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International Journal of Fuzzy Systems, Vol. 10, No. 1, March 2008
Adaptive Fuzzy Tracking Control of a Class of Stochastic Nonlinear Systems with Unknown Dead-Zone Input Jianjiang Yu, Kanjian Zhang, and Shumin Fei Abstract 1 In this paper, a direct adaptive fuzzy tracking control scheme is presented for a class of stochastic uncertain nonlinear systems with unknown dead-zone input. A direct adaptive fuzzy tracking controller is developed by using the backstepping approach. It is proved that the design scheme ensures that all the error variables are bounded in probability while the mean square tracking error becomes semiglobally uniformly ultimately bounded (SGUUB) in an arbitrarily small area around the origin. Simulation results show the effectiveness of the control scheme. Keywords: Stochastic system; Adaptive control; Fuzzy control; Dead-zone input; backstepping
1. Introduction Recently, the adaptive control of uncertainty nonlinear system has been extensively studied. As a breakthrough in nonlinear control area, a recursive design procedure, backstepping approach, was presented to obtain global stability and asymptotic tracking for a large class of nonlinear system, mostly the strict-feedback system [1,2]. In some backstepping-based control laws ([3]-[8]) for stochastic strict-feedback systems that include a Wiener process have been developed to guarantee stability, known as stability in probability. Combined problem of the control of stochastic strict-feedback nonlinear system with nonlinear uncertainties is firstly studied in [9]. Nonsmooth nonlinear characteristics such as deadzone, backlash, hysteresis are common in actuator and sensors such as mechanical connections, hydraulic actuators and electric servomotors. Dead-zone is one of the most important non-smooth nonlinearities in many industrial processes. Its presence severely limits system performance, and its study has been drawing much interest in the control community for a long time ([2] and the reference therein). In this paper, we consider adaptive fuzzy tracking Corresponding Author: Jianjiang Yu is with the School of Automation, Southeast University, China, and he is also with School of Information Science and Technology, Yancheng Teachers University, China. E-mail:
[email protected] Manuscript accepted 15 March 2008.
controller for a class of stochastic uncertain nonlinear systems with dead-zone input. As in [2], the dead-zone output is represented as a simple linear system with a static time-varying gain and bounded disturbance by introducing characteristic function. The first type fuzzy system is employed to approximate the unknown nonlinear system. Extensive stability analysis proves that all the error variables are bounded in probability while the mean square tracking error becomes semiglobally uniformly ultimately bounded in an arbitrarily small area around the origin. This paper is organized as follows. The preliminaries and problem formulation are presented in Section II. In Section III, a systematic procedure for the synthesis of the adaptive fuzzy tracking controller is developed. In Section IV, simulation example is used to demonstrate the effectiveness of the proposed scheme. Finally, the conclusion is given in Section V.
2. Preliminaries and Problem Formulation The following notation will be used throughout the paper. For a given vector or matrix X , X T denotes its transpose, Tr{X } denotes its trace when X is square, and | X | denotes the Euclidean norm of a veci
tor X . C denotes the set or all functions with continuous i th partial derivatives. K denotes the set of all functions: R+ → R+ , which are continuous, strictly increasing and vanishing at zero, K ∞ denotes the set of all functions which are of class K and unbounded; KL denotes the set of all functions β ( s, t ) : R+ × R+ → R+ which is of class K for each fixed t , and decreases to zero as t → ∞ for fixed s .
Consider the following stochastic nonlinear system:
dx = f ( x)dt + h( x)dω , ∀x ∈ R n
(1)
where x ∈ R n is the state of the system, ω is r − dimensional independent standard Wiener process defined on a probability space (Ω, F , P) . f (⋅) , h(⋅) are locally Lipschitz in x .
© 2008 TFSA
Jianjiang Yu et al.: Adaptive Fuzzy Tracking Control of a Class of Stochastic Nonlinear Systems with Unknown Dead-Zone Input
Definition 1 For any given V ( x) ∈ C 2 , associated with stochastic system (1), we define the differential operator L as follows:
∂V ∂ 2V 1 LV ( x) = f ( x) + Tr{h T ( x) 2 h( x)} ∂x 2 ∂x
(2)
Definition 2 For the stochastic system (stochastic) with f (0) = h(0) = 0 , the equilibrium x(t ) = 0 is (i) globally stable in probability if for any 0 < ε < 1 , there exists a class K function γ (⋅) such that P{| x(t ) |< γ (| x0 |)} ≥ 1 − ε , ∀t ≥ 0, ∀x0 ∈ R n − {0} ; (ii) globally asymptotically stable in probability if ∀ε > 0 , there exists a class KL function β (⋅,⋅) such that P{| x(t ) |< β (| x0 |, t ) ≥ 1 − ε , ∀t ≥ 0, ∀x0 ∈ R n − {0}.
