Indirect Adaptive Fuzzy and Impulsive Control of Nonlinear Systems

Transcription

1 International Journal of Automation an Computing 7(4), November 200, DOI: 0007/s Inirect Aaptive Fuzzy an Impulsive Control of Nonlinear Systems Hai-Bo Jiang School of Mathematics, Yancheng Teachers University, Yancheng , PRC Abstract: The problem of inirect aaptive fuzzy an impulsive control for a class of nonlinear systems is investigate Base on the approximation capability of fuzzy systems, a novel aaptive fuzzy an impulsive control strategy with supervisory controller is evelope With the help of a supervisory controller, global stability of the resulting close-loop system is establishe in the sense that all signals involve are uniformly boune Furthermore, the aaptive compensation term of the upper boun function of the sum of resiual an approximation error is aopte to reuce the effects of moeling error By the generalize Barbalat s lemma, the tracking error between the output of the system an the reference signal is prove to be convergent to zero asymptotically Simulation results illustrate the effectiveness of the propose approach Keywors: Fuzzy control, aaptive control, impulsive control, generalize Barbalat s lemma Introuction In the past few ecaes, fuzzy logic control of nonlinear systems has receive consierable attention [] Among various kins of fuzzy methos, aaptive fuzzy control of nonlinear systems using universal function approximators has been an attractive subject of research, an many results have been reporte [2 5] There are two istinct approaches that have been formulate in the esign of a fuzzy aaptive control system: irect an inirect schemes In the irect scheme, the fuzzy system is use to approximate an unknown ieal controller [0] On the other han, the inirect scheme uses fuzzy systems to estimate the plant ynamics an then synthesizes a control law base on these estimates [2, 7, 8, ] Using the approximation capability of the first-type fuzzy system, which is the linear function of ajustable parameters, the esign schemes of some stable irect aaptive fuzzy controllers were propose in [0] In [0], the tracking error convergence epens upon the assumption that the approximation error shoul be squareintegrable The supremum of the optimal approximation error is assume to be known or boune in [2, 4, 8, 9] In [6, 7], a new scheme of inirect aaptive fuzzy control for a class of nonlinear systems has been propose, an the scheme oes not require the optimal approximation error to be square-integrable or the supremum of the optimal approximation error to be known In [7], a robust aaptive fuzzy controller was esigne for nonlinear system using estimation of bouns for approximation errors On the other han, base on the approximation capability of multilayer neural networks, several aaptive control strategies have been propose in [8 20] In the real worl, there are natural phenomena that characteristically change their own state abruptly at certain moments Consequently, it is natural to assume that these Manuscript receive February 0, 2009; revise November 24, 2009 This work was supporte by Natural Science Founation of Jiangsu Province of China (No BK200292, No BK200293), an Natural Science Founation of the Jiangsu Higher Eucation Institutions of China (No 09KJB5008, No 08KJD50008) perturbations act instantaneously, that is, in the form of impulses [2 23] In