fuzzy neural network sliding-mode position controller for induction servo motor drive

Fuzzy neural network sliding-mode position controller for induction servo motor drive

R.-J.Wai and F.-J.Lin

Abstract: A sliding-mode controller with an integral-operation switching surface is adopted to control the position of an induction servo motor drive. Moreover, to relax the requirement for the bound of uncertainties, a fuzzy neural network (FNN) sliding-mode controller is investigated, in which the FNN is utilised to estimate the bound of uncertainties real-time. The theoretical analyses for the proposed FNN sliding-mode controller are described in detail. In addition, to guarantee the convergence of tracking error, analytical methods based on a discrete-type Lyapunov function are proposed to determine the varied learning rates of the FNN. Simulation and experimental results show that the proposed FNN sliding-mode controller provides high-performance dynamic characteristics and is robust with regard to plant parameter variations and external load disturbance. Furthermore, comparing with the sliding-mode controller, smaller control effort results, and the chattering phenomenon is much reduced by the proposed FNN sliding-mode controller.

1 Introduction

Since the variable structure control strategy using the sliding mode can offer many good properties [l, 21, such as insensitivity to parameter variations, external disturbance rejection, and fast dynamic response, the sliding-mode control has been studied by many researchers for the control of the AC motor drive systems in the past decade [3-71. The motion of the control system employing sliding-mode control can be described as two modes: reaching and sliding modes. The reaching mode means the control mode before the states of the system reach the designed sliding surface, and in which there is a control action toward the sliding surface. Once the states of the controlled system enter the sliding mode, the dynamics of the system are determined by the choice of sliding hyperplanes and are independent of uncertainties and external disturbances. Furthermore, in the design of sliding-mode controller, the bound of the uncertainties, which include unknown dynamics, parameter variations and external load disturbance, must be available. However, the bound of the uncertainties is difficult to obtain in advance for practical applications in industry. Moreover, to satisfy the existence condition of the sliding mode, a conservative control law with large control effort usually results. Therefore, Karakasoglu and Sundareshan [8] proposed a novel scheme for integrating a neural network approach with an adaptive implementation of variable structure control for robotic manipulators; Lin and Chiu [9] proposed an adaptive fuzzy sliding-mode control system, in which a fuzzy inference mechanism is used to estimate the upper bound of uncertainties for a PM synchronous motor drive.

0 IEE, 1999 IEE Proceedkgs online no. 19990290 DOL 10.1049/ipepa:19990290 Paper fmt received 30th July and in revised form 5th November 1998 The authors are with the Department of Electrical Engineering, Chung Yuan Christian University, Chung Li 32023, Taiwan

IEE Proc.-Elrctr. Power Appl.. Vol 146, No. 3, May 1999

Recently much research has been done on applications of FNN systems, which combine the capability of fuzzy rea- soning in handling uncertam information [9-121 and the capability of neural networks in learning from processes [13-15], in the control fields to deal with nonlinearities and uncertainties of the control systems [16-201. For instance, Chen and Teng [17] proposed a model reference control structure using an FNN controller, which is trained on-line using an FNN identifier with adaptive learning rates; Zhang and Morris [19] described a technique for the modeling of nonlinear systems using an FNN topology; Wai and Lin [20] introduced an FNN controller with adaptive learning rates to control a nonlinear mechanism.

In ths study, first, a sliding-mode controller with an integral-operation switching surface [2 11 is extended and applied to control the rotor position of an indirect field-oriented [22, 231 induction servo motor drive. In the sliding mode position controller, when the sliding mode occurs, the system dynamic behaves as a robust state feedback control system. Then, an FNN sliding mode position controller is investigated, in which the FNN is utilised to estimate the bound of uncertainties real-time for the position control system. The inputs of the FNN are the switching surface and its derivative, and the output of the FNN is the estimated bound of uncertainties. If the uncertainties are absent, once the switching surface is reached initially, a very small positive estimated value of bound of uncertainties would be sufficient to keep the trajectory on the switching surface, and the amplitude of chattering is small. However, when the uncertainties are present, deviations from the switching surface will require a continuous updating of the estimated value produced by the FNN to steer the system trajectory quickly back into the switching surface [SI. Though the true value of the bound of uncertainties cannot be obtained by the FNN, a less conservative control gain results in minimum control effort according to the switching surface and its derivative. Furthermore, the varied learning rates of the FNN, which are determined based on the convergence analyses using a discrete-type

291

e; + Y

e

3-phase 220v 60Hz

q pmf? p controller

rectifier

I RCCC

6 es, &.+ er

digital filter -, and dO,/dt

I

limiter

er

4 1

4THIJjd-l speed

controller co-ordinate translator

field-weakening control

.* sinlcos

generator * 6 s

Tr id', -

I

Fig. 1 DCM = DC machine ISM = induction servo motor RCCC = ramp comparison current control

System configuration of direc t fEU-oriented lnductwn motor servo drive

Lyapunov function, are used in the online training of the FNN to alleviate heavy computation requirements of the identifier [15, 1 1 for the o d n e identification of the Jaco- bian of the system.

2 Indirect field-oriented induction motor drive

The block diagram of an indirect field-oriented inductiori servo motor drive system is shown in Fig. 1 [22, 231, which consists of an induction servo motor loaded with a DC machine, a ramp comparison current-controlled PWM voltage source inverter, an indirect field-oriented mechanism, a coordinate translator, a unit vector (cos 6, + j sin e,, where 6, is the position of rotor flux) generator, a speed control loop, and a position control loop. The induction servo motor used in this drive system is a three-phase Y-connected two-pole 8OOW 60Hz 12OVf5.4A type.