⎧dx1 = x2 dt ⎪dx = x dt 3 ⎪ 2 ⎪⎪# ⎨ ⎪dxn−1 = xn dt ⎪dxn = [ f ( x) + g ( x)u ]dt + hT ( x)dω ⎪ ⎪⎩ y = x1
19
(6)
Dead-zone: ⎧ g r (v) if ⎪ u = D (v ) = ⎨ 0 if ⎪ g (v) if ⎩ l
v ≥ br , bl < v < br ,
(7)
v ≤ bl ,
where x = ( x1 , " , xn )T , y ∈ R are state variables and output, respectively. ω is r − dimensional independent standard Wiener process defined on a probability space (Ω, F , P) . f ( x), g ( x) are unknown smooth functions. u ∈ R is the output of Lemma1[9] Consider the stochastic nonlinear sys- the dead-zone. v(t ) ∈ R is the input to the deadtem (1). If there exists a positive definite, radially zone, b and b are the unknown parameters of l r unbounded, twice continuously differentiable Lyathe dead-zone. The reference signal is y d (t ) , and punov function V : R n → R , and constants c1 > 0 , y d (t ), y d(1) (t ),", y d( n ) (t ) are smooth and bounded. c > 0 such that 2
LV ( x) = −c1V ( x) + c 2
(3) Assumption 1 There exist positive constants g , g 0 1
then (i) the system has a unique solution almost such that 0 < g 0 ≤ g ( x) ≤ g1 , ∀x ∈ Ω x ⊂ R n , where surely and (ii) the system is bounded in probability. Ω x is a compact set. Remark 1 In the paper [9], the g (x) must be in[10] Lemma2 Let f (x) be a continuous function dependent from x n . In this paper, we removed this defined on a compact set Ω . Then for any constant restriction. ε > 0 , there exists a fuzzy logic system such that For the unknown dead-zone input, we make the sup | f ( x) − θ T ξ ( x) |≤ ε . (4) following assumptions. x∈Ω Define the ideal estimation parameter as
θ ∗ = arg min[sup | f ( x) − θ T ξ ( x) |], θ ∈Θ
x∈Ω
(5)
Assumption 2 The dead-zone output, u , is not available.
Assumption 3 The dead-zone parameters, br and bl , where Θ and Ω denote the sets of suitable are unknown bounded constants, but their signs are bounds on θ and x . known, i.e., br > 0 and bl < 0.
Consider a SISO stochastic nonlinear systems in Assumption 4 The functions, g l (v) and g r (v) , the following form: are smooth, and there exist unknown positive conPlant: stants, k l 0 , kl1 , k r 0 , and k r1 such that
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International Journal of Fuzzy Systems, Vol. 10, No. 1, March 2008
zi = xi − α i −1 ( z[ i −1] , yd[i −1] ),
′
0 < kl 0 ≤ g l (v) ≤ kl1 , ∀v ∈ (−∞, bl ] ′
0 < k r 0 ≤ g r (v) ≤ k r1 , ∀v ∈ [br ,+∞)
i = 1, 2,", n
(9)
where α i −1 ( z[ i −1] , y d[ i −1] ) to be given in the follow-
T β 0 ≤ min{kl 0 , k r 0 } is a known positive constant, ing steps, α 0 = y d , α n = u , z[ i −1] = [ z1 , " , z i −1 ] , ′
where gl (v) =
dg l ( z ) z =v dz
|
′
and g r (v) =
dg r ( z ) z =v dz
|
.
y d[ i −1] = [ y d , y d ,", y di −1 ]T , z = z[ n ] .
