Additive-state-decomposition-based tracking control for TORA benchmark
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 for example, it is decomposed into two subsystems: and , where and , respectively. The lower-order subsystem decomposition satisfies By contrast, the proposed additive state decomposition satisfies 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 .
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 and the initial value in (14), (1b), (1c), (1d). The unknown dimensionless disturbance Fd is generated by an autonomous LTI system (2) with the parameters as follows: The objective here is to design a controller u such that the output as meanwhile keeping the other states bounded.
The parameters of the observer (13) are chosen as . In (16), the parameters of are chosen as and
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)
- et al.
Stabilization and tracking via output feedback for a nonlinear benchmark system
Automatica
(1998) - et al.
Controlled passage through resonance in mechanical systems
Journal of Sound and Vibration
(2011) - et al.
Design of self-tuning fuzzy sliding mode control for TORA system
Expert Systems with Applications
(2007) - et al.
Global stabilization of the oscillating eccentric rotor
Nonlinear Dynamics
(1996) - et al.
Flexible backstepping design for tracking and disturbance attenuation
International Journal of Robust and Nonlinear Control
(1998) - et al.
Global output-feedback tracking for a benchmark nonlinear system
IEEE Transactions on Automatic Control
(2000) - et al.
Trajectory tracking by TP model transformation: case study of a benchmark problem
IEEE Transactions on Industrial Electronics
(2007) - et al.
Control design for the nonlinear benchmark problem via the output regulation method
Journal of Control Theory and Applications
(2004) - et al.
Experimental output regulation for a nonlinear benchmark system
IEEE Transactions on Control Systems Technology
(2007) Output regulation for the TORA benchmark via rotational position feedback
Automatica
(2011)
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
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