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Observer–based indirect model reference fuzzy control system with application to control of chaotic systems Mojtaba Ahmadieh Khanesar, Okyay Kaynak, Mohammad Teshnehlab

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S0016-0032(12)00293-1 http://dx.doi.org/10.1016/j.jfranklin.2012.11.014 FI1613

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Cite this article as: Mojtaba Ahmadieh Khanesar, Okyay Kaynak and Mohammad Teshnehlab, Observer–based indirect model reference fuzzy control system with application to control of chaotic systems, Journal of the Franklin Institute, http://dx.do i.org/10.1016/j.jfranklin.2012.11.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Observer–Based Indirect Model Reference Fuzzy Control System with Application to Control of Chaotic Systems Mojtaba Ahmadieh Khanesara , Mohammad Teshnehlabb, Okyay Kaynakc a

Dept. of Control and Electrical Engineering, Semnan University, Semnan, Iran e-mail: [email protected]. b Control Engineering K. N. Toosi University of Tech., Tehran, Iran e-mail: [email protected]. c Dept. of Electrical and Electronics Engineering, 34342, Istanbul, Turkey, e-mail: [email protected]

Abstract This paper proposes a novel, observer–based, indirect model reference fuzzy control approach for nonlinear systems, expressed in the form of a Takagi Sugeno (TS) fuzzy model. Based on this model, an adaptive observer based, indirect model reference fuzzy controller is developed to deal with external disturbances. In contrast to what is seen in the literature on adaptive observer based TS fuzzy control systems, the proposed method is robust in the existence of bounded external disturbances and it is capable of tracking a reference signal rather than just regulation. The proposed method is simulated on the control of Chua’s circuit and it is shown that it is capable of controlling this chaotic system with high performance. Keywords: Fuzzy control, TS fuzzy model, Adaptive observer, Indirect model reference fuzzy controller 1. Introduction Fuzzy adaptive controllers are generally applicable to plants that are mathematically poorly modeled and/or when there is not enough expert knowledge to specify all the parameters of the control system. Conceptually, fuzzy adaptive systems combine partial expert knowledge with numerical measurements from sensors. They have been successfully applied in different industrial control applications. Despite their learning capabilities and practical implementations, the earlier fuzzy adaptive systems have suffered from the lack of a rigorous stability analysis. To ensure the stability of fuzzy adaptive controllers, different classical control approaches have been utilized. The fusion of fuzzy systems with classical control approaches makes it possible to benefit from Preprint submitted to Journal of The Franklin Institute

December 11, 2012

general function approximation properties of fuzzy systems as well as their power to use expert knowledge and the well established stability proof of classical control systems. For example in [1] and [2], sliding mode fuzzy controllers are proposed, and in [3] and [4], a fuzzy-identificationbased back-stepping controller and a model reference controller with an adaptive parameter estimator based on Takagi Sugeno (TS) fuzzy models are proposed, respectively. Using a model system to generate the desired response is one of the most important adaptive control schemes [5] studied in such hybrid approaches, and to date, different fuzzy model reference approaches have been proposed. The indirect model reference fuzzy controllers described in [4], [6], [7], [8] and [9] and the direct model reference fuzzy controllers for TS fuzzy models described in [10], [11], [12], and [13] can be cited as some examples. Most of the model reference fuzzy controller schemes existing in the literature assume that the full state measurement of the plant is available [4, 6, 7, 8, 9, 10, 11, 12, 13]. However, in some practical applications, state variables are not accessible for sensing devices or the sensor is expensive, and the state variables are just partially measurable. In such cases it is very essential to design an observer to estimate the states of the system. There has been a tremendous amount of activity on the design of nonlinear observers using fuzzy models. Existing nonlinear fuzzy observers are designed based on some approaches like LMI [14, 15, 16], SPR Lyapunov function [17, 18] and adaptive methods [19]. The TS fuzzy system is one of the most popular fuzzy systems in model based fuzzy control. A dynamical TS fuzzy system describes a highly nonlinear dynamical system in terms of locally linear TS fuzzy systems. The overall fuzzy system is achieved as a fuzzy blending of these locally linear systems [20]. Using this approach it is possible to deal with locally linear dynamical systems rather than the original nonlinear dynamical system. The design of fuzzy observers using TS fuzzy models is considered in a number of different papers. In [19] TS fuzzy model indirect adaptive fuzzy observer and controller design is considered. However the proposed method considers only the regulation of TS fuzzy model and its tracking control is not addressed. In this paper, a novel indirect model reference TS fuzzy controller that incorporates a novel fuzzy observer is designed for a TS fuzzy model subject to bounded external disturbances. Not only can the proposed indirect model reference fuzzy controller regulate the states of the system 2

under control but also it can make the system track a desired trajectory. The proposed method is then used to control chaotic systems. It is shown that using the proposed approach it is possible to make the chaotic system follow the reference model. The proposed method is used for the control of Chua’s chaotic system. The simulation results show that the proposed controller is a very effective method for the control of chaotic dynamical systems. 2. The fuzzy Takagi-Sugeno modeling The basic idea behind the fuzzy (TS) modeling is to describe a nonlinear system by some fuzzy locally linear IF-THEN rules. The overall nonlinear model of the system is achieved by a fuzzy blending of the linear system models. In [21], it is proven that the TS fuzzy models are universal approximators. Consider the following nonlinear dynamical system.   x˙ = Ax + B f (x) + g(x)u + d(x, t) y = Cx

