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Journal of Intelligent & Fuzzy Systems 26 (2014) 2567–2576 DOI:10.3233/IFS-130928 IOS Press
Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems using bacterial foraging optimization Alireza Alfia,b,∗ , Ali Akbarzadeh Kalata,b and Mohammad Hassan Khoobanc a Faculty
of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood, Iran and Artificial Intelligence Research Center, Shahrood University of Technology, Shahrood, Iran c Department of Electrical Engineering, Sarvestan Branch, Islamic Azad University, Sarvestan, Iran b Automation
Abstract. This paper introduces a novel hybrid control strategy, namely an optimal adaptive fuzzy sliding mode (OAFSM) control scheme, to realize the synchronization of general uncertain chaotic systems in master-slave configuration. The proposed controller not only guaranties the stability and robustness against the lumped uncertainties caused by unmodeled dynamics and external disturbances, but also significantly reduces the control chattering inherent in conventional sliding mode control. The chaos synchronization is obtained by optimal proper choice of the control parameters including the sliding surface and the reaching law parameters. To achieve this, a bio-mimetic algorithm namely bacterial foraging optimization algorithm (BFOA) is employed. An illustrative example is given to demonstrate the validity and confirm the performance of the proposed scheme. Keywords: Chaos synchronization, sliding mode control, fuzzy systems, lyapunov stability theory, uncertain nonlinear systems, bacterial foraging optimization algorithm
1. Introduction Chaos synchronization problem means making two systems namely master and slave oscillate in a synchronized manner [1]. Since it has many potential application fields in secure communication and information processing, chaos synchronization has been developed extensively in the past two decades. To synchronize chaotic systems, a lot of control strategies have been introduced such as backstepping [2], pinning control [3], sliding mode control (SMC) [4, 5], impulsive control [6], and optimal control [7, 8]. ∗ Corresponding author. Alireza Alfi. Tel./Fax: +98 273 3300250; E-mail: a
[email protected].
The parameter deviation between the master and slave systems is commonly considered as an uncertainty. This is why the robustness of the controller is paid so much attention in recent years. However, when there are both parameter uncertainties and the external disturbances in the master and salve systems, it will be more difficult to realize chaos synchronization [9]. SMC is one of the most well-known strategies as a robust nonlinear control technique which guarantees the stability and robustness of the system [10, 11]. Nevertheless, this strategy can offer strong robustness to system parameter uncertainties and external disturbances, but it inherits a discontinuous control action and hence chattering phenomena will take place when the system operates near the sliding surface. Generally,
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A. Alfi et al. / Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems
there is a trade-off between the chattering phenomenon and robustness. For these above reasons, to reduce the chattering in SMC, adaptive fuzzy control, as a promising way of approaching nonlinear control problems, has attracted many researchers in recent years [12]. To improve the robustness of the system, different schemes have been also proposed regarding the design of the adaptive fuzzy sliding mode controller (AFSMC) [13, 14], which integrates the SMC into the adaptive fuzzy controller design approach. Based on the synchronization error between master and slave systems, first a sliding surface must be defined. Thus, the synchronizing problem becomes to design a control law that forces the state error vector to a sliding surface in finite time and to remain on this surface. Based on this, the parameters of switching function and the reaching law are significant factors to impact system performance. Accordingly, we face with choosing an arbitrary set of the coefficients for the error dynamic. It is unclear how the locations of coefficients relate to the controller performance. It seems that a control engineer attempts several different coefficients by trial and error until a desirable performance is obtained. Therefore, there