Mathematical Problems in Engineering

Volume 2016, Article ID 3796486, 9 pages

http://dx.doi.org/10.1155/2016/3796486

## ADRC Method for Noncascaded Integral System Based on the Total Derivative of Composite Functions of Several Variables

^{1}Merchant Marine College, Shanghai Maritime University, Shanghai 201306, China^{2}Institute of Power Plant and Automation, Shanghai Jiao Tong University, Shanghai 200030, China

Received 31 May 2016; Revised 13 October 2016; Accepted 30 October 2016

Academic Editor: Rafael Morales

Copyright © 2016 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 standard ADRC controller usually selects the canonical plant in the form of cascaded integrators. However, the condition variables of practical system do not necessarily have the cascaded integral relationship. Therefore, this paper proposes a method of total derivative of composite functions of several variables and a structure, which can convert the state space system of noncascaded integral form into the cascaded integral form. In this way, the converted system can be directly controlled with ADRC. Meanwhile, the control of Chen chaotic system is discussed in detail to show the conversion and the controller design. The control performances under different levels of complication and different strengths of disturbance are comparably researched. The converted system achieves significantly better control effects under ADRC than that of the PID. This converting method solves the control problem of some noncascaded integral systems in both theory and application and greatly expands the application scope of the standard ADRC method.

#### 1. Introduction

The Active Disturbance Rejection Control (ADRC) method has begun to attract more and more attention and have been widely used in many areas in recent years [1–23], because of its novel concept, simple implementation, and superior performance in feedback control. The ADRC theory was introduced by Han [24, 25] and significantly improved both the control quality and the control accuracy where PID can be used [25].

However, the ADRC selects the canonical plant as the cascaded integral form. The existing standard ADRC is only available to cascaded integral systems that satisfy the so-called matching conditions, such as the motion control system [10]. ADRC uses extended state observer (ESO) to estimate the object states and its disturbances, which has cascaded integral character. This character makes the ADRC more suitable to exert its unique control effect for cascaded integral systems. However, the canonical form of the cascaded integral system was usually misconstrued as the only form in which ADRC can be used. In fact, the practical system is not always the cascaded integral form. Thus, it is necessary to study how to design ADRC for other forms of system which does not satisfy the matched conditions or replace the PID for the control of state space systems.

At present, the research situation in this area is as follows: () some applications are used as converting method. A converting method is differential geometry [26, 27]. This method is based on the model of the controlled object and combines the nonlinear state conversion and linearization. Huang and Xue used algebraic substitution to obtain one of the partial derivatives, but the second-order state is not converted [27], so it is not a strict conversion method. For the multiorder state space system, Huang and Xue also used the same method, which also has the same situation [27]. Ramírez-Neria et al. used the decoupling property of object model and naturally decomposed it into a cascaded connection of two independent blocks [28], which simplifies the observer design in ADRC. () Some applications are only limited to linear system where the controlled object can be converted into cascaded integral system, that is, the controlled object of differential equation, rational proper fraction, or state space form, such as Refs [12, 13, 29, 30]. So, the application scope is limited. () Some applications are only limited to controlled object system of cascaded expression form; for example, the nonlinear ADRC has been successfully applied to the fast tool servosystem [14, 31]. This is a cascaded integral system of two stages. So, the application scope is even more limited. () A lot of applications are reduced to a pure cascaded integral plant with closed-loop feedback. This is just the original function of ADRC. It does not treat object form before controller design; for example, Wu et al. used ESO and linear ADRC feedback to control a fast tool servosystem [10]. Feng and Guo adopted the ADRC control approach to stabilize a system described by the partial differential equation with corrupted output feedback [32]. Yang, Sun, Zhang, and Huang et al. used similar methods and had the similar situations [7, 8, 15, 16]. () In [33], the nonminimum phase system is actually a noncascaded integral system. The reference presents a combined feedforward and model-assisted ADRC strategy. It has only one tuning parameter but needs a nominal model.

However, the actual controlled systems have various forms. The system needs to be converted to a typical paradigm of ADRC to adapt to its control and better play its control effect when not necessarily conforming to the typical cascaded integral case. Thus, there may be a lot of styles to be converted, and the conversion method may not be the same. To the best of our knowledge, the state space system is one of the most widely expressed forms in practical systems. If the system, expressed as state space form or as state space form, is converted into cascaded integral system suitable for ADRC application, it will be highly representative.

In this paper, the method of total derivative of composite functions of several variables and the structure is used to convert system, expressed as state space form or as state space form, into cascaded integral system suitable for ADRC application. Since this conversion will not be influenced by the accuracy of the object model estimation, for state space object system of noncascaded integral form with unknown object model, it can still realize the conversion and its ADRC control. The error of the object model can be seen as an internal or external disturbance for ADRC controller and then be estimated and compensated by the ESO. This is also a great advantage of ADRC controller. The conversion for state space controlled object with noncascaded integral form has a certain degree of representation and it can solve the conversion problem of many systems. Also, the feedback control effect of ADRC for many noncascaded integral systems can be greatly improved. Thus, the application scope of ADRC method is also greatly expanded.

#### 2. ADRC Method for Cascaded Integral System

The so-called cascaded integral system is a closed state feedback system, described by state equation

In it, , are the state variables of controlled system; is the control input; is the system output. is nonlinear and possibly unknown, and is nonsingular. If the ADRC feedback control system of above states is expressed with structure graph, the controlled object is series form of integral part of , as shown within the gridlines in Figure 1. This kind of cascaded integral system is the typical form suitable for ADRC method.