(1)

where xT (t) = [x1 , x2 , · · · , xn ] are state variables, u ∈ R is the control signal, A ∈ Rn×n , C ∈ R1×n and B ∈ Rn×1 are known state matrices of the system. f (x) : Rn → R and g(x) : Rn → R are unknown continuous nonlinear functions. d(x, t) is a bounded external disturbance acting on the system for which we have |d(x, t)| < D. The TS fuzzy model can be expressed either in IF–THEN form or in Input–Output form. The IF–THEN representation of TS fuzzy model describing the nonlinear system of (1) is as:   Rulei : IF y is Mi THEN x˙ = Ax + B aTi x + bi u + d(x, t)

(2)

where ai ∈ Rn and bi ∈ R are unknown system parameters. On the other hand, the Input/Output form can be written as:

⎤ ⎡ l l ⎥⎥ ⎢⎢⎢ hi (y)bi u + d(x, t)⎥⎥⎥⎦ x˙ = Ax + B ⎢⎢⎣ hi (y)aTi x + i=1

y = Cx where hi (y) =

l

Mi

i=1

Mi

i=1

(3)

and Mi (y) is the grade of membership function of y in Mi and l is the number

of rules. The summations that appear in (3) can be expressed by the nonlinear functions, f (x) and 3

g(x) as indicated below. f (x) =

l

hi (y)aTi x

(4)

i=1

g(x) =

l

hi (y)bi

(5)

i=1

3. The design of observer and its stability analysis In this section, an indirect adaptive fuzzy observer for a nonlinear system subject to external disturbances is proposed. The adaptation law for the estimation of the parameters of the nonlinear system is derived. Using an appropriate Lyapunov function, the stability of the proposed observer and the adaptation laws are analyzed. 3.1. The structure of the proposed observer The structure of the proposed observer is: ⎤ ⎡ l l ⎥⎥ ⎢⎢⎢ hi (y)bˆ i u + ρsgn(y − yˆ)⎥⎥⎥⎦ + L(y − yˆ ) x˙ˆ = Aˆx + B ⎢⎢⎣ hi (y)ˆaTi xˆ + i=1

i=1

yˆ = Cˆx

(6)

where xˆ ∈ Rn , aˆ i ∈ Rn and bˆ i ∈ R are the estimated values for x, ai and bi , respectively. In addition ρ is an adaptive parameter, designed to compensate the effects of the external disturbances. Using (6) and (3) the error dynamics of the observer (e = x − xˆ ) is obtained as: ⎡ l ⎤ l ⎢⎢⎢ ⎥⎥ T T e˙ = (A − LC)e + B ⎢⎢⎣ hi (y)(ai x − aˆ i xˆ ) + hi (y)(bi − bˆ i )u + d(x, t) − ρsgn(y − yˆ )⎥⎥⎥⎦ i=1

(7)

i=1

Assumption 1. (A, C) is detectable. Assumption 2. There exists a positive definite function P1 such that [19]: (A − LC)T P1 + P1 (A − LC) = −Q1 ,

BT P1 = C

(8)

where Q1 is a positive definite matrix. Remark 1. The existence of a positive definite matrix P1 is a sufficient condition for satisfying assumption 2 and there is no need to find P1 in order to design the controller. It can be shown that the sufficient condition for the existence of P1 is that (A, C) are detectable and CB  0 [5]. 4

3.2. The stability analysis of the proposed observer To analyze the stability of the proposed observer the following Lyapunov function is considered: l l 1 T 1 ˜2 1 bi + (ρ − ρ∗ )2 a˜ i a˜ i + V1 = e P1 e + γ1 i=1 γ2 i=1 γ3 T

(9)

in which γ1 , γ2 and γ3 are positive design parameters, a˜ i = ai − aˆ i and b˜ i = bi − bˆ i . The time derivative of the Lyapunov function (V1 ) is: l l 2 ˜ ˙˜ 2 2 T˙ T T ˙ bi bi + ρ(ρ a˜ i a˜ i + ˙ − ρ∗ ) V1 = e˙ P1 e + e P1 e˙ + γ1 i=1 γ2 i=1 γ3

(10)

Using the error dynamics of (7) and the equation (8) the above equation can be rewritten as: V˙ 1 = −eT Q1 e + 2eT P1 B

l

hi (y)aTi e

i=1

⎡ l ⎤ l ⎢⎢⎢ ⎥⎥ T T hi (y)b˜ i u + d(x, t) − ρsgn(y − yˆ )⎥⎥⎥⎦ + 2e P1 B ⎢⎢⎣ hi (y)˜ai xˆ + i=1

i=1

l l 2 ˙˜ ˜ 2 2 ˙T ˙ − ρ∗ ) bi bi + ρ(ρ a˜ i a˜ i + + γ1 i=1 γ2 i=1 γ3

(11)

