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A self-tuning fuzzy logic design for perturbed time-delay systems with nonlinear input
Expert Systems with Applications 36 (2009) 5304–5309
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A self-tuning fuzzy logic design for perturbed time-delay systems with nonlinear input Fang-Ming Yu * Department of Computer Science and Information Engineering, St. John’s University, Taipei 251, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Fuzzy control Tuning Perturbed time-delay systems
a b s t r a c t In this study, a method of self-tuning single-input fuzzy logic controller (ST-SIFLC) is proposed to efficiently eliminate the chattering problem of perturbed time-delay systems with nonlinear input. This work has the tuning scheme derived from only two rules of fuzzy inference that result in an ultimate equation from defuzzification. This tuning algorithm is convenient and easy to utilize. The control method provides a simple way to achieve asymptotic stability of the perturbed time-delay system. Other attractive features of the method include the fuzzy rules been greatly reduced in spite of the number of state variables, as well as the insensitivity to the perturbation, and the behavior of control input is dramatically improved better than previous report. Simulation results are presented to demonstrate the power of the method. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy control has gained a considerable attraction among a variety of engineering systems over past two decades. The algorithm of fuzzy control consists of a set of heuristic decision rules and can be regarded as a nonmathematical control algorithm, in contrast to a conventional feedback control algorithm. Fuzzy logic controllers (FLC’s) can effectively deal with a plant with complex mathematical models or existing performance limitations for conventional linear control scheme. Most work in fuzzy control field uses the error e and change-of-error e_ as fuzzy input variables. FLC’s are sufficient for simple second order plants since we can assign two closed-loop poles arbitrarily. However, in the case of complex high order plants, all states of a system are required as fuzzy input variables for implement state feedback FLC’s. It needs huge number of control rules and much effort to implement. This fact motivates the design of simpler FLC’s. In recent years, the fuzzy sliding mode control (FSMC) (Glower & Munighan, 1997; Kim & Lee, 1995) has been widely used for reducing the number of rules in the rule base. Several authors have suggested using the composite state to obtain a fuzzy sliding mode controller. The advantage of such controllers is that the number of rules is reduced from mn to m2 in Hwang and Lin (1992), Hwang and Tomizuka (1994). Obviously, the FSMC is one of the reducible fuzzy rules methods. In this paper, we propose a design method with single-input fuzzy logic controller (SIFLC) and self-tuning scheme, which satisfies the reaching condition ss_ < 0, and only * Tel.: +886 921 821129. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.06.129
use the variable Ds, a signed distance to the sliding surface, as a fuzzy input variable. Thereby, the total number of rules of the system could be greatly reduced. Simultaneously, in light of the Lyapunov stability theorem, the system can thus be ensured to be asymptotically stable. The rule table is then constructed in a one-dimensional space and will be quickly organized and easily calculated. Furthermore, in order to get a better response of the system than that of pervious work (Hsu, 1998; Yu, Chung, & Chen, 2003), a new kind of self-turning algorithm for improving system performance by adjusting the scaling factor is proposed. The input scaling factor in a fuzzy control system is commonly used to conduct proper transformations between the real input data and pre-specified universe of discourses of the fuzzy input variables in the system. Theoretically, the scaling factor is a constant parameter. Nevertheless, it is also applied to a fine-tuning performance of the system in a similar way to the tuning of PID controller. Unlike the fuzzy tuner (Woo, Chung, & Lin, 2000), which works by tuning of error with respect to the derivation of gains, we derived a tuning scheme with only two rules of fuzzy inference and an ultimate equation from the result of defuzzification, this tuning algorithm is simple and convenient to use. Moreover, most work of robust control has concentrated on the systems with uncertainties and/or external disturbances. The assumption of linear input is that the system model is indeed linearizable. However, in practice, due to physical limitation, there do exist nonlinearities in the control input and their effect can not be ignored in realization. In addition, to obtain a more practical system model, the information of delay time should not be discarded. Thereby, it is necessary to develop a new robust control method to
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deal with the perturbed time-delay systems with input nonlinearity. In this study, an adaptive fuzzy logic controller is derived through variable structure control (VSC) (Hsu, 1998). Although the method of variable structure control can solve the above-mentioned system, the drawback of the variable structure control is their chattering owing to the sliding control law that has to be discontinuous across the sliding surface. Moreover, if it is filtered at the output of the process, it may excite unmodeled high frequency modes, and may degrade the performance of the system and may even lead to instability. Herein, the fuzzy control principle is used to overcome this drawback. Mostly, the FSMC method is applied in the nonlinear system such as the inverted pendulum system (Hwang & Lin, 1992) and the tracking control of a nonlinear system (Lin & Chen, 2002). The FSMC method can be used to control the above-mentioned systems (the inverted pendulum system, tracking control system) very well, but they are SISO systems with a linear input, not a nonlinear input which cannot be ignored in reality. Therefore, in this work, we will adopt the FSMC together with selftuning and single-input fuzzy logic scheme to design a robust controller for the perturbed time-delay system with a nonlinear input. This control method provides a simple way to achieve asymptotic stability of the perturbed time-delay system. Meanwhile, by using the self-tuning scheme, the behavior of control input can be dramatically improved better than previous report. The rest of this paper is divided into five sections. In Section 2, some properties of the perturbed nonlinear time-delay system with nonlinearities are reviewed. In Section 3, a robust variable structure controller is derived from those properties. In Section 4, a self-tuning single-input fuzzy logic controller (ST-SIFLC) is developed and presented. In Section 5, a simulation example is illustrated to demonstrate the features of the ST-SIFLC. Finally, we conclude with Section 6. 2. System description
xðtÞ ¼ hðtÞ;
sd 6 t < 0;
Ad ¼ bg
kgk 6 b2
ð3Þ
f ¼ bd kdk 6 b3 where h, g, and d have appropriate dimensions. The above matching conditions (3) are for simplicity of derivation. Assumption 3. The nonlinear input U(u) applied to the system satisfies the following property:
u UðuÞ P a u2
ð4Þ
where a > 0 and U(0) = 0. Remark 1. In general, if the series nonlinearity is inside sector [a1 a2], i.e.,
a2 u UðuÞ P u UðuÞ P a1 u2 ; where U(u) denotes a sector bounded function, a2 is often called the gain margin, and a1 is called the gain reduction tolerance. So, we have a2 ? 1 and a1 = a in Assumption 3. It is seen that if u increases, then U(u) increases and vice versa (Hsu, 1998). 3. Variable structure control design (Hsu, 1998) First, the following lemmas will be employed to derive the variable structure controller. Lemma 1 (Hsu, 1998). If the nonlinear input satisfies the property as indicated in Eq. (4), there exists a continuous function /():R+ ? R+, /(0) = 0, and /(p) > 0 for p > 0. Therefore, if ju(t)j = /(q), then
a u2 P q /ðqÞ; 8q P 0
ð5Þ
For the perturbed time-delay systems with nonlinear input (1), the switching surface is defined as
sðtÞ ¼ c T xðtÞ ¼ 0
A general description of perturbed time-delay dynamical systems with nonlinear input (Hsu, 1998) is given in the form of
_ xðtÞ ¼ ðA þ DAÞxðtÞ þ bUðuÞ þ Ad xðt sd Þ þ f ðtÞ
DA ¼ bh khk 6 b1
ð1Þ
where c 2 R is a constant vector. Lemma 2 (Hsu, 1998). The motion of the sliding mode (7) is asymptotically stable, if the following condition holds:
ss_ < 0;
where x(t), b, and f(t) 2 Rn is the state variables, the input vector, and the disturbance vector, respectively. u 2 R is the control input of the system, A 2 Rn n is the state matrix, Ad is the delay term matrix including the uncertainty, and U(u):R ? R is a continuous function of nonlinear input and U(0) = 0. DA is the bounded uncertainty matrix of A, sd represents a nonzero time-delay, and h(t) is a continuous vector-valued initial function. For dealing with the study of system (1), the following assumptions are taken.