7 ISM drive svstem

I I

Fig. 2 SirnplfEd control system block diagmm

By using the field-oriented technique, the induction motor drive shown in Fig. 1 can be reasonably represented

298

by the block diagram of rotor position control shown in Fig. 2, in which

T, = Ktii, (1)

( 3 ) 1

J s + B H p ( s ) = ~

where T, is the electric torque; Kt is the torque constant; iqs* is the torque current command; id* is the flux current command, np is the number of pole pairs; L, is the'magnetising inductance per phase; Lr is the rotor inductance per phase; J is the moment of inertia; B is the damping coefficient; s is the Laplace operator. Moreover, in Fig. 2, e,* and U,* represent the commands of position and speed of rotor; 6, and U, represent the position and speed of rotor; TL represents the torque of external load disturbance. The unit vector used in the transformation matrix is generated by using the measured rotor position 6, and the integration of the following estimated slip angular velocity:

(4) 2;s

Tr2is W,l = -

where T, is the rotor time-constant. The block diagram of the computer control system for

the indirect field-oriented induction motor servo drive is shown in Fig. 3. The control algorithms are realised in a Pentium computer. To reduce the calculation burden of the

IEE Proc-Elect,. Power Appl.. Vol. 146, No. 3, May 1999

control computer

I I

Fig.3 DCM = DC machine ISM = induction servo motor

Computer-controlled &tion s m o motor drive system with vector processor

CPU and to increase the accuracy of the three-phase command current, the coordinate transformation is implemented by an AD2S100 AC vector processor [24]. The curve fitting technique based on step response technique is applied to find the drive model offline at the nominal condition with TL = ONm. The results are

- Kt = 0.384 Nm/A J = 9.64 x Nms2 -

- B = 1.134 x Nms/rad ( 5 )

The '-' symbol represents the system parameter in the nominal condition.

ISM drive system TL

I I I I I I Induction servo motor drive with s l h g mode position controller Fig. 4

3 Sliding mode position controller

The sliding mode position controller is shown in Fig. 4, where the state variables are defined as follows:

x1 ( t ) = e: - e,(t)

k l ( t ) = -e&) = -w,(t) = - 2 2 ( t )

(6)

(7) Then the induction servo motor drive system can be represented in the following statespace form:

(8) The above equation can be represented as:

X ( t ) = A,X(t) + B,U(t) + D P T ~ (9) where

Consider eqn. 9 with uncertainties:

X ( t ) = (A , + AA,) X ( t ) + (B, + AB,) U ( t ) + (D , + AD,) T L

(10) where hAP, LIB, and ADp are the associated uncertainties. Reformulate eqn. 10, then

X ( t ) = A,X( t ) + B, ( U ( t ) + L ( t ) ) (11) where L(t) is the lumped uncertainty and is defined by

L( t ) = B:AA,X(t) + B:AB,U(t)

where Bpi L? (BpTB,)-' BpT is the pseudo inverse. The switching surface mth integral-operation for the sliding mode position controller is given by Shyu and Shieh [21]:

.+ B,+(D, + AD,)TL

S ( t ) = C X ( t ) - ( A , + B,K)X(T)~ .T = 0

(12) [ k 1

where C E Rlx2 and is set as a positive constant matrix, and K is a state feedback gain matrix.

From eqn. 12, if the states of the system represented by eqn. 11 reach the switching surface eqn. 12, namely S(t) = S(t) = 0, then the equivalent dynamics of eqn. 1 1 is gov- erned by the following equation:

From eqn. 13, the position error x l ( t ) will converge to zero exponentially if the pole of eqn. 13 is designed to locate on the left-hand plane. Thus, the system dynamics will behave as a state feedback control system. Since the pair (Ap, Bp) in eqn. 9 is controllable, the closed-loop eigenvalues, (Ap + B$O, in the sliding mode can be arbitrary assigned by K. Moreover, from eqns. 11, 12 and 13, in the sliding mode, S(t) = 0, the controlled system is insensitive to the lumped uncertainty.

Based on the developed switching surface, a switching control law which guarantees the reachability and existence of the sliding mode is then proposed in the following:

X ( t ) = (A, + B,K)X( t ) (13)

U ( t ) = K X ( t ) - f sgn(S ( t ) ) (14) IEE Proc.-Electr. Power Appl., Vol. 146, No. 3, May 1999 299

where sgn(.) is a sign function defined as

+1 if S ( t ) > 0 -1 if S( t ) < 0

and the control gain f is set as IL(t)l 5 f . The condition for the reachability and existence of a sliding mode is [l]

s+o lim SS < o (15) Differentiate S(t) and multiply with S(t), then

S ( t ) S ( t ) = S ( t ) { C X ( t ) - C(A, + B,K)X( t ) }

= S( t ) {C [ A p X ( t ) + B p ( W + W)l

- C(A, + ~ , K ) X ( t ) } (16)

Replacing the control input U(t) with eqn. 14, the above equation can be rewritten as

S( t )S ( t ) = S ( t ) { C [ A p X ( t )

+%(KX - f sgn(S(t)) + W)l - C(A, + B , K ) X ( t ) )

= S ( t ) { - CB,f sgn(S(t)) + CB,L(t)}

= - CB,{ f lS(t)l - S( t )L ( t ) }

L - C%{ f ' IS(t)I - I S M . IW} = - CB,IS(t)l{ f - lM)l} L 0

(17) The existence condition of the sliding mode can be satisfied using the position controller eqn. 14 with IL(t)l 5J: Since the selection of the control gainfhas a si&icant effect on the system performance, an FNN is utilised to estimate the bound of uncertainties real-time for the sliding mode position control system.

