Fuzzy sliding-mode control with rule adaptation for nonlinear systems
Article _____________________________ Fuzzy sliding-mode control with rule adaptation for nonlinear systems Lon-Chen Hung and Hung-Yuan Chung Department of Electrical Engineering, National Central University, Jhong-Li, Tao-Yuan 320, Taiwan, Republic of China E-mail: [email protected]
Abstract: A fuzzy sliding-mode control with rule adaptation design approach with decoupling method is proposed. It provides a simple way to achieve asymptotic stability by a decoupling method for a class of uncertain nonlinear systems. The adaptive fuzzy sliding-mode control system is composed of a fuzzy controller and a compensation controller. The fuzzy controller is the main rule regulation controller, which is used to approximate an ideal computational controller. The compensation controller is designed to compensate for the difference between the ideal computational controller and the adaptive fuzzy controller. Fuzzy regulation is used as an approximator to identify the uncertainty. The simulation results for two cart–pole systems and a ball– beam system are presented to demonstrate the effectiveness and robustness of the method. In addition, the experimental results for a tunnelling robot manipulator are given to demonstrate the effectiveness of the system.
Keywords: fuzzy, sliding-mode control, decoupling, adaptive, robot
1. Introduction Fuzzy logic control, as one of the most useful approaches for utilizing expert knowledge, has undergone extensive research in the past decade (Lim & Hiyama, 1991; Lau & Wong, 1998; Lee, 2002; Pham & Chen, 2002; Leung et al., 2003). Fuzzy logic control is generally applicable to plants that are mathematically poorly modelled and where experienced operators are available for providing qualitative guidance. Although achieving much practical success, fuzzy control has not been viewed as a rigorous science, due to a lack of formal synthesis techniques which guarantee the very basic requirements of global stability and acceptable performance (Wang, 1994; Kazemian, 2001). In stability analysis (Filev & Yager, 1994; Chai & Tong, 1999), it is commonly assumed that the mathematical 226
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model of the plant is known; this assumption contradicts the very basic premise of fuzzy control systems, i.e. to control processes that are poorly modelled from a mathematical viewpoint. Based on fuzzy systems which are capable of approximating, with arbitrary accuracy, any real continuous function on a compact set, a globally stable adaptive controller is first synthesized from a collection of fuzzy IF–THEN rules (Lee, 1990; Wu et al., 2003). The fuzzy system, used to approximate an optimal controller, is adjusted by an adaptive law based on a Lyapunov function synthesis approach. In recent years, there have been attempts to design a fuzzy system based on the sliding-mode control law (Hwang & Lin, 1992; Palm, 1994; Kim & Lee, 1995; Glower & Munighan, 1997; Chen, 2001; Pham & Chen, 2002; Liang & Su, 2005). These authors have shown that the
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boundary layer can be reached in finite time and the ultimate boundedness of states is obtained asymptotically even though there exists some disturbance of dynamic uncertainties of the system. Palm (1994) showed the analogy between a simple and a sliding-mode controller with a boundary layer. Hwang and Li (1992) proposed a fuzzy sliding-mode controller and opened a way of designing for higher order nonlinear systems. Kim and Lee (1995) used some fuzzy control rules to construct reaching control under the assumption that equivalent control already exists. The Liang and Su (2005) design of a new two-input, one-output fuzzy sliding-mode controller is based on its distinct capability of improving the transient behaviour of the system during the reaching phase as well as the guaranteed steady-state tracking precision while in the boundary layer. In most studies, the fuzzy controller of secondorder systems is designed on a phase plane built by the error and change of error that are produced from the states and change of states. For example, in a cart–pole system only the pole subsystem is considered, ignoring the cart subsystem, and it is thus impossible to achieve a good control around the set point. In this study, a decoupling fuzzy controller design is proposed. This controller guarantees some properties, such as the robust performance and stability properties. Further, a class of fourth-order nonlinear systems is investigated. Lo and Kuo (1998) proposed a decoupled fuzzy sliding-mode control (DFSMC) to cope with the above issue. However, the method focuses on giving fuzzy rules, under the assumption that equivalent control exists. An adaptive decoupling fuzzy sliding-mode control (ADFSMC) design scheme is presented through width adaptation for a class of fourthorder uncertain under-actuated systems. Each subsystem, which decouples into two secondorder systems, is said to have main and subcontrol purposes. Two sliding surfaces are constructed through the state variables of the decoupling subsystem. We define the main and subtarget conditions for these sliding surfaces and introduce an intermediate variable from the sub-sliding surface condition. The proposed
adaptation law, which results from the direct adaptive approach, is used to appropriately determine the width of the unknown system variables. And the membership functions in the THEN part vary with the width adaptation of consequence. An adaptive law is then used to tune width in the THEN part to appropriately determine the distribution of each membership function. The proposed control is that the structure of the fuzzy controller does not need to be changed while using a common design procedure and the computing time may be reduced considerably. A tuning methodology is derived to tune the consequence parts of the fuzzy rules. The online tuning algorithm is derived in the Lyapunov sense; thus, the stability of the control system can be guaranteed. Furthermore, to relax the necessity for the uncertain bound of the compensation controller, an estimation mechanism is investigated. The uncertain bound is investigated so that the chattering phenomena of the control efforts can be decreased. To illustrate the effectiveness of the proposed design method, a comparison between a DFSMC and the proposed ADFSMC is made. The proposed ADFSMC has the following advantages. (1) With it we can control most complex systems without knowing the exact mathematical models. (2) The dynamic behaviour of the controlled system can be approximately represented by a decoupling sliding surface. (3) The proposed control approach has an advantage over the former model-based control schemes in that it does not require prior knowledge of the dynamic nonlinear system. (4) The ADFSMC not only can increase the robustness to system uncertainties but also can decrease the chattering phenomenon in the conventional sliding-mode controller. The paper is divided into five sections. In Section 2, the systems and the problem formulation are described. In Section 3, the ADFSMC is presented. In Section 4, the proposed controller is used to control a cart–single-pendulum system, a cart–double-pendulum system and a ball–beam system, and we show how the controller is synthesized as a Borland Cþþ Builder
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code and integrated into the software executed by the computer that controls a tunnelling robot manipulator system. Finally, we conclude with Section 5.
s1 ¼ c1 ðx1 zÞ þ x2
2. System description Consider a second-order nonlinear system, which can be represented by the following state-space model in canonical form:
x_ 2 ðtÞ ¼ f ðxÞ þ bðxÞu þ dðtÞ
¼ ½c1 1½x1 x2 T c1 z ¼ cT x12 c1 z
ð3Þ
and then the other switching line as
x_ 1 ðtÞ ¼ x2 ðtÞ
s2 ¼ c 2 x 3 þ x 4
ð1Þ
yðtÞ ¼ x1 ðtÞ where x ¼ [x1 x2]T is the state vector, f(x) and b(x) are nonlinear functions, u is the control input and d(t) is an external disturbance. The disturbance is assumed to be bounded as follows: |d(t)|rD(t). For this kind of second-order system, we can use many kinds of control methods, such as proportional integral derivative control, sliding-mode control, fuzzy control etc. A control law u can be easily designed to make the secondorder system (1) arrive at our control goal. However, for such nonlinear models as an inverted pendulum system, the system dynamic representation is generally not in an exact canonical form. Rather, it has the following form: x_ 1 ðtÞ ¼ x2 ðtÞ
ð4Þ
In the design of a decoupling sliding-mode controller, an equivalent control is first given so that the states can stay on the sliding surface. Thus, in sliding motion, the system dynamic is independent of the original system and a stable equivalent control system is achieved. The equivalent control can be obtained by letting s_ 1 equal zero, i.e. _ þ x_ 2 s_1 ¼ c1 ðx_ 1 zÞ ¼ c1 x2 c1 z_ þ f1 þ b1 u þ d1 ¼ 0
ð5Þ
The decoupling sliding-mode control input is chosen as follows for a Lyapunov function candidate: 1 V ¼ s1 s21 2
ð6Þ
Taking the time derivative of (6), we have
x_ 2 ðtÞ ¼ f1 ðxÞ þ b1 ðxÞu1 þ d1 ðtÞ x_ 3 ðtÞ ¼ x4 ðtÞ
ð2Þ
V_ ¼ s1 s_ 1 ¼ s1 ðc1 x2 c1 z_ þ f1 þ b1 u þ d1 Þ
x_ 4 ðtÞ ¼ f2 ðxÞ þ b2 ðxÞu2 þ d2 ðtÞ T
where x ¼ [x1 x2 x3 x4] is the state vector, f1(x), f2(x) and b1(x), b2(x) are nonlinear functions, u1, u2 are the control inputs and d1(t), d2(t) are external disturbances. The disturbances are assumed to be bounded as follows: |d1(t)|rD1(t), |d2(t)| rD2(t). One can use (2) effectively to design u1 and u2; however, this approach is only utilized to control a subsystem of (2). For example, if the model is a cart–pole system, we only control either the pole or the cart of a system such as (2). Hence, the idea of decoupling 228
is employed to design a control u to govern the whole system. In equation (2), we first define one switching line as
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ð7Þ
It can easily be shown from (7) that the decoupling sliding-mode controller u1 can be divided into an equivalent control input and a reaching control input which, if it has the following form, will be negative: u ¼ ueq K sgnðs1 =F1 Þ
ð8Þ
where K > D1(t)=|b1(t)|. K is a positive constant. Then the system is controlled in such a way that the state always moves towards the sliding surface and hits it. Thus, the trajectory is always forced to move towards the sliding surface. But
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(8) will show high-frequency switching near the sliding surface (s1 ¼ 0) due to the sgn function involved. Thus, in order to reduce the chattering phenomenon, we replace sgn(s1) with sat(s1) as follows: u ¼ ueq K satðs1 =F1 Þ
ð9Þ
where K > D1(t)=|b1(t)|. Hence, in the sliding motion, an equivalent controller will be ueq ¼
1 ðc1 x2 þ c1 z_ f1 þ z_ 1 þ ls1 Þ b1
ð10Þ
Substituting equation (10) into equation (5), we obtain s_ 1 þ ls1 ¼ 0
ð11Þ
where l is a positive value. The sliding surface on the phase plane can be defined as (3). The control objective is to drive the system state to the original equilibrium point. The switching line variables s1 and s2 are reduced to zero gradually at the same time by an intermediate variable z. In equation (3), z is a value transferred from s2; it has a value proportional to s2 and has the range proper to x1. Equation (3) denotes that the control objective of u1 is changed from x1 ¼ 0, x2 ¼ 0 to x1 ¼ z, x2 ¼ 0 (Lo & Kuo, 1998). Because the controller u ¼ u1 is used to govern the whole system, the bound of x1 can be guaranteed by letting jzjrZupper
0 < Zupper < 1
ð12Þ
where Zupper is the upper bound of abs(z). Equation (12) implies that the maximum absolute value of x1 will be limited. Summarizing what we have mentioned above, z can be defined as z ¼ satðs2 =Fz Þ Zupper
0 < Zupper < 1 ð13Þ
where Fz is the boundary layer of s2 to smooth z, Fz transfers s2 to the proper range of x1, and the definition of the sat(.) function is sgnðjÞ if jjjZ1 ð14Þ satðjÞ ¼ j if jjj< 1
Notice that z is a decaying oscillation signal because Zupper is a factor less than one. Remark. Decouple the whole system into subsystems A and B. System A contains x1, x2 and its corresponding sliding surface s1 ¼ c1 x1 þ x2. System B contains x3, x4 and its corresponding sliding surface s2 ¼ c2x3 þ x4. By doing this, the main control objective is to keep the states of system A moving toward the surface s1 ¼ 0 and converging to x1 ¼ x2 ¼ 0 asymptotically, and a subtarget is to keep the states of system B moving toward the surface s2 ¼ 0 and sliding to x3 ¼ x4 ¼ 0 asymptotically. Because the main target is to stabilize system A, it is reasonable to consider the information from system B as secondary and this secondary information must be reflected through a mechanism to the primary. An intermediate variable z, which represents secondary information, is incorporated into s1 and therefore the sliding surface is modified to take the form c1(x1 z) þ x2. This modification reflects the fact that the main target is changed from x1 ¼ 0, x2 ¼ 0 to x1 ¼ z, x2 ¼ 0, where x1 is a function of z. Notice that the subtarget of system B, s2 ¼ 0, is embedded in the main target through the variable z. Both primary and secondary targets can be controlled simultaneously. z represents a decaying oscillation signal whose value is assigned through (13). Hence, consider equation (3). If s1 ¼ 0, then x1 ¼ z, x2 ¼ 0. Since z is a value transferred from s2, when s2 ! 0, then z ! 0 and x1 ! 0. From equation (4), if the condition s1 ! 0 holds, the control objective can be achieved. Moreover, the choice of c1 and c2 has a strong influence on the behaviour in the transient state of the system. Appropriate choice of c1 and c2 is necessary for achieving a favourable transient response. Hence, control implementation can be started with zero initial fuzzy rules to eliminate the trial-and-error process for designing fuzzy control rules and has been developed with rule bases that do not need to be defined a priori.