ð6Þ
n
8t > 0
ð7Þ
To fulfill the condition stated in Eq. (7), the desired variable structure control is suggested by
uðtÞ ¼
sc T b /ðx; tÞ ksc T bk
ð8Þ
where
/ðx; tÞ ¼
r
a
f½kðc T bÞ1 c T Ak þ b1 kxk þ b2 kxd k þ b3 g;
r>1
ð9Þ
Assumption 1. For the nominal part of the perturbed time-delay system with nonlinearities indicated in Eq. (1), matrix pair (A, b) is completely controllable (Hsu, 1998).
The following theorem shows that the proposed variable structure control in Eq. (8) drives the perturbed system with nonlinear input into the sliding mode s(t) = 0.
Assumption 2. The matrix Ad and perturbed matrices DA, f satisfy the following rank conditions (Luo & De La Sen, 1993).
Theorem 1 (Hsu, 1998). Consider the perturbed system (1) subjected to Assumptions 1 and 2, and inequality of (3). If the variable structure control input u(t) in Eq. (1) is given as that indicated by (8), then the system trajectories asymptotically converge to the sliding mode (6).
. rank½DA ¼ rank½DA..b . rank½DAd ¼ rank½DAd ..b . rank½f ¼ rank½f ..b
ð2Þ
Based on Assumption 2, there exist vectors h, g, and scalar d such that the following matching conditions hold (Shyu & Yan, 1994).
4. Self-tuning single-input fuzzy logic controller (ST-SIFLC) In this section, a self-tuning single-input fuzzy logic controller (ST-SIFLC) is proposed and expressed as following.
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By taking the time derivative of both sides of (6), we obtain
4.1. Single-input fuzzy logic controller (SIFLC) In this section, the single-fuzzy input controller for the perturbed time-delay system with nonlinear input (1) is proposed. Also, the idea of Choi, Kwak, and Kim (1999) named the signed distance is used and the feasibility of the present approach will be demonstrated. The switching line for a second order system (1) is defined by:
s : x_ þ c1 x ¼ 0
ð10Þ
First, we introduce a new variable called the signed distance. Let _ be the intersection point of the switching line and the line Aðx; xÞ perpendicular to the switching line from an operating point Bðx1 ; x_ 1 Þ, as illustrated in Fig. 1. Then, d1, the distance between _ and Bðx1 ; x_ 1 Þ can be expressed by the following equation: Aðx; xÞ
jx_ 1 þ c1 x1 j d1 ¼ ½ðx x1 Þ2 þ ðx_ x_ 1 Þ2 1=2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c21
s_ ¼ cT x_ ¼ c T Ax þ c T bUðuÞ þ c T bðhx þ gxd þ dÞ
Then, multiplying both sides of the above equation by s leads to
ss_ ¼ sc T Ax þ sc T bUðuÞ þ sc T bðhx þ gxd þ dÞ
Here, we consider that c b > 0 for any x. In (17), it is seen that s_ increases as U(u) increases and vice versa. In (18), it is seen that if s > 0, then decreasing U(u) assures s_ to be negative making ss_ negative, and that if s < 0, then increasing U(u) assures s_ to be positive making ss_ negative. From Assumption 3, we know that U(u) P au for u > 0 and U(u) 6 au for u < 0, which implies that if u is positive then U(u) is positive. So, it is seen that if s > 0, then decreasing u assures s_ to be negative making ss_ negative, and that if s < 0, then increasing u assures s_ to be positive making ss_ negative. Now, we choose a Lyapunov-like function
V¼
1 2 D 2 s
ð19Þ
Without loss of generality, Eq. (11) can be rewritten as follows:
Then
jx_ þ c1 xj d ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c21
V_ ¼ Ds D_ s ¼
The signed distance ds is defined for an generally point Bðx1 ; x_ 1 Þ as follows:
x_ þ c1 x jx_ þ c1 xj s ds ¼ sgnðsÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ c1 1 þ c1 1 þ c21 where
sgnðsÞ ¼
ð13Þ
ss_ 1 þ c2n1 þ þ c22 þ c21
ð20Þ
Hence, it is seen that if s > 0, then Ds > 0, decreasing u assures s_ to be negative which makes ss_ negative so that V_ < 0 and that if s < 0, then Ds < 0, increasing u assures s_ to be positive making ss_ negative so that V_ < 0. So we can ensure that the system is asymptotically stable. From the above relation, we conclude that
u / Ds 1
for s > 0
ð14Þ
1 for s < 0
Similarly, For the nth order system (1), it is different from that of the switching hyperplane, i.e., Eq. (6) can be represented as
s ¼ xðn1Þ þ cn1 xn2 þ þ c2 x_ þ c1 x ¼ 0:
ð15Þ
Hence, the general signed distance Ds is changed to be
Ds ¼
ð18Þ
T
ð11Þ
ð12Þ
ð17Þ
xðn1Þ þ cn1 xn2 þ þ c2 x_ þ c1 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c2n1 þ þ c22 þ c21
s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c2n1 þ þ c22 þ c21
ð16Þ
. x
ð21Þ
Hence, the fuzzy rule table can be established on a one-dimensional space of variable Ds as shown in Table 1 instead of a two-dimensional space of x and x_ or a n-dimensional space of n variables. The control action can be determined by variable Ds only. Hence, we can easily add or modify rules for fine control. For implementation, a triangular type membership function is chosen for the aforementioned fuzzy variables, as shown in Fig. 2 Remark 2. The rule table can be established on a one-dimensional space like Table 1. It is seen that the total number of rules is greatly reduced compared with conventional FLCs. In other words, a traditional fuzzy system has two states to be controlled and the range of each state variable is divided into seven fuzzy sets. The number of rules forming the knowledge database is 72 = 49. With the present method, only one variable Ds needs to be fuzzified and only seven rules are necessary to establish the knowledge database.