4

The major advantage of a sliding-mode controller is its insensitive to parameter variations and external load disturbance once on the switching surface. Large control gainfis often required to minimise the time required to reach the switching surface from the initial state, and the selection of the control gain f relative to the magnitude of uncertainties to keep the trajectory on the sliding surface. However, the parameter variations of the system are difficult to measure, and the exact value of the external load disturbance is also difficult to know in advance for practical applications. Therefore, usually a conservative control law with large control gain f is selected. Although using a conservative constant control gain results in a simple implementation of the sliding-mode controller, it will yield unnecessary deviations from the switchmg surface, causing a large amount of chattering [8]. Therefore, an F" is adopted in ths study to facilitate adaptive control gain adjustment.

The control block diagram of the I?" sliding-mode controller is shown in Fig. 5, where the inputs of the FNN are S(t) and its derivative, and the output of the FNN is Kr. If the uncertainties are absent, once the switching surface is reached initially, a very small positive value of Kr would be sufficient to keep the trajectory on the switching surface, and the amplitude of chattering is small. However, when the uncertainties are present, deviations from the switching

FNN sliding mode position controller

300

surface will require a continuous updating of Kf produced by the FNN to steer the system trajectory quickly back into the switching surface. Though the true value of the lumped uncertainty cannot be obtained by the FNN, a less conservative control gain is resulted to achieve minimum control effort (i,,*) according to the S(t) and its derivative.

~~ ~

Fig. 5 I&twn servo motor drive with FNN sliding mode position controller

The adjustment of Kf is stop when the output error between the position command and the actual plant is zero. If the output error e - 0 as t -+ CO implies S and S - 0 as t - 03. Replacingfby Krin eqn. 14, the following equation can be obtained:

U ( t ) = K X ( t ) - K f s g n ( S ( t ) ) (18)

4. I Description of FNN A four-layer FNN, as shown in Fig. 6, which comprises an input (the i layer), a membership (the j layer), a rule (the k layer) and an output layer (the o layer), is adopted to implement the FNN. The signal propagation and the basic function in each layer is introduced below.

Y:

. . . . . .

inout

output 0

layer

k rule layer

membership . layer I

Fig. 6 Structure offour-layer FNN

Layer I: input layer For every node i in this layer, the net input and the net output are represented as

y,1 = f,'(neta) = net:, i = 1 , 2 (19) where xll = S(t) and x2' = s(t).

net: = x:,

IEE Proc.-Electr. Power Appl., Vol. 146, No. 3, May 1999

Layer 2: membership layer In th s layer each node performs a membershp function. The Gaussian function is adopted as the membershp function. For thejth node

(x: - m. - ) 2

(oijI2 net; = - 23 , yj" = fj2(net'$) = exp(net5)

j = 1,. . . ,n (20) where inij and q, are, respectively, the mean and the standard deviation of the Gaussian function in thejth term of the ith input linguistic variable x; to the node of layer 2, and n is the total number of the linguistic variables with respect to the input nodes. Layer 3: rule layer Each node k in this layer is denoted by n, which multiplies the input signals and outputs the result of the product. For the kth rule node

k = 1,. . . , I (21) where xi3 represents the jth input to the node of layer 3; W J ) , the weights between the membership layer and the rule layer, are assumed to be unity; I = (n/$ is the number of rules with complete rule connection if each input node has the same linguistic variables. Layer 4: output layer The single node o in ths layer is labelled as 1x1, which com- putes the overall output as the absolute value of the s m - mation of all incoming signals

where the connecting weight wk4, is the output action strength of the 0th output associated with the kth rule; xk4 represents the kth input to the node of layer 4; 1.1 is the absolute value, and yf Kf

4.2 Online learning algorithm The central part of the learning algorithm for an FNN con- cerns how to recursively obtain a gradient vector in which each element in the learning algorithm is defined as the derivative of an energy function with respect to a parameter of the network using the chain rule. Since the gradient vector is calculated in the direction opposite to the flow of the output of each node [16-20], the method is generally referred as the backpropagation learning rule. To describe the online learning algorithm of the FNN using the super- vised gradient decent method, first the energy function E is chosen as

(23) 1 2 1

2 2 E = - (0; - 19,) = -e2

where e denotes the output error between the position command and the actual plant. If the output error e - 0 as t - M implies S and S - 0 as t - m. Then, the learning algorithm based on the back-propagation method is described below. h y e r 4: The error term to be propagated is given by

dE 86, dU ay: 6; = -- - dnet: aE - [ 80, dU dy; anet:

and the weight is updated by the amount

IEE Proc -Electr Power Appl , Vol 146, N o 3, Muy 1999

= [-%--I dE ay: (+ anet: ay/," anet: dwko

= q,i5;2: (25) where qbv is the learning-rate parameter of the connecting weights of the FNN. The weights of the output layer are updated according to the following equation

w2,(N + 1) = w;,(N) + AW;, (26) where N denotes the number of iterations. Layer 3: Since the weights in this layer are unified, only the error term needs to be calculated and propagated.

dE anet; 6; --

= gw;, (27) Layer 2: The multiplication operation is done in this layer. The error term is computed as follows:

aE 6 ? = -- &et'$

k

and the update law of mij is

(29) where rlm is the learning-rate parameter of the mean of the Gaussian functions. The update law of q, is

1 (30)

where v0 is the learning-rate parameter of the standard deviation of the Gaussian functions. The mean and standard deviation of the hidden layer are updated as follows:

m,,(N + 1) = mz,(N) + Am,,

0 2 3 ( N + 1) = g 2 3 ( N ) + Am23 (31)