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If the perfect control law does not obtain, it is not possible to implement (9). To overcome such a problem, a novel approach of the rule adaptation of fuzzy control is proposed to estimate an equivalent control input ueq. This will be proposed in Section 3.
this term is to force the system state to slide on the sliding surface. Another is the corrective control uc that drives the states towards the sliding surface. Thus the control law can be represented as
3. Design of ADFSMC
where uADFSMC is the approximate equivalent control and the corrective control uc is designed to stabilize the states of the control system around a preselected uncertainty bound. Now, the rule base of ADFSMC is constructed as
In this section, we show how to develop an adaptive fuzzy sliding-mode controller for obtaining equivalent control through rule adaptation in the decoupling system. Then, we construct corrective control to guarantee the system’s stability. The proposed ADFSMC has online self-tuning fuzzy rules without the trial-and-error process for finding appropriate fuzzy rules. If the state trajectory can be forced to slide on the sliding surface, then a stable equivalent control system is achieved. However, if the function f1 is unknown (for simplicity, we assume b1 is known), there is no way to yield equivalent control ueq. In this paper, a set of fuzzy rule bases is applied to approximating (10). Motivated by the principle of decoupling sliding-mode control, the control law consists of two parts. One is the estimated decoupling sliding component uADFSMC that is constructed by an adaptive mechanism, as shown in Figure 1. The effect of
u ¼ uADFSMC þ uc
ð15Þ
(i; j)th rule: If s1 is Ai and s_1 is Bj then uADFSMC is uk where i 2 I ¼ fp; p þ 1; . . . ; 1; 0; 1; ð16Þ
. . . ; p 1; pg j 2 J ¼ fq; q þ 1; . . . ; 1; 0; 1;
ð17Þ
. . . ; q 1; qg k 2 K ¼ fr; r þ 1; . . . ; 1; 0; 1;
ð18Þ
. . . ; r 1; rg
The rule base of ADFSMC is constructed as in this study; triangular types and singletons,
x3 S2 SMC x4
Adaptive Law (39)
Z x1 S1 SMC
Adaptive Fuzzy Controller
x2
Corrective Controller (40),(41)
uADFSMC +
u Plant +
y
uc
Figure 1: Block diagram of the ADFSMC system. 230
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respectively, are used to define the membership functions of the IF part and the THEN part, which are shown in Figures 2–4. The membership functions of the IF part and the THEN part are arranged to have the same width and boundary in the universe of discourse [ p; p], [ q; q] and [ U; U] respectively. The rule is defined in the following analytical form (Chen, 2001): k ¼ ½si ð1 sÞj
s 2 ½0; 1
ð19Þ
where k is an operation that takes an integer which is the nearest to k and s is a rule regulating factor. Obviously, by properly adjusting s, the value of k will be changed by (19), which indirectly determines which uk should be taken into account. So the controller is expected to provide different control actions corresponding to different s. According to (16)–(18), it is easily found that (19) is constrained by rrsi þ ð1 sÞjrr
ð20Þ
A− p
Ap +1
A0
Ap − 1
u−r
ur
uADFSMC
−U
U
0
Figure 4: The membership functions of the THEN part.
Considering the extreme case, i ¼ p and j ¼ q, the following condition shows that s should be satisfied:
sr
rq pq
ð21Þ
It also shows that r ¼ p > q satisfies the condition s 2 [0, 1]. Define each membership function of the THEN part as (Chen, 2001) ð22Þ
where h ¼ U=r and kr is the modified function of k and is represented as
p
B0
kr ¼ si ð1 sÞj Considering the case pirs1rpiþ1 qi r s_1 rqiþ1 , four rules are fired:
Bq − 1
ð23Þ and
Bq
ði; jÞ
s1 0
ur−1
h
Figure 2: The membership functions of the input variable s1 of the IF part.
−q
u0
uk ¼ kr h
0
B− q B− q+ 1
u−r+1
Ap
s1 −p
ði; j þ 1Þ
ði þ 1; jÞ
ði þ 1; j þ 1Þ
ð24Þ
Therefore, we can get a crisp output through the defuzzification strategy
q
Figure 3: The membership functions of the input variable s_ 1 of the IF part. c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
uADFSMC ¼
O X
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ð25Þ
231
where
where O ¼ w1 u1 þ w2 u2 þ w3 u3 þ w4 u4
y ¼ ½w1 ði þ jÞ þ w2 ði þ j 1Þ
¼ w1 ½si ð1 sÞjh þ w2 ½sði þ 1Þ
þ w3 ði þ j 1Þ þ w4 ði þ jÞh
ð1 sÞjh þ w3 ½si ð1 sÞðj þ 1Þh
ð31Þ
The control law for the ADFSMC system is assumed to have the following form:
þ w4 ½sði þ 1Þ ð1 sÞð1 þ jÞh ¼ fw1 ½sði þ jÞ j þ w2 ½sði þ j 1Þ j
u ¼ uADFSMC þ uc
ð32Þ
þ w3 ½sði þ j 1Þ ðj þ 1Þ þ w4 ½sði þ jÞ ð1 þ jÞgh
X ¼ w1 þ w2 þ w3 þ w4
ð26Þ
ð27Þ
and
where uADFSMC is the approximate equivalent control and the corrective control uc is designed to stabilize the states of the control system around a preselected uncertainty bound. Substituting equation (32) into equation (2), we obtain x_ 2 ¼ f1 ðxÞ þ b1 ðxÞu
w1 ¼ min½mAi ðs1 Þ; mBj ðs_1 Þ w2 ¼ min½mAiþ1 ðs1 Þ; mBj ðs_1 Þ
¼ f1 ðxÞ þ b1 ðxÞðuADFSMC þ uc Þ ð28Þ
¼ c1 x2 þ c1 z_ þ b1 ðxÞðuADFSMC þ uc ueq Þ
w3 ¼ min½mA ðs1 Þ; mBjþ1 ðs_1 Þ
ð33Þ or, equivalently,
w4 ¼ min½mAiþ1 ðs1 Þ; mBjþ1 ðs_1 Þ The main aim is to derive an adaptive law to adjust the regulating factor s such that the estimated equivalent control uADFSMC can be optimally approximated to the equivalent control of the decoupling sliding-mode control under the situation of an unknown function f1. For simplicity, a triangular type membership function is chosen for the aforementioned fuzzy variables. The online tuning algorithm of parameters is proposed to adjust the consequent parameters for monitoring the system control performance. Suppose there exists an optimal regulating factor s* which is a constant such that the uADFSMC has minimum approximation error: e ¼ uADFSMC uADFSMC
ð29Þ
Then from (19)–(21) we have uADFSMC 232
uADFSMC
1 ¼ ðs s Þy X
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x_ 12 ¼ Ac x12 þ Bc ðuADFSMC þ uc ueq Þ þ Pz_ þ Qs1
ð34Þ
where " x12 ¼ ½ x1 Bc ¼ ½ 0 P¼½0
x2
T
Ac ¼
0
1
0
c1
#
b1 ðxÞ T c1 T
Q¼½0
l T
Equation (5) can be rewritten s_D ¼ cT x12 c1 z ¼ cT Ac x12 þ cT Bc ðuADFSMC þ uc ueq Þþ cT Pz þ cT Qs1 c1 z_ ¼ b1 ðuADFSMC þ uc ueq Þ lsD ð35Þ
ð30Þ
where c ¼ ½ c1
1 T and sD ¼ s1 F1 sat(s1=F1).
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
The above properties of the boundary layer concept are exploited in the design of the ADFSMC system, our goal being rule adaptation as soon as the boundary layer is reached. This approach aims to avoid the possibility of unbounded growth. As described above we develop an adaptive fuzzy control law with rule regulation in an attempt to accomplish this objective. The adjustment method for the fuzzy rule is local, and therefore only the specific operation domain is adapted without detrimental effects on other operating areas. In this paper, a method is proposed to guarantee the closed-loop stability by the Lyapunov function. In order to analyse the overall system stability, we introduce assumptions as follows. Assumption 1: b1(x) in (2) is differentiable, i.e.
(32), in which the adaptation laws of the fuzzy controller are designed as (39) and the corrective controller is designed as (40) with the adaptive bound estimation shown in (41), then we can guarantee the asymptotic stability of the closedloop system. The fuzzy rule adaptive laws are given by
ð40Þ
K_^ ¼ Z2 jsD j
ð41Þ
V¼
ð36Þ
Assumption 2: There exist optimal values for the fuzzy rule such that 1 @b1 1 ¼ K u _ s u þ ð37Þ x eq D ADFSMC 2 @x
1 2 1 2 1 2 C þ L s þ 2jb1 j D 2Z1 2Z2
ð42Þ
^ s , L ¼ K^ K and Z1, Z2 are where C ¼ s positive constants. The variation of this function (42) with respect to time is sD s_D 1 2 @b1 1 _ þ 1 L L_ V_ ¼ þ sD sgnðb1 Þ 1 x_ þ C C Z1 Z2 jb1 j 2 @x
where the uncertainty bound K is a positive constant.
