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Approximation-Based Discrete-Time Adaptive Position Tracking Control for Interior Permanent Magnet Synchronous Motors Jinpeng Yu, Peng Shi, Senior Member, IEEE, Haisheng Yu, Bing Chen, and Chong Lin, Senior Member, IEEE

Abstract—This paper considers the problem of discrete-time adaptive position tracking control for a interior permanent magnet synchronous motor (IPMSM) based on fuzzy-approximation. Fuzzy logic systems are used to approximate the nonlinearities of the discrete-time IPMSM drive system which is derived by direct discretization using Euler method, and a discrete-time fuzzy position tracking controller is designed via backstepping approach. In contrast to existing results, the advantage of the scheme is that the number of the adjustable parameters is reduced to two only and the problem of coupling nonlinearity can be overcome. It is shown that the proposed discrete-time fuzzy controller can guarantee the tracking error converges to a small neighborhood of the origin and all the signals are bounded. Simulation results illustrate the effectiveness and the potentials of the theoretic results obtained. Index Terms—Backstepping, discrete-time, fuzzyapproximation, permanent magnet synchronous motor (PMSM).

I. I NTRODUCTION N recent years, permanent magnet synchronous motor (PMSM) has received increased attention for high performance electric drive applications because of its considerable advantages. Especially for the interior PMSM (IPMSM) with many attractive characteristics such as wide-speed operation range, high-power density, large torque to inertia ratio, and free from maintenance [1], it is suitable for many applications such as electric vehicle drive system. Nevertheless, it is still a challenging problem to control the IPMSM drive systems to get the perfect dynamic performance because their dynamic models are usually multivariable, coupled, and highly

I

Manuscript received February 26, 2014; revised July 21, 2014; accepted August 18, 2014. This work was supported in part by the Australian Research Council under Grant DP140102180, in part by the National Key Basic Research Program (973), China, under Grant 2011CB710706 and Grant 2012CB215202, in part by the 111 Project under Project B12018, in part by the Natural Science Foundation of China under Grant 61104076, Grant 61174131, and Grant 61174033, and in part by the China Postdoctoral Science Foundation under Grant 2014T70620, Grant 2013M541881, and Grant 201303062. This paper was recommended by Associate Editor M. J. Er. J. Yu, H. Yu, B. Chen, and C. Lin are with the School of Automation Engineering, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]; [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia, and also with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2351399

nonlinear and they are also very sensitive to external load disturbances and parameter variations [2]. In order to achieve high performance of IPMSMs, many researchers have aimed to develop nonlinear control methods for the IPMSM and various algorithms have been proposed including nonlinear feedback linearization control [3], fuzzy logic control [4], [5], adaptive backstepping control [2], neural network control [6], sliding mode control [7], and disturbance-observer-based control [8]. However, most of those methods above were developed for nonlinear continuous-time systems, namely, nonlinear discretetime control design techniques for PMSM drive system have not been discussed to the same degree. The discrete-time approach is regarded as typically superior to the continuoustime emulation approach in terms of stability and achievable performances [9], [10]. This has motivated an interesting research activity in the design of controllers based on the discrete-time model of the system and there already exist good publications about control methods for discrete-time motor systems [11]–[14]. A discrete-time Takagi–Sugeno (T-S) fuzzy speed regulator considering the nonlinearity of PMSM was proposed in [15], but the proposed T-S fuzzy model of PMSM is only for the case of Ld = Lq (Ld and Lq are the d − q axis stator inductance of the PMSM), and for the most threephase IPMSM with Ld = Lq , the T-S fuzzy speed regulator will be invalid because Ld = Lq will add the coupling nonlinearity of the IPMSM and make the control design difficult. Backstepping-based adaptive fuzzy control provides an effective way to design control system with parameter uncertainty, particularly those systems in which the uncertainties do not satisfy matching conditions [16]–[29]. And there already have some excellent contributions for nonlinear discretetime systems combined backstepping [30]–[33]. For nonlinear discrete-time systems, the control problem is more complex because of the couplings nonlinearity among subsystems, inputs, and outputs [34]–[36], which will result in the additional complexities and difficulties for the controller design and closed-loop stability analysis [37]. Besides the difficulty of input couplings, the noncausal problem is another difficulty that is to be solved which caused by the future information in the virtual controller via backstepping [36]. If we continue the process to construct the real controller by use of the virtual controller, we will end up with a controller which is infeasible again because it contains more

