International Journal of Automation and Computing 8(4), November 2011, 403-410

DOI: 10.1007/s11633-011-0597-x

Adaptive Iterative Learning Control for NonlinearTime-delay Systems with Periodic Disturbances

Using FSE-neural Network

Chun-Li Zhang Jun-Min LiDepartment of Applied Mathematics, Xidian University, Xi′an 710071, PRC

Abstract: An adaptive iterative learning control scheme is presented for a class of strict-feedback nonlinear time-delay systems, withunknown nonlinearly parameterised and time-varying disturbed functions of known periods. Radial basis function neural network andFourier series expansion (FSE) are combined into a new function approximator to model each suitable disturbed function in systems.The requirement of the traditional iterative learning control algorithm on the nonlinear functions (such as global Lipschitz condition) isrelaxed. Furthermore, by using appropriate Lyapunov-Krasovskii functionals, all signs in the closed loop system are guaranteed to besemiglobally uniformly ultimately bounded, and the output of the system is proved to converge to the desired trajectory. A simulationexample is provided to illustrate the effectiveness of the control scheme.

Keywords: Adaptive control, iterative learning control (ILC), time-delay systems, Fourier series expansion-neural network, periodicdisturbances.

1 Introduction

During the past decades, iterative learning control (ILC)has been shown to be one of the most effective controlstrategies in dealing with repeated tracking control or pe-riodic disturbance rejection for nonlinear dynamic systems.ILC is a trajectory-tracking improvement technique for sys-tems performing a prescribed task. The ILC system im-proves its control performance by a self-tuning process with-out using an accurate system model, and can be applied topractical applications, such as the control of robotics, servomotors, etc.

Based on Lyapunov theory[1], the ILC is incorporatedwith the adaptive control, and an adaptive ILC has beendeveloped. The design method on adaptive learning lawis divided into three aspects: time domain[1−3], iterativedomain[4, 5], and mixed domain[6]. A novel adaptive ILCmethod is proposed via Lyapunov technique in [1], by in-corporating continuous learning along each iteration, andit has shown that, the assumptions on minimum phase andrelative degree for ILC can be relaxed. In [2], an adap-tive ILC scheme is presented for uncertain robotic systems.Based on the universal adaptive scheme and high gain con-cepts, a full proof of convergence of the adaptive ILC schemeis given for linear systems in [3]. In [4], by a robust ILCwith a dead-zone scheme, the tracking system converges tothe error bound within finite iterations. In [7], a hybridadaptive iterative learning control for time-varying nonlin-ear systems is proposed.

Practically speaking, time delays are frequently encoun-tered in many fields of science and engineering, includingin communications network, manufacturing systems, bio-logical systems, and economics. In [8], a novel adaptive

Manuscript received August 14, 2010; revised January 24, 2011This work was supported by National Natural Science Foundation

of China (No. 72103676) (partially supported by the FundamentalResearch Funds for the Central Universities).

neural network (NN) output-feedback regulation algorithmfor a class of nonlinear time-varying time-delay systems isproposed. In [9], a novel output-feedback adaptive learningcontrol approach is developed for a class of linear time-delaysystems. In [10], an open-closed-loop proportional-integral-derivative (PID) type learning algorithm is studied for theiterative learning control of linear time-delay systems. In[11], a two-dimensional system theory based on ILC meth-ods have been discussed for linear continuous multi-variablesystems with time delays in state or with time delays in in-put. Necessary and sufficient conditions are given for con-vergence of the proposed ILC rules. In [12], a new ILCalgorithm for a class of linear dynamic systems with timedelay is proposed using the holding mechanism, and theconvergence of the proposed algorithm is given.

In [13], an adaptive iterative learning control scheme ispresented for a class of strict-feedback nonlinear time-delaysystems. The NN is introduced for iterative learning con-trol. However, the proposed algorithm in the paper is notapplied in strict-feedback nonlinear time-delay systems withunknown non-linearly parameterised and time-varying dis-turbed functions. In [14, 15], the adaptive NN tracking con-trol problem for the strict-feedback nonlinear time-delaysystems with an unknown non-linearly parameterised, andtime-varying disturbed function is considered. However,Chen[14] proposed an adaptive backstepping dynamic sur-face control method, and the systems in the paper did notconcern time-delay issues. Zhu et al.[16] proposed an adap-tive iterative learning control for a class of strict feedbacknonlinear time-varying systems, but the systems in the pa-per also did not concern time-delay terms and unknownnonlinearly parameterised terms. Adaptive iterative learn-ing control for time-delay systems with unknown functionsaffected by the unmeasured time-varying disturbances, tothe best of the author′s knowledge, has not been reportedin the literature at present.

Motivated by the above discussion, an adaptive itera-

404 International Journal of Automation and Computing 8(4), November 2011

tive learning control design approach is proposed for a classof time-delay systems with unknown functions affected bythe unmeasured time-varying disturbances. We adopt twoFourier series expansion (FSE) neural networks to approx-imate unknown smooth functions with time-varying dis-turbances, and unknown time delay functions with time-varying disturbances. The restricting condition on the non-linear functions (such as global Lipschitz condition) is re-laxed.

