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Observer-Based Adaptive Neural Network Control for Nonlinear Stochastic Systems With Time Delay Qi Zhou, Peng Shi, Senior Member, IEEE, Shengyuan Xu, and Hongyi Li
Abstract— This paper considers the problem of observer-based adaptive neural network (NN) control for a class of singleinput single-output strict-feedback nonlinear stochastic systems with unknown time delays. Dynamic surface control is used to avoid the so-called explosion of complexity in the backstepping design process. Radial basis function NNs are directly utilized to approximate the unknown and desired control input signals instead of the unknown nonlinear functions. The proposed adaptive NN output feedback controller can guarantee all the signals in the closed-loop system to be mean square semi-globally uniformly ultimately bounded. Simulation results are provided to demonstrate the effectiveness of the proposed methods. Index Terms— Adaptive control, backstepping, surface control, fuzzy control, nonlinear systems.
dynamic
I. I NTRODUCTION
I
N THE past decade, research on stochastic systems has received considerable attention due to the fact that stochastic disturbance exists in many practical systems and is often the source of instability [1]–[5]. On the other hand, controller design for nonlinear systems has always been an important topic since practical stochastic system models are largely nonlinear. Many nonlinear stochastic control results and methods, such as sliding mode control [6] and T-S fuzzy control [7], have been reported in the literature. In recent years, the well-known backstepping method has been applied to solve the problem of nonlinear stochastic systems (see [8]). The backstepping design for nonlinear stochastic single-input single-output (SISO) systems was investigated in [9] and [10] by introducing quartic Lyapunov functions. More recently, the results have been extended to output feedback control design for stochastic nonminimum-phase nonlinear systems [11], large-scale stochastic nonlinear systems [12], and high-order nonlinear stochastic systems [13]–[15]. In [16], the results have also been extended to adaptive neural network (NN)
Manuscript received May 21, 2011; revised September 18, 2012; accepted September 26, 2012. This work was supported in part by the National Key Basic Research Program, China, under Grant 2012CB215202, the Qing Lan Project, the 111 Project under Grant B12018, the National Natural Science Foundation of China under Grant 61174058, Grant 60974071, and Grant 61203002, and the Engineering and Physical Sciences Research Council, U.K., under Grant EP/F029195. Q. Zhou and S. Xu are with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide 5005, Australia, and also with the School of Engineering and Science, Victoria University, Melbourne 3052, Australia (e-mail: [email protected]). H. Li is with the College of Information Science and Technology, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TNNLS.2012.2223824
output-feedback stabilization for a class of nonlinear stochastic systems. However, there still exists the issue of explosion of complexity in the backstepping design procedure because the backstepping design requires repeated differentiation of virtual controllers, which makes the complexity of controller grow dramatically as the order of the system increases. In order to avoid the above-mentioned problem, the dynamic surface control (DSC) technique was first introduced in [17] for a class of strict-feedback nonlinear system with unknown functions. Then, this approach was extended to solve a class of nonlinear systems with periodic disturbances in [18]. More recently, the DSC approach was further extended to solve the problem of output-feedback adaptive control of stochastic nonlinear systems in [19]. On the other hand, it is well known that the method of fuzzy logic control or NN control is useful to approximate unknown nonlinear functions in systems (see [20]–[35] and the references therein). Recently, the NN approximation approach has been utilized to handle the nonlinear stochastic systems control design problems [36]–[39]. Since the general nonlinear function f (x) can be approximated by an NN as f (x) = θ T ϕ(x)+ε(x), the method can also be used in stochastic nonlinear systems. However, it should be noted that the number of adaptation laws depends on the number of the NN nodes, and therefore the number of parameters to be estimated will increase when the NN nodes increase, which will lead to an unacceptably large learning time. This problem was considered in [40] by considering the norm of ideal weighting vector in fuzzy logic systems as the estimation parameter instead of the elements of weighting vector, and was also studied in [41] by developing a direct adaptive method for a class of nonlinear systems. It is well known that time delays are frequently encountered in engineering systems, and they usually become the sources of instability and degrade the performance of the systems [42]–[47]. Therefore, how to control nonlinear stochastic timedelay systems is still an important and practical topic. In [48] and [49], the observer-based control problem for nonlinear stochastic systems with time delay was solved. However, there exists a common weakness in the work of [48] and [49] in that the time delays are only present in the system output. Time delays in the state variables were not considered, which affect the stability and performance of the systems. Motivated by the above observation, in this paper the problem of observer-based adaptive control for nonlinear stochastic time-delay systems is investigated. The main contributions of this paper are summarized as follows. 1) The DSC approach is successfully applied to nonlinear stochastic systems, which
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avoids repeated differentiating the virtual controller by introducing a first-order filter in each step of backstepping design procedure. 2) A direct adaptive radial basis function NNs (RBF NNs) control method is proposed, where the number of adaptation parameters is only one, which significantly reduces the computation burden. Furthermore, the developed controller design is independent of any prior knowledge of NNs. 3) The time delays exist not only in the system output but also in the state variables, which makes the developed results in this paper more applicable. It is proven that the controller designed in this paper guarantees all the signals in the closed-loop to be mean square semi-globally uniformly ultimately bounded (M-SGUUB). Finally, simulation results are given to illustrate the effectiveness of the proposed control algorithm. The rest of this paper is organized as follows. The problem formulation and preliminaries, stochastic stability, NN approximation, and output feedback DSC design procedure for SISO strict-feedback nonlinear stochastic time-delay systems are presented in Section II. A simulation example is provided in Section III to demonstrate the effectiveness of the proposed design scheme. The conclusions are drawn in Section IV. II. P ROBLEM F ORMULATION AND P RELIMINARIES In this section, some preliminaries and assumptions are first formulated in Section II-A, and then the definition of stochastic stability is presented in Section II-B. In order to approximate the unknown nonlinear functions, RBF NN are presented in Section II-C. By using the backstepping approach, an output feedback DSC design procedure is formulated in Section II-D. A. Problem Formulation Consider the nonlinear stochastic system dx i (t) = (x i+1 (t) + f i (x¯i (t)) + h i (x¯i (t − τi ))) dt 1≤i ≤n +gi (y)T dw, dx n (t) = (u + f n (x (t)) + h n (x (t − τn ))) dt +gn (y)T dw y (t) = x 1 (t)
εp 1 |x| p + q |y|q p qε
where ε > 0, p > 1, q > 1, and ( p − 1)(q − 1) = 1.
B. Stochastic Stability Consider the stochastic system dx (t) = f (x (t)) dt + g (x (t)) dw
(2)
where x ∈ R n is the system state, w is an r -dimentional standard Wiener process, and f : R n → R n , g : R n → R n are locally Lipschitz functions and satisfy f (0) = g(0) = 0. Definition 1: For any given V (x) ∈ C 2 , associated with the stochastic system (2), the infinitesimal generator L is defined as follows: 1 ∂2V ∂V f (x) + Tr g (x)T g LV (x) = (x) ∂x 2 ∂x2 where Tr(A) is the trace of a matrix A. Definition 2 [36]: The solution process {x(t), t ≥ t0 } of stochastic system (2) with initial condition x 0 ∈ S0 (some compact set containing the origin) is said to be M-SGUUB if for any desired escape risk ε (0 < ε < 1), it is bounded with probability 1 − ε in some compact set S(ε) ⊃ S0 , i.e., inf x0∈S0 P{τ S(ε) = ∞} ≥ 1−ε. The hitting time τ S(ε) is defined as the first time that the trajectories of the state variable reach the boundary of S(ε). Lemma 2 [36]: Consider the stochastic system (2). If there exists a positive-definite radially unbounded twice continuously differentiable Lyapunov function V : R n → R, and constants c1 > 0, c2 > 0, and r0 > (c2 /c1 ) such that for some ε (0 < ε < 1) and x 0 ∈ S0 := {x ∈ R n |V (x) ≤ r0 } r0 LV (x) ≤ −c1 V (x)+c2 , ∇x ∈ S (ε) = x ∈ R n V (x) ≤ ε holds, then, for t ∈ [t0 , τ S(ε)], there is a unique solution of (2) and the system is bounded with probability 1 − ε in S(ε). C. NN Approximation
(1)
where x¯i (t) = [x 1 (t), x 2 (t), . . . , x i (t)]T ∈ R i , i = 1, 2, . . . , n − 1, and x(t) = [x 1(t), x 2 (t), . . . , x n (t)]T ∈ R n denote state vectors of the system; u ∈ R and y ∈ R are the input and output of the system, respectively. f i (.) (i = 1, 2, . . . , n) stands for the unknown smooth system function with f i (0) = 0. h i (.) is the unknown smooth nonlinear timedelay functions, τi is the unknown constant delay, and τm is the upper bound of τi . gi (.) is the unknown vector-valued smooth functions with gi (0) = 0. w is an independent r -dimentional standard Wiener process. Lemma 1 [50] (Young’s Inequality): For ∀(x, y) ∈ R 2 , the following inequality holds: xy ≤
Remark 1: Some restrictions on the delay terms are presented in this assumption, which are essential for the controller design in the sequel. In addition, compared with [48] and [49], the time delays considered in this paper exist not only in the system output but also in the state variables.
In this paper, RBF NN will be used to approximate the unknown smooth nonlinear functions. For any continuous unknown smooth nonlinear function f (Z ) over a compact set Z ⊂ R q , there exists a RBF NN W ∗T S(Z ) such that for a desired level of accuracy ε f (Z ) = W ∗T S (Z ) + δ (Z ) , where
|δ (Z )| ≤ ε
(3)
W∗
is the ideal constant weight vector and defined by ∗ T sup f (Z ) − W S (Z ) . W = arg min W ∈R N
Z ∈ Z
δ(Z ) is the approximation error, W = [w1 , . . . , w N ]T is the weight vector and S(Z ) = [s1 (Z ), . . . , s N (Z )]T is the basis function vector with N being the number of the RBF NN nodes and N > 1. RBF si (Z ) = exp[(−(Z − μi )T (Z − μi )/ηi2 )], i = 1, 2, . . . , N, where μi = [μi1 , μi2 , . . . , μin ]T is the center of the receptive field and ηi is the width of the Gaussian function.
