed a sliding mode control law to handle matched disturbances for the fractional-order systems with state delay [9]. control is widely used in disturbance rejection of both the integer-order and fractional-order systems. To calculate the performance, Moze et al. established a bounded real lemma for commensurate fractional-order systems [10]. However, even for the simplest state-feedback stabilization problem, it is not so easy to use parametrization techniques to obtain the feedback gain since the corresponding LMI involves two complex matrix variables. The state-feedback control problem for fractional-order systems was first discussed by Zhuang and Zhong [11]. Shen and Lam [12] addressed the state- feedback suboptimal control problem for fractional-order linear systems. By introducing a new real matrix variable, the feedback gain is decoupled with complex matrix variables and further parameterized by the new matrix variable. Due to the time-delay terms and the shortcoming of fractional direct Lyapunov theorem, to our best knowledge, there is no work relating to the control problem of fractional-order time-delay systems in the existing literature. Fortunately, the equivalent-input-disturbance (EID) [13, 14] is another effective approach to reject both matched and unmatched disturbances for integer-order linear systems [15].

The main objective of this paper is to extend EID disturbance rejection method to fractional-order time-delay systems. The main contribution of this paper is twofold. First, a modified state observer is used to reconstruct the state of the time-delay system in the EID-based control scheme. Second, by using direct Lyapunov approach, a novel monochromatic Lyapunov function was constructed and a stability condition as well as a design algorithm for the control system is derived using LMI techniques.

The rest of this paper is organized as follows. Preliminaries and problem formulation are provided in Section 2. In Section 3, an LMI-based stability condition and controller design algorithm are presented. A numerical example is illustrated to show the validity and superiority of the proposed method in Section 4, and a conclusion follows in Section 5.

#### 2. Preliminaries and Problem Formulation

In this section, some basic definitions and properties (for more details see [1, 2]) are introduced, which will be used in the following sections.

The definitions for fractional derivative commonly used are Grunwald-Letnikov (GL), Riemann-Liouville (RL), and Caputo (C) definition. The advantage of Caputo approach is that the initial conditions for fractional differential equations with Caputo derivative take on the same form as those for integer-order ones [3]. In this paper, we adopt the Caputo definition for fractional derivative.

*Definition 1 (Podlubny [1]). *The Caputo derivative is defined by where is the well-known Gamma function which is defined by and is the first integer which is not less than , that is,

*Definition 2 (see [16]). *Let be the impulse response of a linear system. The diffusive representation (or frequency weighting function) of is called with the following relation: while the diffusive representation of is introduced as

*Remark 3 (see [16]). *It is worth noting that for the fractional-order integral operator can be written as where denote convolution operator and

Lemma 4 (see [16]). *The fractional-order nonlinear differential equation due to the continuous frequency distributed model of the fractional integrator can be expressed aswhere is the same as in Definition 2.*

In this paper, we consider the following factional-order time-delay system:where , , , and are the state of the plant, the control input, the output, and an external disturbance, respectively. , and are constant matrices of appropriate dimensions. is a given continuous initial state vector, and is a positive time-delay.

For convenience of discussion, we assume that has full row rank and its singular-value decomposition iswhere is a semipositive definite matrix and and are unitary matrices.

The definition of the EID-based disturbance rejection is defined as follows.

*Definition 5 (She et al. [13]). *For a controlled system, let the input be zero. A signal, , on the control input channel is called an EID of the disturbance , if it produces the same effect on the output as the disturbance does for all .

In the following, we consider the EID-based disturbance rejection for fractional-order time- delay system in Figure 1. The controllable and observable plant with Caputo definition of the fractional-order derivative is described by time-delay system (7).