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future information. The conventional approach to solve the noncausal problem is to transform the system model into a predictor form, which will add the complexity of the control process [38]. The third main shortcoming for the existing control for nonlinear discrete-time systems is that there are too many adaptive parameters needed to be tuned online, especially, for nonlinear multiinput-multioutput (MIMO) systems, the learning time tends to be unacceptably large. Consequently, it is very important to design an adaptive controller with fewer adaptive parameters to lighten the online computation burden. Motivated by the above facts, in this paper, a discrete-time adaptive position tracking control for IPMSM is proposed based on fuzzy-approximation. Compared with the existing results for IPMSM, the main contributions of this paper are that: 1) an accurate approximate discrete-time IPMSM position tracking system model is derived by direct discretization using the Euler method; 2) a discrete-time adaptive fuzzy position tracking controller which overcomes the coupling nonlinearity because of Ld = Lq is proposed; 3) without transforming the system model into a predictor form [38], the noncausal problem for nonlinear discrete-time IPMSM drive system combined backstepping is overcome; and 4) the number of adaptive parameters is considerably reduced to two only. As a result, the computational burden of the scheme is alleviated, which will render the designed scheme more suitable for practical applications. The proposed fuzzy control method provides a systematic approach to stability analysis and controller design for the underlying systems. Simulation results demonstrate the effectiveness of the proposed novel control scheme. The rest of this paper is organized as follows. Section II gives mathematical model of the IPMSM drive system and preliminaries. Discrete-time adaptive fuzzy controller design for the IPMSM drive system via backstepping is presented in Section III. The stability analysis of the closed-loop system is given in Section IV. In Section V, simulation studies are performed to demonstrate the effectiveness of the proposed scheme. Section VI concludes this paper, followed by the Appendix which gives another set of discrete-time controllers for a comparison. II. M ATHEMATICAL M ODEL OF IPMSM D RIVE S YSTEM AND P RELIMINARIES In this section, some preparatory knowledge of a IPMSM will be first introduced. To obtain the mathematical model of a IPMSM, the following assumptions are made. Assumption 1 [39]: Saturation and iron losses are neglected although it can be taken into account by parameter changes. Assumption 2 [39]: The back electromotive force is sinusoidal. The model of IPMSM in the d −q frame can be represented by the following continuous-time nonlinear equations: θ˙ (t) = ω (t) 3np  B iqs (t) − ω (t) ω˙ (t) = 2J J 3np (Ld − Lq ) 1 + ids (t) iqs (t) − TL 2J J

Rs iqs (t) − Lq 1 + uqs (t) Lq R ˙ids (t) = − s ids (t) + Ld

˙iqs (t) = −

np Ld np  ω (t) − ω (t) ids (t) Lq Lq

np  1 ω (t) iqs (t) + uds (t) Ld Ld

where TL , θ , and ω denote the load torque, rotor position, and rotor angular velocity. ids and iqs stand for the d − q axis currents. uds and uqs are the d − q axis voltages. np denotes the pole pairs, the stator resistance Rs , Ld , and Lq are the d −q axis stator inductance, the rotor inertia J, the viscous friction coefficient B, and the magnetic flux . By use of Euler method, we can obtain the discrete-time dynamic model of IPMSM drivers as follows: θ (k + 1) = θ (k) + t ω (k)   3np  B ω (k + 1) = t iqs (k) + 1 − t ω (k) 2J J 3np (Ld − Lq ) 1 + t ids (k) iqs (k) − t TL 2J J   np  Rs iqs (k + 1) = 1 − t iqs (k) − t ω (k) Lq Lq np Ld 1 − t ω (k) ids (k) + t uqs (k) Lq Lq   np Lq Rs ids (k + 1) = 1 − t ids (k) + t ω (k) iqs (k) Ld Ld 1 + t uds (k) Ld where t is the sampling period. For simplicity, the following notations are introduced: x1 (k) = θ (k), x2 (k) = ω (k), x3 (k) = iqs (k) 3np  3np (Ld − Lq ) x4 (k) = ids (k), a1 = , a2 = 2J 2J B 1 Rs a3 = , a4 = , b1 = J J Lq np  np Ld 1 b2 = , b3 = , b4 = Lq Lq Lq np Lq Rs 1 c1 = , c2 = , c3 = . Ld Ld Ld Making use of these notations, the discrete-time dynamic model of IPMSM drivers can be redescribed by the following equations: x1 (k + 1) = x1 (k) + t x2 (k) x2 (k + 1) = a1 t x3 (k) + a2 t x3 (k) x4 (k) + (1 − a3 t ) x2 (k) − a4 t TL x3 (k + 1) = (1 − b1 t ) x3 (k) − b2 t x2 (k) − b3 t x2 (k) x4 (k) + b4 t uqs (k) x4 (k + 1) = (1 − c1 t ) x4 (k) + c2 t x2 (k) x3 (k) + c3 t uds (k).

