Robust ISS-satisficing variable universe indirect fuzzy control for chaotic systems

Chaos, Solitons and Fractals 39 (2009) 28–38 www.elsevier.com/locate/chaos

Robust ISS-satisficing variable universe indirect fuzzy control for chaotic systems Jiang Wang *, Wenjie Si, Huiyan Li School of Electrical Engineering and Automation, Tianjin University, 300072 Tianjin, PR China Accepted 2 January 2007

Abstract In the conventional robust input to state stable (ISS)-satisficing control system, all parameters of the system must be known beforehand, so the application area is limited. In this paper, an attempt is made to create a bridge between two important design techniques, i.e., the robust ISS-satisficing control strategy and the fuzzy control strategy, and the new control method we first proposed has both the inverse optimality of robust ISS-satisficing control and the robust and predictive performance of fuzzy control. By control Lyapunov method, the overall closed-loop system is shown to be stable. In this work, we combine these two control methods, make them learn from the other’s strong points, offset its weakness. The simulation results are given to confirm the control algorithm is feasible and performances well. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction Since fuzzy logic is universal approximator, the adaptive control schemes of nonlinear system that incorporate the techniques of fuzzy logic are used to identify or control the nonlinear dynamic system. According to Universal Approximation Theorem, for any given real continuous function f ðxÞ on a compact subset U  RN and arbitrary e > 0, there exists a fuzzy system yðxÞ, such that max jjf ðxÞ  yðxÞk < e. But there is an error between the exact nonlinear system and xU an approximate model, which deteriorates the stability and control performance. Therefore an adaptive fuzzy system basing on-line tune fuzzy rules, which can incorporate with the expert information systematically, has been proposed [2,12,13,15,17–19,23,25,26,28,29]. Thus, the approximate error is decreased. The indirect adaptive fuzzy method has been developed to estimate the unknown parameters of nonlinear dynamic system successfully [16,20,22,24,27]. Some articles [7–10] have shown the design approaches of fuzzy controller, but these approaches depend on operation experience. In this work, an indirect adaptive variable universe fuzzy method is investigated to estimate the parameters of the plant. By adjusting scaling gains (contraction–expansion factors), the universe is changed automatically and the rules are dynamically adjusted. The robust ISS-satisficing control method, which has the advantages of robustness, inverse optimality and the compact framework, is developed by Curtis and Beard [1,5]. The extension of Lyapunov’s second method to dynamical systems with inputs was first introduced by Artstein [3] and Sontag [4] with the introduction of control Lyapunov functions *

Corresponding author. Tel.: +86 22 27402293; fax: +86 22 27401101. E-mail address: [email protected] (J. Wang).

0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.146

J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

29

(clf), ulteriorly, Sontag’s formula proves constructively that the existence of a control Lyapunov function implies asymptotic stabilizability. The robust ISS-satisficing control method also can be regarded as the ISSv control with Kal men-type gain margins of 12 ; 1 . It is the ISS version of Sontag’s formula by completely parameterizing all continuous ISS control laws that can be generated by a known ISS-clf. We can get many benefits by using the robust ISS-satisficing method to design control systems, such as robustness and inverse optimality, and, withal, besides these two, there is a main advantage, that is it gives the control designer maximum flexibility in choosing a control law that is guaranteed to be ISS [1]. But when the parameters of the plant are not known by the designer, this effective method will be out of considering. Now in this paper, the authors propose a new version of the robust ISS-satisficing method which could get over the problem. The rest of this paper is organized as follows. In Section 2, the problem formulation is presented. In Section 3, a variable universe indirect fuzzy method is expounded. In Section 4, robust ISS-satisficing control basing variable universe indirect fuzzy method is designed. In Section 5, simulation examples to demonstrate the performance of the proposed method are provided. Lastly, in Section 6, we make a conclusion of the advocated design methodology.z