Table 1 Rule table for SIFLC Ds u
NB PB
NM PM
NS PS
ZE ZE
PS NS
x
.
d
B ( x 1, x 1 )
. A (x , x ) . x + c1 x = 0 Fig. 1. Derivation of a signed distance.
Fig. 2. Fuzzy variable of triangular type.
PM NM
PB NB
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Ds (t )
~ Ds
K1
Fuzzy Control Rules
5. Simulation
U
K2
u(t) Consider a perturbed time-delay system with series nonlinearity, which is described by Eq. (1) (Hsu, 1998). It is shown below that
Fig. 3. The block diagram of the SIFLC.
_ xðtÞ ¼ ðA þ DAÞxðtÞ þ bUðuÞ þ Ad xðt sd Þ þ f ðtÞ ¼ AxðtÞ þ bUðuÞ þ b½hxðtÞ þ gxd þ d
4.2. Self-tuning scheme and scaling factor In the light of the self-tuning single-input fuzzy logic controller (ST-SIFLC) design for perturbed time-delay systems with nonlinear input, the single-input fuzzy logic controller (SIFLC), determined by signed distance, is shown in Fig. 3 where Ds and u are the input and output of the single-input fuzzy logic controller, respectively. e s , which is a fuzzThe input of the proposed fuzzy controller is D ified variable corresponding to Ds. The output of the fuzzy controller is U, which is the fuzzified variable corresponding to u. All the e s and U range from 1 to 1. Thus, the universes of discourse of D range of nonfuzzy variables Ds and u must be scaled to fit the unie s and U with scaling facverse of discourse of a fuzzified variable D tors K1 and K2, respectively, namely,
e s ¼ K 1 Ds ðtÞ D
ð22Þ
uðtÞ ¼ K 2 U
ð23Þ
From the consequence given above, we have proven that the SIFLC is ensured to be asymptotically stable. In this subsection we will introduce the self-tuning scheme because the fuzzy system performance is sensitive to the scaling factors K1. If large values of K1 are available the system steady tracking error will be reduced but the overshoot may increase because a large K1 provides a large control gain. Conversely, if small values of K1 are chosen, the tracking accuracy will be degraded, leading to a small control gain. So we propose to decrease K1 when the error is large, and vice versa. The fuzzy rule for tuning K1 can be formalized as If error is large (EL) then scaling factor is small (CS), If error is small (ES) then scaling factor is large (CL).
lCL ¼
1; K 1 ¼ K 1;max 0; K 1 –K 1;max
;
lCS ¼
A¼
ð24Þ
K 1 –K 1;min
K 1 ¼ K 1;min þ DK 1 = coshðDs Þ
K1
b¼
0 ; 1
x¼
x1
x2
;
xd ¼
x1 ðt sd Þ
x2 ðt sd Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 þ l2 cos x2d x21d þ x22d þ l3 cos x2
For all VSC, FSMC, and ST-SIFLC controllers, the switching surface is taken as
sðtÞ ¼ 2x1 þ x2 Obviously, cTb > 0. This meets the condition cTb > 0 in (18). In addition, the following initial values are arbitrarily chosen, that is,
x1 ð0Þ ¼ 1;
x2 ð0Þ ¼ 1
For the purpose of simulation, the following numerical value are employed:
d ¼ 1:0;
l1 ¼ 0:04;
l2 ¼ 0:3;
c ¼ 0:3
l3 ¼ 0:6;
5.1. VSC method
The FSMC method (Yu et al., 2003), designed to cope with the perturbed time-delay system with a nonlinear input, has been pre-
S
s (t )
K1
ð25Þ s& (t )
Remark 3. Intuitively, we may combine the SIFLC and self-tuning scheme to get better results in terms of system performance. It is demonstrated in Fig. 4 and named self-tuning single-input fuzzy logic controller (ST-SIFLC), Moreover, we can ensure that the system is asymptotically stable, creating the rule table from Eq. (21).
Fuzzy Control Rules
2 3
;
hx þ gxd þ d ¼ l1 eð1þsin x1 Þ
where DK1 = K1,max K1,min.
~ Ds
1
UðuÞ ¼ ðdebsin uc þ c cos uÞu; d > c > 0
and using the max-min inference for defuzzification the value of K1 can be found as
Ds (t )
0
5.2. FSMC method
1; K 1 ¼ K 1;min 0;
where xd = x(t sd), khk 6 b1, kgk 6 b2 and kdk 6 b3. From Eq. (3), we also obtain DA = bh, Ad = bg, and f = bd. The corresponding parameters of the illustrated system are given as follows:
For the computer simulation, the corresponding parameters of VSC method are r = 1.1, a = 0.7, b1 = 0.3, b2 = 0.3, b3 = 0.3, respectively. The delay time constants are taken as sd = 0.2.
By defining the following membership function:
lES ¼ 2=½expðDs Þ þ expðDs Þ ¼ 1= coshðDs Þ lEL ¼ 1 lES
ð26Þ
U
Self-tuning Fig. 4. The block diagram of the ST-SIFLC.
K2
u(t)
S&
K2
Fuzzy Control Rules
U
u(t) K3
Fig. 5. The block diagram of the FSMC (Yu et al., 2003).
