Elsevier

Journal of Sound and Vibration

Volume 332, Issue 20, 30 September 2013, Pages 4829-4841
Journal of Sound and Vibration

Additive-state-decomposition-based tracking control for TORA benchmark

https://doi.org/10.1016/j.jsv.2013.04.033Get rights and content

Abstract

In this paper, a new control scheme, called additive-state-decomposition-based tracking control, is proposed to solve the tracking (rejection) problem for rotational position of the translational oscillator with a rotational actuator (TORA, a nonlinear nonminimum phase system). By the additive state decomposition, the tracking (rejection) task for the considered nonlinear system is decomposed into two independent subtasks: a tracking (rejection) subtask for a linear time invariant (LTI) system, leaving a stabilization subtask for a derived nonlinear system. By the decomposition, the proposed tracking control scheme avoids solving regulation equations and can tackle the tracking (rejection) problem in the presence of any external signal (except for the frequencies at±1) generated by a marginally stable autonomous LTI system. To demonstrate the effectiveness, numerical simulation is given.

Introduction

The tracking (rejection) problem for a nonlinear benchmark system called translational oscillator with a rotational actuator (TORA) and also known as rotational–translational actuator (RTAC) has received a considerable amount of attention these years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Due to its special nonminimum phase and nonlinearity, the TORA is often of independent interest to sever as a benchmark for different nonlinear control methods. Some results were presented to concern the tracking (rejection) problem for general external signals [2], [4], [5]. However, the proposed control methods cannot achieve asymptotic disturbance rejection. By taking this into account, the nonlinear output regulation theory was applied to track (reject) external signals generated by an autonomous system. The considered external signals are often constant or periodic due to some periodic operations and vibrations, which can often be modeled as an autonomous system approximately. In this case, asymptotic disturbance rejection can be achieved. With different measurement, the tracking (rejection) problem for translational displacement of the TORA was investigated [6], [7], [8]. Readers can refer to [8] for details. Based on the same benchmark system, some other work was also presented to concern the tracking (rejection) problem for rotational position [9], [10]. In this problem, the rotational actuator may be considered as a rotational antenna, which is required to point a given direction. For the two types of tracking (rejection) problems, regulator equations have to be solved and then the resulting solutions will be further used in the controller design. However, the difficulty of constructing and solving regulator equations will increase as the complexity of external signals increases.

To overcome these drawbacks above, the tracking (rejection) problem for rotational position of the TORA as [9], [10] is revisited by a new control scheme called additive-state-decomposition-based tracking control, which is based on the additive state decomposition.1 The proposed additive state decomposition is a new type of decomposition different from the lower-order subsystem decomposition. Concretely, taking the system ẋ(t)=f(t,x),xRn for example, it is decomposed into two subsystems: ẋ1(t)=f1(t,x1,x2) and ẋ2(t)=f2(t),x1,x2), where x1Rn1 and x2Rn2, respectively. The lower-order subsystem decomposition satisfies n=n1+n2andx=x1x2.By contrast, the proposed additive state decomposition satisfies n=n1=n2andx=x1+x2.In our opinion, lower-order subsystem decomposition aims to reduce the complexity of the system itself, while the additive state decomposition emphasizes the reduction of the complexity of tasks for the system.

By following the philosophy above, the original tracking (rejection) task is “additively” decomposed into two independent subtasks: an output tracking (rejection) subtask for a linear time invariant (LTI) system and a state stabilization subtask for a derived nonlinear system. Since tracking (rejection) subtask only needs to be achieved on an LTI system, the complexity of external signals can be handled easier by the transfer function method. It is proved that the designed controller can tackle the tracking (rejection) problem for rotational position of the TORA in the presence of any external signal (except for the frequency at±1) generated by a marginally stable autonomous LTI system.

This paper is organized as follows. In Section 2, the problem is formulated and the additive state decomposition is recalled briefly first. In Section 3, an observer is proposed to compensate for nonlinearity; then the resulting system is “additively” decomposed into two subsystems; sequently, controllers are designed for them. In Section 4, numerical simulation is given. Section 5 further discusses the proposed method. Section 6 concludes this paper.