2. Plant and the framework of satisficing control We are described as following system x_ ¼ A þ Bðf ðxÞ þ gðxÞu þ g1 ðxÞdÞ y ¼ CT x 2 0 60 6 6. . A¼6 6. 6 40 0

1 0 .. . 0 0

0 1 .. . 0 0

0  0 0  0 .. .. .. . . . 0 0 0 0 0 0

3 0 07 7 7 ... 7; 7 7 15 0

2 3 0 607 6 7 6.7 .7 B¼6 6 . 7; 6 7 405 1

2 3 1 607 6 7 6 7 07 C¼6 6 7 6 .. 7 4.5 0

ð1Þ

where f(x), g(x) and g1 ðxÞ are unknown but bounded functions, u 2 R, y 2 R are control input and output of the system, respectively. d Denotes external bounded disturbance which is unknown. Let x ¼ ½x1 x2    xn  ¼ ½x_x    xðn1Þ  2 Rn be the state vector of the system, which is assumed to be available. Let gðxÞ–0; Rn ! Rnm for x in certain controllability region U c 2 Rn . Definition 1 [6]. The ISS-clf V is said to satisfy the small control property if there exist a control law ac ðxÞ continuous in Rn such that V Tx f þ V Tx gac þ jV Tx g1 jq1 ðjxjÞ < 0

8x–0

ð2Þ

where q satisfies the following requirement: A function C1 function V: Rn ! R is said to be an ISS-clf for system (1) if V is positive definite, radially unbounded, and if there exists a class K1 function q such that the following implication holds 8x–0 and 8d 2 Rr : jxj P qðjdjÞ ) infm V Tx ðf þ gu þ g1 dÞ < 0 u2R

ð3Þ

Additionally we find Lemma 1 useful. Lemma 1 [1]. A pair ðV ; qÞ, where V ðxÞ is positive defined and radially unbounded, and q 2 K 1 satisfies Eq. (3) if and only if: V Tx g ¼ 0 ) V Tx f þ jV Tx g1 jq1 ðjxjÞ < 0

8x–0

ð4Þ

Theorem 1 [1]. Given an ISS-clf V satisfying the small control property, system (1) will be input to state stable if and only if u ¼ k is a continuous control law with kð0Þ ¼ 0 and such that V Tx f þ V Tx gk þ jV Tx g1 jq1 ðjxjÞ < 0 8x–0

ð5Þ

30

J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

Definition 2 [1]. We say that a control law is input to state stabilizing with respect to V ðISSV Þ, if it satisfies the requirements of Theorem 1. Now, we give ISSV control law in the following theorem: Theorem 2 [1]. An arbitrary continuous control law, u ¼ bðxÞgT V x þ nðxÞ (where nðxÞ is orthogonal to gT V x Þ with uð0Þ ¼ 0, is an ISSV control if and only if: bðxÞ >

V Tx f þ jV Tx g1 jq1 ðjxjÞ V Tx ggT V x

ð6Þ

Definition 3 [3]. Let u ¼ bðxÞgT V x þ nðxÞ (where n is in the null space of gT V x Þ be an ISSV control value.   V T f þ jV Tx g1 jq1 ðjxjÞ S gm ðxÞ, ¼ u ¼ bgT V x þ n : b > maxð2 x ; 0Þ V Tx ggT V x

ð7Þ

n here can improve the control performance. Theorem 3 [3]. S gm ðxÞ contains all ISSV points which also have gain margin robust satisficing control if and only if u 2 S gm .