Using the Lipschitz condition [22], the following condition is applicable. l

hi (y)(aTi x − aTi xˆ ) ≤ k f e

(12)

i=1

By using the Lipschitz condition above and the fact that D is an upper bound for the external disturbance and defining ey = y − yˆ , one obtains the following equation. ⎡ l ⎤ l ⎢ ⎥⎥ ⎢ hi (y)b˜ i u⎥⎥⎥⎦ + 2|ey |D V˙ 1 ≤ − [λmin (Q1 ) − 2k f Bλmax (P1 )]e2 + 2eT P1 B ⎢⎢⎢⎣ hi (y)˜aTi xˆ + i=1

i=1

l l 2 ˙T 2 ˙˜ ˜ 2 ˙ − ρ∗ ) bi bi + ρ(ρ a˜ i a˜ i + − 2ρ|ey | + γ1 i=1 γ2 i=1 γ3

(13)

in which λmin (.) and λmax (.) correspond to the smallest and the biggest eigenvalues of the corresponding matrices respectively. Furthermore:

⎡ l ⎤ l ⎢⎢⎢ ⎥⎥ T V˙ 1 ≤ − [λmin (Q1 ) − 2k f Bλmax (P1 )]e + 2e P1 B ⎢⎢⎣ hi (y)˜ai xˆ + hi (y)b˜ i u⎥⎥⎥⎦ + 2|ey |D 2

T

i=1

i=1

l l 2 ˙T 2 ˙˜ ˜ 2 ∗ ∗ ˙ − ρ∗ ) bi bi + ρ(ρ a˜ i a˜ i + − 2ρ|ey | − 2ρ |ey | + 2ρ |ey | + γ1 i=1 γ2 i=1 γ3

5

(14)

Considering D < ρ∗ , we have: ⎡ l ⎤ l ⎢ ⎥⎥ ⎢ V˙ 1 ≤ − [λmin (Q1 ) − 2k f Bλmax (P1 )]e2 + 2eT P1 B ⎢⎢⎢⎣ hi (y)˜aTi xˆ + hi (y)b˜ i u⎥⎥⎥⎦ i=1

− 2ρ|ey | + 2ρ∗ |ey | +

2 γ1

l

a˙˜ Ti a˜ i +

i=1

2 γ2

l i=1

i=1

2 ˙ − ρ∗ ) b˙˜ i b˜ i + ρ(ρ γ3

(15)

The following adaptation laws are proposed for the parameters of the system. a˙ˆ Ti = γ1 hi (y)ey xˆ T − γ1 ν|ey |ˆaTi

(16)

b˙ˆ i = γ2 hi (y)ey u − γ2 ν|ey |(bˆ i − b0 )

(17)

ρ˙ = γ3 |ey | − γ3 ν|ey |ρ

(18)

In (16-18) 0 < ν is the leakage parameter. In addition, in (17), b0 is assumed to be a known positive constant such that b0 < |bi |. Using these adaptation laws, (14) can be rewritten as: V˙ 1 ≤ − [λmin (Q1 ) − 2k f Bλmax (P1 )]e2 − ν

l i=1

+ ν

l

a˜ Ti a˜ i

+ν

l

˜ai − aˆ i |ey | − ν 2

i=1

l

b˜ 2i

i=1

b˜ i − (bˆ i − b0 )|ey |2 − ν(ρ − ρ∗ )2 + ν(ρ − ρ∗ ) − ρ|ey |2

(19)

i=1

Q1 can be chosen such that: k f B <

λmin (Q1 ) 4λmax (P1 )

(20)

so that: λmin (Q1 ) 2 e − ν V˙ 1 ≤ − 2 +ν

l

l i=1

a˜ Ti a˜ i

+ν

l i=1

˜ai − aˆ i |ey | − ν 2

l

b˜ 2i

i=1

b˜ i − (bˆ i − b0 )|ey |2 − ν(ρ − ρ∗ )2 + ν(ρ − ρ∗ ) − ρ|ey |2

(21)

i=1

(21) can be rewritten as: V˙ 1 ≤ −c1 V1 + c2 6

(22)

in which: 

λmin (Q1 ) , νγ1 , νγ2, νγ3 c1 = min 2λmax (P1 )

 (23)

and: c2

=ν

l

˜ai − aˆ i |ey | + ν 2

i=1

l

b˜ i − (bˆ i − b0 )|ey |2 + ν(ρ − ρ∗ ) − ρ|ey |2

(24)

i=1

It follows that aˆ i , ρ, ey and bˆ i are bounded. So that: c2 = sup {ν

l

˜ai − aˆ i |ey | + ν 2

i=1

l

b˜ i − (bˆ i − b0 )|ey |2 + ν(ρ − ρ∗ ) − ρ|ey |2 }

(25)

i=1

t In the next section it will be seen that bˆ i appears in the denumerator of the control signal and hence near zero values of it should be avoided. In order to achieve this goal, the priori knowledge that b0 ≤ |bi | is used and the adaptation law for bˆ i is modified as follows: ⎧ ⎪ ⎪ ⎪ ⎨ ϕi |bˆ i | > b0 or bˆ i = b0 and hi (y)ey usgn(bˆ i ) > 0 b˙ˆ i = ⎪ ⎪ ⎪ ⎩ 0 otherwise

(26)

where: ϕi = γ2hi (y)ey u − γ2 ν|ey |(bˆ i − b0 )

(27)

and sgn(.) is the Signum function. The stability analysis of the system should be reconsidered with ˆ i > b0 or bˆ i = b0 and (y− yˆ )hi (y)usgn(bˆ i ) > 0, the modified adaptation law. In the first case, when |b| the stability analysis is the same as before and we have: V˙ 1 < −c1 V1 + c2 . In the other case, the time derivative of the Lyapunov function for the modified adaptation law is obtained as: λmin (Q1 ) 2 b˜ 2i e − ν a˜ Ti a˜ i + ν ˜ai − aˆ i |ey |2 − ν V˙ 1 ≤ − 2 i=1 i=1 i=1 l