exists an optimization problem in this case. The most elegant and precise numerical methods to solve this problem are gradient descents, which may get trapped at local optimum depending on the initial guess of solution. To achieve a good final result, the proper initial guesses for control variable trajectory are required. To overcome this shortage, many bio-inspired computational methodologies such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) have been proposed and applied to various real optimization problems [15–20]. They can be a promising alternative to traditional techniques. In recent years, a new and rapidly growing subject– Bacterial Foraging Algorithm (BFA) [21] developed by Prof. Passion in 2002 has emerged a novel modern Biomimetic optimization algorithms based on the behavior of Escherichia coli (E. coli) bacteria. Bio-mimetic optimization algorithms such as BFOA are populationbased algorithms, which are developed from simulation the evolutionary process and the behaviors of biology. They employ the direct information fitness instead of individual’s ability to adapt to the environment. These individuals are manipulated over many generations by ways of mimicking social behavior of biology, in an effort to find the optimal solution. Compared with other optimization algorithms, the main advantages of biomimetic optimization algorithms are: (i) they are simple and easy to implementation, because of concerns only
with basic mathematical operations; (ii) they do not demand to meet the requirement of differentiability and convexity for the problem in hand and (iii) they are more compatible and robust, because of the individual components are distributed and autonomous, there is no central control, and the fault of an individual cannot influence solving the problem. Based on aforementioned, the goal of this paper is to introduce an optimal adaptive fuzzy sliding mode control (OAFSMC) design for synchronizing a class of non-identical nonlinear chaotic systems subject to the system uncertainties and external disturbances. To this end, BFOA is employed to determine the appropriate coefficients of the parameters of the sliding surface and the reaching law. It solves perfectly the contradiction between favorable quality and high-frequency chattering existing in traditional SMC. The rest of this paper is organized as follows. Section 2 presents the problem statement. The preliminaries including the basic idea of SMC and AFSMC are given in section 3. BFOA is illustrated in section 4. In Section 5, the proposed OAFSMC controller is designed for chaos synchronization of nonlinear chaotic systems in general master-slave configuration. In section 6, a numerical example is provided to illustrate the effectiveness of the proposed method. It is shown that the proposed controller design can attenuated the chattering phenomenon efficiently whereas the robust performance is ensured. Eventually, section 7 presents conclusions.
2. Problem formulation In order to explore the chaos synchronization problem, the following systems in master-slave configuration are considered. x(t) ˙ = Ax(t) + Bg(x) y(t) ˙ = Ay(t) + B[f (y) + u(t) + d(t)]
(1)
where x = [x, x˙ , . . . , x(n−1) ]T = [x1 , x2 , . . . , xn ]T and y = [y, y˙ , . . . , y(n−1) ]T = [y1 , y2 , . . . , yn ]T are n-dimensional state vectors of the master and slave system, respectively, g(x) and f (y) are unknown but bounded nonlinear functions, u ∈ R is the control input of the slave system and d ∈ R denotes a external bounded disturbance. Finally, A and B are the constant matrices as
A. Alfi et al. / Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems
⎡
0 ⎢0 ⎢ ⎢. ⎢. . A=⎢ ⎢ ⎢. ⎢ .. ⎣ 0
1 0 .. . .. . 0
⎤ 0 ··· 0 1 ··· 0⎥ ⎥ .. . . .. ⎥ ⎥ . . ⎥, . ⎥ .. .. ⎥ . ··· . ⎥ ⎦ ··· ··· 0
⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢.⎥ B = ⎢ .. ⎥ ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 1
s(e) = CT e
(2)
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(6)
where e is the synchronization error given in (3) and C = [c1 , . . . , cn−1 , cn ]T > 0 should satisfy Hurwitz polynomial. It means that all the roots of polynomial cn λn + cn−1 λn−1 + . . . + c1 = 0 are in open left half-plane. Here, λ shows the Laplace operator.
Assumption 1. The disturbance is assumed to be bounded as |d(t)| ≤ α1 where α1 is a positive constant.
Assumption 2. Without loss of generality, assume that cn = 1.