(1)

Remark 1: It should be pointed out that the coupling nonlinear term a2 t x3 (k) x4 (k) (because of Ld = Lq ) within the

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By use of (3), (2) can be rewritten to the following form: 1 2 2 1 (4)  e (k) − e21 (k) 2 t 2 2 with e2 (k) = x2 (k) − α1 (k). Step 2: From the second equation of (1), we can obtain V1 (k) =

e2 (k + 1) = x2 (k + 1) − α1 (k + 1) = a1 t x3 (k) + a2 t x3 (k) x4 (k) − a4 t TL + (1 − a3 t ) x2 (k) − α1 (k + 1). (5) According to (3), we can get Fig. 1.

above model (1) makes the discrete-time IPMSM drive system more complex than the model of PMSM described in [15], which adds the coupling nonlinearity and complexity and will make the backstepping design difficult. The discrete-time fuzzy control system structure for IPMSM is illustrated as Fig. 1. The control objective in this paper is to design an adaptive fuzzy controller such that the state variable x1 (k) follows the given reference signal xd (k) and all the closed-loop signals are bounded. The approximation property of the fuzzy logic systems (FLSs) can be found in [40]. By using the FLSs, given a compact set z = [z1 , z2 , . . . , zn ] ∈ z , the unknown smooth function ϕ(z) can be expressed as ϕ(z) = W T S(z) + ε(z), where W ∈ RN is the optimal paramT is a fuzzy basis eter vector, S(z) = [s1 (z), s2 (z), . . . , sN (z)] n n l function vector with s (z) = i=1 μφ l (zi )/ N l=1 i=1 μφil (zi ), i then S(z) has the following property: λmax [S(z)ST (z)] < 1. And ε(z) ∈ R is the approximation error satisfying |ε(z)| ≤ ε¯ with the constant ε¯ > 0. μφ l (zi ) is the fuzzy membership i function and φil are fuzzy sets in R. III. D ISCRETE -T IME F UZZY C ONTROL FOR IPMSM In this section, we will design an adaptive fuzzy control for the discrete-time IPMSM drive system via backstepping. Step 1: For the reference signal xd , define the tracking error variable as e1 (k) = x1 (k) − xd (k). From the first equation of (1), we can gain e1 (k + 1) = x1 (k + 1) − xd (k + 1) = x1 (k) + t x2 (k) − xd (k + 1). Choose the Lyapunov function candidate as V1 (k) = (1/2)e21 (k), then the difference of V1 (k) is computed by 1 2 1 e1 (k + 1) − e21 (k) 2 2 1 = [x1 (k) + t x2 (k) − xd (k + 1) ]2 2 1 − e21 (k). 2 Construct the virtual control law α1 (k) as V1 (k) =

α1 (k) =

1 [−x1 (k) + xd (k + 1)]. t

1 [−x1 (k + 1) + xd (k + 2)] t 1 [−x1 (k) − t x2 (k) + xd (k + 2)]. = t

α1 (k + 1) =

Discrete-time control system block diagram for IPMSM.

(2)

(3)

(6)

Substituting (6) into (5) leads to e2 (k + 1) = a1 t x3 (k) + a2 t x3 (k) x4 (k) xd (k + 2) − + (2 − a3 t ) x2 (k) t x1 (k) − a4 t TL + . t

(7)

Choose the Lyapunov function candidate as V2 (k) = (1/2)e22 (k) + V1 (k) . Then the difference of V2 (k) is given by 1 2 1 e2 (k + 1) − e22 (k) + V1 (k) 2 2 2 1 f1 (k) + a2 t x3 (k) x4 (k) = 2 1 − e22 (k) + V1 (k) 2

V2 (k) =

(8)

where f1 (k) = a1 t x3 (k) + (2 − a3 t ) x2 (k) + − a4 t TL −

1 x1 (k) t

1 xd (k + 2). t

Construct α2 (k) as

 1 1 − (2 − a3 t ) x2 (k) − α2 (k) = x1 (k) a1 t t  1 xd (k + 2) . + a4 t TL + t

(9)

Using (4) and (9), the difference of V2 (k) can be rewritten to the following form: 1 [a1 t e3 (k) + a2 t x3 (k) x4 (k)]2 2

1 1 1 − 2t e22 (k) − e21 (k) − (10) 2 2 with e3 (k) = x3 (k) − α2 (k). Utilizing the fact that (a1 t e3 (k) + a2 t x3 (k) x4 (k))2 ≤ 2a21 2t e23 (k) + 2a22 2t x32 (k) x42 (k), we can obtain V2 (k) =