the case where the resetting events are efine by a prescribe sequence of times that are inepenent of the system state, the systems are known as timeepenent impulsive systems [2 25] Alternatively, in the case where the resetting events are efine by a manifol in the state space that is inepenent of time, the systems are autonomous an are known as state-epenent impulsive systems [2 25] Recently, the stability of systems with impulsive effect has sparke the interest of many researchers [26, 27] On the other han, impulsive control provies a new viewpoint when the plant has at least one changeable state variable or when the plant has impulsive effects [2] It has been shown that impulsive control or synchronization approach is effective an robust in control or synchronization of chaotic systems [2] an complex ynamical networks [28 33] In [28], some robust aaptive-impulsive synchronization schemes were evelope to synchronize the noes of an uncertain ynamical network where the network coupling is an unknown but boune nonlinear function In [29, 30], the authors propose a new hybri impulsive an switching controller for nonlinear systems Many aaptive control algorithms have been evelope in the literature for both continuous-time an iscrete-time systems However, there are few results on hybri systems Very recently, a new invariant set stability theorem for a class of state-epenent impulsive ynamical systems is propose in [34] Base on the new invariant set stability theorem, Haa et al [35] evelope a irect hybri aaptive control framework for nonlinear uncertain stateepenent impulsive ynamical systems The well-known Barbalat s lemma is an effective tool to analyze the stability of the system The Barbalat s lemma has been use to prove that the approximation error is convergent to zero asymptotically [2, 4, 7 ] The Barbalat s lemma require that the function is ifferentiable on the interval [0, ); thus, it is not suitable to the impulsive ynamical systems [36, 37]

2 H B Jiang / Inirect Aaptive Fuzzy an Impulsive Control of Nonlinear Systems 485 In this paper, we investigate the problem of inirect aaptive fuzzy an impulsive control for a class of nonlinear systems Base on the approximation capability of fuzzy systems, a novel aaptive fuzzy an impulsive control strategy with supervisory controller is evelope By the generalize Barbalat s lemma, the tracking error between the output of the system an the reference signal is prove to be convergent to zero asymptotically The paper is organize as follows In Section 2, the moel of the plant an the basic assumptions an lemmas are given The inirect aaptive fuzzy an impulsive control algorithm is introuce in Section 3 A proof of the stability result is given in Section 4 In Section 5, the esign metho is emonstrate by two examples Section 6 conclues the paper Throughout this paper, R n enotes n-imensional Eucliean space; let R + = [0, ), N = {0,, 2, }; R n m is the set of all n m real matrices; for real symmetric matrices X an Y, the notation X Y (respectively, X > Y ) means that the matrix X Y is positive semi-efinite (respectively, positive efinite); I is an appropriately imensione ientity matrix; an λ min(p ) (λ max(p )) enotes the smallest (largest) eigenvalue of P The vector norm of x R n is Eucliean, ie, x = x T x For a real-value vector function f : R n R n, we efine f(a) = f(a + ) f(a ), where f(a + ) = lim