GB2572458A

GB2572458A - An adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology

Info

Publication number: GB2572458A
Application number: GB1808590.2A
Authority: GB
Inventors: Luo Shaohua; Wu Songli; Hou Zhiwei; Cao Suqun; Trandinh Phung
Original assignee: Huaiyin Institute of Technology
Current assignee: Huaiyin Institute of Technology
Priority date: 2018-03-28
Filing date: 2018-05-24
Publication date: 2019-10-02
Anticipated expiration: 2038-05-24
Also published as: GB2572458B; CN108390606A; CN108390606B

Abstract

An adaptive sliding-mode control method of a permanent magnet synchronous motor is based on dynamic surface technology. The method comprises: establishing a standardized model of the motor with time-varying delay, and applying a single-weight RBF (Radial Basis Function) neural network with approximation ability for coping with unknown disturbances and unknown system dynamics to controller design. An adaptive sliding-mode controller is then designed in the framework of backstepping, and a first-order low-pass filter may be introduced to handle the problem of derivative explosion associated with traditional backstopping control. A rapid terminal sliding-mode surface may be adopted to further improve tracking accuracy and obtain better performance. Finally an adaptive sliding-mode control method may be proposed in the case of merging the rapid terminal sliding-mode surface, the first-order low-pass filter and the RBF neural network.

Description

An Adaptive Sliding-Mode Control Method of the Permanent Magnet Synchronous Motor based on Dynamic Surface Technology

Technical Field

The present invention relates to a permanent magnet synchronous motor, and more particularly relates to an adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology.

Background Art

The permanent magnet synchronous motor is widely applied to robot and machine-tool control systems by virtue of the advantages such as energy saving, high efficiency, small size, simple structure, long life. However, when being affected by uncertain factors such as system parameters disturbances and external disturbances, it will present rich dynamic behaviors such as a limit cycle and a chaotic behavior. The chaotic behavior may directly destroy the stability and reliability of the motor system. For control methods of the permanent magnet synchronous motor, most scholars hardly considered influences caused by parameters variations, uncertain bounded disturbances and unknown gams on the basis of accurate dynamic model. For example, traditional vector control, direct torque control and other methods improve the performance of the permanent magnet synchronous motor, but most of them are established over an engineering foundation. Meanwhile, a complete proof is not given theoretically, and the issue of nonlinear control for the permanent magnet synchronous motor cannot be solved essentially.

In order to better improve static and dynamic performance of the permanent magnet synchronous motor, its control technology based on intelligent control draws more and more attention in view of influence of nonlinear factors. To improve system robustness, an intelligent control scheme which overcomes adverse factors such as parameters time-varying, load disturbances, time delay and system nonlinearity of the permanent magnet synchronous motor is proposed. An adaptive integral sliding-mode controller not only improves the position tracking accuracy of the permanent magnet synchronous motor, but also improves the system robustness. However, a sliding-mode system no longer has the characteristic of order reduction on a sliding-mode surface. Meanwhile, a slidingmode control technology is prone to a buffeting phenomenon. The fact that a fuzzy control theory can perfectly imitate actual control experiences and methods of specialists and skilled workers can achieve the purpose of high-performance control of the permanent magnet synchronous motor. However, its dynamic response cannot be dealt with by using a single-fuzzy controller which is employed to control the motor transmission system with high-precision. Therefore, the satisfactory control performance cannot be obtained.

The single control method is restricted by own inherent characteristics, so hybrid control fusing intelligent control technologies will be a tendency in the permanent magnet synchronous motor. Moreover, to achieve high-quality control, the factors like time delay, chaos and parameters uncertainties cannot be ignored in the controller design of motor system due to requirements of motor safety and actual process. Thus, it is meaning for putting forward an advanced control method to prevent the phenomena of limit cycle, chaos and time-varying delay in the theoretical research and engineering application. Then this method also can suppress external disturbances and improve static and dynamic performance of the permanent magnet synchronous motor.

Summary of the Invention

The objective of the present invention is to provide an adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology, and this method is able to overcome the influence on the system caused by chaotic oscillation and time-varying delay. In addition, the present invention can also improve the operational stability and motion accuracy of the system.

The technical scheme is presented as follows: the control method of the present invention comprises the following steps:

Step 1: establishing a standardized model of the permanent magnet synchronous motor;

Step 2: adopting a single-weight RBF neural network, and applying this network to controller design; and

Step 3: designing an adaptive sliding-mode controller.

The specific content of Step 1 is:

where g - -y_r /(kL) , s = Bt / J , — = s (i -w}-T_Ldt ^u ’ ^L ^di _q . .