V2 (k) ≤ a21 2t e23 (k) + a22 2t x32 (k) x42 (k)

1 1 1 − 2t e22 (k) − e21 (k). − 2 2

(11)

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Step 3: From the third equation of (1), we can obtain e3 (k + 1) = x3 (k + 1) − α2 (k + 1) = f3 (k) + b4 t uqs (k)

(12)

where f3 (z3 (k)) = (1 − b1 t ) x3 (k) − b2 t x2 (k) − b3 t x2 (k) x4 (k) − α2 (k + 1) z3 (k) = [x1 (k), x2 (k), x3 (k), x4 (k), xd (k), xd (k + 1), xd (k + 2), xd (k + 3)]T , and α2 (k + 1) can be obtained from equality (9). Remark 2: The virtual controller α2 (k + 1) contains future information. If we continue to construct the real controller via backstepping, we will end up with a controller containing more future information, and make it possibly infeasible in practice. This drawback was called noncausal problem [38]. The existing result to solve this problem is to transform the systems into a predictor form, which will add the control complexity. In this paper, we use the recursion formula to gain the expression of time k to indicate α2 (k + 1), thus the noncausal problem can be overcome. Choose the Lyapunov function candidate as V3 (k) =

1 2 e (k) + V2 (k). 2 3

V3 (k) =

(13)

(14)

where ε3 is the approximation error. In general, W3 is bounded and unknown. Define W3  = η3 where η3 > 0 is unknown constant. Let ηˆ 3 (k) be the estimate of η3 and η˜ 3 (k) = η3 − ηˆ 3 (k). Now choose the following control law uqs (k) and adaptive law ηˆ 3 (k + 1) as: 1 ηˆ 3 (k) S3 (z3 (k)) uqs (k) = − b4 t ηˆ 3 (k + 1) = ηˆ 3 (k) + γ3 S3 (z3 (k))e3 (k + 1) − δ3 ηˆ 3 (k)

e4 (k + 1) = x4 (k + 1) = (1 − c1 t ) x4 (k) + c3 t uds (k) + c2 t x2 (k) x3 (k).

(18)

P 2 P e4 (k + 1) − e24 (k) + V3 (k) 2 2 2 P f4 (k) + c3 t uds (k) = 2 P − e24 (k) + V3 (k) (19) 2 where f4 (z4 (k)) = (1 − c1 t ) x4 (k) + c2 t x2 (k) x3 (k) and z4 (k) = [x2 (k), x3 (k), x4 (k)]T . Similarly, the FLS W4T S4 (z4 (k)) is utilized to approximate the nonlinear function f4 (z4 (k)) in order to simplify the controller design and V4 (k) =

Remark 3: Noting that f3 (z3 (k)) contains α2 (k + 1) and the nonlinear term b3 t x2 (k)x4 (k), this will make the backstepping design becomes very difficult, and the designed control law uqs (k) will have a complex structure. Hence, we will use FLSs to approximate the nonlinear function f3 (z3 (k)) in order to simplify the structure of the control signal. As shown later, the design procedure of uqs (k) becomes simpler and uqs (k) has a simpler and more practical structure. By use of the approximation property of the FLSs, for any given ε3 > 0, there exists a FLS W3T S3 (z3 (k)) such that f3 (z3 (k)) = W3T S3 (z3 (k)) + ε3

Step 4: Define the tracking error variable as e4 (k) = x4 (k). From the fourth equation of (1), we can obtain

Choose the Lyapunov function candidate as V4 (k) = (P/2)e24 (k) + V3 (k) with P > 0, then the difference of V4 (k) is computed by

Furthermore, differencing V3 (k) yields 1 2 1 e (k + 1) − e23 (k) + V2 (k) 2 3 2 2 1 f3 (z3 (k)) + b4 t uqs (k) = 2 1 − e23 (k) + V2 (k). 2

where γ3 and δ3 are the positive parameters. Furthermore, using equality (11), (14), and (15), (13) can be easily verified that 1 V3 (k) ≤ [η3 S3 (z3 (k)) + ηˆ 3 (k) S3 (z3 (k)) + ε3 ]2 2 1 − e23 (k) + V2 (k) 2 1 ≤ [2η3 S3 (z3 (k)) − η˜ 3 (k) S3 (z3 (k)) + ε3 ]2 2 1 − e23 (k) + V2 (k) 2 ≤ 4η32 S3 (z3 (k))2 + η˜ 32 (k) S3 (z3 (k))2   1 1 2 2 − a1 t e23 (k) − e21 (k) − 2 2

1 1 − 2t e22 (k) + ε32 − 2 + a22 2t x32 (k) x42 (k). (17)