Table 2 Rule table for FSMC (Yu et al., 2003) s=s_
NB
NM
NS
ZE
PS
PM
PB
PB PM PS ZE NS NM NB
ZE PS PM PB PB PB PB
NS ZE PS PM PB PB PB
NM NS ZE PS PM PB PB
NB NM NS ZE PS PM PB
NB NB NM NS ZE PS PM
NB NB NB NM NS ZE PS
NB NB NB NB NM NS ZE
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F.-M. Yu / Expert Systems with Applications 36 (2009) 5304–5309
sented previously. Fig. 5 shows the block diagram of the FSMC (Yu et al., 2003), and the two-dimension rules table are chosen in Table
2. Accordingly, the scaling factors are chosen as K1 = 0.5, K2 = 0.5 and K3 = 50. 5.3. ST-SIFLC method For the proposed adaptive fuzzy logic controller, ST-SIFLC method, the rule base is a one-dimension table only as the SIFLC in Table 1. The scaling factor K1 is variable due to the self-tuning scheme. Furthermore, K1,min = 0.1,K1,max = 1.1 and the other scaling factor is chosen as K2 = 25.
30
a
25
control force U
20
Fig. 6. State variable dynamics of x1: (a) VSC, (b) the result of Yu et al. (2003) and (c) the proposed method of ST-SIFLC.
15 10 5 0 -5 -10 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.35
0.4
0.45
0.5
0.35
0.4
0.45
0.5
time(sec) 30
b
25
control force U
20 15 10 5 0 -5 Fig. 7. State variable dynamics of x2: (a) VSC, (b) the result of Yu et al. (2003) and (c) The proposed method of ST-SIFLC.
-10 0
0.05
0.1
0.15
0.2
0.25
0.3
time(sec)
c
30 25
control force U
20 15 10 5 0 -5 -10
0
0.05
0.1
0.15
0.2
0.25
0.3
time(sec)
Fig. 8. The phase plane between x1 and x2.
Fig. 9. The control force u: (a) VSC, (b) FSMC (Yu et al., 2003) and (c) the proposed method of ST-SIFLC.
F.-M. Yu / Expert Systems with Applications 36 (2009) 5304–5309
The computer simulation sampling step for the perturbed timedelay system with nonlinearities in above three methods are 0.01 s. The responses of state variable x1 for the above three methods are shown in Fig. 6. The ST-SIFLC, as a simpler algorithm than FSMC and VSC, still own the desirable response of x1. As to the state variable dynamics of x2 for the system, which are shown in Fig. 7. We can note that the angular velocity of ST-SIFLC is greatly reduced, as compared to others. Fig. 8 shows the phase plane between x1 and x2. From Fig. 8, it is observed that the sliding surface of ST-SIFLC is modified successively. And the control input u is shown in Fig. 9. It is clear that the control signal of ST-SIFLC is dramatically reduced and all of the chattering has been eliminated. This demonstration shows that the proposed technique serves as a perfect control input behavior that is with high efficiency. Remark 4. In sliding mode control, state trajectories with chattering along a sliding hyperplane are inevitable in practice, because the control input is discontinuous in the vicinity of the sliding hyperplane. Chattering is a serious drawback of sliding mode control and has been an important problem in the field of sliding mode control. Here, we want to demonstrate that the STSIFLC can handle this chattering problem effectively by adjusting the control input near the sliding hyperplane. Simultaneously, the ST-SIFLC with the self-tuning scaling factor to enhance the efficiency for the control input. The simulation results in Fig. 8 show the state trajectory of system response for a perturbed time-delay system with nonlinearities. We can observe that the slope of the sliding surface in ST-SIFLC is modified successively. This is because the effect of tuning scaling factor is similar to tuning the slope of sliding surface. ST-SIFLC has the most compact control signal without redundant force, and eliminates chattering effectively. Remark 5. Because the switching function is defined only by the state variable of x(t), that is, s(t) = cTx(t) = 0, free of the time-delay sd, sd affect the system stability very little. From the simulation result, it is found that the robustness of system is little susceptible to time-delay sd. 6. Conclusions We have shown that the ST-SIFLC developed in this work can be easily applied to perturbed time-delay dynamical systems with nonlinear input as described in Eq. (1). The scheme of this work is to design a single-input fuzzy logic controller with self-tuning scheme to stabilize perturbed time-delay systems with nonlinear input. Several important features of the proposed technique are summarized in the following:
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(1) Without exact mathematical model of the nonlinear systems for the controller design. The only information needed to design the controller is the influences of u with respect to Ds. (2) An inherent chattering problem of sliding mode control can be eliminated effectively without involving sophisticated mathematics. (3) By the Lyapunov stability theorem and the reaching condition of ðV_ ¼ ss_ Þ < 0, we only need to adjust the control input u with relation to Ds to satisfy ss_ < 0, the construction of rule base is simple and clear. Also, the asymptotic system stability for perturbed time-delay systems with nonlinear input can be guaranteed. (4) The minimum numbers of fuzzy rules can be obtained. Because in each fuzzy implication only Ds is taken as input of the fuzzy logic controller in spite of the number of state variables in the system. The ST-SIFLC can be created quickly and the control law can be computed easily. (5) This new kind of self-tuning scheme in fuzzy logic controller, which improves the performance of chattering elimination and increases the efficiency of the control input by adjusting the scaling factor. (6) From the simulation result, it is shown that the ST-SIFLC indeed can be applied to the perturbed time-delay system with a nonlinear input. The robustness for the uncertainty and perturbation is very satisfactory.