1 2

 ; 1 , and we can say one control law u ¼ k is

3. Variable universe indirect adaptive fuzzy approach Let X j ¼ bEj ; Ej cðj ¼ 1; 2; . . . ; nÞ be the universe of the input variable xj ðj ¼ 1; 2; . . . ; nÞ, and Y ¼ ½U ; U  be the universe of the output f ðyÞ. fAjl gð16l6hÞ stand for a fuzzy partition on Xj and fBl gð16l6hÞ stand for a fuzzy partition on Y. For any xj 2 X j , the membership Ajl ðxj Þ which is the true value of ‘‘xj is Ajl’’ is transferred to the resulting consequent parameter yl. If the value is 1, the resulting value of the consequent parameter is certainlyyl. However, Ajl ðxj Þ is not always equal to 1. So, yl is not completely chosen as the resulting consequent parameter value. We choose a ‘‘reliability’’ which is not higher than Ajl ðxj Þ to be a weight multiplied by yl. In the paper, the ‘‘reliability’’ is equal to Ajl ðxj Þ, the resulting output is as follows: h  P n Pj¼1 Ajl ðxj Þy l h X   f ðyÞ, l¼1 h ; Ajl xj ¼ 1 ð8Þ P l¼1 Ajl ðxj Þ l¼1

In this paper, the variable universe fuzzy approach is presented. In the premise that the number of initial control rules is fixed, the universe discourse is changed with the changing error. Thus, the rules are tuned dynamically. The situation of variable universe is shown as Fig. 1 [8]. The transformed universe discourse is denoted as X j ðxj Þ ¼ ½aj ðxj ÞEj aj ðxj ÞEj ; Y ðyÞ ¼ ½bðyÞU bðyÞU , where aj ðxj Þ; bðxÞ are contraction–expansion factors. Generally speaking, we give the following contraction–expansion factor (for detailed reasoning, refer to [8]); aðxÞ ¼ 1  k expðjx2 Þ; k 2 ð0; 1Þ; k > 0. The output of the variable universe indirect fuzzy approach is represented as   h X xj Pnj¼1 Ajl ð9Þ y f^ ðyjbÞ ¼ b aðxj Þ l l¼1 From (9), we can select a reasonable b to optimize the adaptive laws.

Fig. 1. The situation of universe.

J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

31

4. The modified robust ISS-satisficing control For simple, let d in system (1) be zero, then, modify our control law (bases in robust satisficing control law, see Theorem 3): 1 us ¼ bgT V x þ ðy nm þ kT eÞ ð10Þ g  T where b ¼ max 2 V TVggx Tf V x ; 0 ; y nm is the reference input that the output of system (1) is desired to track. Through variable x universe indirect adaptive fuzzy approach, we set f^ and g^ be the approximate factors of the system parameters, i.e., f and g, as follows: f^ ðx=hf Þ ¼ hTf nðxÞ;

nðxÞ ¼

h Y n X



 xj y aðxj Þ l

ð11Þ

 xj p aðxj Þ l

ð12Þ

Ajl

l¼1 j¼1

Similarly, g^ðx=hg Þ ¼ hTg gðxÞ;

gðxÞ ¼

k Y n X

 Bjl

l¼1 j¼1

Then the control law could be rewritten as:  1 n y þ kT e g^ m  T^ where b ¼ max 2 V TVg^xg^Tf V x ; 0 . us ¼ b^ gT V x þ

ð13Þ

x

T^

Now, first we discuss the case of 2 V TVg^xg^Tf V x P 0: x

Under the control of us, the error dynamic equation becomes: en ¼ y nm  f ðxÞ  gðxÞus ¼ y nm  f ðxÞ þ 2gðxÞ

V

V Tx f^ ðxÞ T gðxÞV x x gðxÞ^

g^T ðxÞV x  gðxÞ

1 ðy n þ kT eÞ g^ðxÞ m

ð14Þ

For many systems, such as Chua circuit and Duffing system, they are affine nonlinear systems, and the discourse is developed to deal with such systems, then (14) can be written as the following form: en ¼ y nm  f ðxÞ þ

gðxÞ p 1 gðxÞ  gðxÞÞus f ðxÞ  gðxÞ ðy n þ kT eÞ ¼ k T e þ ðf p ðxÞ  f ðxÞÞ þ ð^ g^ðxÞ g^ðxÞ m

