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Fuzzy Sets and Systems
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A new approach to the design of a fuzzy sliding mode controller
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Graduate School of Engineering Science and Technology (Doctoral Program), National Yunlin University of Science and Technology, Yunlin 640, Taiwan, ROC b Department of Electrical Engineering, National Yunlin University of Science and Technology, No. 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan, ROC
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Received 14 August 2001; received in revised form 31 August 2002; accepted 13 September 2002
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Keywords: Fuzzy control; Sliding mode control; Generalized error; Robot e4ector
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Variable structure controllers with sliding mode control were 1rst proposed in early 1950s [11]. Due to its good robustness to uncertainties, sliding mode control has been accepted as an e;cient method for robust control of uncertain systems. Being limited only by practical constraints on the magnitude of control signals, the sliding mode controller, in principle, can treat a variety of uncertainties as well as bounded external disturbances. A key step in the design of controllers is to introduce a proper transformation of tracking errors to generalized errors so that an nth-order tracking problem can be transformed into an equivalent 1rst-order stabilization problem. Since the equivalent
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1. Introduction
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By de1ning a new generalized error transformation as a complement to the conventional sliding variable, we derived a novel stable sliding mode control scheme on which the design of a new two-input, one-output fuzzy sliding mode controller is based. A signi1cant advantage of this control scheme is its distinct capability of improving the transient behavior of the system during the reaching phase as well as the guaranteed steady-state tracking precision while in the boundary layer. The design procedure was illustrated by a force-adaptive robot e4ector acting on various pro1les of surfaces. As demonstrated from simulations, it appears that a comparatively good performance can be reached by the proposed fuzzy controller. c 2002 Published by Elsevier Science B.V.
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Abstract
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Chi-Ying Lianga , Juhng-Perng Sub;∗
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Corresponding author. Tel.: +886-5-5342601; fax: +886-5-5312065. E-mail address:
[email protected] (J.-P. Su).
c 2002 Published by Elsevier Science B.V. 0165-0114/02/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 4 8 0 - 3
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(n−1)
(FSMC). For a general nth-order nonlinear system, a DFLC uses the state variables: e; e; ˙ : : : ; e as inputs to construct the rule-base, which is impractical for n¿2 due to the curse of dimensionality. In contrast, a FSMC uses only the generalized error s as its input so that the number of fuzzy rules can be remarkably reduced. Nevertheless, FSMCs relate the same value of |s| to the same value of |u|, disregarding the small or big error e. This could be a main drawback of FSMCs. To further improve the performance, some researchers introduced an extra input in addition to the conventional sliding variable s in the construction of fuzzy rules such that a two-input single-output (TISO) FSMC can be obtained. Particularly, Hwang [6] introduced s˙ in the rules together with s to help determine a better control output u. Palm [8] de1ned an additional distance measure so that the region near the origin of the phase plane can be reached by the state vector e at a faster speed than before. This paper presents a new approach to the design of a fuzzy sliding mode controller. By de1ning a new complementary sliding variable to the conventional sliding variable, a novel TISO FLC for general nth-order nonlinear systems will be established. We will provide a new theoretical framework for the construction of a TISO fuzzy sliding mode controller. The proposed control scheme will be shown to result in a closed-loop system satisfying a general sliding condition such that a region containing the hyperplane formed by the intersection of the two surfaces, de1ned by the two sliding variables, is attractive and positively invariant. In addition, we will show that this control scheme will reduce the ultimate bound of tracking error at least by one-half, as compared to the conventional
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1rst-order problem is likely to be simpler to handle, a control law may thus be easily developed to achieve the so-called reaching condition. Unfortunately, an ideal sliding mode controller inevitably has a discontinuous switching function. Due to imperfect switching in practice it will raise the issue of chattering, which is highly undesirable. To suppress chattering, a continuous approximation of the discontinuous sliding control is usually employed in the literature. Although chattering can be made negligible if the width of the boundary layer is chosen large enough, the guaranteed tracking precision will deteriorate if the available control bandwidth is limited. To reach a better compromise between small chattering and good tracking precision in the presence of parameter uncertainties, various compensation strategies have been proposed. For example, integral sliding control [7,2,1], sliding control with time-varying boundary layers [9], etc., were presented. Visible improvement of tracking precision can be observed from simulations; however, these methods seem to su4er from lacking of analytical results justifying a better guaranteed tracking precision given the available control bandwidth and the extent of parameter uncertainty. Besides, the issue of improving the error transient response during the reaching phase appears not to draw much attention in the literature. Recently, some researchers applied the so-called reaching law approach [4,5] to the design of controllers such that the trajectories are forced to approach the sliding surface faster when they are far away from the sliding surface. Although this reaching law increases the reaching speed toward the sliding surface, the behavior of the system dynamics, governed by the transformed 1rstorder equation, can only be measured through the generalized error. Hence, the transient response during the reaching phase may not be remarkably improved. Recently, treating fuzzy logic controllers (FLCs) as variable structure systems, some researchers have provided theoretical frameworks based on the theory of variable structure systems for the analysis and design of fuzzy sliding mode controllers [8,3]. In particular, Palm [8] presented two types of fuzzy controllers: the diagonal form FLC (DFLC) and the fuzzy sliding mode controller
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sliding control. More importantly, it will be shown to signi1cantly improve the tracking performance during the reaching phase as well as in the boundary layer. The basic ideal behind the new sliding control theory in some aspect is very much like the eventual sliding manifold approach, described in [4,5]. To illustrate the e4ectiveness of the design, a robot e4ector control system (Fig. 4) will be used as a simulated example. 2. A new sliding mode control scheme
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Assume that f is not known exactly but can be written as f = fˆ + Of, where fˆ is the nominal part and Of is the uncertain part, which is bounded by a known function F, i.e.,
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In this section, we will present a new sliding mode control scheme to serve as a theoretical basis on which the development of a TISO fuzzy controller can rest. Consider an nth-order SISO nonlinear system described by (n) (n−1) y = f y; y; y ˙ :::; ; t + u: (1)
|Of(x; t)| 6 F(x; t); (n−1)
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˙ : : : ; y ]T . The control object is to 1nd a continuous sliding control where x := [x1 ; x2 ; : : : ; xn ]T = [y; y; u such that the output y of (1) will approximately track a desired signal, yd , which is assumed to be nth-order continuously di4erentiable and all of its derivatives are uniformly bounded. Given the tracking error e(t) := y(t) − yd (t);
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the generalized error n− 1 d + s(t) = e(t); dt
with ¿0, provides a useful transformation such that the original nth-order tracking problem can be transformed into an equivalent 1st-order stabilization problem in s variable. It is easy to show that the control law
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with
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n (k) (n−k) e uˆ = −fˆ + k k=1 s ; vs = −K sat n− 1 (n) xd −
(5)
(6) (7)
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where sat(·) is the saturation function, will result in a closed-loop system satisfying the reaching condition 1 d 2 s 6 −|s| 2 dt
∀|s| ¿
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where = = n−1 . If, instead, the variable of interest involved in transformation (4) is the integral of tracking error, we can obtain the so-called integral sliding mode control. Speci1cally, if transformation (4) is de1ned by n d + ; s= (11) dt t where (t) = e() d, and the control uˆ (6) is modi1ed into n− 1 n + 1 (k) n−k (n) e − n+1 ; (12) uˆ = −fˆ + xd − k + 1 k=0
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n−1 n+1 (k) n−k (n) e − n+1 , and the sliding condition (8) can also be then we have s˙ + s = e + k=0 k+1 satis1ed by controller (5). Next, we will de1ne an additional complementary transformation to the generalized error transformation (11) such that a more meaningful error measure may be obtained. Corresponding to the same parameter ¿0, de1ne n− 1 d d sc = + − : (13) dt dt
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for some ¿0. This controller ensures that starting from any initial state the error trajectories will reach the boundary layer, |s|6, in 1nite time. Whenever |s|6, the guaranteed tracking precision will be [10] (i) e 6 (2 )i ; (10)
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K = F(x; t) +
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(n)
˙ : : : ; ]T We note that if = 1, the subspace de1ned by Sc = { ∈ Rn+1 |sc = dT = 0}, where = [; ; T and d = [d0 ; d1 ; : : : ; dn ] with values of di ’s shown in Table 1, is orthogonal to the sliding surface: S = { ∈ Rn+1 |s = cT = 0}. In other words, the corresponding normal vectors of the two surfaces are perpendicular to each other; that is, dT c = 0. For n = 2, the two perpendicular planes are depicted in Fig. 1. A striking result concerning about the relationship between s and sc is
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s˙c + (s + sc ) = s:˙
(14)
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Table 1 The d-vector ˙
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−1 −1 −1 −1 −1 −1
1 0 −1 −2 −3 −4
1 1 0 −2 −5
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1 2 2 0 .. .