( 3 2 ) The exact calculation of the Jacobian of the plant, 80)

dU, cannot be determined due to the uncertainties of the plant dynamics. Although the identifier [15, 171 can be implemented to calculate the Jacobian of the plant, heavy computation effort is required. To overcome this problem

30 1

5.7 Function evaluation of FNN The simulated results of the learning processes for the FNN sliding-mode controller are given here. The FNN has two, six, nine and one neurons at the input, membership, rule and output layers, respectively. The learning processes of Ks are shown in Fig. 7, where the theoretic value of the lumped uncertainty is the at), and the estimated value is the ICr. In the simulation, suppose that the TL = 1Nm with the nomnal case changing to case 1 is given at 2s. Moreo- ver, a step command with 2n rad is also given at 2% and the control law in eqn. 18 is used. Figs. 7a, 7b, 8a and 8b show the simulated results without the pre-training of the FNN. After the occurrence of the TL and parameter variations, the FNN learns a Kf which is little higher than the

......

2 ...... ....... ....... , , , , . ,

m r

C 3

._

(U 20

7J 10 Q

-

and to increase the online learning rate of the parameters of the FNN, the Jacobian of the plant is replaced by its sign function in this study. Eqn. 24 now becomes

, . * , I , , . , , . , , ,

........ L ...... : ....... : ....... , . I , . I , , . * , I . , ,

........................ i ....... , . , , . , , , , , , I s o , ,

2 -e sgn (z) sgn(s(t)>sgn(nett> (33)

According to the qualitative knowledge of the dynamic behaviour of the plant, the 0, will increase or decrease as U increases or decreases. Therefore, the sgn(d0Jd U) in the FNN sliding mode position controller is set to be +1 for practical implementation.

4.3 Convergence and analyses of FNN Selection of values for the learning-rate parameters has a significant effect on the network performance. To train the FNN effectively, the varied learning rates, whch guarantee the convergence of traclung error based on the analyses of a discrete-type Lyapunov function, are derived in the Appendix.

All the parameters of the membershp functions and connective weights are randomly initialised in the range [0, I]. Then, a pre-training process is implemented to enhance the control performance. The pre-training process, which is aimed at extracting the connective weights and membershp functions from the process of iterative learning, is used to initial the parameters of the FNN to achieve good control characteristics. The effectiveness of the online training FNN based on the varied learning rates with pre-training process for the application of estimating the lumped uncertainty will be demonstrated by the following simulation and experimental results.

5 Simulated results

The control performance of the sliding mode and FNN sliding-mode controllers are compared in ths section by simulated results. To reduce the chattering phenomenon due to the switchng control law, the sign function in eqns. 14 and 18 can be replaced by the following saturation function:

if S( t ) 2 E

sa t (S ( t ) ) = S ( ~ ) / E if S ( t ) < 1 ~ 1 (34) i' -1 if S ( t ) 5 --E

and the boundary layer E is a small positive constant and usually chosen by trial and error. Then the switching control law can be modified to the smooth control law in the following forms for both the sliding mode and FNN sliding-mode controllers:

U ( t ) = K X ( t ) - f sa t (S ( t ) ) U ( t ) = K X ( t ) - KfSUt(S( t ) ) (35)

In addition, the parameters of the control laws are given in the following:

C = [ l 11, K = [ 2 5 0 -401, ~ ~ 0 . 1 (36) Since the sliding mode condition shown in eqn. 15 is not satisfied within the boundary layer, E = 0.1 is chosen as a compromise between the stability and the reduction of chattering. Moreover, the mechanical inertia and damping constants are significantly vaned to allow the transfer function model Hp(s) to be changed from that of a nominal case to:

case 1: J = 5 x 7, B = 5 x B (37)

0 1 2 3 4 5 time, s

40; I I I : I I ! I : 1

....... , , , , * . 20

a

-201 i i i i i ' i i i i I 0 1 2 3 4 5 b time, s

Fig.7 Function evaluation of FNN (i) Command; (ii) rotor position a Command tracking response without pre-training b Control effort without pre-training

a time, s

, . . . 0 I ; )

, , I , , I 4:S(t) i : , , -20 ...... ' .............................. ' ...............................,......

I . , , . . , , , , ,

0 1 2 3 4 5 b time, s

Fig. 8 Function evaluation of FNN (i) Experimental value K j (ii) theoretical value L(t) a Theoretic.and estimated values of lumped uncertainty without pre-training b S(t) and S(t) without pre-training

0); . . 4 ...... L ...... ; ....... : ....... .............................. : ...... 2 ......

2 ...... j ............... i ................... , l.... ...,.......r............. \

0

. . I

. . I

L i : : I , , . , , , , . , , , . . a . , , , I . . , . .

I , . , , , . I ,

I . , I , . I , ,

0 1 2 3 4 5 a time, s

, , . . , I * . , , , , , , .

40 I : : I . , , , ,

...... : ....... ; ....... : ................. ................................. . , .

. . . . -20 -40 I I I I I I I I I

.............................. ;..p...i ............... ............... j ...... I . . l I , I , I . , , , , , , , , . . .

0 1 2 3 4 5 b time, s

Fig.9 Function evaluation of FNN (i) Command; ( i ) rotor position a Command tracking response with pre-training b Control effort with pre-training

302 IEE Proc.-Electr. Power Appi., Vol. 146, No. 3, May 1999

theoretic value. Moreover, in Fig. 8b the S(t) and its derivative become zero when the output error e becomes zero. Since the Kfis obtained after the occurrence of the uncertainties, the performance of the rotor position response is not good. On the other hand, Figs. 9a, 9b, loa and lob show the simulated results with the pre-training of the FNN. In Fig. loa the position error is zero before the step command given at 2s, therefore, the Kf keeps to a constant. After the given of the step command, a new estimated value of the lumped uncertainty can be obtained from the FNN which is also little higher than the theoretic value. Since the FNN has been pre-trained, robust control performance of the rotor position response can be obtained.