Section snippets

Nonlinear benchmark problem

As shown in Fig. 1, the TORA system consists of a cart attached to a wall with a spring. The cart is affected by a disturbance force F. An unbalanced point mass rotates around the axis in the center of the cart, which is actuated by a control torque N. The translational displacement of the cart is denoted by xc and the rotational position of the unbalanced point mass is denoted by θ.

For simplicity, after normalization and transformation, the TORA system is described by the following state-space

Additive-state-decomposition-based tracking control

In this section, an observer is first proposed to compensate for nonlinearity. After the compensation, the resulting nonlinear nonminimum phase tracking system is decomposed into two systems by the additive state decomposition: an LTI system including all external signals as the primary system, leaving the secondary system with a zero equilibrium point. Therefore, the tracking problem for the original system is correspondingly decomposed into two subproblems by the additive state decomposition:

Numerical simulation

In the simulation, set ε=0.2 and the initial value x0=[0000]T in (14), (1b), (1c), (1d). The unknown dimensionless disturbance Fd is generated by an autonomous LTI system (2) with the parameters as follows: S=[0220],Cd=[10]T,w(0)=[00.02]T.The objective here is to design a controller u such that the output y(t)=x3(t)r=0.5 as t meanwhile keeping the other states bounded.

The parameters of the observer (13) are chosen as l1=l2=10. In (16), the parameters of A are chosen as a=1 and K=[0ε12]T.

Discussions

The idea of the proposed additive state decomposition has been implicitly mentioned in existing literature. For example, a commonly used step to transform a tracking problem to a stabilization problem has implicitly used the additive state decomposition, where the reference system presents the primary system, leaving the error dynamics to be the secondary system. The decomposition here is with respect to state. So, we call it “additive state decomposition”. The authors also proposed “additive

Conclusions

In this paper, the tracking (rejection) problem for rotational position of the TORA was discussed. Our main contribution lies in the presentation of a new decomposition scheme, named additive state decomposition, which not only simplifies the controller design but also increases flexibility of the controller design. By the additive state decomposition, the considered system was decomposed into two subsystems in charge of two independent subtasks, respectively: an LTI system in charge of a

Acknowledgment

This work was supported by the National Natural Science Foundation of China (61104012), the 973 Program (2010CB327904), and Specialized Research Fund for the Doctoral Program of Higher Education (20111102120008).

References (25)

  • Z.P. Jiang et al.

    Stabilization and tracking via output feedback for a nonlinear benchmark system

    Automatica

    (1998)
  • A. Fradkov et al.

    Controlled passage through resonance in mechanical systems

    Journal of Sound and Vibration

    (2011)
  • L.-C. Hung et al.

    Design of self-tuning fuzzy sliding mode control for TORA system

    Expert Systems with Applications

    (2007)
  • C.-J. Wan et al.

    Global stabilization of the oscillating eccentric rotor

    Nonlinear Dynamics

    (1996)
  • J. Zhao et al.

    Flexible backstepping design for tracking and disturbance attenuation

    International Journal of Robust and Nonlinear Control

    (1998)
  • Z.P. Jiang et al.

    Global output-feedback tracking for a benchmark nonlinear system

    IEEE Transactions on Automatic Control

    (2000)
  • Z. Petres et al.

    Trajectory tracking by TP model transformation: case study of a benchmark problem

    IEEE Transactions on Industrial Electronics

    (2007)
  • J. Huang et al.

    Control design for the nonlinear benchmark problem via the output regulation method

    Journal of Control Theory and Applications

    (2004)
  • A. Pavlov et al.

    Experimental output regulation for a nonlinear benchmark system

    IEEE Transactions on Control Systems Technology

    (2007)
  • C. Fabio

    Output regulation for the TORA benchmark via rotational position feedback

    Automatica

    (2011)
  • W. Lan et al.

    Adaptive estimation and rejection of unknown sinusoidal disturbances through measurement feedback for a class of non-minimum phase non-linear MIMO systems

    International Journal of Adaptive Control and Signal Processing

    (2006)
  • Y. Jiang, J. Huang, Output regulation for a class of weakly minimum phase systems and its application to a nonlinear...
  • Cited by (27)

    View all citing articles on Scopus
    View full text