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Neural Network-Based Adaptive Dynamic Surface Control for Permanent Magnet Synchronous Motors Jinpeng Yu, Peng Shi, Senior Member, IEEE , Wenjie Dong, Member, IEEE , Bing Chen, and Chong Lin, Senior Member, IEEE Abstract— This brief considers the problem of neural networks (NNs)based adaptive dynamic surface control (DSC) for permanent magnet synchronous motors (PMSMs) with parameter uncertainties and load torque disturbance. First, NNs are used to approximate the unknown and nonlinear functions of PMSM drive system and a novel adaptive DSC is constructed to avoid the explosion of complexity in the backstepping design. Next, under the proposed adaptive neural DSC, the number of adaptive parameters required is reduced to only one, and the designed neural controllers structure is much simpler than some existing results in literature, which can guarantee that the tracking error converges to a small neighborhood of the origin. Then, simulations are given to illustrate the effectiveness and potential of the new design technique.
Index Terms— Backstepping, dynamics surface control (DSC), neural networks (NNs), nonlinear system, permanent magnet synchronous motor (PMSM). I. I NTRODUCTION
I
N THE past few decades, permanent magnet synchronous motors (PMSMs) have attracted considerable attention as its wide industrial applications [1]. However, it is still a challenging problem to control the PMSMs to get desired dynamic performance because their dynamics are usually multivariable, coupled, and highly nonlinear; and very sensitive to external load disturbances and parameter variations. To achieve better performance of PMSMs, much work has devoted to develop nonlinear control methods for PMSM and various algorithms have been proposed, see [2]–[12] and the references therein. In another research front line, the problem of backstepping-based nonlinear adaptive control has been extensively studied for nonlinear systems with parameter uncertainty, particularly, matching conditions not being satisfied [13]–[18]. The conventional backstepping technique has been successfully applied to the control of PMSM drivers recently [19], [20]. But there are some drawbacks in backstepping approach. One problem is that certain functions must be linear in the unknown system parameters [20]. Another limitation is the explosion of complexity caused by the repeated differentiations of virtual input [21]–[24]. Theoretically, the calculation of virtual control derivation is simple, but it can be quite tedious and complicated in practical applications when n is larger than three because the desired
Manuscript received August 22, 2013; revised December 3, 2013 and February 14, 2014; accepted April 6, 2014. This work was supported in part by the Australian Research Council under Grant DP140102180, in part by the National Key Basic Research (973) Program, China under Grant 2011CB710706 and Grant 2012CB215202, in part by the 111 Project under Grant B12018, and in part by the Natural Science Foundation of China under Grant 61104076, Grant 61174131, and Grant 61174033. J. 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, 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]). W. Dong is with the Department of Electrical Engineering, University of Texas-Pan American, Edinburg, TX 78539 USA (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/TNNLS.2014.2316289
controller u will include the derivation of αn , which requires the second derivation of αn−1 and so on. For instance, the backstepping controller in [25] is overly cumbersome, which is the representative of explosion of complexity problem. To overcome this issue, a dynamic surface control (DSC) method was proposed by introducing a firstorder filtering of the virtual input at each step of the conventional backstepping approach [26]–[30]. In addition, neural network (NN) approximation method has been widely used in many applications, mainly by its inherent capability for modeling and controlling highly uncertain, nonlinear, and complex systems [31]–[33]. The radial basis function (RBF) NN is considered in [34] and [35] as a two-layer network, which contains both hidden and output layers. Therefore, NNs can be used to handle uncertain information and be employed to control the systems which are ill-defined or too complex to have a precise mathematical model. The RBF NN has been found as one of the very effective tools in functional approximations, which has wide applications [36]–[38]. In this brief, an NN-based adaptive DSC is proposed to solve the above problems of the conventional backstepping method for PMSM drive systems. The RBF networks are used to approximate the unknown nonlinear functions to solve the first problem of linear in the unknown system parameters, and a DSC technique is proposed to tackle the second problem of explosion of complexity by firstorder filtering technique at each step of the conventional backstepping design [39], [40]. The proposed control scheme not only guarantees the boundedness of all of the signals in the closed-loop system, but also reduces the number of adaptive parameters which alleviates the computational burden. Compared with the existing results on adaptive neural control for PMSMs, the main contributions of this brief are that: 1) the developed approximation-based neural controller has a simpler structure and both the problems of linear in the unknown system parameters and explosion of complexity are considered and 2) the number of adaptive parameters is considerably reduced to only one, which is independent of the number of the neural basis function and system state variables. As a result, the computational burden of the scheme is alleviated, which will render the designed scheme more suitable for practical applications. Finally, the novelty of the new design method is verified by simulations to demonstrate the effectiveness and the robustness against the parameter uncertainties and load disturbances. II. M ATHEMATICAL M ODEL OF THE PMSM D RIVE S YSTEM AND P RELIMINARIES The model of PMSMs can be described in d − q frame through the Park transformation as follows [1]: d dt dω J dt di d Ld dt di q Lq dt
=ω =
3 n p (L d − L q )i d i q + i q − Bω − TL 2
= −Rs i d + n p ωL q i q + u d = −Rs i q − n p ωL d i d − n p ω + u q
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where is the rotor position, ω denotes the rotor angular velocity, i d and i q stand for the d − q axis currents, u d and u q are the d − q axis voltages, Rs is the stator resistance, L d and L q are the d − q axis stator inductors, n p is the pole pair, J means the rotor moment of inertia, B is the viscous friction coefficient, T is the electromagnetism torque, TL is the load torque, and denotes magnet flux linkage. To simplify the above model, the following notations are introduced: x1 = , x2 = ω, x3 = i q , x4 = i d 3n p 3n p (L d − L q ) Rs a1 = , a2 = , b1 = − 2 2 Lq n p Ld n p 1 b2 = − , b3 = − , b4 = Lq Lq Lq n p Lq Rs 1 c1 = − , c2 = , c3 = . Ld Ld Ld Using these notations, the dynamic model of the PMSM can be described by the following differential equations: x˙1 = x2 a1 a2 B T x˙2 = x3 + x3 x4 − x2 − L J J J J x˙3 = b1 x3 + b2 x2 x4 + b3 x2 + b4 u q x˙4 = c1 x4 + c2 x2 x3 + c3 u d .
(1)
In this brief, the RBF NN will be used to approximate the unknown ˆ = φ ∗T P(z), where z ∈ continuous function ϕ(z) : R q → R as ϕ(z) z ⊂ R q is the input vector with q being the NN input dimension, φ ∗ = [∗1 , . . . , ∗n ]T ∈ R n , is the weight vector, n > 1 is the NN node number, and P(z) = [ p1 (z), . . . , pn (z)]T ∈ R n is the basis function vector with pi (z) chosen as the commonly used Gaussian function in the following form: −(z − νi )T (z − νi ) , i = 1, 2, . . . , n pi (z) = exp qi2 where νi = [νi1 , . . . , νiq ]T is the center of the receptive field and qi is the width of the Gaussian function. It has been shown in [41] that, for a given scalar ε > 0, by choosing sufficiently large l, the RBF NN can approximate any continuous function over a compact set z ∈ R q to an arbitrary accuracy as ϕ(z) = φ T P(z) + δ(z) ∀ z ∈ z ⊂ R q , where δ(z) is the approximation error satisfying |δ(z)| ≤ ε and φ is an unknown ideal constant weight vector, which is an artificial quantity required for analytical purpose. Typically, φ is chosen as the value of φ ∗ that minimizes |δ(z)| for all z ∈ z ∗T φ := arg min sup ϕ(z) − φ P(z) . φ∗ ∈ R n
1 and α1d (0) = α1 (0). The purpose of this filter is to generate α1d and its derivative α˙ 1d such that |α1d − α1 | is smaller than a given level (4) 1 α˙ 1d + α1d = α1 , α1d (0) = α1 (0). Define z 2 = x2 − α1d . Using (3) and (4), (2) can be rewritten in the following form: V˙1 = z 1 z˙ 1 = z 1 (z 2 + α1d − x˙d ) = −k1 z 12 + z 1 z 2 + z 1 (α1d − α1 ). Step 2: Differentiating z 2 obtains z˙2 =
V˙2 = −k1 z 12 + z 1 (α1d − α1 ) + z 2 (a1 x3 + f2 )
V˙1 = z 1 z˙ 1 = z 1 (x2 − x˙d ).
