Nonlinear Dyn (2010) 62: 553–559 DOI 10.1007/s11071-010-9742-2
O R I G I N A L PA P E R
Directly adaptive fuzzy control of discrete-time chaotic systems by least squares algorithm with dead-zone Haibo Jiang
Received: 5 January 2010 / Accepted: 17 May 2010 / Published online: 4 June 2010 © Springer Science+Business Media B.V. 2010
Abstract A new design scheme of directly adaptive fuzzy control for a class of discrete-time chaotic systems is proposed in this paper. The T-S fuzzy model is employed to represent the discrete-time chaotic systems. Then a fuzzy controller is designed and the unknown coefficients of the controller are identified by least squares algorithm with dead-zone. By Lyapunov method, all the signals involved in the closed-loop systems are shown to be bounded and the error between the system output and the reference output is proved to converge to a small neighborhood of zero. Simulation results demonstrate the effectiveness of the theoretical results. Keywords Discrete-time chaotic systems · T-S fuzzy model · Adaptive control · Least squares algorithm with dead-zone
1 Introduction Since Ott, Grebogi, and Yorke proposed the famous OGY method for chaos control, many effective methods have been developed [1]. These methods include time-delay feedback [2], adaptive control [3–5], backstepping design [6, 7], impulsive control [8, 9], etc. H. Jiang () School of Mathematics, Yancheng Teachers University, Yancheng 224051, P.R. China e-mail:
[email protected]
Over the past few years, adaptive control of discretetime systems has been studied extensively [10–16]. Some adaptive control schemes for nonlinear discretetime systems have been proposed by using neural networks or B-spline functions [12–14]. In the past few decades, fuzzy logic control of nonlinear systems has received considerable attention [17, 18]. Among various kinds of fuzzy methods, fuzzymodel-based control of nonlinear systems has been an attractive subject of research, and many results have been reported [19–22]. More recently, there has been increasing interest in the application of the T-S fuzzy model to chaotic systems modeling and control. A unified approach to controlling chaos via LMIbased fuzzy control system design was suggested in [23] where the key idea is to use the well-known Takagi–Sugeno (T-S) fuzzy model to represent typical chaos models and then apply some effective fuzzy control techniques. Following the idea of representing chaotic systems via the T-S fuzzy model, some adaptive control methods have been proposed for stabilization or synchronization of discrete-time chaotic systems [24–26]. However, the modeling error and unknown disturbances are not considered in [24–26]. In this paper, a new design scheme of directly adaptive fuzzy control for a class of discrete-time chaotic systems is proposed. The main contribution of our paper includes: (1) the modeling error and unknown disturbances are considered; (2) the least squares algorithm with dead-zone is used to reduce the effect of the modeling error and unknown disturbances; (3) by Lya-
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punov method, all the signals involved in the closedloop systems are shown to be bounded and the error between the system output and the reference output is proved to converge to a small neighborhood of zero. The paper is organized as follows. In Sect. 2, the T-S fuzzy model is employed to represent the discretetime chaotic systems. In Sect. 3, a fuzzy controller is designed and the unknown coefficients of the controller are identified by least squares algorithm with dead-zone. By Lyapunov method, stability analysis is presented in Sect. 4. One numerical example is presented to show the effectiveness of the theoretical results in Sect. 5. Finally, the conclusion is provided in Sect. 6.
where wi (z(k)) , hi z(k) = q i=1 wi (z(k)) i Mj zj (k) , wi z(k) = g
j =1
and Mji (zj (k)) is the grade of membership of zj (k) in Mji . We assume that
wi z(k) ≥ 0,
q
wi z(k) > 0,
i = 1, 2, . . . , r.
i=1
Then we get 2 T-S fuzzy modeling of discrete-time chaotic systems
hi z(k) ≥ 0,
q
hi z(k) = 1,
i = 1, 2, . . . , r.
