Adaptive NN finite-time tracking control for PMSM with full state constraints

Abstract

This paper addresses the tracking control problem for a class of permanent magnet synchronous motors (PMSM) systems with asymmetric full-state constraints. To overcome the difficulty of controller design caused by the state constraint problems of system, a nonlinear transformation function i introduced to transform the state constraint problems into a non-constraint problems. Then, radial basis function neural networks (NN) is employed to approximate the uncertainties in the system. In addition, by combining the techniques of command filter and finite-time control, a novel virtual control signal and modified error compensation signals are proposed to construct the actual control law, which solves the problems of “explosion of complexity” and “singularity”. It is shown that all signals of the closed-loop system are bounded and the tracking error remains in a small neighborhood of the origin in finite time. Finally, the simulations show the performance and feasibility of the proposed control scheme.

Introduction

PMSM has the advantages of small size, high efficiency, low power consumption, light weight, simple structure, stable and reliable operation, which are incomparable to other motors. Therefore, the control design of PMSM has attracted significantly increasing attention from researchers. Nevertheless, the dynamic mathematical models of PMSM are highly nonlinear and multivariable. Besides, the PMSM systems are susceptible to unknown nonlinear factors such as load and friction. In order to overcome these problems, many new nonlinear control techniques have been applied to the PMSM systems in recent years, such as sliding mode variable structure control [1], [2], [3], [4], [5], [6], automatic disturbance rejection technology [7], [8], adaptive control [9], [10], [11], [12], and backstepping control [13], [14], [15]. These methods not only enrich the control theory of the PMSM systems, but also improve its performance to some extent. In particular, backstepping method is widely used because it is easy to be combined with adaptive control and can effectively eliminate the influence of time-varying parameters and external disturbance on system performance. It should be mentioned that the aforementioned results only consider systems with infinite time stability, which means that the system state or tracking error converge to the origin or a desired value as time goes to infinity.

Note that the finite-time control technology has better robust and anti-disturbance performance than the asymptotic control approach. Therefore, many finite-time control methods for nonlinear systems have been developed during the past few years. For example, the authors of [16] devise finite time stability theory with fractional powers of the state variables. Based on finite-time control theory, a novel adaptive control scheme was proposed for PMSM in [17] for the first time. Recently, a fast finite-time practical stability criteria for nonlinear systems based on backstepping technical using fractional powers was developed in [18]. Though the proposed design methods can obtain finite time tracking, it suffers from the “explosion of complexity” and “singularity” problems. In order to overcome the above difficulties, the command filter control method was first introduced in [19]. By the aid of recent theoretical results of adaptive backstepping design with the command filtered control method, many significant developments have been acquired for uncertain nonlinear systems [20], [21], [22]. Recently, by combining backstepping technology with the command filtered technology, [22] studied the fuzzy finite time tracking control problem for PMSM.

It is common knowledge that there exists inevitable state constraints problems in most practical engineering. In the tracking control of PMSM, the inherent properties of the motor restrict the state variables such as current, rotating speed and rotor angular velocity, while the voltage amplitude of the direct current side of the inverter also restricts the voltage variation. For this reason, the state constraints should be considered when designing control law for PMSM systems in the practical application. Up to now, many important constrained control methods have been proposed, such as prescribed performance control [23], [24], funnel control [25], [26] and barrier Lyapunov function (BLF)[27], [28], [29], [30], [31], [32]. The prescribed performance control method ensure the tracking errors converge to a small residual set with the prescribed performance bounds[23], [24]. The authors in [25], [26] proposed funnel control to improve the transient response of the control systems. It should be pointed out that the state constraint problems of system was neglected in the above results. Recently, the BLF becomes an effective control method to solve the constraint problems. In [27], the authors first developed the BLF to deal with the output constraint problems of the strict-feedback nonlinear system. An adaptive controller was designed for a class of non-strict feedback systems with output constraints in[28]. Furthermore, the adaptive control schemes [29], [30] were introduced to deal with the time-varying full state constraints. The adaptive command filtered backstepping control approach combined BLF was first put forward in [31] for PMSM systems. However, the common feature of the aforementioned results is that the virtual controller should satisfy a feasibility condition by the BLF technology. That is, in the design procedure of the traditional backstepping, the virtual controller ${\alpha }_{i-1}$ should be constrained in a pre-given region. It is therefore, difficult to choose the suitable parameters, since they need to be calculated off-line to meet the feasibility condition. Moreover, there will be no optimal design parameters to be identified when the state in a small constrained region[32]. To overcome the aforementioned drawbacks, the system transformation approaches have been introduced in [33]–[35]. By using a nonlinear state-dependent function in [33], the problems of asymmetric full-state constraints were solved without involving feasibility conditions for strict-feedback systems. [34] proposed a robust control scheme to deal with asymmetric time varying state constraints for pure-feedback systems. To prevent the state constraints being violated, the nonlinear state-dependent function was introduced for nonstrict-feedback systems [35]. However, to the best of our knowledge, there are no available studies on the PMSM systems with full-state constraints, which motivates us for this study.

Based on the above discussions, this paper focuses on the problem of adaptive NN finite-time tracking control for PMSM systems with asymmetric time-varying full state constraints. In comparison with the previous literatures, the major contributions of this paper are stated as follows:

• (1) By using the transformation techniques, the original constrained systems are transformed into a new “non constrained” systems. Compared with [27], [28], [29], [30], [31], [32], which converts the original constraints into boundary errors, this paper provides a method to deal with the state constraints directly and eliminates the feasibility conditions of the virtual controllers.

• (2) By fusing the techniques of command filtered backstepping control and finite time control, a novel finite time method is developed to construct the controllers, which avoids the “explosion of complexity” and “singularity” problems [39]. The proposed scheme is able to ensure the boundedness of all closed-loop signals and steer the tracking errors into a small neighborhood of the origin in finite time.

• (3) Compared with [20], [21], [22], [39], a finite-time fast stability adaptive control scheme has been obtained by introducing the terms $-{\sigma }_{i2}{\widehat{\theta }}^{2\beta -1}$ in the adaptive law. Therefore, the system not only has the high tracking precision and convergence speed, but also has the good disturbance-rejection ability.