— = -i_q-i,y+g' + i<_q di_d dt ~ ^ld+1i^W+ud

T_L T_L ! J , u_q - u_q / (kR) , (1) ^Md=^Md^{and u}d represent for standardized //-axis and <7-ax is stator voltages, T_L denotes a standardized load torque, v and g are system parameters.

The specific content of Step 2 is:

introducing new variables = w , x₂ =i_q. x₃ = i_d, and simplifying a mathematical model of the permanent magnet synchronous motor with time-varying delay to get:

= .s x₂ (/)-v Xj (t)-T_L +H₁ ^t,x^t'),x^t-d₁ (/)))

Δ (/) = -x₂ (/) - Xj (/) x₃ (/)+gx_l (/)+u_q (t)+H ₂ (t, x(t),x(t- d₂ (/))) (2)

^.(/) = -x₃ (t) + X; (r)x₂ (t)+u_d (t)+H₃ (f,x(/),x(/-J₃ (/))) where x(/) = [x₄ (/), x₂ (/), x₃ (/)]^ e R³ is a state variable vector, H_j (-),/=1-3 represent for the delayed state perturbations, and d_t (·) denote a time-varying delay and satisfies:

o < d_t (/) < 1₃, d&(/) < 1₄ < 1, i = 1 - 3 (3) where 1₃ and 1₄ represent for unknown upper bounds.

The specific content of Step 3 is:

3.1: defining the first tracking error S] = x_l-y_r where y_r represents for a reference trajectory, and calculating a time derivative of Sj:

^-=sx₂ +f₁(-)-j^+H₁(t,x(t),x(t-d₁(t)')) (4) where / (·) = -s x₄ - T_L .

There exists an RBF neural network such as f₃ (·) = q^x₃ (x₃ (/)) + e₃, where q₃ = q’. Using lemma 1 and hypothesis 1, it can obtain:

S' 1 $< ξ x₂ + e₄ + qlXj (x₄ (/)) - j& + ) S'Hjj (¾ (/))J + Hl₂ (x₄ (/ -d, (/))) (5) where b₃ represents for a positive design constant;

introducing new variables:

/^=/,.-/,., / = 1-3, §/ts=i-s (6) where ./ and /,. represent for estimated values of λ and I _t,i = 1-3 ;

designing virtual control and adaptation laws:

a₂

4(1-/,)^¹^¹^] (xi (/))^(/))5(² (7) /² +h

J^= Q (S₂a₂ -cj ) (8) (9) where k^m^a^g^ and G! represent for design constants, and h denotes a smaller positive constant;

obtaining the following inequality by using lemma 2:

25,.6,.-2576 ,₀ tanhl I < 2 e,._o |5,.|-e,._o5,. tanhl (10) <2x0.2785k = k',i =1,2,3 where e ,₀ represents for an upper bound of e,;

obtaining a _2f by a ₂ using a first-order low-pass filter which has a time constant / ₂:

1₂(&_if +a_2f = a₂,a_2/(0) = a₂(0) (11) defining a filter error as y₂ = a _2f - a ₂, so (12) calculating a derivative of y₂:

+#4(1-/,)^¹^^

51/711 (¾ (/))

7, 2(l-/₄) ^77,,(^(/))

(j²+//)T^ ~ ^'^x' (^X1 ~^{610 tanh +}~ (13) so,

< B₂ J,H_n,y_r,#.,?fy (14) where B₂\S_uS₂,y₂,l _us ,H_n,y_r,^,^ represents for a continuous function, and Χ₂=χ₂-α_2/;

using equation (14) and lemma 1 to get:

- ~Y~+y²2 +-^² (15) using equation (7) to equation (15) to rewrite (5) as follows:

+k +5^+^^+5^+-1- _y ^q L +1_£± Hj₂ (_Xj (/ - d_r (/))) +1 + - krf < (2G-k^S² +±GS² +±Gy² -^₂ξ (¾ (/))¾ (¾ (/)K +| + ^ + ⁴ H² ₂ (¾ (/ -d₂ (/))) +k where G represents for an upper bound of 5 , and satisfies 0 < |σ| < G ;

3.2: obtaining a derivative of S₂ from equation (2):

= Λ (·) + «, +H₂(t,x(t),x(t-d₂(tj))-(fy_f (17)

There is an RBF neural network f₂ (·) = q^T ₂x₂ (x₂ (/)) + e₂ , where q₂ =q₂ . In order to further improve tracking accuracy and obtain better performance, a fast terminal sliding-mode surface is designed as:

^ + aS₂+bS₂ ^q =0 (18) wherep and q represent for positive odd numbers, a and β denote positive constants.