The synchronization problem is to obtain the controller u(t) such that the trajectories of the slave system follow that of the master one. Following this purpose, define the synchronization error e as
The system is controlled in such a way that the state always moves towards the sliding surface and hits it. The sign of the control value must change at the intersection between the state trajectory and sliding surface. The SMC control procedure is contained two phases: (1) the sliding phase, (2) the reaching phase. Based on different phases, two types of control law must be separately designed. If the nonlinear functions f (y) and g(x) are exactly known and free of external disturbance, i.e. d = 0, one can design the following equivalent controller
e = [y − x, y˙ − x˙ , . . . , y(n−1) − x(n−1) ]T = [e, e˙ , . . . , en−1 ]T
(3)
Then the dynamics of synchronization error between master and slave systems given in (2) can be described by e˙ = Ae + B[f (y) − g(x) + u + d]
(4)
To simplify in presentation, define f (z) = f (y)−g(x), which is a bounded nonlinear function. Therefore, the error dynamics becomes e˙ = Ae + B[f (z) + u + d]
ueq = us − f (z)
where us is the sliding mode control input Differentiating (6) with respect to time and using (7) yields
(5)
s˙ (e) = CT e˙ = CT Ae + CT Bus
Based on this, the control problem in this paper is that the two coupled systems given in (1) to be synchronized by designing an appropriate input u such that lim e = 0
t→∞
where * is the Euclidean norm of a vector. In other words, the problem of synchronization between the master and the slave systems can be transformed into the one that how to realize the stabilization of the tracking error given in (5) at the origin.
3. Preliminaries 3.1. Overview of sliding mode controller design In the following, a brief introduction to classical SMC design method is illustrated. In traditional SMC, the states of controlled system are first guided to reside on a switching surface (sliding surface). Define the sliding surface as
(7)
= CT Ae + us
(8)
From assumption 2, it can be seen that CT B = 1. On the other hand, if the system operates in the sliding mode, the controlled system satisfies the following conditions: s(e) = 0
(9)
s˙ (e) = 0
(10)
and
Considering (8) and (10), the equivalent control can be written as us = −CT Ae
(11)
On the contrary, in the reaching phase there exists s(e) = / 0. Thereby, to design the sliding mode controller, we can utilize the following reaching law [15]. s˙ = −ks − q sgn(s)
(12)
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where
A. Alfi et al. / Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems
⎧ 1 if s > 0 ⎪ ⎨ 0 if s = 0 sgn(s) = ⎪ ⎩ −1 if s < 0
∗
β = arg Min
β∈β
Sup |βη(s, s˙ ) − f (z)|
Z∈z
(17)
(13)
where β and z denote the sets of suitable bounds of β and z, respectively.
and k and q are a positive constants such that the sliding condition is satisfied and the sliding mode motion will occur. Using Equations (8) and (12), the sliding mode control will be stated as
Assumption 3. According to the universal approximation theorem [16], β∗ is a bounded quantity so that ∗ β ≤ βM (18)