+ ν

l

l

l

b˜ i − bˆ i |ey |2 − ν(ρ − ρ∗ )2 + ν(ρ − ρ∗ ) − ρ|ey |2 + 2eT P1 B

i=1

l

hi (y)b˜ i u (28)

i=1

so that: V˙ 1 ≤ −c1 V1 + c2 + 2eT P1 B

l i=1

7

hi (y)b˜ i u

(29)

Since eT P1 Bhi (y)b˜ i usgn(bˆ i ) < 0 and b˜ i sgn(bˆ i ) > 0 we have eT P1 B li=1 hi (y)b˜ i u < 0. So that V˙ 1 ≤ −c1 V1 + c2

(30)

It is seen that the same result is obtained in both cases, indicating that all signals are uniformly bounded. Therefore, V1 decreases monotonically until e, bˆ i and aˆ i i = 1, ..., l reach the compact set in which V1 ≤

c2 . c1

This means that e, aˆ i and bˆ i are uniformly bounded. Furthermore it is possible to

choose ν, γ1 γ2 and γ3 so that the residual

c2 c1

is arbitrarily small and consequently any desired level

for the state estimation error can be obtained. The following theorem summarizes the foregoing stability analysis. Theorem 1. If assumptions 1 and 2 are satisfied, the observer given by (6) for the nonlinear dynamical system of (3) with the adaptation laws of (16), (18) and (26) ensures that the state estimation error and the estimated values of the system aˆ i and bˆ i are uniformly bounded. Furthermore, the estimation error can be made to approach an arbitrarily small value by choosing the design constants ν, γ1 , γ2 and γ3 appropriately. 4. The design of the indirect model reference fuzzy controller based on the proposed observer and its stability analysis In this section, the indirect model reference fuzzy controller based on the proposed observer is designed and its stability analysis is presented. 4.1. The dynamics of the observed tracking error The reference model for the plant is considered to be of the following form. x˙ m = Am xm + Br r ym = Cxm

(31)

where xm ∈ Rn , Am ∈ Rn×n is the stable state matrix of the reference model and Br ∈ Rn×1 is the input matrix of the reference system, defined as Br = br B in which br is a scalar design value. It is assumed that Am satisfies the following equation: Am = A − LC + BaTm 8

(32)

in which am ∈ Rn×1 . The observed tracking error is defined as: eˆ m = xˆ − xm

(33)

The dynamical equation of the observer (6) is rewritten as: ⎡ l ⎤ l ⎢⎢⎢ ⎥⎥ T T T hi (y)bˆ i u + ρsgn(ey )⎥⎥⎥⎦ + Ly xˆ˙ = (A − LC + Bam )ˆx + B ⎢⎢⎣ hi (y)(ˆai − am )ˆx + i=1

so that:

(34)

i=1

⎡ l ⎤ l ⎢⎢⎢ ⎥⎥ hi (y)bˆ i u + ρsgn(ey )⎥⎥⎥⎦ + Ly x˙ˆ = Am xˆ + B ⎢⎢⎣ hi (y)(ˆaTi − aTm )ˆx + i=1

(35)

i=1

The dynamics of the observed tracking error can be expressed as: ⎡ l ⎤ l ⎢⎢⎢ ⎥⎥ T T hi (y)bˆ i u + ρsgn(ey )⎥⎥⎥⎦ + Ly − Br r eˆ˙ m = Am eˆ m + B ⎢⎢⎣ hi (y)(ˆai − am )ˆx + i=1

(36)

i=1

4.2. The proposed fuzzy control signal The control signal is proposed as:  T T

l am −ˆai ˆ xˆ + i=1 hi (y)|bi | bˆ i u=

l ˆ i=1 hi (y)|bi |

br r bˆ i

 σy2 eˆmy ρsgn(ey ) − l − l ˆ ˆ e2my + δ) i=1 hi (y)bi i=1 hi (y)bi (ˆ

(37)

in which δ > 0 and has a very small value and eˆ my is defined as: eˆ my = yˆ − ym . This control signal can be decomposed into two terms: an indirect model reference fuzzy term (u f ) and a robust term to ensure the stability of the system (ur ). u f is defined as follows:  T T   T T

l am −ˆai am −ˆai br l ˆ ˆ ˆ h (y)| b | r M (y)| b | x + xˆ + i i i i i=1 i=1 bˆ i bˆ i bˆ i uf = =

l

l ˆ ˆ i=1 hi (y)|bi | i=1 Mi (y)|bi |

br r bˆ i

 (38)

and ur is defined as: ρsgn(ey ) σ(y)2 eˆ my − l ur = − l ˆ ˆ e2my + δ) i=1 hi (y)bi i=1 hi (y)bi (ˆ

(39)

The fuzzy term u f can be written in the form of fuzzy IF–THEN rule. The ith rule of the corresponding fuzzy system is as: Rulei : IF y is Ni THEN u = Ki xˆ + Kri r 9