The rest of the paper is organized as follows. The prob-lem formulation and preliminaries are given in Section 2. InSection 3, an adaptive iterative learning controller designscheme is presented. Section 4 gives the stability analysisof a closed loop system and the main results of this paper.In Section 5, a simulation example is provided to illustratethe effectiveness of the proposed controller. Finally, a con-clusion is given in Section 6.

2 Problem formulation and preliminar-ies

2.1 System description and problem for-mulation

Consider a class of periodically disturbed and non-linearly parameterised time-delay systems with the follow-ing strict-feedback form:

xi,k(t)=xi+1,k(t) + fi(xi,k(t), θi,k(t)) +

hi(yk(t− τ), θi,k(t− τ)) (1 6 i 6 n− 1)

xn,k(t)=uk(t) + fn(xn,k, θn,k(t)) +

hn(yk(t− τ), θn,k(t− τ))

yk(t)=x1,k(t) (1)

where xi,k = [x1,k, x2,k, · · · , xi,k] ∈ Ri, i = 1, 2, · · · , n,uk ∈ R, yk ∈ R are state variables, system input, andoutput, respectively. fi(·), hi(·) (i = 1, 2, · · · , n) are un-known smooth functions, and τ is a known time delayconstant. θi,k = [ϑi1,k, · · · , ϑim,k]T ∈ Ωθ ⊂ Rmi(1 6i 6 n) are unknown and continuously time-varying distur-bances with known period T , that is, θi,k(t + T ) = θi,k(t);θi,k(t− τ + T ) = θi,k(t− τ). Here, k > 1 denotes the indexof iteration and t ∈ [0, T ].

The control objective is to design an adaptive iterativelearning controller for systems (1) in specified time interval[0, T ], such that

1) all the signals in the closed-loop remain semi-globallyuniformly ultimately bounded;

2) the output yk(t) follows a desired trajectory yd com-pletely as k approaches to infinity, where the derivatives ofthe desired trajectory yd up to the n-th order are bounded.

Define Zi,k = [x1,k, x2,k, · · · , xi,k]T ∈ ΩZi,k ⊂ Ri, ΩZi,k

is a known compact set.

2.2 Function approximation using FSE-RBF neural networks and preliminar-ies

In this paper, because the periodic disturbances θi,k(t),θi,k(t − τ) are not used as radial basis function inputs, weconsider the estimates of θi,k(t), θi,k(t − τ). The follow-

ing FSE-RBFNN[14] is used to approximate the unknowncontinuous function. We first employ FSE to estimateθi,k(t), θi,k(t − τ), and then employ the measured systemsignals χi,k and the estimated values of θi,k(t), θi,k(t−τ) asthe RBFNN inputs to approximate some suitable unknownfunctions ~i(χi,k(t), θi,k(t)), ~i(χi,k(t− τ), θi,k(t− τ)).

Without loss of generality, we consider an unknown func-tion ~(χ, θ(t)), where χ ∈ Ωχ ⊂ Rl is a measured signalwith Ωχ a compact set, and θ(t) = [ϑ1(t), · · · , ϑm(t)]T ∈Ωθ ⊂ Rm is an unknown continuous disturbance vector ofknown period T with Ωθ a compact set. On the other hand,the continuous and periodic disturbance vector θ(t) can alsobe expressed by a linearly parameterised FSE as[17]

θ(t) = MTΦ(t) + δθ(t), ‖δθ(t)‖ 6 δθ (2)

where M = [υ1, · · · , υm] ∈ Rq×m is a constant matrix withυi ∈ Rq being a vector consisting of the first q coefficientsof the FSE of ϑij(t) ( q is an odd integer), δθ(t) is the trun-cation error with the minimum upper bound δθ > 0, whichcan be arbitrarily decreased by increasing q, and Φ(t) =[φ1(t), · · · , φq(t)]

T with φ1(t) = 1, φ2j(t) =√

2 sin(2πjt/T )and φ2j+1(t) =

√2 cos(2πjt/T ), j = 1, · · · , (q−1)/2, whose

derivatives up to the n-order are smooth and bounded.On the other hand, if θ(t) is measured, the unknown

continuous function ~(χ, θ(t)) can be approximated over thecompact set Ω = Ωχ × Ωθ by an RFBNN as follows[18]:

~(χ, θ(t)) = WTS(χ, θ(t)) + δ~(χ, θ(t)), |δ~(χ, θ(t))| 6 δ~(3)

where S(χ, θ(t)) = [s1(χ, θ(t)), · · · , sp(χ, θ(t))]T is aknown smooth vector-valued function with the compo-nent sj(χ, θ(t)) = exp[−‖z − µj‖2/η2] (j = 1, · · · , p),here z = [χT, θT(t)]T, µj ∈ Ω is a constant, that iscalled the center of sj(χ, θ(t)), and η > 0 is a realnumber that is called the width of sj(χ, θ(t)). The op-timal weight vector W = [ω1, · · · , ωp]T is defined asW = arg minW∈Rp sup(χ,θ(t))∈Ω|~(χ, θ(t))− WTS(χ, θ(t))|,and δ~(χ, θ(t)) is the inherent NN approximation error withthe minimum upper bound δ~ > 0, which can be decreasedby increasing the NN node number p.

