Elsevier

ISA Transactions

Volume 110, April 2021, Pages 63-70
ISA Transactions

Research article
Nonlinear disturbance observer based sliding mode control for a benchmark system with uncertain disturbances

https://doi.org/10.1016/j.isatra.2020.10.032Get rights and content

Highlights

  • A sliding mode control scheme, along with a nonlinear disturbance observer, is investigated for a benchmark system.

  • The composite control scheme is devised for the control issue of the TORA system subject to uncertainties/disturbances.

  • The chattering phenomenon can be alleviated to some extent.

Abstract

Taking uncertain disturbances into account, the control problem of the translational oscillator with a rotational actuator (TORA) is considered in this paper. Motivated by the desire to attenuate influences from unknown disturbances/uncertainties, the disturbance-observer-based methodology is utilized. Specifically, a system model with a cascade structure for the TORA system is obtained after a series of coordinate transformations. Then, inspired by the backstepping control technique, for the first subsystem, a virtual control variable is introduced and an error-signal-based system is designed. Next, an estimator is presented and used to estimate unknown disturbances, based upon which a sliding mode control approach is devised to ensure the underactuated TORA system is asymptotically stable at the origin. Finally, the applicability and robustness of the designed control method are examined via three group numerical simulation tests.

Introduction

Underactuated systems, such as cranes [1], [2], cart pendulum [3], quadrotors [4], [5], etc., are widely utilized in extensive practical places, because they own many advantages, which include simple structure, high flexibility, low energy consumption, and so on [6]. However, the control problem for these systems is challenging due to their special feature that the degrees of freedom to be controlled exceed the number of actuators. Consequently, abundant theoretical control design methodologies for control problems of underactuated mechanical systems have been proposed during the past few decades.

The translational oscillator with a rotational actuator (TORA), which consists of an actuated rotor and a non-actuated translational cart, is one of the underactuated mechanical systems. This system possesses several attractive features both from the educational viewpoint and from the research viewpoint. From the educational viewpoint, the dynamics are simple and can be used to understand control theory for educational purposes. In addition, from the research viewpoint, this system is nonlinear and underactuated, which can be utilized as a nonlinear benchmark example to examine the feasibility and efficiency of advanced control methodologies proposed recently. Therefore, studying the control problems of underactuated TORA systems has important practical and theoretical significance.

During the past few decades, lots of researchers have studied the control problems of underactuated TORA systems. In particular, in [7], several cascade and passivity-based control methods are developed to address the stabilization issue of the underactuated TORA system. In general, existing reported control methodologies for the TORA control can be roughly divided into two types. One of the categories mainly focuses on passivity/energy-based control. Specifically, taking the unmeasurable velocity signal into consideration, based on the work in [7], a virtual velocity variable is devised and used to substitute the actual velocity variable and a controller that uses feedback of only position measurements is developed in [8]. But they all do not consider the control amplitude-saturated constraint situation. To deal with this drawback, Sun et al. [9] design an amplitude-restricted controller for the multiple-TORA systems. In addition, taking the variation of system parameters into account, Wu and Gu [10] employ an adaptive control law, which could avoid unwanted unwinding behavior. In [11], consider the control issue of multiple-TORA systems subject to parametric uncertainties, a nonlinear coupling control method is presented. Apart from aforementioned control methods, intelligent control methodology is also used for the TORA control due to its superiority [12]. The other category is mainly in light of the cascade model of the underactuated TORA system. Ref. [13] applies the dynamic surface control technique and a nonlinear control law is exploited for the control issue of this benchmark system. Furthermore, in [14], adaptive control, sliding mode control and fuzzy control are used, and an intelligent control scheme is proposed for the TORA control, wherein the adaptive scheme is utilized for the parameter tuning of the switching surface.

It is noted that all the above-mentioned results do not consider uncertain/unknown disturbances when making control design. In order to deal with this issue, many methodologies with fine robustness have been used for the control problem of the benchmark TORA system. Specifically, taking the air friction and spring damping into consideration, in [15], a robust H control algorithm is provided for the control of the benchmark TORA system. In [16], a variable structure control method is reported for the TORA control. Besides, a sliding mode control algorithm with global robustness is given for the stabilization of the benchmark TORA system [17]. However, in order to copy with external uncertain disturbances, a large switching gain needs to be chosen, which will result in severe chattering. To deal with this problem, in [18], fuzzy logic control is used to find the appropriate sliding factor. Unfortunately, fuzzy sets and corresponding fuzzy rules are specially hard to tune for underactuated systems.

