ARTICLE

J. Cent. South Univ. (2019) 26: 1637-1648

DOI: https://doi.org/10.1007/s11771-019-4118-3

Nonlinear cascade control of single-rod pneumatic actuator based on an extended disturbance observer

LI Ai-min(李艾民)¹, MENG De-yuan(孟德远)¹, LU Bo(路波)², LI Qing-yang(李庆阳)¹

1. School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China;

2. National Quality Supervision and Inspection Centre of Pneumatic Products, Ningbo 315500, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Precise position tracking control of the single-rod pneumatic actuator is considered and a nonlinear cascade controller is developed. The proposed controller comprises an extended disturbance observer (EDOB) and a nonlinear robust control law synthesized by the backstepping method. The EDOB is designed to estimate not only the influence of disturbances but also the parameter uncertainties. With the use of parameter and disturbance estimates, the nonlinear cascade controller, which consists of an outer position tracking loop and an inner load pressure loop, is further designed to attenuate the effects of parameter and disturbance estimation errors. The stability of the closed-loop system is proven by means of Lyapunov theory. Extensive comparative experimental results obtained verify the effectiveness of the proposed nonlinear cascade controller and its performance robustness to parameter and external disturbance variations in practical implementation.

Key words: electro-pneumatic servo system; extended disturbance observer; cascade control; robust control; position tracking

Cite this article as: LI Ai-min, MENG De-yuan, LU Bo, LI Qing-yang. Nonlinear cascade control of single-rod pneumatic actuator based on an extended disturbance observer [J]. Journal of Central South University, 2019, 26(6): 1637-1648. DOI: https://doi.org/10.1007/s11771-019-4118-3.

1 Introduction

Electro-pneumatic servo systems are widely used in many applications such as robot manipulators [1], industrial automation equipment [2], and test devices [3], since they possess advantages such as cleanliness, low operating cost, cheapness, high power/weight output, no magnetic field, and no risk of overheating [4, 5]. However, the dynamics of electro-pneumatic servo systems have strong nonlinearities, such as nonlinear relationship between flow and pressure through the control valve, actuator thermodynamics, and friction. Furthermore, electro-pneumatic servo systems are likely to be affected by variations in the physical parameters and unknown external disturbances in industrial applications. These characteristics make controller design of electro- pneumatic servo systems a challenging task.

Since the fixed-gain linear control methods cannot guarantee satisfactory position tracking accuracy of electro-pneumatic servo systems [6], techniques such as gain scheduling [7], optimal control [8], feedback linearization [9], observer based friction compensation [10], model reference adaptive control [11], and self-tuning control [12] were adopted in many studies. However,unmodelled dynamics and external disturbances, which can easily degrade system performance, are not taken into account explicitly in these control strategies. In view of the fact that sliding mode control has been verified to be an effective method to deal with the effects of system uncertainties and external disturbances, it was applied to electro- pneumatic servo systems by lots of researchers during the past decade [13-15]. Furthermore, in order to reduce the chattering in control signal, several types of adaptive-gain sliding mode controllers were proposed [16-18]. As an alternative to higher order sliding mode controllers, the backstepping technique was utilized to develop nonlinear robust controllers for electro-pneumatic servo systems [19-21]. In addition, a cascade controller which consists of three blocks was proposed for a pneumatic cylinder controlled by two servo valves [22, 23]. The disadvantage of this cascade controller is that the requirement of an extra servo valve adds more than 500 dollars per axis of the electro-pneumatic servo system. It should also be noted that because only large nonlinear feedback is employed to attenuate the effects of parameter uncertainties, unmodelled dynamics and external disturbances, the aforementioned robust controllers suffer from quite severe control input chattering when stringent control performance is of major concern. Recently, the adaptive robust control (ARC) method was used for motion control of pneumatic rodless cylinders [24-26]. Unlike the robust controllers proposed in Refs. [17-23], the ARC controller employs the on-line parameter estimation to obtain reliable estimates of some unknown model parameters, and utilizes the sliding mode control technique to deal with other model uncertainties and disturbance. Thus, the position tracking accuracy and robustness of electro-pneumatic servo systems can be further improved. However, the physical model based parameter estimation utilized in the ARC controller is rather complicated for practical implementation.