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D. Output Feedback DSC Design In this subsection, an observer-based controller is designed to guarantee all the signals in the closed-loop system to be SGUUB. First, we design the following observer to estimate the unmeasured states: .
xˆ i = xˆ i+1 + li y − xˆ1 i = 1, . . . , n (4) where xˆn+1 = u. Let x˜ = x − x, ˆ where xˆ = [xˆ1 , . . . , xˆn ], be the observer error, which satisfies the equation d x˜ (t) = (A x˜ (t) + f (x¯ (t)) + h (x¯ (t − τ ))) dt+ g (y (t))T dw where
⎡
A f (x¯ (t)) h (x¯ (t − τ )) g (y)
⎤ −l1 ⎢ ⎥ = ⎣ ... In−1 ⎦ −ln 0 . . . 0 T = f 1 (x 1 (t)) · · · f n (x n (t)) T = h 1 (x 1 (t − τ1 )) · · · h n (x n (t − τn )) T = g1 (y (t)) · · · gn (y (t))
(5)
d x˜ (t) = (A x˜ (t)+ f (x¯ (t))+h (x¯ (t − τ ))) dt+g (y (t))T dw
dy (t) = xˆ2 (t) + x˜2 (t) + f 1 (y (t)) + h 1 (y (t − τ1 )) dt +g1 (y (t))T dw
dxˆ2 (t) = xˆ3 (t) + l1 x˜1 (t) dt .. .
i = 2, 3, . . . , n
where αi f is the output of the first-order filter with αi−1 as the input. Ito’s differentiation rule yeilds
dz 1 = xˆ2 (t) + x˜2 (t) + f 1 (y (t)) + h 1 (y (t − τ1 )) dt +g1 (y (t))T dw
dz i = xˆi+1 (t) + li x˜1 (t) − α˙ i f dt,
i = 2, . . . , n.
Assumption 1: For 1 ≤ i ≤ n, there exist positive unknown smooth functions il (¯z l + α¯ l f ) such that
l=1
1 3 z θˆ , 2ai2 i
i = 1, . . . , n − 1
(6)
where X 1 = x 1 , X i = (x˜1 , x¯ˆi , α¯ i f , α˙ i f )T , i = 2, . . . , n − 1, with x¯ˆi = (xˆ1 , xˆ2 , . . . , xˆi )T . Remark 3: From Assumption 1, we have z 13 h 1 (x 1 (t − τ1 )) 1 3 4 4 ≤ 4 z 14 (t − τ1 ) 11 (z l (t − τ1 )) + c33 z 14 4 4c3
(7)
where c3 is a positive constant. Now we are ready to present the main result of this paper. Theorem 1: Consider the stochastic nonlinear time-delay system in (1) with (4). If a control law is chosen as 1 3 z θˆ 2an2 n
with the intermediate virtual control signals αi described as (6) and the adaptive law defined as
where y and xˆ i , (i = 1, . . . , n) are available for the controller design. In order to avoid the problem of explosion of complexity, DSC approach is introduced in this part, and the backstepping design is, based on the change of coordinates, as follows:
i
The backstepping design procedure contains n steps. In each step, a virtual control function αˆ i should be developed using an appropriate Lyapunov function Vi , and then the real control law u will be designed. To begin with the backstepping design procedure, let us define a constant 2 θ = max Ni Wi∗ : i = 0, 1, 2, . . . , n .
u=−
dxˆn (t) = (u + ln x˜1 (t)) dt
|h i (x¯i )| ≤
g (y (t)) = y g¯ (y) = [y g¯ 1 (y) , . . . , y g¯ n (y)] .
αi (X i ) = −
Then the entire system can be expressed as
z1 = y z i = xˆi − αi f ,
where z¯l = [z 1 , . . . , z l ]T , α¯ l f = [α1 f , α2 f , . . . , αl f ]T , α1 f = 0. Remark 2: Since g(y(t)) is a smooth function and g(0) = 0, g(y(t)) can be expressed as
From the definition, we know θ is an unknown constant, and we define θˆ as the estimate of θ . And the feasible virtual control signal is designed as
and A is a strict Hurwitz matrix. So there exists a matrix P > 0 satisfying the following equation: A T P + P A = −I.
3
|z l | il z¯l + α¯ l f
θ˙ˆ =
n r 6 z − k0 θˆ 2 i 2a i i=1
(8)
where positive constants ai (i = 1, . . . , n), r , and k0 are design parameters, then the closed-loop system can be guaranteed to be M-SGUUB with probability 1 − ε in (ε). Proof: Step 1: Consider the Lyapunov function 2 1 a T 1 x˜ P x˜ + z 14 + θ˜ 2 VZ1 = 2 4 2r t 1 VQ 1 = 4 z 14 (τ ) l14 (z 1 (τ )) dτ 4c3 t −τ1
n i t
a4 z l4 (τ ) il4 z¯l (τ ) + α¯ l f (τ ) dτ + 2 i=1 l=1 t −τi
V1 = V Z 1 + VQ 1 where a > 0 and θ˜ = θ − θˆ .
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According to Definition 1, one has LV1 = −a x˜ T P x˜ x ˜ 2 + a x˜ T P x˜ 2 x˜ T P ( f + h)
Using Lemma 1, the following inequalities can be obtained: 2aTr g (z 1 ) 2P x˜ x˜ T P + x˜ T P x˜ P g (z 1 )T ≤ 2an g (z 1 ) 2P x˜ x˜ T P + x˜ T P x˜ P g (z 1 )T √ ≤ 6an nz 12 g (z 1 ) 2 P 2 x ˜ 2 √ √ 3a 2 n n 4 ˜ 4 (12) ≤ z 1 g (z 1 ) 4 + 3n nc12 P 4 x 2 c1 3 4 1 3 4 1 ˜ 4. z 13 x˜2 ≤ c23 z 14 + 4 x˜24 ≤ c23 z 14 + 4 x (13) 4 4 4c2 4c2
3 + z 12 g1 (z 1 )T g1 (z 1 ) 2 +2aTr g (z 1 ) 2P x˜ x˜ T P + x˜ T P x˜ P g (z 1 )T
+z 13 xˆ2 + x˜2 + f1 (z 1 ) + h 1 (z 1 (t − τ1 )) 1 1 − θ˜ θ˙ˆ + 4 z 14 l14 (z 1 ) r 4c3 1 4 4 − 4 z 1 (t − τ1 ) 11 (z 1 (t − τ1 )) 4c3 + −
a4 2 a4 2
× il4
i n
Applying inequalities (10)–(13) and (7)–(9), we have
z l4 il4 z¯l + α¯ l f
1 LV1 ≤ − x ˜ 4 + z 13 xˆ 2 + f¯1 − θ˜ θ˙ˆ r i n
a4 4 4 + z l il z¯l + α¯ l f 2
i=1 l=1 i n
z l4 (t − τi )
i=2 l=2
i=1 l=1
z¯l (t − τi ) + α¯ l f (t − τi ) .
(9)
As f ( f1 (x) ¯ , . . . , fn (x)) ¯ T , and f i (x) ¯ , i = 1, 2, . . . , n is an unknown function, by Lemma 1, for any given εi0 > 0, ∗T S (X ) such that there exists RBF NN Wi0 i0 0 ∗T Wi0 S0 (X 0 ) + δi0 (X 0 )
f i (X 0 ) = |δi0 (X 0 )| ≤ εi0
where X 0 = x, X 0 ∈ X 0 = {X 0 |x ∈ x } and x is defined as a compact set through which the state trajectories may travel. Therefore f (X 0 ) = W0∗T S0 (X 0 ) + δ0 (X 0 ) δ0 (X 0 ) ≤ ε0 . As S0T S0 ≤ N0 is used and N0 is the dimension of S0 , and according to the definition of θ , we know W0∗ 4 S04 ≤ θ 2 . Therefore, the following inequality holds when X 0 ∈ X 0 : 2a x˜ T P x˜ x˜ T P f = 2a x˜ T P x˜ x˜ T P W0∗T S0 (X 0 ) + δ0 (X 0 ) 4 a4 3 x P 8 W0∗ S04 ˜ 4+ 2 2 3 a4 4 P 8 δ04 ˜ + + x 2 2 a4 a4 = 3 x ˜ 4 + P 8 θ 2 + P 8 ε04 . (10) 2 2 According to Assumption 1, the following inequality holds: ≤
2a x˜ T P x˜ x˜ T Ph ≤ 2a x ˜ 3 P 2 h i n z l (t − τi ) ≤ 2a x ˜ 3 P 2 i=1 l=1
× il z¯l (t − τi ) + α¯ l f (t − τi ) ≤
a4 a4 3 P 8 ε04 − z 14 + P 8 θ 2 + 2 2 4
n i a4 4 z l (t − τi ) 2 i=1 l=1
× il4 z¯l (t − τi ) + α¯ l f (t − τi ) 8 3n (n + 1) P 3 x ˜ 4. + (11) 4
where 8 3n (n + 1) P 3 4 √ 1 −3n nc12 P 4 − 4 4c2 √ 3 4 3a 2 n n 3 34 g¯ (z 1 ) 4 c2 + c33 + f¯1 (X 1 ) = f 1 (z 1 ) + 4 4 c12 1 3 4 + 4 11 (z 1 ) + g¯ 1T (z 1 ) g¯ 1 (z 1 ) 2 4c3 n a4 4 3 + 11 (z 1 ) + z1. 2 4
= aλmin (P) − 3 −
i=1
Now, take the intermediate control signal αˆ 1 (X 1 ) as
αˆ 1 (X 1 ) = − k1 z 1 + f¯1 where k1 > 0. Then, we have
1 ˜ 4 + z 13 xˆ2 − αˆ 1 − k1 z 14 − θ˜ θ˙ˆ LV1 ≤ − x r a4 a4 3 P 8 θ 2 + P 8 ε04 − z 14 + 4 2 2 n i
a4 4 4 + z l il z¯l + α¯ l f . 2
(14)
i=2 l=2
However, αˆ 1 (X 1 ) is an unknown nonlinear function as it contains f 1 (z 1 ), which cannot be implemented in practice. Therefore, according to (3), for any given constant ε1 > 0, there exists RBF NN W1∗T S1 (X 1 ) such that αˆ 1 (X 1 ) = W1∗T S1 (X 1 ) + δ1 (X 1 ) |δ1 (X 1 )| ≤ ε1 where X 1 ∈ X 1 = {X 1 |x ∈ x }.