x a + f(x), f(a ) = lim x a f(x) 2 Problem formulations Consier the following n-th-orer nonlinear system ẋ i = x i+, i =,, n ẋ n = f(x) + g(x)u + (t) () y = x where x = (x, x 2,, x n) T R n is the state vector, u is the control input, f is an unknown continuous function, g is the unknown function control gain, an (t) enotes the external isturbance The control objective is to force the output y to track a given boune reference signal y Define e = [y y, ẏ ẏ,, y (n ) y (n ) ] T (2) Then, the aim is to esign an aaptive fuzzy controller u an an impulsive controller U k such that the resulting close-loop system is globally stable in the sense that all signals involve are uniformly boune, an the tracking error between the output of the system y an the given boune reference signal y is convergent to zero asymptotically, ie, e 0 as t + Uner the aaptive an impulsive control, the controlle state of system () satisfies the following system: ẋ i = x i+, i =,, n ẋ n = f(x) + g(x)u + (t), (3) y = x x(t k ) = U k (e(t k )), k =, 2, where x(t k ) = x(t + k ) x(t k ), x(t+ k ) = lim t t + x(t) an k x(t k ) = lim t t x(t) Without loss of generality, we as- k sume that lim t t + x(t) = x(t k ), which means that the solution x(t) is right continuous at time t k The impulsive k controller U k is to be esigne later, an the impulsive time instants t k satisfy 0 < t < t 2 < < t k < t k <, lim k t k = From (2) an (3), we obtain the following error ynamical system { ė i = e i+, i =,, n ė n = y (n) f(x) g(x)u (t), (4) e(t k ) = U k (e(t k )), k =, 2, In orer to esign stable aaptive fuzzy an impulsive control, we make the following assumptions, which have been use in many papers, for example, [7, 8,, 6, 7] Assumption There exists a known positive continuous function F (x) such that f(x) F (x), x R n Assumption 2 There exist two positive constants g 0 an g such that 0 < g 0 g(x) g, x R n Assumption 3 There exists a positive constant M such that [y, ẏ,, y (n) ]T M Assumption 4 There exists a positive constant D such that (t) D, t 0 Since fuzzy system can use not only the sensor s igital information but also the expert s language information, in this paper, we consier the case where the fuzzy rule base consists of M rules in the following form [] : R l : If x is A l an an x n(t) is A l n, then y is B l, l =, 2,, M By using the fuzzy inference metho with singleton fuzzification, prouct inference, an center average efuzzification, the crisp output of fuzzy system can be expresse as follows: M y l [ n µ A l (x l= i= i i)] y = = θ M n T ξ(x) µ A l (x i i) l= i= where θ = (y,, y M ) T, ξ(x) = (p (x),, p M (x)) T, x = (x, x 2,, x n) T, p l (x) = ( n i= µ A l (xi))/( M n i l= i= µ A l (xi)), l =, 2,, M, are i calle fuzzy basis functions, an the membership function of fuzzy set A l i is taken as µ A l (x i i) = exp( ((x i a l i)/b l i) 2 ) Accoring to [], the above fuzzy system is a universal approximator Assumption 5 For any continuous function h(x) over a compact set Ω x, there exist an ieal ajustable parameter vector θ an positive constant ε such that θ T ξ(x) h(x) ε, x Ω x The ieal parameter vector θ is an artificial quantity require for analytical purpose θ is efine as follows: θ = arg min θ Rm{ sup h(x) θ T ξ(x) } x Ω x Before proceeing, we recall some preliminaries that will be use throughout the proofs of our main results