¼
sD s_D 1 @b1 1 _ 1 ^ þ L K_^ þ s_D sgnðb1 Þ 1 x_ þ C s Z1 Z2 jb1 j 2 @x
¼
1 1 @b1 _ þ s2D sgnðb1 Þ 1 x_ sD ðcT x_ 12 c1 zÞ jb1 j 2 @x
This uncertainty bound cannot be measured for practical applications. Therefore, a bound estimation is developed to observe the bound of approximation error: ^ K X ¼ KðtÞ
uc ¼ K sgnðb1 Þ satðs1 =F1 Þ
Proof: Choose the Lyapunov function as
where 1 @b1 @b1 @b1 ; 1 ; :::; 1 @x1 @x2 @x4
ð39Þ
where Z1 and Z2 are positive constants. Moreover, the system states converge to the sliding surface asymptotically.
d 1 dx _ b ðxÞ ¼ Db1 ¼ Db1 1 ðxÞ 1 ðxÞðxÞ dt 1 dt
Db1 1 ðxÞ ¼
1 ^_ ¼ s Z sD sgnðb1 Þ y X 1
ð38Þ
^ is the estimated uncertainty bound. where KðtÞ Theorem 1: Considering the dynamic system described by (2) and the sliding surface (3) for the bounded continuous desired state trajectory, if the ADFSMC law is designed as c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
þ
¼
1 1 ^_ þ L K_^ Cs Z1 Z2
l 2 sD þ sD sgnðb1 Þ uADFSMC þ uc ueq jb1 j 1 @b1 þ sD 1 x_ þuADFSMC uADFSMC 2 @x þ
1 1 ^_ þ L K_^ Cs Z1 Z2 Expert Systems, September 2006, Vol. 23, No. 4
233
¼
l 2 s þ sD sgnðb1 Þ jb1 j D 1 @b1 1 uADFSMC ueq þ sD x_ 2 @x
þ sD sgnðb1 ÞðuADFSMC uADFSMC Þ 1 _ 1 ^ þ L K_^ þ sD sgnðb1 Þ uc þ C s Z1 Z2 l 2 1 s þ jsD jK þ sD sgnðb1ÞCyjsD jK^ r jb1 j D X 1 _ 1 ^ þ L K_^ þ Cs Z1 Z2 l 2 1 _ 1 ^ þ sD sgnðb1 Þ y s ¼ sD þ C T jb1 j Z1 X 1 _^ þL ð43Þ K jsD j Z2 By selecting appropriate values for F1, (43) implies that V_ is negative semidefinite: l 2 s V_ r jb1 j D
ð44Þ
If |sD|rF1, sD ¼ 0, then V ¼ 0 and V_ ¼ 0. If |sD| > F1, then sD ¼ s1 F1 sat(s1=F1) and s_D ¼ s_1 has the same sign as s1. From the algorithm, we have s1 s_1 < 0; therefore V_ ¼ s1 s_1 < 0. Then, for all t Z 0, V_ r 0 holds. This is thus a monotonic non-increasing function. Since V_ r 0, limt!1 V exists, i.e. V(1) exists. Thus sD is bounded, as is s. Since the continuous function of the closure set is bounded, then xi is bounded and so is s_D . As a consequence sD is uniformly continuous, as is V_ ¼RsD s_D . Since V(t) is bounded and t limt!1 0 V_ dt ¼ Vð1Þ Vð0Þ exists, then according to the Barbalat lemma (Slotine & Li, 1991) we have limt!1 V_ ¼ 0 and we obtain limt!1 s1 ¼ 0. To sum up the ADFSMC control law with ^ adjusted by (36), the method parameter vector s can find appropriate fuzzy rules in fuzzy control implementation and the online adjust rule also has the effect of improving the stability property. In particular, only one regulating factor is needed to tune the control rules. This will lead to less computational time and easier design than 234
Expert Systems, September 2006, Vol. 23, No. 4
conventional design approaches. The results show that the rule regulation of ADFSMC is stable in the sense of the Lyapunov function candidate. Compared to conventional design procedures of fuzzy sliding-mode controllers, the proposed controller design employs the expert knowledge of the sliding-mode to accomplish a control task that is easier and more efficient. Hence, the dead-zone concepts of nonlinear adaptive control are applied in this fuzzy rule regulation equation to improve the smoothness and robustness of the ADFSMC controller.
4. Computer simulations and experimental results In this section, we shall demonstrate that the ADFSMC is applicable to a cart–single-pendulum system, a cart–double-pendulum system, a ball–beam system (Lo & Kuo, 1998) and a tunnelling robot manipulator system to verify the theoretical development. 4.1. Case A: Cart–single-pendulum system The structure of a cart–single-pendulum system is illustrated in Figure 5. It consists of a straightline rail, a cart, a pendulum and a driving unit. The cart can move left or right on the rail. The pendulum is hinged at the centre of the cart and can rotate around the pivot in the same vertical
u
x
Figure 5: Structure of an inverted pendulum system.
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
plane with the rail. Its dynamics are described below: x_ 1 ¼ x2 x_ 2 ¼
mt g sin x1 mp L sin x1 cos x1 x22 þ u cos x1 Lð43 mt mp cos2 x1 Þ
simulation results. It is found that the pole and the cart can be stabilized to the equilibrium point. Further, the performance and robustness of the proposed control is better than those achieved by Lo and Kuo (1998) and Chen et al. (2002).
þd
x_ 4 ¼
4 3
ð45Þ
mp Lx22 sin x1 þ mp g sin x1 4 2 3 mt mp cos x1
þ
3ð43
cos x1
4 uþd mt mp cos2 x1 Þ
where x1 ¼ y is the angle of the pole with respect to the vertical axis; x2 ¼ y_ is the angular velocity of the pole with respect to the vertical axis; x3 ¼ x is the position of the cart; x4 ¼ x_ is the velocity of the cart; and mt ¼ mc þ mp. In what follows, we define the following variables: s1 ¼ c1 ðy zÞ þ y_ ¼ c1 ðx1 zÞ þ x2 s2 ¼ c2 x þ x_ ¼ c2 x3 þ x4
4.2. Case B: Cart–double-pendulum system As shown in Figure 9, the cart–double-pendulum system considered here consists of a straight-line rail, a cart, a double pendulum and a driving unit. The cart can move left or right on the rail. The pendulum is hinged on the centre of the cart and can rotate around the pivot in the same vertical plane with the rail. Pole 1 is the pole
0.8 proposed control DFSM control SFSM control
0.6 0.4 0.2
ð46Þ ð47Þ
theta(rad)
x_ 3 ¼ x4
0 −0.2 −0.4 −0.6
and
−0.8
0 < Zu < 1
−1
ð48Þ
In the simulation, the following specifications are used: mp ¼ 0.05 kg, mc ¼ 1 kg, L ¼ 0.5 m, g ¼ 9.8 m=s2, c1 ¼ 5, c2 ¼ 0.5, F1 ¼ 5, Fz ¼ 15, Zu ¼ 0.9425, d ¼ 0.087sin(t), Z1 ¼ 2, Z2 ¼ 2. The initial values are y ¼ 601, y_ ¼ 0 , x ¼ 0, x_ ¼ 0. In this simulation, it is impossible to regulate the cart position and pole angle simultaneously. If we want to regulate the cart position error, the pole will fall down. The sign of the pole angle is the same as that of the cart position; both the pole angle error and its velocity are zero, but the cart position is still in an undesired location; the cart position error and its velocity have the same sign. We produce the results for the ADFSMC assuming unknown nonlinear continuous functions f1 of the subsystem, with a DFSMC assuming known nonlinear continuous functions f1 of the subsystem. Figures 6–8 show the
−1.2
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 6: Angular evolution of the pole.
1.2 proposed control DFSM control SFSM control
1 0.8 0.6 z
z ¼ satðs2 =Fz Þ Zu
0.4 0.2 0 −0.2
0
2
4
6
8
10 12 time(s)
14
16
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Figure 7: System intermediate variable z of the cart–single-pendulum system.
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is the angle of pole 2 with respect to the vertical axis; x4 ¼ y_ 2 is the angular velocity of pole 2 with respect to the vertical axis; x5 ¼ x is the position of the cart; and x6 ¼ x_ is the velocity of the cart. In what follows, we define the following variables:
12 proposed control DFSM control SFSM control
10 8
x(m)
6 4 2 0
s1 ¼ c1 ðy zÞ þ y_ ¼ c1 ðx1 zÞ þ x2
−2 −4
2
0
4
6
8
10 12 time(s)
14
16
18
20
s2 ¼ c2 x þ x_ ¼ c2 x3 þ x4
Figure 8: Position evolution of the cart.
sz ¼ cz x5 þ x6 z ¼ satðzp Þ Zu
2
sz =Fz zp ¼ s2 =F2
1
ð51Þ
ð52Þ 0 < Zu < 1 if s2 r jsq j if s2 > jsq j
ð53Þ
where sq is the threshold value of s2. In the simulation, the following specifications are used: l1 ¼ 1 m, l2 ¼ 1 m, m1 ¼ 1 kg, m2 ¼ 1 kg, mc ¼ 1 kg, L ¼ 0.5 m, g ¼ 9.8 m=s2, c1 ¼ 5, c2 ¼ 1, cz ¼ 0.5, F1 ¼ 3, F2 ¼ 3, Fz ¼ 15, sq ¼ 0.01,
u
Zu ¼ 0.4712, d ¼ 0.08sin(t), Z1 ¼ 2, Z2 ¼ 2. The initial values are y1 ¼ 301, y2 ¼ 101, €y1 ¼ y_ 1 ¼ 0 ,
x
Figure 9: Structure of a cart–double-pendulum system. connected to the cart and pole 2 is above pole 1. The system’s dynamics are represented by x_ 1 ¼ x2 x_ 2 ¼ f1 þ b1 u þ d x_ 3 ¼ x4 x_ 4 ¼ f2 þ b2 u þ d
ð49Þ
x_ 6 ¼ f3 þ b3 u where x1 ¼ y1 is the angle of pole 1 with respect to the vertical axis; x2 ¼ y_ 1 is the angular velocity of pole 1 with respect to the vertical axis; x3 ¼ y2 Expert Systems, September 2006, Vol. 23, No. 4
€ y2 ¼ y_ 2 ¼ 0 , x€ ¼ x_ ¼ 0. We produce the results for the ADFSMC assuming unknown nonlinear continuous functions f1 of the subsystem, with a DFSMC assuming known nonlinear continuous functions f1 of the subsystem. Figures 10–12 show the simulation result. It is found that the pole and the cart can be stabilized to the equilibrium point. Further, the performance and robustness of the proposed control is better than those achieved by Lo and Kuo (1998) and Lin and Mon (2005). 4.3. Case C: Ball–beam system
x_ 5 ¼ x6
236
ð50Þ
The ball and beam system shown in Figure 13 consists of a beam pivoting at the centre point and a ball free to move along the beam. Let r be the distance of the ball from the origin and y be the angle that the beam makes with the horizontal line. The objective is to design an ADFSMC where the output converges to zero from
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arbitrary initial conditions in a certain region. Its dynamics are described below:
20 proposed control DFSM control SFSM control
15
x_ 1 ¼ x2
10
x(m)
x_ 2 ¼ u þ d ð54Þ
x_ 3 ¼ x4
5 0
x_ 4 ¼ Bðx3 x22 G sin x1 Þ −5
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 12: Position evolution of the cart. 10 proposed control DFSM control SFSM control
8 6 r(m)
where x1 ¼ y is the angle of the pole with respect to the vertical axis; x2 ¼ y_ is the angular velocity of the pole with respect to the vertical axis; x3 ¼ r is the position of the cart; x4 ¼ r_ is the velocity of the cart; B ¼ MR2=(Jb þ MR2); Jb is the moment of inertia of the ball; M is the mass of the ball; R is the radius of the ball; and g is the acceleration due to gravity. The centre of rotation is assumed to be frictionless and the ball is free to roll along the
4 2
0.6
0
proposed control DFSM control SFSM control
0.5 0.4
−2
theta1(rad)
0.3
0
2
4
6
8
10 12 time(s)
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Figure 13: Structure of a ball–beam system.