ð15Þ

where f p ¼ 2f^ . f ðxÞ; gðxÞ are unknown, so we define the optimal parameter vectors hf ; hg for which the variable universe fuzzy controllers can approximate the parameters f ðxÞ; gðxÞ optimally.

 hf ¼ arg min sup kf^ ðxjhf Þ  f ðxÞk hf 2Xhf

hg



¼ arg min

x2XX

ð16Þ



sup k^ gðxjhf Þ  gðxÞk

hf 2Xhg

x2XX

where XX ; Xhf and Xhg denote the sets of desired bounded on x; hf and hg , respectively. Let the minimum approximate error be denoted by x ¼ ðf^ ðxjhf Þ  f ðxÞÞ þ ð^ gðxjhg Þ  gðxÞÞ

ð17Þ

The error dynamic Eq. (15) can be rewritten as e_ ¼ Ae þ bf½f p ðxÞ  f  ðxÞ þ ½^ gðxÞ  g ðxÞus þ xg ¼ Ae þ b½ð2hf  hf ÞT nðxÞ þ ðhg  hg ÞT gðxÞus þ x T

T

T

T

e_ ¼ e A þ n ðxÞð2hf  where

hf ÞbT

T

þ g ðxÞðhg 

hg ÞbT us

ð18Þ T T

þx b

32

J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

2

0 6 0 6 6 . . A¼6 6 . 6 4 0 k n

1 0 .. . 0 k n1

0 1 .. . 0 k n2

0 0 .. . 0 k n3

  .. .  

0 0 .. . 0 k 2

3 0 0 7 7 .. 7 . 7 7; 7 1 5 k 1

2 3 0 607 6 7 6.7 .7 B¼6 6.7 6 7 405 1

Let us choose a Lyapunov function 1 1 ~T ~ 1 ~T ~ V ¼ eT P e þ h hf þ h hg 2 2c1 f 2c2 g

ð19Þ

where ~hf ¼ 2hf  hf ; ~hg ¼ hg  hg ; P is a positive definite matrix, satisfying: AT P þ PA ¼ Q

ð20Þ

where Q is an arbitrary positive definite matrix with order of n  n. Differentiate (19) with respect to time, then 1 1 1 ~_ T ~ 1 ~T ~_ 1 ~_ T ~ 1 ~T ~_ h hf þ h hg hf hf þ hg hg þ V_ ¼ e_ T P e þ eT P e_ þ 2 2 2c1 2c1 f 2c2 2c2 g

ð21Þ

From (18) and (20), (21) can be represented as: 1 1 1 1 1 1 T V_ ¼ eT AT Pe þ nT ðxÞð2hf  hf ÞbT Pe þ gT ðxÞðhg  hg ÞbT us Pe þ xT bT Pe þ eT PAe þ eT Pbð2hf  hf Þ nðxÞ 2 2 2 2 2 2 1 1 1 ~_ T ~ 1 ~T ~_ 1 ~_ T ~ 1 ~T ~_ h hf þ h hg hf hf þ hg hg þ þ eT Pbðhg  hg ÞT gðxÞus þ eT Pbx þ 2 2 2c1 2c1 f 2c2 2c2 g 1 1 1 ¼  eT Qe þ eT Pbx þ ð2hf  hf ÞT ð2h_ f þ c1 eT PbnðxÞÞ þ ðhg  hg ÞT ðh_ g þ c2 eT PbgðxÞus Þ ð22Þ 2 c1 c2 Since  12 eT Qe is negative, and the minimum approximate error and the tracking error are very small, so a good strategy for selecting the adapting laws is to make the last two items in (22) be zero: 1 h_ f ¼  c1 eT PbnðxÞ 2 h_ g ¼ c2 eT PbgðxÞus

ð23Þ ð24Þ

Then we consider the case of

T^ 2 V TVg^xg^Tf V x x

< 0:

According to (13), the control law turns into us ¼

1 ðy n þ kT eÞ g^ðxÞ m

ð25Þ

and the adaptive law of f^ is shown by (23), then the problem focuses on what the adaptive law of g^ turns to be. The following proof will give the answer: en ¼ y nm  f ðxÞ  gðxÞ

1 gðxÞ  gðxÞÞus ðy n þ kT eÞ ¼ kT e  f ðxÞ þ ð^ g^ðxÞ m

ð26Þ

(26) can be rewritten as follows: gðxjhg Þ  g^ðxsjhg Þus þ xg: e_ ¼ Ae þ bff^ ðxjhf Þ þ ½^ where A; b are the same as (18), while Choose Lyapunov function as

hf ; hg

ð27Þ

is shown in (16), and x is determined by (17).

1 1 ðhg  hg Þ V ¼ eT Pe þ 2 2c2 Differentiate it, and the following equations will be obtained: 1 1 1 1 ðhg _ hg ÞT ðhg  hg Þ þ ðhg  hg ÞT ðhg _ hg Þ V_ ¼ e_ T Pe þ eT P e_ þ 2 2 2c2 2c2 1 1 ¼ ðeT AT  nT ðxÞhf bT þ gT ðhg  hg ÞbT us þ xT bT ÞPe þ eT P ðAe  bhf T nðxÞ þ bðhg  hg ÞT gðxÞus þ bxÞ 2 2

ð28Þ

J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

1 1 1 ðhg _ hg ÞT ðhg  hg Þ þ ðhg  hg ÞT ðhg _ hg Þ ¼  eT Qe þ eT Pbðx  hf nðxÞÞ 2c2 2c2 2 1 þ ðhg  hg ÞT ðhg þ c2 eT PbgðxÞus Þ c2

33

þ

ð29Þ

where P ; Q satisfy the condition proposed in (20). Since e approaches to zero and the second item in (29) can be considered to be very small, so the rational adaptive law of g^ could be h_ g ¼ c2 eT PbgðxÞus

ð30Þ

this is the same as (24). h Consider the bounded problem of the adaptive laws. Assume that the constraint set Xh is specified as Xh ¼ ½h=khk 6 N h 

ð31Þ

where N h is a positive constraint, then, the parameter update laws in (23), (24), (30) can be modified as: If an element hgi in hg equals to e (set to be very small), then use:  c2 eT Pbgi ðxÞus if eT Pbgi ðxÞus < 0 h_gi ¼ 0 if eT Pbgi ðxÞus P 0 where gi ðxÞ is the ith component of gðxÞ. Else use: ( c2 eT PbgðxÞus if khg k < N b or ðkhg k ¼ N b and eT PbhTg gðxÞus P 0Þ _hg ¼ Pr ojðc2 eT PbgðxÞus Þ if khg k ¼ N b and eT PbhTg gðxÞu < 0

ð32Þ

ð33Þ

where Pr ojðc2 eT PbgðxÞus Þ ¼ c2 eT PbgðxÞus þ For the adaptive law hf of f^ : ( c1 eT PbnðxÞ _hf ¼ Pr ojðc1 eT PbnðxÞÞ

c2 eT Pbhg hTg gðxÞus khg k2

if khf k < N f or ðkhf k ¼ N f and eT PbhTf nðxÞ P 0Þ if khf k ¼ N f and eT PbhTf nðxÞ < 0

Pr ojðc1 eT PbnðxÞÞ ¼ c1 eT PbnðxÞ þ c1 eT Pb

hf hTf nðxÞ khf k

ð34Þ

ð35Þ ð36Þ

5. Example 5.1. Chua chaotic circuit The typical Chua’s chaotic circuit in Fig. 2 consists of one linear resistor (R), two capacities (C 1 C 2 Þ, one inductor and one piecewise linear resistor (g) [2,14,21]. The dynamic equation of Chua’s chaotic circuit is as followings:

Fig. 2. Chua chaotic circuit.