References Choi, B. J., Kwak, S. W., & Kim, B. K. (1999). Design of a single-input fuzzy logic controller and its properties. Fuzzy Sets and Systems, 106, 299–308. Glower, J. S., & Munighan, J. (1997). Design fuzzy controllers from a variable structures standpoint. IEEE Transactions on Fuzzy Systems, 5(1), 138–144. Hsu, K. C. (1998). Adaptive variable structure control design for uncertain timedelayed systems with nonlinear input. Dynamics and Control, 8, 341–354. Hwang, G. C., & Lin, S. C. (1992). A stability approach to fuzzy control design for nonlinear systems. Fuzzy Sets and Systems, 48, 279–287. Hwang, Y. R., & Tomizuka, M. (1994). Fuzzy smoothing algorithms for variable structure systems. IEEE Transactions on Fuzzy Systems, 2, 277–284. Kim, S. W., & Lee, J. J. (1995). Design of a fuzzy controller with fuzzy sliding surface. Fuzzy Sets and Systems, 71, 359–367. Lin, W. S., & Chen, C. S. (2002). Robust adaptive sliding mode control using fuzzy modeling for a class of uncertain MIMO nonlinear systems. Control Theory and Applications, IEE Proceedings Part D, 149, 193–201. Luo, N., & De La Sen, M. (1993). State feedback sliding mode control of a class of uncertain time delay systems. IEE Proceeding Part D, 140, 261–274. Shyu, K., & Yan, J. (1994). Variable-structure model following adaptive control for systems with time-varying delay. Control Theory and Advanced Technology, 10, 513–521. Woo, Z. W., Chung, H. Y., & Lin, J. J. (2000). A PID type fuzzy controller with selftuning scaling factors. Fuzzy Sets and Systems, 115, 321–326. Yu, F. M., Chung, H. Y., & Chen, S. Y. (2003). Fuzzy sliding mode controller design for uncertain time-delayed system with nonlinear input. Fuzzy Sets and Systems, 140, 359–374.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A self-tuning fuzzy logic design for perturbed time-delay systems with nonlinear input Fang-Ming Yu * Department of Computer Science and Information Engineering, St. John’s University, Taipei 251, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Fuzzy control Tuning Perturbed time-delay systems
a b s t r a c t In this study, a method of self-tuning single-input fuzzy logic controller (ST-SIFLC) is proposed to efficiently eliminate the chattering problem of perturbed time-delay systems with nonlinear input. This work has the tuning scheme derived from only two rules of fuzzy inference that result in an ultimate equation from defuzzification. This tuning algorithm is convenient and easy to utilize. The control method provides a simple way to achieve asymptotic stability of the perturbed time-delay system. Other attractive features of the method include the fuzzy rules been greatly reduced in spite of the number of state variables, as well as the insensitivity to the perturbation, and the behavior of control input is dramatically improved better than previous report. Simulation results are presented to demonstrate the power of the method. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy control has gained a considerable attraction among a variety of engineering systems over past two decades. The algorithm of fuzzy control consists of a set of heuristic decision rules and can be regarded as a nonmathematical control algorithm, in contrast to a conventional feedback control algorithm. Fuzzy logic controllers (FLC’s) can effectively deal with a plant with complex mathematical models or existing performance limitations for conventional linear control scheme. Most work in fuzzy control field uses the error e and change-of-error e_ as fuzzy input variables. FLC’s are sufficient for simple second order plants since we can assign two closed-loop poles arbitrarily. However, in the case of complex high order plants, all states of a system are required as fuzzy input variables for implement state feedback FLC’s. It needs huge number of control rules and much effort to implement. This fact motivates the design of simpler FLC’s. In recent years, the fuzzy sliding mode control (FSMC) (Glower & Munighan, 1997; Kim & Lee, 1995) has been widely used for reducing the number of rules in the rule base. Several authors have suggested using the composite state to obtain a fuzzy sliding mode controller. The advantage of such controllers is that the number of rules is reduced from mn to m2 in Hwang and Lin (1992), Hwang and Tomizuka (1994). Obviously, the FSMC is one of the reducible fuzzy rules methods. In this paper, we propose a design method with single-input fuzzy logic controller (SIFLC) and self-tuning scheme, which satisfies the reaching condition ss_ < 0, and only * Tel.: +886 921 821129. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.06.129
use the variable Ds, a signed distance to the sliding surface, as a fuzzy input variable. Thereby, the total number of rules of the system could be greatly reduced. Simultaneously, in light of the Lyapunov stability theorem, the system can thus be ensured to be asymptotically stable. The rule table is then constructed in a one-dimensional space and will be quickly organized and easily calculated. Furthermore, in order to get a better response of the system than that of pervious work (Hsu, 1998; Yu, Chung, & Chen, 2003), a new kind of self-turning algorithm for improving system performance by adjusting the scaling factor is proposed. The input scaling factor in a fuzzy control system is commonly used to conduct proper transformations between the real input data and pre-specified universe of discourses of the fuzzy input variables in the system. Theoretically, the scaling factor is a constant parameter. Nevertheless, it is also applied to a fine-tuning performance of the system in a similar way to the tuning of PID controller. Unlike the fuzzy tuner (Woo, Chung, & Lin, 2000), which works by tuning of error with respect to the derivation of gains, we derived a tuning scheme with only two rules of fuzzy inference and an ultimate equation from the result of defuzzification, this tuning algorithm is simple and convenient to use. Moreover, most work of robust control has concentrated on the systems with uncertainties and/or external disturbances. The assumption of linear input is that the system model is indeed linearizable. However, in practice, due to physical limitation, there do exist nonlinearities in the control input and their effect can not be ignored in realization. In addition, to obtain a more practical system model, the information of delay time should not be discarded. Thereby, it is necessary to develop a new robust control method to