According to lemma 1 and hypothesis 1, it yields

S₂t§^ < .S'₂1 q₂x₂ (x₂ (/)) + e₂ + u_q -(&_if +aS₂+b S₂ ^q -I—-—-—-S₂H^ (x₂ (/))

V 4(1 -/ ₄) +-^^(^(/-^(/)))

Z?2 (19) where b₂ represents for a positive design constant; selecting a t/-axis control law and an adaptation law:

- e₂₀ tanh (20) ~ 2 ο 2 z

2a₂ where m₂, a₂ and g₂ represent for design constants;

(21)

3.3: defining the third error function S₃:

(22) according to equation (2) and equation (22), obtaining:

(23) where /₃(-) =

An RBF neural network is employed as f₃ = q₃x₃ (x₃ (/)) + e₃, where q₃ = q’. With lemma 1 and hypothesis 1, one has

b.

(24) where b₃ represents for a positive design constant;

designing a J-axis control law as:

-e,„ tanh — ³⁰ I k

b.

(25) where k₃ represents for a positive constant;

selecting a corresponding adaptation law:

la.

-m (26) where m,.a, and g. represent for design constants.

The said lemma 1, lemma 2 and hypothesis 1 are:

Lemma 1: The Young's inequality yz <by² + — _z ² exists; Ab

Lemma2:For u&R and k > 0 , the following relation 0 < |m|-m tanh — < 0.2785½ holds; and j

Hypothesis 1: There exist smooth functions 77^(-) and TZ_i2 (·) such that:

= 1,2,3 where x,. (/) = U (/), x₂ (/) ,L , x₍. (/)]^r represents for the vector of the i -order state, and a (0) = 0.

Beneficial effects: The present invention presents an adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology. It includes: firstly use the Lyapunov theory being favorable for the stability analysis to construct the controller; secondly, adopt an adaptive RBF neural network with single-weight to estimate unknown dynamic items with any small error, then reduce requirements for accurate modeling of system and restrain effect on external environment of the system and parameters disturbances; introduce a first-order low-pass fdter to handle the problem of derivative explosion of traditional backstepping control, then reduce the difficulty of controller design; adopt a rapid terminal sliding-mode surface to further improve tracking accuracy as well as obtain better performance. Finally, an adaptive sliding-mode control method is proposed by merging the backstepping, rapid terminal sliding-mode surface, first-order low-pass filter and RBF neural network. The present invention not only reduces requirements for accurate modeling of system and accurate measurement of parameters, but also simplifies controller design. Meanwhile, this invention overcomes effects on a system caused by external disturbances, chaotic oscillation and time-varying delay, and improves the operational stability and motion accuracy of the system.

Brief Description of the Drawings

Figure 1 shows strange attractor;

Figure 2 shows chaotic time series;

Figure 3 shows the working principle of the traditional permanent magnet synchronous motor;

Figure 4 depicts the working principle of the present invention;

Figure 5 depicts the trajectory tracking of the present invention under the reference trajectory 5sin (2ΐ+3/2π);

Figure 6 shows the performance test of the present invention under the reference trajectory 5sin (2ί+3/2π);

Figure 7 shows the performance comparison of the present invention for the reference trajectory 5sin (2ί+3/2π).

Detailed Description of the Invention

The present invention will be further described below with reference to the drawings.

A control method provided by the present invention comprises the following steps:

Step 1: Establish a standardized model of the permanent magnet synchronous motor.

The mathematical model of the permanent magnet synchronous motor is written as:

di,

R- L_d y_r _ 1 _

--I--—Ml,—-w-i--u dt L„ ^q L^d L„ L„ ^q

1) ^did ^R ~ dt L_d ^d L_d _ —^ud ^Ld where i_d represents for the o'-ax is current (A), z means the t/-axis current (A), w denotes the velocity of the rotor (rad/s), t is the time (s), u_d represents for the d-axis voltage (V), u means the t/-axis voltage (V), T_L denotes the load torque (Nm), R is the stator coil resistance (Ω), L_d represents for the d-axis winding inductance (H), L_q means the t/-axis winding inductance (H), y_r denotes the permanent magnet flux (Wb), B is the viscous damping coefficient (N/rad/s), J represents for the polar moment of inertia (kgm²), and n_p means the number of pole pairs.

By solving the control problem of the permanent magnet synchronous motor with smooth air gap, it is found that the winding inductances of d-axis and the t/-axis are equal, that is, L = L_d =L_q holds. Define the time scale t =L/ R, the normalized time t = T /1 , and the scalar k = B / (n_pty _r). In the meantime, proportional state variables ω, i_d and i are given as follows:

w = wt , i_d = i_d Ik , i = i Ik 2) where ω, i_d and i represent for the normalized angular speed, the normalized t/-axis and d-axis current, respectively.

