Chattering-free and fast-response sliding mode controller

D.Q.Zhang and S.K.Panda

Abstract: A new chattering-free and fast-response sliding-mode controller with a smooth control law (SMCS) is proposed. Unlike the conventional sliding-mode controller (SMC), or sliding- mode controller with boundary layer (SMCB), which obeys the variable-structure control principle by adopting a switching control term, the proposed SMCS uses a continuously varying term instead which takes the distance of the system state from the sliding surface into account. As a result, chattering is eliminated, and the control performance is improved in contrast to the popularly used SMCB in terms of system response, robustness, adaptability and maximum steady- state error. Both theoretical analysis and simulation studies have been carried out to verify the superiority of SMCS over SMCB. A step-by-step systematic design procedure for SMCS is presented and applied to speed control of a PMSM drive system.

1 Introduction

Sliding-mode control has found many applications in recent years [l]; the most salient feature of this control technique is its robustness to the system uncertainties and disturbances in the so-called sliding mode. The original idea of the sliding-mode control is simple: design a control law with varying control structures, then force the trajec- tory of the system state to a certain predefined surface, known as sliding surface, through an appropriate switching of the control structures such that the system state will stay there afterwards and thus the system dynamics are only determined by the dynamics of the sliding surface.

However, the digital implementation of the sliding-mode control has the chattering problem. Chattering occurs when the control input switches discontinuously across the boundary, and it is undesirable because it involves high control activity and may excite high-frequency dynamics [2]. To eliminate chattering, various methods such as sliding-mode controller with boundary layer (SMCB) [3], fuzzy sliding-mode controller (FSMC) [4], sliding-mode controller (SMC) with sliding sector [5] etc. have been proposed. In all these works the basic idea is just to smooth the control action across the sliding surface while preser- ving the traditional variable-structure control law. To improve the system response of the traditional SMC and SMCB, Wang and Lee [6] proposed a two-phase variable- structure controller which incorporates the distance of the system state from the sliding surface into the controller design. The rationale behind this method is that the

IEE, 1999 IEE Proceedings online no. 19990518 D0I:I 0.1049/ip-cta: 199905 18 Paper first received 22nd January 1998 and in revised form 2nd November 1998 D.Q. Zhang is with the Servo Electronics Group, Data Storage Institute, 10 Kent Ridge Crescent, Singapore 119260 E-mail: dpzhang@dsi.nus.edu.sg S.K. Panda is with the Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 E-mail: eleskp@nus.edu.sg

IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999

switching control action in SMC and SMCB is usually not strong enough for chattering attenuation so an extra distance-dependent varying term is included which helps reduce the hitting time. However, chattering may still occur under certain operating conditions.

In this paper a new chattering-free and fast-response sliding-mode controller with a smooth control law is proposed. Borrowing the idea from the two-phase vari- able-structure controller, the proposed controller also takes the distance of the system state from the sliding surface into account. But unlike the traditional SMC or two-phase variable-structure controller which obeys the variable structure-control principle by adopting a switching control term, the proposed SMCS uses a unified continuously varying term instead. As a result, not only is chattering completely eliminated, but system response, robustness, adaptability and steady-state performance are all improved as well by choosing the new term appropriately. Compared with other sliding-mode control strategies, the proposed SMCS is believed to be one of the most practical methods ever proposed.

2 Traditional approach

Consider the SISO nonlinear system

x(")(t) =f(x,t) + U + d(t) (1)

where U E R is the control input, x(t) E R is the output, x = (x, i , . . . , x ( " - ~ ) ) ~ E Rn is the state vector, and d(t) is the disturbance. Assume the nonlinear function f(x, t) is not exactly known, but

A x , = g(x, 4 + t ) (2)

where g(x,t) is known and Af(x,t) is unknown but bounded by a known function F; that is, IAf(x,t)l 5 I? If the disturbance d(t) has also the upper bound D, the control objective is to determine a feedback control signal U such that the state x of the closed-loop s stem will

presence of model uncertainties and disturbances, i.e. follow the desired state xd = (xd,Xd, . . . ,x:- ( 7 ' ) T in the