In the last decade, disturbance observer-based control (DOBC) technique has received a great deal of attention and interest because of its faster response in handling the disturbances, ease of implementation, and less conservativeness [27, 28]. In these studies, parameter uncertainties, unmodelled dynamics as well as external disturbances were generally treated together as the lumped disturbance. Furthermore, the influence of the lumped disturbance was estimated by an observer and then suppressed by a single control action. The effectiveness of the disturbance observer has been validated through applications to many engineering fields, for example, electro- hydraulic systems [29-34], 6-DOF parallel manipulator [35], 2-DOF robotic arm [36], pneumatic muscle [37], and servo motors [38]. However, due to the fact that rather severe parameter uncertainties normally exist in the controlled systems, the achievable performances of disturbance observer-based control strategies are not better than that of robust adaptive controller or adaptive robust controller in most cases.

In order to overcome the ARC problem and inspired by the work of DOBC, a nonlinear cascade controller based on an extended disturbance observer (EDOB) is developed for position tracking control of the single-rod pneumatic actuator. In contrast to most of the existing disturbance observer, the proposed EDOB estimates the parameter uncertainties and the influence of unmodelled dynamics and external disturbances separately. Based on the parameter and disturbance estimate, a nonlinear cascade controller is synthesized by making use of backstepping method. The stability of the closed-loop system is proven by means of Lyapunov theory. The paper is organized as follows: Section 2 gives the dynamic models and problem statement; Section 3 presents the nonlinear cascade controller; Section 4 gives the experimental results to verify the proposed controller; and Section 5 draws the conclusions.

2 Dynamic models and problem statement

As shown in Figure 1, the electro-pneumatic servo system considered in this paper consists of a single-rod actuator (FESTO DNC-32-500-PPV) and a 5/3-way proportional directional control valve (PDCV, FESTO MPYE-5-1/8-HF-010B). The motion of the piston-rod-load assembly can be expressed as

      (1)

where x,andare the piston position, velocity and acceleration, respectively; m is the normal mass of external load plus piston-rod assembly and slider;

Figure 1 Schematic of electro-pneumatic servo system

p_a and p_b denote the absolute pressures of actuator chambers, p₀ is the ambience pressure; A_a and A_b are the cross section areas of piston chambers; A_r is the cross section area of rod; b is the total load and cylinder viscous friction coefficient; F_f is the unknown friction coefficient;is a continuous function which is always chosen as= is utilized as the smooth approximation for the usual static discontinuous Coulomb friction force [24]; d represents unmodelled dynamics and external disturbances.

As shown in Ref. [39], the actuator pressure dynamic can be described by

  (2)

where R is the gas constant; γ is the ratio of specific heat capacities; V_a and V_b are the volumes in the actuator chambers; T_s is the ambient temperature; T_a and T_b are the gas temperatures in the actuator chambers; andare the mass flow rates entering the actuator chambers; andare the mass flow rates leaving the actuator chambers; and are the heat transfer between the air inside the actuator chambers and outside environment; n_p is the polytropic index of expansion or compression; p_s is the supply pressure; V_a0 and V_b0 are the dead volumes of the actuator chambers; L is the stroke of the actuator. The heat transfer between the air inside the actuator chambers and outside environment can be expressed as

           (3)

where h is the thermal conductivity coefficient between air and inner walls of the actuator chambers and D is the piston diameter.

According to Ref. [24], the mass flow rates entering or leaving the actuator chambers can be calculated by

⁼⁽⁴⁾

where is the mass flow rate; u is the PDCV’s input signal; A(u) is the effective flow area of PDCV [40]; C_d is the discharge coefficient; p_u and p_d are the upstream and downstream pressures; p_r is the critical pressure ratio when choked flow occurs; T_u is the temperature of air which is entering the PDCV; λ is the minimal pressure ratio when laminar flow occurs, typically a value of λ=0.99 is used.