However, since θ(t) is unknown, by replacing θ(t) in (3)with (2), we have

~(χ, θ(t)) = WTS(χ, MTΦ(t) + δθ) + δ~. (4)

Based on (4), we construct a novel FSE-RBFNN-basedapproximator

G(χ, t) = WTS(χ, MTΦ(t)) (5)

to model the unknown function ~(χ, W, M, θ(t)) as

~(χ, θ(t)) = WTS(χ, MTΦ(t)) + δ(χ, t) (6)

where

δ(χ, t) = δ~ + WTS(χ, MTΦ(t) + δθ)−WTS(χ, MTΦ(t)).(7)

Assumption 1. On the compact set ΩZi,k , the ideal NNweights W , M are bounded by ‖W‖ 6 wm, ‖M‖ 6 ma,with wm, ma being positive constants.

Lemma 1[14]. For (χ, W, M, θ(t)) ∈ Ω, the approxima-tion error δ(χ, t) in (7) satisfies

|δ(χ, t)| 6 δ (8)

C. L. Zhang and J. M. Li / Adaptive Iterative Learning Control · · · 405

where δ denotes the minimum upper bound of δ(χ, t), whichcan be arbitrarily decreased by increasing the values of pand q.

In general, parameters W and M are unknown and needto be estimated in controller design. Let W and M bethe estimates of W and M , respectively, and let the weightparameter estimation errors be W = W − W and M =M −M .

Lemma 2[14]. For approximator (5), the estimation errorcan be expressed as

WTS(χ, MTΦ(t))− WTS(χ, MTΦ(t)) =

WT(S(χ, MTΦ(t))− S′MTΦ(t)) +

WTS′MTΦ(t) + d (9)

where S′ = [s′1, s′2, · · · , s′p] ∈ Rm×p with s′i =

(∂si(χ, ω))/∂ω|ω=MΦ(t), i = 1, · · · , p, and the residual termd is bounded by

|d| 6 ‖M‖F ‖Φ(t)WTS′‖F +‖W‖‖S′MTΦ(t)‖+|W |1. (10)

In the following, we let ‖ · ‖ denote the 2-norm. In orderto design the controller, we give a definition called series-convergence sequence.

Definition 1[16]. Series-convergence sequence ∆k de-fined as ∆k = q/km, where k = 1, 2, 3, · · · ; q and m areconstants and satisfy q > 0 ∈ R, m > 2 ∈ N.

From this definition, we obtain that k →∞, ∆k → 0.Lemma 3[16]. For a given sequence q/km, where k =

1, 2, · · · ; m > 2 is a positive integer, the following inequalityis established: limk→∞

∑ki=1(q/im) 6 2q.

3 Adaptive iterative learning controllerdesign

Here, we present the design steps for system (1). Forclarity and conciseness, Step 1 is described with detailedexplanations, while Step i and Step n are simplified, withthe relevant equations and the explanations being omitted.

Step 1. Define z1,k = x1,k − yd. Its derivative is

z1,k(t)=x2,k + f1(x1,k(t), θ1,k(t)) +

h1(yk(t− τ), θ1,k(t− τ))− yd(t). (11)

By viewing x2,k as a virtual control input, iff1(x1,k(t), θ1,k(t)), h1(yk(t− τ), θ1,k(t− τ)) are known, andwe choose α1,k

∗ , x2,k as control input for the z1,k-subsystem in the above equation, and consider the Lya-punov function candidate V11,k = (1/2)z2

1,k, whose deriva-tive is

V11,k = z1,k · z1,k =

z1,k[α∗1,k + f1(x1,k(t), θ1,k(t)) +

h1(yk(t− τ), θ1,k(t− τ))− yd(t)]. (12)

Let us choose feedback controller α∗1,k as follows:

α∗1,k =−a11z1,k − [f1(x1,k(t), θ1,k(t)) +

h1(yk(t− τ), θ1,k(t− τ))− yd(t)] (13)

where a11 > 0. Substituting (13) into (12), we can giveV11,k = −a11z

21,k 6 0. Therefore, z1,k is asymptotically

stable.