0.2 0.1 0
−0.1 −0.2 −0.3
0
2
4
6
8
10 12 time(s)
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Figure 10: Angular evolution of pole 1. 1 proposed control DFSM control SFSM control
0.8 0.6 theta2(rad)
0.4 0.2 0
−0.2 −0.4 −0.6 −0.8
0
2
4
6
8
10 12 time(s)
14
16
Figure 11: Angular evolution of pole 2.
18
20
beam. It is required that the ball remains in contact with the beam and that rolling occurs without slipping. The objective is to keep the ball close to the centre of the beam close to the horizontal position. In the simulation, the following specifications are used: B ¼ 0.7143, Jb ¼ 2 10 6, M ¼ 0.05 kg, R ¼ 0.01 m, g ¼ 9.8 m=s2, c1 ¼ 5, c2 ¼ 0.5, F1 ¼ 5, Fz ¼ 15, Zu ¼ 0.9425, d ¼ 0.08 sin(t), Z1 ¼ 2, Z2 ¼ 2. The initial values are x1 ¼ y ¼ 601, x2 ¼ y_ ¼ 00 , x3 ¼ 10, x4 ¼ r_ ¼ 0. We produce the results for the ADFSMC assuming unknown nonlinear continuous functions f1 of the subsystem, with a DFSMC assuming known nonlinear continuous functions f1 of the subsystem. Figures 14–16 show the simulation result. It is found that the ball– beam system can be stabilized to the equilibrium point and that y and r both converge to zero.
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Further, the proposed control performance and robustness are better than those achieved by Lo and Kuo (1998) and Chen et al. (2002).
1 proposed control DFSM control SFSM control
0.8 0.6 0.4 z
4.4. Case D: Tunnelling robot manipulator system
0.2 0 −0.2 −0.4
0
2
4
6
8
10 12 time(s)
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16
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Figure 15: System intermediate variable z of the ball–beam system. 1.2 proposed control DFSM control SFSM control
1 0.8 theta(rad)
Robot manipulations in a tunnel for civil engineering (Figure 17) form the hardware of the system (Chou et al., 2003). The hardware of the robot arm (Figure 18) contains (1) the forward and back motion of the x axis, range 0–29.4 cm; (2) the swirl motion of the y axis, range þ 781 to –781; (3) the sliding motion of the R axis, range 0–20 cm. Each axis includes both a limited switch and a home-tested sensor switch. The home-tested sensor allows users to return to the initial position, write programs and measure exactly. Both the limited switch and the hometested sensor switch work by magnetic induction. A supersonic sensor is hung on the R axis to detect the surface of the tunnel. The wooden semicircular tunnel is used to simulate an unsmooth real tunnel for three-axis robot manipulation. There is a 10 cm 10 cm 3.2 cm concavity for the experiment. The aspects of the interfaces are as follows: (1) two encoder=DAC interfaces; (2) an analog input–output interface; (3) a digital input–output interface. The utilization of a D=A card makes u become an analog signal. The position control of the y axis and x axis means that the motion of the robot arm is from its initial position to the set point. A comparison of the proposed control with the fuzzy sliding-
0.6 0.4 0.2 0 − 0.2 − 0.4
0
2
4
6
8
10 12 time(s)
14
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Figure 16: Angular evolution of the beam.
10 proposed control DFSM control SFSM control
8
Figure 17: A real robot manipulator for civil engineering in a tunnel.
r(m)
6 4 2 0 −2
0
2
4
6
8
10 12 time(s)
14
16
Figure 14: Position evolution of the ball. 238
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18
20
mode control (Chou et al., 2003) for the same plant is shown in Figures 19 and 20, respectively. It is clear that the proposed method not only reduces overshoot to near zero but also maintains a small setting time and steady error. Experimental results showed that the proposed c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
20
proposed control FSM control
Translational motion R axis
x (cm)
15 Rotational motion axis
10 5 0
Forward and Backward motion x axis
theta (degree)
proposed control FSM control
0
2
4
6
8
10 12 time (s)
14
16
18
2
4
6
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10 12 time (s)
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Figure 20: The proposed controller for the x axis is moved from 0 cm to 15 cm.
Figure 18: Diagram of the three-axis robot.
50 45 40 35 30 25 20 15 10 5 0
0
20
Figure 19: The response of the position controller for the y axis is moved from 01 to 401. control determines the robot position accurately and constructs tunnels precisely.
5. Conclusions An ADFSMC design scheme is presented through width adaptation for fourth-order nonlinear systems. The proposed adaptation law, which results from the direct adaptive approach, is used to appropriately determine the width of the unknown system variables. The membership functions in the THEN part vary with the width adaptation. An adaptive law is then used to tune width in the THEN part to appropriately determine the distribution of each membership function. The fuzzy rules of DFSMC should be preconstructed by trial-and-error tuning. By the ADFSMC, the fuzzy rules can be learned online by an adaptive law and the stability of the proposed ADFSMC system can be guaranteed. This control strategy establishes the appropriate fuzzy rules by continuous online learning in-
stead of a trial-and-error process. It simplifies the implementation of a fuzzy controller. The response of the system converges faster than results achieved in previous work. Simulation results have been provided to demonstrate the robustness and effectiveness of the proposed control system. Moreover, the experimental results have demonstrated that the proposed control scheme is effective and valid for real-world engineering applications.
References CHAI, T. and S. TONG (1999) Fuzzy direct adaptive control for a class of nonlinear systems, Fuzzy Sets and Systems, 103 (3), 379–387. CHEN, J.Y. (2001) Rule regulation of fuzzy slidingmode controller design: direct adaptive approach, Fuzzy Sets and Systems, 120 (1), 159–168. CHEN, S.Y., F.M. YU and H.Y. CHUNG (2002) Decoupled fuzzy controller design with single-input fuzzy logic, Fuzzy Sets and Systems, 129 (3), 335–342. CHOU, W.H., C.N. HUNG and H.Y. CHUNG (2003) A fuzzy sliding controller with gray predictor for the motion of a robot manipulator, Proceedings of the 25th National Symposium on Automatic Control, 805–810. FILEV, D.P. and R.R. YAGER (1994) On the analysis of fuzzy logic controllers, Fuzzy Sets and Systems, 68 (2), 39–66. GLOWER, J.S. and J. MUNIGHAN (1997) Design fuzzy controllers from a variable structures standpoint, IEEE Transactions on Fuzzy Systems, 5 (1), 138–144. HWANG, G.C. and S.C. LIN (1992) A stability approach fuzzy control design for nonlinear systems, Fuzzy Sets and Systems, 48 (3), 279–287.
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KAZEMIAN, H.B. (2001) Study of learning fuzzy controllers, Expert Systems, 18 (4), 186–193. KIM, S.W. and J.J. LEE (1995) Design of a fuzzy controller with fuzzy sliding surface, Fuzzy Sets and Systems, 71 (2), 359–367. LAU, H. and T.N. WONG (1998) A fuzzy expert system for complex closed-loop control: a non-mathematical approach, Expert Systems, 15 (2), 98–109. LEE, C.C. (1990) Fuzzy logic in control systems: fuzzy logic controller – Part I, IEEE Transactions on Systems, Man and Cybernetics, 20, 404–418. LEE, M.R. (2002) Expert system for nuclear power plant accident diagnosis using a fuzzy inference method, Expert Systems, 19 (4), 201–207. LEUNG, R.W.K., H.C.W. LAU and C.K. KWONG (2003) On a responsive replenishment system: a fuzzy logic approach, Expert Systems, 20 (1), 20–32. LIANG, C.Y. and J.P. SU (2005) A new approach to the design of a fuzzy sliding-mode controller, Fuzzy Sets and Systems, 139 (1), 111–124. LIM, C.M. and T. HIYAMA (1991) Application of fuzzy logic control to a manipulator, IEEE Transactions on Robotics and Automation, 7 (5), 688–691. LIN, C.M. and Y.J. MON (2005) Decoupling control by hierarchical fuzzy sliding-mode controller, IEEE Transactions on Control Systems Technology, 13 (4), 593–598. LO, J.C. and Y.H. KUO (1998) Decoupled fuzzy sliding-mode control, IEEE Transactions on Fuzzy Systems, 6 (3), 426–435. PALM, R. (1994) Robust control by fuzzy sliding-mode, Automatic, 30 (9), 1429–1437. PHAM, T.T. and G. CHEN (2002) Some applications of fuzzy logic in rule-based expert systems, Expert Systems, 19 (4), 208–223. SLOTINE, J.J.E. and W. LI (1991) Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall. WANG, L.X. (1994) Adaptive Fuzzy Systems and Control Design and Stability Analysis, Englewood Cliffs, NJ: Prentice Hall, 140–154.
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WU, W.Z., W.X. ZHANG and H.Z. LI (2003) Knowledge acquisition in incomplete fuzzy information systems via the rough set approach, Expert Systems, 20 (5), 280–286.
The authors Lon-Chen Hung Lon-Chen Hung was born in Kaoshiung, Taiwan, Republic of China. He is currently pursuing a PhD degree in electrical engineering from the National Central University, Taiwan. His research interests include sliding-mode control, fuzzy theory and control, and neural network applications.
Hung-Yuan Chung Hung-Yuan Chung was born in Ping-Tung, Taiwan, Republic of China. He received a PhD in electrical engineering from the National Cheng Kung University, Tainan, Taiwan, in 1987. In August 1987 he joined the Department of Electrical Engineering at the National Central University, Chung-li, Taiwan, as an associate professor. He has been a professor since August 1992. His research and teaching interests include system theory and control, adaptive control, fuzzy control, neural network applications and microcomputer-based control applications.