By using variable transformation, the normalized model of the permanent magnet synchronous motor can be written as:

•-^- = -ⁱq-ⁱd^w+s^w+u<1 ³) dt_d dt - ¹d⁺¹g^W+Ud where g = -y_r/(kL) , s -Bt /J , T_L=t¹T_Lu , u =u l(kR) , u_d=u_d/(kR), u_q and u_ddenote normalized i/-axis and J-axis stator voltages, T_L is the standardized load torque, s and g denote system parameters.

It is obvious that the permanent magnet synchronous motor has a highly nonlinear characteristic due to the coupling effect between speed and currents. Once system parameters slip into a certain range, chaotic oscillation will occur. Figures 1-2 show the strange attractor and chaotic time series of the permanent magnet synchronous motor under the condition of σ=5.46, ;.'=20. u_q =u_d = 0, '/',=(). ω(0)=-5, i (0) = 0.01, and i_d (0) = 20, and the chaotic oscillation of the permanent magnet synchronous motor possesses features such as aperiodic, random, sudden or intermittent morbid oscillation.

In the engineering application, system parameters perturbations within a certain scope is inevitable due to external factors such as temperature, voltage oscillation and material wear. System parameters are regarded as unknown items in the controller design. Meanwhile, time-varying delay, which may be caused by low pass fdter, communication media, hysteresis control inverter and computation time of the microprocessor program, is often encountered in the motor driving system. If no actions are taken, time delay can give rise to voltage and current distortion, and then destroy the stability of the permanent magnet synchronous motor. Therefore, it should consider these factors in chaos control of the permanent magnet synchronous motor.

The principle block diagram of the permanent magnet synchronous motor is shown in Figure 3. The overall system consists of the permanent magnet synchronous motor, space vector pulse width modulation (SVPWM), power source rectifier, voltage source inverter (VSI), automatic current regulator (ACR), encoder, speed and position tracking controller. A multi-loop cascade control structure with a speed loop and two current loops is presented. PI controller, which is used to control the d-axis current error of the vector controlled drive, is adopted in the <7-axis current loop. However, in real applications, when the permanent magnet synchronous motor suffers from the effects of the time-varying delay, back electromotive forces, torque ripples, parameters variations and unmolded dynamics, a PI control effect is very limited.

Step 2: To apply an RBF neural network to controller design, it makes a brief introduction of it which has great power in coping with unknown disturbance and dynamics. It can approximate any smooth function f_n (X): Rⁿ -> R as /,(.V) = 7’x(.V) 4) where XeDcR is the input vector, q = &R^l is the weight vector, />1 is the node number of neuron, and x(X) = [Xj(X),x₂(Y),L .x,(.V)|' &R^l is a basic function vector, with x_; (X) being chosen as the commonly used Gaussian functions which have following forms:

:, (X) = exp i = l,2,L ,/

5) where m_i = \ιη_Λ,ιη_!2,Τ .///,,]⁷ is the center of a region of acceptance, and s_: is the width of the

Gaussian function. For nonlinear function f(X), there exists a RBF neural network such that f(X)=q^Tx(X)+e

6) where e denotes the approximation error, and q is the optimal parameter vector and is defined as:

q* = arg min

7) where W is a compact set for q . There exists a constant e₀ such that 0 < |e| < e₀.

Introduce new symbols as =w , x₂ = i, x₃ = i_d. Then the mathematical model of the permanent magnet synchronous motor with time-varying delay can be simplified as = 5 x₂ (/)-5 X! +Hi {t,x(t^,x{t-d_l (/)))

A(0 ^{= _X}2 (4^_X1 (0^X3 (4⁺£^xi (4 ^{+ m}? 0)⁺ Ύ (/,x(/),x(/-ri₂ (/))) 8)

3§.(i) = -x₃ (/) + X; (/)x₂ (/) + «_rf 0) ^{+ i}T (fx(/),x(/-ri₃ (/))) where x(/) = [x₃ (/), x₂ (/), x₃ (t)]^r e R' is the state variable vector, H_i (·), i = 1-3 are the delayed state perturbations, and d_t (·) denotes the time-varying delay satisfying ο < d, (t) < 1₃, d&(/) < 1₄ < 1, i = 1 - 3 9) where 1₃ and 1₄ are unknown upper bounds.

Lemma 1: The Young's inequality yz <by² + — z² holds.

4Z>

Lemma 2: For u&R and k > 0 , the following relation 0<|M|-Mtanh — < 0.2785½ holds.

\_ A: j

Assumption 1: There exist smooth functions Η_Λ(·} and 77,₂(·) such that:

\h, d, (t)))| < Η,_Ί (^ (/)) H_i2 (/-/(/))),/ = 1,2,3 where x, (/) = | x, (/), x₂ (/) ,L , x, (/)]^r is the vector of the i -order state, H_i2 (0) = 0.

