New Results of Global Exponential Stabilization for BLDCMs System

Abstract

The global exponential stabilization for brushless direct current motor (BLDCM) system is studied. Four linear and simple feedback controllers are proposed to realize the global stabilization of BLDCM with exponential convergence rate; the control law used in each theorem is less conservative and more concise. Finally, an example is given to demonstrate the correctness of the proposed results.

Fengxia Tian 1 and Fangchao Zhen 1 and Guopeng Zhou 1 and Xiaoxin Liao 1,2

Academic Editor:Zenghui Wang

1, Institute of Engineering and Technology, Hubei University of Science and Technology, Xianning 437100, China
2, College of Automation, Huazhong University of Science and Technology, Wuhan 430074, China

Received 16 March 2015; Accepted 27 May 2015; 25 October 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A brushless direct current motor (BLDCM) is a synchronous electric motor which is powered by direct current electricity [1]. With the development of electronic technology and control technology, the BLDCM is used widely in motion control applications. The major advantage of BLDCM is the elimination of the physical contact between the mechanical brushes and the commutators [2]. However, it is found that, with certain values of system parameters, the BLDCM exhibits chaotic behavior, such as aperiodic, random, sudden, or intermittent morbid oscillations of the motor angle speed, which can seriously destroy the stable operation of the motor and can even induce industrial drive system collapse [3-8]. Therefore, studying the stability of BLDCM is a significant job and has been receiving considerable attention.

There has been a lot of work done in studying stability and designing controller of BLDCMs [1, 2, 9-13]. For instance, Dadras et al. studied the stability of chaotic uncertain BLDCMs by feedback linearization and sliding-mode control methods [1]. Ge and Chang studied the chaos synchronization and parameters identification of single time-scale BLDCMs by adaptive control and backstepping design method [2]. Ren and Chen proposed a piecewise quadratic state feedback method to control chaos in the BLDCM [9]. Meng et al. designed a fuzzy controller with time-delay feedback to control the chaos movement in the BLDCM [10]. Uyaroglu and Cevher studied sliding-mode control for single time-scale BLDCM [13]. But the controllers proposed in these papers are nonlinear or multiple, and the stability is uniform or asymptotic, which is a weaker property than exponential stability. So, searching a lower-dimensional linear control for the global exponential stabilization of chaotic BLDCM is significant.

Recently, Wei et al. studied the global exponential stabilization for chaotic BLDCMs based on Lyapunov-like theorem [14], although Wei et al. present the method to address the problem about global exponential stability of the BLDCM system firstly. It is innovative to generalize stability and asymptotic stability to global exponential stability and generalize nonlinear feedback control law to linear feedback control law which is easier to implement. There is only one theorem and the control law is not concise. In this paper, we further improved the results of Wei et al. and gave four theorems. The control law used in each theorem is less conservative and more concise, and the proofs are much more simple.

The rest of this paper is organized as follows. In Section 2, system model is presented. In Section 3, the main results of global exponential stabilization for BLDCM system are derived. In Section 4, one example is presented to demonstrate the effectivity of the proposed method. The final section is the conclusion.

2. Preliminaries

Consider the mathematical model of a BLDC motor as follows [3-7]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denote quadrature-axis current, direct-axis current, and angle speed, respectively. [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are system parameters, which determine the type of the dynamical regime of the motor. Consider the case that the motor is running freely under no loading conditions; that is, [figure omitted; refer to PDF] , so system (1) becomes [figure omitted; refer to PDF]

System (2) has chaotic behaviors, with a chaotic attractor shown in Figure 1.

Figure 1: The typical chaotic attractor of the BLDCM at [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

From Figure 1 one can see that the BLDCM is not stable and shows some chaos characteristics, which led to the unstable operation of the motor which is not acceptable in real applications. Therefore, studying stabilization for chaotic BLDCM is significant and this paper will work on it.

Now, some necessary definitions [15-18] that will be used in this paper are introduced in the following. Consider a general nonautonomous system of differential equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the state vector, [figure omitted; refer to PDF] are continuous functions with respect to [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Furthermore, [figure omitted; refer to PDF] is a radially unbounded and positive definite Lyapunov function.

Definition 1.

The trivial solution of system (3) is globally exponentially stable if, for any [figure omitted; refer to PDF] , there exist two constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , such that, for any solution of system (3), it holds [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Lemma 2.

If there exists a Lyapunov function [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , then the trivial solution of system (3) is globally exponentially stable.

Proof.

[figure omitted; refer to PDF] , then [figure omitted; refer to PDF] So the trivial solution of system (3) is globally exponentially stable.

3. Main Results

Theorem 3.

Add a linear feedback controller [figure omitted; refer to PDF] to the second equation of system (2), and then the system can be described as [figure omitted; refer to PDF] or add a linear feedback controller [figure omitted; refer to PDF] to the third equation of system (2), and then the system can be described as [figure omitted; refer to PDF]

If [figure omitted; refer to PDF] is negative definite, which means [figure omitted; refer to PDF] then system (7) is globally exponentially stable at the equilibrium point [figure omitted; refer to PDF] .

And if [figure omitted; refer to PDF] is negative definite, which means [figure omitted; refer to PDF] then system (8) is globally exponentially stable at the equilibrium point [figure omitted; refer to PDF] .

Proof.

Construct a radially unbounded and positive definite Lyapunov function: [figure omitted; refer to PDF]

[figure omitted; refer to PDF] Because [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is negative definite. So [figure omitted; refer to PDF] is negative definite.