(40)

in which: Ki =

aTm − aˆ Ti bˆ i br Kri = bˆ i

(41) (42)

and Ni is the grade of the membership function for the control system and is defined as: Ni = |bˆ i |Mi . 4.3. Stability analysis of the proposed controller The substitution of the control signal of (37) into (36) results in: e˙ˆ m = Am eˆ m + Ly − B

σy2 eˆmy eˆ 2my + δ

(43)

To analyze the stability of the proposed observer based controller the following Lyapunov function is considered. V = e P1 e + T

eˆ Tm P2 eˆ m

l l 1 T 1 ˜T ˜ 1 bi bi + (ρ − ρ∗ )2 + a˜ i a˜ i + γ1 i=1 γ2 i=1 γ3

(44)

Considering (9), (44) can be rewritten as: V = V1 + eˆ Tm P2 eˆ m

(45)

The time derivative of the Lyapunov function is obtained as: V˙ = V˙ 1 + e˙ˆ Tm P2 eˆ m + eˆ Tm P2 e˙ˆ m

(46)

Substituting (43) into (46) gives: V˙ = V˙ 1 + eˆ Tm ATm P2 eˆ m + eˆ Tm P2 Am eˆ m + 2ˆeTm P2 Ly − 2

eˆ Tm P2 Bσy2 eˆ my eˆ 2my + δ

(47)

It is assumed that there exists a positive definite matrix P2 such that: ATm P2 + P2 Am = −Q2

(48)

P 2 B = CT

(49)

where Q2 is a positive definite matrix. It should be noted that (48) is not a restrictive condition and that it is possible to take Am = A − LC so that the same positive definite matrices as in (8) can be used to satisfy (48). 10

With the use of (48) it is obtained that: σδy2 − 2σy2 V˙ = V˙ 1 − eˆ Tm Q2 eˆ m + 2ˆeTm P2 Ly + 2 2 eˆ my + δ

(50)

Since for any α > 0 we have: 2ˆeTm P2 Ly ≤ α(ˆeTm P2 L)2 +

1 2 y α

(51)

we get: 1 σδy2 V˙ ≤ V˙ 1 − eˆ Tm Q2 eˆ m + α(ˆeTm P2 L)2 + y2 − 2σy2 + 2 2 α eˆ my + δ

(52)

By choosing σ as: 1 <σ 2α

(53)

1 αLT P2 P2 L ≤ λmin (Q2 ) 2

(54)

1 σδy2 V˙ ≤ V˙ 1 − λmin (Q2 )ˆem 2 + 2 2 2 eˆ my + δ

(55)

V˙ ≤ −c3 V + c4

(56)

and Q2 such that:

we obtain:

Furthermore:

in which:



λmin (Q2 ) λmin (Q1 ) , , νγ1, νγ2 , νγ3 c3 = min 2λmax (P2 ) 2λmax (P1 )

 (57)

and: c4

=ν

l

˜ai − aˆ i |ey | + ν 2

i=1

l i=1

σδy2 b˜ i − (bˆ i − b0 )|ey |2 + ν(ρ − ρ∗ ) − ρ|ey |2 + 2 2 eˆ my + δ

(58)

So that: c4 = sup {ν

l i=1

˜ai − aˆ i |ey | + ν 2

l i=1

σδy2 2 ∗ 2 ˜ ˆ (}59) bi − (bi − b0 )|ey | + ν(ρ − ρ ) − ρ|ey | + 2 2 eˆ my + δ

t 11

Using lemma 1 as in appendix, we have:   c4 −c3 t c4 V(t) ≤ V(0) − e + c3 c3

(60)

Up to now the stability of the observed tracking error is considered but not the tracking error of the system which is the main concern in this paper. For this purpose, the following Lyapunov function is considered. V2 = e P3 e + T

eTm P4 em

l l 1 T 1 ˜T ˜ 1 bi bi + (ρ − ρ∗ )2 + a˜ i a˜ i + γ1 i=1 γ2 i=1 γ3

(61)

in which em is the tracking error, defined as: em = x − xm and P3 and P4 are two positive definite matrices. Since eˆ m = xˆ − xm and e = x − xˆ we have: l l 1 T 1 ˜T ˜ 1 bi bi + (ρ − ρ∗ )2 a˜ i a˜ i + V2 = e P3 e + (ˆem + e) P4 (ˆem + e) + γ1 i=1 γ2 i=1 γ3 T

T

(62)

and further: V2 = eT P3 e + eˆ Tm P4 eˆ m + eT P4 e + 2ˆeTm P4 e +

l l 1 T 1 ˜T ˜ 1 bi bi + (ρ − ρ∗ )2 a˜ i a˜ i + γ1 i=1 γ2 i=1 γ3

(63)

Considering the fact that for any β > 0 we have: 1 2ˆeTm P4 e ≤ βˆeTm P4 eˆ m + eT P4 e β

(64)

It is therefore obtained that: l l 1 1 T 1 ˜T ˜ 1 V2 ≤ eT P3 e + eˆ Tm P4 eˆ m + eT P4 e + βˆeTm P4 eˆ m + eT P4 e + bi bi + (ρ − ρ∗ )2 (65) a˜ i a˜ i + β γ1 i=1 γ2 i=1 γ3

and: l l 1 1 T 1 ˜T ˜ 1 a˜ i a˜ i + bi bi + (ρ − ρ∗ )2 V2 ≤ eT (P3 + P4 + P4 )e + eˆ Tm (P4 + βP4 )ˆem + β γ1 i=1 γ2 i=1 γ3