Step 3: Design of adaptive sliding-mode controller.

To reduce the computational time, the number of weight vectors of the RBF neural network is reduced by series transformation. So there exists the following transform / (Φ, (t (0) = < (^K (Ϋ (0)t (t (0) + y 10) where 1₁ (/) = (/)¾ (/)|| , a,>0, z > 1, /^(/)=/,(/)-1,(/) holds, and /,(/) denotes the estimation of 1₁ (/).

The whole procedure can be divided into three steps.

3.1: Define the first tracking error as S) = Xj - y. where y. denotes the reference trajectory. Then, the time derivative of S) is obtained $=sx₂ + /(·) - jfc + 7/ι (/, x(/),x(/-/(/))) 11) where / (·) = x₄ - T_L . There exists a RBF neural network such that / (·) = 1/,+, (x, (/)) + e₃, where q, = q*. By using the Lemma 1 and Assumption 1, substituting 12) into 11) yields

S'i$< / x₂ + + q{x₄ (x₄ (/)) - + ) /yj (¾ (/)) j + H² ₂ (x₄ (/ - / (/)))

12) where b, is a positive design constant.

Introduce new variables as follow:

/,.-/,, / = 1-3, §/a=J-s 13) where and /, are the estimator of λ and I _t,i = 1-3 .

Then, the virtual control with adaptive laws is designed as follow:

a₂

14)

2a[

-m $= Q (S_la₂ -q

15)

16) where and G, are design constants, and h is a smaller positive constant.

Using Lemma 2, the following inequality holds

25,.e ,-25,.e _/0 tanhl I < 2 e,._o |5,.|-e_i0S_i tanhl <2x0.2785k =k',i=l,2,3 where e ,₀ is the upper bound of e,.

a ₂ is obtained by using a first-order low-pass filter which has a time constant 1₂:

t + a_2f = a ₂,a _2f(0) = a ₂(0)

The filter error is defined as y₂=a_2f-a₂. One has

17)

18)

I⁹)

The derivative of y₂ is computed by

T₂t 2 .2 ^b!

^b!

20)

Then, it has

A+— ^b₂ t 2

21) where B₂\S_l,S₂,y₂,l ,H_n,y_r, is a continuous function with S₂ = x₂ - a _2f.

Using 21) and Lemma 1, it follows that λΑ - ~~+yl +-^2² 22)

With the help of 14)- 22) and Lemma 1,12) can be rewritten as follows

+k y²2 ^-^2^+ L2± h² (Xj (/ - d_r (/))) +1 + - k_tS² < (iG-k^s² +hw; +±Gy² ₂ (^xi(0)¾ (¾(OK +|+^- ¹23) + ⁴ H² ₂ (¾ (/ - </ (/))) +k where G denotes an upper bound of 5 satisfying 0 < |σ| < G .

3.2: From 8), the derivative of is given by ^2= Λ (’) + ¹¹ q + 7^2 (O’^X “A (0))^_24)

There is a RBF neural network such that f₂ (·) = q₂x₂ (x₂ (/)) + e₂, where q₂ = q₂. In order to further improve the tracking precision and attain a good performance, a fast terminal sliding mode surface is (FTSM) designed as follows ^+aS₂+bS?^lq =025) where p and q are the positive odd number, and a. β denote are the positive constant.

According to Lemma 1 and Assumption 1, one has

S₂t§^ < S₂ q₂x₂ (x₂ (0) ^{+ e}2^+uq~d&if + aS 2+b S2^q -I—-—-—-S2H^ (x₂ (/)) I 4(1-t₄) + ^^2^(1-^(0))

Z?2 where b₂ is a positive design constant.

Then, the q-axis control law with an adaptive law is chosen as —₂S₂x^T2 (x2 (0)^x2 (¾ (0)^{- e}?o tanhi ^²¹¹^² [ + (&if -aS ₂-b S₂ ^q

26)

27)

28) where in,, a₂ and g₂ are design constants.

3.3: Define the third error function S', to be

29)

From 8) and 29), it follows that

30) where f₃ (·) = -x₃ + x₃x₂. There exists a RBF neural network such that f₃ = q[x₃ (x₃ (t)) + e₃, where q₃ = q₃. Using Lemma 1 and Assumption 1, it obtains:

31) where b₃ is a positive design constant.

Then, the d-axis control law is designed as:

-e_3n tanh — ³⁰ ( k

b.

32) where k₃ is a positive constant.

And the adaptive law is chosen as follows la:

-m

33) where in,, a, and g₃ are design constants.