(15)

(16)

f4 (z4 (k)) = W4T S4 (z4 (k)) + ε4

(20)

where ε4 > 0 is the approximation error. Similar to W3 , W4 is also unknown and bounded. Define W4  = η4 where η4 > 0 is unknown constant. Let ηˆ 4 (k) be the estimate of η4 and η˜ 4 (k) = η4 − ηˆ 4 (k) . Now choose the following control law uds (k) and adaptive law ηˆ 4 (k + 1) as: 1 ηˆ 4 (k) S4 (z4 (k)) c3 t ηˆ 4 (k + 1) = ηˆ 4 (k) + γ4 S4 (z4 (k))e4 (k + 1) − δ4 ηˆ 4 (k) uds (k) = −

(21)

(22)

where γ4 and δ4 are the positive parameters. Remark 4: From controllers (15) and (21), it is clearly seen that the proposed controllers have simpler structure. This means that the proposed fuzzy-approximation-based discretetime adaptive controllers are easy to be implemented in

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practical engineering. For this, the Appendix gives another set of approximation-based discrete-time controllers for a comparison. Remark 5: Note that a similar approximation-based discretetime controller is constructed for induction motor in [11] where the controllers (A.1) listed in the Appendix contain two groups of adaptive parameters. Comparing our proposed discrete-time controllers (15) and (21) with the controllers (A.1), it can be seen that the adaptive parameter in our proposed discrete-time controllers (15) and (21) are scalars, while the adaptive parameter in the controllers (A.1) are vectors. Thus, the number of the adaptive parameters in our proposed scheme is reduced to two only, which is much less than the number of the adaptive parameters in [11]. As a result, the computational burden of the scheme is dramatically reduced, which will render our designed scheme more suitable for practical applications. Using equality (17), (20), and (21), it can be shown that P [η4 S4 (z4 (k)) + ηˆ 4 (k) S4 (z4 (k)) + ε4 ]2 2 P − e24 (k) + V3 (k) 2 P ≤ [2η4 S4 (z4 (k)) − η˜ 4 (k) S4 (z4 (k)) + ε4 ]2 2 P − e24 (k) + V3 (k) 2 ≤ 4Pη42 S4 (z4 (k))2 + Pη˜ 42 (k) S4 (z4 (k))2

V4 (k) ≤

+ 4η32 S3 (z3 (k))2 + η˜ 32 (k) S3 (z3 (k))2  

1 1 2 2 − a1 t e23 (k) − 1 − 2t e22 (k) − 2 2   1 P − a22 2t x32 (k) e24 (k) − e21 (k) − 2 2 2 2 + ε3 + Pε4 . (23) Remark 6: It can be observed that the fuzzy-approximationbased adaptive tracking control scheme is proposed and the problems of the coupling nonlinearity because of Ld = Lq and noncausal issue for backstepping of IPMSM drive system can be overcome without transforming the system model into a predictor form [38]. From the above analysis, we now present our first main result in this paper as follows. Theorem 1: Consider system (1) satisfying assumptions 1 and 2 and the given reference signal xd . Then under the action of the fuzzy-approximation-based adaptive discrete-time controllers (15), (21) and the adaptive laws (16) and (22), the tracking error of the closed-loop controlled system will converge to a sufficiently small neighborhood of the origin and all the closed-loop signals will be bounded. The detailed proof is given in Section IV.

5

where γ3 and γ4 are positive parameters. Furthermore, differencing V (k) yields 1 2 V (k) = V4 (k) + [η˜ (k + 1) − η˜ 32 (k) ] 2γ3 3 P 2 η˜ 4 (k + 1) − η˜ 42 (k) . + 2γ4

(25)

As defined before, we can obtain η˜ 32 (k + 1) − η˜ 32 (k) = η32 + ηˆ 32 (k + 1) − 2η3 ηˆ 3 (k + 1) − η˜ 32 (k).

(26)

Using (16), we can get ηˆ 32 (k + 1) = [ηˆ 3 (k) + γ3 S3 (z3 (k))e3 (k + 1) − δ3 ηˆ 3 (k) ]2 = (1 − δ3 )2 ηˆ 32 (k) + 2(1 − δ3 )γ3 S3 (z3 (k))e3 (k + 1) ηˆ 3 (k) (27) + γ32 e23 (k + 1) S3 (z3 (k))2 η3 ηˆ 3 (k + 1) = η3 [ηˆ 3 (k) + γ3 S3 (z3 (k))e3 (k + 1) − δ3 ηˆ 3 (k) ] = (1 − δ3 )η3 ηˆ 3 (k) + γ3 S3 (z3 (k))e3 (k + 1) η3 .