Because f1(x1,k(t), θ1,k(t)) and h1(yk(t− τ), θ1,k(t− τ))are unknown, the desired feedback control α∗1,k cannot beimplemented in practice. Instead, two FSE-RBF-neuralnetworks are adopted to approximate the unknown smoothfunctions f1(x1,k(t), θ1,k(t)) and h1(yk(t − τ), θ1,k(t − τ)),i.e.,

f1(x1,k(t), θ1,k(t))=WT11,kS11(Z1,k, MT

1,kΦ1(t)) +

δ11,k,

h1(yk(t− τ), θ1,k(t− τ))=WT12,kS12(yk(t− τ),

MT1,kΦ1(t− τ)) + δ12,k (14)

where Z1,k = [x1,k]T ⊂ R1, W11,k and W12,k

are the optimal weight vectors of f1(x1,k(t), θ1,k(t))and h1(yk(t − τ), θ1,k(t − τ)), respectively. M1,k =[v11,k, v12,k, · · · , v1m,k] ∈ Rq×m is a constant matrix, withv1i,k ∈ Rq being a vector consisting of the first q coeffi-cients of the FSE of ϑ1i,k (q is an odd integer), and Φ1(t) =[φ1(t), · · · , φq(t)]

T with φ1(t) = 1, φ2j(t) =√

2 sin(2πjt/T )and φ2j+1(t) =

√2 cos(2πjt/T ), j = 1, · · · , (q − 1)/2,

whose derivatives up to n-order are smooth and bounded;Φ1(t − τ) = [φ1(t − τ), · · · , φq(t − τ)]T with φ1(t − τ) =1, φ2j(t − τ) =

√2 sin(2πj(t − τ)/T ) and φ2j+1(t − τ) =√

2 cos(2πj(t − τ)/T ), j = 1, · · · , (q − 1)/2, whose deriva-tives up to n-order are also smooth and bounded. Theneural reconstruction error e1,k = δ11,k + δ12,k is bounded,i.e., there exists a constant ε1,k > 0 such that |e1,k| 6 ε1,k.Throughout this paper, we shall define the reconstructionerrors as ei,k = δi1,k + δi2,k, where i = 1, · · · , n. Like in thecase of e1,k, ei,k is also bounded, i.e., |ei,k| 6 εi,k. BecauseW11,k, W12,k, and M1,k are unknown, let W11,k, W12,k, andM1,k be the estimates of W11,k, W12,k, and M1,k, respec-tively.

According to the main result stated in [19], any real-valued continuous function can be arbitrarily closely ap-proximated by a network of RBF types over a compact set.The compactness of set Ω0

Z1,kis a must to guarantee the

feasibility of neural network approximation, which is shownin the following lemma.

Lemma 4[20]. Set Ω0Z1,k

is a compact set.Define error variable z2,k = x2,k − α1,k, β1,k =

η1,k tanh(η1,kz1,k/∆k) and choose the virtual control

α1,k =−a11z1,k − WT11,kS11(Z1,k, MT

1,kΦ1(t))−WT

12,kS12(yk(t− τ), MT1,kΦ1(t− τ)) +

yd(t)− β1,k. (15)

By Lemma 2, z1,k can be obtained by

z1,k = z2,k + α1,k + f1(x1,k(t), θ1,k(t)) + h1(yk(t− τ),

θ1,k(t− τ))− yd(t) =

z2,k − a11z1,k − WT11,kS11(Z1,k, MT

1,kΦ1(t))−WT

12,kS12(yk(t− τ), MT1,kΦ1(t− τ)) +

WT11,kS11(Z1,k, MT

1,kΦ1(t))− β1,k +

WT12,kS12(yk(t− τ), MT

1,kΦ1(t− τ)) + e1,k =

z2,k − a11z1,k − WT11,k(S11(Z1,k, MT

1,kΦ1(t))−S′11,kMT

1,kΦ1(t))− WT11,kS

′11,kMT

1,kΦ1(t)−WT

12,k(S12(yk(t− τ), MT1,kΦ1(t− τ))−

406 International Journal of Automation and Computing 8(4), November 2011

S′12,kMT

1,kΦ1(t− τ)) + e1,k − d11,k − d12,k −WT

12,kS′12,kMT

1,kΦ1(t− τ)− β1,k (16)

where W11,k = W11,k − W11,k, W12,k = W12,k − W12,k,V1,k = M1,k −M1,k.

Throughout this paper, we shall define (·) = (·) −(·). And S

′11,k = [s

′11,k1, s

′11,k2, · · · , s

′11,kp] ∈ Rm×p

with s′11,ki = (∂s

′11,ki(Z1,k, θ1,k))/∂θ1,k|θ1,k=MT

1,kΦ1(t), i =

1, · · · , p; S′12,k = [s

′12,k1, s

′12,k2, · · · , s

′12,kp] ∈ Rm×p

with s′12,ki = (∂s

′12,ki(yk(t − τ), θ1,k(t − τ)))/∂θ1,k(t −

τ)|θ1,k(t−τ)=MT1,k

Φ1(t−τ), i = 1, · · · , p. The residual terms

−d11,k and −d12,k are bounded by

| − d11,k|6 ‖M1,k‖F ‖Φ1(t)WT11,kS

′11,k‖F +

‖W11,k‖‖S′11,kMT

1,kΦ1(t)‖+ |W11,k|1 6m1a‖Φ1(t)W

T11,kS

′11,k‖F +

w11m‖S′11,kMT

1,kΦ1(t)‖+ w11m;

