Abstract
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.
1. Introduction
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 [1]. The sliding mode control (SMC) method, proposed by Utkin [2, 3] in 1977, is the main control strategy of VSC. Because of its design simplicity and strong robustness, it has been widely studied, vigorously developed, and gradually popularized and applied in practical engineering such as motor and power system control, robot control, and aircraft and satellite attitude control since it was firstly proposed [4–9]. To further optimize the design process and the comprehension performance of traditional SMC system, one new method based on lines cluster approaching theory using a set of interacting lines cluster was put forward by Liu et al. [10, 11]. Based on LCAMC method, Liu et al. designed a basic switch control method for linear time-invariant (LTI) system with single input and equivalent disturbances, and a smoothing method was used to reduce the system buffeting.
Assume that there is a -dimensional linear system with inputs; the traditional sliding mode control method uses reference elements (a line or surface about state variables) to divide the trajectory of the system into reaching phase and sliding mode phase artificially. Accessibility condition makes the trajectory of the system trace to reference elements selected, and the convergence of the sliding mode motion leads it to stable zero point. As we can see, only by trajectory tracking of conference elements it is absolutely impossible to guarantee the system asymptotic convergence. Expanding the amount of reference elements to and moreover taking intersecting state variable lines whose intersection is stable zero point, then what remain to do are making reasonable system trajectory convergence strategy by using the reference elements and designing control law to meet the convergence strategy. Because of the nonlinear switching including in the control law in this paper, the design method still belongs to variable structure control.
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 [12]. The typical structure of ADRC consisted of tracking differentiator (TD), extended state observer (ESO), and nonlinear state error feedback control law (NLSEF). In contrast to existing model-based designs, ADRC method requires little information about the plant. It addresses both the discrepancy between the plant and the model and the external disturbance as a generalized disturbance that is estimated by ESO and is effectively compensated for in the control law. Reference [13] presents the analysis of the stability and tracking characteristics of a particular class of linear ESO and the associated feedback control system linear ADRC for nonlinear time varying systems that are largely unknown. Similar analysis of disturbance rejection problem for Markovian jump nonlinear systems or Markovian jump systems with multiple disturbance and lossy measurements is shown in [14, 15].
Although ADRC method is widely used and combined with other control methods, ADRC and SMC compound control is relatively rare. References [16, 17] present the comparison between ADRC and SMC method in tracking performance, disturbing resistance, and the robustness, and [18] connects ADRC and SMC method to control the water level of steam generator (SG) where SMC replaces the nonlinear state feedback control error rate of ADRC. To the best of the author’s knowledge, the problem of connecting ADRC method with the LCAMC method has not been investigated.
The remainder of this paper is organized as follows. Section 2 introduces the line cluster approaching theory and the principle of ADRC. Section 3 puts forward new kinds of SMC control law based on line cluster approaching theory (LCAMC and ESO-LCAMC) and the design guidelines and mathematical proofs are also given, respectively. Section 4 takes the single input linear time-invariant (LTI) system as example and designs two kinds of control scheme, one based on LCAMC and the other one on ESO-LCAMC. A numerical example together with simulation results is given in Section 5. Finally, we conclude the paper in Section 6.
2. Theory of Lines Cluster Approaching and ADRC
2.1. Theory of Lines Cluster Approaching
In a -dimensional linear space, the coordinate system of which can be noted by , there always exist lines uniquely intersecting at a certain point . These lines are defined as lines cluster, which can be expressed bywhere and are all real constants.
Matrix must be nonsingular to ensure that the lines cluster uniquely intersects at point . Then the solution of (1) can be expressed by , and it is consistent with the coordinate of point , , , .
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 . Furthermore, because matrix is nonsingular, we can get vector , which means the trajectory will tend to point .
For conventional SMC, we usually choose sliding mode plants, that is, , and then we design control law to satisfy that(1)system trajectory approaches each sliding mode plane, that is, ;(2)all the sliding mode motions are convergent, that is, , on sliding mode planes.
For the control of LCAMC, we directly choose intersecting planes, that is, , and then we design control law to merely satisfy that system trajectory approaches each line, that is, and . Therefore, the design method of LCAMC is simpler than that of conventional SMC, and it also has many more advantages [10].
2.2. Theory of ADRC
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 1.
The tracking differentiator (TD) not only traces the reference input signal and arranges the expected transition process but also softens the change of in order to reduce the overshoot of the system output.
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 , the output of the system is and the external disturbance is . Only ESO is used in the paper and its function model can be described as follows:where is a specific piecewise function [19].