us = −CT Ae − ks − q sgn(s)
(14)
However, as the function f (z) is unknown, the control law (14) is generally inapplicable. A sound solution is that fuzzy inference system is utilized to approximate unknown functions. To this end, in the next subsection we introduce the adaptive fuzzy sliding mode control using the fuzzy inference system. Furthermore, the adaptive laws to adjust corresponding parameters will be derived. 3.2. Adaptive fuzzy sliding mode control It is evident that the fuzzy systems can uniformly approximate nonlinear continuous functions to an arbitrary accuracy. Here, the following lemma is presented. Lemma 1. For any given continuous function P(x) on a compact set U ∈ Rn and an arbitrary >0, there exists a fuzzy inference systems such that ˆ < ε [16]. Sup P(x) − P(x) x∈U
To approximate the nonlinear function f (z), we consider r µj (z)αi j=1 η(z) = r fˆ (z) = βη(z); (15) µj (z) j=1
where αi and µi (z) are the weight center and firingstrength of rule i, respectively. To approximate f (z) over compact set z , the sliding variables s and s˙ are utilized as input signals (z = [s, s˙ ]) to design approximator fˆ (z) = βη(z). To this end, the following approximator with arbitrary accuracy is defined. f (z) = β∗ η(s, s˙ ) + ε (16) In (16), is the minimum approximator error for optimal estimator and β∗ is the optimal constant factor defined as follows:
let the minimum approximation error as β˜ = βˆ − β∗
(19)
where βˆ denotes the estimated of β. Accordingly, the real control input control can be considered as ˆ u(t) = us (t) − βη(s, s˙ )
(20)
Then the error dynamics is given by ˆ e˙ = Ae + B[f (z) + us − βη(s, s˙ ) + d] ˆ s˙ ) + ω] = Ae + B[us − βη(s,
(21)
where ω = f (z) + d is the bounded lumped uncertainty, |ω| ≤ ωM . Thereby, we have ˜ s˙ (e) = −ks − q sgn(s) − βη(s, s˙ ) + ω
(22)
Theorem 1. Consider the error dynamic given in (5) with unknown nonlinear bounded functions g(x) and f (y). Construct the control law given in (20) with the following adaptation law as βˆ˙ = γsη − kc γ |s| βˆ
(23)
where kc and γ are given positive constants. Then the synchronization error e(t) is uniformly ultimately bounded. Proof. Choose the following Lyapunov function candidate 1 1 ˜2 V = s2 + (24) β 2 2γ Then, the time derivative of the Lyapunov function along is given by 1 V˙ = s˙s + β˜ βˆ˙ γ
(25)
According to (18), we have ˜ β˜ + β∗ ) β˜ βˆ = β( 2 ∗ 2 ≥ β˜ − β˜ β ≥ β˜ − β˜ βM
(26)
A. Alfi et al. / Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems
Choosing the adaptive law given in (23) and substituting (22) into the above equation, it yields V˙ = s −ks − q sgn(s) − β˜ η(s, s˙ ) + ω 1 + β˜ γs η(s, s˙ ) − kc γ |s| βˆ γ = −ks2 − q |s| − kc |s| β˜ βˆ + sω = − |s| k |s| + q + kc β˜ βˆ − ω sgn(s)
(27)
Considering (26) and |ω| ≤ ωM , we obtain 2 V˙ ≤ − |s| k |s| + q + kc β˜ − β˜ βM − ωM (28) Rewriting (28) as 2 k |s| + q + kc β˜ − β˜ βM − ωM
βM = k |s| + q + kc β˜ − 2
2 β2 − kc M − ωM (29) 4
whenever 2 |s| > (kc βM /4 + ωM )/k
or
β˜ > βM /2 + β2 /4 + ωM /kc M
(30)
(31)
V˙ ≤ 0. According to the standard Lyapunov theorem, it concludes that e is uniformly ultimately bounded (UUB). This completes the proof.