Perform stability analysis on a system:

To do squaring operation, these relationships and —c.&o +—c.s² hold.

¹ 2 ¹

Consider the first Lyapunov function candidate as

34)

Then, the derivative of V₃ can be obtained

1&< (2G-k₃)S² +±GS² +±Gy²-^₂S₃(¾ (0)¾ (¾ (OK ⁺ ^ ⁺ ^~r ^{+ y2} ^² (¾ (0)¾ (¾ (OK ₂s₃-—

8i ^b!

= (2G-k.)S² +-GS² +-+-^- + fl- —₊ 1gX₂ ² + -B²

2 2 ( t₂ 4 / 4 2 g, 2 ⁺ ' ⁺^2.^C'^{S 2 +} b ^{4 H}'² (^X‘ ^_d'

Choose the second Lyapunov function candidate as

F=qk+—36) ² I gl J

Then the following inequality holds j&<^L₊L__aS ²2-bS2^p,«^+l2 +^^fifr-i/₂(/)W37) ² 2 2 ^{2 2} 2 g2 ² 2 g2 ² b₂ ²²' ^{1V 2 V 777}

Define the third Lyapunov function candidate as

Then the time derivative of V3 is given by

1&< -k₃S² +1+1-1^+--1 ² +1ς1±_η ² i_x (t-d₃ (t\\\+k'39) ^{3 3 3} 2 2 2 g3 ³ 2 g3 ³ b₃ ³² 1 ^{1 v 3 v 777}

For chaos control of the permanent magnet synchronous motor 8) with chaotic oscillation, unknown dynamics and delay, controllers with adaptive laws 15), 16), 28) and 33) are designed as 27) and 32) when the Assumption 1 holds. Then, by selecting the rational parameters such as £,,/ = 1,3, a,,/=1-3, /),,/ = 1-3, g,,/=1-3, /₂, m,,/=1-3, Q, q, q, p, a and b , all signals in the closed-loop system are uniformly ultimately bounded, and tracking errors can rapidly converge to a neighborhood of zero.

Proof: Define the Lyapunov function candidate

Differentiating V can obtain ^(2G-k₃)S²3 +|GS2²-k3S² ₃ +^l--L + lGjy₂ ²-aS²-bS^p/^ ⁺ 2^CA^{2 +} 4^{B2 +}^^^±H^²(^X(^t~^di^) ^{+ 3k 41}) < -2«₀Γ + b₀ where/)₀ = |qs² +^-1 +Υ^- + ^ + \β², +yJ-1±h² ₂ ^x(t-d_i (1))) + 3/% , a₀ = min^,^,, Z Z ,₌₁ g,. ,₌₁ Z Z 4 ,₌₁ 0,.

,1111 l·;,.—m.. —m-,.— ³ 2 2 2 2

Furthermore, 41) implies that

1-(0+ 1 ^r’(0)-— |^ — + F'(0)- V>/₀ 42) a_Q ( a_Q J a_Q

Namely, all signals in the closed-loop system are bounded. Especially, one has — S₃ < — + J '(0).

a₀ which implies that lim.S,² < 2 —.

a₀

The above analysis illustrates that tracking errors, i.e., S₃,S₂,S₃ depend on a₀ and b₀. Because a₀ and b₀ are unknown, it is difficult to obtain an explicit estimation of tracking errors. From definitions of a₀ and b₀, it can be seen that reducing a^i =1-3 , increasing g,U=l-3 and Q can theoretically result in smaller tracking errors.

Numerical simulation is carried out to validate the feasibility of the proposed scheme. The simulation is done under the initial conditions x₃ (0) = -5, x₂ (0) = 0.01 and x₃ (0) = 20 . The corresponding controller parameters are chosen as k_t = k₂ = k₃ =15 , Q = 20 , q = 0.02 , g_t = g₃ = 3 , g₂ = 20 , m_l=m₂=m₃= 0.02, a₃ = a₃ = 20, a₂ = 6 , t₂ =0.01, Λ =0.01, a =10, b =5 , p = 5, q = 7, /^0) = 0, /^(0) = 272, /^(0) = 0.339, ./(0) = 8, e₁₀ = e₂₀ = e₃₀ = 0.1, b₃=b₂=b₃ =Q.5 , /₃=0.1, /₄ =0.15.