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J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

  1 1 ðV C2  V C1 Þ  gðV C1 Þ V_ C1 ¼ C1 R   1 1 ðV C1  V C2 Þ þ iL V_ C2 ¼ C2 R 1 i_L ¼ ðV C1  R0 iL Þ L

ð37Þ

where voltages V C1 ; V C2 and current iL are stable variables. R0 is the interior resistor of the inductor and g denotes the nonlinear resistor, which is a function of the voltages across the two terminals of C1. Here g is defined as a cubic function as in (38): gðV C1 Þ ¼ aV C1 þ cV 3C1

ða < 0; c > 0Þ

ð38Þ

The system can be rewritten as: z_ ¼ GzðtÞ þ Hg

ð39Þ T

T

where z ¼ ½ z1 z2 z3  ¼ ½ V C1 V C2 iL  ; 3 2 2 13  C11 R C11 R 0  C1 7 6 1 7 6 1 1 7  H ¼ G¼6 4 0 5; C2 R C2 5 4 C2 R 0 0  L1  RL0 The obtained state space equations are not in the standard form defined in (1). Therefore, we need to perform linear transformation to transform into the form of (1). Define z ðtÞ ¼ T 1 zðtÞ, where T is a transformation matrix. The transformed system can be obtained as: z_  ðtÞ ¼ T 1 GT z ðtÞ þ T 1 Hg ¼ G z ðtÞ þ H  g 

1



ð40Þ

1

where G ¼ T GT ; H ¼ T H , 3 2 RþR 0 0 C 2 þL  RR  C11  C1 C2 RL C 1 C 2 RL 7 6 R0 1 0 7 T ¼6 5; 4  C1 C2 RL  C1 C2 R 1 0 0 C 1 C 2 RL

2

0

6 G ¼ 4

0 1 C 1 C 2 RL

0

3

0

1

C 1 RþC 2 R0 þC 1 R0 C 1 C 2 RL

C 1 C 2 RR0 þC 2 LþC 1 L C 1 C 2 RL

7 5

1

Choose the parameters as following: R ¼ 1:428;

R0 ¼ 0;

C 1 ¼ 1;

C 2 ¼ 9:5;

L ¼ 1:39;

a ¼ 0:8;

c ¼ 0:044;

After simple manipulations, we can get the transformed system as follows: z_ 1 ¼ z2 z_ 2 ¼ z3 z_ 3

 3 14  168  1 2 28  7 ¼ z1  z2 þ z3  z1 þ z2 þ z3 1485 9025 38 45 321 95

ð41Þ

We will design a robust satisficing controller to force the transformed system to track the given reference signal. For convenience, we let x replace z* in (41). Therefore, the closed-loop system (41) can be represented as 3 2 3 2 3 2 3 2 x1 0 1 0 0 x_ 1 7 6 7 6 7 6 7 6 _  ¼ 0 0 1 þ x x 5 4 2 5 4 0 5ðf þ gu þ dÞ 4 25 4 0 0 0 1 x_ 3 x3 2 3 ð42Þ x1 6 7 y ¼ ½ 1 0 0   4 x2 5 x3  28 3 14 168 where f ¼ 1805 x1  9025 x2 þ 381 x3  452  321 x1 þ 957 x2 þ x3 ; g ¼ 1; d is the bounded external disturbance. If u = 0, (42) is chaotic system. The phase trajectory of x1 x2 x3 is showing in Fig. 3.g ¼ 1; is a constant, so it is no need to develop a fuzzy approximator to approach it. Although f in (42) is well defined since the Chua’s circuit is well specified, we suppose it to be an unknown parameter of the system, and design a variable universe fuzzy approximator to replace it as well as a robust satisficing controller to make the output y track the reference signal, i.e. y r ðtÞ ¼ cosðtÞ.

J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

35

0.6 0.4

x3

0.2 0 -0.2 -0.4 -0.6 4 2 0

x2

-2 -4

-40

-30

-20

-10

0

10

20

30

40

x1

Fig. 3. The phase plane without control.