Based on Assumption 4, the dead-zone (5) can be Step 1 The derivation of z1 is rewritten as follows as shown [2]: dz1 = dx1 − y d dt = ( z2 + α ( z1 , yd , y d ) − y d )dt. (10) T u = D(v) = K (t )Φ(t )v + d (v), (8) Consider the following Lyapunov function canwhere didate Φ (t ) = [ϕ r (t ), ϕ l (t )]T , ⎧1 if ϕ r (t ) = ⎨ ⎩0 if
v(t ) > bl , v(t ) ≤ bl ,
⎧1 if v(t ) > br , ⎩0 if v(t ) ≤ br ,
ϕ l (t ) = ⎨
K (t ) = [ g r' (ξ r (v(t ))), g l' (ξ l (v(t )))]T , ⎧− g r (ξ r (v))br if v ≥ br , ⎪⎪ ′ ′ d (v) = ⎨− [ g l (ξ l (v)) + g r (ξ r (v))]v if bl < v < br , ′ ⎪ ⎪⎩− g l (ξ l (v))bl , if v ≤ bl ′
the time derivative of
V1 =
z12 , 2
V1
is
(11)
dV1 = z1 ( z2 + α1 ( z1 , yd , y d ) − y d )dt. Select virtual control α 1 ( z1 , y d , y d ) as
α1 ( z1 , yd , y d ) = −k1 z1 + y d ,
(12)
with design constant k1 > 0 , it is easy to get
dV1 ≤ ( z1 z2 − k1 z12 )dt.
(13)
Step k (2 ≤ k ≤ n − 1) A similar procedure is emand | d (v) |≤ p , p is an unknown positive con- ployed recursively for each step k . The derivation of α k −1 is stant with p ∗ = (k r1 + k l1 ) max{br ,−bl } . ∗
∗
k −1 ∂α k −1 ∂α dz j + ∑ (kj−)1 yd( j +1) )dt j =1 ∂z j j =0 ∂y d
k −1
dα k −1 = (∑
Remark 2 There are many results for the case of linear dead-zone outside the deadband, but equation (8) is to capture the most realistic situation. As then the derivation of z is k shown in [2], we know that K T (t )Φ(t ) ∈ k −1 k −1 ∂α k −1 ∂α [β0 , kl1 + kr1 ] ⊂ (0,+∞) . dz k = [ z k +1 + α k − ∑ dz j − ∑ (k j−)1 y d( j +1) ]dt , j =1 ∂z j j = 0 ∂y d The control objective is to design an adaptive (14) fuzzy controller v(t ) for the system (1) such that the output y follows the specified desired trajec- choose the Lyapunov candidate functions and the tory y d with guaranteeing that the system is virtual control laws as follows, bounded in probability. 1 Vk = Vk −1 + zk2 , (15) 2 3. Control design and stability analysis
In this section, we will use backstepping to design an adaptive fuzzy controller. First, we introduce the error variables
k −1 ∂α k −1 ∂α dz j + ∑ (k j−)1 y d( j +1) , j =1 ∂z j j = 0 ∂y d (16)
k −1
α k = − k k z k − z k −1 + ∑
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Jianjiang Yu et al.: Adaptive Fuzzy Tracking Control of a Class of Stochastic Nonlinear Systems with Unknown Dead-Zone Input
and we get
~ 1~ V = Vn + θ T Γ −1θ , 2
k
dVk ≤ ( z k z k +1 − ∑ k j z )dt. 2 j
(17)
j =1
and choose the adaptive law as
θˆ = Γ [ g 0ξ ( x) z n3 − σ (θˆ − θ 0 )],
Step n The derivation of z n is n −1 ∂α n −1 ∂α dz j − [∑ (k j−)1 y d( j +1) ]dt , j =1 ∂z j j = 0 ∂y d
n −1
(18)
LV ≤
zn4 , 4
1 2
k 3 LVn ≤ − k1 z12 − " − n z n4 + z n2 h T h + z n [ z n −1 + 2 2 (20) kn 3 2 z n + z n ( f ( x) + g ( x)u − α n −1 )] 2
Consider the following inequalities: 1 9 3 2 T 4 z n h ( x)h( x) ≤ + ζ h( x) z n4 ∀ζ > 0 2 ζ 16 (21) 1 1 3ξ 3 4 4 z n z n −1 ≤ + ( ) z n −1 z n ∀ξ > 0 ξ 4 4
So, n −1
k LVn ≤ + − ∑ k j z 2j − n z n4 + z n3 { f ( x) + g ( x)u ζ ξ j =1 2 1
− α n −1 + [
k 9 1 3ξ 4 ζ h( x) + ( ) 3 z n4−1 + n ]z n } 16 4 4 2 (22)
Then use the first type fuzzy systems to approxi4 mate f ( x) = g 0−1 ( f ( x ) − α n −1 + 169 ζ h( x) z n ) ， f ( x ) = θ ∗T ξ ( x ) + ε f ,
where ε f
ζ
+
1
− 2μ1V + z n3 {g 0θˆ T ξ ( x) + g ( x)u
ξ 1 3ξ
(23)
is approximation error, according to
|| θ ∗ − θ 0 || 2 + g 0 ε M2 . It is easy to see μ1 > 0
and μ 20 > 0 . Choose the control law v as follows: v=−
1
β0
(−ur − u f ).