2/j, H₂, H₃, d₃, d₂ and d₃ are set to be:

H_l ^t,x(t),x^t-d_i (/))) = e^x'^ sin(x₂ (/)) (x₄ (/-<ή(/))) d₁ (/) = 0.05 + 0.05sin(2/)

H₂ (t,x(t),x(t-d₂ (/))) = e^X1^ sin(xj (/))(x₄ {t-d₂ (/))) d₂ (/) = 0.05 + 0.05cos(2/)

H₃ (t,x(/),x(/-d₃ (/))) = e^X1^ sin(x₃ (/))(x, (/-d₃ (/))) , d₃ (/) = 0.05 + 0.05sin(2/ + 0.3) 43)

Choose ^Hn (^xi (0) = ^e*¹⁽ⁱ⁾ ’ ^Hn (¾ (l - (0)) = (¾ (l - (^Z)))² , ^H2l (*2 (0) = ’ ^,(^(/-^(/))) = (^(/-^(/)))³, f/31 (x3 (/)) = e^Ia(i), TV32 (x_x (/-cZ3(/))) = (x_x (/-c/₃ (/)))³ 44)

The expression of external disturbances is given us il.5Mw,0 < / < 4

T_L = ^L l3.0Mw,/>4

45)

Furthermore, the RBF neural network, whose Gauss basis width \,. equals 1.5, contains nine nodes with the center m, distributed evenly in the field of [-5, 5], Note that a further increase of node number does not significantly improve the system performance.

The state trajectory 5sin (2ΐ+3/2π) of the permanent magnet synchronous motor is depicted in Figure 5. It can be seen that the tracking error of the rotor velocity converges to zero quickly. The tracking error between a desired trajectory and an actual trajectory is less than ±0.05 Rad/s. So a conclusion can be drawn that the chaotic oscillation of the permanent magnet synchronous motor is thoroughly inhibited, and the trajectory tracking performance is very high. The results of performance test are given in Figure 6(a)-(f) for different σ and γ. A change of system parameters of the permanent magnet synchronous motor docs not lead to performance degradation, and an ability to resist parameters disturbances is good.

To show the superiority of the presented scheme, a comparison task is executed for the permanent magnet synchronous motor without time delay. Then, an exponential approach sliding mode (EASM) surface is employed t^ + aS ₂+bsignS₂=t) 46)

Figure 7 shows the comparing results. It is obvious that the presented scheme (fast terminal slidingmode control method) has higher precision tracking, and requires less control input and current. Therefore, the presented scheme is superior to the EASM scheme.

Claims

1. An adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology, which is characterized by comprising the following steps: Step 1: establishing a standardized model of the permanent magnet synchronous motor;

Step 3: designing an adaptive sliding-mode controller.

2. According to the adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology of claim 1, it is characterized in that the specific content of Step 1 is:

setting the standardized model of the permanent magnet synchronous motor as:

— = s (i -w\-T_L dt ! ^L di — = -ι,-ι^+Ά^ + ιι, (1) dt ~ ¹d⁺¹ _q ^w+ud where g = -y_r/(kL), s - Bt / J , T_L=t²T_L/J , u t(kR) , u_d=u_d/{kR), u_q and u_d represent for standardized t/-axis and //-axis stator voltages, T, means a standardized load torque,

5 and g denote system parameters.

3. According to the adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology of claim 1, it is characterized in that the specific content of Step 2 is: introducing new variables Xj = w , x₂ =i_q. x₃ = i_d, and simplifying the mathematical model of the permanent magnet synchronous motor with time-varying delay to get:

.^(/) = .s x₂ (t)-s Xj (t)-T_L +H₁ (t,x(t),x(t-d₁ (/))) A(0 ⁼ ~^x2 (0 ^{_ x}i (0 ^x3 (0 ⁺ 8^xi (t) + u_q (t) + H₂ (t,x(t),x(t- d₂ (/))) (2)

^.(/) = —x₃ (t) + X; (t)x₂ (0 ^{+ M}rf 0) ^{+ i}T (t,x(/),x(/-ri₃ (/))) where x(t) = [x₃ (t), x₂ (t), x₃ (t)]^r e R³ represents for a state variable vector, 22,(-),/=1-3 denote the delayed state perturbations, and ¢/,. (·) mean a time-varying delay and satisfy:

o < d_j (/) < 1₃, d&(/) < 1₄ < 1, i = 1 - 3 (3) where 1₃ and 1₄ represent for unknown upper bounds.

4. According to the adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology of claim 1, it is characterized in that the specific content of Step 3 is:

3.1: defining the first tracking error / = x_l-y_r where y. represents for a reference trajectory, and calculating a time derivative of /:

$=sx₂ + /(·) - fc + H_l (/,x(/),x(/-d_l (/))) (4) where / (·) = x₄ - T_L .