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From the definition of θ and α1 , we have
Similarly, we have
−z 13 αˆ 1 = −z 13 W1∗T S1 (X 1 ) − z 13 δ1 (X 1 ) N1 6 W ∗ 2 + 1 a 2 + 3 z 4 + 1 ε4 z ≤ 1 1 2 1 4 1 4 1 2a12 1 6 1 3 1 ≤ z 1 θ + a12 + z 14 + ε14 2 4 4 2a12 1 z 13 α1 = − 2 z 16 θˆ 2a1
(15)
+
n i
2
˜ 4+ LVm ≤ − x
m−1
m−1
z i3 z i+1 +
i=1
(17)
z l4 il4 z¯l + α¯ l f
(18)
i=2 l=2
z i3 χi+1 −
i=1
m−1 1 ˜ r 6 ˙ˆ + θ z i − θ + m−1 r 2ai2 i=1 m−1 4 χi+1 3 − − χi+1 Bi+1 (X i ) κi+1
(16)
where the inequality S1T S1 ≤ N1 is used and N1 is the dimension of S1 . Then, substituting (16) and (17) into (14) yields
˜ 4 + z 13 xˆ2 − α1 − k1 z 14 LV1 ≤ − x r 6 ˙ 1 + θ˜ z 1 − θˆ + 1 r 2a12 a4
5
m−1
ki z i4
i=1
(19)
i=1
+
a4 2
n
i
z l4 il4 z¯l + α¯ l f
i= m+1 l=m+1
3 4 3 +z m xˆm+1 + f¯m − z m 4
(20)
where f¯m (X m ) = lm x˜1 − α˙ m f + z m
n
3 a4 4 + lm z¯ m + α¯ m f . 4 2 i=m
where 1 =
a4 a4 1 1 P 8 θ 2 + P 8 ε04 + a12 + ε14. 2 2 2 4
Take the intermediate control signal αˆ m (X m ) as
By the definition of xˆ2 = z 2 + α2 f , (18) can be rewritten as
LV1 ≤ − x ˜ 4 + z 13 z 2 + α2 f − α1 − k1 z 14 r 6 ˙ 1 ˜ ˆ + θ z 1 − θ + 1 r 2a12 n i
a4 4 4 + z l il z¯l + α¯ l f . 2
˜ + LVm ≤ − x
α2 f (0) = α1 (0).
B2 (X 1 ) =
χ2 + B2 (X 1 ) κ2
4
˜ + LV1 ≤ − x −k1 z 14 +
z 13 z 2
+
z 13 χ2
1 + θ˜ r
r 6 ˙ˆ z1 − θ 2a12
+
m−1
z i3 χi+1
i=1
m−1 r 1 ˙ 4 6 − ki z i + θ˜ z i − θˆ r 2ai2 i=1 i=1 m−1 4 χi+1 3 − χi+1 Bi+1 (X i ) + m−1 − κi+1 m
+
a4 2
n
i
z l4 il4 z¯l + α¯ l f
i= m+1 l=m+1
3 4 3 xˆm+1 − αˆ m − z m +z m . 4
1 3 2 ˆ ˙ˆ z 1 z˙ 1 θ + 2 z 13 θ. 2 2a1 2a1
Then, it implies
z i3 z i+1
i=1
Let χ2 = α2 f − α1 be the output error of this filter; then one has α˙ 2 f = −(χ2 /κ2 ) and
where
m−1 i=1
In order to avoid repeatedly differentiating α1 , a new state variable α2 f is introduced, and let α1 pass through a firstorder filter with time constant κ2 to obtain α2 f as
χ˙ 2 = α˙ 2 f − α˙ 1 = −
where km > 0. Then, adding and subtracting αˆ m (X m ) in (20) yields 4
i=2 l=2
κ2 α˙ 2 f + α2 f = α1 ,
αˆ m = − km z m + f¯m
(21)
Similarly, αˆ m (X m ) can be approximated by the RBF NN Wm∗T Sm (X m ) as
i n
a4 4 4 z l il z¯l + α¯ l f + 1 . 2 i=2 l=2
Step m: (2 ≤ m ≤ n − 1). Choose the Lyapunov function candidate 1 4 1 Vm = Vm−1 + z m + χm4 . 4 4
αˆ m (X m ) = Wm∗T Sm (X m ) + δm (X m ) |δm (X m )| ≤ εm where X m ∈ X m = {X m |x ∈ x }. And we can obtain 1 6 1 2 3 4 1 4 z m θ + am + zm + εm 2 2am 2 4 4 1 3 6 ˆ θ. zm αm = − 2 z m 2am
3 −z m αˆ m ≤
(22) (23)
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Then, by substituting (22) and (23) into (21), we have ˜ 4+ LVm ≤ − x
m−1
z i3 z i+1 +
m−1
where f¯n (X n ) = ln x˜1 − α˙ n f + z n
z i3 χi+1
i=1 m r 1 ˙ 4 6 − ki z i + θ˜ z i − θˆ r 2ai2 i=1 i=1 m−1 4 χi+1 3 − χi+1 Bi+1 (X i ) − κi+1 i=1
3 xˆm+1 − αm +m + z m
a4 2
n
i
3 a4 4 + nn z¯l + α¯ l f . 4 2
Take the intermediate control signal αˆ n (X n ) as αˆ n = −(kn z n + f¯n ), where kn > 0; then, adding and subtracting αˆ n (X n ) in (26) yields
i=1
m
+
LVn ≤ − x ˜ 4+
n−1
z i3 z i+1 +
i=1
z l4 il4 z¯l + α¯ l f
(24)
i= m+1 l=m+1
where
n−1
z i3 χi+1 −
i=1
n
ki z i4
i=1
n−1 r
3 1 ˙ 6 + θ˜ z i − θˆ + z n3 u − αˆ n − z n4 2 r 4 2ai i=1 n−1 4 χi+1 3 − − χi+1 Bi+1 (X i ) + n−1 . (27) κi+1 i=1
m =
a4 2
P 8 θ 2 +
a4
P 8 ε04 +
2
m 1
2
ai2 +
i=1
m 1
4
εi4 .
i=1
Next, introduce a new variable αm+1 f , and let αm pass through a first-order filter with the constant κm+1 to obtain αm+1 f κm+1 α˙ m+1 f + αm+1 f = αm ,
αm+1 f (0) = αm (0) . (25)
Then define χm+1 = αm+1 f − αm as the output error of this filter. We have α˙ m+1 f = −(χm+1 /κm+1 ) and χ χ˙ m+1 = α˙ m+1 f − α˙ m = − m+1 + Bm+1 (X m ) κm+1 where Bm+1 (X m ) =
1 3 ˙ˆ 3 2 ˆ z m z˙ m θ + 2 z m θ. 2 2am 2am
Substituting (25) into (24) yields ˜ 4+ LVm ≤ − x
m
z i3 z i+1 +
m
z i3 χi+1
i=1
i=1 m 1 r 6 ˙ 4 ˜ ˆ − ki z i + θ z −θ r 2ai2 i i=1 i=1 m−1 4 χi+1 3 − χi+1 Bi+1 (X i ) + m − κi+1 m
i=1
+
a4 2
n
i
z l4 il4 z¯ l + α¯ l f .
i= m+1 l=m+1
Step n: Consider the following Lyapunov function: 1 1 Vn = Vn−1 + z n4 + χn4 . 4 4 Similarly, we obtain LVn = − x ˜ 4+
n−1 i=1
z i3 z i+1 +
n−1 i=1
z i3 χi+1 −
αˆ n (X n ) = Wn∗T Sn (X n ) + δn (X n ) |δn (X n )| ≤ εn where X n ∈ X n = {X n |x ∈ x }. Now, define the closed-loop state variables x c = (x, ˜ z¯ n , χ¯n , θ˜ , α¯ n f )T , where χ¯ n = (χ1 , . . . , χn ) and the initial ˜ 0 ), z¯ n (t0 ), χ¯ n (t0 ), θ˜ (t0 ), α¯ n f (t0 ))T . For condition is x 0c = (x(t ¯ n /c), ¯ n and c¯ are given below, some constant c0 > ( ¯ where let 0 = {x c |Vn ≤ c0 }, (ε) = {x c |Vn ≤ (c0 /ε)}, and the initial value of Lyapunov function Vn (t0 ) ≤ c0 . Then define the approximation region with respect to parameter ε, (0 < ε < 1) as n t 1 1 z i4 + 4 z 14 (τ ) l14 (z 1 (τ )) dτ X n (ε) = X n 4 4c 3 t −τ1 i=1 i n
a4 t 4 c0 4 . z l (τ ) il z¯l (τ ) + α¯ l f (τ ) dτ ≤ + 2 ε t −τi i=1 l=1
Therefore, X n (t0 ) ∈ X n (ε) and x c (t0 ) ∈ 0 ⊂ (ε). If x c (t) ∈ (ε), then X n (t) ∈ X n (ε), which means τε ≤ τ X n , where τε is the first time x c reaches the boundary of (ε) and τ X n is the first time X n reaches the boundary of X n (ε) . Following a similar procedure and by the definition of u, we have 1 6 1 3 1 z θ + an2 + z n4 + εn4 (28) −z n3 αˆ n ≤ 2an2 n 2 4 4 1 z n u = − 2 z n6 θˆ . (29) 2an Then, by substituting (28) and (29) into (27), when x c (t) ∈ (ε), for any t ∈ [t0 , τε ]
n−1 i=1
ki z i4
n−1
1 ˜ r 6 ˙ˆ + θ z i − θ + n−1 + z n3 u + f¯n 2 r 2ai i=1 n−1 4 χi+1 3 3 − − χi+1 Bi+1 (X i ) − z n4 (26) κi+1 4 i=1
Similar to the above steps, αˆ n (X n ) can be approximated by the RBF NN Wn∗T Sn (X n ) as
LVn ≤ − x ˜ 4+
n−1 i=1
z i3 z i+1 +
n−1
z i3 χi+1
i=1 n r 1 ˙ 4 6 ˜ ˆ − ki z i + θ zi − θ r 2ai2 i=1 i=1 n−1 4 χi+1 3 − χi+1 Bi+1 (X i ) + n − κi+1 n
i=1
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7
3
where n =
a4 2
P 8 θ 2 +
a4 2
P 8 ε04 +
n 1
2
ai2 +
i=1
n 1
4
εi4 .
n−1
z i3 z i+1 +
i=1
+
k0 ˜ ˆ θθ − r
n−1 4 χi+1 i=1
κi+1
n−1
z i3 χi+1 −
i=1
n
ki z i4 + n
3 − χi+1 Bi+1 (X i ) .