(2)
Construct the virtual control law α1 as α1 = −k1 z 1 + x˙d
with k1 > 0 being a design control gain. Next, introduce a new state variable α1d . Let α1 pass through a first-order filter with time constant
(7)
where θˆ is the estimation of the unknown constant θ which will be specified later. Define a new state variable α2d . Let α2 pass through a first-order filter with time constant 2 to obtain α2d as 2 α˙ 2d + α2d = α2 , α2d (0) = α2 (0)
(10)
and define z 3 = x3 − α2d 2 i=1
1 ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 ) + l22 2
1 1 + 2 z 22 φ2 2 − θˆ P2T P2 + a1 z 2 z 3 + ε22 . 2 2l2
(11)
Step 3: Differentiating z 3 results in the following equation: z˙3 = x˙3 − α˙ 2d = b1 x3 + b2 x2 x4 + b3 x2 + b4 u q − α˙ 2d . Choose the following Lyapunov function candidate V3 = V2 + (1/2)z 32 . Furthermore, differentiating V3 yields V˙3 = −
(3)
(6)
where f2 (Z 2 ) = z 1 + a2 x3 x4 − Bx2 − TL − J α˙ 1d and Z 2 = [x1 , x2 , x3 , x4 , xd , x˙d ]T . Remark 1: It should be mentioned that the system parameters B, TL , and J may be unknown in the PMSM drive system, then they cannot be used to construct the control signal unless we specify their corresponding adaptation laws. To avoid this trouble, we will employ the NNs to approximate the nonlinear function f2 and the indeterministic parameters will be considered. According to the RBF NN approximation property, for a given ε2 > 0, there exists a RBF NN φ2T P2 (Z 2 ) such that f2 = φ2T P2 (Z 2 ) + δ2 (Z 2 ), where δ2 (Z 2 ) is the approximation error satisfying |δ2 | ≤ ε2 . Consequently, we can show the following inequality:
1 z 2 f2 = z 2 φ2T P2 + δ2 ≤ 2 z 22 φ2 2 P2T P2 2l2 1 2 2 2 + l 2 + z 2 + ε2 . (8) 2 Then the virtual control α2 is constructed as
1 1 1 − k2 z 2 − z 2 − 2 z 2 θˆ P2T P2 (9) α2 = a1 2 2l2
V˙2 = −
In this section, we will present an adaptive DSC for PMSMs based on backstepping. Step 1: For the reference signal xd , we define the tracking error variable as z 1 = x1 − xd . From the first subsystem of (1), the error dynamic system is computed by z˙ 1 = x2 − x˙d . Choose a Lyapunov function candidate as V1 = (1/2)z 12 , then the time derivative of V1 is given by
a1 a2 B T x3 + x3 x4 − x2 − L − α˙ 1d . J J J J
Now, choose the Lyapunov function candidate as V2 = V1 + J /2z 22 . Obviously, time derivative of V2 can be expressed as
z∈z
III. A DAPTIVE DSC FOR PMSM S
(5)
2
as
1 ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 ) + l22 2
i=1 z2 1 + 22 (φ2 2 − θˆ )P2T P2 + ε22 + z 3 ( f 3 + b4 u q ) 2 2l2
(12)
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where f3 (Z 3 ) = b1 x3 + b2 x2 x4 + b3 x2 + a1 z 2 − α˙ 2d and Z 3 = Z 2 . Remark 2: It should be noted that f3 contains the derivative of α2d and the nonlinear term b2 x2 x4 , this will make the backstepping design become very difficult and the designed u q will have a complex structure. To solve this problem, we will use NNs to approximate the nonlinear function f 3 . Similarly, for given ε3 > 0, there exists φ3T P3 (Z 3 ) such that 1 2 1 1 1 z 3 φ3 2 P3T P3 + l32 + z 32 + ε32 . 2 2 2 2 2l3
z3 f3 ≤
(13)
2
3 1
2
li2 +
i=1
+
i=2
1 ε2 + z 32 + z 3 b4 u q . 2 i 2
(14)
i=1
where f4 (Z 4 ) = c1 z 4 + c2 x2 x3 and Z 4 = Z 2 . Similarly, for a given ε4 > 0, there exists φ4T P4 (Z 4 ) satisfying 1 2 1 1 1 z 4 φ4 2 P4T P4 + l42 + z 42 + ε42 . 2 2 2 2 2l4
(16)
Substituting (16) into (15) gives ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
i=1 3 4 4 1 2 1 2 1 2 2 − θ)P ˆ T Pi + + l ε z (φ + i i 2 i 2 i 2l 2 i i=2 i i=2 i=2 1 1 + 2 z 42 φ4 2 P4T P4 + z 42 + c3 z 4 u d . (17) 2 2l 4
We design u d as ud =
1 1 1 − k4 z 4 − z 4 − 2 z 4 θˆ P4T P4 c3 2 2l4
(21)
1 2 1 2 1 ˜2 θ y + y + 2 1 2 2 2r1
(22)
4
ki z i2 +
4 1 i=2
2 ⎡
(li2 + εi2 ) +
2
yi y˙i + z 1 y1
i=1
⎤ 4 1 ˜ ⎣ r1 2 T ˙ + a1 z 2 y2 + θ − z P Pi + θˆ ⎦ . r1 2l 2 i i i=2
(23)
i
According to (23), the corresponding adaptive law is chosen as follows:
ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
3 3 3 1 2 1 2 1 2 ˆ T Pi + z i (φi 2 − θ)P li + ε + i 2 2 2 i 2l i=2 i i=2 i=2 + z 4 ( f 4 (Z 4 ) + c3 u d ) (15)
3
(20)
with D2 = 1/a1 (k2 +1/2+(1/2)l22 θˆ P2T P2 )˙z 2 +(1/2)a1 l22 θ˙ˆ P2T P2 z 2 + ˆ Z˙ 2 ∂ P2 (Z 2 )/∂ Z 2 P2 (Z 2 )z 2 . 1/a1 l22 θ( Choose the following Lyapunov function candidate:
i=1
Step 4: At this step, we will construct the control law u d . Define z 4 = x4 and choose V4 = V3 + (1/2)z 42 . Then, we have
V˙4 ≤ −
y2 = α2d − α2 , θ˜ = θˆ − θ.