(27)
where u f = −θˆT ξ ( x) tanh(
z nθˆT ξ ( x)
δ
(28)
),
1 3ξ k 1 g2 ur = − g 0−1[ ( )3 zn4−1 + n + zn2 + 1 zn2 ]zn , (29) 4 4 2 2 4 with δ is a positive constant. Then z n3 ( g 0θˆ T ξ ( x) + g ( x)u f ) z θˆ T ξ ( x) )) ≤ z n2 g 0 (| z nθˆ T ξ ( x) | − z nθˆ T ξ ( x) tanh( n
δ
1 ≤ κ + z n6 , 3
where κ = 23 (0.2785 g 0δ ) 3 / 2 is a positive constant,
z n3 g ( x)d (v) ≤
g12 6 z n + p ∗2 4
From (26), it holds true that
LV ≤ −2 μ1 + μ 2 Assumption 1, there exist ε M > 0 such that | ε f |< ε M . Let θˆ is the estimation of θ ∗ , and with μ 2 = μ 20 + κ + p ∗2 . ~ θ = θˆ − θ ∗ .
We define the overall Lyapunov function as
(26)
(19) where μ1 = min1≤i≤n {(1 − ε i )k i , σλmin (Γ)} , μ 20 =
then
1
1
k 1 + [ ( ) 3 z n4−1 + n + z n2 ]z n } + μ 20 4 4 2 4
choose Lyapunov candidate as
Vn = Vn −1 +
(25)
with gain matrix Γ > 0 . So, we get
dz n = [ f ( x) + g ( x)u ]dt + h T ( x)dω −∑
(24)
(30)
The main result on the asymptotic stability of the closed-loop system is summarized in the following theorem.
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International Journal of Fuzzy Systems, Vol. 10, No. 1, March 2008
Theorem For the stochastic uncertain nonlinear systems (1) satisfying Assumptions 1-4, with control law (27) and adaptive law (25), then the tracking error is bounded and the mean square tracking error enters inside the region
Ω = { y (t ) ∈ R | E[( y (t ) − y d (t )) ] ≤ 2
2μ 2
μ1
It is easy to get g0 = 1, g1 = 3, β0 = 1 . In the simulation, the fuzzy membership functions are defined as
μ F ( x) = exp[ − ( x4+ 2 ) ] , μ F ( x) = exp[ − ( x4+1) ] , 2
μ F ( x) = exp[ − (4x ) ] , 2
, ∀t ≥ T1 }
2
2 j
1 j
3 j
μ F ( x) = exp[ − ( x4−2) ] , 2
4 j
μ F ( x) = exp[ − ( x4−1) ] . 2
5 j
wherein it remains for all time thereafter, and the We choose the virtual control law and the control variable T1 will be given later. law as Proof From (30) and Lemma 1, it is easy to get α1 = −k1 z1 + y d , (37) that the system is bounded in probability and the mean value of the Lyapunov function satisfies z θˆ T ξ ( x) 1 3ξ v = −θˆ T ξ ( x) tanh( 2 ) − 10[ ( ) 3 d δ 4 4 [ E (V )] ≤ −2μ1E[V (t )] + μ 2 . (31) (38) 2 dt k 2 1 2 g1 2 + + z2 + z 2 ]z 2 , So, 2 2 4
E[V (t )] ≤ e − 2 μ1t Vn (0) +
μ2 , 2 μ1
∀t ≥ 0
(32)
θˆ = Γ [ξ ( x ) z 23 − σ (θˆ − θ 0 )],
there exists a time T1
T1 = max{0,
2μ V (0) 1 ln[ 1 n ]}. 2 μ1 μ2
such that
E[( y (t ) − y d (t )) 2 ] ≤ 2 E[V (t )] ≤
where z1 = y − yd = x1 − yd , z 2 = x2 − α1 , k1 = k 2 = 2 , δ = 0.2 , ξ = 2 . The adaptive law
2μ 2
μ1
.