0 1 2 3 4 5 a time, s

...... ...... ....... ....... ......

...... ....... ....... ...... . . . . . . . . . , I , . I I I I I , . -401 I I 1 ' I I I I

0 1 2 3 4 5 b time, s

Fig. 10 Function evuluution ofFNN (i) Experimental value Kr; (ii) theoretical value L(r) a Theoretical and estimated values of lumped uncertainty with pre-training b S(t) and S(t) with pre-training

0 2 4 6 8 10 a time, s

100 : : ! ! ! ! ! ! ! . . . . . . . . . , , . . ,_ ... L. ..... ..... .:. ...... L. ..... ... ...... ...... L ..... ..i.. .... . .

. . . . -50 .............. 4 ............... f ....... j ....................... .......

. I . . , , , , . , , , , I . , . . # , . . I *

-1 00 I I I I I I I I !

0 2 4 6 8 10 b time, s

Fig. 11 Sbnulation results o sluing mode controller: nominal case with T - 0 using switching control Jw . . ati)Command; (ii) rotor position b Control effort

...... ...... ......

...... ...... ........ , , : ....... .

, , . . -2 0 2 4 6 8 10 a time, s

, . , . , , I I I . . . . . . . , . cn L ..... ...... ....... ; ....... L ...... :L ...... ...... L ....... L ...... J ......

-50 ...... ....... .... ....... y ...... ...... , . I . . I , , I

, I . , I S # I 1 -1oo---L-' ' ' ' 1 I I

0 2 4 6 8 10 b time, s

a (i) Command; (ii) rotor position b Control effort

IEE Proc.-Electr. Power Appl.. Vol. 146, No. 3, May 1999

8 6

U 4

0 -2

2 2

............. ......,....... . ......,....... .............. ............. : (ii): ! ......,....... .............. ......*...... ..........................

' ............... 1 1 1 1 1 1 1 , , , . . . , *

0 2 4 6 8 10 a time, s

0 2 4 6 8 10 b time, s Sirmlation results of slding mode controlh: nominal care with TL Fi

= !?king smooth control U (i) Command; (ii) rotor position b Control effort

13

.....

...... ......

.......

0 2 4 6 8 10 a time, s

I . a

...... ' ...................................................... , , , . . . .

-100 0 2 4 6 8 10 b time, s Sinnhtion results of sliding mode controller TL = INm givm ut Is Fi .14

wi$ nominal care chmgmg to case 1 ut 5s using m o t h control law a (i) Command; (ii) rotor position b Control effort

...... ......

...... .......

..... .......

-2 0 2 4 6 8 10 a time, s

601 ! ! ' ' ' ' ' ' I . . . , . .

a

0 2 4 6 8 10 b time, s Simulation results of FNN sliding mode controller: nominal case Fi .I5

wiz TL = 0 usmg switching control law a (i) Command; (ii) rotor position b Control effort

5.2 Sliding mode and FNN sliding-mode controllers In the simulation two cycles of step rotor position commands (step command = 2n rad) are given periodically. Since the theoretic value of the lumped uncertainty is close to 10 from the results shown in Fig. loa, the control gainf = 20 is chosen in the simulation. The control performance of the switchng control law and the smooth control law are compared in Figs. 11-1 8 for both the sliding mode and FNN sliding-mode controllers. The rotor position responses of the sliding mode and FNN sliding-mode controllers using the switching control law at the nominal case with external load disturbance TL = 0 are shown in

303

Figs. l l a and 15a. Favorable tracking responses can be obtained by both types of controllers. The associated control efforts (torque current commands) of the sliding mode and FNN sliding-mode controllers are shown in Figs. 116 and 156. The chattering phenomena in the control efforts are serious due to the sign function in the switching control law, and the amplitude of chattering of the FNN sliding- mode controller is smaller than that of the sliding-mode controller. Now, the disturbance torque with 1Nm is given at Is, and the nominal Hp(s) is changed to case 1 at 5s. The rotor position responses of the sliding mode and FNN sliding-mode controllers using the switching control law are shown in Figs. 12a and 16a, respectively. The robust control performance of the sliding-mode and FNN sliding- mode controllers both in the command tracking and load regulation are obvious. The associated control efforts are shown in Figs. 12b and 166. The chattering phenomena in the control efforts are also serious due to the sign function in the switching function, and the amplitude of chattering of the F" sliding-mode controller is also smaller than that of the sliding-mode controller.

4 ....

....... ...... U

.......

-2 0 2 4 6 8 10 a time, s

! ! ! ! ! ! ! ! ! , , , , , , , , , , , , . , ,

...... 1 I ....... : ...... :L ...... 1 ..... .................,....... , . , , , ,

..... .< ..... ....... : ....... ; ....... ~ ............. ......., ...... ........ -mmm U

. . 0 -20 .............. i ................... ...r......l.......,....... -40 I l l I l l I

. . . , , . I . , , , .

............. , . , . , ,

0 2 4 6 8 10 b time, s Simulation results of FNN slidmg mode controller: TL = 1Nm given Fig. 1 6

at I s with nominal caye c h g m g to case I at 5s using witching control law a (i) Command; (ii) rotor position b Control effort

......

-2 0 2 4 6 8 10 a time, s

...... ......

, , I .

...... I , I a -20 .......