Introduce y1 , y2 , and θ˜ as
V˙ ≤ −
3 3 1 2 1 2 1 2 2 − θˆ )P T P + + z (φ + l ε . i i i i 2 i 2 i 2l 2 i=2 i i=2 i=2
z 4 f 4 (Z 4 ) ≤
(19)
where r1 is a positive constant. By differentiating V and taking (19)– (22) into account, one has
ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
3
2
4 1 2 T ˆ z P P (θ − θ). 2 i i i 2l i=2 i
(li2 + εi2 ) +
V = V4 +
i=1 3
V˙4 ≤ −
4 1 i=2
Furthermore, using (14), it can be verified that 3
ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
α1d − α1 − α˙ 1 1 y1 y1 = − + k1 z˙1 − x¨d = − + D1 1 1 y2 y˙2 = − + D2 2
3 1
Now, the control input u q is designed as
1 1 1 − k3 z 3 − z 3 − 2 z 3 θˆ P3T P3 . uq = b4 2 2l3
V˙3 ≤ −
4
y˙1 = α˙ 1d − α˙ 1 = −
1 1 + 2 z 22 (φ2 2 − θˆ )P2T P2 + 2 z 32 φ3 2 P3T P3 2l2 2l3
i=2
V˙4 ≤ −
Then, we have ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
i=1
+
and define θ = max{φ2 2 , φ3 2 , φ4 2 }. Furthermore, combining (17) with (18) results in
y1 = α1d − α1 ,
Thus, substituting (13) into (12) it follows that: V˙3 ≤ −
3
(18)
4 r1 2 T z P P − m 1 θˆ θ˙ˆ = 2 i i i 2l i=2 i
(24)
where m 1 and li for i = 2, 3, and 4 are positive constants. Remark 3: By combining the RBF networks approximation and DSC technique, the controller designed has a simpler structure and the problems of linear in the unknown system parameters and explosion of complexity are overcome. Also, the number of adaptive parameters is reduced to only one, while four adaptive parameters are required in [42]. This will alleviate the computational burden and render the designed scheme more effective and suitable in practical applications. Remark 4: It can be clearly seen that the proposed controllers (14) and (18) have simpler structure. This means that the proposed NNs-based adaptive DSCs are easy to be implemented in real world applications. To show this, a comparison between the proposed controllers and the classical ones will be given in Section V. Now, we are ready to present our first main result in this brief. Theorem 1: Consider (1) and the given reference signal xd . Then under the action of the NNs-based adaptive DSCs (14), (18), and the adaptive law (24), 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.
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IV. S TABILITY A NALYSIS To address the stability analysis of the resulting closed-loop system, substituting (24) into (23), one has V˙ ≤ −
4 i=1
ki z i2 +
4 1 i=2
2
(li2 + εi2 ) +
+ z 1 y1 + a1 z 2 y2 −
2
yi y˙i
i=1
m1 ˜ ˆ θ θ. r1
(25)
According to [29] and [30], |Di | has a maximum Di M on compact set |i |, i = 1, 2, i.e., |Di | ≤ Di M . Therefore, we have y2 y2 1 2 2 τ y1 y˙1 ≤ − 1 + |D1M ||y1 | ≤ − 1 + D y + 1 1 2τ 1M 1 2 y2 1 2 2 τ D y + y2 y˙2 ≤ − 2 + 2 2τ 2M 2 2
Fig. 1.
x1 and xd for DSC.
Fig. 2.
x1 and xd for classical backstepping.
Fig. 3.
Tracking error for DSC.
with τ > 0. Using the following inequalities z 1 y1 ≤ 1/4y12 + z 12 , a1 z 2 y2 ≤ a12 /4y22 +z 22 , and −θ˜ θˆ ≤ −θ˜ (θ˜ +θ) ≤ −(1/2)θ˜ 2 +(1/2)θ 2 , (25) can be rewritten in the following form: V˙ ≤ −(k1 − 1)z 12 − (k2 − 1)z 22 −
4
ki z i2 −
i=3
m1 2 θ˜ 2r1
4 m1 2 1 1 1 2 1 2 − − θ + D1M l + εi2 + y12 + 1 4 2τ 2 i 2r1 i=2
2 a1 1 1 2 − − (26) + D2M y22 + τ. 2 4 2τ Choose the design parameters k1 , k2 , and τ such that k1 − 2 ) > 0, and 1 > 0, k2 − 1 > 0, 1/ 1 − (1/4 + (1/2)τ D1M 2 2 1/ 2 − (a1 /4 + (1/2)τ D2M) > 0. Also, we can obtain V˙ ≤ −a0 V + b0 , where a0 = min 2(k1 − 1), 2(k2 − 1)/J, 2k3 , 2(1/ 1− 2 )), 2k , m , 2(1/ − (a 2 /4 + (1/2)τ D 2 )) , and (1/4 + (1/2)τ D1M 4 1 2 1 2M b0 = 4i=2 1/2(li2 + εi2 ) + (m 1 /2)r1 θ 2 + τ . Furthermore, the above inequality implies that
b0 −a0 (t −t0 ) e V (t) ≤ V (t0 ) − a0 b0 b0 + ≤ V (t0 ) + ∀t ≥ t0 . (27) a0 a0 As a result, all z i (i = 1, 2, 3, 4), yi (i = 1, 2), and θ˜ ˜ ≤ belong to the compact set = {(z i , y1 , y2 , θ)|V V (t0 ) + b0 /a0 , ∀t ≥ t0 }. Namely, all the signals in the closed-loop system are bounded. In particular, from (27) we have 2b0 . lim z 2 ≤ t →∞ 1 a0
(28)
Remark 5: It can be seen from the definitions of a0 and b0 that to get a small tracking error we can set r1 large, but li and εi small enough after giving the parameters ki , i , τ , and m 1 . V. S IMULATION R ESULTS In this section, ison between the and the classical given in [42] for parameters:
an example is used to make a comparproposed neural controllers (14) and (18) backstepping controllers (A.1) and (A.2) the PMSM drive system with the following
J = 0.00379 Kgm2 , Rs = 0.68 , L d =0.00315 H, n p =3 L q = 0.00285 H, B = 0.001158 Nm/(rad/s), = 0.1245 H.
The simulation is carried out under the zero initial condition. The reference signals are taken as xd = 0.5 sin(t) + sin(0.5t) and 1.5, 0 ≤ t ≤ 20 TL = 3, t ≥ 20. The RBF NNs are chosen in the following way. The NN φ2T P2 (Z 2 ), φ3T P3 (Z 3 ), and φ4T P4 (Z 4 ) contain eleven nodes with centers spaced evenly in the interval [−10, 10] and widths being equal to 2, respectively. The proposed adaptive neural controllers in this brief are used to control this PMSM motor. The control parameters are chosen as follows: k1 = 60, k2 = 20, k3 = 35, k4 = 25, r1 = 0.01 m 1 = 0.05, l2 = l3 = l4 = 0.5. Also the classical backstepping controllers (A.1) and (A.2) are also utilized to control the systems and the controller parameters ki (i = 1, 2, 3, 4) are chosen as the same as the those in the above adaptive neural controllers. The simulation results for the above two control methods are shown in Figs. 1–10. Note that Figs. 1–4 display the system outputs, the reference signals and the tracking error for both
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5
Fig. 4.
Tracking error for classical backstepping.
Fig. 8.
u d for classical backstepping.
Fig. 5.
u q for DSC.
Fig. 9.
id and iq for DSC.
Fig. 6.
u q for classical backstepping.
Fig. 10.
id and iq for classical backstepping.
VI. C ONCLUSION
Fig. 7.
u d for DSC.
control approaches. In Figs. 1–4, it can be clearly observed that under the actions of controllers (14) and (18) and the traditional backstepping controllers (A.1) and (A.2) in [42], the system outputs follow the desired reference signals well. The control input signals are shown in Figs. 5–8, while Figs. 9 and 10 display the trajectories of i d and i q . From the simulations, it is clearly shown that the proposed adaptive neural DSCs in this brief can trace the reference signal quite well, even though the controllers have much simpler structure than the classical ones, which is more practical to be implemented.
In this brief, the NNs-based adaptive DSC is designed for PMSMs. The proposed control method can overcome not only the problem of linear in the unknown system parameters, but also the explosion of complexity inherent in the backstepping design. It is demonstrated that under such controllers, all the signals of the closedloop system are bounded and the tracking error converges to a small neighborhood of the origin. The effectiveness and robustness of the developed new control scheme against the parameter uncertainties and load disturbances are illustrated by an example and simulation. A PPENDIX A This part is devoted to provide the designed controllers by classical backstepping approach in [42]
1 ∂α2 x2 uq = − k3 z 3 − b1 x3 − b3 x2 − a1 z 2 + b4 ∂ x1 2 ∂α2 (i+1) ∂α2 ˙ˆ ∂α2 ˙ˆ ∂α2 ˙ˆ + x + + B + J T L (i) d ∂ Bˆ ∂ TˆL ∂ Jˆ ∂x
i=0 d ∂α2 a1 a2 B T + x3 + x3 x4 − x2 − L (A.1) ∂ x2 J J J J
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ud = −
1 (k4 z 4 + a2 z 2 x3 + b2 x2 z 3 + c1 z 4 + c2 x2 x3 ). c3
(A.2)
Remark 6: By comparing the NNs-based adaptive DSCs (14) and (18) with the classical backstepping controllers (A.1) and (A.2) given in [42], it can be seen that the classical controllers (A.1) and (A.2) are much more complicated than the DSCs (14) and (18). The number of terms in classical backstepping controllers is much larger. This drawback was called explosion of terms in [21]. R EFERENCES [1] W. Leonhard, Control of Electrical Drives. New York, NY, USA: Springer-Verlag, 1985. [2] R. J. Wai, “Total sliding-mode controller for PM synchronous servo motor drive using recurrent fuzzy neural network,” IEEE Trans. Ind. Electron., vol. 48, no. 5, pp. 926–94, Oct. 2001. [3] W. Shang, S. Zhao, Y. Shen, and Z. Qi, “A sliding mode flux-linkage controller with integral compensation for switched reluctance motor,” IEEE Trans. Magn., vol. 45, no. 9, pp. 3322–3328, Sep. 2009. [4] C. M. Verrelli, “Adaptive learning control design for robotic manipulators driven by permanent magnet synchronous motors,” Int. J. Control, vol. 84, no. 6, pp. 1024–1030, Jun. 2011. [5] C. M. Verrelli, “Synchronization of permanent magnet electric motors: New nonlinear advanced results,” Nonlinear Anal. RWA, vol. 13, no. 1, pp. 395–409, Feb. 2012. [6] Y. Guo, Z. Xi, and D. Cheng, “Speed regulation of permanent magnet synchronous motor via feedback dissipative Hamiltonian realization,” IET Control Theory Appl., vol. 1, no. 1, pp. 281–290, Jan. 2007. [7] Z. Chen, M. Tomita, S. Ichikawa, S. Doki, and S. Okuma, “An extended electromotive force model for sensorless control of interior permanent magnet synchronous motors,” IEEE Trans. Ind. Electron., vol. 50, no. 2, pp. 288–295, Apr. 2003. [8] H. Z. Jin and J. M. Lee, “An RMRAC current regulator for permanent magnet synchronous motor based on statistical model interpretation,” IEEE Trans. Ind. Electron., vol. 56, no. 1, pp. 169–177, Jan. 2009. [9] I. C. Baik, K. H. Kim, and M. J. Youn, “Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique,” IEEE Trans. Control Syst. Technol., vol. 8, no. 1, pp. 47–54, Jan. 2000. [10] J. Zhou and Y. Wang, “Real-time nonlinear adaptive backstepping speed control for a PM synchronous motor,” Control Eng. Pract., vol. 13, no. 10, pp. 1259–1269, Oct. 2005. [11] H. Chaoui and P. Sicard, “Adaptive fuzzy logic control of permanent magnet synchronous machines with nonlinear friction,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1123–1133, Feb. 2012. [12] S. Barkat, A. Tlemçani, and H. Nouri, “Noninteracting adaptive control of PMSM using interval type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 925–936, Oct. 2011. [13] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995. [14] A. Boulkroune and M. M’Saad, “On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign,” Fuzzy Sets Syst., vol. 201, no. 16, pp. 71–85, Aug. 2012. [15] X. P. Liu, G. X. Gu, and K. M. Zhou, “Robust stabilization of MIMO nonlinear systems by backstepping,” Automatica, vol. 35, no. 2, pp. 987–992, May 1999. [16] Q. Zhou, P. Shi, S. Xu, and H. Li, “Observer-based adaptive neural network control for nonlinear stochastic systems with time-delay,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 1, pp. 71–80, Jan. 2013. [17] S. C. Tong, Y. Li, Y. M. Li, and Y. J. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [18] A. Boulkroune, M. M’Saad, and M. Farza, “State and output feedback fuzzy variable structure controllers for multivariable nonlinear systems subject to input nonlinearities,” Int. J. Adv. Manuf. Technol., vol. 71, nos. 1–4, pp. 539–556, Mar. 2014. [19] Y. A.-R. I. Mohamed, “Design and implementation of a robust current control scheme for a PMSM vector drive with a simple adaptive disturbance observer,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 1981–1988, Aug. 2007. [20] C. M. Kwan and F. L. Lewis, “Robust backstepping control of induction motors using neural networks,” IEEE Trans. Neural Netw., vol. 11, no. 5, pp. 1178–1187, Sep. 2000.
[21] S. C. Tong, X. L. He, and H. G. Zhang, “A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1059–1069, Oct. 2009. [22] B. Jiang, Z. Mao, and P. Shi, “H∞ filter design for a class of networked control systems via T-S fuzzy model approach,” IEEE Trans. Fuzzy Syst., vol. 18, no. 1, pp. 201–208, Feb. 2010. [23] A. Boulkroune, M. M’Saad, and M. Farza, “Fuzzy approximationbased indirect adaptive controller for multi-input multi-output non-affine systems with unknown control direction,” IET Control Theory Appl., vol. 6, no. 17, pp. 2619–2629, Nov. 2012. [24] A. Boulkroune, M. M’Saad, and M. Farza, “Adaptive fuzzy tracking control for a class of MIMO non-affine uncertain systems,” Neurocomputing, vol. 93, pp. 48–55, Sep. 2012. [25] J. Hu and J. Zou, “Adaptive backstepping control of permanent magnet synchronous motors with parameter uncertainties,” Control Decision, vol. 21, no. 11, pp. 1264–1269, Nov. 2006. [26] T. P. Zhang and S. S. Ge, “Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form,” Automatica, vol. 44, no. 7, pp. 1895–1903, Jul. 2008. [27] D. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes, “Dynamic surface control for a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. 45, no. 10, pp. 1893–1899, Oct. 2000. [28] P. Li, J. Chen, T. Cai, and G. H. Wang, “Adaptive robust dynamic surface control of pure-feedback systems using self-constructing neural networks,” Int. J. Innov. Comput., Inf. Control, vol. 9, no. 7, pp. 2839–2860, Jul. 2013. [29] S. C. Tong, Y. M. Li, G. Feng, and T. S. Li, “Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 4, pp. 1124–1135, Aug. 2011. [30] D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strictfeedback form,” IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195–202, Jan. 2005. [31] Z. Wu, P. Shi, H. Su, and J. Chu, “Dissipativity analysis for discretetime stochastic neural networks with time-varying delays,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 3, pp. 345–355, Mar. 2013. [32] T. S. Li, R. H. Li, and D. Wang, “Adaptive neural control of nonlinear MIMO systems with unknown time delays,” Neurocomputing, vol. 78, no. 1, pp. 83–88, Feb. 2012. [33] X. Luan, F. Liu, and P. Shi, “Neural network based stochastic optimal control for nonlinear Markov jump systems,” Int. J. Innov. Comput., Inf. Control, vol. 6, no. 8, pp. 3715–3723, Aug. 2010. [34] M. Chen, S. S. Ge, and B. How, “Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812, May 2010. [35] T. P. Zhang, S. S. Ge, and C. C. Hang, “Adaptive neural network control for strict-feedback nonlinear systems using backstepping design,” Automatica, vol. 36, no. 12, pp. 1835–1846, Dec. 2000. [36] Y. J. Liu, S. C. Tong, D. Wang, T. S. Li, and C. L. P. Chen, “Adaptive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO systems,” IEEE Trans. Neural Netw., vol. 22, no. 8, pp. 1328–1334, Aug. 2011. [37] M. Wang, X. Liu, and P. Shi, “Adaptive neural control of pure-feedback nonlinear time-delay systems via dynamic surface technique,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1681–1692, Dec. 2011. [38] R. Wai and H. Chang, “Backstepping wavelet neural network control for indirect field-oriented induction motor drive,” IEEE Trans. Neural Netw., vol. 15, no. 2, pp. 367–382, Mar. 2004. [39] T. S. Li, D. Wang, G. Feng, and S. C. Tong, “A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 915–927, Jun. 2010. [40] T. S. Li, S. C. Tong, and G. Feng, “A novel robust adaptivefuzzy-tracking control for a class of nonlinear multi-input/multi-output systems,” IEEE Trans. Fuzzy Syst., vol. 18, no. 1, pp. 150–160, Feb. 2010. [41] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [42] J. P. Yu, Y. M. Ma, B. Chen, and H. S. Yu, “Adaptive fuzzy backstepping position tracking control for a permanent magnet synchronous motor,” Int. J. Innov. Comput., Inf. Control, vol. 7, no. 4, pp. 1589–1602, Apr. 2011.
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Neural Network-Based Adaptive Dynamic Surface Control for Permanent Magnet Synchronous Motors Jinpeng Yu, Peng Shi, Senior Member, IEEE , Wenjie Dong, Member, IEEE , Bing Chen, and Chong Lin, Senior Member, IEEE Abstract— This brief considers the problem of neural networks (NNs)based adaptive dynamic surface control (DSC) for permanent magnet synchronous motors (PMSMs) with parameter uncertainties and load torque disturbance. First, NNs are used to approximate the unknown and nonlinear functions of PMSM drive system and a novel adaptive DSC is constructed to avoid the explosion of complexity in the backstepping design. Next, under the proposed adaptive neural DSC, the number of adaptive parameters required is reduced to only one, and the designed neural controllers structure is much simpler than some existing results in literature, which can guarantee that the tracking error converges to a small neighborhood of the origin. Then, simulations are given to illustrate the effectiveness and potential of the new design technique.
Index Terms— Backstepping, dynamics surface control (DSC), neural networks (NNs), nonlinear system, permanent magnet synchronous motor (PMSM). I. I NTRODUCTION
I
N THE past few decades, permanent magnet synchronous motors (PMSMs) have attracted considerable attention as its wide industrial applications [1]. However, it is still a challenging problem to control the PMSMs to get desired dynamic performance because their dynamics are usually multivariable, coupled, and highly nonlinear; and very sensitive to external load disturbances and parameter variations. To achieve better performance of PMSMs, much work has devoted to develop nonlinear control methods for PMSM and various algorithms have been proposed, see [2]–[12] and the references therein. In another research front line, the problem of backstepping-based nonlinear adaptive control has been extensively studied for nonlinear systems with parameter uncertainty, particularly, matching conditions not being satisfied [13]–[18]. The conventional backstepping technique has been successfully applied to the control of PMSM drivers recently [19], [20]. But there are some drawbacks in backstepping approach. One problem is that certain functions must be linear in the unknown system parameters [20]. Another limitation is the explosion of complexity caused by the repeated differentiations of virtual input [21]–[24]. Theoretically, the calculation of virtual control derivation is simple, but it can be quite tedious and complicated in practical applications when n is larger than three because the desired
Manuscript received August 22, 2013; revised December 3, 2013 and February 14, 2014; accepted April 6, 2014. This work was supported in part by the Australian Research Council under Grant DP140102180, in part by the National Key Basic Research (973) Program, China under Grant 2011CB710706 and Grant 2012CB215202, in part by the 111 Project under Grant B12018, and in part by the Natural Science Foundation of China under Grant 61104076, Grant 61174131, and Grant 61174033. J. 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, 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]). W. Dong is with the Department of Electrical Engineering, University of Texas-Pan American, Edinburg, TX 78539 USA (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/TNNLS.2014.2316289
controller u will include the derivation of αn , which requires the second derivation of αn−1 and so on. For instance, the backstepping controller in [25] is overly cumbersome, which is the representative of explosion of complexity problem. To overcome this issue, a dynamic surface control (DSC) method was proposed by introducing a firstorder filtering of the virtual input at each step of the conventional backstepping approach [26]–[30]. In addition, neural network (NN) approximation method has been widely used in many applications, mainly by its inherent capability for modeling and controlling highly uncertain, nonlinear, and complex systems [31]–[33]. The radial basis function (RBF) NN is considered in [34] and [35] as a two-layer network, which contains both hidden and output layers. Therefore, NNs can be used to handle uncertain information and be employed to control the systems which are ill-defined or too complex to have a precise mathematical model. The RBF NN has been found as one of the very effective tools in functional approximations, which has wide applications [36]–[38]. In this brief, an NN-based adaptive DSC is proposed to solve the above problems of the conventional backstepping method for PMSM drive systems. The RBF networks are used to approximate the unknown nonlinear functions to solve the first problem of linear in the unknown system parameters, and a DSC technique is proposed to tackle the second problem of explosion of complexity by firstorder filtering technique at each step of the conventional backstepping design [39], [40]. The proposed control scheme not only guarantees the boundedness of all of the signals in the closed-loop system, but also reduces the number of adaptive parameters which alleviates the computational burden. Compared with the existing results on adaptive neural control for PMSMs, the main contributions of this brief are that: 1) the developed approximation-based neural controller has a simpler structure and both the problems of linear in the unknown system parameters and explosion of complexity are considered and 2) the number of adaptive parameters is considerably reduced to only one, which is independent of the number of the neural basis function and system state variables. As a result, the computational burden of the scheme is alleviated, which will render the designed scheme more suitable for practical applications. Finally, the novelty of the new design method is verified by simulations to demonstrate the effectiveness and the robustness against the parameter uncertainties and load disturbances. II. M ATHEMATICAL M ODEL OF THE PMSM D RIVE S YSTEM AND P RELIMINARIES The model of PMSMs can be described in d − q frame through the Park transformation as follows [1]: d dt dω J dt di d Ld dt di q Lq dt
=ω =
3 n p (L d − L q )i d i q + i q − Bω − TL 2
= −Rs i d + n p ωL q i q + u d = −Rs i q − n p ωL d i d − n p ω + u q
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where is the rotor position, ω denotes the rotor angular velocity, i d and i q stand for the d − q axis currents, u d and u q are the d − q axis voltages, Rs is the stator resistance, L d and L q are the d − q axis stator inductors, n p is the pole pair, J means the rotor moment of inertia, B is the viscous friction coefficient, T is the electromagnetism torque, TL is the load torque, and denotes magnet flux linkage. To simplify the above model, the following notations are introduced: x1 = , x2 = ω, x3 = i q , x4 = i d 3n p 3n p (L d − L q ) Rs a1 = , a2 = , b1 = − 2 2 Lq n p Ld n p 1 b2 = − , b3 = − , b4 = Lq Lq Lq n p Lq Rs 1 c1 = − , c2 = , c3 = . Ld Ld Ld Using these notations, the dynamic model of the PMSM can be described by the following differential equations: x˙1 = x2 a1 a2 B T x˙2 = x3 + x3 x4 − x2 − L J J J J x˙3 = b1 x3 + b2 x2 x4 + b3 x2 + b4 u q x˙4 = c1 x4 + c2 x2 x3 + c3 u d .
(1)
In this brief, the RBF NN will be used to approximate the unknown ˆ = φ ∗T P(z), where z ∈ continuous function ϕ(z) : R q → R as ϕ(z) z ⊂ R q is the input vector with q being the NN input dimension, φ ∗ = [∗1 , . . . , ∗n ]T ∈ R n , is the weight vector, n > 1 is the NN node number, and P(z) = [ p1 (z), . . . , pn (z)]T ∈ R n is the basis function vector with pi (z) chosen as the commonly used Gaussian function in the following form: −(z − νi )T (z − νi ) , i = 1, 2, . . . , n pi (z) = exp qi2 where νi = [νi1 , . . . , νiq ]T is the center of the receptive field and qi is the width of the Gaussian function. It has been shown in [41] that, for a given scalar ε > 0, by choosing sufficiently large l, the RBF NN can approximate any continuous function over a compact set z ∈ R q to an arbitrary accuracy as ϕ(z) = φ T P(z) + δ(z) ∀ z ∈ z ⊂ R q , where δ(z) is the approximation error satisfying |δ(z)| ≤ ε and φ is an unknown ideal constant weight vector, which is an artificial quantity required for analytical purpose. Typically, φ is chosen as the value of φ ∗ that minimizes |δ(z)| for all z ∈ z ∗T φ := arg min sup ϕ(z) − φ P(z) . φ∗ ∈ R n
1 and α1d (0) = α1 (0). The purpose of this filter is to generate α1d and its derivative α˙ 1d such that |α1d − α1 | is smaller than a given level (4) 1 α˙ 1d + α1d = α1 , α1d (0) = α1 (0). Define z 2 = x2 − α1d . Using (3) and (4), (2) can be rewritten in the following form: V˙1 = z 1 z˙ 1 = z 1 (z 2 + α1d − x˙d ) = −k1 z 12 + z 1 z 2 + z 1 (α1d − α1 ). Step 2: Differentiating z 2 obtains z˙2 =
V˙2 = −k1 z 12 + z 1 (α1d − α1 ) + z 2 (a1 x3 + f2 )
V˙1 = z 1 z˙ 1 = z 1 (x2 − x˙d ).
(2)
Construct the virtual control law α1 as α1 = −k1 z 1 + x˙d
with k1 > 0 being a design control gain. Next, introduce a new state variable α1d . Let α1 pass through a first-order filter with time constant
(7)
where θˆ is the estimation of the unknown constant θ which will be specified later. Define a new state variable α2d . Let α2 pass through a first-order filter with time constant 2 to obtain α2d as 2 α˙ 2d + α2d = α2 , α2d (0) = α2 (0)
(10)
and define z 3 = x3 − α2d 2 i=1
1 ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 ) + l22 2
1 1 + 2 z 22 φ2 2 − θˆ P2T P2 + a1 z 2 z 3 + ε22 . 2 2l2
(11)
Step 3: Differentiating z 3 results in the following equation: z˙3 = x˙3 − α˙ 2d = b1 x3 + b2 x2 x4 + b3 x2 + b4 u q − α˙ 2d . Choose the following Lyapunov function candidate V3 = V2 + (1/2)z 32 . Furthermore, differentiating V3 yields V˙3 = −
(3)
(6)
where f2 (Z 2 ) = z 1 + a2 x3 x4 − Bx2 − TL − J α˙ 1d and Z 2 = [x1 , x2 , x3 , x4 , xd , x˙d ]T . Remark 1: It should be mentioned that the system parameters B, TL , and J may be unknown in the PMSM drive system, then they cannot be used to construct the control signal unless we specify their corresponding adaptation laws. To avoid this trouble, we will employ the NNs to approximate the nonlinear function f2 and the indeterministic parameters will be considered. According to the RBF NN approximation property, for a given ε2 > 0, there exists a RBF NN φ2T P2 (Z 2 ) such that f2 = φ2T P2 (Z 2 ) + δ2 (Z 2 ), where δ2 (Z 2 ) is the approximation error satisfying |δ2 | ≤ ε2 . Consequently, we can show the following inequality:
1 z 2 f2 = z 2 φ2T P2 + δ2 ≤ 2 z 22 φ2 2 P2T P2 2l2 1 2 2 2 + l 2 + z 2 + ε2 . (8) 2 Then the virtual control α2 is constructed as
1 1 1 − k2 z 2 − z 2 − 2 z 2 θˆ P2T P2 (9) α2 = a1 2 2l2
V˙2 = −
In this section, we will present an adaptive DSC for PMSMs based on backstepping. Step 1: For the reference signal xd , we define the tracking error variable as z 1 = x1 − xd . From the first subsystem of (1), the error dynamic system is computed by z˙ 1 = x2 − x˙d . Choose a Lyapunov function candidate as V1 = (1/2)z 12 , then the time derivative of V1 is given by
a1 a2 B T x3 + x3 x4 − x2 − L − α˙ 1d . J J J J
Now, choose the Lyapunov function candidate as V2 = V1 + J /2z 22 . Obviously, time derivative of V2 can be expressed as
z∈z
III. A DAPTIVE DSC FOR PMSM S
(5)
2
as
1 ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 ) + l22 2
i=1 z2 1 + 22 (φ2 2 − θˆ )P2T P2 + ε22 + z 3 ( f 3 + b4 u q ) 2 2l2
(12)
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where f3 (Z 3 ) = b1 x3 + b2 x2 x4 + b3 x2 + a1 z 2 − α˙ 2d and Z 3 = Z 2 . Remark 2: It should be noted that f3 contains the derivative of α2d and the nonlinear term b2 x2 x4 , this will make the backstepping design become very difficult and the designed u q will have a complex structure. To solve this problem, we will use NNs to approximate the nonlinear function f 3 . Similarly, for given ε3 > 0, there exists φ3T P3 (Z 3 ) such that 1 2 1 1 1 z 3 φ3 2 P3T P3 + l32 + z 32 + ε32 . 2 2 2 2 2l3
z3 f3 ≤
(13)
2
3 1
2
li2 +
i=1
+
i=2
1 ε2 + z 32 + z 3 b4 u q . 2 i 2
(14)
i=1
where f4 (Z 4 ) = c1 z 4 + c2 x2 x3 and Z 4 = Z 2 . Similarly, for a given ε4 > 0, there exists φ4T P4 (Z 4 ) satisfying 1 2 1 1 1 z 4 φ4 2 P4T P4 + l42 + z 42 + ε42 . 2 2 2 2 2l4
(16)
Substituting (16) into (15) gives ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
i=1 3 4 4 1 2 1 2 1 2 2 − θ)P ˆ T Pi + + l ε z (φ + i i 2 i 2 i 2l 2 i i=2 i i=2 i=2 1 1 + 2 z 42 φ4 2 P4T P4 + z 42 + c3 z 4 u d . (17) 2 2l 4
We design u d as ud =
1 1 1 − k4 z 4 − z 4 − 2 z 4 θˆ P4T P4 c3 2 2l4
(21)
1 2 1 2 1 ˜2 θ y + y + 2 1 2 2 2r1
(22)
4
ki z i2 +
4 1 i=2
2 ⎡
(li2 + εi2 ) +
2
yi y˙i + z 1 y1
i=1
⎤ 4 1 ˜ ⎣ r1 2 T ˙ + a1 z 2 y2 + θ − z P Pi + θˆ ⎦ . r1 2l 2 i i i=2
(23)
i
According to (23), the corresponding adaptive law is chosen as follows:
ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
3 3 3 1 2 1 2 1 2 ˆ T Pi + z i (φi 2 − θ)P li + ε + i 2 2 2 i 2l i=2 i i=2 i=2 + z 4 ( f 4 (Z 4 ) + c3 u d ) (15)
3
(20)
with D2 = 1/a1 (k2 +1/2+(1/2)l22 θˆ P2T P2 )˙z 2 +(1/2)a1 l22 θ˙ˆ P2T P2 z 2 + ˆ Z˙ 2 ∂ P2 (Z 2 )/∂ Z 2 P2 (Z 2 )z 2 . 1/a1 l22 θ( Choose the following Lyapunov function candidate:
i=1
Step 4: At this step, we will construct the control law u d . Define z 4 = x4 and choose V4 = V3 + (1/2)z 42 . Then, we have
V˙4 ≤ −
y2 = α2d − α2 , θ˜ = θˆ − θ.
Introduce y1 , y2 , and θ˜ as
V˙ ≤ −
3 3 1 2 1 2 1 2 2 − θˆ )P T P + + z (φ + l ε . i i i i 2 i 2 i 2l 2 i=2 i i=2 i=2
z 4 f 4 (Z 4 ) ≤
(19)
where r1 is a positive constant. By differentiating V and taking (19)– (22) into account, one has
ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
3
2
4 1 2 T ˆ z P P (θ − θ). 2 i i i 2l i=2 i
(li2 + εi2 ) +
V = V4 +
i=1 3
V˙4 ≤ −
4 1 i=2
Furthermore, using (14), it can be verified that 3
ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
α1d − α1 − α˙ 1 1 y1 y1 = − + k1 z˙1 − x¨d = − + D1 1 1 y2 y˙2 = − + D2 2
3 1
Now, the control input u q is designed as
1 1 1 − k3 z 3 − z 3 − 2 z 3 θˆ P3T P3 . uq = b4 2 2l3
V˙3 ≤ −
4
y˙1 = α˙ 1d − α˙ 1 = −
1 1 + 2 z 22 (φ2 2 − θˆ )P2T P2 + 2 z 32 φ3 2 P3T P3 2l2 2l3
i=2
V˙4 ≤ −
Then, we have ki z i2 + z 1 (α1d − α1 ) + a1 z 2 (α2d − α2 )
i=1
+
and define θ = max{φ2 2 , φ3 2 , φ4 2 }. Furthermore, combining (17) with (18) results in
y1 = α1d − α1 ,
Thus, substituting (13) into (12) it follows that: V˙3 ≤ −
3
(18)
4 r1 2 T z P P − m 1 θˆ θ˙ˆ = 2 i i i 2l i=2 i
(24)
where m 1 and li for i = 2, 3, and 4 are positive constants. Remark 3: By combining the RBF networks approximation and DSC technique, the controller designed has a simpler structure and the problems of linear in the unknown system parameters and explosion of complexity are overcome. Also, the number of adaptive parameters is reduced to only one, while four adaptive parameters are required in [42]. This will alleviate the computational burden and render the designed scheme more effective and suitable in practical applications. Remark 4: It can be clearly seen that the proposed controllers (14) and (18) have simpler structure. This means that the proposed NNs-based adaptive DSCs are easy to be implemented in real world applications. To show this, a comparison between the proposed controllers and the classical ones will be given in Section V. Now, we are ready to present our first main result in this brief. Theorem 1: Consider (1) and the given reference signal xd . Then under the action of the NNs-based adaptive DSCs (14), (18), and the adaptive law (24), 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.
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IV. S TABILITY A NALYSIS To address the stability analysis of the resulting closed-loop system, substituting (24) into (23), one has V˙ ≤ −
4 i=1
ki z i2 +
4 1 i=2
2
(li2 + εi2 ) +
+ z 1 y1 + a1 z 2 y2 −
2
yi y˙i
i=1
m1 ˜ ˆ θ θ. r1
(25)
According to [29] and [30], |Di | has a maximum Di M on compact set |i |, i = 1, 2, i.e., |Di | ≤ Di M . Therefore, we have y2 y2 1 2 2 τ y1 y˙1 ≤ − 1 + |D1M ||y1 | ≤ − 1 + D y + 1 1 2τ 1M 1 2 y2 1 2 2 τ D y + y2 y˙2 ≤ − 2 + 2 2τ 2M 2 2
Fig. 1.
x1 and xd for DSC.
Fig. 2.
x1 and xd for classical backstepping.
Fig. 3.
Tracking error for DSC.
with τ > 0. Using the following inequalities z 1 y1 ≤ 1/4y12 + z 12 , a1 z 2 y2 ≤ a12 /4y22 +z 22 , and −θ˜ θˆ ≤ −θ˜ (θ˜ +θ) ≤ −(1/2)θ˜ 2 +(1/2)θ 2 , (25) can be rewritten in the following form: V˙ ≤ −(k1 − 1)z 12 − (k2 − 1)z 22 −
4
ki z i2 −
i=3
m1 2 θ˜ 2r1
4 m1 2 1 1 1 2 1 2 − − θ + D1M l + εi2 + y12 + 1 4 2τ 2 i 2r1 i=2
2 a1 1 1 2 − − (26) + D2M y22 + τ. 2 4 2τ Choose the design parameters k1 , k2 , and τ such that k1 − 2 ) > 0, and 1 > 0, k2 − 1 > 0, 1/ 1 − (1/4 + (1/2)τ D1M 2 2 1/ 2 − (a1 /4 + (1/2)τ D2M) > 0. Also, we can obtain V˙ ≤ −a0 V + b0 , where a0 = min 2(k1 − 1), 2(k2 − 1)/J, 2k3 , 2(1/ 1− 2 )), 2k , m , 2(1/ − (a 2 /4 + (1/2)τ D 2 )) , and (1/4 + (1/2)τ D1M 4 1 2 1 2M b0 = 4i=2 1/2(li2 + εi2 ) + (m 1 /2)r1 θ 2 + τ . Furthermore, the above inequality implies that
b0 −a0 (t −t0 ) e V (t) ≤ V (t0 ) − a0 b0 b0 + ≤ V (t0 ) + ∀t ≥ t0 . (27) a0 a0 As a result, all z i (i = 1, 2, 3, 4), yi (i = 1, 2), and θ˜ ˜ ≤ belong to the compact set = {(z i , y1 , y2 , θ)|V V (t0 ) + b0 /a0 , ∀t ≥ t0 }. Namely, all the signals in the closed-loop system are bounded. In particular, from (27) we have 2b0 . lim z 2 ≤ t →∞ 1 a0
(28)
Remark 5: It can be seen from the definitions of a0 and b0 that to get a small tracking error we can set r1 large, but li and εi small enough after giving the parameters ki , i , τ , and m 1 . V. S IMULATION R ESULTS In this section, ison between the and the classical given in [42] for parameters:
an example is used to make a comparproposed neural controllers (14) and (18) backstepping controllers (A.1) and (A.2) the PMSM drive system with the following
J = 0.00379 Kgm2 , Rs = 0.68 , L d =0.00315 H, n p =3 L q = 0.00285 H, B = 0.001158 Nm/(rad/s), = 0.1245 H.
The simulation is carried out under the zero initial condition. The reference signals are taken as xd = 0.5 sin(t) + sin(0.5t) and 1.5, 0 ≤ t ≤ 20 TL = 3, t ≥ 20. The RBF NNs are chosen in the following way. The NN φ2T P2 (Z 2 ), φ3T P3 (Z 3 ), and φ4T P4 (Z 4 ) contain eleven nodes with centers spaced evenly in the interval [−10, 10] and widths being equal to 2, respectively. The proposed adaptive neural controllers in this brief are used to control this PMSM motor. The control parameters are chosen as follows: k1 = 60, k2 = 20, k3 = 35, k4 = 25, r1 = 0.01 m 1 = 0.05, l2 = l3 = l4 = 0.5. Also the classical backstepping controllers (A.1) and (A.2) are also utilized to control the systems and the controller parameters ki (i = 1, 2, 3, 4) are chosen as the same as the those in the above adaptive neural controllers. The simulation results for the above two control methods are shown in Figs. 1–10. Note that Figs. 1–4 display the system outputs, the reference signals and the tracking error for both
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5
Fig. 4.
Tracking error for classical backstepping.
Fig. 8.
u d for classical backstepping.
Fig. 5.
u q for DSC.
Fig. 9.
id and iq for DSC.
Fig. 6.
u q for classical backstepping.
Fig. 10.
id and iq for classical backstepping.
VI. C ONCLUSION
Fig. 7.
u d for DSC.
control approaches. In Figs. 1–4, it can be clearly observed that under the actions of controllers (14) and (18) and the traditional backstepping controllers (A.1) and (A.2) in [42], the system outputs follow the desired reference signals well. The control input signals are shown in Figs. 5–8, while Figs. 9 and 10 display the trajectories of i d and i q . From the simulations, it is clearly shown that the proposed adaptive neural DSCs in this brief can trace the reference signal quite well, even though the controllers have much simpler structure than the classical ones, which is more practical to be implemented.
In this brief, the NNs-based adaptive DSC is designed for PMSMs. The proposed control method can overcome not only the problem of linear in the unknown system parameters, but also the explosion of complexity inherent in the backstepping design. It is demonstrated that under such controllers, all the signals of the closedloop system are bounded and the tracking error converges to a small neighborhood of the origin. The effectiveness and robustness of the developed new control scheme against the parameter uncertainties and load disturbances are illustrated by an example and simulation. A PPENDIX A This part is devoted to provide the designed controllers by classical backstepping approach in [42]
1 ∂α2 x2 uq = − k3 z 3 − b1 x3 − b3 x2 − a1 z 2 + b4 ∂ x1 2 ∂α2 (i+1) ∂α2 ˙ˆ ∂α2 ˙ˆ ∂α2 ˙ˆ + x + + B + J T L (i) d ∂ Bˆ ∂ TˆL ∂ Jˆ ∂x
i=0 d ∂α2 a1 a2 B T + x3 + x3 x4 − x2 − L (A.1) ∂ x2 J J J J
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ud = −
1 (k4 z 4 + a2 z 2 x3 + b2 x2 z 3 + c1 z 4 + c2 x2 x3 ). c3
(A.2)
Remark 6: By comparing the NNs-based adaptive DSCs (14) and (18) with the classical backstepping controllers (A.1) and (A.2) given in [42], it can be seen that the classical controllers (A.1) and (A.2) are much more complicated than the DSCs (14) and (18). The number of terms in classical backstepping controllers is much larger. This drawback was called explosion of terms in [21]. R EFERENCES [1] W. Leonhard, Control of Electrical Drives. New York, NY, USA: Springer-Verlag, 1985. [2] R. J. Wai, “Total sliding-mode controller for PM synchronous servo motor drive using recurrent fuzzy neural network,” IEEE Trans. Ind. Electron., vol. 48, no. 5, pp. 926–94, Oct. 2001. [3] W. Shang, S. Zhao, Y. Shen, and Z. Qi, “A sliding mode flux-linkage controller with integral compensation for switched reluctance motor,” IEEE Trans. Magn., vol. 45, no. 9, pp. 3322–3328, Sep. 2009. [4] C. M. Verrelli, “Adaptive learning control design for robotic manipulators driven by permanent magnet synchronous motors,” Int. J. Control, vol. 84, no. 6, pp. 1024–1030, Jun. 2011. [5] C. M. Verrelli, “Synchronization of permanent magnet electric motors: New nonlinear advanced results,” Nonlinear Anal. RWA, vol. 13, no. 1, pp. 395–409, Feb. 2012. [6] Y. Guo, Z. Xi, and D. Cheng, “Speed regulation of permanent magnet synchronous motor via feedback dissipative Hamiltonian realization,” IET Control Theory Appl., vol. 1, no. 1, pp. 281–290, Jan. 2007. [7] Z. Chen, M. Tomita, S. Ichikawa, S. Doki, and S. Okuma, “An extended electromotive force model for sensorless control of interior permanent magnet synchronous motors,” IEEE Trans. Ind. Electron., vol. 50, no. 2, pp. 288–295, Apr. 2003. [8] H. Z. Jin and J. M. Lee, “An RMRAC current regulator for permanent magnet synchronous motor based on statistical model interpretation,” IEEE Trans. Ind. Electron., vol. 56, no. 1, pp. 169–177, Jan. 2009. [9] I. C. Baik, K. H. Kim, and M. J. Youn, “Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique,” IEEE Trans. Control Syst. Technol., vol. 8, no. 1, pp. 47–54, Jan. 2000. [10] J. Zhou and Y. Wang, “Real-time nonlinear adaptive backstepping speed control for a PM synchronous motor,” Control Eng. Pract., vol. 13, no. 10, pp. 1259–1269, Oct. 2005. [11] H. Chaoui and P. Sicard, “Adaptive fuzzy logic control of permanent magnet synchronous machines with nonlinear friction,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1123–1133, Feb. 2012. [12] S. Barkat, A. Tlemçani, and H. Nouri, “Noninteracting adaptive control of PMSM using interval type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 925–936, Oct. 2011. [13] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995. [14] A. Boulkroune and M. M’Saad, “On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign,” Fuzzy Sets Syst., vol. 201, no. 16, pp. 71–85, Aug. 2012. [15] X. P. Liu, G. X. Gu, and K. M. Zhou, “Robust stabilization of MIMO nonlinear systems by backstepping,” Automatica, vol. 35, no. 2, pp. 987–992, May 1999. [16] Q. Zhou, P. Shi, S. Xu, and H. Li, “Observer-based adaptive neural network control for nonlinear stochastic systems with time-delay,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 1, pp. 71–80, Jan. 2013. [17] S. C. Tong, Y. Li, Y. M. Li, and Y. J. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [18] A. Boulkroune, M. M’Saad, and M. Farza, “State and output feedback fuzzy variable structure controllers for multivariable nonlinear systems subject to input nonlinearities,” Int. J. Adv. Manuf. Technol., vol. 71, nos. 1–4, pp. 539–556, Mar. 2014. [19] Y. A.-R. I. Mohamed, “Design and implementation of a robust current control scheme for a PMSM vector drive with a simple adaptive disturbance observer,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 1981–1988, Aug. 2007. [20] C. M. Kwan and F. L. Lewis, “Robust backstepping control of induction motors using neural networks,” IEEE Trans. Neural Netw., vol. 11, no. 5, pp. 1178–1187, Sep. 2000.
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