Define the state variables and the electro-pneumatic servo system total model can be written as

  (5)

The control objective is to design an EDOB based nonlinear cascade controller such that a given desired trajectory x_d can be tracked by the output of system (5) y=x₁, while guaranteeing a prescribed transient and final tracking accuracy. x_d is assumed to be of at least second-order differentiable. The parametric uncertainties due to unknown b and F_f and unknown external disturbances will be explicitly considered in this paper. Since the extent of parametric uncertainties and external disturbances can be predicted, it is assumed that and

3 Controller design

To deal with the position tracking control problem of the electro-pneumatic system, a nonlinear cascade controller based on an EDOB is developed as shown in Figure 2. Different from previous studies on disturbance observer, parameter uncertainties are no longer considered as a part of disturbance and it will be estimated separately in the proposed EDOB. With the use of parameter and disturbance estimates, the nonlinear cascade controller, which consists of an outer position tracking loop and an inner load pressure loop, is designed by employing the recursive backstepping methodology. Moreover, robust control law is adopted in the nonlinear cascade controller to deal with the effects of parameter and disturbance estimation errors.

3.1  Extended disturbance observer

Although the state x₂ is measurable, it can also be estimated by the following observer:

 (6)

where and are the estimation of x₂, b, F_f and d, respectively; denotes the estimation error of x₂, i.e.,is the observer gain. Defining the estimation errors of b, F_f and d as and , the dynamics of is then given by

             (7)

Based on the above state observer of x₂, the following extended disturbance observer is used to estimate the parameter uncertainties and the influence of external disturbances as well as unmodelled dynamics separately:

                    (8)

              (9)

                   (10)

where and are the observer gains; χ₁, χ₂ and χ₃ are extra corrector terms to be synthesized later to ensure the closed-loop system stability. In Eqs. (8)-(10), the projection mapping is

          (11)

where ζ is a symbol that can be replaced by b, F_f and d.

Figure 2 Block diagram of control system

Using similar arguments about projection mapping as in Ref. [41], one can easily prove that the above extended disturbance observer has the following properties:

P1 (12)

P2  (13)

P3                (14)

P4  (15)

Define a positive semi-definite function V₁as

        (16)

Differentiating V₁ and noting Eqs. (7)-(10) yields

                              (17)

3.2  Nonlinear cascade controller

Step 1: Design of outer position tracking loop

Define the position tracking error as then a sliding surface is defined as follows:

                (18)

where c₀ is a positive feedback gain. Obviously, e₁ is guaranteed to converge to a small value or zero by making s converging to a small value or zero. Therefore, the following design is to make s→0. The derivative of s can be written as

                        (19)

Consideringas a control variable, a virtual control law p_Ld is designed for Eq. (19) as follows:

             (20)

where c₁ is a positive feedback gain.

Define a positive semi-definite function V₂as

                          (21)

where w₁>0 is a weighting factor. Letbe the error variable denoting the difference between the actual and virtual controls of Eq. (19), i.e., Differentiating V₂ and noting Eqs. (5), (17), (19) and (20), one obtains

                   (22)

Therefore, the extra corrector terms χ₁, χ₂ and χ₃ of the extended disturbance observer are chosen as

                           (23)

                      (24)

                            (25)

Substituting Eqs. (23)-(25) into Eq. (22) gives

       (26)

Step 2: Design of inner load pressure loop

Differentiating and substituting the last two equations of Eq. (5) results in

 (27)

where α=A_b/A_a.

Similarly, consider q_L as a control variable, a virtual control law q_Ld is designed for Eq. (27) as follows:

       (28)

where c₂ is a positive feedback gain, w₂>0 is a weighting factor. Once q_Ld is obtained, the desired effective flow area of PDCV A(u) can be calculated by

      (29)

Thus, the PDCV’s input signal u can be further obtained according to the graph “effective flow area vs input signal” (see Ref. [40]). Substituting Eq. (28) into Eq. (27) gives

                   (30)

Consider the following Lyapunov function candidate:

                         (31)

Differentiating V₃ and noting Eqs. (26) and (30) yields

                (32)

which can be further simplified, in view of Eqs. (13)-(15), as

            (33)

Thus, the stability of the closed-loop system consisting of the extended disturbance observer and the nonlinear cascade controller is guaranteed and all system signals are bounded.