3 486 International Journal of Automation an Computing 7(4), November 200 Definition [2] Let V : R + R n R +, then V is sai to belong to class ν 0 if ) V is continuous in each of the sets [t k, t k ) R n, an for each x R n, k =, 2,, lim (t,y) (t + V (t, y) =,x) k V (t + k, x) exists; 2) V is locally Lipschitzian in x R n Let ẋ = f(t, x), then we have the following generalize erivative of a Lyapunov function V (t, x) Definition 2 [2] For V ν 0, (t, x) [t k, t k ) R n, we efine D + V (t, x) = lim sup [V (t + h, x + hf(t, x)) V (t, x)] h 0+ h [36, 37] Lemma (Generalize Barbalat s Lemma) Let {x k } be a sequence of number, which satisfies 0 x 0 < x < x 2 < < x k < x k < an lim k x k = There exist two positive constants λ, λ 2 such that λ = inf k {x k x k } > 0, λ 2 = sup k {x k x k } < The function f(x) is efine on [0, + ) an ifferentiable on the interval [x k, x k ), k =, 2, Suppose f(x) an f (x) are uniformly boune for k =, 2, on the interval [x k, x k ), that is, there exist two positive constants M 0 an M such that for every k =, 2,, x [x k, x k ), it hols that f(x) M 0, f (x) M If the improper integral + x 0 f(x)x converges, then lim x + f(x) = 0 3 Inirect aaptive fuzzy an impulsive controller esign In this section, base on approximation capability of fuzzy systems, a novel aaptive fuzzy an impulsive control strategy with supervisory controller is evelope Let K = [K n, K n,, K ] T R n be such that all roots of polynomial h(s) = s n + K s n + + K n = 0 are in the open left-half complex plane Equation (4) can be transforme into ė = Λ ce + b c[y (n) f(x) g(x)u + K T e], e(t k ) = U k (e(t k )), k =, 2, (5) where Λ c = K n K n K n 2 K 2 K 0 0 b c = 0 Because Λ c is stable, there always exists a positive efinite matrix P = P T R n n satisfying the Lyapunov equation Λ T c P + P Λ c + Q = 0 (6) where Q R n n is an arbitrary positive efinite matrix specifie by the esigner If both functions f(x) an g(x) are known, an there is no external isturbance an impulsive control, then the control law u = [ f(x) + y(n) + K T e] g(x) applie to system () can result in e (n) + K e (n ) + + K ne = 0 which implies that lim t e (t) = 0 However, in many practical problems, f(x) an g(x) are not well-known, so the ieal controller cannot be implemente In this situation, the approximation by fuzzy logic systems is employe to treat this stable control esign problem Define a compact set Ω x = {x x M x}, where M x > M is a parameter to be efine later Let f(x, θ f ), g(x, θ g) be the approximation of first-type fuzzy logic system on the compact set Ω x to f(x), g(x), respectively, ie, f(x, θ f ) = θ T f ξ f (x), g(x, θ g) = θ T g ξ g(x) where θ f = (θ f,,, θ f,mf ) T an θ g = (θ g,,, θ g,mg ) T are ajustable parameter vectors, m f an m g are the number of rules in the fuzzy system, an ξ f (x) = (ξ f, (x),, ξ f,mf (x)) T an ξ g(x) = (ξ g,(x),, ξ g,mg (x)) T are fuzzy basis function vectors Define Ω θf = {θ f θ f M θf } Ω θg = {θ g θ g M θg } where M θf an M θg are two positive constants, specifie by the esigner Let us efine the optimal parameter estimators θ f an θ g as follows: θ f = arg min θ f Ω f [ sup x Ω x f(x, θ f ) f(x) ] θ g = arg min θ g Ω g [ sup x Ω x g(x, θ g) g(x) ] Inspire by [, 6, 7], we aopt the following aaptive fuzzy control law: u = u c + u s + u w (7) u c = g(x, ˆθ [ f(x, ˆθ f ) + y (n) + K T e], g) (8) u c(t k ) = 0, k =, 2, (9) u s = I sign(e T P b c) g 0 [ f(x, ˆθ f ) + F (x)+ (0) g(x, ˆθ g)u c + g u c ], u s(t k ) = 0, k =, 2, () I =, if V = et P e > 2 V 0, if V V (2) u w = g 0 sign(e T P b c)(ˆε w + D), (3) u w(t k ) = 0, k =, 2, (4) where u c is calle equivalent controller, u s is calle supervisory controller, V is chosen such that 0 < V