i=1
Many discrete-time chaotic systems can be described by the well-known T-S fuzzy model [23, 24]. The following T-S fuzzy model is used to represent a class of discrete-time chaotic systems: Plant Rule i: IF z1 (k) is M1i and z2 (k) is M2i and, . . . , and zg (k) is Mgi THEN y(k + 1) = li1 y(k) + · · · + lin y(k − n + 1) + u(k) + ηi (k),
i = 1, 2, . . . , r,
where y(k) ∈ R is the system output, u(k) ∈ R is the system input term, ηi (k) represent the modeling error and unknown disturbances which are small bounded, Mji are fuzzy sets, z(k) = [z1 (k), z2 (k), . . . , zg (k)]T are some measurable system variables, li1 , . . . , lin , pi are coefficients of the ith subsystem and r is the number of inference rules. By applying a standard fuzzy inference method, that is, a singleton fuzzifier, product fuzzy inference, and center-average defuzzifier, the following fuzzy global dynamic model is obtained: y(k + 1) =
r
hi z(k) li1 y(k) + · · ·
Remark 1 Three kinds of chaotic systems, such as Henon chaos model, Ushio chaos model, Lozi chaos model, etc., have been considered and expressed by the T-S fuzzy model in [24].
3 Adaptive control system design Using the idea of parallel distribute compensation (PDC), we design the following controller: Plant Rule i: IF z1 (k) is M1i and z2 (k) is M2i and, . . . , and zg (k) is Mgi THEN u(k) = ai1 y(k) + · · · + ain y(k − n + 1) + bi1 yr (k + 1),
i=1
+ lin y(k − n + 1) + u(k) + ηi (k) .
The control objective is to force the system output y(k + 1) to follow the specified trajectory yr (k + 1). Define tracking error e(k + 1) = y(k + 1) − yr (k + 1). The output y(k + 1) can be described as y(k + 1) = f (y(k), . . . , y(k − n + 1), u(k), η1 (k), . . . , ηr (k)), where f is a smooth function with f (0, 0, . . . , 0) = 0.
(1)
i = 1, 2, . . . , r,
where ai1 , . . . , ain , bi1 are unknown coefficients of the ith subsystem to be identified later.
Directly adaptive fuzzy control of discrete-time chaotic systems by least squares algorithm
By applying a standard fuzzy inference method, the following fuzzy global dynamic model is obtained: u(k) =
r
hi z(k) ai1 y(k) + · · · + ain y(k − n + 1)
i=1
+ bi1 yr (k + 1) .
(2)
Let θi = [ai1 , . . . , ain , bi1 ] , φi (k) = hi z(k) y(k), . . . , hi z(k) y(k − n + 1), T hi z(k) yr (k + 1) , T T θ = θ1T , . . . , θrT , φ(k) = φ1T (k), . . . , φrT (k) , T
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Choose the following least squares algorithm with dead-zone to identify the unknown coefficients of the controller [12]: θˆ (k + 1) = θˆ (k) + α(k)β(k)P (k)φ(k)e(k + 1),
(5)
P (k + 1) = P (k) − α(k)β(k)P (k)φ(k)φ T (k)P (k), (6) P (0) = σ I, 1 , 1 + α(k)φ T (k)P (k)φ(k) √ ν, if β(k)|e(k + 1)| ≥ μd0 , α(k) = √ 0, if β(k)|e(k + 1)| < μd0 , β(k) =
(7) (8)
where σ > 0, μ > 1, ν > 0 are constants specified by the designer.
then we get u(k) = φ T (k)θ.
(3)
4 Stability analysis
In order to design stable adaptive control, we make the following assumption.
In order to analyze the stability of the system, we need the following lemma which is given in [13, 16].
Assumption 1 For the system (1), there exists a constant vector θ ∗ such that if u(k) = u∗ (k) = φ T (k)θ ∗ , then we have
Lemma 1 The nonlinear system (1) is said to be contradictorily stable, if
u(k − 1) ≤ k1 + k2 max y(i) , 0≤i≤k
yr (k + 1) − y ∗ (k + 1) = d(k), where y ∗ (k + 1) = f (y(k), . . . , y(k − n + 1), u∗ (k), η1 (k), . . . , ηr (k)), |d(k)| ≤ d0 , d0 is a small positive constant. Remark 2 By using analogical method proposed in [14], we may easily prove the existence of u∗ (k), which is called implicit desired feedback control (IDFC) for the system (1) with ηi (k) = 0, i = 1, 2, . . . , r. But it is difficult to find the IDFC u∗ (k) for the systems (1) when ηi (k) = 0, i = 1, 2, . . . , r. Therefore, we make the above assumption under the condition that ηi (k), i = 1, 2, . . . , r, are small bounded. From Assumption 1 and (1), we obtain e(k + 1) = yr (k + 1) − y(k + 1) = φ (k)θ˜ (k) + d(k), T
where θ˜ (k) = θ ∗ − θˆ (k), θˆ (k) is the estimate of θ .