(28)

Substituting (27) and (28) into (26) gives us η˜ 32 (k + 1) − η˜ 32 (k) = η32 + (1 − δ3 )2 ηˆ 32 (k) + γ32 e23 (k + 1) S3 (z3 (k))2 − 2(1 − δ3 )η3 ηˆ 3 (k) + 2(1 − δ3 )γ3 S3 (z3 (k))e3 (k + 1) ηˆ 3 (k) − η˜ 32 (k) − 2γ3 S3 (z3 (k))e3 (k + 1) η3 .

(29)

Then, using S3 (z3 (k))2 ≤ 1 and according to Young’s inequality, we have 2γ3 S3 (z3 (k))e3 (k + 1) ηˆ 3 (k) ≤ γ32 e23 (k + 1) + ηˆ 32 (k) −2S3 (z3 (k))e3 (k + 1) η3 ≤ e23 (k + 1) + η32 γ32 e23 (k + 1) S3 (z3 (k))2 ≤ γ32 e23 (k + 1)

(30)

−2η3 ηˆ 3 (k) ≤ η32 + ηˆ 32 (k). Substituting (14) and (15) into (12) leads to e3 (k + 1) = W3T S3 (z3 (k)) + ε3 + b4 t uqs (k). Then, we can obtain e23 (k + 1) ≤ [η3 S3 (z3 (k)) + ηˆ 3 (k) S3 (z3 (k)) + ε3 ]2 ≤ [2η3 S3 (z3 (k)) − η˜ 3 (k) S3 (z3 (k)) + ε3 ]2  2 ≤ 2η3 − η˜ 3 (k) + ε3 ≤ 8η32 + 2η˜ 32 (k) + 2ε32 .

(31)

Substituting (30) and (31) into (29) yields IV. S TABILITY A NALYSIS To address the stability analysis of the resulting closed-loop system, choose the Lyapunov function candidate as 1 2 P 2 η˜ 3 (k) + η˜ (k) V (k) = V4 (k) + 2γ3 2γ4 4

(24)

η˜ 32 (k + 1) − η˜ 32 (k) ≤ (16γ32 − 8γ32 δ3 + 9γ3 − δ3 + 2)η32 + (δ32 − 4δ3 + 3)ηˆ 32 (k) + (4γ32 − 2γ32 δ3 + 2γ3 − 1)η˜ 32 (k) + (4γ32 − 2γ32 δ3 + 2γ3 )ε32 .

(32)

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Similarly, we can get

η˜ 42 (k + 1) − η˜ 42 (k) ≤ 16γ42 − 8γ42 δ4 + 9γ4 − δ4 + 2 η42

+ δ42 − 4δ4 + 3 ηˆ 42 (k)

+ 4γ42 − 2γ42 δ4 + 2γ4 − 1 η˜ 42 (k)

+ 4γ42 − 2γ42 δ4 + 2γ4 ε42 . (33) Then substituting (23), (32), and (33) into (25), we have     1 P − a22 2t x32 (k) e24 (k) − − a21 2t e23 (k) V (k) ≤ − 2 2

1 1 1 − 2t e22 (k) − e21 (k) − 2 2

1 2 δ3 − 4δ3 + 3 ηˆ 32 (k) + β3 + 2γ3

+ 4γ32 − 2γ32 δ3 + 4γ3 − 1 η˜ 32 (k)

P 2 δ4 − 4δ4 + 3 ηˆ 42 (k) + 2γ4

+ β4 + 4γ42 − 2γ42 δ4 + 4γ4 − 1 η˜ 42 (k) where β3 = (4γ32 − 2γ32 δ3 + 4γ3 )ε32 + (16γ32 − 8γ32 δ3 + 17γ3 − δ3 + 2)η32 and β4 = (4γ42 − 2γ42 δ4 + 4γ4 )ε42 + (16γ42 − 8γ42 δ4 + 17γ4 − δ4 + 2)η42 are bounded. Define x32 (k) ≤ M, where M is positive constant. Furthermore     1 P − a22 2t M e24 (k) − − a21 2t e23 (k) V (k) ≤ − 2 2

1 1 1 − 2t e22 (k) − e21 (k) − 2 2

1 2 δ3 − 4δ3 + 3 ηˆ 32 (k) + β3 + 2γ3

+ 4γ32 − 2γ32 δ3 + 4γ3 − 1 η˜ 32 (k)

