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Based on lines cluster approaching theory and inspired by the traditional exponent reaching law method, a new control method, lines cluster approaching mode control (LCAMC) method, is designed to improve the parameter simplicity and structure optimization of the control system. The design guidelines and mathematical proofs are also given. To further improve the tracking performance and the inhibition of the white noise, connect the active disturbance rejection control (ADRC) method with the LCAMC method and create the extended state observer based lines cluster approaching mode control (ESO-LCAMC) method. Taking traditional servo control system as example, two control schemes are constructed and two kinds of comparison are carried out. Computer simulation results show that LCAMC method, having better tracking performance than the traditional sliding mode control (SMC) system, makes the servo system track command signal quickly and accurately in spite of the persistent equivalent disturbances and ESO-LCAMC method further reduces the tracking error and filters the white noise added on the system states. Simulation results verify the robust property and comprehensive performance of control schemes.

The history of variable structure control (VSC) can be traced to 20th century 50s and it has been developing for more than 60 years. It has formed a relatively independent research branch and become an important design method of the automation control system [

Assume that there is a

To further optimize the performance of the control law, the observation of the disturbance is very important. The active disturbance rejection control (ADRC), a method pointing at nonlinear uncertain system, is proposed by Han [

Although ADRC method is widely used and combined with other control methods, ADRC and SMC compound control is relatively rare. References [

The remainder of this paper is organized as follows. Section

In a

Matrix

Assume one trajectory in the aforesaid linear space is synchronously approaching any line of lines cluster, which leads to the formation of LCAMC and can be mathematically explained by

The equations above can be shortened for

For conventional SMC, we usually choose

system trajectory approaches each sliding mode plane, that is,

all the sliding mode motions are convergent, that is,

For the control of LCAMC, we directly choose

ADRC consists of three main parts, that is, TD, ESO, and NLSEF. Consider a second order servo control system and design a second order ADRC controller. The structure is shown in Figure

Diagram of ADRC control structure.

The tracking differentiator (TD) not only traces the reference input signal

The nonlinear state error feedback control law (NLSEF) determines the control law by calculating the difference of expansion state observed by ESO and transition process arranged by TD.

The extended state observer (ESO) is the central part of ADRC. It adopts the method of dual channel compensation, having a dynamic observation of the output position information and its differential, and expands the disturbance of the system into a new order and then provides real-time estimation and compensation. It is conceived to estimate not only the external disturbance but also the plant dynamics. Among the disturbance estimators, ESO requires the least amount of plant information.

As described above, assuming the control quantity is

Consider a

Because most of the servo systems are second order or cascade of second order, we choose a second order LTI system as the research object for the simplicity.

To solve the stability problem of system (

Then, define a lines cluster vector

So the next step will focus on the design of control law to guarantee

Liu et al. [

The control law can lead system (

The parameter matrix

Assume the system (

With control law (

Choose a Lyapunov function:

Differentiating (

Design the control law using the exponent reaching law:

Substituting (

Absolutely

Through variable substitution, transformation system can be obtained as follows:

Substituting (

Because the system (

Multiplying (

Divide (

Compared with the control law (

In order to reduce the buffeting caused by the switching function, a common practice is to replace it by the saturation function as follows:

As it is introduced above, TD, ESO, and NLSEF are all components of ADRC. TD traces the signal input and NLSEF determines the control law by nonlinear feedback function, which are all completed by the LCAMC. But the control law (

On the other hand, in the practical engineering the system states detected by the sensors always contain lots of noise, which will seriously affect the performance of the controller, even to the extent that it would damage the stability of the system. To solve the problem above [

As said before, the second order LTI system (

Then design the ESO filter equation as

As to linear ADRC, parameter tuning of nonlinear ADRC is difficult. But the nonlinear ADRC behaves much more effectively than the linear ADRC [

The nonlinear function

Up to now, a complete control scheme has been established.

LCAMC method is the main part of the control law, but the scheme still needs supplement. The introduction of ESO realizes the observation of both the disturbance and the states of the system. This optimization solves the problem of both the interference compensation and the signal filtering, which greatly enhance the performance of the control law.

The simulation is carried out on a second order LTI servo control system as system (

The tracking problem of system (

System (

Because the simple LCAMC method cannot realize the observation of the disturbance

Inference 1 constitutes the basic framework for the application of LCAMC in servo control system, as shown in Figure

Diagram of LCAMC control scheme.

The control law (

Design the ESO controller based on the LCAMC method according to the analysis above. The final control output is described as follows:

The basic framework for the application of ESO-LCAMC in servo control system is shown in Figure

Diagram of ESO-LCAMC control scheme.

In this section, two comparisons of simulation are designed. One is the proposed LCAMC with those of conventional SMC, in the aspect of tracking performance, degree of chatting, and width of frequency band. The other one is the proposed LCAMC with the optimized ESO-LCAMC in the aspect of tracking performance and the inhibition of the system noise. The computer simulation is carried out on a LTI servo system as (

Impose the equivalent disturbance at control input and choose it as

The parameters in LCAMC (

Because the LCAMC method does not have the observation of disturbance, choose parameter

In order to make a fair analysis of the two methods, the traditional SMC method is also designed based on reaching law approach as follows:

Similarly, the parameters in (

Here parameter

Select the position tracking input as a sinusoidal signal whose frequency is 1 HZ and amplitude is 1 V. The comparisons with respect to state response under LCAMC method and SMC method are all illustrated in Figures

Error curve of position tracking (1 HZ).

Controller output (1 HZ).

Phase trajectory (1 HZ).

Controller output with saturation function (1 HZ).

Error curve of position tracking (8 HZ).

Figure

Observe the controller output in Figure

From Figures

Figure

Bring the smoothing scheme into the controller to reduce the degree of the buffeting further. Apply the saturation function (

Change frequency of the input signal into 8 HZ to take a further study of the tracking performance. Figure

The ESO controller takes the system state

Still select the position tracking input as a sinusoidal signal whose frequency is 1 HZ and amplitude is 1 V. The comparisons between LCAMC method and ESO-LCAMC method are all illustrated in Figures

Error curve of position tracking (without white noise).

Error curve of position tracking (with white noise).

Observation of disturbance and system states (with white noise).

Figure

Then the white noise is added to the system states feedback. From Figure

Moreover, Figure

From Figures

After theoretical analysis and numerical simulations above, we can surely conclude that LCAMC method optimizes the tracking performance and convergence properties of traditional SMC method by making reaching phase and sliding mode phase into one. It not only simplifies the process of design but also makes the tracking trajectory smoother.

Despite the improvement of LCAMC method, buffeting problem still exists in the control scheme and seriously limits its practical application in engineering. Bringing the saturation function into the controller exactly decreases the degree of buffeting, but it also weakens the convergence performance of the system [

Through the extended state observer, ESO-LCAMC method goes one step further. It has a much smaller tracking error and a much steadier error curve than the proposed LCAMC method. The ability to filter and tracking system states also enhance the robustness and practical application of the control method. By connecting the ADRC method and LCAMC method, a complete control scheme is established.

Furthermore, the control scheme has many parameters such as

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (91216304, 61403355).