| − d12,k|6 ‖M1,k‖F ‖Φ1(t− τ)WT12,kS

′12,k‖F +

‖W12,k‖‖S′12M

T1,kΦ1(t− τ)‖+ |W12,k|1 6

m1a‖Φ1(t− τ)WT12,kS

′12,k‖F +

w12m‖S′12M

T1,kΦ1(t− τ)‖+ w12m

then |e1,k − d11,k − d12,k| 6 η1,k, where

η1,k = ε1,k + m1a‖Φ1(t)WT11,kS

′11,k‖F +

w11m‖S′11,kMT

1,kΦ1(t)‖+ w11m +

m1a‖Φ1(t− τ)WT12,kS

′12,k‖F +

w12m‖S′12M

T1,kΦ1(t− τ)‖+ w12m.

Throughout this paper, we shall define the residual terms−di1,k and −di2,k as bounded by

| − di1,k|6mia‖Φi(t)WTi1,kS

′i1,k‖F +

wi1m‖S′i1,kMT

i,kΦi(t)‖+ wi1m;

| − di2,k|6mia‖Φi(t− τ)WTi2,kS

′i2,k‖F −

wi2m‖S′i2,kMT

i,kΦi(t− τ)‖+ wi2m

then |ei,k − di1,k − di2,k| 6 ηi,k, where

ηi,k = εi,k + mia‖Φi(t)WTi1,kS

′i1,k‖F +

wi1m‖S′i1,kMT

i,kΦi(t)‖+ wi1m +

mia‖Φi(t− τ)WTi2,kS

′i2,k‖F +

wi2m‖S′i2,kMT

i,kΦi(t− τ)‖+ wi2m

with mia, wi1m, and wi2m, i = 1, · · · , n, being positive con-stants.

Consider the following Lyapunov-Krasovskii functioncandidate:

V1,k =1

2z21,k +

1

2WT

11,kΓ−111 W11,k +

1

2WT

12,kΓ−112 W12,k +

1

2MT

1,kΓ−113 M1,k (17)

where Γ1j = ΓT1j > 0 (j = 1, 2, 3) are adaptive gain matri-

ces. The derivative of V1,k is

V1,k = z1,k · z1,k + WT11,kΓ−1

11˙

W11,k + WT12,kΓ−1

12˙

W12,k +

MT1,kΓ−1

13˙

M1,k =

z1,kz2,k − a11z21,k + WT

11,kΓ−111 [

˙W11,k −

Γ11(S11(Z1,k, MT1,kΦ1(t))− S

′11,kMT

1,kΦ1(t))z1,k] +

z1,k(e1,k − d11,k − d12,k)− z1,kβ1,k +

WT12,kΓ−1

12 [˙

W12,k − Γ12(S12(yk(t− τ),

MT1,kΦ1(t− τ))− S

′12,kMT

1,kΦ1(t− τ))z1,k] +

MT1,kΓ−1

13 [˙

M1,k − Γ13(Φ1(t)W11,kS′11,k +

Φ1(t− τ)W12,kS′12,k)z1,k] (18)

for z1,k ∈ Ω0Z1,k

.Then consider the following adaptation laws:

˙W11,k = ˙W11,k =

Γ11[(S11(Z1,k, MT1,kΦ1(t))−

S′11,kMT

1,kΦ1(t))z1,k]

˙W12,k = ˙W12,k =

Γ12[(S12(yk(t− τ), MT1,kΦ1(t− τ))−

S′12,kMT

1,kΦ1(t− τ))z1,k]

˙M1,k = ˙M1,k =

Γ13[(Φ1(t)W11,kS′11,k +

Φ1(t− τ)W12,kS′12,k)z1,k]. (19)

Substituting (19) into (18), we obtain

V1,k 6 z1,kz2,k − a11z21,k + |z1,kη1,k| − z1,kβ1,k 6

z1,kz2,k − a11z21,k + δ∆k. (20)

The coupling term z1,k, z2,k will be canceled in the nextstep.

Remark 1. A property of hyperbolic tangent functionis used in deducing previous inequality (20). For ∀ ∆ > 0and a ∈ R, there exists the following inequality 0 6 |a| −a tanh(a/∆) 6 δ∆, where δ is a constant and satisfies δ =e−(δ+1).

Step iii (2666 iii 666 nnn − 111). The derivative of zi,k = xi,k −αi−1,k is

zi,k =xi+1,k + fi(xi,k(t), θi,k(t)) + hi(yk(t− τ),

θi,k(t− τ))− αi−1,k.