0 −1
i=1
(30)
−2 −3
3 1 4 z i3 z i+1 ≤ z i4 + z i+1 4 4 3 4 1 4 3 z i χi+1 ≤ z i + χi+1 4 4 3 4 43 4 1 3 3 χi+1 Bi+1 ≤ π Bi+1 χi+1 + 4 4π 4 1 1 (31) θ˜ θˆ = θ˜ θ − θ˜ ≤ − θ˜ 2 + θ 2 2 2 where π > 0 is a design constant and Bi+1 is a continuous function. Therefore, there exists a positive constant Ni+1 such that |Bi+1 | ≤ Ni+1 . Substituting (31) into (30), one has n 7 4 LVn ≤ − x ˜ 4− ki − z 4 i i=1 n−1 1 1 3 4 43 4 χi+1 + − − π 3 Bi+1 ki+1 4 4 i=1
k0 ˜ 2 ¯n θ + 2r k 1 ¯ n = n + 0 θ 2 + . 2r 4π 4 −
Let >0 and denote 2 7 c = min , , 4 ki − aλ2max (P) 4 1 1 3 4 43 3 4 − − π Bi+1 , k0 i = 1, . . . , n. ki+1 4 4
2 1
By utilizing Lemma 1, we have
Then one has
1
i=1
By the definition of θˆ˙ , we can get ˜ 4+ LVn ≤ − x
x
¯ n. LVn ≤ −c Vn − VQ 1 + Then, there exists a positive constant c¯ such that ¯ n , ∇x c ∈ (ε). ¯ n + LVn ≤ −cV ¯ n /c) ¯ and τε ≤ τ X n , it can be concluded Considering c0 > ( that all the signals in the closed-loop system are M-SGUUB with probability 1 − ε in (ε), i.e., inf x0c ∈0 P {τε = ∞} ≥ 1−ε, which are the desired results and the proof is completed. Remark 4: In the process of controller design, the DSC technique is employed to make the control scheme simple. That is because the repeated differentiation of virtual control αi is replaced by α˙ i+1 f and αi+1 f is defined by a first-order
0
Fig. 1.
10
20
30 Time(sec)
40
50
60
Trajectory of system output y.
filter with αi as input. Therefore, the repeated differentiation of αi is avoided. Remark 5: By using the method of direct adaptive NN control, there is only one adaptive parameter θˆ in the virtual control αi and true control u. Therefore, the computation burden is greatly reduced. In addition, it is not necessary to a priori know the centers of the receptive field and the width of the Gaussian functions, which makes the obtained theory more useful in practice. III. S IMULATION R ESULTS In this section, a simulation example is presented to demonstrate the effectiveness of the proposed adaptive NN control method. Consider the following stochastic nonlinear time-delay system: x 13 (t − τ1 ) 2 dt dx 1 = x 2 − 0.3x 1 + x 1 + 1 + x 12 (t − τ1 ) +x 1 sin (x 1 ) dw x 14 (t − τ2 ) sin (x 2 ) 0.2 dt + dx 2 = u + 0.1x 2 sin 1 + x 12 1 + x 12 (t − τ2 ) x1 + dw 1 + x 12 y = x1 where the nonlinear functions are f 1 (x 1 (t)) = −10x 1 + x 12 , f 2 (x¯2 (t)) = 0.1x 2 sin 0.2/(1 + x 12 ) , nonlinear time-delay terms are defined as h 1 (x¯1 (t − τ1 )) = (x 13 (t − τ1 )/ 1 + x 12 (t − τ1 )), and h 2 (x¯2 (t − τ2 )) = (x 14 (t − τ2 ) sin (x 2 )/ 1 + x 12 (t − τ2 )), the initial states are chosen as x 1 (0) = 0.2, x 2 (0) = 0.5, and the observer is designed as .
xˆ 1 = xˆ2 + l1 y − xˆ1 .
xˆ 2 = u + l2 y − xˆ1 . According to Theorem 1, the virtual control function α1 and the true control law u are chosen, respectively, as 1 1 α1 = − 2 z 13 θˆ , u = − 2 z 23 θˆ (32) 2a1 2a2
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80
1000
x2
800
40
700
20
600
0
500
−20
400 300
−40
200
−60 −80
100 0
Fig. 2.
10
20
30 Time(sec)
40
50
0
60
Trajectory of system state x2 .
Fig. 5.
80
α
2f
0
10
20
30 40 Time(sec)
50
60
Adaptive parameter θˆ .
In the simulation, the time delays are chosen as τ1 = τ2 = 0.5, therefore the upper bound of time delay is chosen as τm = 0.5, and the design parameters are chosen as l1 = l2 = 100, a1 = a2 = 0.21, r = 22.5, k0 = 0.01, and κ2 = 0.1. The simulation results are illustrated in Figs. 1–5, respectively. Figs. 1 and 2 show the system output y and state variable x 2 . Fig. 3 illustrates a new state variable of a first-order filter. Fig. 4 depicts the trajectory of input u, while Fig. 5 illustrates the trajectory of adaptive parameter θˆ .
60 40 20 0 −20 −40 −60
IV. C ONCLUSION
−80 0
10
20
30 Time(sec)
40
50
60
Trajectory of a state variable of a first-order filter α2 f .
Fig. 3.
3000
u
2000 1000 0 −1000 −2000 −3000
Fig. 4.
θˆ
900
60
In this paper, the problem of adaptive NN DSC was investigated for a class of nonlinear stochastic strict-feedback systems with unknown time delays. By using the backstepping approach, a direct adaptive NN control scheme was proposed. In addition, by combining the method of output-feedback control and DSC, the results obtained could be applied to solve the control problem for systems with unmeasurable states, and the problem of explosion of complexity could be avoided. In addition, as the number of the online adaptive parameters was only one, the computation burden could be reduced accordingly. Therefore, it is convenient to implement this algorithm in practical systems. Finally, a simulation example was presented to illustrate the effectiveness of the method proposed. R EFERENCES
0
10
20
30 Time(sec)
40
50
60
Trajectory of control input u.
where z 1 = y, z 2 = xˆ2 − α2 f . The adaptive law is given as θ˙ˆ =
2 r 6 z − k0 θˆ . 2 i 2a i i=1
(33)
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Qi Zhou received the B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, respectively. She is currently pursuing the Ph.D. degree with the international collaboration program, University of Portsmouth, Portsmouth, U.K., and the Nanjing University of Science and Technology, Nanjing, China. Her current research interests include fuzzy control, stochastic control, and robust control.
Peng Shi (M’95–SM’98) received the B.Sc. degree in mathematics from the Harbin Institute of Technology, Harbin, China, the M.Eng. degree in systems engineering from Harbin Engineering University, Harbin, the Ph.D. degree in electrical engineering from the University of Newcastle, Callaghan, Australia, the Ph.D. degree in mathematics from the University of South Australia, Adelaide, Australia, and the D.Sc. degree from the University of Glamorgan, Pontypridd, U.K. He was a Lecturer with the University of South Australia and a Senior Scientist with the Defence Science and Technology Organisation, Edinburgh, Australia. He was a Professor with the University of Glamorgan, Victoria University, Melbourne, Australia, and is currently with the University of Adelaide, Adelaide. He has authored or co-authored widely in these areas. His current research interests include system and control theory, computational and intelligent systems, and operational research. Dr. Shi is a fellow of the Institution of Engineering and Technology, U.K., and the Institute of Mathematics and its Applications, U.K. He is on the editorial board of a number of international journals, such as the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE T RANSACTIONS ON S YSTEMS , M AN AND C YBERNETICS -PART B, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS .
Shengyuan Xu received the B.Sc. degree from the Hangzhou Normal University, Hangzhou, China, the M.Sc. degree from Qufu Normal University, Qufu, China, and the Ph.D. degree from the Nanjing University of Science and Technology, Nanjing, China, in 1990, 1996, and 1999, respectively. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, from 1999 to 2000. From 2000 to 2002, he was a Post-Doctoral Researcher in CESAME with the Université catholique de Louvain, Louvain-La-Neuve, Belgium, and the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, respectively. From 2002 to 2004, he was a William Mong Young Researcher and an Honorary Associate Professor with the Department of Mechanical Engineering, University of Hong Kong. Since 2002, he has been with the School of Automation, Nanjing University of Science and Technology as a Professor. His current research interests include robust filtering and control, singular systems, time-delay systems, neural networks, and multidimensional systems and nonlinear systems. Dr. Xu was a recipient of the National Excellent Doctoral Dissertation Award from the Ministry of Education of China in 2002. He received a grant from the National Science Foundation for Distinguished Young Scholars of China, in 2006. He was awarded a Cheung Kong Professorship from the Ministry of Education of China in 2008. He is a member of the Editorial Boards of the Multidimensional Systems and Signal Processing, and the Circuits Systems and Signal Processing.
Hongyi Li received the B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, respectively, and the Ph.D. degree in intelligent control from the University of Portsmouth, Portsmouth, U.K., in 2012. He is currently with the College of Information Science and Technology, Bohai University. His current research interests include fuzzy control, robust control, and their applications in suspension systems.
