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Enhanced adaptive fuzzy sliding mode control for uncertain nonlinear systems
Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
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Enhanced adaptive fuzzy sliding mode control for uncertain nonlinear systems Mehdi Roopaei a,*, Mansoor Zolghadri b,1, Sina Meshksar c a b c
School of Electrical and Computer Engineering, Shiraz University, P.O. Box 71955-177, Shiraz, Iran Centre for Computational Intelligence, De Montfort University, Leicester LE1 9BH, UK School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 9 August 2008 Received in revised form 23 January 2009 Accepted 23 January 2009 Available online 5 February 2009 PACS: 05.45.a Keywords: Fuzzy sliding mode control Nonlinear dynamics
a b s t r a c t In this article, a novel Adaptive Fuzzy Sliding Mode Control (AFSMC) methodology is proposed based on the integration of Sliding Mode Control (SMC) and Adaptive Fuzzy Control (AFC). Making use of the SMC design framework, we propose two fuzzy systems to be used as reaching and equivalent parts of the SMC. In this way, we make use of the fuzzy logic to handle uncertainty/disturbance in the design of the equivalent part and provide a chattering free control for the design of the reaching part. To construct the equivalent control law, an adaptive fuzzy inference engine is used to approximate the unknown parts of the system. To get rid of the chattering, a fuzzy logic model is assigned for reaching control law, which acting like the saturation function technique. The main advantage of our proposed methodology is that the structure of the system is unknown and no knowledge of the bounds of parameters, uncertainties and external disturbance are required in advance. Using Lyapunov stability theory and Barbalat’s lemma, the closed-loop system is proved to be stable and convergence properties of the system is assured. Simulation examples are presented to verify the effectiveness of the method. Results are compared with some other methods proposed in the past research. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The Variable Structure System (VSS) research was originated in early 1950s for single input systems with high order differential equations [1–3]. VSS was not very popular among control engineers prior to 1970s due to the absence of a systematic design procedure and high oscillation chattering in the control input. Although many scientists had conducted much research in the area of VSS, it was not until 1977 that the VSS concept was fully appreciated. The variable structure control with sliding mode was introduced to control engineers by Utkin [4]. The Sliding Mode Control (SMC) was originally developed for variable structure systems in the continuous domain. In his survey paper, Utkin presented a thorough description of the SMC theory in continuous time. Later, Slotine and Li [5] discussed continuous SMC in more detail. More recently, the research in the field of discrete time SMC has attracted many researchers [6]. SMC is an efficient tool to control complex high-order dynamic plants operating under uncertainty conditions due to its order reduction property and low sensitivity to disturbances and plant parameter variations. In SMC, the states of the controlled system are first guided to reside on a designed surface (i.e., the sliding surface) in state space and then keeping them there with a shifting law (based on the system states). * Corresponding author. Tel.: +989177137538. E-mail addresses: [email protected] (M. Roopaei), [email protected] (M. Zolghadri), [email protected] (S. Meshksar). 1 On sabbatical leave from Shiraz University. 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.01.029
M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
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There has been a wide variety of applications of SMC in areas such as robotics, power control, aerospace, and process control [7–11]. The most prominent property of the SMC is its insensitivity to parameter variations and external disturbances. However, its major drawback in practical applications is the chattering problem. Numerous techniques have been proposed to eliminate this phenomenon in SMC [12,13]. Conventional methods used to eliminate the chattering are to replace the relay control by a saturating approximation [14], integral sliding control [15–17] and Boundary Layer technique [5]. The boundary layer approach was introduced to eliminate the chattering around the switching surface and the control discontinuity within this thin boundary layer. If systems uncertainties are large, the sliding-mode controller would require a high switching gain with a thicker boundary layer to eliminate the higher resulting chattering effect. However, if we continuously increase the boundary layer thickness, we are actually reducing the feedback system to a system without sliding mode. To tackle these difficulties, fuzzy logic controllers are often used to deal with the discontinuous sign function in the reaching phase of SMC [18–33]. Recently, AFSMC methods are also used for this purpose, which is shown to be quite effective [34–36]. Fuzzy Logic Control (FLC) has been an active research topic in automation and control theory since the work of Mamdani [37] based on the fuzzy sets theory of Zadeh [38]. The basic concept of FLC is to utilize the qualitative knowledge of a system for designing a practical controller. Generally, FLC is applicable to plants that are ill-modeled, but qualitative knowledge of an experienced operator is available. It is particularly suitable for those systems which have uncertain or complex dynamics. Generally, in contrast to a conventional feedback control algorithm, there is a fuzzy control algorithm consists of a set of heuristic decision rules that can be represented as a non-mathematical control algorithm. This algorithm proves to be very effective especially when the precise model of the system under control is not available or expensive to prepare. The principle of SMC has been introduced in designing fuzzy logic controllers to guarantee the stability. This combination (i.e., FSMC) provides the mechanism to design robust controllers for nonlinear systems with uncertainty [39–41]. Designing adaptive fuzzy controllers by the integration of fuzzy logic and the SMC [42–44] for ensuring stability and consistent performance is a well known research topic. Many new algorithms have been proposed based on the integration of these control methods [45,46]. These approaches are similar in the aspect that they directly approximate the sliding mode control law by fuzzy approximators. One main advantage of this control scheme is its insensitivity to modeling uncertainty and external disturbances. Many AFSMC schemes have been proposed to eliminate the chattering using a fuzzy sliding surface in the reaching condition of the SMC [45–49]. In this article, a novel AFSMC algorithm is proposed for a class of continuous time unknown nonlinear systems. To design the hitting part of the SMC, a fuzzy controller is used. This will reduce the chattering and improve the robustness. An AFSMC is used (as equivalent control part of SMC) to approximate the unknown parts of the uncertain system. We provide the proof that closed-loop system is globally stable in the Lyapunov sense and the system output can track the reference signal in the presence of modeling uncertainties and external disturbances. The rest of this paper is organized as follows. Section 2 presents the system definitions and introduces the classical SMC design method. Sections 3 and 4 present the design method for the reaching and equivalent control parts, respectively. Section 5 presents the proof of asymptotic stability for the proposed method. Section 4 provides the simulation results and finally, conclusions are given in section 7.
2. System description and traditional SMC Consider nonlinear systems whose dynamical equations can be expressed in the canonical form [50]
x_ i ¼ xiþ1 ; 1 6 i 6 n 1; x_ n ¼ f ðx; tÞ þ dðtÞ þ gðx; tÞuðtÞ;
ð1Þ
where xðtÞ ¼ ½x1 ðtÞx2 ðtÞ xn ðtÞT 2 Rn is the state vector, f(x, t) and g(x, t) are two unknown functions belong to Rn ! R space. u(t), and dðtÞ 2 R are the control input and the external disturbance, respectively. Assumption 1 ([5,51,52]). The unknown functions f(x, t), g(x, t) and d(t) satisfy the following conditions:
jf ðX; tÞj 6 F < 1; 0 < g min 6 gðX; tÞ 6 g max < 1 and jdðtÞj 6 b; n
ð2Þ
n
where X 2 U X R , UX is a compact set defined as: U X ¼ fX 2 R : kXk 6 mx < 1g and F, b, g min and g max are unknown constants. The control problem is to force the system to track an n-dimensional desired vector Xd(t) (i.e. the nth-order tracking probðn1Þ lem of state xd(t) as discussed in [50]), X d ðtÞ ¼ ½xd1 ðtÞxd2 ðtÞ xdn ðtÞ ¼ ½xd ðtÞx_ d ðtÞ xd ðtÞ 2 Rn , which belong to a class of continuous functions in the interval [t0, 1]. The tracking error is defined as ðn1Þ
_ x_ d ðtÞ xðn1Þ ðtÞ xd EðtÞ ¼ XðtÞ X d ðtÞ ¼ ½xðtÞ xd ðtÞxðtÞ
_ eðn1Þ ðtÞ ¼ ½e1 ðtÞe2 ðtÞ en ðtÞ: ðtÞ ¼ ½eðtÞeðtÞ
ð3Þ
The control goal considered in this paper is that for any given target Xd(t), a SMC is designed such that the resulting state response of the tracking error vector satisfies the following condition:
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lim kEðtÞkt!1 ¼ limt!1 kXðtÞ X d ðtÞk ! 0;
ð4Þ
where kk denotes the Euclidean norm of a vector. In the following, we present a brief introduction to classical SMC design method. In traditional SMC, a switching surface representing the desired system dynamics is constructed as
S ¼ en þ
n1 X
c i ei :
ð5Þ
i¼1
The switching surface parameters {ci, i = 1, . . . , n 1} are chosen based on the following two criteria. First, the values are chosen to stabilize the system during sliding mode. Routh-Hurwitz criterion [5] is used to determine the range of coefficients cis’ that produce stable dynamics. That is, all the roots of the following characteristic polynomial describing the sliding surface have negative real parts with desirable pole placement:
PðkÞ ¼ kn þ cn1 kn1 þ þ c2 k þ c1 :
ð6Þ
Second, the values are chosen such that the system has fast and smooth response. When the closed loop system is in the sliding mode, it satisfies s_ ¼ 0 and the equivalent control law is obtained by
" # n1 X 1 ðnÞ ueq ¼ ci eiþ1 þ xd ðtÞ : f ðX; tÞ dðtÞ gðX; tÞ i¼1
ð7Þ
According to the Lyapunov stability theory [50], a Lyapunov function is defined as
V¼
1 2 s : 2
The derivative of
ð8Þ
v is
V_ ¼ ss_ ¼ s e_ n þ
n1 X
! ci e_ i :
ð9Þ
i¼1
In the above equation, if V_ is negative for all s – 0, then the so-called reaching condition [50] is satisfied. That is, the control u is designed to guarantee that the states are hitting on the sliding surface S = 0. In the traditional SMC, the reaching control law is selected as ur = kwuw and the overall control u is determined by
u ¼ ueq þ ur ¼ ueq þ kw uw ;
ð10Þ
where kw is the switching gain, which is positive and uw is obtained by
uw ¼ sgnðsÞ:
ð11Þ
It can be easily concluded that V_ 6 kw j s j and the states of the system are on the sliding surface. The error converges to zero once the states are on s = 0. The terms f(X, t), g(X, t) and d(t) are unknown in (7). In this paper, we use an adaptive fuzzy system to approximate the equivalent part. However, the sign function in overall control u will cause the control input to produce the chattering phenomenon. To overcome this problem, we use a fuzzy system to model the sign function. The overall scheme that we present in this paper to overcome the above-mentioned problems is based on the combination of the methods presented in [45] and [46].