171

e = xd - x = 0. Usually the sliding takes the general form as follows:

surface of an SMC

+ il(n-’)e = o (3)

where e =xd - x and il is a positive constant. Starting from the initial condition e(0) = 0, the tracking problem x = xd is equivalent to keeping the system states at the sliding surface s = 0 for all t L 0. However, if the state is outside the sliding surface s=O, to drive the state to the sliding surface we choose the control law such that

SS 5 -qlsl (4) where q is a positive constant, and eqn. 4 is called the sliding condition. The sliding condition guarantees that JsI will decrease if x is not on the sliding surface s = 0, i.e. the state trajectory will move towards the surface. To summar- ise, we have the following lemma:

Lemma I : Consider the nonlinear system (eqn. 1) and let the sliding surface be defined as in eqn. 3. If we design a controller that satisfies the sliding condition (eqn. 4), then

- the system state will reach the sliding surface from any initial place within a finite time - once the state is on the sliding surface it will remain there and the system dynamics are then governed only by that of the sliding surface - in the sliding mode the tracking error e(t) will converge to zero.

Therefore the problem of designing a SMC becomes formulating a control law U such that the sliding condition of eqn. 4 is satisfied. The traditional sliding-mode control law takes the form

U = ueq + B sgn(s) ( 5 )

where ueq is the so-called equivalent control term, and /3 sgn(s) is the switching control term. Since the equivalent control is defined as the average control in the sliding mode, neglecting the system uncertainties and disturbances [2 ] , so let d(t) = 0 and A f(x, t ) = 0, the equivalent control ueq is the control input which makes

S = O (6 ) The switching control is intended to control the unknown parts of the system, where p is a positive constant whose magnitude depends on that of the system uncertainties and disturbances, and

1 i f s > O -1 i f s < O sgn(s) = (7)

For a second-order nonlinear system (n = 2), for instance, the sliding surface is

s = e + ilf? = i d + h d -x- h = 0 (8) which is a straight line in the x - x phase plane. According to eqn. 6, the equivalent control is thus

ueq = -g(x, t ) + i d + 22 (9)

P m v + D + F (10)

In this way, with the SMC as given by eqn. 5, if

then the sliding condition of eqn. 4 is guaranteed. The main disadvantage of the SMC is the drastic change of the control signal across the sliding surface, which leads to

172

4 us

thinnest

Fig. 1 Sliding-mode control with boundavy layer

chattering in the digital implementation. The larger the magnitude of p, the more severe the chattering will be, thus the rule of thumb for SMC design is to assign a relatively small value to /J but still satifying eqn. 10. Then corre- spondingly a narrow boundary layer can be introduced near the sliding surface so as to smooth out the control beha- viour [3], as shown in Fig. 1 The introduction of the boundary layer corresponds to substituting the function sgn(s) in eqn. 5 by a saturation function sat(s/@), so eqn. 5 becomes

U = ueq + p sat@/@) (1 1)

where O is called the thickness of the boundary layer and the saturation function is defined as

Clearly, if the system state is outside the boundary layer, i.e. if ls/@l > 1, the control law (eqn. 11) is equivalent to the control law of eqn. 5 , which guarantees that the sliding condition (eqn. 4) is always satisfied. While inside the boundary layer the control law (eqn. 11) becomes a smooth varying function and the sliding condition may still hold good, depending on the value of p and the system uncer- tainties. Assume that eqn. 4 holds until Is1 2 Omin but does not hold good while Is1 < Omin, then amin is defined as the thickness of the thinnest boundary layer and the absolute value of the corresponding switching control is defined as f l m i n . From Fig. 1, while 0 ? Qmin and f l 2 f l m i n ,

Since the sliding condition is satisfied outside the thinnest boundary layer, then according to lemma 1 the state vector x will enter the thinnest boundary layer within a finite time and will stay inside it afterwards, but there might be a steady-state tracking error appearing whose value is deter- mined by the following lemma [2].