4 Experimental results

The electro-pneumatic servo system as shown in Figure 1 is built to test the performance of the proposed controller. Figure 3 shows the picture of the experimental setup. A pulley with attached weight is used to simulate the external disturbance. Three pressure sensors (FESTO SDET-22T-D10- G14-I-M12) are used to obtain the actuator chamber pressures and the tank pressure. A magnetostrictive linear position sensor (MTS RPS0500MD601V810 050) is used to directly measure the piston position and velocity. The system physical parameters are presented in Table 1. The control algorithms are programmed in Simulink on the host computer and implemented on the electro-pneumatic servo system through a dSPACE DS1103 control system. The sampling period is set to 1 ms.

To better verify the effectiveness of the proposed controller, the following three control algorithms are compared:

C1 The proposed EDOB based nonlinear cascade controller is presented in Sections 2 and 3. After trial and error, the parameters of the controller are chosen according to Table 1.

C2 Nonlinear robust controller. This control algorithm is the same as C1 but without using parameter estimation and disturbance observation (i.e., set γ₁=γ₂=γ₃=0 in C1).

C3 ARC. This is the adaptive robust controller, which can be easily designed following the procedure [24].

Figure 3 Photo of experimental setup

Table 1 Parameters of system and proposed controller

Three performance indices are utilized to qualify the control algorithms for a comparison.

1) Maximum absolute value of position tracking error is used as a measure of the transient performance, which is defined as

                         (34)

2) Root-mean-square value of position tracking error is used as an index of the average tracking performance, which is defined as

                       (35)

where T denotes the total running time.

3) Maximal absolute value of position tracking error during the last 10 s is used as an index of the stead-state tracking accuracy, which is defined as

                     (36)

The above three controllers are first commanded to track the reference trajectory x_d=0.125sinπt, and the tracking errors are plotted in Figure 4. The corresponding performance indices are collected in Table 2. As seen, and of C1 and C3 are much smaller than that of C2. It is because the influence of the parameter uncertainties and disturbances in electro-pneumatic system is estimated and compensated properly in C1 and C3, while C2 just uses strong nonlinear robust feedback to suppress the influence of the parameter variations and disturbances. Moreover, and of C1 are close to that of C3. This indicates the effectiveness of the proposed EDOB, which is much simpler than the model-based adaptive laws used in C3. Figure 5 shows the parameter and disturbance estimations of the EDOB. It can be seen that the estimates all converge and stay close to constant values quickly. Due to the fact that the modeling error in actuator pressure dynamic is not treated explicitly, C1 has a much bigger transient tracking error than C3. However, C1 can realize the tracking error improvement within the first cycle and thus is still acceptable in practical applications.

Figure 4 Tracking errors for a 0.5 Hz sinusoidal trajectory without external load and added mass for trajectory 0.125sin(πt):

Table 2 Experimental results in terms of performance indices

Figure 5 Parameters and disturbance estimation of proposed EDOB for 0.5 Hz sinusoidal trajectory motion without external load and added mass

The PDCV’s input signals of three controllers are given in Figure 6, which shows that the control input chattering of C1 is slightly bigger than that of C3. The chamber pressures of the proposed controller are omitted because they are all bounded as assumed.

To further verify the effectiveness of the proposed controller, the trajectory x_d=0.05sin(1.25πt)+0.05sin(πt)+0.05sin(0.5πt) is considered. The tracking error of C1 and the system’s steady-state response are presented in Figures 7 and 8, respectively. The maximum tracking error e_1M of C1 is about 2.12 mm, while the final steady-state tracking error e_1F is about 1.58 mm.