(66)

If P3 and P4 are taken as: 1 P1 = P3 + P4 + P4 β P2 = P4 + βP4 12

(67)

(68)

Figure 1: The observer based indirect model reference fuzzy control scheme

it is obtained that:

  c4 −c3 t c4 V2 ≤ V ≤ V(0) − e + c3 c3

(69)

in which c4 and c3 are defined as in (59) and (57). According to the above equation, given any μ>

c4 c3

value

there exists a time T μ such that for all t ≥ T μ we have V2 (t) ≤ μ. Furthermore, the residual c4 c3

can be chosen to have an arbitrarily small value by the appropriate choice of γ1 , γ2 , γ3 and

ν. The following theorem summarizes the foregoing stability analysis. Theorem 2. If assumptions 1 and 2 are satisfied, the control signal given by (37) which is composed of a fuzzy part (38) and a robust part (39), together with the observer given by (6) for the nonlinear dynamical system of (3) with the adaptation laws of (16), (18) and (26) ensures that 13

the tracking error, the state estimation error and the estimated values of the system aˆ i and bˆ i are uniformly bounded. Furthermore, the tracking error and the state estimation error can approach an arbitrarily small value by choosing the design constants ν, γ1 , γ2 and γ3 appropriately. The overall scheme of the observer based indirect model reference fuzzy controller for the plant (3) is shown in Fig. 1. 5. Simulation results In this section we use a well-known chaotic system of Chua’s circuit to depict the design procedure and verify the effectiveness of the proposed algorithm. The control of the nonlinear chaotic Chua’s circuits is an important topic for numerous practical applications since this circuit exhibits a wide variety of nonlinear dynamic phenomena such as bifurcations and chaos [23]. This chaotic circuit possesses the properties of simplicity and universality, and has become a standard prototype for investigation of chaos. In this section we will use the proposed method to control the nonlinear chaotic Chua’s circuit. 5.1. Dynamical equations of Chua’s circuit The modified Chua’s circuit is described by the following dynamical system [24]: 1 x˙1 = p(x2 − (2x31 − x1 )) + u1 + d(x, t), 7 x˙2 = x1 − x2 + x3 + u2 , x˙3 = −q.x2 + u3 ,

(70)

in which u1 , u2 and u3 are the external inputs and d(x, t) is the bounded external disturbance. Considering q = − 100 and u2 = u3 = 0 [24], one obtains the state space equations of the system as: 7 1 x˙1 = p(x2 − (2x31 − x1 )) + u1 + d(x, t), 7 x˙2 = x1 − x2 + x3 , 100 .x2 , x˙3 = − 7 in which p = 10 [24]. 14

(71)

5.2. Control of Chua’s circuit The dynamical equation of Chua’s circuit (71) can be viewed as the nonlinear dynamical system in the form of (1) in which ⎤ ⎡ ⎡ ⎤ ⎥⎥⎥ ⎢⎢⎢ 0 0 ⎢⎢⎢ 1 ⎥⎥⎥ 0 ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢ ⎥⎥⎥   ⎥⎥⎥ ⎢⎢⎢ ⎢ ⎥ A = ⎢⎢⎢ 1 −1 1 ⎥⎥⎥ B = ⎢⎢⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ C = 1 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢ ⎥⎥⎥ ⎦ ⎣ ⎣ ⎦ 100 0 − 7 0 0

(72)

1 f (x) = p(x2 − (2x31 − x1 )), g(x) = 1 7

(73)

and

In order to model f (x) in the interval of [-2, 2], TS membership functions labeled as about(-2), about(-1), about(0), about(1) and about(2) are considered. These labels correspond to fuzzy membership functions as: exp(−(x1 + 2)2 /0.422), exp(−(x1 + 1)2 /0.422 ), exp(−x21 /0.422 ), exp(−(x1 − 1)2 /0.422 ) and exp(−(x1 − 2)2 /0.422 ), respectively. The following rules for the TS fuzzy model are considered. Rule 1 If x1 is about(−2) then x˙ = Ax + B(aT1 x + b1 u) Rule 2 If x1 is about(0) then x˙ = Ax + B(aT2 x + b2 u) Rule 3 If x1 is about(1) then x˙ = Ax + B(aT3 x + b3 u) Rule 4 If x1 is about(1) then x˙ = Ax + B(aT4 x + b4 u) Rule 5 If x1 is about(2) then x˙ = Ax + B(aT5 x + b5 u) in which A, B and C are defined as in (72) and: a1 = a5 =



− 237 p

T p 0

, a2 = a4 =



− 57 p

T p 0

, a3 =

T

 1 p 7

p 0

, b1 = b2 = b3 = [1, 0, 0]T (74)

are the initial values for the TS fuzzy system. The control scheme for the system is the same as Fig. 1. The gain of the observer L is taken as: L=



T 5 1 0

,

(75)

and Q1 as Q1 = 10I3×3 in which I3×3 is the identity matrix. The positive definite matrix P1 which

15

is the solution of (8) is obtained as: ⎡ ⎤T ⎢⎢⎢ 1 ⎥⎥⎥ 0 0 ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥ P1 = ⎢⎢⎢ 0 76.43 −5 ⎥⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎣ ⎦ 0 −5 5.7

(76)