-40 0 2 4 6 8 10 b time, s Swnulution results of FNN slidmg mode controller: nominal case Fi . I 7

w i z TL = 0 i i n g mooth control law a (i) Command; (ii) rotor position b Control effort

Since the chattering phenomenon in sliding mode control will wear the bearing mechanism and excite the unmodelled dynamics, the sign function is replaced by the saturation function in eqn. 34 to reduce the chattering. The rotor position responses for the sliding mode and FNN sliding-mode controllers using the smooth control law at the nominal case with external load disturbance TL = 0 are shown in Figs. 13a and 170, respectively. Favourable tracking responses can be obtained. The associated control efforts of

304

the sliding mode and FNN sliding-mode controllers are shown in Fig. 13b and eqn. 17b. The chattering phenomenon in the control effort of the sliding-mode controller is much reduced, and there is no chattering in the control effort of the FTW sliding-mode controller. Now, the disturbance torque with 1 Nm is given at 1 s, and the nominal Hp(s) is changed to case 1 at 5s. The rotor position responses of the sliding mode and FNN sliding-mode controllers are shown in Figs. 14a and 18a, respectively. The robust control performance of the sliding mode and FNN sliding-mode controllers both in the command traclung and load regulation are obvious. The associated control efforts are shown in Figs. 14b and 1%. The chattering phenomenon in the control effort of the sliding-mode controller is also much reduced, and there is also no chattering in the control effort of the FNN sliding-mode controller. From the simulated results shown in Figs. l l a to 186, one can conclude that the both the sliding mode and FNN sliding- mode controllers with smooth control law can reduce or remove the chattering phenomenon. Therefore, only the results of the smooth control law are presented in the experimentation.

0 2 4 6 8 10 a time, s

...... ...... a

, . ,

0 2 4 6 8 10 b time, s Simulation results of FNN slidhg mode controller: TL = I Nm given Fig. 1 %

at I s with nominal case changing to case 1 at 5s wmg m o t h control law U (i) Command; (ii) rotor position b Control effort

parameters initialisation

I10 initialistion

interrupt interval setting

enable interrupt

monitor data

disable interrupt

-1

1

4

1

1

calculate slip angular

velocity

calculate rotor

flux position

$"SI

1 8, output Be to vector processor

1 ms

interface

calculate [*IT+ , fuzzy neural

network

sliding mode position

controller

'7' by DAC

to vector processor

online learning algorithm to update

parameters of FNN

1 ms Fig. 19 Flowcharts of FNN sliding mode control system

IEE Proc-Electr. Power Appl. , Vol. 146, No. 3. May 1999

6 Experimental results

The software flowcharts of the FNN sliding mode control system in the experimentation are shown in Fig. 19. In the main program, parameters and input/output (I/O) initialisation are proceeded first. Next, the interrupt intervals for the interrupt service routines (ISRs) are set. After enabling the interrupt, the main program is used to monitor the control data. The interrupt service routine ISRl with lms sampling rate is used for the field-oriented mechanism. The interrupt service routine ISR2 with lms sampling rate is used for the encoder interface. and the execution of the FNN sliding- mode controller. Moreover, the online learning algorithm is implemented to update the parameters of the network.

2n rad

0 rad

a

b Fig.20 tion: sluiing mode controller wing smooth control labc, Time scale: I ddiv U (i) Command; (ii) rotor position h Control effort

E,xprinmtul results of command trucking at rwminul inertiu condi-

a

I h I +

Fig. 21 tion FNN sl&g mode controller usmg m t h control kM, Time scale 1 sidiv U (I) Command, (11) rotor position h Control effort

Ex r m t u l results of c o m d truckmg ut nomwl mertu C O ~ I -

Some experimental results are provided here to demon- strate the effectiveness of the proposed FNN sliding-mode controller. Two conditions of rotor inertia are tested here; one is the nominal inertia condition, and the other increases the rotor inertia to approximate five times the

IEE Pro<.-Electr. Power Appl. , Vol. 146. No. 3, Muy 1999

nominal value. In addition, a limiter is added to limit the control effort in the experimentation. First, the measured rotor position responses and the associated control efforts due to a periodic step-command change (2n rad) at the nominal inertia condition of the sliding-mode controller (f = 20) and the FNN sliding-mode controller using the smooth control law are shown in Figs. 20a and b; Figs. 21a and b are the results of the sliding-mode controller, and Figs. 21a and b are the results of the FNN sliding- mode controller. Favourable tracking responses are obtained for both types of controllers. The chattering phenomena are much reduced in the control efforts due to the saturation function in the smooth control law, and the control effort of the FNN sliding-mode controller is smaller than that of the sliding-mode controller due to the smaller real-time estimated value of the lumped uncertainty. Next, the measured rotor position responses due to a step-load disturbance with 1 Nm of the sliding mode and FNN sliding-mode controllers using the smooth control law are shown in Figs. 220 and b, respectively. Identical and

1

a i I i 1 I i

b Fig.22 inertk condition Time scale: 0.5ddiv ( i ) Command: (ii) rotor position U Sliding mode controller using smooth control law h FNN sliding mode controller using smooth control law

Exprimentul results qf s tep- Id clisturhuntr ( I Nm) at nominul

b

Fig.23 vuriution: slia'mg mode controller using m o t h tmtrol hw. Time scale: 1 ddiv U (I) Command; (ii) rotor position h Control effort

E~primmtul results of command trucking at the condition ojinerticr

305

favourable load regulation responses are obtained for both types of controllers. Now, the rotor inertia is varied. The measured responses of a periodic step-command change and the step-load disturbance for both types of controllers using the smooth control law are shown in Figs. 23u, 236, 24u, 246, 2% and 25b. Owing to the sliding mode mechanism, the position responses in command tracking and load regulation are robust for both the sliding mode and FNN sliding-mode controllers. Moreover, the resulted control effort and chattering phenomenon of the FNN sliding- mode controller are smaller than those of the sliding-mode controller owing to the smaller estimated value of the lumped uncertainty.