P 2 δ4 − 4δ4 + 3 ηˆ 42 (k) + β4 + 2γ4

+ 4γ42 − 2γ42 δ4 + 4γ4 − 1 η˜ 42 (k) . By choosing a suitable parameter P and sampling period t , we can get (P/2) − a22 2t M > 0, (1/2) − a21 2t > 0 and 1 − 2t > 0. If we choose the design parameters as follows: δi2 − 4δi + 3 < 0, 4γi2 − 2γi2 δi + 4γi − 1 < 0 for i = 3, 4. Then V (k) ≤ 0

β3 once the error |e3 (k) | > and |e4 (k) | > γ3 (1−2a21 2t )

Pβ4 . That implies that the signal ei (k)(i = 2 2 γ4 (P−2a2 t M)

1, 2, 3, 4) is bounded in a compact set [38]. Subtracting η3 from both sides of (16), we can obtain −η˜ 3 (k + 1) = −η˜ 3 (k) + γ3 S3 (z3 (k))e3 (k + 1) − δ3 ηˆ 3 (k). Noting that η3 = η˜ 3 (k) + ηˆ 3 (k)

Fig. 2.

Position tracking performance.

then η˜ 3 (k + 1) = (1 − δ3 )η˜ 3 (k) − γ3 S3 (z3 (k))e3 (k + 1) + δ3 η3 . Choose a suitable δ3 and let 0 < 1 − δ3 < 1. Noting ||S3 (z3 (k))||, e3 (k), δ3 η3 are bounded and according to [32, Lemma 1], η˜ 3 (k) must be bounded in a compact set. Similarly, η˜ 4 (k) must be bounded in a compact set. So, the boundedness of η˜ 3 (k) and η˜ 4 (k) are obtained. Then the input uqs and uds are bounded. This can guarantee that all the signals including ei (k)(i = 1, 2, 3, 4), ηˆ 3 (k), ηˆ 4 (k) are bounded and lim x1 (k) − xd (k)  ≤ σ where σ is small positive constant. k→∞

V. S IMULATION R ESULTS To illustrate the effectiveness of the proposed results, the simulation is run for IPMSM with the parameters [5] J = 0.00379Kgm2 , Rs = 0.68, Ld = 0.00315H np = 3, Lq = 0.00285H,  = 0.1245H B = 0.001158Nm/(rad/s). The control objective is to design a controller such that x1 (k) tracks the reference signal xd (k) effectively. The reference signal is chosen as xd (k) = 2 cos(t kπ/2). We introduce load torque disturbances in order to assess the motor recovery ability under our proposed controllers and the load torque is given as follows:  1.5, 0 ≤ k ≤ 2000 TL = 3, k ≥ 2000. The initial values of the states are chosen as x1 (0) = x2 (0) = x3 (0) = x4 (0) = 0.The sampling period is chosen as t = 0.0055s. The values of the design parameters were selected as δ3 = 0.39, δ4 = 0.29, γ3 = 0.3, and γ4 = 0.7. Remark 7: As for the discrete-time control system, the select of sampling period t is a critical issue. If the sampling period t is too large, the sample accuracy would be poor and will bring down the control system performance. Decreasing t will gain more precisely discrete-time dynamic model of IPMSMs, but it will add system control burden such as the

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Fig. 3.

Tracking error.

Fig. 4.

q-axis voltage uqs .

Fig. 5.

Fig. 6.

Adaptive law ηˆ 3 (k).

Fig. 7.

Adaptive law ηˆ 4 (k).

Fig. 8.

q-axis current iqs .

7

d-axis voltage uds .

computation burden. We choose a suitable value of t according to the control performance and system control burden in this paper. Simulation results in Figs. 2–9 are obtained by use of the proposed scheme. The trajectories of x1 (k) and xd (k) are given in Fig. 2, where the solid line represents x1 (k) and the dashed

line represents xd (k). The dynamics of the tracking error is shown in Fig. 3. It can be observed that under the actions of controllers (15) and (21), the system output follows the desired reference signal well and the tracking error is bounded in a compact set although there is coupling nonlinearity among

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Fig. 9.

d-axis current ids .