3. Design of LCAMC and ESO-LCAMC Method
3.1. The Overall Design Scheme
Consider a -dimensional LTI system:where is the state vector of the system; is the control input; is the equivalent disturbance; is constant matrix; is constant vector.
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 (4) in the equivalent disturbance, consider state variables of the system representing coordinates of -dimensional space and choose lines intersecting to the origin of coordinate. That is,
Then, define a lines cluster vector ; that is,
is a simple expression for (5). According to lines approaching theory, when the trajectory of system will tend to any line of the lines cluster. At this time the reaching mode is formed. That is to say, the asymptotic stability of the system is guaranteed.
So the next step will focus on the design of control law to guarantee and thus make sure that the system is asymptotically stable. According to variable structure control system, the control law consisted of two parts; one is a linear function of state vector and the other is a nonlinear function having switching item which provides robustness. The LCAMC in this paper has the same characteristic.
3.2. Design of LCAMC Control Law
Liu et al. [10, 11] put forward a basic switch control law based on LCAMC method:
The control law can lead system (4) to asymptotic stability, but it has a few disadvantages.
The parameter matrix is hard to solve especially when the state equation has a high order. Because of the intractability, it is difficult to optimize the control law just by some rules we wanted. So it is to analyze the effect of parameters in the control law. Inspired by the traditional exponent reaching law method, a new optimized control law was put forward to solve the problems above.
Theorem 1. Assume the system (4) is controllable and the disturbance is norm-bounded; the control law is designed aswhere is positive, , , and is the symmetric positive definite matrix.
With control law (8), system (4) will possess global asymptotic stability.
Proof. Choose a Lyapunov function:Differentiating (9) about time getsDesign the control law using the exponent reaching law:Substituting (11) in (10) getsAbsolutely always exists, and holds if and only if occurs, which means that system (4) has global asymptotic stability.
Through variable substitution, transformation system can be obtained as follows:
Substituting (11) in (13) gets
Because the system (4) is controllable, , , are not zero.
Multiplying (14) by gets
Divide (15) by ; then get formula (8).
Compared with the control law (7), control law (8) improves the parameter simplicity and the structure optimization of the scheme. Similar to the typical SMC method based on the reaching law, parameter mainly affects the speed of the convergence and parameter mainly determines the robustness of the system. Parameter matrix influences the shape of the approaching curve. These parameters must be adjusted carefully before the application. For example, the bigger the is, the better robustness the system will get, but if is too large, the system will grow a severe buffeting.
3.3. Smoothing Scheme Design
In order to reduce the buffeting caused by the switching function, a common practice is to replace it by the saturation function as follows:where is the thickness of the boundary layer. The scheme designed in this section makes the control law smooth but the degree of system trajectory approaching the lines cluster selected becomes weak. That is to say, the robustness of system deteriorates, which leads to a worse performance in the system state convergence.
3.4. The Observation of Disturbance and System States
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 (8) based on LCAMC contains a disturbance of the system and absolutely the observation and the tracking of the disturbance will greatly influence the accuracy of 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 [20], bring the ESO, the central part of ADRC into the controller.
As said before, the second order LTI system (4) can be described by another way as follows:
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 [21]. Here nonlinear function has the superiority. It is more effective than a linear function in suppressing the steady-state error and its convergence speed is greatly accelerated so the error of decay time is much smaller. It is used in nonlinear ADRC commonly; therefore the parameter can be tuned as a fixed value according to the practical experience.
The nonlinear function is defined as
and are used to estimate the state variables, and estimate the real-time summation of all the model uncertainty and external disturbance of the object; that is, . , , and are all adjustable parameters.
3.5. Summary
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.
4. Design of Simulation Structure
The simulation is carried out on a second order LTI servo control system as system (4), and the parameters arewhere is the measured angular position, is the measured angular velocity, is the equivalent moment of inertia, and is the equivalent damping ratio.
and represent the angular position tracking error and the angular velocity tracking error. Through variable substitution, system (4) can be transformed intowhere is the angular acceleration. Lines cluster vector (6) will be rewritten as expressions of error vector ; that is,
4.1. Design of LCAMC Structure
The tracking problem of system (4) can be transformed into the zero state stability of system (21). Firstly design the LCAMC control law of system (21); then transform it into the control law of system (4) according to (23). Based on the design idea above, the following inference by the new control law of Theorem 1 is given.
Inference 1. Transform the control law of Theorem 1 into
System (4) will possess global asymptotic stability.