3. FLC design for the reaching control part Fuzzy control (FC) has supplanted conventional technologies in many applications. One major feature of fuzzy logic is its ability to express the amount of ambiguity in human thinking. Thus, when the mathematical model of the process does not exist, or exists but with uncertainties, FC is an alternative way to deal with the unknown process. However, the huge amount of fuzzy rules for high-order systems makes the analysis complex. Therefore, much attention has focused on the FSMC. In this paper, in order to eliminate the chattering problem, fuzzy inference engine is used for reaching phase and fuzzy sliding mode control methodology is proposed. The main advantage of this method is that the robust behavior of the system is guaranteed. The second advantage of the proposed scheme is that the performance of the system in the sense of removing chattering is improved in comparison with the same SMC technique without using FLC. The configuration of our FSMC control scheme is shown in Fig. 1; it contains an equivalent control part and a two-inputsingle-output FSMC. The equivalent control part is the same as that in (7) and the reaching law is selected as [46,53]
ur ¼ kafs ufs ;
ð12Þ
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a
NB
-1
NM
-2/3
NS
-1/3
ZE
0
PS
PM
PB
1/3
2/3
1
b
ufs
NB
-1
NM
NS
ZE
PS
-2/3
-1/3
0
1/3
PM
2/3
PB
1
. s, s
_ Fig. 1. Fuzzy sets assigned to (a) output variable (ufs) and (b) input variables (S and SÞ.
Table 1 Rule-base of FSMC. ufs
S
S_
PB PM PS ZE NS NM NB
PB
PM
PS
ZE
NS
NM
NB
NB NB NB NB NM NS ZE
NB NB NB NM NS ZE PS
NB NB NM NS ZE PS PM
NB NM NS ZE PS PM PB
NM NS ZE PS PM PB PB
NS ZE PS PM PB PB PB
ZE PS PM PB PB PB PB
where kafs is the normalization factor of output variable, and ufs is the output of the FSMC, determined by the normalized _ The fuzzy control rules can now be represented as the mapping of the input linguistic variables S and S_ values of S and S. to output linguistic variable ufs as follows:
_ ufs ¼ FSMCðS; SÞ:
ð13Þ
The membership functions of the linguistic terms Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Medium (PB) and Positive Big (PB) that are assigned to the inputs (s and s_ Þ and the output (ufs) are shown in Fig. 1. The fuzzy rule table is designed as in Table 1 [53]. 4. AFLS design for the equivalent control part To approximate the unknown part (i.e., f(X, t)) of the equivalent control (7), an Adaptive Fuzzy Logic System (AFLS) is introduced in this section. A typical rule in the rule-base of the AFLS is as follows: ðmÞ
R^f
m m ^ : IF x1 is F m x1 and and xn is F xn ; THEN f ðxjhf Þ is F ^f ;
ð14Þ
m where m = 1, 2, . . . , Q is the index of the rule, F m xi ði ¼ 1; . . . ; nÞ denote the fuzzy sets assigned to xi (i = 1, . . . , n), and F ^f denotes ^ the fuzzy singletons assigned to the output f ðXjhf Þ of the fuzzy model. Using the singleton fuzzifier, product inference, and weighted average deffuzifier [52], the outputs of the fuzzy model ^f ðXjhf Þ can be expressed as
^f ðXjh Þ ¼ hT nðXÞ; f f
ð15Þ
½F ^1f ; F ^2f ; . . . ; F ^Qf T
1
2
Q
T
where hf ¼ is the adjustable parameters vector and n(X) = [n (X), n (X), . . . , n (X)] is the vector of fuzzy basis functions [52] defined as
Qn j F xi ðxi Þ hi¼1 i; Qn j Q j¼1 i¼1 F xi ðxi Þ
ni ðXÞ ¼ P
j ¼ 1; 2; . . . ; Q;
ð16Þ
where F jxi ðxi Þ denotes the membership function value of xi in F jxi . By the similar procedure, the function g(X,t) can be approximated as follows:
g^ðXjhg Þ ¼ hTg nðXÞ:
ð17Þ
The fuzzy approximators (15) and (17) can construct the functions f and g with good accuracy only when the following assumption holds. Assumption 2 [51] . The parameter hf belongs to the compact set Xf that is defined as Xf ¼ fhf 2 RQ jkhf k 6 mf g and Xg ¼ fhg 2 RQ jkhg k 6 mg g, where mf and mg are finite positive constants.
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u eq Equivalent Control
Slave System
x Master System
y
u fs Sliding Function
ur
FSMC(S)
Adaptive Filter
k afs
Fig. 2. Block diagram of the AGFSMC scheme.
