Chaos control of single time-scale brushless DC motor with sliding mode control method

Turkish Journal of Electrical Engineering & Computer Sciences http://journals.tubitak.gov.tr/elektrik/ Research Article Turk J Elec Eng & Comp Sci (2013) 21: 649 – 655 c TÜBİTAK  doi:10.3906/elk-1111-59 Chaos control of single time-scale brushless DC motor with sliding mode control method Yılmaz UYAROĞLU,∗ Barış CEVHER Department of Electrical and Electronics Engineering, Faculty of Engineering, Sakarya University, 54187 Serdivan, Turkey Received: 28.11.2011 • Accepted: 24.01.2012 • Published Online: 03.05.2013 • Printed: 27.05.2013 Abstract: In this paper, the sliding mode control (SMC) scheme of single time-scale brushless DC motor (BLDCM) is investigated. The SMC method consists of 2 sections. To simplify the directive of the stability of the controlled single time-scale BLDCM in the sliding mode, first a special type of PI switching surface is adopted. Second, the SMC controller is obtained to guarantee the occurrence of the PI switching surface. The effectiveness of the theoretical analysis is evaluated by numerical simulations. Thus, the numerical results are used to show the verification and trustworthiness of the proposed method. Key words: Brushless DC motor, sliding mode control, chaos, nonlinear system 1. Introduction Chaos is defined as an aperiodic long-time behavior arising in a deterministic dynamical system that exhibits a sensitive dependence on initial conditions [1]. A key element of deterministic chaos is the sensitive dependence of the trajectory on the initial conditions. This, the basic characteristic of chaotic behavior, is due to the internal structure of the systems. However, chaotic behavior may lead to undesirable effects and may need to be controlled. Many researchers have endeavored to find new ways to suppress and control chaos more efficiently. So far, many researchers have presented different types of controllers and control methods, e.g., linear state error feedback control [2,3], sinusoidal state error feedback control [4], variable substitution (or replacing variable) control [5,6], variable structure control [7,8], nonlinear feedback control [9], active control [10], and adaptive control [11] have been successfully applied to chaotic systems. In recent years, brushless motors have been used as a viable choice for motion control applications, such as in electric propulsion, robotics, or aerospace. The advantages of brushless motors when compared with conventional DC motors have caused increasing interest. This interest was provided with the elimination of the physical contact between the mechanical brushes and commutators. Among the numerous types of brushless motors, the brushless DC motor (BLDCM) is the one with the highest potential in high-performance applications. Therefore, BLDCMs are widely used in industrial applications. Hemati, Ge and Chang, and Ge et al. [12–15] used the bifurcation theory to study BLDCMs. Their studies showed that these kinds of machines experience chaotic oscillations. These undesirable chaotic oscillations need to be eliminated. Sliding mode control (SMC) is especially preferred by many researchers due to its capability to tolerate disturbances and ∗ Correspondence: 649 UYAROĞLU and CEVHER/Turk J Elec Eng & Comp Sci dynamic model uncertainties [16–23]. Thus, the SMC controller is designed as a nonlinear controller to eliminate the undesirable chaotic oscillations. Herein, the control of the chaotic system of a BLDCM is investigated by SMC terms. This paper presents the chaos control of a single time-scale BLDCM by means of SMC. This work is organized as follows. In Section 2, the mathematical model of a single time-scale BLDCM is given. In Section 3, the chaos control of a single time-scale BLDCM chaotic system is investigated. In Section 4, numerical simulations are provided to confirm the validity of the method. Finally, in Section 5, the conclusions are given. 2. Mathematical model of single time-scale BLDCM A BLDCM is an electromechanical system. The equations for the electrical and mechanical dynamics of a BLDCM were described by Hemati [12] and Ge and Chang [13]. The system equations for the BLDCM are transformed via Park’s transformation and take the following form: ⎫ i̇q = L1q [−Riq − nw (Ld id + kt ) + vq ] ⎬ i̇d = L1q [−Rid − nwLd iq + vq ] ⎭ , (1) and the electromagnetic torque can be rewritten as follows: T (iq , id ) = n [kt iq + (Ld − Lq ) iq id ] , (2) where i q , i d are the quadrature axis and direct axis currents; vq , vd are the quadrature axis and direct axis voltages; Lq , Ld are the fictitious inductance on the quadrature axis and direct axis; R is the winding resistance; √ n is the number of permanent pole pairs; and kt = 3/2ke , where ke is the permanent magnet flux constant. The system equations are transformed to a compact form through a single time-scale transformation [14,15]. Hence, the equations in compact forms, with a greatly reduced number of parameters, were obtained by Ge et al. [15]. After this transformation by Ge et al. [15], the system equations of the BLDCM take the following form: ⎫ ẋ1 = vq − x1 − x2 .x3 + ρ.x3 ⎪ ⎬ ẋ2 = vd − δ.x2 + x1 .x3 , (3) ⎪ ⎭ ẋ3 = σ (x1 − x3 ) + η.x1 .x2 − TL where vq = 0.168, ρ = 60, vd = 20.66, δ = 0.875, η = 0.26, T L = 0.53, and σ = 4.55. In Figure 1, the phase portraits of the BLDCM system, given in Eq. (3), are presented. Figure 1a shows x 1 – x 3 and Figure 1b shows x 1 – x 2 . In Figure 2, the time responses of the state variables of the BLDCM system are given. 3. SMC design for chaos control of a single time-scale BLDCM The proposed BLDCM chaotic system is described in Eq. (3). Thus, the controlled chaotic system of the single time-scale BLDCM is attained as follows: ẋ1 = vq − x1 − x2 .x3 + ρ.x3 + u1 ẋ2 = vd − δ.x2 + x1 .x3 + u2 ẋ3 = σ (x1 − x3 ) + η.x1 .x2 − TL + u3 650 ⎫ ⎪ ⎬ ⎪ ⎭ , (4) UYAROĞLU and CEVHER/Turk J Elec Eng & Comp Sci 20 15 75 (a) (b) 70 10 5 65 -5 x2 x3 0 60 -10 55 -15 -20 50 -25 -30 -15 -10 -5 0 5 10 45 -15 -10 -5 0 5 10 x1 x1 Figure 1. σ = 4.55 for the phase portraits of the uncontrolled BLDCM system: a) x 1 – x 3 and b) x 1 – x 2 . 80 x1 x2 x3 60 x1, x2, x3 40 20 0 -20 -40 0 5 10 15 20 25 t 30 35 40 45 50 55 Figure 2. σ = 4.55 for the time series of the uncontrolled BLDCM system. where u1 , u2 , and u3 are control signals. e = x − xd , (5) where e = [e1 e2 e3 ]T is the tracking error vector. The error dynamics may be written as below: ė = ẋ − ẋd = Ax + Bg + Bu − ẋd, (6) where A is the system matrix, B is the control matrix, and g represents the system nonlinearities plus the parametric uncertainties in the system. The control problem is to get the state x = [x 1 x 2 x 3 ] T to track a specific time-varying state x d = [x d1 x d2 x d3 ] T in the presence of nonlinearities. ⎡ −1 0 ρ ⎤ ⎡ ⎢ B=⎣ 0 ⎢ A=⎣ 0 −δ ⎥ 0 ⎦; σ 0 −σ 1 0 0 0 ⎤ ⎡ ⎥ ⎢ 1 0 ⎦; g = ⎣ 0 1 vq − x2 .x3 vd + x1 .x3 ⎤ ⎥ ⎦ −TL + η.x1 .x2 651 UYAROĞLU and (...truncated)