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deal with the perturbed time-delay systems with input nonlinearity. In this study, an adaptive fuzzy logic controller is derived through variable structure control (VSC) (Hsu, 1998). Although the method of variable structure control can solve the above-mentioned system, the drawback of the variable structure control is their chattering owing to the sliding control law that has to be discontinuous across the sliding surface. Moreover, if it is filtered at the output of the process, it may excite unmodeled high frequency modes, and may degrade the performance of the system and may even lead to instability. Herein, the fuzzy control principle is used to overcome this drawback. Mostly, the FSMC method is applied in the nonlinear system such as the inverted pendulum system (Hwang & Lin, 1992) and the tracking control of a nonlinear system (Lin & Chen, 2002). The FSMC method can be used to control the above-mentioned systems (the inverted pendulum system, tracking control system) very well, but they are SISO systems with a linear input, not a nonlinear input which cannot be ignored in reality. Therefore, in this work, we will adopt the FSMC together with selftuning and single-input fuzzy logic scheme to design a robust controller for the perturbed time-delay system with a nonlinear input. This control method provides a simple way to achieve asymptotic stability of the perturbed time-delay system. Meanwhile, by using the self-tuning scheme, the behavior of control input can be dramatically improved better than previous report. The rest of this paper is divided into five sections. In Section 2, some properties of the perturbed nonlinear time-delay system with nonlinearities are reviewed. In Section 3, a robust variable structure controller is derived from those properties. In Section 4, a self-tuning single-input fuzzy logic controller (ST-SIFLC) is developed and presented. In Section 5, a simulation example is illustrated to demonstrate the features of the ST-SIFLC. Finally, we conclude with Section 6. 2. System description
xðtÞ ¼ hðtÞ;
sd 6 t < 0;
Ad ¼ bg
kgk 6 b2
ð3Þ
f ¼ bd kdk 6 b3 where h, g, and d have appropriate dimensions. The above matching conditions (3) are for simplicity of derivation. Assumption 3. The nonlinear input U(u) applied to the system satisfies the following property:
u UðuÞ P a u2
ð4Þ
where a > 0 and U(0) = 0. Remark 1. In general, if the series nonlinearity is inside sector [a1 a2], i.e.,
a2 u UðuÞ P u UðuÞ P a1 u2 ; where U(u) denotes a sector bounded function, a2 is often called the gain margin, and a1 is called the gain reduction tolerance. So, we have a2 ? 1 and a1 = a in Assumption 3. It is seen that if u increases, then U(u) increases and vice versa (Hsu, 1998). 3. Variable structure control design (Hsu, 1998) First, the following lemmas will be employed to derive the variable structure controller. Lemma 1 (Hsu, 1998). If the nonlinear input satisfies the property as indicated in Eq. (4), there exists a continuous function /():R+ ? R+, /(0) = 0, and /(p) > 0 for p > 0. Therefore, if ju(t)j = /(q), then
a u2 P q /ðqÞ; 8q P 0
ð5Þ
For the perturbed time-delay systems with nonlinear input (1), the switching surface is defined as
sðtÞ ¼ c T xðtÞ ¼ 0
A general description of perturbed time-delay dynamical systems with nonlinear input (Hsu, 1998) is given in the form of
_ xðtÞ ¼ ðA þ DAÞxðtÞ þ bUðuÞ þ Ad xðt sd Þ þ f ðtÞ
DA ¼ bh khk 6 b1
ð1Þ
where c 2 R is a constant vector. Lemma 2 (Hsu, 1998). The motion of the sliding mode (7) is asymptotically stable, if the following condition holds:
ss_ < 0;
where x(t), b, and f(t) 2 Rn is the state variables, the input vector, and the disturbance vector, respectively. u 2 R is the control input of the system, A 2 Rn n is the state matrix, Ad is the delay term matrix including the uncertainty, and U(u):R ? R is a continuous function of nonlinear input and U(0) = 0. DA is the bounded uncertainty matrix of A, sd represents a nonzero time-delay, and h(t) is a continuous vector-valued initial function. For dealing with the study of system (1), the following assumptions are taken.
ð6Þ
n
8t > 0
ð7Þ
To fulfill the condition stated in Eq. (7), the desired variable structure control is suggested by
uðtÞ ¼
sc T b /ðx; tÞ ksc T bk
ð8Þ
where
/ðx; tÞ ¼
r
a
f½kðc T bÞ1 c T Ak þ b1 kxk þ b2 kxd k þ b3 g;
r>1
ð9Þ
Assumption 1. For the nominal part of the perturbed time-delay system with nonlinearities indicated in Eq. (1), matrix pair (A, b) is completely controllable (Hsu, 1998).
The following theorem shows that the proposed variable structure control in Eq. (8) drives the perturbed system with nonlinear input into the sliding mode s(t) = 0.
Assumption 2. The matrix Ad and perturbed matrices DA, f satisfy the following rank conditions (Luo & De La Sen, 1993).
Theorem 1 (Hsu, 1998). Consider the perturbed system (1) subjected to Assumptions 1 and 2, and inequality of (3). If the variable structure control input u(t) in Eq. (1) is given as that indicated by (8), then the system trajectories asymptotically converge to the sliding mode (6).
. rank½DA ¼ rank½DA..b . rank½DAd ¼ rank½DAd ..b . rank½f ¼ rank½f ..b
ð2Þ
Based on Assumption 2, there exist vectors h, g, and scalar d such that the following matching conditions hold (Shyu & Yan, 1994).
4. Self-tuning single-input fuzzy logic controller (ST-SIFLC) In this section, a self-tuning single-input fuzzy logic controller (ST-SIFLC) is proposed and expressed as following.
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By taking the time derivative of both sides of (6), we obtain
4.1. Single-input fuzzy logic controller (SIFLC) In this section, the single-fuzzy input controller for the perturbed time-delay system with nonlinear input (1) is proposed. Also, the idea of Choi, Kwak, and Kim (1999) named the signed distance is used and the feasibility of the present approach will be demonstrated. The switching line for a second order system (1) is defined by:
s : x_ þ c1 x ¼ 0
ð10Þ
First, we introduce a new variable called the signed distance. Let _ be the intersection point of the switching line and the line Aðx; xÞ perpendicular to the switching line from an operating point Bðx1 ; x_ 1 Þ, as illustrated in Fig. 1. Then, d1, the distance between _ and Bðx1 ; x_ 1 Þ can be expressed by the following equation: Aðx; xÞ
jx_ 1 þ c1 x1 j d1 ¼ ½ðx x1 Þ2 þ ðx_ x_ 1 Þ2 1=2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c21