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brief description of each of these processes will be described. A detailed description can be traced in [17]. 4.1. Chemotaxis This step simulates the movement of an E. coli bacterium in BFO algorithm. An E. coli can move in two different ways: It can swim for a period of time in the same direction or it may tumble, and alternate between these two modes of operation for the entire lifetime. The following equation represents this movement. θ i (j + 1, k, l) = θ i (j, k, l) + C(i)
(i)
T (i) (i)
(32)
where θ i (j, k, l) indicated the position of ith bacterium at jth chemotaxis, kth reproduction, and lth elimination and dispersal, respectively. Moreover, C(i) and (i) are the movement vector length and direction random vector, respectively. 4.2. Swarming The explanation of Chemotaxis step is for the cases while bacteria behaved separately, i.e. without producing signal for other bacteria, however there is an exchange of signals between the bacteria here (through absorbing materials). Hence, the group movement for all bacteria is defined as follows: Jcc (θgm (j, k, l), θ(j, k, l))
4. BFOA The BFO algorithm was first proposed by Passino [17] based on the search and optimal foraging strategies of the E. coli bacteria. Nowadays BFO has been successfully utilized in some optimal problems such as automatic voltage regulator [18], harmonic estimation [19], active power filter for load compensation [20] and optimization of real power loss and voltage stability and distribution static compensator [21]. The idea in this algorithm was adopted from biological and physical living behavior of E. coli bacteria existing in human intestine. Chemotaxis is basically a behavior to earn a living that performs a type of optimization in which bacteria try to reach the nutrients and avoid noxious materials and find a way to exit the neutral and noxious nutrient environment. The bacterial swarm proceeds through four steps namely chemotaxis, swarming, reproduction and elimination-dispersal. A
=
s
i jcc (θgm (j, k, l), θ i (j, k, l))
i=1
=
s i=1
=
s
i jcc (θgm (j, k, l), θ i (j, k, l))
−dattract exp (−ωattract )
i=1
+
P
(θmgm
i 2 − θm )
m=1
s hrepellant exp −ωrepellant i=1
×
P
(θmgm
i 2 − θm )
(33)
m=1
where Jcc (θ(j, k, l)) is the cost function value to be added to the actual cost function (to be minimized) to present a time varying cost function in jth chemotac-
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A. Alfi et al. / Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems
Fig. 1. The general flowchart of BFOA.
tic, kth reproduction, and lth elimination stage, θgm is the location of the global minimum bacterium. Moreover, θmgm represents the mth parameter of the global minimum bacteria, S is the total number of bacteria, P is the number of variables to be optimized, which are present in every bacterium θ = [θ1 , . . . θP ]T is a point in the P dimensional search domain. Finally, dattract , hrepellant , ωattract and ωrepellant are different parameters that should be chosen appropriately. 4.3. Reproduction After Nc chemotactic steps, the reproduction step is taken. Suppose that Nre be the number of reproduction steps and the number of population members denoted by S be a positive even integer number. The least healthy bacteria (Sr = S2 ) ultimately die while each of the Sr healthier bacteria (those yielding higher value of fitness function) asexually split into two bacteria which are placed in the same location. So, the number of bacteria is always S. Using this strategy, the swarm size remains constant. 4.4. Elimination and dispersal Although swimming prepares the environment for local foraging or speeds up convergence in the repro-
duction process, but by swimming and reproduction, a large space cannot be adequate for searching the global optimal solution. In BFO, the dispersal event occurs after a definite number of the reproduction processes. First, a bacterium with regard to a Ped prearranged probability is chosen to move and disperse to another position in the environment. These events can effectively prevent trapping in the local optima. Second, Ped is defined for each bacterium which is the probability of elimination and dispersal while Ned presents the number of elimination and dispersal event. Moreover, it is assume that the frequency of reproduction is more than the elimination and dispersal event and the frequency of the moving steps is more than the frequency of the reproduction steps. As a result, many regeneration steps take place before elimination and dispersion. Furthermore, many movement steps occur before reproduction. Figure 1 illustrates the overall flowchart of BFOA.
5. The proposed OAFSMC When applying AFSMC into the system, it is unclear how to determine the parameters k and q given in (22). It seems that one endeavors some different values by trial and error until a desirable performance is obtained. To conquer this deficiency and achieve an
A. Alfi et al. / Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems
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Fig. 2. Time series of x1 and x2 .
optimal performance, OAFSMC is introduced by utilizing bio-mimetic optimization algorithm namely BFOA to determine an appropriate parameters. To accelerate system convergence and to improve the robustness, the parameters related to the switching function and reaching law are optimized via BFOA. The proposed controller solves perfectly the challenge between good quality and high-frequency chattering existing in SMC. The problem being investigated, two significant topics must be considered. The first one is to define an appropriate performance criterion. The performance criterion assesses the parameters values which must be optimized and returns corresponding fitness value with consideration of the user-defined performance specifications. Let consider the following performance criterion. J=
M
e2 (i)
(34)
i=1
where e is the synchronization error and M is the number of given sampling steps. The second one is that how the optimal values should be represented as bacteria. Since each bacterium must be a candidate solution, each bacterium is considered as a n-dimensional vector. As the BFOA progresses and the bacteria pass through the search space, the parameters associated with each bacterium are employed to simulate the control performance.