1
Observer-Based Adaptive Neural Network Control for Nonlinear Stochastic Systems With Time Delay Qi Zhou, Peng Shi, Senior Member, IEEE, Shengyuan Xu, and Hongyi Li
Abstract— This paper considers the problem of observer-based adaptive neural network (NN) control for a class of singleinput single-output strict-feedback nonlinear stochastic systems with unknown time delays. Dynamic surface control is used to avoid the so-called explosion of complexity in the backstepping design process. Radial basis function NNs are directly utilized to approximate the unknown and desired control input signals instead of the unknown nonlinear functions. The proposed adaptive NN output feedback controller can guarantee all the signals in the closed-loop system to be mean square semi-globally uniformly ultimately bounded. Simulation results are provided to demonstrate the effectiveness of the proposed methods. Index Terms— Adaptive control, backstepping, surface control, fuzzy control, nonlinear systems.
dynamic
I. I NTRODUCTION
I
N THE past decade, research on stochastic systems has received considerable attention due to the fact that stochastic disturbance exists in many practical systems and is often the source of instability [1]–[5]. On the other hand, controller design for nonlinear systems has always been an important topic since practical stochastic system models are largely nonlinear. Many nonlinear stochastic control results and methods, such as sliding mode control [6] and T-S fuzzy control [7], have been reported in the literature. In recent years, the well-known backstepping method has been applied to solve the problem of nonlinear stochastic systems (see [8]). The backstepping design for nonlinear stochastic single-input single-output (SISO) systems was investigated in [9] and [10] by introducing quartic Lyapunov functions. More recently, the results have been extended to output feedback control design for stochastic nonminimum-phase nonlinear systems [11], large-scale stochastic nonlinear systems [12], and high-order nonlinear stochastic systems [13]–[15]. In [16], the results have also been extended to adaptive neural network (NN)
Manuscript received May 21, 2011; revised September 18, 2012; accepted September 26, 2012. This work was supported in part by the National Key Basic Research Program, China, under Grant 2012CB215202, the Qing Lan Project, the 111 Project under Grant B12018, the National Natural Science Foundation of China under Grant 61174058, Grant 60974071, and Grant 61203002, and the Engineering and Physical Sciences Research Council, U.K., under Grant EP/F029195. Q. Zhou and S. Xu are with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide 5005, Australia, and also with the School of Engineering and Science, Victoria University, Melbourne 3052, Australia (e-mail: [email protected]). H. Li is with the College of Information Science and Technology, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TNNLS.2012.2223824
output-feedback stabilization for a class of nonlinear stochastic systems. However, there still exists the issue of explosion of complexity in the backstepping design procedure because the backstepping design requires repeated differentiation of virtual controllers, which makes the complexity of controller grow dramatically as the order of the system increases. In order to avoid the above-mentioned problem, the dynamic surface control (DSC) technique was first introduced in [17] for a class of strict-feedback nonlinear system with unknown functions. Then, this approach was extended to solve a class of nonlinear systems with periodic disturbances in [18]. More recently, the DSC approach was further extended to solve the problem of output-feedback adaptive control of stochastic nonlinear systems in [19]. On the other hand, it is well known that the method of fuzzy logic control or NN control is useful to approximate unknown nonlinear functions in systems (see [20]–[35] and the references therein). Recently, the NN approximation approach has been utilized to handle the nonlinear stochastic systems control design problems [36]–[39]. Since the general nonlinear function f (x) can be approximated by an NN as f (x) = θ T ϕ(x)+ε(x), the method can also be used in stochastic nonlinear systems. However, it should be noted that the number of adaptation laws depends on the number of the NN nodes, and therefore the number of parameters to be estimated will increase when the NN nodes increase, which will lead to an unacceptably large learning time. This problem was considered in [40] by considering the norm of ideal weighting vector in fuzzy logic systems as the estimation parameter instead of the elements of weighting vector, and was also studied in [41] by developing a direct adaptive method for a class of nonlinear systems. It is well known that time delays are frequently encountered in engineering systems, and they usually become the sources of instability and degrade the performance of the systems [42]–[47]. Therefore, how to control nonlinear stochastic timedelay systems is still an important and practical topic. In [48] and [49], the observer-based control problem for nonlinear stochastic systems with time delay was solved. However, there exists a common weakness in the work of [48] and [49] in that the time delays are only present in the system output. Time delays in the state variables were not considered, which affect the stability and performance of the systems. Motivated by the above observation, in this paper the problem of observer-based adaptive control for nonlinear stochastic time-delay systems is investigated. The main contributions of this paper are summarized as follows. 1) The DSC approach is successfully applied to nonlinear stochastic systems, which
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avoids repeated differentiating the virtual controller by introducing a first-order filter in each step of backstepping design procedure. 2) A direct adaptive radial basis function NNs (RBF NNs) control method is proposed, where the number of adaptation parameters is only one, which significantly reduces the computation burden. Furthermore, the developed controller design is independent of any prior knowledge of NNs. 3) The time delays exist not only in the system output but also in the state variables, which makes the developed results in this paper more applicable. It is proven that the controller designed in this paper guarantees all the signals in the closed-loop to be mean square semi-globally uniformly ultimately bounded (M-SGUUB). Finally, simulation results are given to illustrate the effectiveness of the proposed control algorithm. The rest of this paper is organized as follows. The problem formulation and preliminaries, stochastic stability, NN approximation, and output feedback DSC design procedure for SISO strict-feedback nonlinear stochastic time-delay systems are presented in Section II. A simulation example is provided in Section III to demonstrate the effectiveness of the proposed design scheme. The conclusions are drawn in Section IV. II. P ROBLEM F ORMULATION AND P RELIMINARIES In this section, some preliminaries and assumptions are first formulated in Section II-A, and then the definition of stochastic stability is presented in Section II-B. In order to approximate the unknown nonlinear functions, RBF NN are presented in Section II-C. By using the backstepping approach, an output feedback DSC design procedure is formulated in Section II-D. A. Problem Formulation Consider the nonlinear stochastic system dx i (t) = (x i+1 (t) + f i (x¯i (t)) + h i (x¯i (t − τi ))) dt 1≤i ≤n +gi (y)T dw, dx n (t) = (u + f n (x (t)) + h n (x (t − τn ))) dt +gn (y)T dw y (t) = x 1 (t)
εp 1 |x| p + q |y|q p qε
where ε > 0, p > 1, q > 1, and ( p − 1)(q − 1) = 1.
B. Stochastic Stability Consider the stochastic system dx (t) = f (x (t)) dt + g (x (t)) dw
(2)
where x ∈ R n is the system state, w is an r -dimentional standard Wiener process, and f : R n → R n , g : R n → R n are locally Lipschitz functions and satisfy f (0) = g(0) = 0. Definition 1: For any given V (x) ∈ C 2 , associated with the stochastic system (2), the infinitesimal generator L is defined as follows: 1 ∂2V ∂V f (x) + Tr g (x)T g LV (x) = (x) ∂x 2 ∂x2 where Tr(A) is the trace of a matrix A. Definition 2 [36]: The solution process {x(t), t ≥ t0 } of stochastic system (2) with initial condition x 0 ∈ S0 (some compact set containing the origin) is said to be M-SGUUB if for any desired escape risk ε (0 < ε < 1), it is bounded with probability 1 − ε in some compact set S(ε) ⊃ S0 , i.e., inf x0∈S0 P{τ S(ε) = ∞} ≥ 1−ε. The hitting time τ S(ε) is defined as the first time that the trajectories of the state variable reach the boundary of S(ε). Lemma 2 [36]: Consider the stochastic system (2). If there exists a positive-definite radially unbounded twice continuously differentiable Lyapunov function V : R n → R, and constants c1 > 0, c2 > 0, and r0 > (c2 /c1 ) such that for some ε (0 < ε < 1) and x 0 ∈ S0 := {x ∈ R n |V (x) ≤ r0 } r0 LV (x) ≤ −c1 V (x)+c2 , ∇x ∈ S (ε) = x ∈ R n V (x) ≤ ε holds, then, for t ∈ [t0 , τ S(ε)], there is a unique solution of (2) and the system is bounded with probability 1 − ε in S(ε). C. NN Approximation
(1)
where x¯i (t) = [x 1 (t), x 2 (t), . . . , x i (t)]T ∈ R i , i = 1, 2, . . . , n − 1, and x(t) = [x 1(t), x 2 (t), . . . , x n (t)]T ∈ R n denote state vectors of the system; u ∈ R and y ∈ R are the input and output of the system, respectively. f i (.) (i = 1, 2, . . . , n) stands for the unknown smooth system function with f i (0) = 0. h i (.) is the unknown smooth nonlinear timedelay functions, τi is the unknown constant delay, and τm is the upper bound of τi . gi (.) is the unknown vector-valued smooth functions with gi (0) = 0. w is an independent r -dimentional standard Wiener process. Lemma 1 [50] (Young’s Inequality): For ∀(x, y) ∈ R 2 , the following inequality holds: xy ≤
Remark 1: Some restrictions on the delay terms are presented in this assumption, which are essential for the controller design in the sequel. In addition, compared with [48] and [49], the time delays considered in this paper exist not only in the system output but also in the state variables.
In this paper, RBF NN will be used to approximate the unknown smooth nonlinear functions. For any continuous unknown smooth nonlinear function f (Z ) over a compact set Z ⊂ R q , there exists a RBF NN W ∗T S(Z ) such that for a desired level of accuracy ε f (Z ) = W ∗T S (Z ) + δ (Z ) , where
|δ (Z )| ≤ ε
(3)
W∗
is the ideal constant weight vector and defined by ∗ T sup f (Z ) − W S (Z ) . W = arg min W ∈R N
Z ∈ Z
δ(Z ) is the approximation error, W = [w1 , . . . , w N ]T is the weight vector and S(Z ) = [s1 (Z ), . . . , s N (Z )]T is the basis function vector with N being the number of the RBF NN nodes and N > 1. RBF si (Z ) = exp[(−(Z − μi )T (Z − μi )/ηi2 )], i = 1, 2, . . . , N, where μi = [μi1 , μi2 , . . . , μin ]T is the center of the receptive field and ηi is the width of the Gaussian function.