From (15) and by the assumptions (1) and (2), we can also assume that j ^f ðXjhÞ j6 F and 0 < g min 6 g^ðX j hÞ 6 g max < 1. In practical systems, external disturbance d(t)is unknown. Thus, fuzzy models are used to construct the equivalent control law by replacing (7) with the following equation [52]:
" # n1 X 1 ðnÞ ^ ueq ¼ ci eiþ1 þ xd ðtÞ : f ðXjhf Þ g^ðXjhg Þ i¼1
ð18Þ
The overall control u is chosen as
u ¼ ueq þ ur ¼ ueq þ kafs ufs ; " # n1 X 1 ðnÞ ci eiþ1 þ xd ðtÞ þ kafs ufs : u ¼ ueq þ ur ¼ ^f ðXjhf Þ g^ðXjhg Þ i¼1
ð19Þ ð20Þ
In (19), hf, hg and kafs are updated as follows:
h_ f ¼ cf snðXÞ; h_ g ¼ c snðXÞu; g
ð21Þ
k_ afs ¼ ck jsj; where in the above relation cf, cg and ck are positive and arbitrary constants. The block diagram of our proposed AFSMC scheme is shown in Fig. 2. As shown, it uses a fuzzy inference approximator to construct the equivalent control part (17) and a two-input-single-output FSMC to construct the hitting control part using (12) and (13). 5. Stability analysis In the following theorem, the proposed scheme (20) will be proved to be able to drive the nonlinear system (1) onto the sliding surface s(t) = 0. That is, the reaching condition sðtÞs_ ðtÞ < 0 is guaranteed. Theorem 1. Assume that the uncertain nonlinear system (1) is controlled by uðtÞ in (20), where ueq is (18) and, ur is (12). Then, the error state trajectory converges to the sliding surface sðtÞ ¼ 0 (see Appendix-A for proof of theorem). 6. Simulation results In this section, the performance of the proposed approach is evaluated for a well-known benchmark problem. In the first experiment, we give the simulation results of our proposed adaptive fuzzy controller in a stabilizing problem. In the second, the performance of the schemes is investigated in a tracking problem. Furthermore, we also compare our method with the classical SMC. All the simulation procedures are implemented within MATLAB with the step size 0.001. 6.1. Experiment 1: stabilizing problem The inverted pendulum system is shown in Fig. 3. The state equations can be expressed by
where
x_ 1 ¼ x2 ; x_ 2 ¼ f þ gu þ d;
ð22Þ
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M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681 mLx2 sinðx Þ cosðx1 Þ
f ¼
1 g r sinðx1 Þ 2 mþM cosðx1 Þ L 43 mmþM
m cosðx1 Þ
;
g ¼ mþM cosðx1 Þ L 43 mmþM
and d ¼ 1 þ sinðp2 tÞ. The objective is to generate an appropriate actuator force u to control the motion of the cart, such that the pole can be balanced in the vertical position. The parameters values used for the cart and pole system are given in Table 2. The sliding surface is designed as: s = e2 + e1. In the simulations, the initial conditions values are selected as: T Xð0Þ ¼ ½ p=4 0 . If we require that jx1j 6 p/6 and jx2j 6 p/6, then the bounds bmax and bmin can be calculated as jg(x1, x2)j 6 1.46 = bmax and jg(x1, x2)j P 1.12 = bmin. Also, since j f ðx1 ; x2 Þ j6 15:78 þ 0:0366x22 , we can set F = 16. To construct the equivalent control part, ^f ðX; tÞ and g^ðX; tÞ were selected as (15) and (17), respectively. In simulations, we assigned the following Gaussian membership functions to xi (i = 1, 2) over the intervals [3, 3]:
lNB ðxÞ ¼ 1=ð1 þ expð5ðx þ 2ÞÞÞ; lNM ðxÞ ¼ expððx þ 1:5Þ2 Þ; lNS ðxÞ ¼ expððx þ 0:5Þ2 Þ; lPS ðxÞ ¼ expððx 0:5Þ2 Þ; lPM ðxÞ ¼ expððx 1:5Þ2 Þ; lPB ðxÞ ¼ 1=ð1 þ expð5ðx 2ÞÞÞ:
ð23Þ
To cover the entire space, we used 36 fuzzy rules. The initial values of hf(0) and hg(0) were chosen randomly in the interval [2, 2]. In the next step, the vector of fuzzy basic functions, n(X) = [n1(X), n2(X), . . . , nQ(X)]T, were constructed by (16). The learning rates were set to cf, cg and ck. The hitting control part ufs was specified using (13) and the value of kafs = 0.5 were used in simulations. Fig. 4 displays the state trajectories of system (22) with the control law (20). In this figures, one can see that the system states converge to zero rapidly as the system starts, and hence the asymptotical stabilization of the unified chaotic system is achieved. The control signal is also shown in Fig. 5. The trajectories and controller signal of system (22) by applying classical SMC are demonstrated in Figs. 6 and 7, respectively. 6.2. Experiment 2: tracking problem Let us consider the system (22) again. In this experiment, disturbance is considered as: d = 0.1(1 + sin(0.25t)) and the de T sired reference trajectories as: X d ðtÞ ¼ ½xd1 ðtÞ; xd2 ðtÞT ¼ p9 ðsinðtÞ þ 0:2 sinð3tÞÞ; p9 ðcosðtÞ þ 0:6 cosð3tÞÞ . The sliding surface is T s = e2 + e1. In the following simulations, the initial conditions values are selected as: Xð0Þ ¼ ½ p=4 0 The trajectories of the states’ system x1(t) and x2(t) are plotted in Figs. 8 and 9, respectively. As it can be seen, system (22) tracks the desired signals Xd(t) quickly. In Fig. 10, it is demonstrated that there is no chattering in the control signal of the proposed method. Figs. 8–10 show the simulation results for our proposed and conventional controller. The chattering can be reduced using various methods, in order to have a fair comparison between classical and our proposed SMC, saturation function is used. The parameters of various schemes were tuned manually in a way that have approximately the same control effort. In order to have a quantitative comparison of tracking error, in Table 3, we have reported the Integral Absolute Error (IAE) of different schemes. The sliding surface s = e2 + e1 was used for both methods and the thickness of the saturation function was specified as 0.1 for the conventional SMC method. The results of Figs. 3–5 and Table 1 clearly show that our proposed method has comparable performance in comparison with the classical SMC.
Table 2 The parameters values and functions used for the cart and pole system. gr (acceleration due to gravity) 2
9.81 m/s
L (half-length of the pole)
M (mass of the cart)
m (mass of the pole)
0.5
1.0 kg
0.1 kg
Table 3 Integral Absolute Error (IAE) values for various methods. IAE
Method Our method
Classical SMC
Tracking error
1.8475e + 003
1.6408e + 003
M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
Fig. 3. The inverted pendulum system.
1 State1 State 2 0.5
States
0
-0.5
-1
-1.5
-2
0
1
2
3
4
5
6
7
8
9
10
Time(second) Fig. 4. States trajectories: our proposed methodology.
0 controller Signal -5
Controller Signal
3676
-10
-15
-20
-25
-30
0
1
2
3
4
5
6
7
8
Time(second) Fig. 5. Controller signal: our proposed methodology.
9
10
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0.8 State 1 State 2
0.6 0.4
States
0.2 0 -0.2 -0.4 -0.6 -0.8
0
1
2
3
4
5
6
7
8
9
10
Time,sec Fig. 6. States trajectories: classical SMC.
0 Controller Signal
Control signal
-5
-10
-15
-20
-25
0
1
2
3
4
5
6
7
8
9
10
Time,sec
Tracking State and Desired Signal
a
1 Tracking State:1 Desired State:1
0.8 0.6 0.4 0.2 0 -0.2 -0.4 0
2
4
6
8
10
12
Time(Second)
14
16
18
20
b
0.8
Tracking State and Desired Signal
Fig. 7. Controller signal: classical SMC.
0.6
Tracking State:2 Desired Signal:2
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
0
2
4
6
8
10
12
Time(Second)
Fig. 8. Tracking state and desired signal – classical SMC – (a) state 1 and (b) state 2.
14
16
18
20
M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
Tracking State and Desired Signal
a
1 Tracking State:1 Desired Siognal:1
0.8 0.6 0.4 0.2 0 -0.2 -0.4 0
2
4
6
8
10
12
14
16
18
b
0.8
Tracking State and Desired Signal
3678
0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0
20
Tracking State:2 Desired Signal:2
2
4
6
Time(second)
8
10
12
14
16
18
20
Time(Second)
Fig. 9. Tracking state and desired signal – our proposed method – (a) state 1 and (b) state 2.
a
b
Controller Signal
5
0
-5
Controller Signal
Controller Signal
0
Controller Signal
5
-10 -15 -20
-5 -10 -15
-25 -20 -30 -35
-25 0
2
4
6
8
10
12
Time(Second)
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
Time(second)
Fig. 10. Controller signal – (a) our proposed method and (b) classsical SMC.
In our method, the simulation results show that the trajectory of error dynamics converges to S = 0 and the error vector also converges to the origin. Overall, the method performs well and the nonlinear system is indeed achieving stabilization even in presence of uncertainties and external disturbance. Table 3 shows that the classical SMC has better performance rather than our proposed methodology. It is expectable because in our scheme, it is assumed that the structure of the system is unknown while in the classical SMC it is considered as a known term. The main contribution of our proposed method comparing to the classical SMC is that not only the chattering phenomenon is eliminated but the asymptotical stability of the system is guaranteed as well. Our methodology has superior application rather than [46], because in problem formulation considered in the current article, the structure of the system is assumed unknown meanwhile the bound of uncertainty and external disturbances is not required in advance. To overcome the problems caused by unknown structures and external disturbance, in [45] a fuzzy identifier was used to make the approximated equivalent control part, however the reaching part is similar to the classical SMC. The chattering problem which is appeared can be omitted by using the saturation function. As mentioned above, in methods that eliminate the chattering such as boundary layer and saturation function and,. . ., the asymptotical stability of the system would not guarantee. Besides, it is noticeable that the bounds of unknown structures and disturbance were assumed to be known where the mentioned condition in our proposed methodology is relaxed. 7. Conclusion In this paper, an adaptive fuzzy sliding mode control method was proposed for a class of uncertain nonlinear systems. Using the framework of sliding mode control, we proposed two fuzzy systems for reaching and equivalent parts of SMC. One of the fuzzy systems was used to handle uncertainty/disturbance and the other to eliminate the chattering of the control signal. The main advantage of our proposed methodology is that the structure of the system is unknown and no knowledge of the bounds of parameters, uncertainties and external disturbance are required in advance. We proved that the proposed method is globally stable in the Lyapunov sense. Using a system having modeling uncertainties and disturbance, we showed
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that the our proposed scheme is capable of tracking a desired signal with a good accuracy and has comparable performance with other recent methods proposed in the literature.
