Mathematical Problems in Engineering

Volume 2018, Article ID 3905879, 10 pages

https://doi.org/10.1155/2018/3905879

## An ADRC Method for Noncascaded Integral Systems Based on Algebraic Substitution Method and Its Structure

^{1}Lab of Intelligent Control and Computation, Shanghai Maritime University, Shanghai 201306, China^{2}Department of Electrical, Computer, and Biomedical Engineering, University of Rhode Island, Kingston, RI 02881, USA

Correspondence should be addressed to Zhijian Huang; nc.ude.utmhs@gnauhjz

Received 11 August 2017; Accepted 28 May 2018; Published 14 June 2018

Academic Editor: R. Aguilar-López

Copyright © 2018 Zhijian Huang 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

The Active Disturbance Rejection Control (ADRC) prefers the cascaded integral system for a convenient design or better control effect and takes it as a typical form. However, the state variables of practical system do not necessarily have a cascaded integral relationship. Therefore, this paper proposes an algebraic substitution method and its structure, which can convert a noncascaded integral system of PID control into a cascaded integral form. The adjusting parameters of the ADRC controller are also demonstrated. Meanwhile, a numerical example and the oscillation control of a flexible arm are demonstrated to show the conversion, controller design, and control effect. The converted system is proved to be more suitable for a direct ADRC control. In addition, for the numerical example, its control effect for the converted system is compared with a PID controller under different disturbances. The result shows that the converted system can achieve a better control effect under the ADRC than that of a PID. The theory is a guide before practice. This converting method not only solves the ADRC control problem of some noncascaded integral systems in theory and simulation but also expands the application scope of the ADRC method.

#### 1. Introduction

The Active Disturbance Rejection Control (ADRC) has begun to be used in many areas recently [1–23]. This theory was first proposed by* Han* [24, 25]. The central idea is that the internal dynamic and external disturbance of a controlled system can be estimated and compensated in real time with a tracking differentiator (TD), extended state observer (ESO), nonlinear state error feedback (NLSEF), compensator, etc. Thus, the ADRC may promote the control quality and speed where PID is used [25].

For the ADRC, its ESO is a cascaded integral form, its TD tracks the system state and derivative, and its NLSEF is based on the ESO and TD. Thus, these characteristics make the ADRC suitable for a cascaded integral system. This is because the system order, system variables, and known states are explicit to the ESO and TD for the system in a cascaded integral form. Then, the ADRC usually selects the cascaded integral system as a typical form for an easy design or better control effect. For example, in 2015,* Shao* thought that the ADRC was available to a cascaded integral system, such as a motion control system [10]. In 2006,* Gao* also thought that if a plant model was in a cascaded integral form, the ESO could be established, and the ADRC could has a full state feedback from the ESO [26].

However, in practical control systems, there are many cases of noncascaded integral forms. When necessary, a converting method is needed to get the cascaded integral form. At present, the research in this field is as follows: Some scholars adopted a converting method. The Differential Geometry is one of them [27]. This method combines a nonlinear state conversion and linearization using its object model*.* Also in 2014,* Huang* converted a two-order state space form into the cascaded integral system by a mathematic transform [28]. In 2014,* Huang* used the same method to convert a multiorder state space system [28]. In 2014,* Ramírez-Neria* utilized the decoupling property of the object model and decomposed it into the cascaded connection of two independent blocks [29]. Some ADRC applications were limited to the control system with an implicit cascaded integral form, such as the differential equation, rational proper fraction, or state space form, as shown in [12, 13, 26, 30]. Some ADRC applications were limited to the control system with an explicit cascaded integral system. For example, the nonlinear ADRC has been applied to the fast tool servo systems [14, 31], which are cascaded integral systems with two stages.

As the real control systems have various forms, there are many styles to be converted, and their converting method may also be different. In practical application, the PID feedback control is the most widely used. If the PID control object of a noncascaded integral form can be converted into the cascaded integral system, it would be highly representative.

Therefore, an algebraic substitution method and its structure are proposed in this paper to convert the noncascaded integral system of a PID control object into the cascaded integral form. The adjusting parameters of the ADRC controller are also demonstrated. A numerical example and the oscillation control of a flexible arm are simulated to show the conversion and ADRC control effect. The converted system is proved to be more suitable for a direct ADRC control. The ADRC can achieve a nonovershoot tracking control while satisfying the rapidity under disturbances. In addition, the control effect of the numerical example is compared under the periodical, white noise and inaccurate model disturbances during a step input response while the controller and its parameters are kept invariable. The results show that the converted system can achieve a better control effect for the ADRC than that of a PID.

Thus, this paper presents an approach, which can transform a noncascaded integral system into the cascaded integral form for an easier and better ADRC control. The converted ADRC control system has a good antidisturbance and adaptive effect. The theory is a guide before practice. This converting method solves the ADRC control problems of some noncascaded integral systems in theory and simulation. It also expands the application scope of the ADRC method.

#### 2. The ADRC Control Method

For a continuous system, the ADRC control method adopts the following four steps:

Arranging a transient process for the control reference with the TD:

In it, , are the system state and its first derivative, respectively; is the control reference; is the filtering factor; is the time ruler; sign is a sign function; and is a constructed nonlinear function shown in (2) [16, 32].

Estimating the system states and total disturbance of the controlled object with the ESO:

In it, is the error between the estimated state and system output; , , and are the estimations of the system states ; , , and are gain coefficients; is the control signal; is the system output; and are the parameters of function; and are the outputs of function; is another constructed nonlinear function shown in (4) [16, 32].

The nonlinear state error feedback with the NLSEF:

In it, and are the errors between the estimated states and system states; is an intermediate control output before compensation; and are gain coefficients; are the parameters of function.

The disturbance compensation with a compensator:

In it, is the amplification factor of the control signal; is a final control signal.

#### 3. The Algebraic Substitution Method and Its Structure for the ADRC

*Remark 1. *According to (1) and (3), if , , , , and their counterparts , , , and of the controlled system are explicit, this system is in a cascaded integral form. Then, the TD, ESO, and NLSEF of the ADRC controller can be easily designed.

The cascaded integral system is a closed state feedback system, and its archetype can be described ideally in In it, are the state variables of the controlled system; are the unknown system functions; is the control signal; is the system output; and the coefficient . This kind of cascaded integral system is a typical form suitable for the ADRC.

Then, a method is needed to construct an object system and convert the noncascaded integral system into the above cascaded integral form. The two-order control system in (8) is taken for example.In it, is the control signal; is the system output; is the system state; and is the system disturbance. is set as the control reference. Thus, the error between the system state and control reference is

Then, the proposed algebraic substitution method and its structure adopt the following six steps as shown in Figure 1: