Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain Shaohua Luo Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 033135 (2014); doi: 10.1063/1.4895810 View online: http://dx.doi.org/10.1063/1.4895810 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Composite adaptive fuzzy control for synchronizing generalized Lorenz systems Chaos 22, 023144 (2012); 10.1063/1.4721901 Adaptive gain fuzzy sliding mode control for the synchronization of nonlinear chaotic gyros Chaos 19, 013125 (2009); 10.1063/1.3072786 Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control Chaos 18, 033133 (2008); 10.1063/1.2980046 Loss analysis of permanent-magnet synchronous motor using three-dimensional finite-element method with homogenization method J. Appl. Phys. 103, 07F126 (2008); 10.1063/1.2833313 A nonlinear controller design for permanent magnet motors using a synchronization-based technique inspired from the Lorenz system Chaos 18, 013111 (2008); 10.1063/1.2840779
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CHAOS 24, 033135 (2014)
Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain Shaohua Luo School of Automation, Chongqing University, Chongqing 400044, China and College of Mechanical Engineering, Hunan University of Arts and Science, Hunan 415000, China
(Received 8 April 2014; accepted 5 September 2014; published online 17 September 2014) This paper is concerned with the problem of adaptive fuzzy dynamic surface control (DSC) for the permanent magnet synchronous motor (PMSM) system with chaotic behavior, disturbance and unknown control gain and parameters. Nussbaum gain is adopted to cope with the situation that the control gain is unknown. And the unknown items can be estimated by fuzzy logic system. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. Finally, the numerical simulations indicate that the proposed scheme can suppress the chaos of C 2014 PMSM and show the effectiveness and robustness of the proposed method. V AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895810]
In this study, an adaptive fuzzy dynamic surface control approach is developed based on the Nussbaum gain for chaotic PMSM. Nussbaum gain is adopted to cope with unknown control gain and fuzzy logic system is employed to estimate the unknown items. The closed-loop system achieves asymptotic stability by dynamic surface controller. The simulation results are indicated the effectiveness and robustness of the proposed scheme.
I. INTRODUCTION
Recently, permanent magnet synchronous motor is of great interest, particularly for the engineering applications in the low-medium power range, since it has many advantages, such as compact size, high torque/weight ratio, high torque/ inertia ratio, and absence of rotor losses.1 Unfortunately, the PMSM drive system appears chaos which degrades the performance of PMSM in the working condition.2 The chaotic behavior is undesirable since it can completely destroy the stabilization of the motor. Hence, many people have worked on some methods to identify, suppress, synchronize and even induce the occurrence of the chaos in various systems.3–5 The OGY method is a basic methodology for controlling chaos6,7 but finding a reasonable parameter is not often simple. A neuro-fuzzy controller (NFC) is suitable for control of systems with uncertainties and nonlinearities.8,9 Meanwhile, the NFC can also achieve self-learning; however, it is unsuitable for on-line learning real-time control due to the drawback of time consuming.10,11 In view of this, a unified fuzzy approach is presented for controlling chaos in nonlinear partial differential systems which is represented by TakagiSugeno fuzzy model.21 In Ref. 12, a position tracking control method based on adaptive fuzzy backstepping is presented for the induction motors with unknown parameters. However, the backstepping suffers from the “explosion of complexity” caused by the repeated differentiation of virtual 1054-1500/2014/24(3)/033135/9/$30.00
control functions, and the explosion becomes more significant as the order increases.13 To avoid explosion of the differentiation items on the virtual function, the dynamic surface control by introducing a first-order filter at each recursive step of the backstepping design procedure is developed by Swaroop et al.14 By incorporating DSC into a neural-network-based adaptive control design framework, Wang and Huang15 proposed a backstepping-based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. However, disturbances is not involved in the model and unknown control gain is assumed to be equal to one. Zhang and Ge16 further studied the control design for some special nonlinear systems with unknown dead-zone using the DSC method. In their works, it is assumed there exists positive constants which satisfy the constrained condition. But it is very difficult to define the boundedness of control gain in real practice. Incidentally, the PMSM is known to exhibit chaotic behavior under certain conditions. Whether the latter methods can suppress the chaos oscillation in PMSM needs for further research since DSC with radial basis function (RBF) has been pioneered by the work of Wang. Furthermore, the unknown control gain is assumed to be known in most papers, but the value of control gain is usually unknown in the reality. The Nussbaum gain theory is adopted to cope with this kind of problem.17,18 Then, the unknown parameters can be estimated effectively with the Nussbaum gain adaptive method. The attention of the electric drives community has been recently paid to the “sensorless” control problem of PMSM in which only stator current and voltage measurements are available for feedback.22 However, complicated computation process is required. To the best of our knowledge, the combination between adaptive DSC, Nussbaum gain and fuzzy has been seldom applied in the control of chaos for the PMSM system yet. In this paper, an adaptive fuzzy dynamic surface control method is developed based on the Nussbaum gain for PMSM
24, 033135-1
C 2014 AIP Publishing LLC V
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033135-2
Shaohua Luo
wherein the unknown parameters and control gain, disturbance and chaos are presented. The whole process of the system design is performed in a step-by-step manner. For each step, a first-order filter is employed to gain the information of derivative of the virtual control function. Then, the explosion of complexity for backstepping is well solved. At last, the simulation results are presented to demonstrate the effectiveness and robustness of the proposed scheme in the chaotic PMSM system. II. DYNAMICS OF THE PMSM
The model of the permanent magnet synchronous motor with smooth air-gap in the well-known (d-q) frame is described as follow:19,20 8 dh > > > ¼x > > dt > > > dx > < ¼ rðiq xÞ T~L dt (1) diq > > ~ ¼ i i x þ cx þ u > q d q > > dt > > did > > : ¼ id þ iq x þ u~d ; dt where r and c mean the system operating parameters, which are positive. T~L , u~q , and u~d stand for the d q load torque and axis voltages, respectively. h, x, id , and iq represent the state variables, which denote angle, angle speed and the d q axis currents, respectively. According to Eq. (1), the external inputs are set to zero, namely, T~L ¼ u~q ¼ u~d ¼ 0. Then, the PMSM system becomes an unforced system as follows: 8 dh > > > ¼x > > dt > > > dx > < ¼ rðiq xÞ dt (2) di > q > ¼ i i x þ cx > q d > > dt > > did > > : ¼ id þ iq x: dt It is clear that the dynamic model of PMSM is highly nonlinear due to the coupling between the speed and the currents. As we know, r and c of the system parameters have uncertainties, which are influenced by the real work conditions. In addition, the PMSM is experiencing chaotic behavior when the parameters lie in a certain range. The typical chaotic attractor in 3-D space is shown in Figure 1.
FIG. 1. The chaotic attractor with parameters r ¼ 5.45 and c ¼ 20.
Chaos 24, 033135 (2014)
In order to eliminate chaos, use ud and uq as the manipulated variables which are desirable for the real application. Then, the model of PMSM driver system can be rewritten by the following differential equations: 8 dh > > > ¼x > > dt > > > dx > < ¼ rðiq xÞ dt (3) diq > > ¼ i i x þ cx þ u > q d q > > dt > > did > > : ¼ id þ iq x þ ud : dt For simplicity, the following symbols are introduced: x1 ¼ h;
x2 ¼ x;
x3 ¼ i q ;
x4 ¼ id :
By using these symbols, the dynamic model of PMSM with bounded disturbances can be represented as follows 8 x_ 1 ¼ x2 þ d1 > > < x_ 2 ¼ rðx3 x2 Þ þ d2 (4) x_ ¼ x3 x2 x4 þ cx2 þ uq þ d3 > > : 3 x_ 4 ¼ x4 þ x2 x3 þ ud þ d4 ; where di , i ¼ 1; …, 4, mean the disturbances. In order to facilitate the design, the following assumptions are given. Assumption 1. The gain function r is unknown and bounded. Moreover, it is assumed that jrj G. Assumption 2. The unknown disturbance terms di satisfy jdi ðÞj < Di ; 1; …; 4, and Di is the unknown positive constant. Assumption 3. The desired trajectory yr is continuous, :: and ½yr ; y_ r ; yr T 2 Nr with the known compact set Nr ¼ :: T :: f½yr ; y_ r ; yr : y2r þ y_ 2r þ y2r Br g R3 , whose size Br is a known positive constant. Because the sign of control is unknown, the Nussbaum function is employed in this paper. Obviously, a function NðgÞ is called a Nussbaum-type function if it has the following properties:17,18 ð 1 s lim sup N ðgÞdg ¼ þ1; (5) s!þ1 s 0 ð 1 s N ðgÞdg ¼ 1: (6) lim inf s!þ1 s 0 2 Here, a Nussbaum function NðgÞ ¼ eg cos p2 g is adopted. Then following lemma regarding to the property of Nussbaum gain is used in the controller design and theorem proof. Lemma 1. Let Vð:Þ and gð:Þ be smooth functions defined on ½0 1Þ with VðtÞ 0, and Nð:Þ be an even smooth Nussbaum-type function. If the following inequality holds: Ðt Ðt c s _ 1 ds; (7) _ c1 s ds þ ec1 t 0 ge VðtÞ c0 þ ec1 t 0 gðsÞNðgÞge where constant c1 > 0, gðtÞ is a nonzero constant and Ð t c0 denotes some suitable constant, then VðtÞ, gðtÞ and 0 gðsÞ _ must be bounded on ½0 1Þ. NðgÞgds
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033135-3
Shaohua Luo
Chaos 24, 033135 (2014)
III. ADAPTIVE FUZZY DYNAMIC SURFACE CONTROLLER DESIGN
where the design parameter k1 > 0. Step 2: Let a2 pass through a first-order filter with a time constant s2 to obtain a2f
A. Fuzzy logic system
The singleton fuzzifier, product inference, and the centerdefuzzifier are adopted to deduce the following fuzzy rules: Ri : IF x1 is Fi1 and…and xn is Fin THEN y is Bi ði ¼ 1; 2; …; NÞ; where x ¼ ½x1 ; x2 ; …; xn T 2 Rn , and y 2 R are the input and output of the fuzzy system, respectively, Fij and Bi are fuzzy sets in R. The fuzzy inference engine performs a mapping from fuzzy sets in Rn to a fuzzy set in R based on the IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point x ¼ ½x1 ; x2 ; …; xn T 2 Rn into a fuzzy set Ax in R. The defuzzifier maps a fuzzy set in R to a crisp point in R. Since the strategy of singleton fuzzification, center-average defuzzification and product inference is used, the output of the fuzzy system can be written as N P
hj
j¼1
yð xÞ ¼
n Q i¼1
N Q n P j¼1 i¼1
lFj ðxi Þ i
;
(8)
a2f ð0Þ ¼ a2 ð0Þ:
(13)
Define the second dynamic surface is chosen as S2 ¼ x2 a2f .Then, the time derivative of S2 is computed by S_ 2 ¼ rx3 þ f1 þ d2 a_ 2f ;
(14)
where f1 ¼ rx2 . Notice that the uncertain parameter r appears in the expression of Eq. (14). This will make the traditional control methods become troubled. To avoid this trouble, the Nussbaum gain is adopted to cope with the control gain r, and fuzzy logic system is employed to approximate the function f1 with little error. According to the description above, for any given e1 > 0, there is a fuzzy hT1 n1 such that f1 ¼ hT1 n1 þ d1 ;
lFj ðxi Þ
(15)
i
where hj is the point at which fuzzy membership function lBj ðhj Þ achieves its maximum value, and it’s further assumed that lBj ðhj Þ ¼ 1. n Q l j ðxi Þ F
s2 a_ 2f þ a2f ¼ a2 ;
where d1 is the approximation error and satisfies jd1 j < e1 . Substituting Eq. (15) into Eq. (14), one has T T S_ 2 rx3 þ ^h 1 n1 ~h 1 n1 þ q1 a_ 2f :
(16)
T
Remark 1: The estimation error is expressed as T T h ¼ h^ h~ .
and h¼½h1 ;h2 ;…;hn T , then the fuzzy logic system above can be formulated as
Choose the virtual function a3 and the updated laws as T q2 (17) a3 ¼ N ðgÞ K2 S2 þ ^h 1 n1 þ S2 1 a_ 2f ; 2c1 T q2 g_ ¼ K2 S2 þ ^h 1 n1 þ S2 1 a_ 2f S2 ; (18) 2c1
i¼1 Let pj ð xÞ¼ P N hQ
i
n
j¼1
i, nðxÞ¼½n1 ðxÞ;n2 ðxÞ;…;nn ðxÞ
l j ðxi Þ i¼1 F
1
i
yðxÞ ¼ hT nðxÞ:
(9)
If all memberships are taken as Gaussian functions, then the following lemma holds. Lemma 2. Let f ðxÞ be a continuous function defined on a compact set X. Then, there exists a fuzzy logic system in the form (9) such that sup jf ðxÞ yðxÞj d; (10) x2X
where dðxÞ is the approximation error, satisfying jdðxÞj < e. Assumption 4. There exists a smooth and bounded function qi ðÞ which satisfies 0 < jei þ Diþ1 j qi ; i ¼ 1; 2; 3. B. Controller design
The overall design procedure of an adaptive fuzzy dynamic surface controller based on the Nussbaum gain includes four steps as mentioned in Eq. (4). Next, it will be given the procedure of the design. Step 1: For the reference signal yr , the first dynamic surface is chosen as S1 ¼ x1 yr . Obviously, the following equation exists: S_ 1 ¼ x2 þ d1 y_ r :
(11)
Then, the virtual control law a2 is selected as follows: a2 ¼ k1 S2 þ y_ r ;
(12)
1
_ h^ 1 ¼ c1 ðn1 S2 m1 ^h 1 Þ;
(19)
where k2 , c1 , c1 , and m1 are the positive constant, and g is the variable of Nussbaum function. Step 3: Filter a3 through the following first-order filter s3 a_ 3f þ a3f ¼ a3 ;
a3f ð0Þ ¼ a3 ð0Þ;
(20)
where s3 is a time constant. The third dynamic surface is defined as S3 ¼ x3 a3f :
(21)
Taking into consideration (4), the derivative of S3 is given by S_ 3 ¼ f2 þ uq þ d2 a_ 3f ;
(22)
where f2 ¼ x2 x4 x3 þ cx2 . It clearly can be seen that f2 is a nonlinear function with uncertain parameter c. To facilitate the controller design, the fuzzy logic system will be employed to approximate the function f2 . According to the description above, there exists a fuzzy logic system such that
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033135-4
Shaohua Luo
Chaos 24, 033135 (2014)
f2 ¼ hT2 n2 þ d2 ;
(23)
_
ud ¼ k4 S4 h T3 n3 S4
where d2 is the approximation error and satisfies jd2 j < e2 . Substituting Eq. (23) into Eq. (22) gives S_ 3
^h T n2 2
~h T n2 2
þ q2 þ uq a_ 3f :
(24)
Remark 2. The estimation error is expressed as h~ 2 ¼ ^h 2 h2 . Then, the q-axis control law and adaptive law are chosen as follows: _
uq ¼ K3 S3 h T2 n2 S3
q22 þ a_ 3f ; 2c2
_ _ h2 ¼ C2 ðn2 S3 r2 h2 Þ;
(25)
_
(26)
where k3 , c2 , C2 , and r2 are the positive constant. Step 4: At this step, the d-axis control law will be constructed. Choose the fourth dynamic surface as S4 ¼ x4 :
__ _ h3 ¼ C3 ðn3 S4 r3 h3 Þ;
In this part, it will be given the stability analysis for the closed-loop system. First, define the filter error yi ¼ aif ai . Then, the following differential equations are given as a_ if ¼
where f3 ¼ x4 þ x2 x3 . By similar manipulations previously done in steps 2 and 3, the fuzzy logic system hT3 n3 is utilized to approximate the nonlinear item f3 again. For any given d3 > 0, one has f3 ¼ hT3 n3 þ d3 ;
(29)
where d3 is the approximation error and satisfies jd3 j < e3 . Thus, it follows immediately from substituting Eq. (29) into Eq. (28) that T T S_ 4 ^h 3 n3 ~h 3 n3 þ q3 þ ud :
(30)
(32)
IV. STABILITY ANALYSIS
(27)
(28)
(31)
where k4 , c3 , C3 and r3 are the positive constant. Untill now, the whole design process of the controller of the PMSM is already finished, and the configuration of the proposed control system is depicted in Figure 2. Note that the overall system consists of the PMSM with load, space vector pulse width modulation (SVPWM), voltage-source inverter, field-orientation mechanism and controller. The rotor angle and angular velocity can be gotten from the position and speed sensor. The currents id and iq can be calculated from ia and ib by Clarke and Park transformations.
Then, the derivative of S4 is given by S_ 4 ¼ f3 þ d4 þ ud ;
q23 ; 2c3
yi ; si
i ¼ 2; 3:
(33)
With Eqs. (11)–(13) and (33), one has S_ 1 S2 þ y2 k1 S1 þ D1 :
(34)
By similar manipulations previously done in Eq. (34), it can be deduced as follows: S_ 2 rðS3 þ y3 Þ þ rN ðgÞ þ 1 _ q21 T _ K2 S2 þ h 1 n1 þ S2 a 2f 2c1 q2 T h~ 1 n1 þ q1 S2 1 K2 S2 ; 2c1
(35)
q2 T S_ 3 ~h 2 n2 þ q2 S3 2 K3 S3 ; 2c2
(36)
At the present stage, the control law ud and the related parameter adaptive law are designed as
FIG. 2. Block diagram of the PMSM.
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033135-5
Shaohua Luo
Chaos 24, 033135 (2014)
q2 T S_ 4 ~h 3 n3 þ q3 S4 3 K4 S4 : 2c3
(37)
Differentiating y2 gives y_ 2 ¼ a_ 2f a_ 2 ¼
y2 k1 S_ 1 þ y€r : s2
(38)
Then, one has y_ 2 þ y2 B2 ðS1 ; S2 ; y2 ; yr ; y_ r ; y€r Þ: s2
(39)
Using Young’s inequality and Eq. (39), there exists y2 y_ 2
y22 1 þ y22 þ B22 ; s2 4
(40)
where B2 ðS1 ; S2 ; y2 ; yr ; y_ r ; y€r Þ is the continuous function. Similarly, one is transformed as _ y_ þ y3 B3 ðS1 ; S2 ; S3 ; h 1 ; y2 ; y3 ; yr ; y_ ; y€ Þ r r 3 s 3 2 3 0 1 _ T 6 @N ðgÞ B K2 S2 þ h 1 n1 7 C 6 7 C 2 6 @g g_ B 7 q @ A 1 6 7 _ þ S a 2 2f 6 7 2c1 6 7 0 1 6 7: ¼ 6 2 7 _ _ @n _ q 1 1 6 T T _ _ x_ 2 þ S 2 B K2 S 2 þ h 1 n 1 þ h 1 C7 6 @x2 2c1 C 7 6 þN g B C7 ð ÞB 6 @ A7 q1 @q1 4 5 þS2 x_ 2 €a 2f c1 @x2 (41) Using Young’s inequality and Eq. (41), there exists y3 y_ 3
y23 s3
1 þ y23 þ B23 ; 4
For any given p > 0, the closed sets can be defined as follows: 8
> P1 ¼ 8ðS1 Þ : S21 2p > 9 > X > 2 > > > _ > > > > 2 > > = < S1 ; S2 ; h1 ; y2 : Si > > > > > i¼1 P ¼ > 2 > > > > 1 ~T ~ > > > 2 > > > þ y þ 2p h h ; : > 1 2 1 > > C 1 > 9 8 > X > 3 > > _ _ > > > > 2 > > < Si > > > S1 ; …; S3 ; h1 ; h 2 ; y2 ; y3 : = < i¼1 : (47) P ¼ > 3 2 X X > 3 > > > 1 T > > 2 > > ~h ~h i 2p > > > > > i ; : þ yi þ > C > i i¼2 i¼1 > 9 8 > > 4 > X > > _ _ > > > > 2 > > > ; …; S ; h ; …; h ; y ; y S : S > > > 1 4 1 3 2 3 i = < > > > i¼1 > P4 ¼ > 3 3 > X X > > > 1 ~T ~ > > > 2 > > > þ y þ h i h i 2p > > > i ; : : C i i¼2 i¼1 Theorem 1: Suppose that the control law in Eqs. (25) and (31) with adaptive law in Eqs. (18), (19), (26) and (32) is applied to the PMSM (4) stands, and if there exists a positive constant p that the initial condition satisfies V4 2p and the design constants ki ; ci ; Ci and ri are rationally chosen, then the closed-loop control system is semi-global uniformly ultimately bounded, and the output tracking error converges to a neighborhood of zero. Proof: Choose the Lyapunov function candidate as ! 4 3 2 X X 1 X 1 ~T ~ 2 2 S þ yi þ h hi : V¼ (48) 2 i¼1 i C i i¼2 i¼1 i Its time derivative along (27) is
(42)
_
where B3 ðS1 ; S2 ; S3 ; h 1 ; y2 ; y3 ; yr ; y_ r ; y€r Þ is the continuous function. Using Young’s inequality, a calculation is performed to generate the following inequality: 1 1 S1 S_ 1 S22 þ y22 þ ð3 k1 ÞS21 þ D21 : 4 4
(43)
Repeating the previous procedure, it can be deduced as follows:
V_ ¼
4 3 2 X X X 1 ~ T __ h i hi ð3 k1 ÞS21 Si S_ i þ yi y_ i þ C i¼1 i¼2 i¼1 i 1 2 1 2 G k3 S23 þ 2 k2 þ S 2 þ 4 4 3 X y2i 2 k4 S 4 þ þ jyi Bi j si i¼2 2 X 1 1 ci þ D21 þ y22 þ G2 y23 þ GN ðgÞ þ 1 g_ þ 2 4 4 i¼1 T_
1 1 S2 S_ 2 GS23 þ Gy23 þ ð2 k2 ÞS22 4 4 c1 T þ GN ðgÞ þ 1 1S2 ~h 1 n1 S2 þ ; 2 c T 2 S3 S_ 3 ~h 2 n2 S3 k3 S23 þ ; 2 c3 T S4 S_ 4 ~h 3 n3 S4 k4 S24 þ : 2 Remark
3.
The
q2 jSiþ1 jqi S2i 2ci i c2i ; i ¼ a_ 2f and ab a2 þ 14 b2 .
inequalities
are
1; 2; 3, 1 ¼ K2 S2 þ
(44) (45) (46)
defined _
h T1 n1
þ
q2 S2 2c11
as
T_
r1 ~h 1 h 1 r2 ~h 2 h 2 ð3 k1 ÞS21 1 2 1 2 G k3 S23 þ 2 k2 þ S 2 þ 4 4 3 X y2 k4 S24 þ i þ jyi Bi j si i¼2 1 1 þ y22 þ G2 y23 þ GN ðgÞ þ 1 g_ 4 4 2 X ci 1 þ D21 r1 k~h 1 k2 þ r2 k~h 2 k2 þ 2 2 i¼1 1 þ r1 kh1 k2 þ r2 kh2 k2 : 2
(49)
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033135-6
Shaohua Luo
Chaos 24, 033135 (2014)
P P yB T T Remark 4. r1 ~h 1 h 1 12 r1 k~h 1 k2 þ 12 r1 kh1 k2 , r2 ~h 2 h 2 12 r2 k~h 2 k2 þ 12 r2 kh2 k2 and 3i¼2 jyi Bi j 3i¼2 i2ai þ a. _
Define s12 ¼ 14 þ
M22 2a
_
2
and s13 ¼ G4 þ
M32 2a ,
2 2
then the following inequality can be obtained:
2 X 1 2 1 2 ci 1 ~ 2 2 _ r1 kh 1 k þ r2 k~h 2 k2 G k3 S23 k4 S24 þ GN ðgÞþ1 g_ þ V ð3 k1 ÞS1 þ 2 k2 þ S2 þ 2 2 4 4 i¼1 ! ! B2 y2 M2 B 2 y2 M 2 1 þ 1 þ 22 2 2 þ 1 þ 32 3 3 þ a þ r1 kh1 k2 þ r2 kh2 k2 2a 2a 2 M2 M3 1 ¼ ð3 k1 ÞS21 þ ð2 k2 ÞS22 k3 S23 k4 S24 þ GN ðgÞ þ 1 g_ r1 k~h 1 k2 þ r2 k~h 2 k2 2 ! ! B2 y2 M2 B 2 y2 M 2 1 1 þ 1 þ 22 2 2 þ 1 þ 32 3 3 þ a þ S22 þ G2 S22 þ d; 2a 2a 4 4 M2 M3 where d ¼
P2
ci i¼1 2
(50)
þ D21 þ a þ 12 r1 kh1 k2 þ r2 kh2 k2 : Then, one has 1 1 V_ 2a0 V þ GN ðgÞ þ 1 g_ þ S22 þ G2 S23 þ d: 4 4
(51)
Define b ¼ d=2a0 and substitute it into the last-written equation. Meanwhile multiplying e2a0 t at the both ends of Eq. (51) gives 2a0 t 1 2 2a0 t 1 2 2 2a0 t d ðVe2a0 t Þ _ þ S2 e þ G S3 e : de2a0 t þ GN ðgÞ þ 1 ge 4 4 dt
(52)
Then, the integral can be done on [0 t]. Ðt _ 2a0 s ds þ I; 0 VðtÞ b þ Vð0Þ þ e2a0 t 0 ðGNðgÞ þ 1Þge
(53)
Ð t t 1
where I ¼ e2a0 0 4 S22 e2a0 s þ 14 G2 S23 e2a0 s Þds. In addition, there exists ð 1 2 1 2 2 t 2a0 s G2 sup S22 ðsÞ þ S23 ðsÞ : S2 þ G S3 e ds I e2a0 t sup 8 s2½o;t 4 s2½o;t 4 0
(54)
i By Lyapunov function, it is proved that all the signals Sj ; yj and ~h j in the closed-loop system are uniformly ultimately bounded. Consequently, xi ; ai and aif are also uniformly ultimately bounded.
V. NUMERICAL SIMULATION AND ANALYSIS
In this part, in order to illustrate the effectiveness of the proposed controller, the simulation will be conducted to suppress chaos in the PMSM under the initial condition of x1 ð0Þ ¼ 0:5, x2 ð0Þ ¼ x3 ð0Þ ¼ x4 ð0Þ ¼ 0. The control parameters of the proposed controller are chosen as follows: k1 ¼ 5, k2 ¼ 18, k3 ¼ 100, k4 ¼ 60, u1 ¼ 0.3, u2 ¼ 0.3, u3 ¼ 0.3, r1 ¼ 0.8, r2 ¼ 0.8, r3 ¼ 0.8, s1 ¼ 0.02, s2 ¼ 0.02, c1 ¼ 4, c2 ¼ 4, c3 ¼ 4, g (0) ¼ 1. And the fuzzy membership function are:
lF1i lF4i lF7i
lF10 i
2 ð x þ 5Þ ; ¼ exp 2 2 ð x þ 2Þ ; ¼ exp 2 " # 2 ð Þ x1 ; ¼ exp 2 " # 2 ð x 4 Þ ; ¼ exp 2
2 2 ðx þ 4Þ ð x þ 3Þ lF2i ¼ exp ; lF3i ¼ exp ; 2 2 2 ðx þ 1Þ x2 lF5i ¼ exp ; lF6i ¼ exp ; 2 2 " # " # 2 2 ð Þ ð Þ x2 x3 lF8i ¼ exp ; lF9i ¼ exp ; 2 2 " # 2 ð x 5 Þ lF11 : ¼ exp i 2
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033135-7
Shaohua Luo
Chaos 24, 033135 (2014)
FIG. 3. The trajectory tracking of sin (t) þ 0.5sin (2t). (a) The rotor angle tracking. (b) The rotor angle tracking error. (c) The rotor velocity tracking. (d) The rotor velocity tracking error.
FIG. 4. The adaptive analysis of sin (2t). (a) The rotor angle tracking error. (b) The rotor velocity tracking error. (c) q axis current. (d) q axis voltage. (e) d axis voltage. (f) Nussbaum gain.
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033135-8
Shaohua Luo
Chaos 24, 033135 (2014)
FIG. 5. Robustness analysis for the external disturbance. (a) The rotor angle tracking error. (b) Adaptation of parameter g. (c). q axis voltage. (d) d axis voltage.
A. Analysis of trajectory tracking
Figures 3(a) and 3(c) show the result of trajectory tracking of rotor angle and angular velocity of proposed controller, respectively. Figures 3(b) and 3(d) show the steady state error of rotor angle and angular velocity of proposed controller, which equal to 60.018Rad and 60.03 Rad/s with short response time, respectively. From these figures, it can be see clearly that the proposed system successfully escapes from its chaotic behavior within 1s and tracks the reference signals with good performance. B. Adaptive analysis for parameter variations
The curves of the rotor angle tracking error, velocity tracking error, q axis current, q, d-axis voltages and Nussbaum gain are given in Figures 4(a)–4(f). According to these figures, it is obvious that the proposed system can estimate unknown items with short response time. In other words, it has good robustness to the parameter variations.
permanent magnet synchronous motor system to control chaotic behavior. Fuzzy logic systems are employed to approximate the unknown nonlinear functions, and an adaptive Nussbaum gain is used to cope with the uncertain control gain of system. The proposed controller overcomes the shortage of backstepping and guarantees that the tracking error converges to within a small neighborhood of the origin and all the closed-loop signals are bounded. Simulation results are provided to illustrate the effectiveness and robustness. ACKNOWLEDGMENTS
This work was supported in part by the Major State Basic Research Development Program 973 (No. 2012CB215\\202), the National Natural Science Foundation of China (No. 61134001) and Key Laboratory of Dependable Service Computing in Cyber Physical Society (Chongqing University), Ministry of Education. 1
C. Robustness analysis for disturbance
The expression is given as follows: di ¼ 0:02 x22 sin5 ð2tÞ; i ¼ 1 4:
(55)
In order to check the performance of the controller against the disturbance, simulation results are provided to verify it in the PMSM. In Figures 5(a)–5(d), by contrasting one case without external disturbance (ED) and the other’s with ED, two kinds of curves highly coincide in whole process. It indicates that the proposed system owns good robustness. VI. CONCLUSION
Based on the Nussbaum gain, an adaptive fuzzy dynamic surface control approach is developed for the
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N. E. Mitrakis, J. B. Theocharis, and V. Petridis, “A multilayered neurofuzzy classifier with self-organizing properties,” Fuzzy Sets Syst. 159, 3132–3159 (2008). 10 C. Zhang, H. Liu, S. Chen, and F. Wang, “Application of neural networks for permanent magnet synchronous motor direct torque control,” J. Syst. Eng. Electron. 19, 555–561 (2008). 11 C. Elmas, O. Ustun, and H. Sayan, “A neuro-fuzzy controller for speed control of a permanent magnet synchronous motor drive,” Expert Syst. Appl. 34, 657–664 (2008). 12 J. Yu, B. Chen, and H. Yu, “Position tracking control of induction motors via adaptive fuzzy backstepping,” Energy Convers. Manage. 51, 2345–2352 (2010). 13 P. P. Yip and J. K. Hedrick, “Adaptive dynamic surface control: a simplified algorithm for adaptive backstepping control of nonlinear systems,” Int. J. Control 71, 959–979 (1998). 14 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 45, 1893–1899 (2000). 15 D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form,” IEEE Trans. Neural Netw. 16, 195–202 (2005).
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T. Zhang and S. Ge, “Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form,” Automatica 44, 1895–1903 (2008). 17 J. Lei, X. Wang, and Y. Lei, “A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters,” Commun. Nonlinear Sci. Numer. Simulat. 14, 3439–3448 (2009). 18 A. Zhou, G. Ren, C. Shao, and Y. Liang, “Synchronization of chaotic systems by using limited nussbaum gain method,” Res. J. Appl. Sci., Eng. Technol. 4(20), 4007–4011 (2012), see http://www.maxwellsci.com/print/ rjaset/v4-4007-4011.pdf. 19 Z. Li, J. B. Park, Y. H. Joo, B. Zhang, and G. Chen, “Bifurcations and chaos in a permanent-magnet synchronous motor,” IEEE Trans. Circuits Syst. I 49, 383–387 (2002). 20 H. Ren and D. Liu, “Nonlinear feedback control of chaos in permanent magnet synchronous motor,” IEEE Trans. Circuits Syst. II 53, 45–50 (2006). 21 N. Vaseg and F. Khella, “Takagi-Sugeno fuzzy modeling and chaos control of partial differential systems,” Chaos 23, 042101 (2013). 22 T. Chou, T. Liu, and T. Cheng, “Observer-based speed tracking control for sensorless permanent magnet synchronous motors with unknown load torque,” IET Electr. Power Appl. 6, 62–72 (2012).
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CHAOS 24, 033135 (2014)
Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain Shaohua Luo School of Automation, Chongqing University, Chongqing 400044, China and College of Mechanical Engineering, Hunan University of Arts and Science, Hunan 415000, China
(Received 8 April 2014; accepted 5 September 2014; published online 17 September 2014) This paper is concerned with the problem of adaptive fuzzy dynamic surface control (DSC) for the permanent magnet synchronous motor (PMSM) system with chaotic behavior, disturbance and unknown control gain and parameters. Nussbaum gain is adopted to cope with the situation that the control gain is unknown. And the unknown items can be estimated by fuzzy logic system. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. Finally, the numerical simulations indicate that the proposed scheme can suppress the chaos of C 2014 PMSM and show the effectiveness and robustness of the proposed method. V AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895810]
In this study, an adaptive fuzzy dynamic surface control approach is developed based on the Nussbaum gain for chaotic PMSM. Nussbaum gain is adopted to cope with unknown control gain and fuzzy logic system is employed to estimate the unknown items. The closed-loop system achieves asymptotic stability by dynamic surface controller. The simulation results are indicated the effectiveness and robustness of the proposed scheme.
I. INTRODUCTION
Recently, permanent magnet synchronous motor is of great interest, particularly for the engineering applications in the low-medium power range, since it has many advantages, such as compact size, high torque/weight ratio, high torque/ inertia ratio, and absence of rotor losses.1 Unfortunately, the PMSM drive system appears chaos which degrades the performance of PMSM in the working condition.2 The chaotic behavior is undesirable since it can completely destroy the stabilization of the motor. Hence, many people have worked on some methods to identify, suppress, synchronize and even induce the occurrence of the chaos in various systems.3–5 The OGY method is a basic methodology for controlling chaos6,7 but finding a reasonable parameter is not often simple. A neuro-fuzzy controller (NFC) is suitable for control of systems with uncertainties and nonlinearities.8,9 Meanwhile, the NFC can also achieve self-learning; however, it is unsuitable for on-line learning real-time control due to the drawback of time consuming.10,11 In view of this, a unified fuzzy approach is presented for controlling chaos in nonlinear partial differential systems which is represented by TakagiSugeno fuzzy model.21 In Ref. 12, a position tracking control method based on adaptive fuzzy backstepping is presented for the induction motors with unknown parameters. However, the backstepping suffers from the “explosion of complexity” caused by the repeated differentiation of virtual 1054-1500/2014/24(3)/033135/9/$30.00
control functions, and the explosion becomes more significant as the order increases.13 To avoid explosion of the differentiation items on the virtual function, the dynamic surface control by introducing a first-order filter at each recursive step of the backstepping design procedure is developed by Swaroop et al.14 By incorporating DSC into a neural-network-based adaptive control design framework, Wang and Huang15 proposed a backstepping-based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. However, disturbances is not involved in the model and unknown control gain is assumed to be equal to one. Zhang and Ge16 further studied the control design for some special nonlinear systems with unknown dead-zone using the DSC method. In their works, it is assumed there exists positive constants which satisfy the constrained condition. But it is very difficult to define the boundedness of control gain in real practice. Incidentally, the PMSM is known to exhibit chaotic behavior under certain conditions. Whether the latter methods can suppress the chaos oscillation in PMSM needs for further research since DSC with radial basis function (RBF) has been pioneered by the work of Wang. Furthermore, the unknown control gain is assumed to be known in most papers, but the value of control gain is usually unknown in the reality. The Nussbaum gain theory is adopted to cope with this kind of problem.17,18 Then, the unknown parameters can be estimated effectively with the Nussbaum gain adaptive method. The attention of the electric drives community has been recently paid to the “sensorless” control problem of PMSM in which only stator current and voltage measurements are available for feedback.22 However, complicated computation process is required. To the best of our knowledge, the combination between adaptive DSC, Nussbaum gain and fuzzy has been seldom applied in the control of chaos for the PMSM system yet. In this paper, an adaptive fuzzy dynamic surface control method is developed based on the Nussbaum gain for PMSM
24, 033135-1
C 2014 AIP Publishing LLC V
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033135-2
Shaohua Luo
wherein the unknown parameters and control gain, disturbance and chaos are presented. The whole process of the system design is performed in a step-by-step manner. For each step, a first-order filter is employed to gain the information of derivative of the virtual control function. Then, the explosion of complexity for backstepping is well solved. At last, the simulation results are presented to demonstrate the effectiveness and robustness of the proposed scheme in the chaotic PMSM system. II. DYNAMICS OF THE PMSM
The model of the permanent magnet synchronous motor with smooth air-gap in the well-known (d-q) frame is described as follow:19,20 8 dh > > > ¼x > > dt > > > dx > < ¼ rðiq xÞ T~L dt (1) diq > > ~ ¼ i i x þ cx þ u > q d q > > dt > > did > > : ¼ id þ iq x þ u~d ; dt where r and c mean the system operating parameters, which are positive. T~L , u~q , and u~d stand for the d q load torque and axis voltages, respectively. h, x, id , and iq represent the state variables, which denote angle, angle speed and the d q axis currents, respectively. According to Eq. (1), the external inputs are set to zero, namely, T~L ¼ u~q ¼ u~d ¼ 0. Then, the PMSM system becomes an unforced system as follows: 8 dh > > > ¼x > > dt > > > dx > < ¼ rðiq xÞ dt (2) di > q > ¼ i i x þ cx > q d > > dt > > did > > : ¼ id þ iq x: dt It is clear that the dynamic model of PMSM is highly nonlinear due to the coupling between the speed and the currents. As we know, r and c of the system parameters have uncertainties, which are influenced by the real work conditions. In addition, the PMSM is experiencing chaotic behavior when the parameters lie in a certain range. The typical chaotic attractor in 3-D space is shown in Figure 1.
FIG. 1. The chaotic attractor with parameters r ¼ 5.45 and c ¼ 20.
Chaos 24, 033135 (2014)
In order to eliminate chaos, use ud and uq as the manipulated variables which are desirable for the real application. Then, the model of PMSM driver system can be rewritten by the following differential equations: 8 dh > > > ¼x > > dt > > > dx > < ¼ rðiq xÞ dt (3) diq > > ¼ i i x þ cx þ u > q d q > > dt > > did > > : ¼ id þ iq x þ ud : dt For simplicity, the following symbols are introduced: x1 ¼ h;
x2 ¼ x;
x3 ¼ i q ;
x4 ¼ id :
By using these symbols, the dynamic model of PMSM with bounded disturbances can be represented as follows 8 x_ 1 ¼ x2 þ d1 > > < x_ 2 ¼ rðx3 x2 Þ þ d2 (4) x_ ¼ x3 x2 x4 þ cx2 þ uq þ d3 > > : 3 x_ 4 ¼ x4 þ x2 x3 þ ud þ d4 ; where di , i ¼ 1; …, 4, mean the disturbances. In order to facilitate the design, the following assumptions are given. Assumption 1. The gain function r is unknown and bounded. Moreover, it is assumed that jrj G. Assumption 2. The unknown disturbance terms di satisfy jdi ðÞj < Di ; 1; …; 4, and Di is the unknown positive constant. Assumption 3. The desired trajectory yr is continuous, :: and ½yr ; y_ r ; yr T 2 Nr with the known compact set Nr ¼ :: T :: f½yr ; y_ r ; yr : y2r þ y_ 2r þ y2r Br g R3 , whose size Br is a known positive constant. Because the sign of control is unknown, the Nussbaum function is employed in this paper. Obviously, a function NðgÞ is called a Nussbaum-type function if it has the following properties:17,18 ð 1 s lim sup N ðgÞdg ¼ þ1; (5) s!þ1 s 0 ð 1 s N ðgÞdg ¼ 1: (6) lim inf s!þ1 s 0 2 Here, a Nussbaum function NðgÞ ¼ eg cos p2 g is adopted. Then following lemma regarding to the property of Nussbaum gain is used in the controller design and theorem proof. Lemma 1. Let Vð:Þ and gð:Þ be smooth functions defined on ½0 1Þ with VðtÞ 0, and Nð:Þ be an even smooth Nussbaum-type function. If the following inequality holds: Ðt Ðt c s _ 1 ds; (7) _ c1 s ds þ ec1 t 0 ge VðtÞ c0 þ ec1 t 0 gðsÞNðgÞge where constant c1 > 0, gðtÞ is a nonzero constant and Ð t c0 denotes some suitable constant, then VðtÞ, gðtÞ and 0 gðsÞ _ must be bounded on ½0 1Þ. NðgÞgds
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033135-3
Shaohua Luo
Chaos 24, 033135 (2014)
III. ADAPTIVE FUZZY DYNAMIC SURFACE CONTROLLER DESIGN
where the design parameter k1 > 0. Step 2: Let a2 pass through a first-order filter with a time constant s2 to obtain a2f
A. Fuzzy logic system
The singleton fuzzifier, product inference, and the centerdefuzzifier are adopted to deduce the following fuzzy rules: Ri : IF x1 is Fi1 and…and xn is Fin THEN y is Bi ði ¼ 1; 2; …; NÞ; where x ¼ ½x1 ; x2 ; …; xn T 2 Rn , and y 2 R are the input and output of the fuzzy system, respectively, Fij and Bi are fuzzy sets in R. The fuzzy inference engine performs a mapping from fuzzy sets in Rn to a fuzzy set in R based on the IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point x ¼ ½x1 ; x2 ; …; xn T 2 Rn into a fuzzy set Ax in R. The defuzzifier maps a fuzzy set in R to a crisp point in R. Since the strategy of singleton fuzzification, center-average defuzzification and product inference is used, the output of the fuzzy system can be written as N P
hj
j¼1
yð xÞ ¼
n Q i¼1
N Q n P j¼1 i¼1
lFj ðxi Þ i
;
(8)
a2f ð0Þ ¼ a2 ð0Þ:
(13)
Define the second dynamic surface is chosen as S2 ¼ x2 a2f .Then, the time derivative of S2 is computed by S_ 2 ¼ rx3 þ f1 þ d2 a_ 2f ;
(14)
where f1 ¼ rx2 . Notice that the uncertain parameter r appears in the expression of Eq. (14). This will make the traditional control methods become troubled. To avoid this trouble, the Nussbaum gain is adopted to cope with the control gain r, and fuzzy logic system is employed to approximate the function f1 with little error. According to the description above, for any given e1 > 0, there is a fuzzy hT1 n1 such that f1 ¼ hT1 n1 þ d1 ;
lFj ðxi Þ
(15)
i
where hj is the point at which fuzzy membership function lBj ðhj Þ achieves its maximum value, and it’s further assumed that lBj ðhj Þ ¼ 1. n Q l j ðxi Þ F
s2 a_ 2f þ a2f ¼ a2 ;
where d1 is the approximation error and satisfies jd1 j < e1 . Substituting Eq. (15) into Eq. (14), one has T T S_ 2 rx3 þ ^h 1 n1 ~h 1 n1 þ q1 a_ 2f :
(16)
T
Remark 1: The estimation error is expressed as T T h ¼ h^ h~ .
and h¼½h1 ;h2 ;…;hn T , then the fuzzy logic system above can be formulated as
Choose the virtual function a3 and the updated laws as T q2 (17) a3 ¼ N ðgÞ K2 S2 þ ^h 1 n1 þ S2 1 a_ 2f ; 2c1 T q2 g_ ¼ K2 S2 þ ^h 1 n1 þ S2 1 a_ 2f S2 ; (18) 2c1
i¼1 Let pj ð xÞ¼ P N hQ
i
n
j¼1
i, nðxÞ¼½n1 ðxÞ;n2 ðxÞ;…;nn ðxÞ
l j ðxi Þ i¼1 F
1
i
yðxÞ ¼ hT nðxÞ:
(9)
If all memberships are taken as Gaussian functions, then the following lemma holds. Lemma 2. Let f ðxÞ be a continuous function defined on a compact set X. Then, there exists a fuzzy logic system in the form (9) such that sup jf ðxÞ yðxÞj d; (10) x2X
where dðxÞ is the approximation error, satisfying jdðxÞj < e. Assumption 4. There exists a smooth and bounded function qi ðÞ which satisfies 0 < jei þ Diþ1 j qi ; i ¼ 1; 2; 3. B. Controller design
The overall design procedure of an adaptive fuzzy dynamic surface controller based on the Nussbaum gain includes four steps as mentioned in Eq. (4). Next, it will be given the procedure of the design. Step 1: For the reference signal yr , the first dynamic surface is chosen as S1 ¼ x1 yr . Obviously, the following equation exists: S_ 1 ¼ x2 þ d1 y_ r :
(11)
Then, the virtual control law a2 is selected as follows: a2 ¼ k1 S2 þ y_ r ;
(12)
1
_ h^ 1 ¼ c1 ðn1 S2 m1 ^h 1 Þ;
(19)
where k2 , c1 , c1 , and m1 are the positive constant, and g is the variable of Nussbaum function. Step 3: Filter a3 through the following first-order filter s3 a_ 3f þ a3f ¼ a3 ;
a3f ð0Þ ¼ a3 ð0Þ;
(20)
where s3 is a time constant. The third dynamic surface is defined as S3 ¼ x3 a3f :
(21)
Taking into consideration (4), the derivative of S3 is given by S_ 3 ¼ f2 þ uq þ d2 a_ 3f ;
(22)
where f2 ¼ x2 x4 x3 þ cx2 . It clearly can be seen that f2 is a nonlinear function with uncertain parameter c. To facilitate the controller design, the fuzzy logic system will be employed to approximate the function f2 . According to the description above, there exists a fuzzy logic system such that
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033135-4
Shaohua Luo
Chaos 24, 033135 (2014)
f2 ¼ hT2 n2 þ d2 ;
(23)
_
ud ¼ k4 S4 h T3 n3 S4
where d2 is the approximation error and satisfies jd2 j < e2 . Substituting Eq. (23) into Eq. (22) gives S_ 3
^h T n2 2
~h T n2 2
þ q2 þ uq a_ 3f :
(24)
Remark 2. The estimation error is expressed as h~ 2 ¼ ^h 2 h2 . Then, the q-axis control law and adaptive law are chosen as follows: _
uq ¼ K3 S3 h T2 n2 S3
q22 þ a_ 3f ; 2c2
_ _ h2 ¼ C2 ðn2 S3 r2 h2 Þ;
(25)
_
(26)
where k3 , c2 , C2 , and r2 are the positive constant. Step 4: At this step, the d-axis control law will be constructed. Choose the fourth dynamic surface as S4 ¼ x4 :
__ _ h3 ¼ C3 ðn3 S4 r3 h3 Þ;
In this part, it will be given the stability analysis for the closed-loop system. First, define the filter error yi ¼ aif ai . Then, the following differential equations are given as a_ if ¼
where f3 ¼ x4 þ x2 x3 . By similar manipulations previously done in steps 2 and 3, the fuzzy logic system hT3 n3 is utilized to approximate the nonlinear item f3 again. For any given d3 > 0, one has f3 ¼ hT3 n3 þ d3 ;
(29)
where d3 is the approximation error and satisfies jd3 j < e3 . Thus, it follows immediately from substituting Eq. (29) into Eq. (28) that T T S_ 4 ^h 3 n3 ~h 3 n3 þ q3 þ ud :
(30)
(32)
IV. STABILITY ANALYSIS
(27)
(28)
(31)
where k4 , c3 , C3 and r3 are the positive constant. Untill now, the whole design process of the controller of the PMSM is already finished, and the configuration of the proposed control system is depicted in Figure 2. Note that the overall system consists of the PMSM with load, space vector pulse width modulation (SVPWM), voltage-source inverter, field-orientation mechanism and controller. The rotor angle and angular velocity can be gotten from the position and speed sensor. The currents id and iq can be calculated from ia and ib by Clarke and Park transformations.
Then, the derivative of S4 is given by S_ 4 ¼ f3 þ d4 þ ud ;
q23 ; 2c3
yi ; si
i ¼ 2; 3:
(33)
With Eqs. (11)–(13) and (33), one has S_ 1 S2 þ y2 k1 S1 þ D1 :
(34)
By similar manipulations previously done in Eq. (34), it can be deduced as follows: S_ 2 rðS3 þ y3 Þ þ rN ðgÞ þ 1 _ q21 T _ K2 S2 þ h 1 n1 þ S2 a 2f 2c1 q2 T h~ 1 n1 þ q1 S2 1 K2 S2 ; 2c1
(35)
q2 T S_ 3 ~h 2 n2 þ q2 S3 2 K3 S3 ; 2c2
(36)
At the present stage, the control law ud and the related parameter adaptive law are designed as
FIG. 2. Block diagram of the PMSM.
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033135-5
Shaohua Luo
Chaos 24, 033135 (2014)
q2 T S_ 4 ~h 3 n3 þ q3 S4 3 K4 S4 : 2c3
(37)
Differentiating y2 gives y_ 2 ¼ a_ 2f a_ 2 ¼
y2 k1 S_ 1 þ y€r : s2
(38)
Then, one has y_ 2 þ y2 B2 ðS1 ; S2 ; y2 ; yr ; y_ r ; y€r Þ: s2
(39)
Using Young’s inequality and Eq. (39), there exists y2 y_ 2
y22 1 þ y22 þ B22 ; s2 4
(40)
where B2 ðS1 ; S2 ; y2 ; yr ; y_ r ; y€r Þ is the continuous function. Similarly, one is transformed as _ y_ þ y3 B3 ðS1 ; S2 ; S3 ; h 1 ; y2 ; y3 ; yr ; y_ ; y€ Þ r r 3 s 3 2 3 0 1 _ T 6 @N ðgÞ B K2 S2 þ h 1 n1 7 C 6 7 C 2 6 @g g_ B 7 q @ A 1 6 7 _ þ S a 2 2f 6 7 2c1 6 7 0 1 6 7: ¼ 6 2 7 _ _ @n _ q 1 1 6 T T _ _ x_ 2 þ S 2 B K2 S 2 þ h 1 n 1 þ h 1 C7 6 @x2 2c1 C 7 6 þN g B C7 ð ÞB 6 @ A7 q1 @q1 4 5 þS2 x_ 2 €a 2f c1 @x2 (41) Using Young’s inequality and Eq. (41), there exists y3 y_ 3
y23 s3
1 þ y23 þ B23 ; 4
For any given p > 0, the closed sets can be defined as follows: 8
> P1 ¼ 8ðS1 Þ : S21 2p > 9 > X > 2 > > > _ > > > > 2 > > = < S1 ; S2 ; h1 ; y2 : Si > > > > > i¼1 P ¼ > 2 > > > > 1 ~T ~ > > > 2 > > > þ y þ 2p h h ; : > 1 2 1 > > C 1 > 9 8 > X > 3 > > _ _ > > > > 2 > > < Si > > > S1 ; …; S3 ; h1 ; h 2 ; y2 ; y3 : = < i¼1 : (47) P ¼ > 3 2 X X > 3 > > > 1 T > > 2 > > ~h ~h i 2p > > > > > i ; : þ yi þ > C > i i¼2 i¼1 > 9 8 > > 4 > X > > _ _ > > > > 2 > > > ; …; S ; h ; …; h ; y ; y S : S > > > 1 4 1 3 2 3 i = < > > > i¼1 > P4 ¼ > 3 3 > X X > > > 1 ~T ~ > > > 2 > > > þ y þ h i h i 2p > > > i ; : : C i i¼2 i¼1 Theorem 1: Suppose that the control law in Eqs. (25) and (31) with adaptive law in Eqs. (18), (19), (26) and (32) is applied to the PMSM (4) stands, and if there exists a positive constant p that the initial condition satisfies V4 2p and the design constants ki ; ci ; Ci and ri are rationally chosen, then the closed-loop control system is semi-global uniformly ultimately bounded, and the output tracking error converges to a neighborhood of zero. Proof: Choose the Lyapunov function candidate as ! 4 3 2 X X 1 X 1 ~T ~ 2 2 S þ yi þ h hi : V¼ (48) 2 i¼1 i C i i¼2 i¼1 i Its time derivative along (27) is
(42)
_
where B3 ðS1 ; S2 ; S3 ; h 1 ; y2 ; y3 ; yr ; y_ r ; y€r Þ is the continuous function. Using Young’s inequality, a calculation is performed to generate the following inequality: 1 1 S1 S_ 1 S22 þ y22 þ ð3 k1 ÞS21 þ D21 : 4 4
(43)
Repeating the previous procedure, it can be deduced as follows:
V_ ¼
4 3 2 X X X 1 ~ T __ h i hi ð3 k1 ÞS21 Si S_ i þ yi y_ i þ C i¼1 i¼2 i¼1 i 1 2 1 2 G k3 S23 þ 2 k2 þ S 2 þ 4 4 3 X y2i 2 k4 S 4 þ þ jyi Bi j si i¼2 2 X 1 1 ci þ D21 þ y22 þ G2 y23 þ GN ðgÞ þ 1 g_ þ 2 4 4 i¼1 T_
1 1 S2 S_ 2 GS23 þ Gy23 þ ð2 k2 ÞS22 4 4 c1 T þ GN ðgÞ þ 1 1S2 ~h 1 n1 S2 þ ; 2 c T 2 S3 S_ 3 ~h 2 n2 S3 k3 S23 þ ; 2 c3 T S4 S_ 4 ~h 3 n3 S4 k4 S24 þ : 2 Remark
3.
The
q2 jSiþ1 jqi S2i 2ci i c2i ; i ¼ a_ 2f and ab a2 þ 14 b2 .
inequalities
are
1; 2; 3, 1 ¼ K2 S2 þ
(44) (45) (46)
defined _
h T1 n1
þ
q2 S2 2c11
as
T_
r1 ~h 1 h 1 r2 ~h 2 h 2 ð3 k1 ÞS21 1 2 1 2 G k3 S23 þ 2 k2 þ S 2 þ 4 4 3 X y2 k4 S24 þ i þ jyi Bi j si i¼2 1 1 þ y22 þ G2 y23 þ GN ðgÞ þ 1 g_ 4 4 2 X ci 1 þ D21 r1 k~h 1 k2 þ r2 k~h 2 k2 þ 2 2 i¼1 1 þ r1 kh1 k2 þ r2 kh2 k2 : 2
(49)
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Shaohua Luo
Chaos 24, 033135 (2014)
P P yB T T Remark 4. r1 ~h 1 h 1 12 r1 k~h 1 k2 þ 12 r1 kh1 k2 , r2 ~h 2 h 2 12 r2 k~h 2 k2 þ 12 r2 kh2 k2 and 3i¼2 jyi Bi j 3i¼2 i2ai þ a. _
Define s12 ¼ 14 þ
M22 2a
_
2
and s13 ¼ G4 þ
M32 2a ,
2 2
then the following inequality can be obtained:
2 X 1 2 1 2 ci 1 ~ 2 2 _ r1 kh 1 k þ r2 k~h 2 k2 G k3 S23 k4 S24 þ GN ðgÞþ1 g_ þ V ð3 k1 ÞS1 þ 2 k2 þ S2 þ 2 2 4 4 i¼1 ! ! B2 y2 M2 B 2 y2 M 2 1 þ 1 þ 22 2 2 þ 1 þ 32 3 3 þ a þ r1 kh1 k2 þ r2 kh2 k2 2a 2a 2 M2 M3 1 ¼ ð3 k1 ÞS21 þ ð2 k2 ÞS22 k3 S23 k4 S24 þ GN ðgÞ þ 1 g_ r1 k~h 1 k2 þ r2 k~h 2 k2 2 ! ! B2 y2 M2 B 2 y2 M 2 1 1 þ 1 þ 22 2 2 þ 1 þ 32 3 3 þ a þ S22 þ G2 S22 þ d; 2a 2a 4 4 M2 M3 where d ¼
P2
ci i¼1 2
(50)
þ D21 þ a þ 12 r1 kh1 k2 þ r2 kh2 k2 : Then, one has 1 1 V_ 2a0 V þ GN ðgÞ þ 1 g_ þ S22 þ G2 S23 þ d: 4 4
(51)
Define b ¼ d=2a0 and substitute it into the last-written equation. Meanwhile multiplying e2a0 t at the both ends of Eq. (51) gives 2a0 t 1 2 2a0 t 1 2 2 2a0 t d ðVe2a0 t Þ _ þ S2 e þ G S3 e : de2a0 t þ GN ðgÞ þ 1 ge 4 4 dt
(52)
Then, the integral can be done on [0 t]. Ðt _ 2a0 s ds þ I; 0 VðtÞ b þ Vð0Þ þ e2a0 t 0 ðGNðgÞ þ 1Þge
(53)
Ð t t 1
where I ¼ e2a0 0 4 S22 e2a0 s þ 14 G2 S23 e2a0 s Þds. In addition, there exists ð 1 2 1 2 2 t 2a0 s G2 sup S22 ðsÞ þ S23 ðsÞ : S2 þ G S3 e ds I e2a0 t sup 8 s2½o;t 4 s2½o;t 4 0
(54)
i By Lyapunov function, it is proved that all the signals Sj ; yj and ~h j in the closed-loop system are uniformly ultimately bounded. Consequently, xi ; ai and aif are also uniformly ultimately bounded.
V. NUMERICAL SIMULATION AND ANALYSIS
In this part, in order to illustrate the effectiveness of the proposed controller, the simulation will be conducted to suppress chaos in the PMSM under the initial condition of x1 ð0Þ ¼ 0:5, x2 ð0Þ ¼ x3 ð0Þ ¼ x4 ð0Þ ¼ 0. The control parameters of the proposed controller are chosen as follows: k1 ¼ 5, k2 ¼ 18, k3 ¼ 100, k4 ¼ 60, u1 ¼ 0.3, u2 ¼ 0.3, u3 ¼ 0.3, r1 ¼ 0.8, r2 ¼ 0.8, r3 ¼ 0.8, s1 ¼ 0.02, s2 ¼ 0.02, c1 ¼ 4, c2 ¼ 4, c3 ¼ 4, g (0) ¼ 1. And the fuzzy membership function are:
lF1i lF4i lF7i
lF10 i
2 ð x þ 5Þ ; ¼ exp 2 2 ð x þ 2Þ ; ¼ exp 2 " # 2 ð Þ x1 ; ¼ exp 2 " # 2 ð x 4 Þ ; ¼ exp 2
2 2 ðx þ 4Þ ð x þ 3Þ lF2i ¼ exp ; lF3i ¼ exp ; 2 2 2 ðx þ 1Þ x2 lF5i ¼ exp ; lF6i ¼ exp ; 2 2 " # " # 2 2 ð Þ ð Þ x2 x3 lF8i ¼ exp ; lF9i ¼ exp ; 2 2 " # 2 ð x 5 Þ lF11 : ¼ exp i 2
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033135-7
Shaohua Luo
Chaos 24, 033135 (2014)
FIG. 3. The trajectory tracking of sin (t) þ 0.5sin (2t). (a) The rotor angle tracking. (b) The rotor angle tracking error. (c) The rotor velocity tracking. (d) The rotor velocity tracking error.
FIG. 4. The adaptive analysis of sin (2t). (a) The rotor angle tracking error. (b) The rotor velocity tracking error. (c) q axis current. (d) q axis voltage. (e) d axis voltage. (f) Nussbaum gain.
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033135-8
Shaohua Luo
Chaos 24, 033135 (2014)
FIG. 5. Robustness analysis for the external disturbance. (a) The rotor angle tracking error. (b) Adaptation of parameter g. (c). q axis voltage. (d) d axis voltage.
A. Analysis of trajectory tracking
Figures 3(a) and 3(c) show the result of trajectory tracking of rotor angle and angular velocity of proposed controller, respectively. Figures 3(b) and 3(d) show the steady state error of rotor angle and angular velocity of proposed controller, which equal to 60.018Rad and 60.03 Rad/s with short response time, respectively. From these figures, it can be see clearly that the proposed system successfully escapes from its chaotic behavior within 1s and tracks the reference signals with good performance. B. Adaptive analysis for parameter variations
The curves of the rotor angle tracking error, velocity tracking error, q axis current, q, d-axis voltages and Nussbaum gain are given in Figures 4(a)–4(f). According to these figures, it is obvious that the proposed system can estimate unknown items with short response time. In other words, it has good robustness to the parameter variations.
permanent magnet synchronous motor system to control chaotic behavior. Fuzzy logic systems are employed to approximate the unknown nonlinear functions, and an adaptive Nussbaum gain is used to cope with the uncertain control gain of system. The proposed controller overcomes the shortage of backstepping and guarantees that the tracking error converges to within a small neighborhood of the origin and all the closed-loop signals are bounded. Simulation results are provided to illustrate the effectiveness and robustness. ACKNOWLEDGMENTS
This work was supported in part by the Major State Basic Research Development Program 973 (No. 2012CB215\\202), the National Natural Science Foundation of China (No. 61134001) and Key Laboratory of Dependable Service Computing in Cyber Physical Society (Chongqing University), Ministry of Education. 1
C. Robustness analysis for disturbance
The expression is given as follows: di ¼ 0:02 x22 sin5 ð2tÞ; i ¼ 1 4:
(55)
In order to check the performance of the controller against the disturbance, simulation results are provided to verify it in the PMSM. In Figures 5(a)–5(d), by contrasting one case without external disturbance (ED) and the other’s with ED, two kinds of curves highly coincide in whole process. It indicates that the proposed system owns good robustness. VI. CONCLUSION
Based on the Nussbaum gain, an adaptive fuzzy dynamic surface control approach is developed for the
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