6. Simulations To demonstrate the validity of proposed ideas, numerical simulation via Matlab software is presented. To solve the system of deferential, Fourth-order RungeKutta method is utilized with sampling time 0.01 Sec. In BFOA, we take S = 8, Nc = 5, Ns = 3, Nre = 8, Ned = 3, Ped = 0.25, dattract = 0.01, ωattract = 0.04,
hrepellant = 0.01 and ωrepellant = 10 [20]. Moreover, kc and γ are set to 0.01 and 10, respectively. We assumed that the master-slave systems are constructed similarly to the Duffing system. The master and slave are as follows: The master system: x˙ 1 = x2 x˙ 2 = −1.1x1 − 0.4x2 − x13 − 2.1 cos (1.8t) (35) The slave system: y˙ 1 = y2 y˙ 2 = 1.8y1 − 0.1y2 − y13 −1.1 cos (0.4t) + u(t) + d(t)
(36)
where d(t) = 0.2 sin(2t) with |d(t)| ≤ 0.2 = α1 . In order to see the chaotic motion of the master and slave systems, let us take different initial conditions for master and slave systems x(0) = (0, 0)T and y(0) = (0.2, 0.2)T , respectively. The states trajectories correspond to the master and slave system are represented in Figs. 2–4. Here, the goal is to achieve synchronization between the master and slave systems via control input u(t), which is provided by the proposed OAFSMC. The obtained parameters k and q are 6.89 and 0.95, respectively. The membership functions for s and s˙ are defined the same by five Gaussian membership functions uniformly distributed on the interval [–5, 5]. µF1 (s) = exp[−(s + 2)2 ], µF2 (s) = exp[−(s + 1)2 ], µF3 (s) = exp[−s2 ], µF4 (s) = exp[−(s − 1)2 ], µF5 (s) = exp[−(s − 2)2 ].
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Fig. 3. Time series of y1 and y2 .
Fig. 4. Phase plane trajectory of the master and slave systems.
Simulation results are shown in Figs. 5–8. It can be concluded that good synchronization performance has been achieved whereas there is no chattering phenomenon. Figure 5 depicts the time series of the master and slave states using the proposed OFSMC. Figure 6 describes the corresponding synchronization errors using the proposed controller. It can be seen that the synchronization errors go to zero asymptotically. It means that synchronization occurs exactly between master and slave, i.e. the state of xi and the state of yi , i = 1, 2. Time series of the control effort is represented in Fig. 7. Referring to Fig. 7, there is no high-frequency chattering phenomenon in the control input signal existing in SMC. The synchronization performance given in (34) is illustrated in Fig. 8. Consequently, the proposed controller solves perfectly the challenge between good quality and high-frequency chattering existing in SMC.
7. Conclusions This paper investigated the synchronization of chaotic systems with unmodeled dynamics and external disturbances. To this end, OAFSMC scheme, applying BFOA to determine the appropriate coefficients for the sliding surface, was introduced. The proposed controller not only attenuates the uncertainties caused by
Fig. 5. Time series of the master and slave states using the proposed OAFSMC.
the unmodeled dynamics and external disturbances, but also significantly reduces the control chattering inherent in the structure of SMC. The performance of the proposed controller was evaluated through the simula-
A. Alfi et al. / Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems
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Fig. 6. Time series of synchronization errors using the proposed OAFSMC. [4]
[5]
[6]
[7]
Fig. 7. Time series of the control effort using the proposed OAFSMC.
[8]
[9]
[10]
[11]
[12] Fig. 8. The synchronization performance given in (34). [13]
tion results. According to these simulations, the control structure can be successfully applied to the synchronization problem of uncertain non-identical systems.
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