Appendix A. Proof of Theorem 1 Consider the following Lyapunov function:
V¼
1 2 1 T 1 T 1 ^ 2; U Uf þ U Ug þ ðkafs kÞ s þ 2 2cf f 2cg g 2ck
ðA:1Þ
where Uf ¼ hf hf , Ug ¼ hg hg . cf, cg and ck are arbitrary positive constants. To derive the adaptive law for adjusting hf and hg, we first define the optimal parameter vectors of hf and hg as follows:
"
#
hf ¼ arg min sup j f ðXjhf Þ f ðX; tÞ j ; hf 2Xf
X2U X
"
ðA:2Þ
#
hg ¼ arg min sup j gðXjhg Þ gðX; tÞ j : hg 2 Xg
ðA:3Þ
X2U X
Define the minimum approximation error as
w ¼ ½f ðX; tÞ ^f ðXjhf Þ þ ½gðX; tÞ g^ðXjhg Þu:
ðA:4Þ
Lemma 1. If assumption 1 and 2 are satisfied, then w 2 L1. Proof. From assumptions 1 and 2, we have
j w j6j f ðX; tÞ ^f ðXjhf Þ j þ j gðX; tÞ g^ðXjhg Þ jj uðtÞ j6j f ðX; tÞ j þ j ^f ðXjhf Þ j þ½j gðX; tÞ j þ j g^ðXjhg Þ j j uðtÞ j T 6 hT f knðXÞkþ j f ðX; tÞ j þ½j gðX; tÞ j þhg knðXÞk j uðtÞ j6 mf þ F þ ½mg þ g max ðmax j uðtÞ jÞ 6 q t
ðA:5Þ
where q = mf + F + [mg + gmax](maxtju(t)j). Since the control input u is designed as a bounded signal, Therefore, wmust be bounded (i.e., w 2 L1). The time derivative of (A.1) is
1 1 ^ k_ _ f þ 1 UT U _ ðkafs kÞ V_ ¼ ss_ þ UTf U afs g g þ
cf
cg
" ¼ s e_ n þ
n1 X
ck
# ci e_ i þ
i¼1
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
" ¼ s f ðX; tÞ þ gðX; tÞu þ dðtÞ
ðnÞ xd ðtÞ
þ
1
ck
n1 X
^ k_ ðkafs kÞ afs #
ci e_ i
i¼1
þ
1
UTf U_ f þ
cf
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
( ¼s
½f ðX; tÞ ^f ðXjhf Þ þ ^f ðXjhf Þ þ ½gðX; tÞ g^ðXjhg Þu þ g^ðXjhg Þu:
þdðtÞ
ðnÞ xd ðtÞ
þ
n1 X
) ci e_ i
þ
i¼1
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
( 6 sw þ s ^f ðXjhf Þ þ ½^f ðXjhf Þ ^f ðXjhf Þ þ ½g^ðX; tÞ g^ðXjhg Þu þ g^ðXjhg Þu: ðnÞ
þdðtÞ xd ðtÞ þ
n1 X i¼1
) ci e_ i
þ
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
( 6 sw s ½^f ðXjhf Þ ^f ðXjhf Þ þ ^f ðXjhf Þ þ ½g^ðX; tÞ g^ðXjhg Þu þ g^ðXjhg Þu:
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M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
ðnÞ
þdðtÞ xd ðtÞ þ
n1 X
) ci e_ i
þ
i¼1
(
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
6 sw s UTf nðXÞ þ UTg nðXÞu þ ^f ðXjhf Þ þ g^ðXjhg Þu: þdðtÞ
ðnÞ xd ðtÞ
þ
n1 X
) ci e_ i
i¼1
þ
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
ðA:6Þ
The designed control law is consisting of two parts as follows:
" # n1 X 1 ðnÞ ^ ueq ¼ ci eiþ1 þ xd ðtÞ ; f ðXjhf Þ g^ðXjhg Þ i¼1 ur ¼ kafs ufs : The overall controller is chosen as: u = ueq + ur. Using the proposed controller the above mentioned inequality becomes
n o 1 1 ^ k_ ; _ f þ 1 UT U _ 6 sw s UTf nðXÞ þ UTg nðXÞu þ dðtÞ þ kafs ufs þ UTf U ðkafs kÞ afs g g þ
cf
cg
^ k ^ j s j þb j s j þUT snðXÞ þ 1 h_ 6 q j s j j s j ðkafs kÞ f f
cf
ck
! T g
þU
1 snðXÞu þ h_ g
cg
! þ
1
ck
^ k_ ; ðkafs kÞ afs
h_ f ¼ cf snðXÞ; h_ g ¼ c snðXÞu; g
k_ afs ¼ ck j s j : ^ can be chosen in such a way that the value of q þ b k ^ remains negative (i.e., q þ b k ^ ¼ g It is clear that the scalar k where g > 0) [53]. Therefore
^ j s j6 g j s j : V_ 6 ðq þ b kÞ
ðA:7Þ
Using Barbalat’s lemma [5], it can be concluded that s; s_ 2 L1 and s approaches zero as t ? 1. It means, the system is stable and the error asymptotically converges to zero. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
Emelyanov SV. Variable structure control systems. Moscow: Nauka; 1967. Utkin VI. Sliding modes and their applications in variable structure systems. Moscow: Nauka; 1974. Itkis U. Control systems of variable structure systems. New York: Wiley; 1976. Utkin VI. Variable structure systems with sliding modes. IEEE Trans Automat Contr 1977;22:212–22. Slotine JE, Li W. Applied nonlinear control. Englewood Cliffs (NJ): Prentice-Hall; 1991. Bandyopadhyay B, Janardhanan S. Discrete-time sliding mode control. Berlin: Springer; 2006. Zhang DQ, Guo GX. Discrete-time sliding mode proximate time optimal seek control of hard disk drives. IEEE Proc-Control Theory Appl 2000;147:440–6. Zhang Y, Changxi J, Utkin VI. Sensorless sliding mode control of induction motors. IEEE Trans Ind Electron 2000;47:1286–97. Liu TS, Lee WS. A repetitive learning method based on sliding mode for robot control. IEEE Trans ASME 2000;22:40–8. Jafarov EM, Tasaltin R. Robust sliding-mode control for the uncertain MIMO Aircraft Model F-18. IEEE Trans Aerospace Electron Syst 2000;36:1127–41. Salamci M, Ozgoren MK. Sliding mode control with optimal sliding surfaces for missile autopilot design. J Guid Control Dynam 2000;23:719–27. Slotine JJ, Sastry SS. Tracking control of non-linear systems using sliding surfaces, with application to a robot arm. Int J Control 1983;38(2):465–92. Young KD, Utkin VI, Ozguner U. A control engineer’s guide to sliding mode control. IEEE Trans Contr Syst Technol 1999;7(3). Hung JY, Gao W, Hung J-C. Variable structure control: a survey. IEEE Trans Ind Electron 1993;40:2–22. Baik IC, Kim KH, Kim HS, Moon GW, Youn MJ. Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding control with sliding load torque observer. In: IEEE PESC’96 Record, vol. 2; 1996. p. 1242–47. Chern TL, Wu YC. Design of brushless DC position servo systems using integral variable structure approach. IEE Proc Electron Power Appl 1993;140:27–34. Lee JH, Ko JS, Chun SK, Lee JJ, Youn MJ. Design of continuous sliding mode controller for BLDDM with prescribed tracking performance. In: Conference Record, IEEE PESC’92; 1992. p. 770–75. Hwang GC, Lin SC. A stability approach to fuzzy control design for nonlinear systems. Fuzzy Set Syst 1992;48:79–287. Ishingame A, Furukawa T, Kawamoto S, Taniguchi T. Sliding mode controller design based on fuzzy inference for nonlinear systems. IEEE Trans Ind Electron 1993;40:64–70. Kandel A, Langholz G, editors. Fuzzy control systems. Boca Raton (FL): CRC Press; 1994. Kim SW, Lee J-J. Design of a fuzzy controller with fuzzy sliding surface. Fuzzy Set Syst 1995;71:359–67. Kung C-C, Liao CC. Fuzzy sliding mode control design for tracking control of nonlinear system. In: Proc of the American control conference, Baltimore, MD; 1994. p. 180–84. Kung C-C, Lin S-C. Fuzzy controller design: a sliding mode approach. In: Tzafestas S-G, Venetsanopoulos A-N, editors. Fuzzy reasoning in information, decision and control system. London: Kluwer Academic Publishers; 1994. p. 277–306. Lee CC. Fuzzy logic in control system: fuzzy logic controller Part I & II. IEEE Trans Syst Man Cybernet 1990;20:404–35. Li THS, Tsai CY. Parallel fuzzy sliding mode control of the cart-pole system. In: Proc of the IEEE IECON’95, Orlando, USA; 1995. p. 1468–73.