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Abstract: A fuzzy sliding-mode control with rule adaptation design approach with decoupling method is proposed. It provides a simple way to achieve asymptotic stability by a decoupling method for a class of uncertain nonlinear systems. The adaptive fuzzy sliding-mode control system is composed of a fuzzy controller and a compensation controller. The fuzzy controller is the main rule regulation controller, which is used to approximate an ideal computational controller. The compensation controller is designed to compensate for the difference between the ideal computational controller and the adaptive fuzzy controller. Fuzzy regulation is used as an approximator to identify the uncertainty. The simulation results for two cart–pole systems and a ball– beam system are presented to demonstrate the effectiveness and robustness of the method. In addition, the experimental results for a tunnelling robot manipulator are given to demonstrate the effectiveness of the system.
Keywords: fuzzy, sliding-mode control, decoupling, adaptive, robot
1. Introduction Fuzzy logic control, as one of the most useful approaches for utilizing expert knowledge, has undergone extensive research in the past decade (Lim & Hiyama, 1991; Lau & Wong, 1998; Lee, 2002; Pham & Chen, 2002; Leung et al., 2003). Fuzzy logic control is generally applicable to plants that are mathematically poorly modelled and where experienced operators are available for providing qualitative guidance. Although achieving much practical success, fuzzy control has not been viewed as a rigorous science, due to a lack of formal synthesis techniques which guarantee the very basic requirements of global stability and acceptable performance (Wang, 1994; Kazemian, 2001). In stability analysis (Filev & Yager, 1994; Chai & Tong, 1999), it is commonly assumed that the mathematical 226
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model of the plant is known; this assumption contradicts the very basic premise of fuzzy control systems, i.e. to control processes that are poorly modelled from a mathematical viewpoint. Based on fuzzy systems which are capable of approximating, with arbitrary accuracy, any real continuous function on a compact set, a globally stable adaptive controller is first synthesized from a collection of fuzzy IF–THEN rules (Lee, 1990; Wu et al., 2003). The fuzzy system, used to approximate an optimal controller, is adjusted by an adaptive law based on a Lyapunov function synthesis approach. In recent years, there have been attempts to design a fuzzy system based on the sliding-mode control law (Hwang & Lin, 1992; Palm, 1994; Kim & Lee, 1995; Glower & Munighan, 1997; Chen, 2001; Pham & Chen, 2002; Liang & Su, 2005). These authors have shown that the
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boundary layer can be reached in finite time and the ultimate boundedness of states is obtained asymptotically even though there exists some disturbance of dynamic uncertainties of the system. Palm (1994) showed the analogy between a simple and a sliding-mode controller with a boundary layer. Hwang and Li (1992) proposed a fuzzy sliding-mode controller and opened a way of designing for higher order nonlinear systems. Kim and Lee (1995) used some fuzzy control rules to construct reaching control under the assumption that equivalent control already exists. The Liang and Su (2005) design of a new two-input, one-output fuzzy sliding-mode controller is based on its distinct capability of improving the transient behaviour of the system during the reaching phase as well as the guaranteed steady-state tracking precision while in the boundary layer. In most studies, the fuzzy controller of secondorder systems is designed on a phase plane built by the error and change of error that are produced from the states and change of states. For example, in a cart–pole system only the pole subsystem is considered, ignoring the cart subsystem, and it is thus impossible to achieve a good control around the set point. In this study, a decoupling fuzzy controller design is proposed. This controller guarantees some properties, such as the robust performance and stability properties. Further, a class of fourth-order nonlinear systems is investigated. Lo and Kuo (1998) proposed a decoupled fuzzy sliding-mode control (DFSMC) to cope with the above issue. However, the method focuses on giving fuzzy rules, under the assumption that equivalent control exists. An adaptive decoupling fuzzy sliding-mode control (ADFSMC) design scheme is presented through width adaptation for a class of fourthorder uncertain under-actuated systems. Each subsystem, which decouples into two secondorder systems, is said to have main and subcontrol purposes. Two sliding surfaces are constructed through the state variables of the decoupling subsystem. We define the main and subtarget conditions for these sliding surfaces and introduce an intermediate variable from the sub-sliding surface condition. The proposed
adaptation law, which results from the direct adaptive approach, is used to appropriately determine the width of the unknown system variables. And the membership functions in the THEN part vary with the width adaptation of consequence. An adaptive law is then used to tune width in the THEN part to appropriately determine the distribution of each membership function. The proposed control is that the structure of the fuzzy controller does not need to be changed while using a common design procedure and the computing time may be reduced considerably. A tuning methodology is derived to tune the consequence parts of the fuzzy rules. The online tuning algorithm is derived in the Lyapunov sense; thus, the stability of the control system can be guaranteed. Furthermore, to relax the necessity for the uncertain bound of the compensation controller, an estimation mechanism is investigated. The uncertain bound is investigated so that the chattering phenomena of the control efforts can be decreased. To illustrate the effectiveness of the proposed design method, a comparison between a DFSMC and the proposed ADFSMC is made. The proposed ADFSMC has the following advantages. (1) With it we can control most complex systems without knowing the exact mathematical models. (2) The dynamic behaviour of the controlled system can be approximately represented by a decoupling sliding surface. (3) The proposed control approach has an advantage over the former model-based control schemes in that it does not require prior knowledge of the dynamic nonlinear system. (4) The ADFSMC not only can increase the robustness to system uncertainties but also can decrease the chattering phenomenon in the conventional sliding-mode controller. The paper is divided into five sections. In Section 2, the systems and the problem formulation are described. In Section 3, the ADFSMC is presented. In Section 4, the proposed controller is used to control a cart–single-pendulum system, a cart–double-pendulum system and a ball–beam system, and we show how the controller is synthesized as a Borland Cþþ Builder
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code and integrated into the software executed by the computer that controls a tunnelling robot manipulator system. Finally, we conclude with Section 5.
s1 ¼ c1 ðx1 zÞ þ x2
2. System description Consider a second-order nonlinear system, which can be represented by the following state-space model in canonical form:
x_ 2 ðtÞ ¼ f ðxÞ þ bðxÞu þ dðtÞ
¼ ½c1 1½x1 x2 T c1 z ¼ cT x12 c1 z
ð3Þ
and then the other switching line as
x_ 1 ðtÞ ¼ x2 ðtÞ
s2 ¼ c 2 x 3 þ x 4
ð1Þ
yðtÞ ¼ x1 ðtÞ where x ¼ [x1 x2]T is the state vector, f(x) and b(x) are nonlinear functions, u is the control input and d(t) is an external disturbance. The disturbance is assumed to be bounded as follows: |d(t)|rD(t). For this kind of second-order system, we can use many kinds of control methods, such as proportional integral derivative control, sliding-mode control, fuzzy control etc. A control law u can be easily designed to make the secondorder system (1) arrive at our control goal. However, for such nonlinear models as an inverted pendulum system, the system dynamic representation is generally not in an exact canonical form. Rather, it has the following form: x_ 1 ðtÞ ¼ x2 ðtÞ
ð4Þ
In the design of a decoupling sliding-mode controller, an equivalent control is first given so that the states can stay on the sliding surface. Thus, in sliding motion, the system dynamic is independent of the original system and a stable equivalent control system is achieved. The equivalent control can be obtained by letting s_ 1 equal zero, i.e. _ þ x_ 2 s_1 ¼ c1 ðx_ 1 zÞ ¼ c1 x2 c1 z_ þ f1 þ b1 u þ d1 ¼ 0
ð5Þ
The decoupling sliding-mode control input is chosen as follows for a Lyapunov function candidate: 1 V ¼ s1 s21 2
ð6Þ
Taking the time derivative of (6), we have
x_ 2 ðtÞ ¼ f1 ðxÞ þ b1 ðxÞu1 þ d1 ðtÞ x_ 3 ðtÞ ¼ x4 ðtÞ
ð2Þ
V_ ¼ s1 s_ 1 ¼ s1 ðc1 x2 c1 z_ þ f1 þ b1 u þ d1 Þ
x_ 4 ðtÞ ¼ f2 ðxÞ þ b2 ðxÞu2 þ d2 ðtÞ T
where x ¼ [x1 x2 x3 x4] is the state vector, f1(x), f2(x) and b1(x), b2(x) are nonlinear functions, u1, u2 are the control inputs and d1(t), d2(t) are external disturbances. The disturbances are assumed to be bounded as follows: |d1(t)|rD1(t), |d2(t)| rD2(t). One can use (2) effectively to design u1 and u2; however, this approach is only utilized to control a subsystem of (2). For example, if the model is a cart–pole system, we only control either the pole or the cart of a system such as (2). Hence, the idea of decoupling 228
is employed to design a control u to govern the whole system. In equation (2), we first define one switching line as
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ð7Þ
It can easily be shown from (7) that the decoupling sliding-mode controller u1 can be divided into an equivalent control input and a reaching control input which, if it has the following form, will be negative: u ¼ ueq K sgnðs1 =F1 Þ
ð8Þ
where K > D1(t)=|b1(t)|. K is a positive constant. Then the system is controlled in such a way that the state always moves towards the sliding surface and hits it. Thus, the trajectory is always forced to move towards the sliding surface. But
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(8) will show high-frequency switching near the sliding surface (s1 ¼ 0) due to the sgn function involved. Thus, in order to reduce the chattering phenomenon, we replace sgn(s1) with sat(s1) as follows: u ¼ ueq K satðs1 =F1 Þ
ð9Þ
where K > D1(t)=|b1(t)|. Hence, in the sliding motion, an equivalent controller will be ueq ¼
1 ðc1 x2 þ c1 z_ f1 þ z_ 1 þ ls1 Þ b1
ð10Þ
Substituting equation (10) into equation (5), we obtain s_ 1 þ ls1 ¼ 0
ð11Þ
where l is a positive value. The sliding surface on the phase plane can be defined as (3). The control objective is to drive the system state to the original equilibrium point. The switching line variables s1 and s2 are reduced to zero gradually at the same time by an intermediate variable z. In equation (3), z is a value transferred from s2; it has a value proportional to s2 and has the range proper to x1. Equation (3) denotes that the control objective of u1 is changed from x1 ¼ 0, x2 ¼ 0 to x1 ¼ z, x2 ¼ 0 (Lo & Kuo, 1998). Because the controller u ¼ u1 is used to govern the whole system, the bound of x1 can be guaranteed by letting jzjrZupper
0 < Zupper < 1
ð12Þ
where Zupper is the upper bound of abs(z). Equation (12) implies that the maximum absolute value of x1 will be limited. Summarizing what we have mentioned above, z can be defined as z ¼ satðs2 =Fz Þ Zupper
0 < Zupper < 1 ð13Þ
where Fz is the boundary layer of s2 to smooth z, Fz transfers s2 to the proper range of x1, and the definition of the sat(.) function is sgnðjÞ if jjjZ1 ð14Þ satðjÞ ¼ j if jjj< 1
Notice that z is a decaying oscillation signal because Zupper is a factor less than one. Remark. Decouple the whole system into subsystems A and B. System A contains x1, x2 and its corresponding sliding surface s1 ¼ c1 x1 þ x2. System B contains x3, x4 and its corresponding sliding surface s2 ¼ c2x3 þ x4. By doing this, the main control objective is to keep the states of system A moving toward the surface s1 ¼ 0 and converging to x1 ¼ x2 ¼ 0 asymptotically, and a subtarget is to keep the states of system B moving toward the surface s2 ¼ 0 and sliding to x3 ¼ x4 ¼ 0 asymptotically. Because the main target is to stabilize system A, it is reasonable to consider the information from system B as secondary and this secondary information must be reflected through a mechanism to the primary. An intermediate variable z, which represents secondary information, is incorporated into s1 and therefore the sliding surface is modified to take the form c1(x1 z) þ x2. This modification reflects the fact that the main target is changed from x1 ¼ 0, x2 ¼ 0 to x1 ¼ z, x2 ¼ 0, where x1 is a function of z. Notice that the subtarget of system B, s2 ¼ 0, is embedded in the main target through the variable z. Both primary and secondary targets can be controlled simultaneously. z represents a decaying oscillation signal whose value is assigned through (13). Hence, consider equation (3). If s1 ¼ 0, then x1 ¼ z, x2 ¼ 0. Since z is a value transferred from s2, when s2 ! 0, then z ! 0 and x1 ! 0. From equation (4), if the condition s1 ! 0 holds, the control objective can be achieved. Moreover, the choice of c1 and c2 has a strong influence on the behaviour in the transient state of the system. Appropriate choice of c1 and c2 is necessary for achieving a favourable transient response. Hence, control implementation can be started with zero initial fuzzy rules to eliminate the trial-and-error process for designing fuzzy control rules and has been developed with rule bases that do not need to be defined a priori.