Because the simple LCAMC method cannot realize the observation of the disturbance , the output of the LCAMC method is simplified as
Inference 1 constitutes the basic framework for the application of LCAMC in servo control system, as shown in Figure 2.
4.2. Design of ESO-LCAMC Structure
The control law (26) includes parameter which obviously needs to be observed. Extended state observer based lines cluster approaching mode control (ESO-LCAMC) is a hybrid controller, which not only solves the stability of the system but also improves the robustness by observing the state of the system and integrated disturbance and then getting the generalized state error in order to realize the feedforward compensation of the disturbance term.
Design the ESO controller based on the LCAMC method according to the analysis above. The final control output is described as follows:
is the output of LCAMC controller. ESO controller exports the observation of both the disturbance and the system state with the system states feedback and imported.
The basic framework for the application of ESO-LCAMC in servo control system is shown in Figure 3.
5. Simulation
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 (4), whose parameters are
Impose the equivalent disturbance at control input and choose it as . Considering the actual situation, the control input is constrained in 10 V.
5.1. Comparison of LCAMC and SMC
The parameters in LCAMC (8) are chosen as
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 (31) are chosen almost the same:
Here parameter is much bigger because of , and the difference between (31) and (11) must be compensated.
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 4–8.
Figure 4 shows the tracking error of both methods with the input signal 1 HZ. From the figure we can see that all the schemes have good convergence properties. Despite the existence of equivalent disturbance, the tracking speed of servo system is relatively rapid and the control accuracy is relatively high. But it is also clear to see that SMC method has a severe buffeting and by comparison LCAMC method is relatively smooth.
Observe the controller output in Figure 5 and the difference of the two methods becomes more obvious. The controller output of SMC method is buffeting seriously; the LCAMC method has a certain degree of buffeting but it is much smoother.
From Figures 4 and 5, the chatting problem in traditional SMC is reduced in a large part using LCAMC when the parameters are in the same condition.
Figure 6 gives one reason to explain the difference. From the analysis of the angle of phase trajectory, LCAMC method converges faster and smoother than traditional SMC method. Because the convergence of SMC method has two processes, reaching phase and sliding mode phase, but LCAMC method makes the two into one.
Bring the smoothing scheme into the controller to reduce the degree of the buffeting further. Apply the saturation function (16) and choose the parameter as 0.05. Figure 7 shows that buffeting degree of both methods decreases. But the smoothness by the saturation function is based on the weakness of the tracking performance.
Change frequency of the input signal into 8 HZ to take a further study of the tracking performance. Figure 8 shows that LCAMC method still has a good tracking performance, and the tracking error is still small, but SMC method becomes much worse which means that LCAMC method has a faster convergence property.
5.2. Comparison of ESO-LCAMC and LCAMC
The ESO controller takes the system state and LCAMC controller output as input and export as the observation of the disturbance and , as the observation of system states , . The parameters in ESO controller are chosen as follows:
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 9–11.
Figure 9 shows the tracking error of system variable , without white noise. We can see that all the two control schemes possess good convergence properties, but with the observation of disturbance, ESO-LCAMC method has a much smaller tracking error and a much steadier error curve. That means the introduction of ESO decreases the tracking error of LCAMC a step further.
Then the white noise is added to the system states feedback. From Figure 10 we can see that the position tracking error curve of LCAMC method produces severe buffeting because of the uncertainty of white noise. ESO-LCAMC method filters the system states through their observation and thus gets smoother system states feedback. Figure 10 shows the great difference between the two methods.
Moreover, Figure 11 shows the observation of disturbance and system states , . From the figure we can see that ESO controller not only has a good estimation of the system disturbance but also tracks and filters the system state primely.
From Figures 10 and 11 we can see that ESO-LCAMC method not only compensates for the disturbance but also has a strong ability to resist the interference of white noise. This kind of supplement improves the robustness of original system.
6. Conclusion
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 [22]. Methods used to suppress the chatting in SMC have been relatively mature, such as fuzzy method [23, 24], adaptive method [25], and other methods [26, 27]. Connecting them with LCAMC method to further reduce the chatting is one of the future research contents.
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 , , , , . All the parameters have great influence on the convergence performance and robustness of the system and it is hard to adjust them to the best [28, 29]. Nonlinear ADRC parameter tuning with fewer human participations and LCAMC control strategy applied for complicated systems such as nonlinear time varying system [13], Markovian jump systems [14, 15], or switched linear systems [30, 31] are other future research contents, which will be deeply studied and explored in future works.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research was supported by the National Natural Science Foundation of China (91216304, 61403355).