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The control input is also composed of two terms, namely,
with uˆ de1ned in (12) and vc de1ned by s + sc ; vc = −Ksat
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u = uˆ + vc
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Fig. 1. Two perpendicular hyperplanes: s = 0; sc = 0.
where K is chosen as in (9). From Eqs. (1), (14), (15), (12) and (16), we have ss˙ + sc s˙c = (s + sc )(s˙ − sc )
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6 − (s + sc )2 + (Of)(s + sc ) + vc (s + sc )
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6 − (s + sc )2 − |s + sc |
(18)
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for |s + sc |¿. From de1nitions (11) and (13) of s and sc , respectively, one has n− 1 d s + sc = + e: 2 dt There will generate a stable sliding mode on the surface: s + sc = 0.
Remark 1. Let s1 = sc + s=2 and s2 = sc − s=2 . Clearly, s1 = −˙s2 . By noting that s˙sc + sc s˙ =
(sc2 − s2 ) + (s˙s + sc s˙c ), it is easy to establish the following relationships: s1 s˙1 = 12 (s˙s + sc s˙c ) +( =4)(sc2 −s2 ) = 12 (s˙s +sc s˙c )+ 2 s1 s2 and s2 s˙2 = −(1=4 )(sc2 −s2 ) = −s1 s2 . From (18), it follows, for |s + sc |¿,
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s1 s˙1 + 2 s2 s˙2 = 12 (ss˙ + sc s˙c )
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6 −2 s12 − |s1 |:
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s1 s˙1 6 −2 s12 − |s1 | for s1 s2 6 0
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where = = n−1 . If we compare it with the previous results, shown in (10), the guaranteed steadystate tracking precision is reduced by one-half. To illustrate the performance of the complementary sliding control and compare it with those of the other methods, we will consider a simple example from [10], given as follows:
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Therefore, we can conclude from (20)–(22) that any initial states start o4 the boundary layer will move toward the hyperplane, s1 = 0; s2 = 0, and will reach the boundary layer, |s + sc |6, in 1nite time. When s12 + 2 s22 is considered as an error measure, Eq. (20) suggests that the approaching speed of error-state trajectories to the origin will be faster than that of the conventional sliding mode control. Furthermore, whenever in the boundary, |s + sc |6, a similar argument as made in [10] can be used to justify (i) e 6 1 (2 )i ; (23) 2
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s2 s˙2 6 − 12 |s2 | for s1 s2 ¿ 0: 11
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The function a(t) is unknown but veri1es 16a(t)62. Let fˆ = − 1:5y˙ 2 cos 3y and F = 0:5y˙ 2 | cos 3y|. The desired signal is given by yd = sin( t=2). With = 20, de1ne s = (d=dt + )2 = e˙ + 40e + 400 e() d, and sc = (d=dt + )(d=dt − ) = e˙ − 400 e() d. Take = 0:1, = 0:1. The complementary sliding control is u = uˆ + vc , where
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yQ + a(t)y˙ 2 cos 3y = u:
uˆ = 1:5y˙ 2 cos 3y + yQ d − 40e˙ − 400e vc = −20s − (0:5y˙ 2 | cos 3y| + 0:1)sat
s + sc 0:1
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The crucial and the most important step of designing a FLC is the construction of a fuzzy rulebase. Conventionally, the error e and the change of error e˙ are used as input variables in rule-base design of a Mamdani-type FLC. Despite the fact that many rule-bases of successful Mandani-type FLCs are designed heuristically mainly by experts’ experiences, a more formal approach to the construction of rule-bases combines an expert’s experience with a good understanding of systems and control theory. This leads to the so-called template rule-base design. The rule-base suggested by MacVicar–Whelan was often used as a template for initial setting of a rule-base. Speci1cally, the rule-base comprises fuzzy rules forming a partition of the (input) phase plane into two semi-planes separated by a sliding surface. Very similar to sliding mode control with boundary layer, the fuzzy values of the control variable on these two planes are opposite in sign, and the magnitude of a
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3. Fuzzy sliding mode control
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Using a(t) = |sin t| + 1 and zero initial condition in the simulation, we obtained the tracking error shown in Fig. 2 using the complementary sliding control (at a sampling rate of 1 kHz). The tracking errors using, respectively, the conventional sliding control (5)–(7) and the sliding control with time-varying boundary layer proposed in [9] were also depicted in Fig. 2. By comparison, the tracking precision is signi1cantly improved by the complementary control. More strikingly, as indicated in Fig. 3, the error trajectory starting from any nonzero initial state, say, e = 0:2; e˙ = 0, approaches to the boundary layer very quickly when the complementary sliding control was employed.