. . . . . . . . . . . . . . . .

- . . . . . . . . . . . . . . . . . . . a

b Fig. 24 variation: FNN sliding mode controller using mooth control law Time scale: lddiv a (i) Command; (ii) rotor position b Control effort

Experimental results of command trackmg at the condition of kr t ia

a

b Fig. 25 tion of inertia variation Time d e : 0.5ddiv (i) Command; (ii) rotor position a Sliding mode controller using smooth control law b FNN sliding mode controller using smooth control law

Experimental results of step-load disturbance ( I Nm) at the condi-

7 Conclusions

In this study, a sliding mode and an FNN sliding-mode controller based on an integral-operation switching surface

306

have been adopted to control the position of a computer- controlled indirect field-oriented induction servo motor drive. From the simulated and experimental results, robust control characteristics with small control effort and much reduced chattering phenomenon can be obtained from thi FNN sliding-mode controller using the smooth control law.

The comparison of the control characteristics of the shu- ing mode and FNN sliding mode controllers are summa- rised in Table 1. From Table 1, robust control performance can be obtained by both the sliding mode and FNN sliding-mode controllers. However, small control effort is resulted and the chattering phenomenon is not existed in the FNN sliding-mode control system. Therefore, the FNN sliding-mode controller is more suitable for the position control system.

Table 1: Comparison of sliding mode and FNN sliding mode controllers

Robust Control Chattering characteristics effort phenomenon Controller

Sliding mode good large exists controller

FNN sliding mode good small does not exist controller

The major contributions of thls study are: (i) extending the sliding-mode controller with an integral- operation switchmg surface to the position control of the induction servo motor drive; (ii) successfully using a sign function to derive the varied learning rates of the FNN based on a discrete-type Lyapu- nov function; (iii) proposing an FNN sliding-mode controller using smcoth control law to reduce the control effort and chattering phenomenon with robust control characteristics.

8

1

2

3

4

5

6

7

8

9

I O

11

12

13

References

ITKIS, U,: ‘Control systems of variable structure’ (John Wiley & Sons, New York, 1976) UTKTN, V.I.: ‘Sliding mode control design principles and applications to electric drives’, IEEE Truns. Ind. Electron., 1993,40, pp. 22-36 NANDAM, P.K., and SEN, P.C.: ‘A comparative study of Luen- berger observer and adaptive observer-based variable structure speed control system using a self-controlled synchronous motor’, IEEE Trans. Ind. Electron., 1990, 31, pp. 127-132 HO, E.Y.Y., and SEN, P.C.: ‘Control dynamics of speed drive systems using sliding mode controllers with integral compensation’, IEEE Trans. Ind. Appl., 1991, 21, pp. 883-892 KARUNADASA, J.P., and RENFREW, A.C.: ‘Design and implementation of microprocessor based sliding mode controller for brushless servo motor’, IEE Proc. B, 1991, 138, pp. 345-363 PARK, M.H., and KIM, K.S.: ‘Chattering reduction in the position control of induction motor using the sliding mode’, ZEEE Trans. Power Electron., 1991, 6, pp. 317-325 CHERN,T.L., and WU,Y.C.: ‘Design of brushless DC position servo systems using integral variable structure approach’, IEE Proc. B, 1993, 140, pp. 27-34 KARAKASOGLU, A., and SUNDARESHAN, M.K.: ‘A recurrent neural network-based adaptive variable structure model-following control of robotic manipulators’, Automatica, 1995, 31, pp. 1495-1507 LIN, F.J., and CHIU, S.L.: ‘Adaptive fuvy sliding-mode control for PM synchronous servo motor drives’, IEE Proc.-Control Theory Appl., 1998, 145, pp. 63-72 LEE, C.C.: ‘Fuzzy logic in control systems: fuzzy logic controller-Part I and Part 11’, IEEE Trans. Syst. Man. Cybern., 1990,20, pp. 404436 WANG, L.X.: ‘Adaptive fuzzy systems and control: design and stability analysis’ (Prentice-Hall, Englewood Cliffs, New Jersey, 1994) YAGER, R.R., and FILEV, D.P.: ‘Essentials of fuzzy modeling and control’ (John Wiley & Sons, New York, 1994) NARENDRA, K.S., and PARTHASARATHY, K.: ‘Identification and control of dynamical systems using neural networks’, IEEE Trans. Neural Netw., 1990, 1, pp. 4 2 7

IEE Proc.-Electr. Power Appl., Vol. 146, No. 3, May 1999

14 FUKUDA, T., and SHIBATA, T.: ‘Theory and applications of neural networks for industrial control systems’, ZEEE Trans. Ind. Elec- tron., 1992, 39, pp. 472-491

15 SASTRY, P.S., SANTHARAM, G., and UNNIKRISHNAN, K.P.: Memory neuron networks for identification and control of dynamical systems’, IEEE Trans. Neural Netw., 1994, 5, pp. 306319

16 HORIKAWA, S., FURUHASHI, T., and UCHIKAWA, Y.: ‘On fuzzy modeling using fuzzy neural networks with the backpropagation algorithm’, IEEE Trans. Neural Netw., 1992,3, pp. 801406

17 CHEN, Y.C., and TENG, C.C.: ‘A model reference control structure using a fuzzy neural network’, Fuzzy Sets Syst., 1995,73, pp. 291-312

18 JANG, T.S.R., and SUN, C.T.: ‘Neural-fuzzy modeling and control’, Proc. IEEE, 1995,83, pp. 3 7 8 4 5

19 ZHANG J., and MORRIS, A.J.: ‘Fuzzy neural networks for no&- ear systems modelling’, IEE Proc-Control Theory Appl,, 1995, 142, pp. 551-556