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D. Output Feedback DSC Design In this subsection, an observer-based controller is designed to guarantee all the signals in the closed-loop system to be SGUUB. First, we design the following observer to estimate the unmeasured states: .
xˆ i = xˆ i+1 + li y − xˆ1 i = 1, . . . , n (4) where xˆn+1 = u. Let x˜ = x − x, ˆ where xˆ = [xˆ1 , . . . , xˆn ], be the observer error, which satisfies the equation d x˜ (t) = (A x˜ (t) + f (x¯ (t)) + h (x¯ (t − τ ))) dt+ g (y (t))T dw where
⎡
A f (x¯ (t)) h (x¯ (t − τ )) g (y)
⎤ −l1 ⎢ ⎥ = ⎣ ... In−1 ⎦ −ln 0 . . . 0 T = f 1 (x 1 (t)) · · · f n (x n (t)) T = h 1 (x 1 (t − τ1 )) · · · h n (x n (t − τn )) T = g1 (y (t)) · · · gn (y (t))
(5)
d x˜ (t) = (A x˜ (t)+ f (x¯ (t))+h (x¯ (t − τ ))) dt+g (y (t))T dw
dy (t) = xˆ2 (t) + x˜2 (t) + f 1 (y (t)) + h 1 (y (t − τ1 )) dt +g1 (y (t))T dw
dxˆ2 (t) = xˆ3 (t) + l1 x˜1 (t) dt .. .
i = 2, 3, . . . , n
where αi f is the output of the first-order filter with αi−1 as the input. Ito’s differentiation rule yeilds
dz 1 = xˆ2 (t) + x˜2 (t) + f 1 (y (t)) + h 1 (y (t − τ1 )) dt +g1 (y (t))T dw
dz i = xˆi+1 (t) + li x˜1 (t) − α˙ i f dt,
i = 2, . . . , n.
Assumption 1: For 1 ≤ i ≤ n, there exist positive unknown smooth functions il (¯z l + α¯ l f ) such that
l=1
1 3 z θˆ , 2ai2 i
i = 1, . . . , n − 1
(6)
where X 1 = x 1 , X i = (x˜1 , x¯ˆi , α¯ i f , α˙ i f )T , i = 2, . . . , n − 1, with x¯ˆi = (xˆ1 , xˆ2 , . . . , xˆi )T . Remark 3: From Assumption 1, we have z 13 h 1 (x 1 (t − τ1 )) 1 3 4 4 ≤ 4 z 14 (t − τ1 ) 11 (z l (t − τ1 )) + c33 z 14 4 4c3
(7)
where c3 is a positive constant. Now we are ready to present the main result of this paper. Theorem 1: Consider the stochastic nonlinear time-delay system in (1) with (4). If a control law is chosen as 1 3 z θˆ 2an2 n
with the intermediate virtual control signals αi described as (6) and the adaptive law defined as
where y and xˆ i , (i = 1, . . . , n) are available for the controller design. In order to avoid the problem of explosion of complexity, DSC approach is introduced in this part, and the backstepping design is, based on the change of coordinates, as follows:
i
The backstepping design procedure contains n steps. In each step, a virtual control function αˆ i should be developed using an appropriate Lyapunov function Vi , and then the real control law u will be designed. To begin with the backstepping design procedure, let us define a constant 2 θ = max Ni Wi∗ : i = 0, 1, 2, . . . , n .
u=−
dxˆn (t) = (u + ln x˜1 (t)) dt
|h i (x¯i )| ≤
g (y (t)) = y g¯ (y) = [y g¯ 1 (y) , . . . , y g¯ n (y)] .
αi (X i ) = −
Then the entire system can be expressed as
z1 = y z i = xˆi − αi f ,
where z¯l = [z 1 , . . . , z l ]T , α¯ l f = [α1 f , α2 f , . . . , αl f ]T , α1 f = 0. Remark 2: Since g(y(t)) is a smooth function and g(0) = 0, g(y(t)) can be expressed as
From the definition, we know θ is an unknown constant, and we define θˆ as the estimate of θ . And the feasible virtual control signal is designed as
and A is a strict Hurwitz matrix. So there exists a matrix P > 0 satisfying the following equation: A T P + P A = −I.
3
|z l | il z¯l + α¯ l f
θ˙ˆ =
n r 6 z − k0 θˆ 2 i 2a i i=1
(8)
where positive constants ai (i = 1, . . . , n), r , and k0 are design parameters, then the closed-loop system can be guaranteed to be M-SGUUB with probability 1 − ε in (ε). Proof: Step 1: Consider the Lyapunov function 2 1 a T 1 x˜ P x˜ + z 14 + θ˜ 2 VZ1 = 2 4 2r t 1 VQ 1 = 4 z 14 (τ ) l14 (z 1 (τ )) dτ 4c3 t −τ1
n i t
a4 z l4 (τ ) il4 z¯l (τ ) + α¯ l f (τ ) dτ + 2 i=1 l=1 t −τi
V1 = V Z 1 + VQ 1 where a > 0 and θ˜ = θ − θˆ .
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According to Definition 1, one has LV1 = −a x˜ T P x˜ x ˜ 2 + a x˜ T P x˜ 2 x˜ T P ( f + h)
Using Lemma 1, the following inequalities can be obtained: 2aTr g (z 1 ) 2P x˜ x˜ T P + x˜ T P x˜ P g (z 1 )T ≤ 2an g (z 1 ) 2P x˜ x˜ T P + x˜ T P x˜ P g (z 1 )T √ ≤ 6an nz 12 g (z 1 ) 2 P 2 x ˜ 2 √ √ 3a 2 n n 4 ˜ 4 (12) ≤ z 1 g (z 1 ) 4 + 3n nc12 P 4 x 2 c1 3 4 1 3 4 1 ˜ 4. z 13 x˜2 ≤ c23 z 14 + 4 x˜24 ≤ c23 z 14 + 4 x (13) 4 4 4c2 4c2
3 + z 12 g1 (z 1 )T g1 (z 1 ) 2 +2aTr g (z 1 ) 2P x˜ x˜ T P + x˜ T P x˜ P g (z 1 )T
+z 13 xˆ2 + x˜2 + f1 (z 1 ) + h 1 (z 1 (t − τ1 )) 1 1 − θ˜ θ˙ˆ + 4 z 14 l14 (z 1 ) r 4c3 1 4 4 − 4 z 1 (t − τ1 ) 11 (z 1 (t − τ1 )) 4c3 + −
a4 2 a4 2
× il4
i n
Applying inequalities (10)–(13) and (7)–(9), we have
z l4 il4 z¯l + α¯ l f
1 LV1 ≤ − x ˜ 4 + z 13 xˆ 2 + f¯1 − θ˜ θ˙ˆ r i n
a4 4 4 + z l il z¯l + α¯ l f 2
i=1 l=1 i n
z l4 (t − τi )
i=2 l=2
i=1 l=1
z¯l (t − τi ) + α¯ l f (t − τi ) .
(9)
As f ( f1 (x) ¯ , . . . , fn (x)) ¯ T , and f i (x) ¯ , i = 1, 2, . . . , n is an unknown function, by Lemma 1, for any given εi0 > 0, ∗T S (X ) such that there exists RBF NN Wi0 i0 0 ∗T Wi0 S0 (X 0 ) + δi0 (X 0 )
f i (X 0 ) = |δi0 (X 0 )| ≤ εi0
where X 0 = x, X 0 ∈ X 0 = {X 0 |x ∈ x } and x is defined as a compact set through which the state trajectories may travel. Therefore f (X 0 ) = W0∗T S0 (X 0 ) + δ0 (X 0 ) δ0 (X 0 ) ≤ ε0 . As S0T S0 ≤ N0 is used and N0 is the dimension of S0 , and according to the definition of θ , we know W0∗ 4 S04 ≤ θ 2 . Therefore, the following inequality holds when X 0 ∈ X 0 : 2a x˜ T P x˜ x˜ T P f = 2a x˜ T P x˜ x˜ T P W0∗T S0 (X 0 ) + δ0 (X 0 ) 4 a4 3 x P 8 W0∗ S04 ˜ 4+ 2 2 3 a4 4 P 8 δ04 ˜ + + x 2 2 a4 a4 = 3 x ˜ 4 + P 8 θ 2 + P 8 ε04 . (10) 2 2 According to Assumption 1, the following inequality holds: ≤
2a x˜ T P x˜ x˜ T Ph ≤ 2a x ˜ 3 P 2 h i n z l (t − τi ) ≤ 2a x ˜ 3 P 2 i=1 l=1
× il z¯l (t − τi ) + α¯ l f (t − τi ) ≤
a4 a4 3 P 8 ε04 − z 14 + P 8 θ 2 + 2 2 4
n i a4 4 z l (t − τi ) 2 i=1 l=1
× il4 z¯l (t − τi ) + α¯ l f (t − τi ) 8 3n (n + 1) P 3 x ˜ 4. + (11) 4
where 8 3n (n + 1) P 3 4 √ 1 −3n nc12 P 4 − 4 4c2 √ 3 4 3a 2 n n 3 34 g¯ (z 1 ) 4 c2 + c33 + f¯1 (X 1 ) = f 1 (z 1 ) + 4 4 c12 1 3 4 + 4 11 (z 1 ) + g¯ 1T (z 1 ) g¯ 1 (z 1 ) 2 4c3 n a4 4 3 + 11 (z 1 ) + z1. 2 4
= aλmin (P) − 3 −
i=1
Now, take the intermediate control signal αˆ 1 (X 1 ) as
αˆ 1 (X 1 ) = − k1 z 1 + f¯1 where k1 > 0. Then, we have
1 ˜ 4 + z 13 xˆ2 − αˆ 1 − k1 z 14 − θ˜ θ˙ˆ LV1 ≤ − x r a4 a4 3 P 8 θ 2 + P 8 ε04 − z 14 + 4 2 2 n i
a4 4 4 + z l il z¯l + α¯ l f . 2
(14)
i=2 l=2
However, αˆ 1 (X 1 ) is an unknown nonlinear function as it contains f 1 (z 1 ), which cannot be implemented in practice. Therefore, according to (3), for any given constant ε1 > 0, there exists RBF NN W1∗T S1 (X 1 ) such that αˆ 1 (X 1 ) = W1∗T S1 (X 1 ) + δ1 (X 1 ) |δ1 (X 1 )| ≤ ε1 where X 1 ∈ X 1 = {X 1 |x ∈ x }.