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Fig. 2. Tracking error responses while in the boundary layer.
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speci1c fuzzy value of the control variable is proportional to the distance between the state vector and the sliding surface. Based on the new sliding mode control scheme described in the previous section, we will present an alternative new design method of a TISO fuzzy sliding mode controller. Taking s and sc as inputs, a rule-base of an FLC according to the MacVicar–Whelan template can be constructed. This TISO FLC will be used to mimic the function of control (16). Therefore, the new control law will be
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Fig. 3. Tracking error responses during the reaching phase.
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IF s = LS i and sc = LSci ;
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Rulei :
THEN u = LU i ;
where LS i , LSci are linguistic values of s and sc , respectively, in the ith-fuzzy rule. By appropriate tuning of parameters of the FLC, it is possible to obtain a high-performance control without further information about the uncertainties of the system. We will show by simulation that the proposed control law is quite e4ective in a robot e4ector control system [8], described in the next section. It is worth noting that this novel stable sliding mode control scheme described in the previous section is derived based on the two sliding variables s, sc providing a theoretical framework for us to construct a stable TISO fuzzy sliding mode controller. By contrast, the TISO FSMC presented in [8] did not give a rigorous proof of the stability of the closed-loop system. Besides, using the distance measure d in addition to the conventional sliding variable the construction of the rule table seems to be tedious and involves a very trial-and-error process. On the other hand, the rule-base of the
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where uˆ is given in (12), and Kf (s; sc ) is the output of the FLC. Each fuzzy rule of a TISO FSMC can be expressed as
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Coordinate base
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Fig. 4. Simpli1ed model of the robot e4ector acting upon a surface.
new TISO fuzzy controller proposed in this paper is a familiar diagonal form, which is extremely easy to construct and 1ne tune. These are main points we have made in this paper.
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4. Numerical example: a robot e"ector system
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Force-adaptive manipulators have found many industrial applications, such as robot-guided grinding of castings, welding tasks, etc. Keeping a constant desired force between the robot e4ector and the surface is one of major tasks of the manipulator, and we will examine this control problem in this section. A robot e4ector system described in [8] was used as an illustrative example. A schematic diagram of the system is shown in Fig. 4. Let the e4ector be modeled as a mass-spring-damper system and the surface as a spring depicted in Fig. 4. The corresponding di4erential equation is [8]
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˜ FF = K1 (y − y0 − D);
F˙ F = K1 (y˙ − y˙ 0 );
FQ F = K1 (yQ − yQ 0 ):
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Let the spring force FF represent the sensor force, which is the actual force between e4ector and surface. Then
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˜ + K2 (y − yc + r + A) = 0: myQ + c(y˙ − y˙ 0 ) − mg + K1 (y − y0 − D)
We obtain the equation for the spring (sensor) force FF : 15 17
c K1 K1 FF + K 1 g − FO − K1 yQ 0 ; FQ F = − F˙ F − m m m where FO = K2 (y − yc + r + A) is the reaction force within the surface. We note that u = −K1 yQ 0
(26)
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is regarded as the input to system (26). We want to design a TISO fuzzy sliding mode controller u for system (26), so that FF can approach a desired constant force Fd as soon as possible. The following model-speci1c parameters: c = 50 kg s−1 ; m = 0:05 kg; K1 = 5000 kg s−2 ;
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FQ F = −1000F˙ F − 100 000FF + 49 000 − 100 000FO + u:
Using the notations given in Section 2, one has (n−1) fˆ y; y; ˙ : : : ; y ; t = −1000F˙ F − 100 000FF + 49 000 − 100 000FO :
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= 1000F˙ F + 100 000FF − 49 000 + 100 000FO − 3 Q − 3 2 ˙ − 3 + Kf (s; sc ); t where (t) = (Fd −FF ) d and Kf (s; sc ) is the output of the TISO FLC designed as follows. Taking s and sc as inputs, we de1ne a set of fuzzy rules
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given in [8], were used in the controller design and simulations. Rewrite Eq. (26) as follows:
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IF s is P and sc is P then K is PB;
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IF s is P and sc is N then K is ZO;
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IF s is Z and sc is P then K is PS;
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IF s is Z and sc is Z then K is ZO; IF s is Z and sc is N then K is NS;
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IF s is N and sc is P then K is ZO; IF s is N and sc is Z then K is NS; IF s is N and sc is N then K is NB;
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where P stands for positive; N stands for negative; PB stands for positive big; PS stands for positive small; both Z and ZO stand for zero; NB stands for negative big, and NS stands for negative small.