Similarly, choose the virtual control

αi,k =−zi−1,k − aiizi,k − WTi1,kSi1(Zi,k, MT

i,kΦi(t))−WT

i2,kSi2(yk(t− τ), MTi,kΦi(t− τ)) +

αi−1,k − βi,k (21)

where

Zi,k = [x1,k, x2,k, · · · , xi,k]T ⊂ Ri,

βi,k = ηi,k tanh[ηi,kzi,k

∆k]

C. L. Zhang and J. M. Li / Adaptive Iterative Learning Control · · · 407

αi−1,k =

i∑j=1

∂αi−1,k

∂xj,k+

i−1∑j=1

(∂αi−1,k

∂Wj1,k

˙Wj1,k +

∂αi−1,k

∂Wj2,k

˙Wj2,k +

∂αi−1,k

∂Mj,k

˙Mj,k

).

Then, we have

zi,k = zi+1,k + αi,k + fi(xi,k(t), θi,k(t)) + hi(yk(t− τ),

θi,k(t− τ))− αi−1,k =

zi+1,k − zi−1,k − aiizi,k − WTi1,k(Si1(Zi,k, MT

i,kΦi(t))−S′i1,kMT

i,kΦi(t))− WTi1,kS

′i1,kMT

i,kΦi(t)−WT

i2,k(Si2(yk(t− τ), MTi,kΦi(t− τ))−

S′i2,kMT

i,kΦi(t− τ))− WTi2,kS

′i2,kMT

i,kΦi(t− τ) +

ei,k − di1,k − di2,k − βi,k, (22)

where zi+1,k = xi+1,k − αi,k.Consider the Lyapunov-Krasovskii function candidate

Vi,k =Vi−1,k +1

2z2

i,k +1

2WT

i1,kΓ−1i1 Wi1,k +

1

2WT

i2,kΓ−1i2 Wi2,k +

1

2MT

i,kΓ−1i3 Mi,k (23)

where Γij = ΓTij > 0 (j = 1, 2, 3) are adaptive gain matri-

ces.Consider the following adaption laws:

˙Wi1,k =Γi1[(Si1(Zi,k, MT

i,kΦi(t))−S′i1,kMT

i,kΦi(t))zi,k];

˙Wi2,k =Γi2[(Si2(yk(t− τ), MT

i,kΦi(t− τ))−ˆS′i2,kMT

i,kΦi(t− τ))zi,k];

˙Mi,k =Γi3[(Φi(t)Wi1,kS

′i1,k +

Φi(t− τ)Wi2,kS′i2,k)zi,k]. (24)

By using (21), (22), and (24), the derivative of Vi,k be-comes

Vi,k 6 zi,kzi+1,k −i∑

j=1

ajjz2j,k + iδ∆k. (25)

Step nnn. This is the final step. The derivative of zn,k =xn,k −αn−1,k is zn,k = uk + fn(xn,k(t), θn,k(t))+ hn(yk(t−τ), θn,k(t− τ))− αn−1,k.

Similarly, choosing the practical control law as

uk =−zn−1,k − annzn,k − WTn1,kSn1(Zn,k, MT

n,kΦn(t))−WT

n2,kSn2(yk(t− τ), MTn,kΦn(t− τ)) +

αn−1,k − βn,k (26)

where Zn,k = [x1,k, x2,k, · · · , xn,k]T ⊂ Ri, βn,k =ηn,k tanh(ηn,kzn,k/∆k).

Then, we have

zn,k =uk + fn(xn,k(t), θn,k(t)) + hn(yk(t− τ),

θn,k(t− τ))− αn−1,k =

−zn−1,k − annzn,k − WTn1,k(Sn1(Zn,k, MT

n,kΦn(t))−

S′n1,kMT

n,kΦn(t))− WTn1,kS

′n1,kMT

n,kΦn(t)−WT

n2,k(Sn2(yk(t− τ), MTn,kΦn(t− τ))−

S′n2,kMT

n,kΦn(t− τ))−WT

n2,kS′n2,kMT

n,kΦn(t− τ) +

en,k − dn1,k − dn2,k − βn,k. (27)

Consider the overall Lyapunov-Krasovskii function can-didate

Vn,k = Vn−1,k +1

2z2

n,k +1

2WT

n1,kΓ−1n1 Wn1,k +

1

2WT

n2,kΓ−1n2 Wn2,k +

1

2MT

n,kΓ−1n3 Mk

n,k.

(28)

Consider the following adaption laws:

˙Wn1,k =Γn1[(Sn1(Zn,k, MT

n,kΦn(t))−S′n1,kMT

n,kΦn(t))zn,k]

˙Wn2,k =Γn2[(Sn2(yk(t− τ), MT

n,kΦn(t− τ))−S′n2,kMT

n,kΦn(t− τ))zn,k]

˙Mn,k =Γn3[(Φn(t)Wn1,kS

′n1,k +

Φn(t− τ)Wn2,kS′n2,k)zn,k]. (29)

By using (26), (27), and (29), the derivative of Vn,k be-comes

Vn,k 6−n∑

j=1

ajjz2j,k + nδ∆k. (30)

4 Stability analysis

The following theorem shows the stability and controlperformance of the closed-loop adaptive system.