Step 1: the feedback matrix is chosen 2 1 Step 2: Q in (3) is selected as Q ¼ 4 0 0 got.

as kT ¼ ½1 2 1; 3 0 0 1 0 5; c ¼ 21:2. By solving Eq. (20), the solution of the Riccati-like P can be 0 1

Step 3: we use H 1 control u1 ¼ 1r bT P e to substitute the kT e in order to gain a better performance ([2,11]), where r ¼ 0:005. Step 4: Select the membership functions of e and e_ as follows: eNB ¼ minð1; maxð0; 3e=2  2ÞÞ eNM ¼ maxð0; minð3e=2 þ 3; 3e=2  1ÞÞ eNS ¼ maxð0; minð3e=2 þ 2; 3e=2ÞÞ eZE ¼ maxð0; minð3e=2 þ 1; 3e=2 þ 1ÞÞ ePS ¼ maxð0; minð3e=2; 3e=2 þ 2ÞÞ ePM ¼ maxð0; minð3e=2  1; 3e=2 þ 3ÞÞ ePB ¼ minð1; maxð0; 3e=2  2ÞÞ e_ NB ¼ minð1; maxð0; 3_e=8  2ÞÞ e_ NM ¼ maxð0; minð3_e=8 þ 3; 3_e=8  1ÞÞ e_ NS ¼ maxð0; minð3_e=8 þ 2; 3_e=8ÞÞ e_ ZE ¼ maxð0; minð3_e=8 þ 1; 3_e=8 þ 1ÞÞ e_ PS ¼ maxð0; minð3_e=8; 3_e=8 þ 2ÞÞ e_ PM ¼ maxð0; minð3_e=8; 3_e=8 þ 3ÞÞ

e_ PB ¼ minð1; maxð0; 3_e=8  2ÞÞ

Step 5: Note that V ðxÞ ¼ 12 xT x is a valid ISS-clf for this system without disturbance because V Tx g ¼ 0 if and only if x ¼ 0 (according to Definition 1). Step 6: Adopt (13) to control the chaotic system and use (33) to on-line tune parameters.

Fig. 4. The trajectory of y and yr with satisficing control.

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J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

Fig. 5. The trajectory of y and yr without satisficing control.

Fig. 6. The output of the controller with satisficing control.

Fig. 7. The output of the controller without satisficing control.

J. Wang et al. / Chaos, Solitons and Fractals 39 (2009) 28–38

37

The trajectories of actual states y and yr with and without satisficing are shown in Figs. 4 and 5, respectively. And, withal, the outputs of these two controllers are given by Figs. 6 and 7. The simulations results indicate the facts that the tracking performance can be guaranteed by our control algorithms. Figs. 6 and 7 show that the new controller costs much less energy than the controller without satisficing, but their tracking performance is almost the same. By the way, in simulation, our control algorithm calculates much faster than the one without satisficing.

6. Conclusion Under the condition of uncertainty or unknown plant parameters, the proposed control technique whose framework is robust ISS-satisficing control combines the adaptive fuzzy approximate method to attenuate the disturbance due to the approximate error in plant. The new control strategy achieves the desired tracking performance and costs much less energy than the approach under the same condition except for satisficing. One distinct advantage of satisficing whose construction is compact is that it gives the designer maximum flexibility in choosing a control law that is guaranteed to be ISS. In the paper, an indirect variable universe fuzzy approach is made to approximate the parameters of the system. The conventional T–S fuzzy approximator must on-line tunes all the consequent parameters. Thus, the realization is difficult. While the variable universe fuzzy approximator only on-line tunes the contraction and expansion factor, so the realization is easy. In case of few fuzzy rules, the variable universe fuzzy controller can generate many rules by transforming the universe of discourse and reduces the dependence on the expertise experience knowledge.

Acknowledgements The authors gratefully acknowledge the support of the NSFC (No. 50537030).

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