Lemma 2: If the sliding condition (eqn. 4) is satisfied outside the thinnest boundary layer with the thickness amin around sliding surface of eqn. 3, it is guaranteed that the steady-state error is

(14) A

Lemma 2 shows that if we are willing to sacrifice precision, that is, from perfect tracking e(t) = 0 to tracking within precision le(t)l 5 Omin/il” - the requirement for control law is reduced from satisfying the sliding condition (eqn. 4) all the time to satisfying it only when the state vector is outside the boundary layer, thus leaving the space and freedom for smoothing the control within the boundary

IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999

layer and eliminating chattering in the digital implementa- tion.

To improve the system response of the SMC, Wang and Lee [6] enhanced the SMC with an extra term in the form of

U = ueq + as + P sgn(s) (15)

where a is a positive constant and s is the predefined sliding function. If we choose p in eqn. 15 as in eqn. 5 , then by applying eqn. 15, the sliding condition becomes

ss 5 -as2 - YlSJ (16) Again for real-time implementation the switching term f l sgn(s) in eqn. 15 needs to be substituted by a saturation function to attenuate chattering, thus the two-phase sliding- mode controller as given by eqn. 15 is modified as

U = ueq + as + /3 sat@/@) (17)

Lemma 3: The hitting time thl of the SMC by eqn. 5 under the sliding condition

ss = -ylsl (18) is greater than the hitting time th2 of the two-phase SMC by eqn. 15 under the sliding condition

ss = -as2 - ylsl (19) Proofi Let thl be the time required for SMC (eqn. 5 ) to hit the surface s = 0. Integrating eqn. 18 between t = 0 and t = thl leads to

s(t = thl) - s(t = 0 ) = 0 - s(t = 0) = -yl(th, - 0) sgn(s)

(20) where s(t = 0) is the initial system state when t = 0. Thus

The hitting time required for two-phase SMC (eqn. 15) can be obtained by solving the differential eqn. 19

Let s(t = th2) = 0, then th2 can be obtained as

while x > 0, since

e" > 1 +x or x > ln(1 +x) (24) Suppose the initial state of Js(t=O)I and y for eqns. 5 and 15 are the same, then apparently

th2 < Thl

thus lemma 3 holds.

Remark I : The hitting time of the SMC depends on the initial system state and the switching control. As the robustness of the SMC is not guaranteed during the hitting phase and the system response relies on the hitting time, it is expected that the hitting time should be minimised.

Remark 2: As the SMC by eqn. 15 has a shorter hitting time than the one by eqn. 5 according to lemma 3, so if the thickness of the boundary layer is chosen small and equal for the SMCB eqns. 17 and 11, the SMCB by eqn. 17 should also have a shorter hitting time than the one by eqn. 11.

IEE Proc.-Conpol Theory Appl., Vol. 146, No. 2, March 1999

3 Sliding mode controller with unified smooth control law

Based on the idea of the two-phase sliding-mode control- ler, the sliding-mode controller with a unified smooth control law can be obtained with a simple form of

U = Ucq + ys (26)

where u,,~ remains the equivalent control term, s is the predefined sliding function, y is a positive constant. Obviously, the proposed SMCS control law is a smooth and continuous function, and the resulting switching term ys always has the same sign with p sat@/@) in SMCB. Fig. 2 shows the switching control term of SMCS and SMCB for the case y = p/@; the slopes of the two control terms inside the boundary layer will be different while y is chosen either bigger or smaller than fila.

The following theorems state the properties of SMCS in contrast to SMCB. For simplicity, we still consider the system of eqn. 1 with n=2 , but the approach can be generalised to high-order systems.

Theorem 1: Consider the nonlinear system (eqn. 1) with n = 2 and assume that the control U is given by eqn. 26. If

y = !2 = p /@ (27)

where @ is the thickness of the boundary layer and f l is given by eqn. 10, it is guaranteed that the maximum tracking error will be the same as in eqn. 14.