s_ ¼ cT x_ ¼ c T Ax þ c T bUðuÞ þ c T bðhx þ gxd þ dÞ
Then, multiplying both sides of the above equation by s leads to
ss_ ¼ sc T Ax þ sc T bUðuÞ þ sc T bðhx þ gxd þ dÞ
Here, we consider that c b > 0 for any x. In (17), it is seen that s_ increases as U(u) increases and vice versa. In (18), it is seen that if s > 0, then decreasing U(u) assures s_ to be negative making ss_ negative, and that if s < 0, then increasing U(u) assures s_ to be positive making ss_ negative. From Assumption 3, we know that U(u) P au for u > 0 and U(u) 6 au for u < 0, which implies that if u is positive then U(u) is positive. So, it is seen that if s > 0, then decreasing u assures s_ to be negative making ss_ negative, and that if s < 0, then increasing u assures s_ to be positive making ss_ negative. Now, we choose a Lyapunov-like function
V¼
1 2 D 2 s
ð19Þ
Without loss of generality, Eq. (11) can be rewritten as follows:
Then
jx_ þ c1 xj d ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c21
V_ ¼ Ds D_ s ¼
The signed distance ds is defined for an generally point Bðx1 ; x_ 1 Þ as follows:
x_ þ c1 x jx_ þ c1 xj s ds ¼ sgnðsÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ c1 1 þ c1 1 þ c21 where
sgnðsÞ ¼
ð13Þ
ss_ 1 þ c2n1 þ þ c22 þ c21
ð20Þ
Hence, it is seen that if s > 0, then Ds > 0, decreasing u assures s_ to be negative which makes ss_ negative so that V_ < 0 and that if s < 0, then Ds < 0, increasing u assures s_ to be positive making ss_ negative so that V_ < 0. So we can ensure that the system is asymptotically stable. From the above relation, we conclude that
u / Ds 1
for s > 0
ð14Þ
1 for s < 0
Similarly, For the nth order system (1), it is different from that of the switching hyperplane, i.e., Eq. (6) can be represented as
s ¼ xðn1Þ þ cn1 xn2 þ þ c2 x_ þ c1 x ¼ 0:
ð15Þ
Hence, the general signed distance Ds is changed to be
Ds ¼
ð18Þ
T
ð11Þ
ð12Þ
ð17Þ
xðn1Þ þ cn1 xn2 þ þ c2 x_ þ c1 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c2n1 þ þ c22 þ c21
s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c2n1 þ þ c22 þ c21
ð16Þ
. x
ð21Þ
Hence, the fuzzy rule table can be established on a one-dimensional space of variable Ds as shown in Table 1 instead of a two-dimensional space of x and x_ or a n-dimensional space of n variables. The control action can be determined by variable Ds only. Hence, we can easily add or modify rules for fine control. For implementation, a triangular type membership function is chosen for the aforementioned fuzzy variables, as shown in Fig. 2 Remark 2. The rule table can be established on a one-dimensional space like Table 1. It is seen that the total number of rules is greatly reduced compared with conventional FLCs. In other words, a traditional fuzzy system has two states to be controlled and the range of each state variable is divided into seven fuzzy sets. The number of rules forming the knowledge database is 72 = 49. With the present method, only one variable Ds needs to be fuzzified and only seven rules are necessary to establish the knowledge database.
Table 1 Rule table for SIFLC Ds u
NB PB
NM PM
NS PS
ZE ZE
PS NS
x
.
d
B ( x 1, x 1 )
. A (x , x ) . x + c1 x = 0 Fig. 1. Derivation of a signed distance.
Fig. 2. Fuzzy variable of triangular type.
PM NM
PB NB
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Ds (t )
~ Ds
K1
Fuzzy Control Rules
5. Simulation
U
K2
u(t) Consider a perturbed time-delay system with series nonlinearity, which is described by Eq. (1) (Hsu, 1998). It is shown below that
Fig. 3. The block diagram of the SIFLC.
_ xðtÞ ¼ ðA þ DAÞxðtÞ þ bUðuÞ þ Ad xðt sd Þ þ f ðtÞ ¼ AxðtÞ þ bUðuÞ þ b½hxðtÞ þ gxd þ d
4.2. Self-tuning scheme and scaling factor In the light of the self-tuning single-input fuzzy logic controller (ST-SIFLC) design for perturbed time-delay systems with nonlinear input, the single-input fuzzy logic controller (SIFLC), determined by signed distance, is shown in Fig. 3 where Ds and u are the input and output of the single-input fuzzy logic controller, respectively. e s , which is a fuzzThe input of the proposed fuzzy controller is D ified variable corresponding to Ds. The output of the fuzzy controller is U, which is the fuzzified variable corresponding to u. All the e s and U range from 1 to 1. Thus, the universes of discourse of D range of nonfuzzy variables Ds and u must be scaled to fit the unie s and U with scaling facverse of discourse of a fuzzified variable D tors K1 and K2, respectively, namely,
e s ¼ K 1 Ds ðtÞ D
ð22Þ
uðtÞ ¼ K 2 U
ð23Þ
From the consequence given above, we have proven that the SIFLC is ensured to be asymptotically stable. In this subsection we will introduce the self-tuning scheme because the fuzzy system performance is sensitive to the scaling factors K1. If large values of K1 are available the system steady tracking error will be reduced but the overshoot may increase because a large K1 provides a large control gain. Conversely, if small values of K1 are chosen, the tracking accuracy will be degraded, leading to a small control gain. So we propose to decrease K1 when the error is large, and vice versa. The fuzzy rule for tuning K1 can be formalized as If error is large (EL) then scaling factor is small (CS), If error is small (ES) then scaling factor is large (CL).
lCL ¼
1; K 1 ¼ K 1;max 0; K 1 –K 1;max
;
lCS ¼
A¼
ð24Þ
K 1 –K 1;min
K 1 ¼ K 1;min þ DK 1 = coshðDs Þ
K1
b¼
0 ; 1
x¼
x1
x2
;
xd ¼
x1 ðt sd Þ
x2 ðt sd Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 þ l2 cos x2d x21d þ x22d þ l3 cos x2
For all VSC, FSMC, and ST-SIFLC controllers, the switching surface is taken as
sðtÞ ¼ 2x1 þ x2 Obviously, cTb > 0. This meets the condition cTb > 0 in (18). In addition, the following initial values are arbitrarily chosen, that is,
x1 ð0Þ ¼ 1;
x2 ð0Þ ¼ 1
For the purpose of simulation, the following numerical value are employed:
d ¼ 1:0;
l1 ¼ 0:04;
l2 ¼ 0:3;
c ¼ 0:3
l3 ¼ 0:6;
5.1. VSC method
The FSMC method (Yu et al., 2003), designed to cope with the perturbed time-delay system with a nonlinear input, has been pre-
S
s (t )
K1
ð25Þ s& (t )
Remark 3. Intuitively, we may combine the SIFLC and self-tuning scheme to get better results in terms of system performance. It is demonstrated in Fig. 4 and named self-tuning single-input fuzzy logic controller (ST-SIFLC), Moreover, we can ensure that the system is asymptotically stable, creating the rule table from Eq. (21).
Fuzzy Control Rules
2 3
;
hx þ gxd þ d ¼ l1 eð1þsin x1 Þ
where DK1 = K1,max K1,min.