4 H B Jiang / Inirect Aaptive Fuzzy an Impulsive Control of Nonlinear Systems 487 M x/(4λ min(p )), ˆθf an ˆθ g are the estimators of θ f an θ g at time t, respectively, u w is an aitional robustifying control term The minimum approximation error is efine as w = (f(x, θ f ) f) + (g(x, θ g) g)u c (5) Let ε w = max x Ωx,ˆθ f Ω θf,ˆθ g Ω θg [(f(x, θ f ) f(x)) + (g(x, θ g) g(x))u c], an ε w is an unknown boune constant Let ˆε w be an estimator of ε w at time t Inspire by [28], we aopt the following impulsive control law: U k (e(t k )) = G k e(t k ), k =, 2, (6) where G k R n n are constant matrices to be esigne later Substituting (7) an (6) into (5), an by a simple calculation, we have ė = Λ ce b cg(x)u s b cg(x)u w + b c[(f(x, ˆθ f ) f(x))+ (g(x, ˆθ g) g(x))u c (t)], e(t k ) = G k e(t k ), k =, 2, (7) Combining (5) an (7), we have ė = Λ ce + b cg(x)u s + b cg(x)u w+ b c[ θ f ξ f (x) + θ gξ g(x)u c + w (t)], (8) e(t k ) = G k e(t k ), k =, 2, where θ f = ˆθ f θ f an θ g = ˆθ g θ g Inspire by [, 6, 7], we choose the aaptation law as follows: ˆθ f = γ e T P b cξ f (x), if ˆθ f < Mθf, or ˆθ f = Mθf an e T P b c ˆθf ξ f (x) 0, γ e T P b cξ f (x) + γ e T ˆθf ˆθT f P b c 2 ξ f (x), ˆθ f if ˆθ f = Mθf an e T P b c ˆθf ξ f (x) < 0, (9) ˆθ f (t k ) = 0, k =, 2, (20) Whenever an element ˆθ g,i of ˆθ g equals ε, use ˆθ g,i = { γ e T P b cξ g,i(x)u c, if e T P b cξ g,i(x)u c < 0, 0, if e T P b cξ g,i(x)u c 0, (2) ˆθ g,i(t k ) = 0, k =, 2, (22) Otherwise, use ˆθ g = γ 2e T P b cξ g(x)u c, if ˆθ g < Mθg, or ˆθ g = Mθg an e T P b c ˆθgξ g(x)u c 0, γ 2e T P b cξ g(x)u c + γ 2e T ˆθg ˆθT P b g c ξ g(x)u c, ˆθ g 2 if ˆθ g = Mθg an e T P b c ˆθgξ g(x)u c < 0 (23) ˆθ g(t k ) = 0, k =, 2, (24) ˆε w = γ 3 e T P b c,, (25) ˆε w(t k ) = 0, k =, 2, (26) where γ > 0, γ 2 > 0, an γ 3 > 0 are gain constants that etermine the rate of aaptation 4 Stability analysis Theorem Consier the nonlinear system () with control laws efine by (7) an (6) Let the parameter vector ˆθ f, ˆθ g, an ˆε w be ajuste by the aaptation law etermine by (9) (26), an let Assumptions 5 be true If there exist two positive constants λ an λ 2 such that λ = inf k {t k t k } > 0, λ 2 = sup k {t k t k } <, an the impulsive matrices G k satisfy the following inequalities: G T k P G k G T k P P G k 0, k =, 2, (27) Then, ) if ˆθ f (0) Ω θf, ˆθg(0) Ω θg, then ˆθ f (t) M θf, ˆθ g(t) M θg, t 0 If M x = 2 λ min(p ) V + M, then the state vector x Ω x = {x x M x} Thus, the overall close-loop fuzzy control system is globally stable in the sense that all of the close-loop signals are boune; 2) the tracking error between the output of the system y an the given boune reference signal y are convergent to zero asymptotically, ie, lim t e (t) = 0 Proof ) Let (t) = ˆθ Vˆθf f T ˆθ f /2 From (9) an (20), we have that if ˆθ f = Mθf an e T P b c ˆθf ξ(x) < 0, then (t) 0; if Vˆθf ˆθ f = Mθf an e T P b c ˆθf ξ(x) 0, then (t) = 0,, hence, Vˆθf ˆθ i Mθi, t 0 From the above analysis, we know that if ˆθ i(0) Ω θi, then ˆθ i(t) Ω θi Using the same metho, we can prove that ˆθ g(t) M θg, t 0 To show ˆθ g ε, we can see from (2) an (22) that if ˆθ g,i = ε, then ˆθg,i 0; hence, we always have ˆθ g,i ε, which guarantees ˆθ g ε 2) Consier the Lyapunov function caniate V = e T P e/2 Taking the Dini erivative of V (t) for t [t k, t k ), we obtain D + V = 2 ėt P e + 2 et P ė 2 et Qe+ e T P b c [ f(x, ˆθ f ) + f(x) + g(x, ˆθ g)u c + g(x)u c + D] e T P b cg(x)u s e T P b cg(x)u w