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Fig. 5. Membership functions of s and sc . NM
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Fig. 6. Membership functions of the output variable.
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Fig. 7. Scaling factors of the fuzzy controller.
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The inputs s and sc have the same membership functions de1ned on the same normalized universe of discourse, shown in Fig. 5, and the output has the membership functions depicted in Fig. 6. The respective scaling factors for inputs and output are shown in Fig. 7. We note that the rule-base of the new TISO fuzzy controller proposed in this paper is a familiar diagonal form, which is extremely easy to construct and make a 1ne tuning. We take = 5; = 0:01 and = 0:1 in all the simulations. To make a comparison of the proposed complementary fuzzy sliding mode control (CFSM) with the conventional sliding mode control (SMC), where the sliding variable is de1ned via (4), we simulate on the robot e4ector system, using these control laws, respectively. Two kinds of surface along which the robot e4ector is supposed to slide are tested. One of which is a Uat surface, and the other is a surface with three di4erent slopes (positive, zero, negative), as shown in Fig. 8. We considered, respectively, two constant desired
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Fig. 8. Pro1le of a surface having three di4erent slopes.
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Fig. 9. Force responses for a Uat surface when Fd = 10 N.
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time (sec) Fig. 10. Force responses for a Uat surface when Fd = 15 N:
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Fig. 11. Force responses for a nonUat surface when Fd = 10 N.
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time (sec) Fig. 12. Force responses for a nonUat surface when Fd = 15 N.
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e4ector forces: Fd = 10 and 15 N. For a Uat surface, Figs. 9 and 10 indicated the responses of the actual e4ector force FF when Fd = 10 and 15 N, respectively. If the uneven surface (Fig. 8) is considered, the dynamic responses of FF are depicted in Figs. 11 and 12, respectively, for Fd = 10 and 15 N cases. From the simulation results, as indicated in these 1gures, the performance of the proposed complementary fuzzy sliding mode controller is exceptional when compared with that of the conventional sliding mode control. Speci1cally, a better guaranteed steady-state tracking precision as asserted in the text has been justi1ed, as can be observed from the simulations. In addition, these results obtained in this paper seem to be comparable with those obtained by using the fuzzy sliding mode controller given in [8].
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By introducing a new generalized error transformation (sliding variable), we have presented a novel sliding mode control scheme in this paper, which serves as a theoretical framework for the design of a fuzzy controller. With this control scheme, we have shown that the ultimate error bound is reduced at least by one-half, when compared with the conventional continuous sliding control. Speci1cally, this control law seems to result in a signi1cantly faster reaching speed of the tracking error, as can be observed from the simulation results of a simple example given in Section 2, where various available alternative methods were applied to provide a comparative study. Based on this novel sliding mode control scheme, we have successfully developed a new TISO FLC, which has the inherent properties of the analytic control law. The design procedure was illustrated by a forceadaptive robot e4ector acting on various pro1les of surfaces. As demonstrated with the simulations, it appears certain that the proposed fuzzy controller will achieve comparatively good performance.
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References
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