~ Ds
1
UðuÞ ¼ ðdebsin uc þ c cos uÞu; d > c > 0
and using the max-min inference for defuzzification the value of K1 can be found as
Ds (t )
0
5.2. FSMC method
1; K 1 ¼ K 1;min 0;
where xd = x(t sd), khk 6 b1, kgk 6 b2 and kdk 6 b3. From Eq. (3), we also obtain DA = bh, Ad = bg, and f = bd. The corresponding parameters of the illustrated system are given as follows:
For the computer simulation, the corresponding parameters of VSC method are r = 1.1, a = 0.7, b1 = 0.3, b2 = 0.3, b3 = 0.3, respectively. The delay time constants are taken as sd = 0.2.
By defining the following membership function:
lES ¼ 2=½expðDs Þ þ expðDs Þ ¼ 1= coshðDs Þ lEL ¼ 1 lES
ð26Þ
U
Self-tuning Fig. 4. The block diagram of the ST-SIFLC.
K2
u(t)
S&
K2
Fuzzy Control Rules
U
u(t) K3
Fig. 5. The block diagram of the FSMC (Yu et al., 2003).
Table 2 Rule table for FSMC (Yu et al., 2003) s=s_
NB
NM
NS
ZE
PS
PM
PB
PB PM PS ZE NS NM NB
ZE PS PM PB PB PB PB
NS ZE PS PM PB PB PB
NM NS ZE PS PM PB PB
NB NM NS ZE PS PM PB
NB NB NM NS ZE PS PM
NB NB NB NM NS ZE PS
NB NB NB NB NM NS ZE
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F.-M. Yu / Expert Systems with Applications 36 (2009) 5304–5309
sented previously. Fig. 5 shows the block diagram of the FSMC (Yu et al., 2003), and the two-dimension rules table are chosen in Table
2. Accordingly, the scaling factors are chosen as K1 = 0.5, K2 = 0.5 and K3 = 50. 5.3. ST-SIFLC method For the proposed adaptive fuzzy logic controller, ST-SIFLC method, the rule base is a one-dimension table only as the SIFLC in Table 1. The scaling factor K1 is variable due to the self-tuning scheme. Furthermore, K1,min = 0.1,K1,max = 1.1 and the other scaling factor is chosen as K2 = 25.
30
a
25
control force U
20
Fig. 6. State variable dynamics of x1: (a) VSC, (b) the result of Yu et al. (2003) and (c) the proposed method of ST-SIFLC.
15 10 5 0 -5 -10 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.35
0.4
0.45
0.5
0.35
0.4
0.45
0.5
time(sec) 30
b
25
control force U
20 15 10 5 0 -5 Fig. 7. State variable dynamics of x2: (a) VSC, (b) the result of Yu et al. (2003) and (c) The proposed method of ST-SIFLC.
-10 0
0.05
0.1
0.15
0.2
0.25
0.3
time(sec)
c
30 25
control force U
20 15 10 5 0 -5 -10
0
0.05
0.1
0.15
0.2
0.25
0.3
time(sec)
Fig. 8. The phase plane between x1 and x2.
Fig. 9. The control force u: (a) VSC, (b) FSMC (Yu et al., 2003) and (c) the proposed method of ST-SIFLC.
F.-M. Yu / Expert Systems with Applications 36 (2009) 5304–5309
The computer simulation sampling step for the perturbed timedelay system with nonlinearities in above three methods are 0.01 s. The responses of state variable x1 for the above three methods are shown in Fig. 6. The ST-SIFLC, as a simpler algorithm than FSMC and VSC, still own the desirable response of x1. As to the state variable dynamics of x2 for the system, which are shown in Fig. 7. We can note that the angular velocity of ST-SIFLC is greatly reduced, as compared to others. Fig. 8 shows the phase plane between x1 and x2. From Fig. 8, it is observed that the sliding surface of ST-SIFLC is modified successively. And the control input u is shown in Fig. 9. It is clear that the control signal of ST-SIFLC is dramatically reduced and all of the chattering has been eliminated. This demonstration shows that the proposed technique serves as a perfect control input behavior that is with high efficiency. Remark 4. In sliding mode control, state trajectories with chattering along a sliding hyperplane are inevitable in practice, because the control input is discontinuous in the vicinity of the sliding hyperplane. Chattering is a serious drawback of sliding mode control and has been an important problem in the field of sliding mode control. Here, we want to demonstrate that the STSIFLC can handle this chattering problem effectively by adjusting the control input near the sliding hyperplane. Simultaneously, the ST-SIFLC with the self-tuning scaling factor to enhance the efficiency for the control input. The simulation results in Fig. 8 show the state trajectory of system response for a perturbed time-delay system with nonlinearities. We can observe that the slope of the sliding surface in ST-SIFLC is modified successively. This is because the effect of tuning scaling factor is similar to tuning the slope of sliding surface. ST-SIFLC has the most compact control signal without redundant force, and eliminates chattering effectively. Remark 5. Because the switching function is defined only by the state variable of x(t), that is, s(t) = cTx(t) = 0, free of the time-delay sd, sd affect the system stability very little. From the simulation result, it is found that the robustness of system is little susceptible to time-delay sd. 6. Conclusions We have shown that the ST-SIFLC developed in this work can be easily applied to perturbed time-delay dynamical systems with nonlinear input as described in Eq. (1). The scheme of this work is to design a single-input fuzzy logic controller with self-tuning scheme to stabilize perturbed time-delay systems with nonlinear input. Several important features of the proposed technique are summarized in the following:
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(1) Without exact mathematical model of the nonlinear systems for the controller design. The only information needed to design the controller is the influences of u with respect to Ds. (2) An inherent chattering problem of sliding mode control can be eliminated effectively without involving sophisticated mathematics. (3) By the Lyapunov stability theorem and the reaching condition of ðV_ ¼ ss_ Þ < 0, we only need to adjust the control input u with relation to Ds to satisfy ss_ < 0, the construction of rule base is simple and clear. Also, the asymptotic system stability for perturbed time-delay systems with nonlinear input can be guaranteed. (4) The minimum numbers of fuzzy rules can be obtained. Because in each fuzzy implication only Ds is taken as input of the fuzzy logic controller in spite of the number of state variables in the system. The ST-SIFLC can be created quickly and the control law can be computed easily. (5) This new kind of self-tuning scheme in fuzzy logic controller, which improves the performance of chattering elimination and increases the efficiency of the control input by adjusting the scaling factor. (6) From the simulation result, it is shown that the ST-SIFLC indeed can be applied to the perturbed time-delay system with a nonlinear input. The robustness for the uncertainty and perturbation is very satisfactory.
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