subsystems and load torque is fluctuant. The trajectories of uqs (k) and uds (k) are shown in Figs. 4 and 5. Furthermore, to demonstrate the adaptive learning performance, the system adaptive laws are demonstrated in Figs. 6 and 7. Boundedness of iqs (k) and ids (k) is illustrated by Figs. 8 and 9. From Figs. 4, 5, 8, and 9, it can be seen that boundedness of uds (k), uqs (k), iqs (k), and ids (k) are verified. The controllers can guarantee the robustness against the system parameter variations and load disturbances. In this simulation, it should be remarked that when the load torque changes, the controllers copes easily with the sudden change on the load torque and provides a fast position tracking response. Moreover, the position tracking error remains small and without overshoot, which produces smooth current signals. VI. C ONCLUSION In this paper, fuzzy-approximation-based adaptive discretetime control approach combined backstepping technique is proposed to solve the position tracking problem for IPMSM drive system. The designed controllers guarantee that the tracking error converges to a small neighborhood of the origin and all the signals to be mean square semi-globally uniformly ultimately bounded. Simulation results are provided to demonstrate the effectiveness and robustness against the system parameter variations and load disturbances. A PPENDIX The designed discrete-time controllers for induction in [11] is as follows:   uα (k) = −ηT S2 (z2 (k)) (A.1) uβ (k) where η is an adjustable weight vector and S(z) = [s1 (z), s2 (z), ..., sm (z)]T is a basis function vector, whose detail definitions can be seen in equality(2) – equality(7) in [11]. R EFERENCES [1] S. H. Li and Z. G. Liu, “Adaptive speed control for permanent-magnet synchronous motor system with variations of load inertia,” IEEE Trans. Ind. Electron., vol. 56, no. 8, pp. 3050–3059, Aug. 2009.

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Jinpeng Yu received the B.Sc. degree in automation from the Qingdao University, Qingdao, China, in 2002, the M.Sc. degree in system engineering from the Shandong University, Jinan, China, in 2006, and the Ph.D. degree in system theory from the Institute of Complexity Science, Qingdao University, in 2011. He is currently a Distinguished Professor with the School of Automation Engineering, Qingdao University. His current research interests include electrical energy conversion and motor control, applied nonlinear control, and intelligent systems. Dr. Yu was the recipient of the Outstanding Graduate Award for Technological Innovation of Shandong Province in 2011.

9

Peng Shi (M’95–SM’98) received the B.Sc. degree in mathematics, the M.E. degree in systems engineering, both from the Harbin Institute of Technology, Harbin, China, the Ph.D. degrees in electrical engineering and in mathematics from the University of Newcastle, Callaghan, NSW, Australia, and the University of South Australia, Adelaide, SA, Australia, respectively, and the D.Sc. degree from the University of Glamorgan, Wales, U.K. He was a Post-Doctorate and a Lecturer with the University of South Australia, a Senior Scientist with the Defence Science and Technology Organization, Edinburgh, SA, Australia, and a Professor with the University of Glamorgan. He is currently a Professor with the University of Adelaide, Adelaide, SA, Australia, and Victoria University, Melbourne, VIC, Australia. His current research interests include system and control theory, computational intelligence, and operational research. Dr. Shi has been on the editorial board for a number of journals, including the IEEE T RANSACTIONS ON C YBERNETICS, the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE T RANSACTIONS ON F UZZY S YSTEMS, the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS -I, the IEEE ACCESS, and Automatica. He is currently the Chair of the Control, Aerospace, and Electronic Systems Chapter, IEEE South Australia Section. He is a fellow of the Institution of Engineering and Technology and the Institute of Mathematics and its Applications.

Haisheng Yu received the B.S. degree in electrical automation from the Harbin University of Civil Engineering and Architecture, Harbin, China, in 1985, the M.S. degree in computer applications from Tsinghua University, Beijing, China, in 1988, and the Ph.D. degree in control science and engineering from the Shandong University, Jinan, China, in 2006. He is currently a Professor with the School of Automation Engineering, Qingdao University, Qingdao, China. His current research interests include electrical energy conversion and motor control, applied nonlinear control, and computer control and intelligent systems.

Bing Chen received the B.A. and M.A. degrees, both in mathematics, from the Liaoning University, Shenyang, China, and the Harbin Institute of Technology, Harbin, China, and the Ph.D. degree in electrical engineering from the Northeastern University, Shenyang, China, in 1982, 1991, and 1998, respectively. He is currently a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, China. His current research interests include nonlinear control systems, robust control, and adaptive fuzzy control.

Chong Lin (SM’06) received the B.Sci. and M.Sci. degrees in applied mathematics from Northeastern University, Shenyang, China, in 1989 and 1992, respectively, and the Ph.D. degree in electrical and electronic engineering from the Nanyang Technological University, Singapore, in 1999. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, China, in 1999. From 2000 to 2006, he was a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. Since 2006, he has been a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, China. His current research interests include systems analysis and control, robust control, and fuzzy control. He has published over 60 research papers and co-authored two monographs.

Approximation-Based Discrete-Time Adaptive Position Tracking Control for Interior Permanent Magnet Synchronous Motors.

This paper considers the problem of discrete-time adaptive position tracking control for a interior permanent magnet synchronous motor (IPMSM) based o...
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