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From the definition of θ and α1 , we have
Similarly, we have
−z 13 αˆ 1 = −z 13 W1∗T S1 (X 1 ) − z 13 δ1 (X 1 ) N1 6 W ∗ 2 + 1 a 2 + 3 z 4 + 1 ε4 z ≤ 1 1 2 1 4 1 4 1 2a12 1 6 1 3 1 ≤ z 1 θ + a12 + z 14 + ε14 2 4 4 2a12 1 z 13 α1 = − 2 z 16 θˆ 2a1
(15)
+
n i
2
˜ 4+ LVm ≤ − x
m−1
m−1
z i3 z i+1 +
i=1
(17)
z l4 il4 z¯l + α¯ l f
(18)
i=2 l=2
z i3 χi+1 −
i=1
m−1 1 ˜ r 6 ˙ˆ + θ z i − θ + m−1 r 2ai2 i=1 m−1 4 χi+1 3 − − χi+1 Bi+1 (X i ) κi+1
(16)
where the inequality S1T S1 ≤ N1 is used and N1 is the dimension of S1 . Then, substituting (16) and (17) into (14) yields
˜ 4 + z 13 xˆ2 − α1 − k1 z 14 LV1 ≤ − x r 6 ˙ 1 + θ˜ z 1 − θˆ + 1 r 2a12 a4
5
m−1
ki z i4
i=1
(19)
i=1
+
a4 2
n
i
z l4 il4 z¯l + α¯ l f
i= m+1 l=m+1
3 4 3 +z m xˆm+1 + f¯m − z m 4
(20)
where f¯m (X m ) = lm x˜1 − α˙ m f + z m
n
3 a4 4 + lm z¯ m + α¯ m f . 4 2 i=m
where 1 =
a4 a4 1 1 P 8 θ 2 + P 8 ε04 + a12 + ε14. 2 2 2 4
Take the intermediate control signal αˆ m (X m ) as
By the definition of xˆ2 = z 2 + α2 f , (18) can be rewritten as
LV1 ≤ − x ˜ 4 + z 13 z 2 + α2 f − α1 − k1 z 14 r 6 ˙ 1 ˜ ˆ + θ z 1 − θ + 1 r 2a12 n i
a4 4 4 + z l il z¯l + α¯ l f . 2
˜ + LVm ≤ − x
α2 f (0) = α1 (0).
B2 (X 1 ) =
χ2 + B2 (X 1 ) κ2
4
˜ + LV1 ≤ − x −k1 z 14 +
z 13 z 2
+
z 13 χ2
1 + θ˜ r
r 6 ˙ˆ z1 − θ 2a12
+
m−1
z i3 χi+1
i=1
m−1 r 1 ˙ 4 6 − ki z i + θ˜ z i − θˆ r 2ai2 i=1 i=1 m−1 4 χi+1 3 − χi+1 Bi+1 (X i ) + m−1 − κi+1 m
+
a4 2
n
i
z l4 il4 z¯l + α¯ l f
i= m+1 l=m+1
3 4 3 xˆm+1 − αˆ m − z m +z m . 4
1 3 2 ˆ ˙ˆ z 1 z˙ 1 θ + 2 z 13 θ. 2 2a1 2a1
Then, it implies
z i3 z i+1
i=1
Let χ2 = α2 f − α1 be the output error of this filter; then one has α˙ 2 f = −(χ2 /κ2 ) and
where
m−1 i=1
In order to avoid repeatedly differentiating α1 , a new state variable α2 f is introduced, and let α1 pass through a firstorder filter with time constant κ2 to obtain α2 f as
χ˙ 2 = α˙ 2 f − α˙ 1 = −
where km > 0. Then, adding and subtracting αˆ m (X m ) in (20) yields 4
i=2 l=2
κ2 α˙ 2 f + α2 f = α1 ,
αˆ m = − km z m + f¯m
(21)
Similarly, αˆ m (X m ) can be approximated by the RBF NN Wm∗T Sm (X m ) as
i n
a4 4 4 z l il z¯l + α¯ l f + 1 . 2 i=2 l=2
Step m: (2 ≤ m ≤ n − 1). Choose the Lyapunov function candidate 1 4 1 Vm = Vm−1 + z m + χm4 . 4 4
αˆ m (X m ) = Wm∗T Sm (X m ) + δm (X m ) |δm (X m )| ≤ εm where X m ∈ X m = {X m |x ∈ x }. And we can obtain 1 6 1 2 3 4 1 4 z m θ + am + zm + εm 2 2am 2 4 4 1 3 6 ˆ θ. zm αm = − 2 z m 2am
3 −z m αˆ m ≤
(22) (23)
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Then, by substituting (22) and (23) into (21), we have ˜ 4+ LVm ≤ − x
m−1
z i3 z i+1 +
m−1
where f¯n (X n ) = ln x˜1 − α˙ n f + z n
z i3 χi+1
i=1 m r 1 ˙ 4 6 − ki z i + θ˜ z i − θˆ r 2ai2 i=1 i=1 m−1 4 χi+1 3 − χi+1 Bi+1 (X i ) − κi+1 i=1
3 xˆm+1 − αm +m + z m
a4 2
n
i
3 a4 4 + nn z¯l + α¯ l f . 4 2
Take the intermediate control signal αˆ n (X n ) as αˆ n = −(kn z n + f¯n ), where kn > 0; then, adding and subtracting αˆ n (X n ) in (26) yields
i=1
m
+
LVn ≤ − x ˜ 4+
n−1
z i3 z i+1 +
i=1
z l4 il4 z¯l + α¯ l f
(24)
i= m+1 l=m+1
where
n−1
z i3 χi+1 −
i=1
n
ki z i4
i=1
n−1 r
3 1 ˙ 6 + θ˜ z i − θˆ + z n3 u − αˆ n − z n4 2 r 4 2ai i=1 n−1 4 χi+1 3 − − χi+1 Bi+1 (X i ) + n−1 . (27) κi+1 i=1
m =
a4 2
P 8 θ 2 +
a4
P 8 ε04 +
2
m 1
2
ai2 +
i=1
m 1
4
εi4 .
i=1
Next, introduce a new variable αm+1 f , and let αm pass through a first-order filter with the constant κm+1 to obtain αm+1 f κm+1 α˙ m+1 f + αm+1 f = αm ,
αm+1 f (0) = αm (0) . (25)
Then define χm+1 = αm+1 f − αm as the output error of this filter. We have α˙ m+1 f = −(χm+1 /κm+1 ) and χ χ˙ m+1 = α˙ m+1 f − α˙ m = − m+1 + Bm+1 (X m ) κm+1 where Bm+1 (X m ) =
1 3 ˙ˆ 3 2 ˆ z m z˙ m θ + 2 z m θ. 2 2am 2am
Substituting (25) into (24) yields ˜ 4+ LVm ≤ − x
m
z i3 z i+1 +
m
z i3 χi+1
i=1
i=1 m 1 r 6 ˙ 4 ˜ ˆ − ki z i + θ z −θ r 2ai2 i i=1 i=1 m−1 4 χi+1 3 − χi+1 Bi+1 (X i ) + m − κi+1 m
i=1
+
a4 2
n
i
z l4 il4 z¯ l + α¯ l f .
i= m+1 l=m+1
Step n: Consider the following Lyapunov function: 1 1 Vn = Vn−1 + z n4 + χn4 . 4 4 Similarly, we obtain LVn = − x ˜ 4+
n−1 i=1
z i3 z i+1 +
n−1 i=1
z i3 χi+1 −
αˆ n (X n ) = Wn∗T Sn (X n ) + δn (X n ) |δn (X n )| ≤ εn where X n ∈ X n = {X n |x ∈ x }. Now, define the closed-loop state variables x c = (x, ˜ z¯ n , χ¯n , θ˜ , α¯ n f )T , where χ¯ n = (χ1 , . . . , χn ) and the initial ˜ 0 ), z¯ n (t0 ), χ¯ n (t0 ), θ˜ (t0 ), α¯ n f (t0 ))T . For condition is x 0c = (x(t ¯ n /c), ¯ n and c¯ are given below, some constant c0 > ( ¯ where let 0 = {x c |Vn ≤ c0 }, (ε) = {x c |Vn ≤ (c0 /ε)}, and the initial value of Lyapunov function Vn (t0 ) ≤ c0 . Then define the approximation region with respect to parameter ε, (0 < ε < 1) as n t 1 1 z i4 + 4 z 14 (τ ) l14 (z 1 (τ )) dτ X n (ε) = X n 4 4c 3 t −τ1 i=1 i n
a4 t 4 c0 4 . z l (τ ) il z¯l (τ ) + α¯ l f (τ ) dτ ≤ + 2 ε t −τi i=1 l=1
Therefore, X n (t0 ) ∈ X n (ε) and x c (t0 ) ∈ 0 ⊂ (ε). If x c (t) ∈ (ε), then X n (t) ∈ X n (ε), which means τε ≤ τ X n , where τε is the first time x c reaches the boundary of (ε) and τ X n is the first time X n reaches the boundary of X n (ε) . Following a similar procedure and by the definition of u, we have 1 6 1 3 1 z θ + an2 + z n4 + εn4 (28) −z n3 αˆ n ≤ 2an2 n 2 4 4 1 z n u = − 2 z n6 θˆ . (29) 2an Then, by substituting (28) and (29) into (27), when x c (t) ∈ (ε), for any t ∈ [t0 , τε ]
n−1 i=1
ki z i4
n−1
1 ˜ r 6 ˙ˆ + θ z i − θ + n−1 + z n3 u + f¯n 2 r 2ai i=1 n−1 4 χi+1 3 3 − − χi+1 Bi+1 (X i ) − z n4 (26) κi+1 4 i=1
Similar to the above steps, αˆ n (X n ) can be approximated by the RBF NN Wn∗T Sn (X n ) as
LVn ≤ − x ˜ 4+
n−1 i=1
z i3 z i+1 +
n−1
z i3 χi+1
i=1 n r 1 ˙ 4 6 ˜ ˆ − ki z i + θ zi − θ r 2ai2 i=1 i=1 n−1 4 χi+1 3 − χi+1 Bi+1 (X i ) + n − κi+1 n
i=1
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7
3
where n =
a4 2
P 8 θ 2 +
a4 2
P 8 ε04 +
n 1
2
ai2 +
i=1
n 1
4
εi4 .
n−1
z i3 z i+1 +
i=1
+
k0 ˜ ˆ θθ − r
n−1 4 χi+1 i=1
κi+1
n−1
z i3 χi+1 −
i=1
n
ki z i4 + n
3 − χi+1 Bi+1 (X i ) .