To provide a comparison about the performance robustness of the above three controllers under parameter variations and external disturbance, a 19.6 N payload is added on the slider and another 19.6 N payload is attached to the pulley. For tracking a sinusoidal trajectory (frequency 0.5 Hz and amplitude 0.125 m), the tracking errors are shown in Figure 9 and the corresponding performance indices are collected in Table 2. The tracking performance of C2 degrades significantly, whereas the tracking errors of C1 and C3 remained almost unchanged, which further illustrates the necessity of parameter uncertainties and disturbance rejection. Figure 10 shows the parameter estimations of the proposed controller in this case.

Figure 6 Control inputs of three controllers for 0.5 Hz sinusoidal trajectory motion without external load and added mass for trajectory 0.125sin(πt):

Figure 7 Tracking errors of C1 for a periodic motion trajectory (period=8 s)

Figure 8 Steady-state tracking response of C1 for a periodic motion trajectory (period=8 s)

Figure 9 Tracking errors for a 0.5 Hz sinusoidal trajectory with external load and added mass for trajectory 0.125sin(πt):

Figure 10 Parameters and disturbance estimation of proposed EDOB for 0.5 Hz sinusoidal trajectory motion with external load and added mass

It should be noted that, the disturbance estimate is about 20 N bigger than the case when there is no weight attached to the pulley, which indicates that the parameter uncertainties and disturbance could be estimated accurately by the proposed EDOB. In this case, C1 achieves almost the same steady state tracking performance as before despite the change of payload and external load force (||e₁||_rms and e_1F of C1 are 0.77 mm and 1.53 mm), demonstrating the effectiveness of the proposed EDOB based nonlinear cascade controller.

5 Conclusions

1) An extended disturbance observer is proposed for the electro-pneumatic servo systems in which the parameter uncertainties and the influence of unmodelled dynamics and external disturbances are both estimated.

2) Based on the EDOB, a nonlinear cascade controller is developed by utilizing the backstepping method for precise position tracking control of the single-rod pneumatic actuator. The controller consists of an outer position tracking loop and an inner load pressure loop, in which the robust control law is employed to further attenuate the effects of parameter and disturbance estimation errors. The stability of the closed-loop system is ensured via the Lyapunov method, which shows that a guaranteed robust transient performance and a prescribed stead-state tracking accuracy are achieved.

3) Comparative experimental results are presented to illustrate the effectiveness of the proposed scheme. Experimental results show that the average tracking error and the final tracking error of the proposed controller are close to that of ARC, which gives us the confidence in our research. Compared to the ARC, the proposed EDOB has the merits of faster response in handling the disturbances as well as parametric uncertainties, and ease of implementation. Since the dynamic compensation type fast adaptation is no longer needed, the backstepping controller design becomes simpler. However, it should be noted that the achievable performances of the proposed controller in this paper is not better than that of ARC. There is still a lot of fundamental research has to be carried out to improve this method.

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(Edited by FANG Jing-hua)

中文导读

基于扩展干扰观测器的单杆气缸非线性级联控制

摘要：为实现对单杆气缸活塞运动轨迹的精确控制，本文提出了一种基于扩展干扰观测器的非线性级联控制方法，利用扩展干扰观测器估计干扰与未知模型参数信息，通过非线性鲁棒控制律抑制参数与干扰估计误差、未建模动态的影响。该级联控制器由内环压力控制回路和外环位置回路两部分组成，分别采用滑模控制理论进行设计，利用Lyapunov理论证明了闭环系统的稳定性。试验表明，所设计的控制器能获得良好的轨迹跟踪控制性能，对干扰和系统参数变化具有较强的性能鲁棒性。

关键词：电-气伺服系统；扩展干扰观测器；级联控制；鲁棒控制；位置跟踪

Foundation item: Project(51505474) supported by the National Natural Science Foundation of China; Project(2015XKMS020) supported by the Fundamental Research Funds for the Central Universities, China; Project(2016T90520) supported by the China Postdoctoral Science Foundation, Project supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions, China

Received date: 2018-06-14; Accepted date: 2018-10-23

Corresponding authors: MENG De-yuan, PhD, Associate Professor; Tel: +86-516-83590718; E-mail: tinydreams@126.com; ORCID: 0000-0003-2149-1948