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Mamdani H. Applications of fuzzy algorithms for simple dynamic plants. Proc IEE 1974;121:1585–8. Palm R. Robust control by fuzzy sliding mode. Automatica 1994;30:1429–37. Qin SJ, Borders G. A multiregion fuzzy logic controller for nonlinear process control. IEEE Trans Fuzzy Syst 1994;2:74–81. Shao S. Fuzzy self-organizing controller and its application for dynamic processes. Fuzzy Set Syst 1988;26:151–64. Suyitno, Fujikawa J, Kobayashi H, Dote Y. Variable structured robust controller by fuzzy logic for servomotors. IEEE Trans Ind Electron 1993;40:80–8. Ting J-S, Li T-HS, Kung F-C. An approach to systematic design of the fuzzy control system. Fuzzy Set Syst 1996;27:151–66. Van Der Waal AJ. Application of fuzzy logic control in industry. Fuzzy Set Syst 1995;74:33–41. Wang L-X. Stable adaptive fuzzy control of nonlinear systems. IEEE Trans Fuzzy Syst 1993;1:146–55. Lin W-S, Chen C-S. Robust adaptive sliding mode control using fuzzy modeling for a class of uncertain MIMO nonlinear systems. IEE Proc Contr Theory Appl 2002;149:193–201. Su JP, Chen TM, Wang CC. Adaptive fuzzy sliding mode control with GA-based reaching laws. Fuzzy Set Syst 2001;120:145–58. Tao CW, Chan M-L, Lee T-T. Adaptive fuzzy sliding mode controller for linear systems with mismatched time-varying uncertainties. IEEE Trans Syst Man Cybernet—Part B 2003;33:283–93. Mamdani EH, Assilian S. An experiment in linguistic synthesis with a fuzzy logic controller. Int J Man–Machine Stud 1975;7(1):1–13. Zadeh LA. Fuzzy set. Inform Control 1965;8:338–53. Yu X, Man Z, Wu B. Design of fuzzy sliding mode control systems. Fuzzy Set Syst 1998;95:295–306. Shih MC, Lu CS. Fuzzy sliding mode position control of a ball screw driven by pneumatic servomotor. Mechtronics 1994;5:421–31. Ting C, Li TS, Kung F. Fuzzy sliding mode control of nonlinear system. In: Proc of the 3rd IEEE conference on Fuzzy systems, IEEE world congress on computational intelligence, vol. 3; 1996. p. 1620–25. Su CY, Stepanenko Y. Adaptive control of a class of nonlinear systems with fuzzy logic. IEEE Trans Fuzzy Syst 1994;2:285–94. Yoo B, Ham W. Adaptive fuzzy sliding mode control of nonlinear system. IEEE Trans Fuzzy Syst 1998;6:315–21. Wang J, Rad AB, Chan PT. Indirect adaptive fuzzy sliding mode control: Part I-Fuzzy switching. Fuzzy Set Syst 2001;122(August):21–30. Kung C-C, Chen T-H. Observer-based indirect adaptive fuzzy sliding mode control with state variable filters for unknown nonlinear dynamical systems. Fuzzy Set Syst 2005;155:292–308. Yau H-T, Chen C-L. Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems. Chaos Solitons Fract 2006;30:709–18. Kim SW, Lee JJ. Design of a fuzzy controller with fuzzy sliding surface. Fuzzy Set Syst 1995;71(3):359–67. Lee H, Kim E, Kang HJ, Park M. A new sliding-mode control with fuzzy boundary layer. Fuzzy Set Syst 2001;120(1):135–43. Berstecher RG, Palm R, Unbehauen HD. An adaptive fuzzy sliding-mode controller. IEEE Trans Ind Electron 2001;48:18–31. Isidori A. Nonlinear control systems. Berlin: Springer; 1995. Wang CH, Liu HL, Lin TC. Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems. IEEE Trans Fuzzy Syst 2002;10:39–49. Wang LX. A course in fuzzy systems and control. Englewood Cliffs (NJ): Prentice-Hall; 1997. Roopaei M, Zolghadri Jahromi M. Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control. Chaos 2008;18:033133.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Enhanced adaptive fuzzy sliding mode control for uncertain nonlinear systems Mehdi Roopaei a,*, Mansoor Zolghadri b,1, Sina Meshksar c a b c
School of Electrical and Computer Engineering, Shiraz University, P.O. Box 71955-177, Shiraz, Iran Centre for Computational Intelligence, De Montfort University, Leicester LE1 9BH, UK School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 9 August 2008 Received in revised form 23 January 2009 Accepted 23 January 2009 Available online 5 February 2009 PACS: 05.45.a Keywords: Fuzzy sliding mode control Nonlinear dynamics
a b s t r a c t In this article, a novel Adaptive Fuzzy Sliding Mode Control (AFSMC) methodology is proposed based on the integration of Sliding Mode Control (SMC) and Adaptive Fuzzy Control (AFC). Making use of the SMC design framework, we propose two fuzzy systems to be used as reaching and equivalent parts of the SMC. In this way, we make use of the fuzzy logic to handle uncertainty/disturbance in the design of the equivalent part and provide a chattering free control for the design of the reaching part. To construct the equivalent control law, an adaptive fuzzy inference engine is used to approximate the unknown parts of the system. To get rid of the chattering, a fuzzy logic model is assigned for reaching control law, which acting like the saturation function technique. The main advantage of our proposed methodology is that the structure of the system is unknown and no knowledge of the bounds of parameters, uncertainties and external disturbance are required in advance. Using Lyapunov stability theory and Barbalat’s lemma, the closed-loop system is proved to be stable and convergence properties of the system is assured. Simulation examples are presented to verify the effectiveness of the method. Results are compared with some other methods proposed in the past research. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The Variable Structure System (VSS) research was originated in early 1950s for single input systems with high order differential equations [1–3]. VSS was not very popular among control engineers prior to 1970s due to the absence of a systematic design procedure and high oscillation chattering in the control input. Although many scientists had conducted much research in the area of VSS, it was not until 1977 that the VSS concept was fully appreciated. The variable structure control with sliding mode was introduced to control engineers by Utkin [4]. The Sliding Mode Control (SMC) was originally developed for variable structure systems in the continuous domain. In his survey paper, Utkin presented a thorough description of the SMC theory in continuous time. Later, Slotine and Li [5] discussed continuous SMC in more detail. More recently, the research in the field of discrete time SMC has attracted many researchers [6]. SMC is an efficient tool to control complex high-order dynamic plants operating under uncertainty conditions due to its order reduction property and low sensitivity to disturbances and plant parameter variations. In SMC, the states of the controlled system are first guided to reside on a designed surface (i.e., the sliding surface) in state space and then keeping them there with a shifting law (based on the system states). * Corresponding author. Tel.: +989177137538. E-mail addresses: [email protected] (M. Roopaei), [email protected] (M. Zolghadri), [email protected] (S. Meshksar). 1 On sabbatical leave from Shiraz University. 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.01.029
M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
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There has been a wide variety of applications of SMC in areas such as robotics, power control, aerospace, and process control [7–11]. The most prominent property of the SMC is its insensitivity to parameter variations and external disturbances. However, its major drawback in practical applications is the chattering problem. Numerous techniques have been proposed to eliminate this phenomenon in SMC [12,13]. Conventional methods used to eliminate the chattering are to replace the relay control by a saturating approximation [14], integral sliding control [15–17] and Boundary Layer technique [5]. The boundary layer approach was introduced to eliminate the chattering around the switching surface and the control discontinuity within this thin boundary layer. If systems uncertainties are large, the sliding-mode controller would require a high switching gain with a thicker boundary layer to eliminate the higher resulting chattering effect. However, if we continuously increase the boundary layer thickness, we are actually reducing the feedback system to a system without sliding mode. To tackle these difficulties, fuzzy logic controllers are often used to deal with the discontinuous sign function in the reaching phase of SMC [18–33]. Recently, AFSMC methods are also used for this purpose, which is shown to be quite effective [34–36]. Fuzzy Logic Control (FLC) has been an active research topic in automation and control theory since the work of Mamdani [37] based on the fuzzy sets theory of Zadeh [38]. The basic concept of FLC is to utilize the qualitative knowledge of a system for designing a practical controller. Generally, FLC is applicable to plants that are ill-modeled, but qualitative knowledge of an experienced operator is available. It is particularly suitable for those systems which have uncertain or complex dynamics. Generally, in contrast to a conventional feedback control algorithm, there is a fuzzy control algorithm consists of a set of heuristic decision rules that can be represented as a non-mathematical control algorithm. This algorithm proves to be very effective especially when the precise model of the system under control is not available or expensive to prepare. The principle of SMC has been introduced in designing fuzzy logic controllers to guarantee the stability. This combination (i.e., FSMC) provides the mechanism to design robust controllers for nonlinear systems with uncertainty [39–41]. Designing adaptive fuzzy controllers by the integration of fuzzy logic and the SMC [42–44] for ensuring stability and consistent performance is a well known research topic. Many new algorithms have been proposed based on the integration of these control methods [45,46]. These approaches are similar in the aspect that they directly approximate the sliding mode control law by fuzzy approximators. One main advantage of this control scheme is its insensitivity to modeling uncertainty and external disturbances. Many AFSMC schemes have been proposed to eliminate the chattering using a fuzzy sliding surface in the reaching condition of the SMC [45–49]. In this article, a novel AFSMC algorithm is proposed for a class of continuous time unknown nonlinear systems. To design the hitting part of the SMC, a fuzzy controller is used. This will reduce the chattering and improve the robustness. An AFSMC is used (as equivalent control part of SMC) to approximate the unknown parts of the uncertain system. We provide the proof that closed-loop system is globally stable in the Lyapunov sense and the system output can track the reference signal in the presence of modeling uncertainties and external disturbances. The rest of this paper is organized as follows. Section 2 presents the system definitions and introduces the classical SMC design method. Sections 3 and 4 present the design method for the reaching and equivalent control parts, respectively. Section 5 presents the proof of asymptotic stability for the proposed method. Section 4 provides the simulation results and finally, conclusions are given in section 7.