Recently, the disturbance-observer-based technique is utilized to address the affect of uncertain disturbances and the estimated value is taken as a feedforward term in the composite control law for compensation [19], [20], [21]. By virtue of the exact disturbance estimation, the composite control methods exhibit superior control performance without requiring large feedback control parameters.

In order to remedy these drawbacks associated with the published works for the TORA control, inspired by the existing reported disturbance-observer-based control techniques [19], [20], [21], this paper will present a composite control method consisting of two parts. One part is a nonlinear disturbance observer, the task of which is to estimate uncertain disturbances. The other part is sliding mode control, whose main task is to regulate the state variables to the origin. More precisely, by using some coordinate transformations, we transform the TORA dynamics into a cascade form. Then, a disturbance estimator is given in light of the new dynamic equations, and a sliding mode control approach is suggested subsequently. Finally, corresponding stability analysis is provided using strict mathematical analysis, the applicability and robustness of the composite control law are examined by carrying out some numerical simulation tests. To sum up, in comparison with the existing reported related works, the merits of the proposed composite control law are summarized as follows:

(1) Uncertain disturbances are taken into account, the disturbance estimation methodology is employed to eliminate the influences of unknown disturbances.

(2) Benefit from the exact disturbance estimation, the control parameter of the switching function can be chosen much smaller than that of conventional sliding mode control approaches. Therefore, to a large extent, the chattering phenomenon can be alleviated.

The rest parts of this paper are segmented into the four sections. In Section 2, the dynamic equations of the TORA system will be provided and transformed, via several coordinate changes, into a cascade structure. The main results are detailed in Section 3. In the next Section, a series of numerical simulation tests are performed to show the excellent performances of the presented controller. The last part is a conclusion in Section 5.

Section snippets

Mathematic model

The structure of the TORA system is displayed in Fig. 1. As illustrated in Fig. 1, M is the cart mass, the stiffness coefficient of the spring is k, an actuated rotor whose mass is m has rotational radius r and moment of inertia I about its center of mass. The dynamic model of the TORA system is represented by the following equations [7], [8], [10]: (m+M)ẍmr(θ̇2sinθθ̈cosθ)+kx=0mrẍcosθ+(I+mr2)θ̈=N+d where x(t) and θ(t) are the cart position and the rotor angular, respectively, N(t)

Main results

In this part, to realize the above-mentioned control goal, a disturbance estimator will be employed to suppress the affects of unknown disturbances, and a sliding mode control scheme will be developed in light of the devised disturbance estimator. Moreover, corresponding convergence analysis will be given.

Simulation results

To illustrate the stabilization and disturbance estimation performances of the devised composite control scheme, a series of numerical simulation tests are performed utilizing MATLAB/Simulink and divided into three groups. First, for comparison purposes, an existing control method is selected. In the subsequent groups, to reduce chattering, the discontinuous term sgn() is replaced by tanh(20). In the second group, to validate the efficiency of the introduced composite control algorithm here,

Conclusions

For the stabilization of a benchmark system, this paper has presented a composite control approach consisting of a sliding mode control scheme and a nonlinear disturbance observer, where unknown disturbances have been taken into account. The boundedness and convergence of all the system state variables have been proven using rigorous theoretical analysis. The effectiveness and robustness of the presented control approach have been examined via a comparison study and a series of simulation tests.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (25)

  • SunN. et al.

    Nonlinear stabilization control of multiple-RTAC systems subject to amplitude-restricted actuating torques using only angular position feedback

    IEEE Trans Ind Electron

    (2017)
  • WuX. et al.

    Adaptive control of the TORA system with partial state constraint

    Trans Inst Meas Control

    (2019)
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    This work was supported by the National Natural Science Foundation of China (61803339) and the Natural Science Foundation of Zhejiang Province, China (LQ18F030011).

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