There is an RBF neural network / (·) = ¢//¾ (x₄ (/)) + e₄, where q_l=q{; using lemma 1 and hypothesis 1, one can obtain:

Si^< Si x₂ + 6i + q[X; (x; (/)) - + (¾ (/)) j + H[₂ (xj (/ -/ (/))) (5) where represents for a positive design constant;

introducing new variables:

/^=/,.-/,., / = 1-3, §/a=]-s (6) where ./ and /,. represent for estimated values of x and I,, i = 1-3 ;

designing virtual control and adaptation laws:

2/

J^= Q (S_la₂ -Cj (8) (9) where k^m^a^g^ and G, represent for design constants, and h denotes a smaller positive constant;

obtaining the following inequality by using lemma 2:

ISy-lSy ,0 tanhl1 < 2 e,₀|5,| -e,₀5, tanhl <2x0.2785½ =^,/=1,2,3 where e _1O represents for an upper bound of e,;

obtaining a _2f by a ₂ using a first-order low-pass filter which has a time constant 1₂:

1+ a _2f = α ₂, a _2/ (0) = a ₂ (0) (10) (11) defining a filter error as y₂ = a _2f - a ₂, so (12) calculating a derivative of y₂:

^+# 4(1-/,)^¹^¹^ 2(1-/,)^¹¹^¹^ (j² ₊//)T^ ~(^X1 ~^{610 tanh+} ~ 4(1-/ )^SlH ¹ (^X1 (13) so

A+— (14) where B₂ ¢$(,52,4/2,/^,7^,4/,.,4^,^ represents for a continuous function, and 5₂ = using equation (14) and lemma 1 to get:

λΑ - ~Y~+y²2 ⁺^^Bi using equation (7) to equation (15) to rewrite (5) as follows:

(15)

+k' +5 f 5² + S2 Ί + 5 f 5² + i y²2 ^-^2^ + H² (¾ (/ - d₂ (/))) +1 + - k₂S² ^/,(^(/)) < (iG-k^Sj +|gS² +±Gy> (¾ (0)¾ (¾ (0)^i +| + ^ + ⁴ 7^12 (¾ (t ~ d₂ (0)) +k where G represents for an upper bound of 5 , and satisfies 0 < |σ| < G ;

3.2: obtaining a derivative of S₂ from equation (2):

= Λ (·) +¹¹ q + ^H2 (/, x (0, x (t - d₂ (0)) - (fy_f (16) (17)

There is an RBF neural network f₂ (·) = q^T ₂x₂ (x₂ (0) +G where q₂ =q₂ . In order to further improve tracking accuracy and obtain better performance, a fast terminal sliding-mode surface is designed as:

^ + aS₂+bSj'^q =0 (18) wherep and q represent for positive odd numbers, a and β denote positive constants;

According to lemma 1 and hypothesis 1, it yields

S₂$j< S₂ i q₂x₂ (x₂ (/)) + e₂ + u_q -+aS₂+bS₂ ^q + ²—jS₂Hj₃ (x₂

+^^(^-^(0)) where b₂ represents for a positive design constant;

selecting a t/-axis control law and an adaptation law:

+ =-77+/7 (x₂(0)x₂(x₂(0)-G_otanhp^'|+4_r -aS₂-bS₂ ^p/q

(20) (21) (22) where m₂, a₂ and g₂ represent for design constants;

3.3: defining the third error function S₃:

S) = x₃ - 2 according to equation (2) and equation (22), obtaining:

-^=/3 (·) + «./ + H₃(t,x(t),x(t-d₃(t))) (23) where f₃ (·) = -x₃ + x₃x₂ ; an RBF neural network is employed as f₃ = q₃x₃ (x₃ (/)) + e₃, where q, = q*; with lemma 1 and hypothesis 1, one has:

S₃^-<S₃ Kx₃(x₃ (/)) + e, +

i-G ^3

77₃ ² ₂ (xi (/-/(/))) (24) where b₃ represents for a positive design constant;

designing a <7-ax is control law as:

(25) where k₃ represents for a positive constant;

selecting a corresponding adaptation law:

2 a:

-m (26) where in,, a, and g₃ represent for design constants.

5. According to the adaptive sliding-mode control method of the permanent magnet synchronous motor based on dynamic surface technology of claim 4, it is characterized in that lemma 1, lemma 2 and hypothesis 1 are:

Lemma 1: The Young's inequality yz <by² + — z² exists;

diLemma 2: For u&R and k > 0 , the following relation 0 < |M|-Mtanh — < 0.2785½ holds; and

1,½ )

Hypothesis 1: There exist smooth functions 77,_x (·) and 77,₂(·) such that:

|t7, (/, x (/), x (/ - / (/)))!< 77,1 (X, (0)^H 12 (-% = 1,2,3 where x, (/) = [Xj (/),x₂ (/),L ,x, (/)]^r represents for the vector of the i -order state, and

77,₂ (0) = 0.