5 488 International Journal of Automation an Computing 7(4), November 200 If V V, then I = Substituting (8) (4) into (8) an using Assumptions 4 lea to D + V 2 et Qe 0 (28) On the other han, when t = t k, V = 2 (e G ke) T P (e G k e) 2 et P e = 2 et (G T k P G k G T k P P G k )e 0 (29) Since V is a continuous function with respect to time t, when sampling perio is sufficiently small, we have V 2 V, t 0 Hence, e(t) 2 λ min(p ) V, t 0 From Assumption 3 an (2), we have x(t) 2 λ min(p ) V +M ie, x Ω x Define the Lyapunov function caniate V = 2 et P e + 2γ θt f θf + 2γ 2 θt g θg + 2γ 3 ε 2 w where ε w = ˆε w ε w Taking the Dini erivative of V (t) for t [t k, t k ), we obtain D + V 2 et Qe g(x)e T P b cu s g(x)e T P b cu w + e T P b c ε w+ γ θt f [ ˆθf + γ e T P b cξ f (x)]+ γ 2 θt g [ ˆθg + γ 2e T P b cξ g(x)u c] + γ 3 ε w ˆεw 2 et Qe + γ θt f [ ˆθf + γ e T P b cξ f (x)]+ γ 2 θt g [ ˆθg + γ 2e T P b cξ g(x)u c]+ γ 3 ε w[ ˆε w γ 3 e T P b c 2 et Qe + I e T P b c θt f ˆθf ˆθT f ˆθ f 2 I 2e T P b c θt g ˆθg ˆθT g ˆθ g 2 ξg(x)uc ξ f (x)+ where I = 0(), if the first (secon) conition of (9) is true, an I 2 = 0(), if the first (secon) conition of (23) is true If the secon conition of (9) is true, then ˆθ f = M θf an θ f T ˆθ f = [ θf 2 ˆθ f 2 θf ˆθ f 2 ]/2 0 If the secon conition of (24) is true, then ˆθ g = M θg an θ g T ˆθ g = [ θg 2 ˆθ g 2 θg ˆθ g 2 ]/2 0 Therefore, D + V 2 et Qe 2 λmin(q)et e 0 (30) On the other han, when t = t k, V = 2 (e G ke) T P (e G k e) 2 et P e = 2 et (G T k P G k G T k P P G k )e 0 (3) Therefore, V is ecreasing on [0, + ), V 0, thus lim t V (t) exists By Cauchy criteria, for every ε > 0, there exists M > a, such that when t > t > M, V (t ) V (t ) < ε Therefore, 2 λmin(q) tk2 t t e T et = tk 2 λmin(q)( t k e T et + + ( t tk t D + V t + t ks D + V t) = t t ks tk2 t e T et) e T et+ t k D + V t + + V (t ) V (t k ) + V (t + k ) V (t k2 ) + + V (t + k s ) V (t ) = V (t ) V (t ) + V (t + k ) V (t k ) + + V (t + k s ) V (t ks ) V (t ) V (t ) < ε By Cauchy criteria, we obtain that (λ + min(q) e T et)/2 is convergent Thus, + e T et 0 0 is convergent From (), an the fact that all of the close-loop signals are boune, we obtain that e an ė are uniformly boune for k =, 2, on the interval [t k, t k ) Thus, e T e an e T ė are uniformly boune for k =, 2, on the interval [t k, t k ) By Lemma, we obtain that lim t + e T e = 0 Thus, lim t + e = 0 5 Simulation results In this section, we give two examples to illustrate the effectiveness of the propose approach All experiments are run in the Matlab environment Example In this example, we use the propose aaptive fuzzy an impulsive controller to control the inverte penulum The ynamic equation of the inverte penulum system is where ẋ = x 2 ẋ 2 = f(x) + g(x)u + (t) y = x (32) f(x) = g sin x (mlx2 2 cos x sin x )(m c + m) [ ] 4 l 3 (m cos2 x )(m c + m) (cos x )(m c + m) g(x) = [ ] 4 l 3 (m cos2 x )(m c + m) an l = 05 m, m = 0 kg, m c = kg, g = 98 ms 2, an (t) = 002 cos(2t) The control objective is to force the output y to track a given boune reference signal y = π sin(t)/30 Define e = [y y, ẏ ẏ] T (33) Then, we shoul esign an aaptive fuzzy controller u an an impulsive controller U k such that the resulting closeloop system is globally stable in the sense that all signals