2. System description and traditional SMC Consider nonlinear systems whose dynamical equations can be expressed in the canonical form [50]
x_ i ¼ xiþ1 ; 1 6 i 6 n 1; x_ n ¼ f ðx; tÞ þ dðtÞ þ gðx; tÞuðtÞ;
ð1Þ
where xðtÞ ¼ ½x1 ðtÞx2 ðtÞ xn ðtÞT 2 Rn is the state vector, f(x, t) and g(x, t) are two unknown functions belong to Rn ! R space. u(t), and dðtÞ 2 R are the control input and the external disturbance, respectively. Assumption 1 ([5,51,52]). The unknown functions f(x, t), g(x, t) and d(t) satisfy the following conditions:
jf ðX; tÞj 6 F < 1; 0 < g min 6 gðX; tÞ 6 g max < 1 and jdðtÞj 6 b; n
ð2Þ
n
where X 2 U X R , UX is a compact set defined as: U X ¼ fX 2 R : kXk 6 mx < 1g and F, b, g min and g max are unknown constants. The control problem is to force the system to track an n-dimensional desired vector Xd(t) (i.e. the nth-order tracking probðn1Þ lem of state xd(t) as discussed in [50]), X d ðtÞ ¼ ½xd1 ðtÞxd2 ðtÞ xdn ðtÞ ¼ ½xd ðtÞx_ d ðtÞ xd ðtÞ 2 Rn , which belong to a class of continuous functions in the interval [t0, 1]. The tracking error is defined as ðn1Þ
_ x_ d ðtÞ xðn1Þ ðtÞ xd EðtÞ ¼ XðtÞ X d ðtÞ ¼ ½xðtÞ xd ðtÞxðtÞ
_ eðn1Þ ðtÞ ¼ ½e1 ðtÞe2 ðtÞ en ðtÞ: ðtÞ ¼ ½eðtÞeðtÞ
ð3Þ
The control goal considered in this paper is that for any given target Xd(t), a SMC is designed such that the resulting state response of the tracking error vector satisfies the following condition:
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lim kEðtÞkt!1 ¼ limt!1 kXðtÞ X d ðtÞk ! 0;
ð4Þ
where kk denotes the Euclidean norm of a vector. In the following, we present a brief introduction to classical SMC design method. In traditional SMC, a switching surface representing the desired system dynamics is constructed as
S ¼ en þ
n1 X
c i ei :
ð5Þ
i¼1
The switching surface parameters {ci, i = 1, . . . , n 1} are chosen based on the following two criteria. First, the values are chosen to stabilize the system during sliding mode. Routh-Hurwitz criterion [5] is used to determine the range of coefficients cis’ that produce stable dynamics. That is, all the roots of the following characteristic polynomial describing the sliding surface have negative real parts with desirable pole placement:
PðkÞ ¼ kn þ cn1 kn1 þ þ c2 k þ c1 :
ð6Þ
Second, the values are chosen such that the system has fast and smooth response. When the closed loop system is in the sliding mode, it satisfies s_ ¼ 0 and the equivalent control law is obtained by
" # n1 X 1 ðnÞ ueq ¼ ci eiþ1 þ xd ðtÞ : f ðX; tÞ dðtÞ gðX; tÞ i¼1
ð7Þ
According to the Lyapunov stability theory [50], a Lyapunov function is defined as
V¼
1 2 s : 2
The derivative of
ð8Þ
v is
V_ ¼ ss_ ¼ s e_ n þ
n1 X
! ci e_ i :
ð9Þ
i¼1
In the above equation, if V_ is negative for all s – 0, then the so-called reaching condition [50] is satisfied. That is, the control u is designed to guarantee that the states are hitting on the sliding surface S = 0. In the traditional SMC, the reaching control law is selected as ur = kwuw and the overall control u is determined by
u ¼ ueq þ ur ¼ ueq þ kw uw ;
ð10Þ
where kw is the switching gain, which is positive and uw is obtained by
uw ¼ sgnðsÞ:
ð11Þ
It can be easily concluded that V_ 6 kw j s j and the states of the system are on the sliding surface. The error converges to zero once the states are on s = 0. The terms f(X, t), g(X, t) and d(t) are unknown in (7). In this paper, we use an adaptive fuzzy system to approximate the equivalent part. However, the sign function in overall control u will cause the control input to produce the chattering phenomenon. To overcome this problem, we use a fuzzy system to model the sign function. The overall scheme that we present in this paper to overcome the above-mentioned problems is based on the combination of the methods presented in [45] and [46].
3. FLC design for the reaching control part Fuzzy control (FC) has supplanted conventional technologies in many applications. One major feature of fuzzy logic is its ability to express the amount of ambiguity in human thinking. Thus, when the mathematical model of the process does not exist, or exists but with uncertainties, FC is an alternative way to deal with the unknown process. However, the huge amount of fuzzy rules for high-order systems makes the analysis complex. Therefore, much attention has focused on the FSMC. In this paper, in order to eliminate the chattering problem, fuzzy inference engine is used for reaching phase and fuzzy sliding mode control methodology is proposed. The main advantage of this method is that the robust behavior of the system is guaranteed. The second advantage of the proposed scheme is that the performance of the system in the sense of removing chattering is improved in comparison with the same SMC technique without using FLC. The configuration of our FSMC control scheme is shown in Fig. 1; it contains an equivalent control part and a two-inputsingle-output FSMC. The equivalent control part is the same as that in (7) and the reaching law is selected as [46,53]
ur ¼ kafs ufs ;
ð12Þ
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a
NB
-1
NM
-2/3
NS
-1/3
ZE
0
PS
PM
PB
1/3
2/3
1
b
ufs
NB
-1
NM
NS
ZE
PS
-2/3
-1/3
0
1/3
PM
2/3
PB
1
. s, s
_ Fig. 1. Fuzzy sets assigned to (a) output variable (ufs) and (b) input variables (S and SÞ.
Table 1 Rule-base of FSMC. ufs
S
S_
PB PM PS ZE NS NM NB
PB
PM
PS
ZE
NS
NM
NB
NB NB NB NB NM NS ZE
NB NB NB NM NS ZE PS
NB NB NM NS ZE PS PM
NB NM NS ZE PS PM PB
NM NS ZE PS PM PB PB
NS ZE PS PM PB PB PB
ZE PS PM PB PB PB PB
where kafs is the normalization factor of output variable, and ufs is the output of the FSMC, determined by the normalized _ The fuzzy control rules can now be represented as the mapping of the input linguistic variables S and S_ values of S and S. to output linguistic variable ufs as follows:
_ ufs ¼ FSMCðS; SÞ:
ð13Þ
The membership functions of the linguistic terms Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Medium (PB) and Positive Big (PB) that are assigned to the inputs (s and s_ Þ and the output (ufs) are shown in Fig. 1. The fuzzy rule table is designed as in Table 1 [53]. 4. AFLS design for the equivalent control part To approximate the unknown part (i.e., f(X, t)) of the equivalent control (7), an Adaptive Fuzzy Logic System (AFLS) is introduced in this section. A typical rule in the rule-base of the AFLS is as follows: ðmÞ
R^f
m m ^ : IF x1 is F m x1 and and xn is F xn ; THEN f ðxjhf Þ is F ^f ;
ð14Þ
m where m = 1, 2, . . . , Q is the index of the rule, F m xi ði ¼ 1; . . . ; nÞ denote the fuzzy sets assigned to xi (i = 1, . . . , n), and F ^f denotes ^ the fuzzy singletons assigned to the output f ðXjhf Þ of the fuzzy model. Using the singleton fuzzifier, product inference, and weighted average deffuzifier [52], the outputs of the fuzzy model ^f ðXjhf Þ can be expressed as
^f ðXjh Þ ¼ hT nðXÞ; f f
ð15Þ
½F ^1f ; F ^2f ; . . . ; F ^Qf T
1
2
Q
T
where hf ¼ is the adjustable parameters vector and n(X) = [n (X), n (X), . . . , n (X)] is the vector of fuzzy basis functions [52] defined as
Qn j F xi ðxi Þ hi¼1 i; Qn j Q j¼1 i¼1 F xi ðxi Þ
ni ðXÞ ¼ P
j ¼ 1; 2; . . . ; Q;
ð16Þ
where F jxi ðxi Þ denotes the membership function value of xi in F jxi . By the similar procedure, the function g(X,t) can be approximated as follows:
g^ðXjhg Þ ¼ hTg nðXÞ:
ð17Þ
The fuzzy approximators (15) and (17) can construct the functions f and g with good accuracy only when the following assumption holds. Assumption 2 [51] . The parameter hf belongs to the compact set Xf that is defined as Xf ¼ fhf 2 RQ jkhf k 6 mf g and Xg ¼ fhg 2 RQ jkhg k 6 mg g, where mf and mg are finite positive constants.