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If the perfect control law does not obtain, it is not possible to implement (9). To overcome such a problem, a novel approach of the rule adaptation of fuzzy control is proposed to estimate an equivalent control input ueq. This will be proposed in Section 3.
this term is to force the system state to slide on the sliding surface. Another is the corrective control uc that drives the states towards the sliding surface. Thus the control law can be represented as
3. Design of ADFSMC
where uADFSMC is the approximate equivalent control and the corrective control uc is designed to stabilize the states of the control system around a preselected uncertainty bound. Now, the rule base of ADFSMC is constructed as
In this section, we show how to develop an adaptive fuzzy sliding-mode controller for obtaining equivalent control through rule adaptation in the decoupling system. Then, we construct corrective control to guarantee the system’s stability. The proposed ADFSMC has online self-tuning fuzzy rules without the trial-and-error process for finding appropriate fuzzy rules. If the state trajectory can be forced to slide on the sliding surface, then a stable equivalent control system is achieved. However, if the function f1 is unknown (for simplicity, we assume b1 is known), there is no way to yield equivalent control ueq. In this paper, a set of fuzzy rule bases is applied to approximating (10). Motivated by the principle of decoupling sliding-mode control, the control law consists of two parts. One is the estimated decoupling sliding component uADFSMC that is constructed by an adaptive mechanism, as shown in Figure 1. The effect of
u ¼ uADFSMC þ uc
ð15Þ
(i; j)th rule: If s1 is Ai and s_1 is Bj then uADFSMC is uk where i 2 I ¼ fp; p þ 1; . . . ; 1; 0; 1; ð16Þ
. . . ; p 1; pg j 2 J ¼ fq; q þ 1; . . . ; 1; 0; 1;
ð17Þ
. . . ; q 1; qg k 2 K ¼ fr; r þ 1; . . . ; 1; 0; 1;
ð18Þ
. . . ; r 1; rg
The rule base of ADFSMC is constructed as in this study; triangular types and singletons,
x3 S2 SMC x4
Adaptive Law (39)
Z x1 S1 SMC
Adaptive Fuzzy Controller
x2
Corrective Controller (40),(41)
uADFSMC +
u Plant +
y
uc
Figure 1: Block diagram of the ADFSMC system. 230
Expert Systems, September 2006, Vol. 23, No. 4
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
respectively, are used to define the membership functions of the IF part and the THEN part, which are shown in Figures 2–4. The membership functions of the IF part and the THEN part are arranged to have the same width and boundary in the universe of discourse [ p; p], [ q; q] and [ U; U] respectively. The rule is defined in the following analytical form (Chen, 2001): k ¼ ½si ð1 sÞj
s 2 ½0; 1
ð19Þ
where k is an operation that takes an integer which is the nearest to k and s is a rule regulating factor. Obviously, by properly adjusting s, the value of k will be changed by (19), which indirectly determines which uk should be taken into account. So the controller is expected to provide different control actions corresponding to different s. According to (16)–(18), it is easily found that (19) is constrained by rrsi þ ð1 sÞjrr
ð20Þ
A− p
Ap +1
A0
Ap − 1
u−r
ur
uADFSMC
−U
U
0
Figure 4: The membership functions of the THEN part.
Considering the extreme case, i ¼ p and j ¼ q, the following condition shows that s should be satisfied:
sr
rq pq
ð21Þ
It also shows that r ¼ p > q satisfies the condition s 2 [0, 1]. Define each membership function of the THEN part as (Chen, 2001) ð22Þ
where h ¼ U=r and kr is the modified function of k and is represented as
p
B0
kr ¼ si ð1 sÞj Considering the case pirs1rpiþ1 qi r s_1 rqiþ1 , four rules are fired:
Bq − 1
ð23Þ and
Bq
ði; jÞ
s1 0
ur−1
h
Figure 2: The membership functions of the input variable s1 of the IF part.
−q
u0
uk ¼ kr h
0
B− q B− q+ 1
u−r+1
Ap
s1 −p
ði; j þ 1Þ
ði þ 1; jÞ
ði þ 1; j þ 1Þ
ð24Þ
Therefore, we can get a crisp output through the defuzzification strategy
q
Figure 3: The membership functions of the input variable s_ 1 of the IF part. c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
uADFSMC ¼
O X
Expert Systems, September 2006, Vol. 23, No. 4
ð25Þ
231
where
where O ¼ w1 u1 þ w2 u2 þ w3 u3 þ w4 u4
y ¼ ½w1 ði þ jÞ þ w2 ði þ j 1Þ
¼ w1 ½si ð1 sÞjh þ w2 ½sði þ 1Þ
þ w3 ði þ j 1Þ þ w4 ði þ jÞh
ð1 sÞjh þ w3 ½si ð1 sÞðj þ 1Þh
ð31Þ
The control law for the ADFSMC system is assumed to have the following form:
þ w4 ½sði þ 1Þ ð1 sÞð1 þ jÞh ¼ fw1 ½sði þ jÞ j þ w2 ½sði þ j 1Þ j
u ¼ uADFSMC þ uc
ð32Þ
þ w3 ½sði þ j 1Þ ðj þ 1Þ þ w4 ½sði þ jÞ ð1 þ jÞgh
X ¼ w1 þ w2 þ w3 þ w4
ð26Þ
ð27Þ
and
where uADFSMC is the approximate equivalent control and the corrective control uc is designed to stabilize the states of the control system around a preselected uncertainty bound. Substituting equation (32) into equation (2), we obtain x_ 2 ¼ f1 ðxÞ þ b1 ðxÞu
w1 ¼ min½mAi ðs1 Þ; mBj ðs_1 Þ w2 ¼ min½mAiþ1 ðs1 Þ; mBj ðs_1 Þ
¼ f1 ðxÞ þ b1 ðxÞðuADFSMC þ uc Þ ð28Þ
¼ c1 x2 þ c1 z_ þ b1 ðxÞðuADFSMC þ uc ueq Þ
w3 ¼ min½mA ðs1 Þ; mBjþ1 ðs_1 Þ
ð33Þ or, equivalently,
w4 ¼ min½mAiþ1 ðs1 Þ; mBjþ1 ðs_1 Þ The main aim is to derive an adaptive law to adjust the regulating factor s such that the estimated equivalent control uADFSMC can be optimally approximated to the equivalent control of the decoupling sliding-mode control under the situation of an unknown function f1. For simplicity, a triangular type membership function is chosen for the aforementioned fuzzy variables. The online tuning algorithm of parameters is proposed to adjust the consequent parameters for monitoring the system control performance. Suppose there exists an optimal regulating factor s* which is a constant such that the uADFSMC has minimum approximation error: e ¼ uADFSMC uADFSMC
ð29Þ
Then from (19)–(21) we have uADFSMC 232
uADFSMC
1 ¼ ðs s Þy X
Expert Systems, September 2006, Vol. 23, No. 4
x_ 12 ¼ Ac x12 þ Bc ðuADFSMC þ uc ueq Þ þ Pz_ þ Qs1
ð34Þ
where " x12 ¼ ½ x1 Bc ¼ ½ 0 P¼½0
x2
T
Ac ¼
0
1
0
c1
#
b1 ðxÞ T c1 T
Q¼½0
l T
Equation (5) can be rewritten s_D ¼ cT x12 c1 z ¼ cT Ac x12 þ cT Bc ðuADFSMC þ uc ueq Þþ cT Pz þ cT Qs1 c1 z_ ¼ b1 ðuADFSMC þ uc ueq Þ lsD ð35Þ
ð30Þ
where c ¼ ½ c1
1 T and sD ¼ s1 F1 sat(s1=F1).
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
The above properties of the boundary layer concept are exploited in the design of the ADFSMC system, our goal being rule adaptation as soon as the boundary layer is reached. This approach aims to avoid the possibility of unbounded growth. As described above we develop an adaptive fuzzy control law with rule regulation in an attempt to accomplish this objective. The adjustment method for the fuzzy rule is local, and therefore only the specific operation domain is adapted without detrimental effects on other operating areas. In this paper, a method is proposed to guarantee the closed-loop stability by the Lyapunov function. In order to analyse the overall system stability, we introduce assumptions as follows. Assumption 1: b1(x) in (2) is differentiable, i.e.
(32), in which the adaptation laws of the fuzzy controller are designed as (39) and the corrective controller is designed as (40) with the adaptive bound estimation shown in (41), then we can guarantee the asymptotic stability of the closedloop system. The fuzzy rule adaptive laws are given by
ð40Þ
K_^ ¼ Z2 jsD j
ð41Þ
V¼
ð36Þ
Assumption 2: There exist optimal values for the fuzzy rule such that 1 @b1 1 ¼ K u _ s u þ ð37Þ x eq D ADFSMC 2 @x
1 2 1 2 1 2 C þ L s þ 2jb1 j D 2Z1 2Z2
ð42Þ
^ s , L ¼ K^ K and Z1, Z2 are where C ¼ s positive constants. The variation of this function (42) with respect to time is sD s_D 1 2 @b1 1 _ þ 1 L L_ V_ ¼ þ sD sgnðb1 Þ 1 x_ þ C C Z1 Z2 jb1 j 2 @x
where the uncertainty bound K is a positive constant.