6 H B Jiang / Inirect Aaptive Fuzzy an Impulsive Control of Nonlinear Systems 489 involve are uniformly boune, an the tracking error between the output of the system y an the given boune reference signal y is convergent to zero asymptotically, ie, e 0 as t + Uner the aaptive an impulsive control, the controlle state of system (32) satisfies the following system: ẋ = x 2, ẋ 2 = f(x) + g(x)u + (t), y = x, x(t k ) = U k (e(t k )) = G ke(t k ), k =, 2, (34) From (33) an (34), we obtain the following error ynamical system: Fig The state x an its esire value y in Example { ė = e 2, ė 2 = y (2) f(x) g(x)u (t), (35) e(t k ) = U k (e(t k )) = G ke(t k ), k =, 2, The esign parameters are F (x) = x 2 2, g 0 = 2, g = 46, V = 0, Mθf = 20, M θg = 5, ε = 005, γ = [ 2, γ 2 =, ] γ 3 = 05, K = [2 ] T, Q = iag{0, 0}, 5 5 P = For simplicity, we consier the equiistant 5 5 impulsive interval t k = t k t k = 005, k =, 2,, an the impulsive matrices G k = 02 iag{, }, k =, 2, The fuzzy systems that are use to approximate f(x) an g(x) are efine by the following rules: Fig 2 The aaptive control input signal u in Example R l f : If x is A i an x 2(t) is A j 2, then f(x, ˆθ f ) is G ij, i =, 2,, 5, l =, 2,, 25 (36) R l g : If x is A i an x 2(t) is A j 2, then g(x, ˆθ g) is H ij, i =, 2,, 5, l =, 2,, 25 (37) where A i, A i, G ij, an H ij are fuzzy sets The membership function of fuzzy set A i p is taken as µ A i p (x p) = exp( ((x p a l i)/b l i) 2 ), p =, 2, i =, 2,, 5, where a = π/6, a = π/2, a 3 = 0, a 4 = π/2, a 5 = π/6, b = b 2 = = b 5 = π/24 The aaptation laws of ˆθ f, ˆθ g, an ˆε are etermine by (9) (26) The parameters ˆθ f, ˆθ g, ˆε, an x(0) are chosen as follows: ˆθf (0)(i) = 8, i =, 2,, 0; ˆθ f (0)(i) = 48, i =, 2,, 25; ˆθ g(0)(i) = 6, i =, 2,, 5; ˆθ g(0)(i) = 32, i = 6, 7,, 25; ˆε(0) = 05, x(0) = [ π/60 0] T A fourth-orer Runge-Kutta metho with step size 0025 is use Simulation results are shown in Figs an 2 From the simulations results, it is shown that the propose approach is effective to control the nonlinear system Remark For simplicity, we can choose the impulsive matrices as r k I, where I is an appropriately imensione ientity matrix If we select r k satisfying r 2 k 2r k 0, then the impulsive matrices r k I satisfy (27) Remark 2 Because the controller contains the switch function sign(x), there exist some high-frequency chattering In the simulation, we have eliminate the chattering problem by replacing the switch function sign(x) by the saturation function sat(x) [6, 7] Example 2 In this example, we use the aaptive fuzzy an impulsive controller to control the Duffing force oscillation system The ynamic equation of the Duffing force oscillation system is ẋ = x 2 ẋ 2 = 0x 2 x cos(t) + u + (t) y = x (38) It can be shown that without control, ie, u 0, an without isturbances, ie, (t) 0, the system is chaotic The control objective is to force the output y to track a