The state matrix of the reference model is set as Am = A − LC. Firstly it is assumed that there is no external disturbance and d(x, t) is set to zero. Using these design parameters, the regulation and the tracking performances of the proposed control scheme are tested. Figures 2(a)-2(d) show the results of the regulation of the system as well as the state estimation performance of the observer. The initial values for the states of the system, the observer and the reference model are considered as: x = [0.5, 0, 0]T , xˆ = [0.4, 0, 0.05]T and xm = [0.6, 0, 0.05] respectively. In addition, the tracking performance of the proposed controller is tested. Figures 3(a)-3(d) depict the tracking performance of the controller and the response of the observer when the reference signal is non zero. The initial values considered for the systems are the same as in the regulation experiments. In order to show the applicability of the proposed approach in the presence of external disturbances, it is assumed that d(x, t) = 2sin(t). Figures 4(a)-4(d) show the tracking performance of the proposed controller in the presence of the stated external disturbance. As can be seen from the figures, the tracking performance of the system under control in the presence of the external disturbance is quite satisfactory. 6. Conclusions and Future Works This paper describes the design of an observer–based indirect model reference adaptive controller for use with nonlinear systems subject to external disturbances. The proposed method adopts a TS fuzzy model to represent the dynamics of the system in hand and the adaptive model reference controller. The main contribution of the current work with respect to the previous studies in the field of model reference fuzzy controllers is that the current approach benefits from an observer and therefore full state measurement is no longer needed. Its additional superiority over the observer based TS adaptive fuzzy controllers seen in the literature is that it is capable of making the system follow a reference model rather than just regulation of the system to zero, even 16

when the system under control is subject to external disturbances. The stability of the approach is automatically accomplished with the derivation of the adaptive law by the Lyapunov theory. Lastly, through the application to a Chua’s circuit, the applicability of the design to the practical problems of control of chaotic systems is verified. It is shown that the current approach is capable of controlling Chua’s circuit with high performance. Designing a fuzzy system based on fuzzy and/or piecewise Lyapunov functions are highly appreciated methods studied in recent years [25, 26]. The controller designed based on these methods are known to be less conservative. The authors find it possible to extend the current work based on these methods in order to relax the conditions found in this paper. Appendix lemma 1. Suppose V(t) ≥ 0 satisfies the inequality ˙ ≤ −c1 V(t) + c2 V(t) where c1 > 0 and c2 > 0 are constants. Then V(t) satisfies   c2 −c1 t c2 V(t) ≤ V(0) − e + c1 c1

(1)

(2)

˙ = −c1 V(t)+c2 −k(t). ˙ ≤ −c1 V(t)+c2 , there exists a function k(t) ≥ 0 such that V(t) Proof. Since V(t) Therefore v(t) satisfies −c1 t



t

e−c1 (t−τ) (c2 − k(τ))dτ 0  t  t −c1 t −c1 (t−τ) e dτ − e−c1 (t−τ) k(τ)dτ = e V(0) + c2 0   0 c2 −c1 t c2 ≤ V(0) − e + c1 c1

V(t) = e

V(0) +

(3) (4) (5)

References [1] J.-Y. Chen, Expert smc-based fuzzy control with genetic algorithms, Journal of the Franklin Institute 336 (1999) 589 – 610.

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[2] Y. Hacioglu, Y. Z. Arslan, N. Yagiz, Mimo fuzzy sliding mode controlled dual arm robot in load transportation, Journal of the Franklin Institute 348 (2011) 1886 – 1902. [3] C.-F. Hsu, C.-M. Lin, Fuzzy-identification-based adaptive controller design via backstepping approach, Fuzzy Sets and Systems 151 (2005) 43 – 57. [4] C.-W. Park, M. Park, Adaptive parameter estimator based on t-s fuzzy models and its applications to indirect adaptive fuzzy control design, Information Sciences 159 (2004) 125 – 139. [5] K. J. Astrom, B. Wittenmark, Adaptive Control, Dover Publications, 2008. [6] Y.-W. Cho, C.-W. Park, M. Park, An indirect model reference adaptive fuzzy control for siso takagi-sugeno model, Fuzzy Sets and Systems 131 (2002) 197 – 215. [7] Y.-W. Cho, C.-W. Park, J.-H. Kim, M. Park, Indirect model reference adaptive fuzzy control of dynamic fuzzystate space model, Control Theory and Applications, IEE Proceedings - 148 (2001) 273 –282. [8] T. Koo, Stable model reference adaptive fuzzy control of a class of nonlinear systems, Fuzzy Systems, IEEE Transactions on 9 (2001) 624 –636. [9] C.-W. Park, Y.-W. Cho, Adaptive tracking control of flexible joint manipulator based on fuzzy model reference approach, Control Theory and Applications, IEE Proceedings - 150 (2003) 198 – 204. [10] M. Khanesar, O. Kaynak, M. Teshnehlab, Direct model reference takagi–sugeno fuzzy control of siso nonlinear systems, Fuzzy Systems, IEEE Transactions on 19 (2011) 914 –924. [11] K. L. Youngwan Cho, Yangsun Lee, E. Kim, A lyapunov function based direct model reference adaptive fuzzy control, Knowledge-Based Intelligent Information and Engineering Systems 3214 (2004) 202–210. [12] Y.-W. Cho, E.-S. Kim, K.-C. Lee, M. Park, Tracking control of a robot manipulator using a direct model reference adaptive fuzzy control, in: Intelligent Robots and Systems, 1999. IROS ’99. Proceedings. 1999 IEEE/RSJ International Conference on, volume 1, pp. 100 –105 vol.1. [13] M. A. Khanesar, M. Teshnehlab, Direct stable adaptive fuzzy neural model reference control of a class of nonlinear systems, in: Innovative Computing Information and Control, 2008. ICICIC ’08. 3rd International Conference on, volume 1, pp. 512 –515. [14] H. H. Choi, Lmi-based nonlinear fuzzy observer-controller design for uncertain mimo nonlinear systems, Fuzzy Systems, IEEE Transactions on 15 (2007) 956 –971. [15] M. Teixeira, E. Assuncao, R. Avellar, On relaxed lmi-based designs for fuzzy regulators and fuzzy observers, Fuzzy Systems, IEEE Transactions on 11 (2003) 613 – 623. [16] K.-Y. Lian, C.-S. Chiu, T.-S. Chiang, P. Liu, Lmi-based fuzzy chaotic synchronization and communications, Fuzzy Systems, IEEE Transactions on 9 (2001) 539 –553. [17] Y.-G. Leu, T.-T. Lee, W.-Y. Wang, Observer-based adaptive fuzzy-neural control for unknown nonlinear dynamical systems, Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on 29 (1999) 583 –591.