Proofi Substituting eqn. 27 into 26,

U = Ueq + p s p (28)

When the state vector x is outside the thinnest boundary layer, that is, Is1 ?Qrnin, then

Plsl/@ = Pmin ISI /@min B m i n (29)

which means that the control U in eqn. 28 can guarantee the sliding condition (eqn. 4) being satisfied, thus lemmas 1 and 2 hold, and the maximum tracking error will remain the same as in eqn. 14.

Theorem 2: Consider the nonlinear system (eqn. 1) with n = 2 and assume that the control U is given by eqn. 26. If

y > a = j?/@ (30)

where @ is the thickness of the boundary layer and f l is given by eqn. 10, it is guaranteed that the maximum tracking error of SMCS by eqn. 26 is smaller than that of SMCB by eqn. 11.

Proo) Assume Qmin is the thinnest boundary layer thick- ness corresponding to a and Prnin, and @' is the one

Fig. 2 Switching control terms of SMCS and SMCB

173

corresponding to y and Pmin, then according to eqns. 30 and 13,

so

@I < omin (32) As the maximum tracking error is proportional to the corresponding thinnest boundary layer thickness according to lemma 2 , so the maximum tracking error of SMCS by eqn. 26 under the condition of eqn. 30 is smaller than that of SMCB by eqn. 11.

Corollary I : For SMCB by eqn. 11, when the switching gain P and the thickness of the corresponding boundary layer @ are given, the maximum tracking error is fixed by eqn. 14 with

(33)

that is, the tracking error depends on the ratio of @ and P. For the proposed SMCS by eqn. 26, the maximum tracking error le(t)l is also bounded by eqn. 14 but

Pmin omin = - Y (34)

then the target error depends only on y .

Corollary 2: For digital implementation of SMCB by eqn. 1 1, the choice of @ is not arbitrary but depends strongly on the value of P for a certain system and condition to avoid chattering. For a certain P, if the chosen @ is not big enough, chattering still occurs. But if @ is too big, since Qmin is proportional to @, from lemma 2 the maximum steady-state error may surpass a certain range. In contrast, the digital implementation of SMCS by eqn. 26 has no such a problem.

Theorem 3: Consider the nonlinear system (eqn. 1) with n = 2. If y , @ and P are chosen such that y = M + P/@, it is guaranteed that the hitting time required by SMCS of eqn. 26 is shorter than that of SMCB by eqn. 11.

Proof If y = CI + b/@ in eqn. 26, the SMCS by eqn. 26 is functionally equivalent to eqn. 17 in satisfying the sliding condition (eqn. 16). Thus according to lemma 3 and remark 2, suppose the initial state of Is(t=O)I and q for SMCB and SMCS are the same, then the hitting time of SMCS is shorter than that of SMCB. Thus theorem 3 holds.

Corollary 3: Consider the nonlinear system (eqn. 1) with n = 2 and assume that the control U is given by eqn. 26. If y , P, and 0 satisfy the definition and relationship as stated in theorem 2, it is guaranteed that the hitting time of SMCS can be reduced by increasing y. Remark 3: In theorem 3 if y is set to PI@., the SMCS by eqn. 26 still has a shorter hitting time than SMCB by eqn. 1 1. This is because inside the boundary layer the switching control terms of eqns. 26 and 11 are the same, while outside the boundary layer the switching control action of SMCS is stronger than that of the SMCB, thus totally the hitting time of SMCS will always be shorter than that of SMCB. Actually, even in case of y < PI@, there is still a good possibility for SMCS to have a better response; this can also be explained by comparing the two switching terms both inside and outside the boundary layer.

Remark 4: In a digital implementation assume a sudden variation of system parameters or disturbances make the system state x escape from the equilibrium point where

174

Is1 = 0 to a certain point x1 in which JsJ # 0. According to theorem 3 and remark 3, as the hitting time required by SMCS (eqn. 26) is shorter than that of SMCB (eqn. 11) under the condition of eqns. 27 or 30, so the deviated operating point under SMCS control goes back to the equilibrium point quicker than that of SMCB in the presence of system uncertainties and disturbances. Thus it can be concluded that SMCS is more robust than SMCB to system uncertainties and disturbances.

