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.

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order [

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 [

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 [

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

In this section, some basic definitions and properties (for more details see [

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 [

The Caputo derivative is defined by

Let

It is worth noting that for the fractional-order integral operator

The fractional-order nonlinear differential equation

In this paper, we consider the following factional-order time-delay system:

For convenience of discussion, we assume that

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

For a controlled system, let the input

In the following, we consider the EID-based disturbance rejection for fractional-order time- delay system in Figure

Configuration of the EID-based time-delay system.

In Figure

To reproduce the state of the time-delay plant (

Unlike the Luenberger-type observer constructed in [

The state-feedback control law

As discussed in [

Observer (

Since the output,

The purpose of this paper is to investigate the design problem of the full-order state observer (

In this section, we will present an LMI-based method for both stability analysis and the parameter design for the proposed control law.

First, set the exogenous signals to be zero, that is,

Define

Set

As we all know, plenty of LMI-based stability and stabilization conditions have been proposed for integer-order time-delay systems. However, by now, there is not a simple and effective approach to stabilize a fractional-order time-delay system according to the direct Lyapunov theorem and LMI technique.

In the following, we will propose a Lyapunov theorem for fractional-order system with delay and get an LMI-based stability condition.

The fractional-order time-delay system (

It follows from Lemma

Let us consider the Lyapunov functions:

The time derivative of

Now, a second Lyapunov-Krasovskii function candidate

Therefore, if

By constructing a novel monochromatic Lyapunov function and using direct Lyapunov approach, a sufficient condition of stability for fractional-order time-delay systems is obtained. It is worth mentioning that the authors in [

Based on the proposed stability condition, we can easily analyze the stability of a fractional-order time-delay system and design a stabilization controller to stabilize a fractional system with delay by many approaches which have been used in integer-order systems.

To obtain the main result in this paper, we need the following lemmas.

For a given matrix

For a given symmetric matrix

Based on the above lemmas, a stabilization result is established for the EID-based disturbance rejection control systems as follows.

The fractional-order time-delay system (

Moreover, the gains of the state-feedback controller and the observer are given as

Applying Lemma

In terms of Lemma

Let

Using Theorem

This section presents a numerical example to demonstrate the validity of the proposed design method.

Assume that the parameters of plant (

A unit step signal is introduced as the reference input at

In terms of Algorithm

A low-pass filter is selected as

By Theorem

To simulate the fractional-order systems, an Oustaloup’s recursive poles/zeros filter [

The system response without control input is shown in Figure

System response without control input.

System response with the proposed method.

For comparison, we used the proposed method in [

System response with the proposed method in [

This paper has presented an EID-based disturbance rejection method for fractional-order time-delay systems. The configuration with a modified state observer and a disturbance estimator was first constructed. Then, by introducing a continuous frequency distributed equivalent model and using direct Lyapunov approach, the sufficient condition for asymptotic stability of the closed-loop fractional-order time-delay system is presented. Based on this stability condition, the parameters of the controller for the closed-loop system can be easily obtained by many approaches which have been used in integer-order systems. Finally, the simulation results demonstrated that the proposed method can reject matched disturbance of fractional-order time-delay systems effectively without knowing any prior information about the disturbances.

The authors declare that they have no competing interests.