7 490 International Journal of Automation an Computing 7(4), November 200 given boune reference signal y = sin(t) Define e = [y y, ẏ ẏ] T (39) Then, we shoul esign an aaptive fuzzy controller u(t) an an impulsive controller U ik (t) such that the resulting close-loop system is globally stable in the sense that all signals involve are uniformly boune, an the tracking error between the output of the system y an the given boune reference signal y is convergent to zero asymptotically, ie, e 0 as t + Uner the aaptive an impulsive control, the controlle state of system (38) satisfies the following system: ẋ = x 2, ẋ 2 = f(x) + g(x)u + (t), y = x, x(t k ) = U k (e(t k )) = G ke(t k ), k =, 2, (40) Fig 3 The state x an its esire value y in Example 2 where f(x) = 0x 2 x 3 +2 cos(t) an (t) = 002 sin(2t) From (39) an (40), we obtain the following error ynamical system: { ė = e 2, ė 2 = y (2) f(x) g(x)u (t), (4) e(t k ) = U k (e(t k )) = G ke(t k ), k =, 2, The esign parameters are F (x) = 2 + x 3, g 0 =, g =, V = 0, Mθf = 30, ε = 00, γ = γ 2 = γ 3 = [ ], K = [2 ] T 5 5, Q = iag{0, 0}, P = For 5 5 simplicity, we consier the equiistant impulsive interval t k = t k t k = 005, k =, 2,, an the impulsive control matrices G k = 02 iag{, }, k =, 2, The fuzzy systems that are use to approximate f(x) are efine by the following rules: R l f : If x is A i an x 2(t) is A j 2, then f(x, ˆθ f ) is G ij, i =, 2,, 6, l =, 2,, 36 (42) where A i, A i, G ij, an H ij are fuzzy sets The membership function of fuzzy set A i p is taken as µ A p (x p) = /( + exp(5(x p + 2))), µ A 2 p (x p) = exp( (x p + 5) 2 ), µ A 3 p (x p) = exp( (x p +05) 2 ), µ A 4 p (x p) = exp( (x p 05) 2 ), µ A 5 p (x p) = exp( (x p 5) 2 ), µ A 6 p (x p) = /( + exp(5(x p 2))), p =, 2 The aaptation laws of ˆθ f an ˆε are etermine by (9) (26) The parameters ˆθ f, ˆε, an x(0) are chosen as follows: ˆθf (0)(i) = 05, i =, 2,, 0; ˆθf (0)(i) = 03, i =, 2,, 25; ˆε(0) = 05, x(0) = [2 2] T A fourth-orer Runge-Kutta metho with step size 0005 is use Simulation results are shown in Figs 3 an 4 From the simulations results, it is shown that the impulsive controller can improve the tracking performance of the close-loop system Fig 4 The aaptive control input signal u in Example 2 6 Conclusions The problem of inirect aaptive fuzzy an impulsive control for a class of nonlinear systems is investigate Base on approximation capability of fuzzy systems, a novel aaptive fuzzy an impulsive control strategy with supervisory controller is evelope References [] G Feng A survey on analysis an esign of moel-base fuzzy control systems IEEE Transactions on Fuzzy Systems, vol 4, no 5, pp , 2006 [2] B S Chen, C H Lee, Y C Chang H tracking esign of uncertain nonlinear SISO systems: Aaptive fuzzy approach IEEE Transactions on Fuzzy Systems, vol 4, no, pp 32 43, 996 [3] Y C Chang Aaptive fuzzy-base tracking control for nonlinear SISO systems via VSS an H approaches IEEE Transactions on Fuzzy Systems, vol 9, no 2, pp , 200 [4] K Fischle, D Schroer An improve stable aaptive fuzzy control metho IEEE Transactions on Fuzzy Systems, vol 7, no, pp 27 40, 999

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class of non-autonomous chaotic systems Journal of Dynamics an Control, vol 6, no 4, pp 32 35, 2008 (in Chinese) [37] H B Jiang Hybri aaptive an impulsive synchronization of uncertain complex ynamical networks by the generalize Barbalat s lemma IET Control Theory an Applications, vol 3, no 0, pp , 2009 Hai-Bo Jiang receive the M Sc egree in computer science from Yangzhou University, PRC in 2006 Now, he is a lecturer at Yancheng Teachers University, PRC His research interests inclue fuzzy control, impulsive control, an multi-agent systems yctcjhb@gmailcom