where 0 ≤ k1 < ∞, 0 ≤ k2 < ∞. To this end, the main result is given as follows. Theorem 1 Under Assumption 1, consider the nonlinear system (1) which is contradictorily stable. The controller is designed as (2). Let the weights in the controller be adjusted by the adaptation law determined by (5)–(8). Then the closed-loop system has the following properties: (1) the tracking error between the system output and the reference output is proved to converge to a small neighborhood of zero, with ra√ dius is μd0 ; (2) all the signals in the direct adaptive system will remain bounded; (3) θ˜ (k) is bounded, and θ˜ (k + 1) − θ˜ (k) converges to zero. Proof (1) From (4), (5), and (7), it is easy to see that
(4)
φ T (k)θ˜ (k + 1) + d(k) = β(k) φ T (k)θ˜ (k) + d(k) = β(k)e(k + 1).
(9)
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From (5) and (9), we obtain
Hence,
θ˜ (k + 1) = θ˜ (k) − α(k)P (k)φ(k) × φ T (k)θ˜ (k + 1) + d(k) .
V (k + 1) ≤V (k) − α(k) β 2 (k)e2 (k + 1) − d 2 (k) . (12)
(10)
Using (A + BC)−1 = A−1 − A−1 B(I + CA−1 B)−1 × CA−1 , we get P −1 (k + 1) = P −1 (k) + α(k)φ(k)φ T (k).
(11)
Inspired by [12], we consider the Lyapunov function candidate V (k) = θ˜ T (k)P −1 (k)θ˜ (k). Then we have
By (8), we get V (k + 1) ≤ V (k). From (8) and (12), we obtain 1 1− α(k)β 2 (k)e2 (k + 1) μ ≤ V (k) − V (k + 1) 1 2 2 2 β (k)e (k + 1) − d (k) − α(k) μ
V (k + 1) = θ˜ T (k + 1)P −1 (k + 1)θ˜ (k + 1) = θ˜ T (k + 1) P −1 (k) + α(k)φ(k)φ T (k) × θ˜ (k + 1)
≤ V (k) − V (k + 1).
= θ (k + 1)P (k)θ˜ (k + 1) 2 + α(k) φ T (k)θ˜ (k + 1) T = θ˜ (k) − α(k)β(k)P (k)φ(k)e(k + 1) × P −1 (k) θ˜ (k) − α(k)β(k)P (k)φ(k) 2 × e(k + 1) + α(k) φ T (k)θ˜ (k + 1) 2 = V (k) + α(k) φ T (k)θ˜ (k + 1) ˜T
(13)
−1
− 2α(k)φ T (k)θ˜ (k) × φ T (k)θ˜ (k + 1) + d(k) + α 2 (k)φ T (k)P (k)φ(k) φ T (k)θ˜ (k + 1) 2 + d(k) 2 = V (k) − α(k) φ T (k)θ˜ (k + 1) − 2α(k)φ (k)θ˜ (k + 1)d(k) − α 2 (k)φ T (k)P (k)φ(k) φ T (k)θ˜ (k + 1) 2 + d(k) 2 ≤ V (k) − α(k) φ T (k)θ˜ (k + 1) T
− 2α(k)φ T (k)θ˜ (k + 1)d(k) = V (k) + α(k)d 2 (k) − α(k) φ T (k)θ˜ (k + 1) 2 + d(k) = V (k) + α(k)d 2 (k) − α(k)β 2 (k)e2 (k + 1) = V (k) − α(k) β 2 (k)e2 (k + 1) − d 2 (k) .
Summing up both sides of this expression from 0 to ∞ yields 1−
∞ 1 α(k)β 2 (k)e2 (k + 1) μ k=0
≤ V (0) = θ˜ T (0)P −1 (0)θ˜ (0) < ∞,
(14)
which implies that α(k)β 2 (k)e2 (k + 1) → 0,
k → ∞.
(15)
Thus, there exists a positive integer k0 such that for all k ≥ k0 , β(k)e(k + 1)