20 WAI, R.J., and LIN, F.J.: ‘A fuzzy neural network controller with adaptive learning rates for nonlinear slider-crank mechanism’, Neuro- computing, I998,20, pp. 295-320

21 SHYU, K.K., and SHIEH, H.J.: ‘A new switching surface sliding- mode speed control for induction motor drive systems’, IEEE Truns. Poiver Electron., 1996, 11, pp. 66&667

22 LEONHARD, W.: ‘Control of electrical drives’ (Springer-Verlag, Belin, 1996)

23 NOVOTW, D.W., and LIPO, T.A.: ‘Vector control and dynamics of AC drives’ (Oxford University Press, New York, 1996)

24 LIN, F.J.: ‘Robust speed-controlled induction-motor drive using EKF and RLS estimators’. IEE Proc-Electr. Power Appl., 1996, 143, pp. 186192

9 Appendix

Theorem I: Let qlv be the learning-rate parameter of the connecting weights of the FNN and let P, ~x be defined as P, 3 mux~lP,(N>II, where P,(N) = dy,4/dwk; and 1 1 . 1 1 is the Euclidean norm in W. The Jacobian of the plant is replaced by its sign function. Then the convergence is guaranteed if qw is chosen as q,” = l/(P,v,,zu)2 = URu, in which R, is the number of rules in the FNN. Prooj.~. Since

ay4 a 4 0 Pw(N) = 2 = sgn(nett) (38)

Thus

l l P w ( N ) I I < JRU (39) Then, a discrete-type Lyapunov function is selected as

(40) 1 2

V ( N ) = -e2(N)

The change in the Lyapunov function is obtained by A V ( N ) = V ( N + 1) - V ( N )

1 2

= - [e2(N + 1) - e 2 ( N ) ] (41)

The error difference can be represented by

In order to prove theorem 2, the following lemmas [I71 will be used. Lemma 1: Let p(z) = z exp(-z2). Then Ip(z)/ < 1, V z E 8. Lemma 2: Let q(z) = z2 exp(-z2). Then /q(z)/ < 1, V z E 8. Theorem 2: Let qm and qo be the learning-rate parameters of the mean and the standard deviation of the Gaussian function for the FNN; let P, ,ax be defined as P, ,M = maxNllP,(N)l, where P,(N) = ay;/amq; let Po max be defined as Po,, = muxdlPAN)II, where PAN) =dy;/dqj; ((.(( is the Euclidean norm in !Rn. The Jacobian of the plant is replaced by its sign function. Then the convergence are guaranteed if q, and qa are chosen as q, = qa =

Proofl According to lemma 1, I[(x? - mq)/qj] exp{-E(x? - rng)/qj]l2}( < 1. Since

1 /&[I w& waxlC2/0~

8Yj” = wto sgn(net:) - amij

(44) Thus,

The error difference can also be represented by

where Awk2 represents a weight change in output layer. Because the Jacobian of the plant is replaced by its sign function, then

ll4N + 1111 = I l e ( N ) [I - v w E ( ~ - ) p w ( N ) l II L Il4W II ( 1 1 - vw EIWPw (NI I1

(43) If qlY is chosen as qlb = l/(Pl,, ,zaJ2 = I/Ru, the term 111 - q,,,PwT(N>Pl,,(N)II in eqn. 43 is less than 1. Therefore, the Lyapunov stability of V > 0 and AV < 0 is guaranteed. The output error between the position command and the actual plant will converge to zero as t -+ W . This completes the proof of the theorem.

IEE Proc -Electr Power Appl Val 146, No 3, M C I ~ 1999

where Amq represents a mean change of the Gaussian function in membership layer. Because the Jacobian of the plant is replaced by its sign function, then

ll4N + 1) II = I l e ( W [1 - v73:T(~)%(N)I II I I l e (Wl l 111 - vmpmT(N)Pm(N)II

(47) If q, is chosen as q, = l/(Pm mx)2 = 1/R, [IW& (2/qj min)]-2, the term 111 - q,PmT(N)P,(N)II in eqn. 47 is less than 1. Therefore, the Lyapunov stability of V > 0 and AV < 0 by eqns. 40 and 41 is guaranteed. The output error between the position command and the actual plant will converge to zero as t - W.

307

According to lemma 2, I[(xf - rn,)/o,I2 exp{-[(x? - m,)/ crJ2>\ c 1. Since

(48)

(49)

Thus,

lIP,(N) II < JR, I W L ma.2 I (a /% mTd

The error difference can also be represented by

e(N+1) = e ( N ) + A e ( N ) = e ( N ) + 1 - 1 Agij - I -

(50) where ACT, represents a standard deviation change of the Gaussian function in membershp layer. Because the Jaco- bian of the plant is replaced by its sign function, then

Ile(N + 1111 = I1 e ( N ) [I - %P,T(N)Pu(N)l I I 5 Ile(” 111 - %~,T(N)Pu(~)(I

(51) If qo is chosen as qo = l/(P0 ,ux)2 = l/R,[Iwk4, ,(LyI (2/q, ,3]-2, the term 111 - q$S,T(N)P#T)II in eqn. 51 is less than 1. Therefore, the Lyapunov stability of V > 0 and AV < 0 by eqns. 40 and 41 is guaranteed. The output error between the position command and the actual plant will converge to zero as t 4. This completes the proof of the theorem.

308 kEE Proc-Electr. Power Appl., Vol. 146, No. 3, May 1999

fuzzy neural network sliding-mode position controller for induction servo motor drive

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