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u eq Equivalent Control
Slave System
x Master System
y
u fs Sliding Function
ur
FSMC(S)
Adaptive Filter
k afs
Fig. 2. Block diagram of the AGFSMC scheme.
From (15) and by the assumptions (1) and (2), we can also assume that j ^f ðXjhÞ j6 F and 0 < g min 6 g^ðX j hÞ 6 g max < 1. In practical systems, external disturbance d(t)is unknown. Thus, fuzzy models are used to construct the equivalent control law by replacing (7) with the following equation [52]:
" # n1 X 1 ðnÞ ^ ueq ¼ ci eiþ1 þ xd ðtÞ : f ðXjhf Þ g^ðXjhg Þ i¼1
ð18Þ
The overall control u is chosen as
u ¼ ueq þ ur ¼ ueq þ kafs ufs ; " # n1 X 1 ðnÞ ci eiþ1 þ xd ðtÞ þ kafs ufs : u ¼ ueq þ ur ¼ ^f ðXjhf Þ g^ðXjhg Þ i¼1
ð19Þ ð20Þ
In (19), hf, hg and kafs are updated as follows:
h_ f ¼ cf snðXÞ; h_ g ¼ c snðXÞu; g
ð21Þ
k_ afs ¼ ck jsj; where in the above relation cf, cg and ck are positive and arbitrary constants. The block diagram of our proposed AFSMC scheme is shown in Fig. 2. As shown, it uses a fuzzy inference approximator to construct the equivalent control part (17) and a two-input-single-output FSMC to construct the hitting control part using (12) and (13). 5. Stability analysis In the following theorem, the proposed scheme (20) will be proved to be able to drive the nonlinear system (1) onto the sliding surface s(t) = 0. That is, the reaching condition sðtÞs_ ðtÞ < 0 is guaranteed. Theorem 1. Assume that the uncertain nonlinear system (1) is controlled by uðtÞ in (20), where ueq is (18) and, ur is (12). Then, the error state trajectory converges to the sliding surface sðtÞ ¼ 0 (see Appendix-A for proof of theorem). 6. Simulation results In this section, the performance of the proposed approach is evaluated for a well-known benchmark problem. In the first experiment, we give the simulation results of our proposed adaptive fuzzy controller in a stabilizing problem. In the second, the performance of the schemes is investigated in a tracking problem. Furthermore, we also compare our method with the classical SMC. All the simulation procedures are implemented within MATLAB with the step size 0.001. 6.1. Experiment 1: stabilizing problem The inverted pendulum system is shown in Fig. 3. The state equations can be expressed by
where
x_ 1 ¼ x2 ; x_ 2 ¼ f þ gu þ d;
ð22Þ
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M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681 mLx2 sinðx Þ cosðx1 Þ
f ¼
1 g r sinðx1 Þ 2 mþM cosðx1 Þ L 43 mmþM
m cosðx1 Þ
;
g ¼ mþM cosðx1 Þ L 43 mmþM
and d ¼ 1 þ sinðp2 tÞ. The objective is to generate an appropriate actuator force u to control the motion of the cart, such that the pole can be balanced in the vertical position. The parameters values used for the cart and pole system are given in Table 2. The sliding surface is designed as: s = e2 + e1. In the simulations, the initial conditions values are selected as: T Xð0Þ ¼ ½ p=4 0 . If we require that jx1j 6 p/6 and jx2j 6 p/6, then the bounds bmax and bmin can be calculated as jg(x1, x2)j 6 1.46 = bmax and jg(x1, x2)j P 1.12 = bmin. Also, since j f ðx1 ; x2 Þ j6 15:78 þ 0:0366x22 , we can set F = 16. To construct the equivalent control part, ^f ðX; tÞ and g^ðX; tÞ were selected as (15) and (17), respectively. In simulations, we assigned the following Gaussian membership functions to xi (i = 1, 2) over the intervals [3, 3]:
lNB ðxÞ ¼ 1=ð1 þ expð5ðx þ 2ÞÞÞ; lNM ðxÞ ¼ expððx þ 1:5Þ2 Þ; lNS ðxÞ ¼ expððx þ 0:5Þ2 Þ; lPS ðxÞ ¼ expððx 0:5Þ2 Þ; lPM ðxÞ ¼ expððx 1:5Þ2 Þ; lPB ðxÞ ¼ 1=ð1 þ expð5ðx 2ÞÞÞ:
ð23Þ
To cover the entire space, we used 36 fuzzy rules. The initial values of hf(0) and hg(0) were chosen randomly in the interval [2, 2]. In the next step, the vector of fuzzy basic functions, n(X) = [n1(X), n2(X), . . . , nQ(X)]T, were constructed by (16). The learning rates were set to cf, cg and ck. The hitting control part ufs was specified using (13) and the value of kafs = 0.5 were used in simulations. Fig. 4 displays the state trajectories of system (22) with the control law (20). In this figures, one can see that the system states converge to zero rapidly as the system starts, and hence the asymptotical stabilization of the unified chaotic system is achieved. The control signal is also shown in Fig. 5. The trajectories and controller signal of system (22) by applying classical SMC are demonstrated in Figs. 6 and 7, respectively. 6.2. Experiment 2: tracking problem Let us consider the system (22) again. In this experiment, disturbance is considered as: d = 0.1(1 + sin(0.25t)) and the de T sired reference trajectories as: X d ðtÞ ¼ ½xd1 ðtÞ; xd2 ðtÞT ¼ p9 ðsinðtÞ þ 0:2 sinð3tÞÞ; p9 ðcosðtÞ þ 0:6 cosð3tÞÞ . The sliding surface is T s = e2 + e1. In the following simulations, the initial conditions values are selected as: Xð0Þ ¼ ½ p=4 0 The trajectories of the states’ system x1(t) and x2(t) are plotted in Figs. 8 and 9, respectively. As it can be seen, system (22) tracks the desired signals Xd(t) quickly. In Fig. 10, it is demonstrated that there is no chattering in the control signal of the proposed method. Figs. 8–10 show the simulation results for our proposed and conventional controller. The chattering can be reduced using various methods, in order to have a fair comparison between classical and our proposed SMC, saturation function is used. The parameters of various schemes were tuned manually in a way that have approximately the same control effort. In order to have a quantitative comparison of tracking error, in Table 3, we have reported the Integral Absolute Error (IAE) of different schemes. The sliding surface s = e2 + e1 was used for both methods and the thickness of the saturation function was specified as 0.1 for the conventional SMC method. The results of Figs. 3–5 and Table 1 clearly show that our proposed method has comparable performance in comparison with the classical SMC.
Table 2 The parameters values and functions used for the cart and pole system. gr (acceleration due to gravity) 2
9.81 m/s
L (half-length of the pole)
M (mass of the cart)
m (mass of the pole)
0.5
1.0 kg
0.1 kg
Table 3 Integral Absolute Error (IAE) values for various methods. IAE
Method Our method
Classical SMC
Tracking error
1.8475e + 003
1.6408e + 003
M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
Fig. 3. The inverted pendulum system.
1 State1 State 2 0.5
States
0
-0.5
-1
-1.5
-2
0
1
2
3
4
5
6
7
8
9
10
Time(second) Fig. 4. States trajectories: our proposed methodology.
0 controller Signal -5
Controller Signal
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-10
-15
-20
-25
-30
0
1
2
3
4
5
6
7
8
Time(second) Fig. 5. Controller signal: our proposed methodology.
