Adaptive synchronization method for chaotic permanent magnet synchronous motor

doi:10.1016/j.matcom.2014.03.005

Mathematics and Computers in Simulation

Volume 101, July 2014, Pages 31-42

https://doi.org/10.1016/j.matcom.2014.03.005 Get rights and content

Highlights

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We develop an adaptive synchronization method for a chaotic permanent magnet synchronous motor (PMSM).
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We show stability and convergence of the system responses.
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We give simulation results to verify that the proposed method is useful for adaptive synchronization of a chaotic PM synchronous motor under model parameter and load torque variations.

Abstract

This paper proposes a simple adaptive synchronization method for a chaotic permanent magnet synchronous motor (PMSM). Convergence of the closed-loop system responses is shown by using a Lyapunov function. The proposed adaptive synchronization method does not require the restrictive assumption of the complete availability of information on the PMSM parameters. Simulation results are given to verify that the proposed method can be successfully used for digital implementation and it gives an effective means for adaptive synchronization of a chaotic PMSM under model parameter variations.

Introduction

Because chaos can be encountered in many fields such as medicine, biology, economy, and engineering, synchronization and control of chaotic systems has been studied by many researchers [1], [2], [3], [5], [6], [8], [9], [10], [11], [13], [12], [17], [20], [23], [25], [26], [28]. Chaos control is to remove chaotic behaviors whereas chaos synchronization tries to make a so-called slave system follow a given chaos system or master system. The authors of [15], [18] have recently showed that a permanent magnet synchronous motor (PMSM) can exhibit chaos when motor parameters are in a certain regime. Electric motors in chaos oscillate randomly and an industrial machine with a PMSM in chaos can malfunction. This is not acceptable in practical situations. On the other hand, synchronization of electric motors is necessary for accurately executing repetitive operations in many industrial applications such as robots. These facts have been an impetus for the development of various analysis and control/synchronization methods for a chaotic PMSM [7], [15], [16], [18], [19], [21], [22], [24], [27], [29]. Almost all the previous PMSM control or synchronization methods are based on the assumption that the load torque and/or PMSM parameter values are accurately known. This assumption is unrealistic because a PMSM is subject to parameter variations, load torque variations, and nonlinearities. None of the previous PMSM control or synchronization methods given in [7], [16], [19], [21], [22], [24], [27], [29] can guarantee stability and convergence of the closed-loop system responses under model parameter and load torque variations. Considering these facts, this paper proposes a simple adaptive synchronization method for a chaotic PMSM. The proposed method does not require any information on the PMSM parameter and load torque values. Via simulations it is verified that the proposed method can be successfully used for synchronization of a chaotic PMSM under inexact information on the PMSM parameter and load torque values.

Section snippets

Problem formulation

A PMSM has permanent magnets on the rotating rotor and armature coils on the static stator. The stator generates a rotating magnetic field. Because the permanent magnet rotor will align with the magnetic field of the stator, some appropriately designed commutation law for the stator can make the rotor move with a desired speed trajectory. To simplify the analysis and synthesis problem relating to such commutation laws, many researchers have represented a PMSM in the rotor reference frame by

Adaptive controller for synchronization of PMSM

Let the response system be given by ${\dot{x}}_{r} = A x_{r} - Bu$ where x_r = [ω_r, β_r, i_r]^T is the response system state and u = [u_q, u_d]^T is the control input vector. By introducing the error $e = x - x_{r} = [\begin{matrix} ω_{e} \\ β_{e} \\ i_{e} \end{matrix}] = [\begin{matrix} ω - ω_{r} \\ β - β_{r} \\ i_{ds} - i_{r} \end{matrix}]$ and by subtracting (10) from (5), the following synchronization error dynamics can be obtained $\dot{e} = Ae + B (u - V_{f})$

Theorem 1

Consider the drive system (1) under the assumptions A1–4. Assume that the signals of the drive system, (ω, i_qs, i_ds, V_qs, V_ds), are bounded. Let the control input variables u_q and u_d be given by the

Simulation

We will consider a PMSM (1) with the nominal parameter values n_p = 1, R_s = 0.99 [Ω], L_s = L_q = L_d = 14.25 [mH], ψ_r = 0.031 [V s/rad], J = 0.000047[kg m²], B = 0.0003 [N m s/rad]. From the parameters the following dynamic equation can be obtained $\begin{matrix} \dot{ω} & = & 659.57 i_{qs} - 344.68 ω - 21277 T_{L} \\ {\dot{i}}_{qs} & = & - 63.158 i_{qs} - 2.1754 ω + 70.175 V_{qs} - ω i_{ds} + d_{q} (t) \\ {\dot{i}}_{ds} & = & - 63.158 i_{ds} + 70.175 V_{ds} + ω i_{qs} + d_{d} (t) \end{matrix}$ which can be rewritten as $\begin{matrix} {\dot{ω}}_{e} & = & β \\ \dot{β} & = & - 344.68 β - 41657 i_{qs} - 1434.9 ω - 659.57 ω i_{ds} + 46286 V_{qs} + 659.57 d_{q} (t) \\ {\dot{i}}_{ds} & = & - 63.158 i_{ds} + 70.175 V_{ds} + ω i_{qs} + d_{d} (t) \end{matrix}$ Fig. 2 shows the overall block diagram of

Conclusion

An adaptive synchronization method was developed for a chaotic PMSM. Unlike the previous PMSM control/synchronization methods given in [7], [16], [19], [21], [22], [24], [27], [29] the proposed method can guarantee stability and convergence of the closed-loop system responses under model parameter and load torque variations. Via simulations, it was verified that the proposed algorithm gives very satisfactory performances in the presence of model parameter and load torque variations.

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the ministry of Education, Science and Technology (2012R1A1A2001439).

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Original ArticlesAdaptive synchronization method for chaotic permanent magnet synchronous motor

Highlights

Abstract

Introduction

Section snippets

Problem formulation

Adaptive controller for synchronization of PMSM

Simulation

Conclusion

Acknowledgements

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Original Articles
Adaptive synchronization method for chaotic permanent magnet synchronous motor