Novel auxiliary saturation compensation design for neuroadaptive NTSM tracking control of high speed trains with actuator saturation☆
Introduction
High speed train (HST) as a green transportation has become one of the most popular ways of transportation during the past decade because of its fast, safe, comfortable and economical nature [1], [2]. Since unmodeled nonlinear dynamics, parameter variations, and external disturbances often exist in HST [2], the controller design for guaranteeing the desired position and velocity tracking performances is really a complex and challenging issue. In the last few years, many effective control methods have been proposed to achieve the train trajectory tracking such as adaptive sliding mode control [1], [2], adaptive backstepping control [3], [4], and adaptive iterative control method [5], [6], etc. A novel nonsingular terminal sliding mode (NTSM) control strategy is developed in [1] to solve the tracking control problem of HST without considering the negative effect of actuator saturation. Adaptive trajectory tracking control schemes for HST are developed by using sliding mode methods [2] and backstepping control methods [3], [4] under the assumption that the resistance coefficients are constant, which cannot reflect the actual situation during the train operation. In addition, it’s known that the tedious and complex design of virtual control laws in backstepping design procedure will lead to the phenomenon of ‘explosion of complexity’ [7], [8], [9]. Iterative learning control strategies for HST are proposed in [5], [6], where an assumption that the zero initial velocity is required throughout the repeated iterations, which will limit the practical implementation of the proposed strategies. Therefore, it is scientifically meaningful to develop new control strategies to guarantee desired trajectory tracking of HST.
On the other hand, it should be noted that most of the above results assume that the actuators of HST provide unlimited actuation, and the actuator saturation is not taken into account. In practice, however, the saturation constraints on traction/braking must be considered due to the physical limitations [10], [11], which come either from an artificial soft limiter or a real hard limiter [6]. If neglected, actuator saturation may lead to performance degradation and even instability [12], [13], [14]. It is therefore interesting and important to consider actuator saturation from both a theoretical and a practical perspective. There are two main types of actuator saturations. One is symmetric linear actuator saturation [10], [11], [12], [13], [14], [15], the other is asymmetric linear actuator saturation [6], [16], [17]. However, the saturation problems for these two types of actuator saturations are addressed under a common assumption that the unsaturated region is linear to the control input. In order to remove this assumption, a type of asymmetric nonlinear actuator saturation (ANAS) is first proposed by Chen et al. [18] but under the assumption that the nonlinear functions of the ANAS are strictly monotonous, which obviously limits its application in practical systems. As a consequence, how to construct a control scheme to guarantee desired tracking performance of HST and simultaneously attenuate the negative effect of ANAS without the monotonous assumption in [18] is a tremendous challenge.
It is well known that radial basis function neural network (RBFNN) has strong approximation characteristics and can approach unknown function with arbitrary precision [19], [20], [21], whereas terminal sliding mode (TSM) control has some superior properties over sliding mode control, such as better tracking precision, stronger robustness to system uncertainty and external disturbance, and fast convergence [22], [23], [24]. Therefore, it is interesting to study the trajectory tracking problem for HST by combining RBFNN with TSM control method. The main contributions of this paper are highlighted as follows:
- (1)
A novel auxiliary saturation compensation system is developed to attenuate the negative effect of ANAS without assuming as in [18] that the nonlinear functions of ANAS are strictly monotonous.
- (2)
Two different chatter-free neuroadaptive nonsingular TSM (NTSM) tracking control schemes without involving the sign function as in [1] are developed to guarantee the uniform ultimate boundedness of all signals. In addition, it is worth mentioning that the requirement of zero initial velocities for the HST and the desired trajectory in [5], [6] is not required in this paper.
- (3)
Similar to Lai et al. [20] and Guo et al. [21], an optimized adaptive learning mechanism, which is independent of the number of neurons, is also developed. Only one adaptive parameter is tuned, which will significantly reduce the computation cost and considerably simplify the control design.
- (4)
The two assumptions made in [26], [27], i.e., assuming matched constraint and requiring satisfy a differential mean value property [26] or a Lipschitz condition [27], are removed in this paper.
Section snippets
System description
Generally, the dynamic motion of a high speed train (HST) can be described by a second-order as follows:where and are resp. the position and velocity of the HST and are resp. abbreviated as x and v in the subsequent formulas. is the control input with F(t) and M being resp. the driving/braking force and the total mass (including passengers) of the HST. w(t) is the external disturbance and bounded by with an
Adaptive NN tracking control design and stability analysis
In this section, two adaptive NN tracking controllers will be constructed by using nonsingular terminal sliding mode (NTSM) control technique and approximation power of RBFNN. In the first scheme, adaptive NN tracking control is developed for the case of saturation free, whereas in the second scheme, a novel auxiliary saturation compensation system is developed to eliminate the negative effect of asymmetric nonlinear actuator saturation (ANAS). In addition, alleviating the computational burden
Simulation studies
A numerical simulation example is carried out in this section to illustrate the effectiveness of the proposed chatter-free neuroadaptive NTSM tracking control schemes in Theorems 1 and 2, and simultaneously to show the advantage of the proposed auxiliary saturation compensation system. The train dynamic model used in simulations is stemmed from a train similar to CRH2-A [6], where the total mass of the HST is and the resistance coefficients a(t), b(t) and c(t) are given as follows:
Conclusion
This paper has proposed two chatter-free neuroadaptive NTSM tracking control strategies for high speed train to guarantee desired tracking performance and realize the uniform ultimate boundedness of all signals. Actuator saturation-free and ANAS saturation are considered simultaneously. It is worthwhile to mention that a novel auxiliary saturation compensation system is constructed to eliminate the negative effect of ANAS without requiring some assumptions as the existing results. Besides, the
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This work was supported by National Natural Science Foundation of China (Grant no. 61773056), Fundamental Research Funds for the Central Universities of USTB (No. FRF-GF-18-026B, 230201606500061, FRF-BD-19-002A), the Open Project Program of Engineering Research Center for Metallurgical Automation and Measurement Technology of Ministry of Education, Wuhan University of Science and Technology (No. MADTOF2019A02), China Postdoctoral Science Foundation (No. 2018T110047, 2017M610046), the National Key Research and Development Program of China (Grant No. 2017YFB1401203), Beijing Key Discipline Development Program (No. XK100080537), and National Natural Science Foundation of China (Grant no. 61573263, 61703314, 61673055, 61673056, and 61603274).