Remark 5: Consider the nonlinear system (eqn. 1) with n = 2 and assume that the control U is given by eqn. 26. If y , P, and @ satisfy the definition and relationship as stated in theorem 2 , it is guaranteed that the robustness of SMCS can be increased by increasing y.

Remark 6: Assume that SMCS is given by eqn. 26, if y is varied under different working conditions, then SMCS becomes adaptive.

4 Speed Control of PMSM drive with SMCS and SMCB

To show the effectiveness of the proposed SMCS and the superiority of SMCS over SMCB as stated in Section 3, both SMCS and SMCB have been implemented for the speed control of a permanent magnet synchronous motor (PMSM) drive system. The adopted nonlinear state equa- tions of the PMSM in the synchronous rotating reference frame can be expressed as follows [7, 81:

de - dt = o r

-?--A do 3P ' - - - - W Tl D - dt 4 J f l q J J did R P dt L, 2

vd - = -- id +-oriq +- LS

where vd, V, are the d- and q-axis voltages, respectively, id and i, are the d- and q-axis currents, respectively, is the flux linkage of the permanent magnet, L, is the stator inductance, R is the stator resistance, J is the moment of inertia, P is the number of poles, D is the viscous friction coefficient, 8 is rotor angle, o, is the rotor speed, and Tl is the load torque. Fig. 3 shows the block diagram of the PMSM drive system. The field-oriented control principle has been applied for the PMSM drive system. The basic idea of the field-oriented control is to force the stator current vector on the d-axis to zero, i.e. idref=O so as to maximise the generated torque. Two sliding-mode control- lers have been formulated, one for the d-axis current control generating the control voltage Vd, the other for the speed control which is responsible for the generation of v,.

Fig. 3 Block diagram of PMSM drive system

IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999

4. I d-axis current sliding-mode control law For ease of implementation, the sliding surface €or the d- axis current control is defined as

(39) s, = idref - id According to the field-oriented control principle idref = 0, so the derivative of eqn. 39 is

did R . P . V, s, = -- = -ld - -orl, - - dt L, 2 L S

(40)

Getting the equivalent control from eqn. 40, the d-axis current sliding-mode control law corresponding to eqn. 26 becomes

P 2 vd = v& + y l s 1 = Rid ---Ur-&iq + (41)

where y1 is a positive constant.

4.2 Speed sliding-mode control law Let wref denote the reference speed and e = uref - or. If a sliding surface is chosen as in eqn. 8 for the speed controller design, then

S2 =e+/Ze (42) Its derivative is

So the speed SMCS corresponding to eqn. 26 is

v, =gJ(g+] +;Afo,+-wrL,id P 2

+ Ri, + y2S2 (44)

where 7 2 is also positive. And the speed SMCB corre- sponding to eqn. 11 is

v, = 4JL, [ ($ - +] +;.,or + P -o,L,i, 3 PAf 2

+ Ri, + p2 sat(S,/@,> (45)

where @ is the thickness of the boundary layer and p2 is a positive constant whose value should be greater than the upper bound of the system uncertainties and disturbances.

4.3 Simulation results To make the comparison sound the d-axis current control- ler (eqn. 41) is used for both SMCS and SMCB speed control schemes where y1 is set to 230 through tuning, and the parameters y2, p2 and @ in the speed controllers are chosen such that eqn. 27 can be satisfied and the two schemes are said to be equivalent. In all the tests, the constant 1 in the sliding surface (eqn. 42) for speed control is set to 5000 and the adopted sampling time is set to 0.2 ms. The applied motor parameters are listed in Table 1.

Table 1: Specification of PMSM

First, the speed tracking response is checked with both schemes under the same condition. Fig. 4 shows the case when there is a speed reference change from 0 to 2000 rev/ min (rated speed) and then from 2000 to 200 revlmin using SMCS and SMCB, the assumed load torque TI is 1 Nm and the moment of inertia J is set to its nominal value, and (y2, /I2, @) are chosen as (4.5 x 450, lo5) according to eqn. 27. In Fig. 4, it can be seen that SMCS has a quicker response than SMCB, which verifies theorem 3. Fixing @ and reducing y2 and p2 by two times, we can get the simulation results under the same working condition as shown in Fig. 5. Comparing Figs. 4 and 5, it is found that the speed response with bigger (y2, p2) pair is quicker than that with smaller (y2 , /I2) pair, but SMCS always has a shorter rise time than SMCB.