¼
sD s_D 1 @b1 1 _ 1 ^ þ L K_^ þ s_D sgnðb1 Þ 1 x_ þ C s Z1 Z2 jb1 j 2 @x
¼
1 1 @b1 _ þ s2D sgnðb1 Þ 1 x_ sD ðcT x_ 12 c1 zÞ jb1 j 2 @x
This uncertainty bound cannot be measured for practical applications. Therefore, a bound estimation is developed to observe the bound of approximation error: ^ K X ¼ KðtÞ
uc ¼ K sgnðb1 Þ satðs1 =F1 Þ
Proof: Choose the Lyapunov function as
where 1 @b1 @b1 @b1 ; 1 ; :::; 1 @x1 @x2 @x4
ð39Þ
where Z1 and Z2 are positive constants. Moreover, the system states converge to the sliding surface asymptotically.
d 1 dx _ b ðxÞ ¼ Db1 ¼ Db1 1 ðxÞ 1 ðxÞðxÞ dt 1 dt
Db1 1 ðxÞ ¼
1 ^_ ¼ s Z sD sgnðb1 Þ y X 1
ð38Þ
^ is the estimated uncertainty bound. where KðtÞ Theorem 1: Considering the dynamic system described by (2) and the sliding surface (3) for the bounded continuous desired state trajectory, if the ADFSMC law is designed as c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
þ
¼
1 1 ^_ þ L K_^ Cs Z1 Z2
l 2 sD þ sD sgnðb1 Þ uADFSMC þ uc ueq jb1 j 1 @b1 þ sD 1 x_ þuADFSMC uADFSMC 2 @x þ
1 1 ^_ þ L K_^ Cs Z1 Z2 Expert Systems, September 2006, Vol. 23, No. 4
233
¼
l 2 s þ sD sgnðb1 Þ jb1 j D 1 @b1 1 uADFSMC ueq þ sD x_ 2 @x
þ sD sgnðb1 ÞðuADFSMC uADFSMC Þ 1 _ 1 ^ þ L K_^ þ sD sgnðb1 Þ uc þ C s Z1 Z2 l 2 1 s þ jsD jK þ sD sgnðb1ÞCyjsD jK^ r jb1 j D X 1 _ 1 ^ þ L K_^ þ Cs Z1 Z2 l 2 1 _ 1 ^ þ sD sgnðb1 Þ y s ¼ sD þ C T jb1 j Z1 X 1 _^ þL ð43Þ K jsD j Z2 By selecting appropriate values for F1, (43) implies that V_ is negative semidefinite: l 2 s V_ r jb1 j D
ð44Þ
If |sD|rF1, sD ¼ 0, then V ¼ 0 and V_ ¼ 0. If |sD| > F1, then sD ¼ s1 F1 sat(s1=F1) and s_D ¼ s_1 has the same sign as s1. From the algorithm, we have s1 s_1 < 0; therefore V_ ¼ s1 s_1 < 0. Then, for all t Z 0, V_ r 0 holds. This is thus a monotonic non-increasing function. Since V_ r 0, limt!1 V exists, i.e. V(1) exists. Thus sD is bounded, as is s. Since the continuous function of the closure set is bounded, then xi is bounded and so is s_D . As a consequence sD is uniformly continuous, as is V_ ¼RsD s_D . Since V(t) is bounded and t limt!1 0 V_ dt ¼ Vð1Þ Vð0Þ exists, then according to the Barbalat lemma (Slotine & Li, 1991) we have limt!1 V_ ¼ 0 and we obtain limt!1 s1 ¼ 0. To sum up the ADFSMC control law with ^ adjusted by (36), the method parameter vector s can find appropriate fuzzy rules in fuzzy control implementation and the online adjust rule also has the effect of improving the stability property. In particular, only one regulating factor is needed to tune the control rules. This will lead to less computational time and easier design than 234
Expert Systems, September 2006, Vol. 23, No. 4
conventional design approaches. The results show that the rule regulation of ADFSMC is stable in the sense of the Lyapunov function candidate. Compared to conventional design procedures of fuzzy sliding-mode controllers, the proposed controller design employs the expert knowledge of the sliding-mode to accomplish a control task that is easier and more efficient. Hence, the dead-zone concepts of nonlinear adaptive control are applied in this fuzzy rule regulation equation to improve the smoothness and robustness of the ADFSMC controller.
4. Computer simulations and experimental results In this section, we shall demonstrate that the ADFSMC is applicable to a cart–single-pendulum system, a cart–double-pendulum system, a ball–beam system (Lo & Kuo, 1998) and a tunnelling robot manipulator system to verify the theoretical development. 4.1. Case A: Cart–single-pendulum system The structure of a cart–single-pendulum system is illustrated in Figure 5. It consists of a straightline rail, a cart, a pendulum and a driving unit. The cart can move left or right on the rail. The pendulum is hinged at the centre of the cart and can rotate around the pivot in the same vertical
u
x
Figure 5: Structure of an inverted pendulum system.
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
plane with the rail. Its dynamics are described below: x_ 1 ¼ x2 x_ 2 ¼
mt g sin x1 mp L sin x1 cos x1 x22 þ u cos x1 Lð43 mt mp cos2 x1 Þ
simulation results. It is found that the pole and the cart can be stabilized to the equilibrium point. Further, the performance and robustness of the proposed control is better than those achieved by Lo and Kuo (1998) and Chen et al. (2002).
þd
x_ 4 ¼
4 3
ð45Þ
mp Lx22 sin x1 þ mp g sin x1 4 2 3 mt mp cos x1
þ
3ð43
cos x1
4 uþd mt mp cos2 x1 Þ
where x1 ¼ y is the angle of the pole with respect to the vertical axis; x2 ¼ y_ is the angular velocity of the pole with respect to the vertical axis; x3 ¼ x is the position of the cart; x4 ¼ x_ is the velocity of the cart; and mt ¼ mc þ mp. In what follows, we define the following variables: s1 ¼ c1 ðy zÞ þ y_ ¼ c1 ðx1 zÞ þ x2 s2 ¼ c2 x þ x_ ¼ c2 x3 þ x4
4.2. Case B: Cart–double-pendulum system As shown in Figure 9, the cart–double-pendulum system considered here consists of a straight-line rail, a cart, a double pendulum and a driving unit. The cart can move left or right on the rail. The pendulum is hinged on the centre of the cart and can rotate around the pivot in the same vertical plane with the rail. Pole 1 is the pole
0.8 proposed control DFSM control SFSM control
0.6 0.4 0.2
ð46Þ ð47Þ
theta(rad)
x_ 3 ¼ x4
0 −0.2 −0.4 −0.6
and
−0.8
0 < Zu < 1
−1
ð48Þ
In the simulation, the following specifications are used: mp ¼ 0.05 kg, mc ¼ 1 kg, L ¼ 0.5 m, g ¼ 9.8 m=s2, c1 ¼ 5, c2 ¼ 0.5, F1 ¼ 5, Fz ¼ 15, Zu ¼ 0.9425, d ¼ 0.087sin(t), Z1 ¼ 2, Z2 ¼ 2. The initial values are y ¼ 601, y_ ¼ 0 , x ¼ 0, x_ ¼ 0. In this simulation, it is impossible to regulate the cart position and pole angle simultaneously. If we want to regulate the cart position error, the pole will fall down. The sign of the pole angle is the same as that of the cart position; both the pole angle error and its velocity are zero, but the cart position is still in an undesired location; the cart position error and its velocity have the same sign. We produce the results for the ADFSMC assuming unknown nonlinear continuous functions f1 of the subsystem, with a DFSMC assuming known nonlinear continuous functions f1 of the subsystem. Figures 6–8 show the
−1.2
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 6: Angular evolution of the pole.
1.2 proposed control DFSM control SFSM control
1 0.8 0.6 z
z ¼ satðs2 =Fz Þ Zu
0.4 0.2 0 −0.2
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 7: System intermediate variable z of the cart–single-pendulum system.
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Expert Systems, September 2006, Vol. 23, No. 4
235
is the angle of pole 2 with respect to the vertical axis; x4 ¼ y_ 2 is the angular velocity of pole 2 with respect to the vertical axis; x5 ¼ x is the position of the cart; and x6 ¼ x_ is the velocity of the cart. In what follows, we define the following variables:
12 proposed control DFSM control SFSM control
10 8
x(m)
6 4 2 0
s1 ¼ c1 ðy zÞ þ y_ ¼ c1 ðx1 zÞ þ x2
−2 −4
2
0
4
6
8
10 12 time(s)
14
16
18
20
s2 ¼ c2 x þ x_ ¼ c2 x3 þ x4
Figure 8: Position evolution of the cart.
sz ¼ cz x5 þ x6 z ¼ satðzp Þ Zu
2
sz =Fz zp ¼ s2 =F2
1
ð51Þ
ð52Þ 0 < Zu < 1 if s2 r jsq j if s2 > jsq j
ð53Þ
where sq is the threshold value of s2. In the simulation, the following specifications are used: l1 ¼ 1 m, l2 ¼ 1 m, m1 ¼ 1 kg, m2 ¼ 1 kg, mc ¼ 1 kg, L ¼ 0.5 m, g ¼ 9.8 m=s2, c1 ¼ 5, c2 ¼ 1, cz ¼ 0.5, F1 ¼ 3, F2 ¼ 3, Fz ¼ 15, sq ¼ 0.01,
u
Zu ¼ 0.4712, d ¼ 0.08sin(t), Z1 ¼ 2, Z2 ¼ 2. The initial values are y1 ¼ 301, y2 ¼ 101, €y1 ¼ y_ 1 ¼ 0 ,
x
Figure 9: Structure of a cart–double-pendulum system. connected to the cart and pole 2 is above pole 1. The system’s dynamics are represented by x_ 1 ¼ x2 x_ 2 ¼ f1 þ b1 u þ d x_ 3 ¼ x4 x_ 4 ¼ f2 þ b2 u þ d
ð49Þ
x_ 6 ¼ f3 þ b3 u where x1 ¼ y1 is the angle of pole 1 with respect to the vertical axis; x2 ¼ y_ 1 is the angular velocity of pole 1 with respect to the vertical axis; x3 ¼ y2 Expert Systems, September 2006, Vol. 23, No. 4
€ y2 ¼ y_ 2 ¼ 0 , x€ ¼ x_ ¼ 0. We produce the results for the ADFSMC assuming unknown nonlinear continuous functions f1 of the subsystem, with a DFSMC assuming known nonlinear continuous functions f1 of the subsystem. Figures 10–12 show the simulation result. It is found that the pole and the cart can be stabilized to the equilibrium point. Further, the performance and robustness of the proposed control is better than those achieved by Lo and Kuo (1998) and Lin and Mon (2005). 4.3. Case C: Ball–beam system
x_ 5 ¼ x6
236
ð50Þ
The ball and beam system shown in Figure 13 consists of a beam pivoting at the centre point and a ball free to move along the beam. Let r be the distance of the ball from the origin and y be the angle that the beam makes with the horizontal line. The objective is to design an ADFSMC where the output converges to zero from
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
arbitrary initial conditions in a certain region. Its dynamics are described below:
20 proposed control DFSM control SFSM control
15
x_ 1 ¼ x2
10
x(m)
x_ 2 ¼ u þ d ð54Þ
x_ 3 ¼ x4
5 0
x_ 4 ¼ Bðx3 x22 G sin x1 Þ −5
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 12: Position evolution of the cart. 10 proposed control DFSM control SFSM control
8 6 r(m)
where x1 ¼ y is the angle of the pole with respect to the vertical axis; x2 ¼ y_ is the angular velocity of the pole with respect to the vertical axis; x3 ¼ r is the position of the cart; x4 ¼ r_ is the velocity of the cart; B ¼ MR2=(Jb þ MR2); Jb is the moment of inertia of the ball; M is the mass of the ball; R is the radius of the ball; and g is the acceleration due to gravity. The centre of rotation is assumed to be frictionless and the ball is free to roll along the
4 2
0.6
0
proposed control DFSM control SFSM control
0.5 0.4
−2
theta1(rad)
0.3
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 13: Structure of a ball–beam system.