Compute the derivative of [figure omitted; refer to PDF] along system (7): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the largest eigenvalue of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

So the zero solution of system (7) is globally exponentially stable.

[figure omitted; refer to PDF] Because [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is negative definite. So [figure omitted; refer to PDF] is negative definite.

Similarly, compute the derivative of [figure omitted; refer to PDF] along system (8): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the largest eigenvector of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

So the zero solution of system (8) is globally exponentially stable.

Therefore, systems (7) and (8) are globally exponentially stable at the equilibrium point [figure omitted; refer to PDF] .

Theorem 4.

Add a linear negative feedback controller [figure omitted; refer to PDF] to the second equation of system (2), and then the system can be described as [figure omitted; refer to PDF] if [figure omitted; refer to PDF] , system (18) is globally exponentially stable at the equilibrium point [figure omitted; refer to PDF] .

Proof.

[figure omitted; refer to PDF] If [figure omitted; refer to PDF] , construct a radially unbounded and positive definite Lyapunov function: [figure omitted; refer to PDF]

Compute the derivative of [figure omitted; refer to PDF] along system (18): [figure omitted; refer to PDF]

So the zero solution of system (18) is globally exponentially stable.

[figure omitted; refer to PDF] If [figure omitted; refer to PDF] , system (18) changes as [figure omitted; refer to PDF]

For the first two equations, construct a radially unbounded and positive definite Lyapunov function: [figure omitted; refer to PDF]

Compute the derivative of [figure omitted; refer to PDF] along system (21): [figure omitted; refer to PDF]

So the zero solution of system (21) is globally exponentially stable on partial variables [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

For the third equation of system (21) [figure omitted; refer to PDF] where we can suppose that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

So the zero solution of system (21) is globally exponentially stable on partial variables [figure omitted; refer to PDF] .

Therefore system (18) is globally exponentially stable on all variables [figure omitted; refer to PDF] at the equilibrium point [figure omitted; refer to PDF] .

Theorem 5.

Add a linear negative feedback controller [figure omitted; refer to PDF] to the third equation of system (2), and then the system can be described as [figure omitted; refer to PDF] if [figure omitted; refer to PDF] , system (25) is globally exponentially stable at the equilibrium point [figure omitted; refer to PDF] .

Proof.

[figure omitted; refer to PDF] If [figure omitted; refer to PDF] , construct a radially unbounded and positive definite Lyapunov function: [figure omitted; refer to PDF]

Compute the derivative of [figure omitted; refer to PDF] along system (25): [figure omitted; refer to PDF]

When [figure omitted; refer to PDF] , the zero solution of system (25) is globally exponentially stable.

[figure omitted; refer to PDF] If [figure omitted; refer to PDF] , the system is [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] is negative definite, [figure omitted; refer to PDF] is negative definite.

For the first two equations, compute the derivative of [figure omitted; refer to PDF] along system (28): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the largest eigenvector of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

When [figure omitted; refer to PDF] , the zero solution of system (25) is globally exponentially stable on partial variables [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

For the third equation of system (28) [figure omitted; refer to PDF]

So the zero solution of system (28) is globally exponentially stable on partial variable [figure omitted; refer to PDF] .

Therefore, system (25) is globally exponentially stable on all variables [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] at the equilibrium point [figure omitted; refer to PDF] .

Theorem 6.

Add linear negative feedback controllers [figure omitted; refer to PDF] , [figure omitted; refer to PDF] to the second and third equations of system (2) simultaneously; then the system can be described as [figure omitted; refer to PDF]

If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that following matrix [figure omitted; refer to PDF] is negative definite. Then system (33) is globally exponentially stable at the equilibrium point [figure omitted; refer to PDF] .

Proof.

Since [figure omitted; refer to PDF] is negative definite, [figure omitted; refer to PDF] is negative definite.

Compute the derivative of [figure omitted; refer to PDF] along system (33): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the largest eigenvalue of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

So when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] system (33) is globally exponentially stable at the equilibrium point [figure omitted; refer to PDF] .

4. Illustrative Example

The parameters of the motor are selected as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and the initial values of the system are chosen as [figure omitted; refer to PDF] .

According to Theorem 3, we may choose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, we can see that it takes short time for the system to be stabilized at origin point and the state trajectories are illustrated in Figures 2 and 3.

Figure 2: The state trajectory of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with the controller [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 3: The state trajectory of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with the controller [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

According to Theorem 4, we may choose [figure omitted; refer to PDF] . Then, we can see that it takes short time for the system to be stabilized at origin point and the state trajectories are illustrated in Figure 4.

Figure 4: The state trajectory of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with the controller [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

According to Theorem 5, we may choose [figure omitted; refer to PDF] . Then, we can see that it takes short time for the system to be stabilized at origin point and the state trajectories are illustrated in Figure 5.

Figure 5: The state trajectory of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with the controller [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

According to Theorem 6, we may choose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, we can see that it takes short time for the system to be stabilized at origin point and the state trajectories are illustrated in Figure 6.

Figure 6: The state trajectory of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with the controllers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

5. Conclusion

In this paper, we investigated the global exponential stabilization for BLDCM system and designed four linear and simple feedback controllers. These controllers realized the global stabilization of BLDCM with exponential convergence rate, and the control law used in each theorem is less conservative and more concise. Finally an example is given to demonstrate the correctness of the proposed results.

Acknowledgments

This work was supported by National Natural Science Foundation of China (61340042), Natural Science Foundation of Hubei Province (2013CFC011), the Project of the Education Department of Hubei Province (T200910, T201009, and XD2014287), and the Project of the Hubei University of Science and Technology (KY13058).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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