To give a full picture of SMCB and SMCS the control input Vq associated with Figs. 4 and 5 are shown in Figs. 6 and 7, respectively, where Vq is limited to 230V for practical consideration. Obviously, in both cases the control action of SMCS during transient period is much stronger than that of SMCB, which explains why SMCS

~~

0 0.01 0.02 0.03 0.04 0.05 006 time t , s

Fig. 4

- _ _ _ SMCS

Speed tracking with SMCS and SMCB - SMCB

0 0.01 0.02 0.03 0.OL 0.05 0.06 time t,s

Fig. 5 ~ SMCB _ _ _ _ SMCS

Speed tracking with SMCS and SMCB with reduced yr and PZ

~

Poles 4 stator resistance 8.1 !2 Stator self inductance 0.03 H moment of inertia 0.093 x 10 - Nms' Damping coefficient 0.095 x 10 ~ Nm.s torque constant 0.89 Nm/A

Rated torque 2.5 Nm rated speed 2000 revlmin

IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999 175

250 r

0 0.01 0.02 0.03 0.OL 0.05 0.06 time t,s

Fig. 6 __ SMCS _ _ _ _ SMCB

Control input Vq ofSMCB and SMCS

2501-

0 0.01 0.02 0.03 0.OL 0.05 0.06 time t,s

Fig. 7 ~ SMCS ___. SMCB

Control input Vq of SMCB and SMCS

has a faster speed response than SMCB in Figs. 4 and 5. Fig. 8 shows the relative magnitude of the two control input components: the equivalent control action U,, and switching control action U,, corresponding to SMCB and SMCS in Fig. 6. The switching control term of SMCS differs much from that of SMCB, thus giving rise to much different equivalent control and total control action.

5000

-2000

-3000

-LOO0

- 50001 I I

0 0.01 0.02 0.03 0 . U 0.05 0.06 time t,s

Fig. 8 (i) switching control of SMCS (ii) switching control of SMCB (iii) equivalent control of SMCB (iv) equivalent control of SMCS

Equivalent control and switching control of SMCB and SMCS

500 r I

300 -

>- 200- E k 100-

3 -100-

&

.- F 0 -

z -200-

c ._

-300-

-LOO - -5001

0 0.005 0.01 0.015 0.02 0.025 time t,s

Fig. 9 Switching control inputs of SMCB with different @ ~ @ = io5 _ _ _ _ @ = io4

In these simulations we have chosen a large @ for SMCB to avoid chattering. But if @ is set small under the same conditions, the generated control input may oscillate. Fig. 9 demonstrates the switching control inputs of SMCB with the thickness of the boundary layer @ = lo5 in contrast to @ = lo4, where the reference speed changes from 0 to 500 rev/min, the load torque is set to 1 Nm and the moment of inertia remains at its nominal value. The corresponding speed responses are shown in Fig. 10. Obviously, the chattering of SMCB is quite serious while @ = lo4, even though the motor can filter the torque ripples to some extent, there are still ripples in the resulting speed response as shown in Fig. 10. For SMCS with a similar response, since the generated control input is always smooth chattering does not appear, which has been verified by simulations.