0.2 0.1 0
−0.1 −0.2 −0.3
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 10: Angular evolution of pole 1. 1 proposed control DFSM control SFSM control
0.8 0.6 theta2(rad)
0.4 0.2 0
−0.2 −0.4 −0.6 −0.8
0
2
4
6
8
10 12 time(s)
14
16
Figure 11: Angular evolution of pole 2.
18
20
beam. It is required that the ball remains in contact with the beam and that rolling occurs without slipping. The objective is to keep the ball close to the centre of the beam close to the horizontal position. In the simulation, the following specifications are used: B ¼ 0.7143, Jb ¼ 2 10 6, M ¼ 0.05 kg, R ¼ 0.01 m, g ¼ 9.8 m=s2, c1 ¼ 5, c2 ¼ 0.5, F1 ¼ 5, Fz ¼ 15, Zu ¼ 0.9425, d ¼ 0.08 sin(t), Z1 ¼ 2, Z2 ¼ 2. The initial values are x1 ¼ y ¼ 601, x2 ¼ y_ ¼ 00 , x3 ¼ 10, x4 ¼ r_ ¼ 0. We produce the results for the ADFSMC assuming unknown nonlinear continuous functions f1 of the subsystem, with a DFSMC assuming known nonlinear continuous functions f1 of the subsystem. Figures 14–16 show the simulation result. It is found that the ball– beam system can be stabilized to the equilibrium point and that y and r both converge to zero.
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
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237
Further, the proposed control performance and robustness are better than those achieved by Lo and Kuo (1998) and Chen et al. (2002).
1 proposed control DFSM control SFSM control
0.8 0.6 0.4 z
4.4. Case D: Tunnelling robot manipulator system
0.2 0 −0.2 −0.4
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 15: System intermediate variable z of the ball–beam system. 1.2 proposed control DFSM control SFSM control
1 0.8 theta(rad)
Robot manipulations in a tunnel for civil engineering (Figure 17) form the hardware of the system (Chou et al., 2003). The hardware of the robot arm (Figure 18) contains (1) the forward and back motion of the x axis, range 0–29.4 cm; (2) the swirl motion of the y axis, range þ 781 to –781; (3) the sliding motion of the R axis, range 0–20 cm. Each axis includes both a limited switch and a home-tested sensor switch. The home-tested sensor allows users to return to the initial position, write programs and measure exactly. Both the limited switch and the hometested sensor switch work by magnetic induction. A supersonic sensor is hung on the R axis to detect the surface of the tunnel. The wooden semicircular tunnel is used to simulate an unsmooth real tunnel for three-axis robot manipulation. There is a 10 cm 10 cm 3.2 cm concavity for the experiment. The aspects of the interfaces are as follows: (1) two encoder=DAC interfaces; (2) an analog input–output interface; (3) a digital input–output interface. The utilization of a D=A card makes u become an analog signal. The position control of the y axis and x axis means that the motion of the robot arm is from its initial position to the set point. A comparison of the proposed control with the fuzzy sliding-
0.6 0.4 0.2 0 − 0.2 − 0.4
0
2
4
6
8
10 12 time(s)
14
16
18
20
Figure 16: Angular evolution of the beam.
10 proposed control DFSM control SFSM control
8
Figure 17: A real robot manipulator for civil engineering in a tunnel.
r(m)
6 4 2 0 −2
0
2
4
6
8
10 12 time(s)
14
16
Figure 14: Position evolution of the ball. 238
Expert Systems, September 2006, Vol. 23, No. 4
18
20
mode control (Chou et al., 2003) for the same plant is shown in Figures 19 and 20, respectively. It is clear that the proposed method not only reduces overshoot to near zero but also maintains a small setting time and steady error. Experimental results showed that the proposed c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
20
proposed control FSM control
Translational motion R axis
x (cm)
15 Rotational motion axis
10 5 0
Forward and Backward motion x axis
theta (degree)
proposed control FSM control
0
2
4
6
8
10 12 time (s)
14
16
18
2
4
6
8
10 12 time (s)
14
16
18
20
Figure 20: The proposed controller for the x axis is moved from 0 cm to 15 cm.
Figure 18: Diagram of the three-axis robot.
50 45 40 35 30 25 20 15 10 5 0
0
20
Figure 19: The response of the position controller for the y axis is moved from 01 to 401. control determines the robot position accurately and constructs tunnels precisely.
5. Conclusions An ADFSMC design scheme is presented through width adaptation for fourth-order nonlinear systems. The proposed adaptation law, which results from the direct adaptive approach, is used to appropriately determine the width of the unknown system variables. The membership functions in the THEN part vary with the width adaptation. An adaptive law is then used to tune width in the THEN part to appropriately determine the distribution of each membership function. The fuzzy rules of DFSMC should be preconstructed by trial-and-error tuning. By the ADFSMC, the fuzzy rules can be learned online by an adaptive law and the stability of the proposed ADFSMC system can be guaranteed. This control strategy establishes the appropriate fuzzy rules by continuous online learning in-
stead of a trial-and-error process. It simplifies the implementation of a fuzzy controller. The response of the system converges faster than results achieved in previous work. Simulation results have been provided to demonstrate the robustness and effectiveness of the proposed control system. Moreover, the experimental results have demonstrated that the proposed control scheme is effective and valid for real-world engineering applications.
References CHAI, T. and S. TONG (1999) Fuzzy direct adaptive control for a class of nonlinear systems, Fuzzy Sets and Systems, 103 (3), 379–387. CHEN, J.Y. (2001) Rule regulation of fuzzy slidingmode controller design: direct adaptive approach, Fuzzy Sets and Systems, 120 (1), 159–168. CHEN, S.Y., F.M. YU and H.Y. CHUNG (2002) Decoupled fuzzy controller design with single-input fuzzy logic, Fuzzy Sets and Systems, 129 (3), 335–342. CHOU, W.H., C.N. HUNG and H.Y. CHUNG (2003) A fuzzy sliding controller with gray predictor for the motion of a robot manipulator, Proceedings of the 25th National Symposium on Automatic Control, 805–810. FILEV, D.P. and R.R. YAGER (1994) On the analysis of fuzzy logic controllers, Fuzzy Sets and Systems, 68 (2), 39–66. GLOWER, J.S. and J. MUNIGHAN (1997) Design fuzzy controllers from a variable structures standpoint, IEEE Transactions on Fuzzy Systems, 5 (1), 138–144. HWANG, G.C. and S.C. LIN (1992) A stability approach fuzzy control design for nonlinear systems, Fuzzy Sets and Systems, 48 (3), 279–287.
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KAZEMIAN, H.B. (2001) Study of learning fuzzy controllers, Expert Systems, 18 (4), 186–193. KIM, S.W. and J.J. LEE (1995) Design of a fuzzy controller with fuzzy sliding surface, Fuzzy Sets and Systems, 71 (2), 359–367. LAU, H. and T.N. WONG (1998) A fuzzy expert system for complex closed-loop control: a non-mathematical approach, Expert Systems, 15 (2), 98–109. LEE, C.C. (1990) Fuzzy logic in control systems: fuzzy logic controller – Part I, IEEE Transactions on Systems, Man and Cybernetics, 20, 404–418. LEE, M.R. (2002) Expert system for nuclear power plant accident diagnosis using a fuzzy inference method, Expert Systems, 19 (4), 201–207. LEUNG, R.W.K., H.C.W. LAU and C.K. KWONG (2003) On a responsive replenishment system: a fuzzy logic approach, Expert Systems, 20 (1), 20–32. LIANG, C.Y. and J.P. SU (2005) A new approach to the design of a fuzzy sliding-mode controller, Fuzzy Sets and Systems, 139 (1), 111–124. LIM, C.M. and T. HIYAMA (1991) Application of fuzzy logic control to a manipulator, IEEE Transactions on Robotics and Automation, 7 (5), 688–691. LIN, C.M. and Y.J. MON (2005) Decoupling control by hierarchical fuzzy sliding-mode controller, IEEE Transactions on Control Systems Technology, 13 (4), 593–598. LO, J.C. and Y.H. KUO (1998) Decoupled fuzzy sliding-mode control, IEEE Transactions on Fuzzy Systems, 6 (3), 426–435. PALM, R. (1994) Robust control by fuzzy sliding-mode, Automatic, 30 (9), 1429–1437. PHAM, T.T. and G. CHEN (2002) Some applications of fuzzy logic in rule-based expert systems, Expert Systems, 19 (4), 208–223. SLOTINE, J.J.E. and W. LI (1991) Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall. WANG, L.X. (1994) Adaptive Fuzzy Systems and Control Design and Stability Analysis, Englewood Cliffs, NJ: Prentice Hall, 140–154.
240
Expert Systems, September 2006, Vol. 23, No. 4
WU, W.Z., W.X. ZHANG and H.Z. LI (2003) Knowledge acquisition in incomplete fuzzy information systems via the rough set approach, Expert Systems, 20 (5), 280–286.
The authors Lon-Chen Hung Lon-Chen Hung was born in Kaoshiung, Taiwan, Republic of China. He is currently pursuing a PhD degree in electrical engineering from the National Central University, Taiwan. His research interests include sliding-mode control, fuzzy theory and control, and neural network applications.
Hung-Yuan Chung Hung-Yuan Chung was born in Ping-Tung, Taiwan, Republic of China. He received a PhD in electrical engineering from the National Cheng Kung University, Tainan, Taiwan, in 1987. In August 1987 he joined the Department of Electrical Engineering at the National Central University, Chung-li, Taiwan, as an associate professor. He has been a professor since August 1992. His research and teaching interests include system theory and control, adaptive control, fuzzy control, neural network applications and microcomputer-based control applications.
c 2006 The Authors. Journal Compilation c 2006 Blackwell Publishing Ltd.