Theorem. Consider the closed-loop system consistingof the plant (1), the controller (26), and the NN weightupdating laws (19), (24), and (29). Assume that there existsufficiently large compact sets Ω0

Zk= Ω0

Z1,k∪ Ω0

Z2,k∪ · · · ∪

Ω0Zn,k

and Ω1 ⊂ R1, such that Zi,k ∈ Ω0Zk

, yk(t − τ) ∈ Ω1

and yd(t − τ) ∈ Ω1. Then, for bounded initial conditions

xn,k(0) = [yd(0), α1,k − yd(0), · · · , αn−1,k − y(n−1)d (0)]T,

Wi1,k(0) = Wi1,k−1(T ), Wi2,k(0) = Wi2,k−1(T ), Mi,k(0) =Mi,k−1(T ) (i = 1, 2, · · · , n; k > 1), we have the followingresults:

1) All signals in the closed-loop system on interval [0, T ]remain semi-globally uniformly ultimately bounded;

2) The output tracking error yk(t) − yd(t) converges tozero as k approaches to infinity completely.

Proof. Let zk = z1,k, z2,k, · · · , zn,k, Vn,k =Vn,k(zk(0), Wn1,k(T ), Wn2,k(T ), Mn1,k(T )). According tothe initial conditions, we have ‖zk(0)‖2 = 0 6 ‖zk(T )‖2.

By (28),

Vn,k 6Vn,k(zk(T ), Wn1,k(T ), Wn2,k(T ), Mn1,k(T )) =

Vn,k(zk(0), Wn1,k(0), Wn2,k(0), Mn1,k(0)) +∫ T

0

Vn,kdt. (31)

408 International Journal of Automation and Computing 8(4), November 2011

Substituting (30) into (31), (31) becomes

Vn,k 6 Vn,k(zk(0), Wn1,k(0), Wn2,k(0), Mn1,k(0)) +∫ T

0

nδ∆kdt−∫ T

0

n∑j=1

ajjz2j,kdt 6

Vn,1(z1(0), Wn1,1(0), Wn2,1(0), Mn1,1(0)) +

nδT (

k∑i=1

∆k)−k∑

i=1

n∑j=1

∫ T

0

ajjz2j,kdt.

(32)

Let V0(k) = Vn,1(z1(0), Wn1,1(0), Wn2,1(0), Vn1,1(0)) +nδT (

∑ki=1 ∆k), (32) becomes

k∑i=1

n∑j=1

∫ T

0

ajjz2j,kdt 6 V0(k)− Vn,k. (33)

Because of limk→∞V0(k) = Vn,1(0) + 2nδTq, that is tosay, V0(k) is bounded, and Vn,k > 0, then

limk→∞

n∑j=1

∫ T

0

ajjz2j,kdt = 0. (34)

From (28), for any k, Vn,k(t) = Vn,k(0)+∫ t

0Vn,k(ς)dς, by

(30), we obtain

Vn,k(t) 6 Vn,k(0)−n∑

j=1

∫ t

0

ajjz2j,kdς + tnδ∆k. (35)

From (34),∑n

j=1

∫ t

0ajjz

2j,kdς is bounded, and by

Definition 1, ∆k is bounded, t ∈ [0, T ], so tnδ∆k isbounded. Because Wi1,k(0) = Wi1,k−1(T ), Wi2,k(0) =Wi2,k−1(T ), Mi,k(0) = Mi,k−1(T ), from (28), for anyk, Vn,k is bounded, Vn,k(0, Wn1,k(0), Wn2,k(0), Mn,k(0)) =Vn−1,k is bounded. From all the above, for any k, Vn,k(t) isbounded, xi,k, ‖Wi1,k‖, ‖Wi2,k‖, ‖Mi,k‖ are bounded. By(26), uk is bounded. By (34), we get limk→∞zi,k = 0, solimk→∞z1,k = 0, that is to say, the output tracking erroryk(t)− yd(t) converges to zero as k approaches to infinity.

¤

5 Simulation example

In this section, a numerical simulation example is usedto illustrate the efficiency of the proposed control approach.Consider the following second-order nonlinear system

x1,k =x2,k + x21,kθ2

1,k(t) + x21,k(t− τ)θ2

1,k(t− τ)

x2,k =uk + x1,kx2,kθ22,k(t) + x2

1,k(t− τ)θ22,k(t− τ)

yk =x1,k (36)

where the unknown time-varying disturbance and time-varying time-delay disturbance are θ1,k(t) = sin(2πt),θ1,k(t − τ) = sin(2π(t − τ)), θ2,k(t) = cos(2πt), andθ2,k(t − τ) = cos(2π(t − τ)), with known periods T = 1.The reference signal is yd(t) = cos(2πt). Based on the con-trol approach developed in Section 3, the control law uk isdesigned as follows:

uk =−z1,k − a22z2,k − WT21,kS21(Z2,k, MT

2,kΦ2(t))−

WT22,kS22(yk(t− τ), MT

2,kΦ2(t− τ)) +

α1,k − β2,k (37)

where z1,k = yk − yd, z2,k = x2,k − α1,k, β1,k =η1,k tanh(η1,kz1,k/∆k), β2,k = η2,k tanh(η2,kz2,k/∆k),α1,k = −a11z1,k − WT