Simulation tests have also been carried out to show the robustness of SMCS and SMCB to load disturbances. Fig. 11 shows the case when there is a sudden load torque change from 0 to rated torque 2.5 Nm, where the reference speed is set to 2000 rev/min the moment of inertia J is set to its nominal value and ( y 2 , p2, 0) = (0.001, 25, 25000). Apparently, from Fig. 11 after the load disturbance at t = 0.02 s , the speed recovers to its original value faster with SMCS than with SMCB. Tests with smaller ( y 2 , 82) pairs have also been carried out with the two schemes. Fig. 12 shows the simulation results under the same condition as in Fig. 11 but with (72, p2, @)=(O.OOOl, 2 . 5 , 25000). Again SMCS has a shorter recovery time than SMCB after the load change. Actually, it is verified by all the

0 0.005 001 0.015 0.02 0.025 time t , s

Fig. 10 Speed responses of SMCB with different @

- @ = io5 _._. @ = io4

IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999 116

2000-

1980

1960- .- E > 1910- E U- 0) 1920- a ul

1900

1880

1860 I 1 I I I

0 0.01 0.02 0.03 0.OL 0.05

-

-

-

time t ,s

Fig. 11 ~ SMCS ___. SMCB

Robushess of SMCS and SMCB to load disturbance

220

200

2 180- >

160

ILO-

2ooo r 1

-

-

-

1960 .-

220

200

2 180- >

160

1LO

time t,s

Fig. 12 Robustness of SMCS and SMCB to load disturbance

_ _ _ _ SMCB ~ SMCS

-

-

-

-

simulations conducted that SMCS always outperforming SMCB under the condition of eqn. 27 in terms of robust- ness.

Once again, the superiority of SMCS over SMCB in terms of robustness can be explained by showing their

1201 I

0 0.01 0.02 0.03 0.OL 0.05 time t,s

Fig. 13

_ _ _ _ SMCB

Control input Vq of SMCB and SMCS - SMCS

*‘Or

1 2 0 L 0 0.01 0.02 0.03 0.01 0.05

time t,s

Fig. 14

_ _ _ _ SMCB

Control input Vq of SMCB and SMCS ~ SMCS

corresponding control input Vq. Figs. 13 and 14 demon- strate the control input trajectories of SMCB and SMCS associated with Figs. 11 and 12, respectively. The control action of SMCS is stronger than that of SMCB in both cases after the sudden load change occurs at t = 0.02 s, thus giving rise to the shorter recovery time with SMCS in contrast to SMCB as shown in Figs. 11 and 12.

5 Conclusion

A new sliding-mode controller called SMCS with chatter- ing-free and fast-response behaviour has been proposed and implemented. Unlike the traditional SMC or SMCB which obeys the variable-structure control principle by adopting a switching control term, the proposed SMCS uses a continuously varying term instead which takes the distance of the system state from the sliding surface into account. As a result, the proposed SMCS has the advantages of chattering-free behaviour, controllable maximum steady- state error, reduced hitting time, better robustness and ease of adaptability, in contrast to SMCB. Both the simulation results and theoretical analysis verify the superiority of the proposed SMCS over SMCB.

6 References

1

2

HUNG, J.Y., GAO, W., and HUNG, J.C.: ‘Variable structure control: a survey’, IEEE Trans., 1993, IE-40, pp. 2-21 SLOTINE, J.J.E., and LI, W. ‘Applied nonlinear control’ (Prentice-Hall, 19911

3

4

S L O h E , J.J.E.: ‘Sliding controller design for non-linear systems’, Int. 1 Control, 1984, 40, pp. 421434 YEUNG, K.S., and CHEN, Y.P.: ‘A new controller design for manip- ulators using the theory of variable structure systems’,-IEEE Trani.,

5 HWANG, G.C., and LIN, C.S., ‘A stability approach to fuzzy control design for non-linear systems’, Fuzzy Sets Sysf., 1992,48, pp. 279-287

6 WANG, W.J., and LEE, J.L.: ‘Hitting time reduction and chattering attenuation in multi-input variable structure systems’, Control Theory Adv. Technol., 1993, 9, pp. 491499

7 KRAUSE, P.C.: ‘Analysis of electric machinery’ (McGraw-Hill, 1986) 8 FUNG, R.F., LIN, F.J., and CHEN, K.W.: ‘Application oftwo-phase VSC

with integral compensation in speed control of a PM synchronous servomotor’, Int. 1 Sys. Sei., 1996, 21, pp. 1265-1273

1988, AC-33, pp. 200-206

IEE Proc-Control Theory Appl., Vol. 146, No. 2, March 1999 177

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