0 −1
i=1
(30)
−2 −3
3 1 4 z i3 z i+1 ≤ z i4 + z i+1 4 4 3 4 1 4 3 z i χi+1 ≤ z i + χi+1 4 4 3 4 43 4 1 3 3 χi+1 Bi+1 ≤ π Bi+1 χi+1 + 4 4π 4 1 1 (31) θ˜ θˆ = θ˜ θ − θ˜ ≤ − θ˜ 2 + θ 2 2 2 where π > 0 is a design constant and Bi+1 is a continuous function. Therefore, there exists a positive constant Ni+1 such that |Bi+1 | ≤ Ni+1 . Substituting (31) into (30), one has n 7 4 LVn ≤ − x ˜ 4− ki − z 4 i i=1 n−1 1 1 3 4 43 4 χi+1 + − − π 3 Bi+1 ki+1 4 4 i=1
k0 ˜ 2 ¯n θ + 2r k 1 ¯ n = n + 0 θ 2 + . 2r 4π 4 −
Let >0 and denote 2 7 c = min , , 4 ki − aλ2max (P) 4 1 1 3 4 43 3 4 − − π Bi+1 , k0 i = 1, . . . , n. ki+1 4 4
2 1
By utilizing Lemma 1, we have
Then one has
1
i=1
By the definition of θˆ˙ , we can get ˜ 4+ LVn ≤ − x
x
¯ n. LVn ≤ −c Vn − VQ 1 + Then, there exists a positive constant c¯ such that ¯ n , ∇x c ∈ (ε). ¯ n + LVn ≤ −cV ¯ n /c) ¯ and τε ≤ τ X n , it can be concluded Considering c0 > ( that all the signals in the closed-loop system are M-SGUUB with probability 1 − ε in (ε), i.e., inf x0c ∈0 P {τε = ∞} ≥ 1−ε, which are the desired results and the proof is completed. Remark 4: In the process of controller design, the DSC technique is employed to make the control scheme simple. That is because the repeated differentiation of virtual control αi is replaced by α˙ i+1 f and αi+1 f is defined by a first-order
0
Fig. 1.
10
20
30 Time(sec)
40
50
60
Trajectory of system output y.
filter with αi as input. Therefore, the repeated differentiation of αi is avoided. Remark 5: By using the method of direct adaptive NN control, there is only one adaptive parameter θˆ in the virtual control αi and true control u. Therefore, the computation burden is greatly reduced. In addition, it is not necessary to a priori know the centers of the receptive field and the width of the Gaussian functions, which makes the obtained theory more useful in practice. III. S IMULATION R ESULTS In this section, a simulation example is presented to demonstrate the effectiveness of the proposed adaptive NN control method. Consider the following stochastic nonlinear time-delay system: x 13 (t − τ1 ) 2 dt dx 1 = x 2 − 0.3x 1 + x 1 + 1 + x 12 (t − τ1 ) +x 1 sin (x 1 ) dw x 14 (t − τ2 ) sin (x 2 ) 0.2 dt + dx 2 = u + 0.1x 2 sin 1 + x 12 1 + x 12 (t − τ2 ) x1 + dw 1 + x 12 y = x1 where the nonlinear functions are f 1 (x 1 (t)) = −10x 1 + x 12 , f 2 (x¯2 (t)) = 0.1x 2 sin 0.2/(1 + x 12 ) , nonlinear time-delay terms are defined as h 1 (x¯1 (t − τ1 )) = (x 13 (t − τ1 )/ 1 + x 12 (t − τ1 )), and h 2 (x¯2 (t − τ2 )) = (x 14 (t − τ2 ) sin (x 2 )/ 1 + x 12 (t − τ2 )), the initial states are chosen as x 1 (0) = 0.2, x 2 (0) = 0.5, and the observer is designed as .
xˆ 1 = xˆ2 + l1 y − xˆ1 .
xˆ 2 = u + l2 y − xˆ1 . According to Theorem 1, the virtual control function α1 and the true control law u are chosen, respectively, as 1 1 α1 = − 2 z 13 θˆ , u = − 2 z 23 θˆ (32) 2a1 2a2
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80
1000
x2
800
40
700
20
600
0
500
−20
400 300
−40
200
−60 −80
100 0
Fig. 2.
10
20
30 Time(sec)
40
50
0
60
Trajectory of system state x2 .
Fig. 5.
80
α
2f
0
10
20
30 40 Time(sec)
50
60
Adaptive parameter θˆ .
In the simulation, the time delays are chosen as τ1 = τ2 = 0.5, therefore the upper bound of time delay is chosen as τm = 0.5, and the design parameters are chosen as l1 = l2 = 100, a1 = a2 = 0.21, r = 22.5, k0 = 0.01, and κ2 = 0.1. The simulation results are illustrated in Figs. 1–5, respectively. Figs. 1 and 2 show the system output y and state variable x 2 . Fig. 3 illustrates a new state variable of a first-order filter. Fig. 4 depicts the trajectory of input u, while Fig. 5 illustrates the trajectory of adaptive parameter θˆ .
60 40 20 0 −20 −40 −60
IV. C ONCLUSION
−80 0
10
20
30 Time(sec)
40
50
60
Trajectory of a state variable of a first-order filter α2 f .
Fig. 3.
3000
u
2000 1000 0 −1000 −2000 −3000
Fig. 4.
θˆ
900
60
In this paper, the problem of adaptive NN DSC was investigated for a class of nonlinear stochastic strict-feedback systems with unknown time delays. By using the backstepping approach, a direct adaptive NN control scheme was proposed. In addition, by combining the method of output-feedback control and DSC, the results obtained could be applied to solve the control problem for systems with unmeasurable states, and the problem of explosion of complexity could be avoided. In addition, as the number of the online adaptive parameters was only one, the computation burden could be reduced accordingly. Therefore, it is convenient to implement this algorithm in practical systems. Finally, a simulation example was presented to illustrate the effectiveness of the method proposed. R EFERENCES
0
10
20
30 Time(sec)
40
50
60
Trajectory of control input u.
where z 1 = y, z 2 = xˆ2 − α2 f . The adaptive law is given as θ˙ˆ =
2 r 6 z − k0 θˆ . 2 i 2a i i=1
(33)
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Qi Zhou received the B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, respectively. She is currently pursuing the Ph.D. degree with the international collaboration program, University of Portsmouth, Portsmouth, U.K., and the Nanjing University of Science and Technology, Nanjing, China. Her current research interests include fuzzy control, stochastic control, and robust control.
Peng Shi (M’95–SM’98) received the B.Sc. degree in mathematics from the Harbin Institute of Technology, Harbin, China, the M.Eng. degree in systems engineering from Harbin Engineering University, Harbin, the Ph.D. degree in electrical engineering from the University of Newcastle, Callaghan, Australia, the Ph.D. degree in mathematics from the University of South Australia, Adelaide, Australia, and the D.Sc. degree from the University of Glamorgan, Pontypridd, U.K. He was a Lecturer with the University of South Australia and a Senior Scientist with the Defence Science and Technology Organisation, Edinburgh, Australia. He was a Professor with the University of Glamorgan, Victoria University, Melbourne, Australia, and is currently with the University of Adelaide, Adelaide. He has authored or co-authored widely in these areas. His current research interests include system and control theory, computational and intelligent systems, and operational research. Dr. Shi is a fellow of the Institution of Engineering and Technology, U.K., and the Institute of Mathematics and its Applications, U.K. He is on the editorial board of a number of international journals, such as the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE T RANSACTIONS ON S YSTEMS , M AN AND C YBERNETICS -PART B, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS .
Shengyuan Xu received the B.Sc. degree from the Hangzhou Normal University, Hangzhou, China, the M.Sc. degree from Qufu Normal University, Qufu, China, and the Ph.D. degree from the Nanjing University of Science and Technology, Nanjing, China, in 1990, 1996, and 1999, respectively. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, from 1999 to 2000. From 2000 to 2002, he was a Post-Doctoral Researcher in CESAME with the Université catholique de Louvain, Louvain-La-Neuve, Belgium, and the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, respectively. From 2002 to 2004, he was a William Mong Young Researcher and an Honorary Associate Professor with the Department of Mechanical Engineering, University of Hong Kong. Since 2002, he has been with the School of Automation, Nanjing University of Science and Technology as a Professor. His current research interests include robust filtering and control, singular systems, time-delay systems, neural networks, and multidimensional systems and nonlinear systems. Dr. Xu was a recipient of the National Excellent Doctoral Dissertation Award from the Ministry of Education of China in 2002. He received a grant from the National Science Foundation for Distinguished Young Scholars of China, in 2006. He was awarded a Cheung Kong Professorship from the Ministry of Education of China in 2008. He is a member of the Editorial Boards of the Multidimensional Systems and Signal Processing, and the Circuits Systems and Signal Processing.
Hongyi Li received the B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, respectively, and the Ph.D. degree in intelligent control from the University of Portsmouth, Portsmouth, U.K., in 2012. He is currently with the College of Information Science and Technology, Bohai University. His current research interests include fuzzy control, robust control, and their applications in suspension systems.