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[18] Y.-G. Leu, W.-Y. Wang, T.-T. Lee, Observer-based direct adaptive fuzzy-neural control for nonaffine nonlinear systems, Neural Networks, IEEE Transactions on 16 (2005) 853 –861. [19] C.-H. Hyun, C.-W. Park, S. Kim, Takagi-sugeno fuzzy model based indirect adaptive fuzzy observer and controller design, Information Sciences 180 (2010) 2314 – 2327. [20] M. Ababneh, A. M. Almanasreh, H. Amasha, Design of digital controllers for uncertain chaotic systems using fuzzy logic, Journal of the Franklin Institute 346 (2009) 543 – 556. [21] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man, and Cybernetics 15 (1985) 116 132. [22] J.-J. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ,, 1991. [23] Y. C. Chang, A robust tracking control for chaotic chua’s circuits via fuzzy approach, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 48 (2001) 889 –895. [24] M. T. Yassen, Adaptive control and synchronization of a modified chua’s circuit system, Applied mathematics and computing 135 (2003) 113–128. [25] J. Qiu, G. Feng, H. Gao, Asynchronous output-feedback control of networked nonlinear systems with multiple packet dropouts: T–s fuzzy affine model-based approach, Fuzzy Systems, IEEE Transactions on 19 (2011) 1014 –1030. [26] J. Qiu, G. Feng, H. Gao, Fuzzy-model-based piecewise static-output-feedback controller design for networked nonlinear systems, Fuzzy Systems, IEEE Transactions on 18 (2010) 919 –934.

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0.6

0.6

x1

x1 m1

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1 0

2

4 6 time (sec)

8

estimated value for x

0.5

the reference signal x

0.5

−0.1 0

10

1

2

(a)

4 6 time (sec)

8

10

(b)

0.08

0.3 x2

0.06

x3

estimated value for x2

estimated value for x3

0.2

0.04

0.1

0.02 0 0 −0.1 −0.02 −0.2

−0.04

−0.3

−0.06 −0.08 0

2

4 6 time (sec)

8

−0.4 0

10

(c)

2

4 6 time (sec)

8

10

(d)

Figure 2: The regulation performance of the proposed observer and the controller when applied to Chua’s chaotic system, (a): The regulation response of Chua’s chaotic system for x1 , (b): The performance of the observer for x1 , (c): The performance of the observer for x2 , (d): The performance of the observer for x3

20

0.8

0.8

x

x

1

1

the reference signal x

0.6

0.6

m1

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6 0

10

20 30 time (sec)

40

estimated value for x

−0.6 0

50

1

10

(a) 0.2

40

50

(b) x

0.15

20 30 time (sec)

x3

1.5 2

estimated value for x3

estimated value for x2

1

0.1 0.5

0.05 0

0

−0.05

−0.5

−0.1 −1

−0.15 −0.2 0

10

20 30 time (sec)

40

−1.5 0

50

(c)

10

20 30 time (sec)

40

50

(d)

Figure 3: The tracking performance of the proposed observer and the controller when applied to Chua’s chaotic system, (a): The tracking response of Chua’s chaotic system for x1 , (b): The performance of the observer for x1 , (c): The performance of the observer for x2 , (d): The performance of the observer for x3

21

0.8

estimated value for x

the reference signal x

0.6

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4 20

40 time (sec)

60

1

0.6

m1

0.4

−0.6 0

x1

0.8

x1

−0.6 0

80

20

(a) 0.2

60

80

(b) x3

1.5

x2

estimated value for x3

estimated value for x2

0.15

40 time (sec)

1

0.1 0.5

0.05 0

0

−0.05

−0.5

−0.1 −1

−0.15 −0.2 0

20

40 time (sec)

60

−1.5 0

80

20

(c)

40 time (sec)

60

80

(d)

Figure 4: The tracking performance of the proposed observer and the controller when applied to Chua’s chaotic system in the presence of external disturbances, (a): The tracking response of Chua’s chaotic system for x1 , (b): The performance of the observer for x1 , (c): The performance of the observer for x2 , (d): The performance of the observer for x3

22