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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.

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 [

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 [

At present, the research situation in this area is as follows: (

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.

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

In it,

The structure diagram of cascaded integral ADRC system expressed in the form of transfer function.

The ADRC method is also shown in Figure

By the structure method of total derivative of composite functions of several variables, the state space object system of noncascaded integral form can be converted into cascaded integral system suitable for ADRC controller. Thus, it extends the application scope of standard ADRC method, no longer limited to the cascaded integral system.

For the case where composite function has several intermediate variables, set

For a given nonlinear time-varying dynamic system with noncascaded integral form, it has three state variables

For the following system of noncascaded integral form, namely, the Chen chaotic system,

The time response and attractor phase space of the Chen chaotic system from initial states

The Chen chaotic system can be controlled by the following self-feedback method, which makes the state variables converge to equilibrium state on zero points.

A controlled object of Chen chaotic system is designed as

So, one of the methods to control Chen chaotic system is to use PID or ADRC controller. The controller generates the appropriate

For the controlled object in (

When the internal/external disturbances are all

Under the same PID controller and its parameter settings, when there is an internal disturbance signal of

Under the same PID controller and its parameter settings, when there is an external disturbance signal of

The PID control effect on Chen chaotic system without disturbing from initial states

The PID control effect on Chen chaotic system from initial states

The PID control effect on Chen chaotic system from initial states

Now, construct (

The adopted ADRC algorithm is shown in [

When the internal/external disturbances are all

Under the same ADRC controller and its parameter settings, when there is an internal disturbance signal of

Under the same ADRC controller and its parameter settings, when there is an external disturbance signal of

The ADRC control effect on Chen chaotic system from initial states

The ADRC control effect on Chen chaotic system from initial states

The ADRC control effect on Chen chaotic system from initial states

From the simulation results, it can be seen that converting control system, expressed as state space form or as state space of noncascaded integral form, into cascaded integral form controlled with ADRC is feasible not only in theory but also in practice. In addition, it is also very easy for the ADRC controller to achieve much better control results than the corresponding PID controller, especially in terms of control speed and overshoot performance, after the system conversion.

This paper first introduces the ADRC method and its typical form of cascaded integral system. The actual application systems have various forms and do not necessarily have the characteristics of the cascaded integral system, especially the most typical state space control system. In this way, this paper presents a kind of method converting the noncascaded integral system into cascaded integral form. Then, the converted system can be controlled directly with ADRC method. Meanwhile, take the Chen chaotic system as an example and convert it into cascaded integral system adopting the method of the total derivative of composite functions of several variables and the structure. Then study comparatively the control effect between the original control system with PID and the converted control system with ADRC, under no disturbance as well as a variety of internal and external disturbance tests, while keeping the controller design and its parameters unchanged. It is very easy for the ADRC controller to achieve much better control results than the corresponding PID controller, especially in the aspect of control speed and overshoot performance. Thus, the research results are shown: (

The authors declare that they have no competing interests.

This work is supported by the NSFC projects of China under Grants nos. 61403250 and 51509151, the Bureau Project of China under Grant no. 2015HT056, and the Science Commission of Shanghai under Grant no. 13510501600.