0 (33) where Γ = 2 I , σ = 2 , θ is randomly taken in the intervals [−1, 1] . The simulation results are shown in Figure 1 and Figure 2. From Figure 1, it can be seen that fairly good tracking performance is obtained. (34)
4. Computer simulation
Consider the following nonlinear systems: ⎧dx1 = x2 dt ⎪ ⎨dx2 = [ x1 x2 + (2 + cos( x2 ))u ]dt + 13 ( x2 + sin x1 )dω ⎪y = x 1 ⎩
(35) where D(v) as follows: ⎧1.5(v − 2) if v ≥ 2, ⎪ u = D (v) = ⎨0 if − 1.5 < v < 2.5, ⎪v + 0.5 if v ≤ −0.5. ⎩
(39)
(36)
with initial conditions x1 (0) = 0.5 , x2 (0) = 0.1 , and the reference signal yd = 1.2(sin t + sin(0.4t )) .
Fig. 1. The response of
x1 .
Jianjiang Yu et al.: Adaptive Fuzzy Tracking Control of a Class of Stochastic Nonlinear Systems with Unknown Dead-Zone Input
23
[4] G. Arslan and T. Basar, “Risk-sensitive adaptive trackers for strict-feedback systems with output measurements,” Automatic Control, IEEE Transactions on, vol. 47, no. 10, pp. 1754-1758, 2002. [5] Y.G. Liu and J.F. Zhang, “Reduced-order observer-based control design for nonlinear stochastic systems,” Systems and Control Letters, vol. 52, no. 2, pp. 123-135, 2004. [6] H.B. Ji and H.S. Xi, “Adaptive output-feedback tracking of stochastic nonlinear systems,” Automatic Control, IEEE Transactions on, vol. 51, no. 2, pp. 355-360, 2006. Fig. 2. Control Signal v .
5. Conclusions In brief, a novel adaptive fuzzy control scheme has been presented for a class of uncertain nonlinear systems with unknown dead-zone input, which is driven by unknown covariance noise inputs. The proposed control scheme ensures that all the error variables are bounded in probability while the mean square tracking error becomes SGUUB in an arbitrarily small area around the origin.
[7] Z.J. Wu, X.J. Xie, and S.Y. Zhang, “Adaptive backstepping controller design using stochastic small-gain theorem,” Automatica, vol. 43, no. 4, pp. 608-620, 2007. [8] S.J. Liu, J.F. Zhang and Z.P. Jiang, “Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems,” Automatica, vol. 43, no.2, pp. 238-251, 2007.
[9] H. E. Psillakis and A. T. Alexandridis, “Adaptive neural tracking for a class of SISO uncertain and stochastic nonlinear systems,” Proceeding of 44th IEEE Conference on Decision Acknowledgment and Control, and the European Control Conference, pp. 7822-7827, 2005. This work is partially supported by the Natural Science Foundation of China (60404006,60574006), par- [10] L.X. Wang, “Stable adaptive fuzzy control of tially by the Natural Science Foundation of the Jiangsu nonlinear systems,” Fuzzy Systems, IEEE Higher Education Institutions of China (Grant Transactions on, vol.1, no. 2, pp. 146-155, No.07KJB510125), and partially by the Natural Science 1993. Foundation of Yancheng Teachers University. References
[1] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and adaptive control design, Wiley, New York, 1995. [2] T.P. Zhang, and S.S. Ge, “Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs,” Automatica, vol. 43, no. 6, pp. 10211033, 2007. [3] H. Deng and M. Krstic, “Stochastic nonlinear stabilization -- I: A backstepping design,” Systems and Control Letters, vol. 32, no. 3, pp. 143-150, 1997.
Jianjiang Yu received the M.S. degree in Computer Applied Technology from Yangzhou University, Yangzhou, PR China in 2004. He is now pursuing his Ph.D. degree in School of Automation, Southeast University, Nanjing, PR China. His current research interests include nonlinear control, fuzzy control and networked control system.