11,kS11(Z1,k, MT1,kΦ1(t)) −

WT12,kS12(yk(t − τ), MT

1,kΦ1(t − τ)) + yd(t) − β1,k,with the adaptive laws

˙W11,k =Γ11[(S11(Z1,k, MT

1,kΦ1(t))−

S′11,kMT

1,kΦ1(t))z1,k]

˙W12,k =Γ12[(S12(yk(t− τ), MT

1,kΦ1(t− τ))−

S′12,kMT

1,kΦ1(t− τ))z1,k]

˙M1,k =Γ13[(Φ1(t)W11,kS

′11,k +

Φ1(t− τ)W12,kS′12,k)z1,k] (38)

˙W21,k =Γ21[(S21(Z2,k, MT

2,kΦ2(t))−

S′21,kMT

2,kΦ2(t))z2,k]

˙W22,k =Γ22[(S22(yk(t− τ), MT

2,kΦ2(t− τ))−

S′22,kMT

2,kΦ2(t− τ))z2,k]

˙M2,k =Γ23[(Φ2(t)W21,kS

′21,k +

Φ2(t− τ)W22,kS′22,k)z2,k]. (39)

In the simulation, the initial condition of the system is setto be x1,k(0) = 1, x2,k(0) = 0. We choose ∆k = q/k2, 1/q =0.01, and the numbers of FSE components as q1 = q2 = 5and the numbers p1 = p2 = 5. The centers of radial basisfunctions (RBFs) evenly cover the compact sets [−1, 1] ×[−1, 1] and [−1, 1]×[−1, 1]×[−1, 1], and the widths of RBFsare set to be η1 = η2 = 0.1. The design parameters area11 = a22 = 5 . The adaptive gains are chosen as Γ11 =Γ12 = 100I, Γ13 = Γ14 = 15I. Taking iteration index k =10, the simulation results are shown in Figs. 1–5.

Fig. 1 The change of ‖z1,k‖ (solid), ‖z2,k‖ (dotted) with itera-

tion index k

C. L. Zhang and J. M. Li / Adaptive Iterative Learning Control · · · 409

Fig. 2 The change of uk with time as iteration index k = 10

Fig. 3 The change of ‖W11,k‖ (solid), ‖W21,k‖ (dotted) with

iteration index k

Fig. 4 The change of ‖M1,k‖ (solid), ‖M2,k‖ (dotted) with it-

eration index k

Fig. 5 ‖W12,k‖ (solid), ‖W22,k‖ (dotted) with iteration index k

It can be seen from Fig. 1 that the tracking error canconverge to zero. Moreover, Figs. 2–5 show that the controlsignals uk(t), ‖W11,k‖, ‖W12,k‖, ‖M1,k‖, ‖W21,k‖, ‖W22,k‖,‖M2,k‖ are bounded on the interval [0, 1]. The simulationresults shown in Figs. 1–5 further confirm the effectivenessof the control scheme developed in this paper.

6 Conclusions

In this paper, an adaptive NN iterative learning controlapproach is presented for a class of strict-feedback nonlin-ear time-delay systems with unknown non-linearly param-eterised and time-varying disturbed function of known pe-riods. Unknown nonlinear vector functions and unknownnonlinear time-delay functions are approximated by twoFSE-RBF-neural networks, respectively, such that the re-quirements on the unknown nonlinear functions and theunknown nonlinear time-delay functions are relaxed. TheNN learning laws and control laws are designed by usingappropriate Lyapunov-Krasovskii functional and backstep-ping technology. Furthermore, based on Lyapunov theory,all signs in the closed loop system are guaranteed to besemi-globally uniformly ultimately bounded and the outputof the system is proved to converge to the desired trajectory.

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Chun-Li Zhang graduated from LinyiNormal University, PRC in 2007. She re-ceived the M. S. degree from Xidian Univer-sity in 2010. She is currently a Ph. D. can-didate at the Department of Applied Math-ematics, Xidian University.

Her research interests include adaptiveiterative learning control, neural network,robust control, and chaos synchronization.

E-mail: gaozhangchunli@yahoo.cn (Cor-responding author)

Jun-Min Li graduated from XidianUniversity, PRC in 1987. He received theM. S. degree from Xidian University in 1990and the Ph. D. degree from Xi′an JiaotongUniversity, PRC in 1997. He is currentlya professor at the Department of AppliedMathematics, Xidian University.

His research interests include adaptivecontrol, learning control, intelligent control,hybrid system control theory, and the net-

worked control systems.E-mail: jmli@mail.xidian.edu.cn