9
10
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0.8 State 1 State 2
0.6 0.4
States
0.2 0 -0.2 -0.4 -0.6 -0.8
0
1
2
3
4
5
6
7
8
9
10
Time,sec Fig. 6. States trajectories: classical SMC.
0 Controller Signal
Control signal
-5
-10
-15
-20
-25
0
1
2
3
4
5
6
7
8
9
10
Time,sec
Tracking State and Desired Signal
a
1 Tracking State:1 Desired State:1
0.8 0.6 0.4 0.2 0 -0.2 -0.4 0
2
4
6
8
10
12
Time(Second)
14
16
18
20
b
0.8
Tracking State and Desired Signal
Fig. 7. Controller signal: classical SMC.
0.6
Tracking State:2 Desired Signal:2
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
0
2
4
6
8
10
12
Time(Second)
Fig. 8. Tracking state and desired signal – classical SMC – (a) state 1 and (b) state 2.
14
16
18
20
M. Roopaei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3670–3681
Tracking State and Desired Signal
a
1 Tracking State:1 Desired Siognal:1
0.8 0.6 0.4 0.2 0 -0.2 -0.4 0
2
4
6
8
10
12
14
16
18
b
0.8
Tracking State and Desired Signal
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0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0
20
Tracking State:2 Desired Signal:2
2
4
6
Time(second)
8
10
12
14
16
18
20
Time(Second)
Fig. 9. Tracking state and desired signal – our proposed method – (a) state 1 and (b) state 2.
a
b
Controller Signal
5
0
-5
Controller Signal
Controller Signal
0
Controller Signal
5
-10 -15 -20
-5 -10 -15
-25 -20 -30 -35
-25 0
2
4
6
8
10
12
Time(Second)
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
Time(second)
Fig. 10. Controller signal – (a) our proposed method and (b) classsical SMC.
In our method, the simulation results show that the trajectory of error dynamics converges to S = 0 and the error vector also converges to the origin. Overall, the method performs well and the nonlinear system is indeed achieving stabilization even in presence of uncertainties and external disturbance. Table 3 shows that the classical SMC has better performance rather than our proposed methodology. It is expectable because in our scheme, it is assumed that the structure of the system is unknown while in the classical SMC it is considered as a known term. The main contribution of our proposed method comparing to the classical SMC is that not only the chattering phenomenon is eliminated but the asymptotical stability of the system is guaranteed as well. Our methodology has superior application rather than [46], because in problem formulation considered in the current article, the structure of the system is assumed unknown meanwhile the bound of uncertainty and external disturbances is not required in advance. To overcome the problems caused by unknown structures and external disturbance, in [45] a fuzzy identifier was used to make the approximated equivalent control part, however the reaching part is similar to the classical SMC. The chattering problem which is appeared can be omitted by using the saturation function. As mentioned above, in methods that eliminate the chattering such as boundary layer and saturation function and,. . ., the asymptotical stability of the system would not guarantee. Besides, it is noticeable that the bounds of unknown structures and disturbance were assumed to be known where the mentioned condition in our proposed methodology is relaxed. 7. Conclusion In this paper, an adaptive fuzzy sliding mode control method was proposed for a class of uncertain nonlinear systems. Using the framework of sliding mode control, we proposed two fuzzy systems for reaching and equivalent parts of SMC. One of the fuzzy systems was used to handle uncertainty/disturbance and the other to eliminate the chattering of the control signal. The main advantage of our proposed methodology is that the structure of the system is unknown and no knowledge of the bounds of parameters, uncertainties and external disturbance are required in advance. We proved that the proposed method is globally stable in the Lyapunov sense. Using a system having modeling uncertainties and disturbance, we showed
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that the our proposed scheme is capable of tracking a desired signal with a good accuracy and has comparable performance with other recent methods proposed in the literature.
Appendix A. Proof of Theorem 1 Consider the following Lyapunov function:
V¼
1 2 1 T 1 T 1 ^ 2; U Uf þ U Ug þ ðkafs kÞ s þ 2 2cf f 2cg g 2ck
ðA:1Þ
where Uf ¼ hf hf , Ug ¼ hg hg . cf, cg and ck are arbitrary positive constants. To derive the adaptive law for adjusting hf and hg, we first define the optimal parameter vectors of hf and hg as follows:
"
#
hf ¼ arg min sup j f ðXjhf Þ f ðX; tÞ j ; hf 2Xf
X2U X
"
ðA:2Þ
#
hg ¼ arg min sup j gðXjhg Þ gðX; tÞ j : hg 2 Xg
ðA:3Þ
X2U X
Define the minimum approximation error as
w ¼ ½f ðX; tÞ ^f ðXjhf Þ þ ½gðX; tÞ g^ðXjhg Þu:
ðA:4Þ
Lemma 1. If assumption 1 and 2 are satisfied, then w 2 L1. Proof. From assumptions 1 and 2, we have
j w j6j f ðX; tÞ ^f ðXjhf Þ j þ j gðX; tÞ g^ðXjhg Þ jj uðtÞ j6j f ðX; tÞ j þ j ^f ðXjhf Þ j þ½j gðX; tÞ j þ j g^ðXjhg Þ j j uðtÞ j T 6 hT f knðXÞkþ j f ðX; tÞ j þ½j gðX; tÞ j þhg knðXÞk j uðtÞ j6 mf þ F þ ½mg þ g max ðmax j uðtÞ jÞ 6 q t
ðA:5Þ
where q = mf + F + [mg + gmax](maxtju(t)j). Since the control input u is designed as a bounded signal, Therefore, wmust be bounded (i.e., w 2 L1). The time derivative of (A.1) is
1 1 ^ k_ _ f þ 1 UT U _ ðkafs kÞ V_ ¼ ss_ þ UTf U afs g g þ
cf
cg
" ¼ s e_ n þ
n1 X
ck
# ci e_ i þ
i¼1
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
" ¼ s f ðX; tÞ þ gðX; tÞu þ dðtÞ
ðnÞ xd ðtÞ
þ
1
ck
n1 X
^ k_ ðkafs kÞ afs #
ci e_ i
i¼1
þ
1
UTf U_ f þ
cf
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
( ¼s
½f ðX; tÞ ^f ðXjhf Þ þ ^f ðXjhf Þ þ ½gðX; tÞ g^ðXjhg Þu þ g^ðXjhg Þu:
þdðtÞ
ðnÞ xd ðtÞ
þ
n1 X
) ci e_ i
þ
i¼1
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
( 6 sw þ s ^f ðXjhf Þ þ ½^f ðXjhf Þ ^f ðXjhf Þ þ ½g^ðX; tÞ g^ðXjhg Þu þ g^ðXjhg Þu: ðnÞ
þdðtÞ xd ðtÞ þ
n1 X i¼1
) ci e_ i
þ
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
( 6 sw s ½^f ðXjhf Þ ^f ðXjhf Þ þ ^f ðXjhf Þ þ ½g^ðX; tÞ g^ðXjhg Þu þ g^ðXjhg Þu:
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ðnÞ
þdðtÞ xd ðtÞ þ
n1 X
) ci e_ i
þ
i¼1
(
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
6 sw s UTf nðXÞ þ UTg nðXÞu þ ^f ðXjhf Þ þ g^ðXjhg Þu: þdðtÞ
ðnÞ xd ðtÞ
þ
n1 X
) ci e_ i
i¼1
þ
1
cf
UTf U_ f þ
1
cg
UTg U_ g þ
1
ck
^ k_ ðkafs kÞ afs
ðA:6Þ
The designed control law is consisting of two parts as follows:
" # n1 X 1 ðnÞ ^ ueq ¼ ci eiþ1 þ xd ðtÞ ; f ðXjhf Þ g^ðXjhg Þ i¼1 ur ¼ kafs ufs : The overall controller is chosen as: u = ueq + ur. Using the proposed controller the above mentioned inequality becomes
n o 1 1 ^ k_ ; _ f þ 1 UT U _ 6 sw s UTf nðXÞ þ UTg nðXÞu þ dðtÞ þ kafs ufs þ UTf U ðkafs kÞ afs g g þ
cf
cg
^ k ^ j s j þb j s j þUT snðXÞ þ 1 h_ 6 q j s j j s j ðkafs kÞ f f
cf
ck
! T g
þU
1 snðXÞu þ h_ g
cg
! þ
1
ck
^ k_ ; ðkafs kÞ afs
h_ f ¼ cf snðXÞ; h_ g ¼ c snðXÞu; g
k_ afs ¼ ck j s j : ^ can be chosen in such a way that the value of q þ b k ^ remains negative (i.e., q þ b k ^ ¼ g It is clear that the scalar k where g > 0) [53]. Therefore
^ j s j6 g j s j : V_ 6 ðq þ b kÞ
ðA:7Þ
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