A New AILC for a Class of Nonlinearly Parameterized Systems with Unknown Delays and Input Dead-Zone

Abstract

This paper presents an adaptive iterative learning control (AILC) scheme for a class of nonlinear systems with unknown time-varying delays and unknown input dead-zone. A novel nonlinear form of deadzone nonlinearity is presented. The assumption of identical initial condition for ILC is removed by introducing boundary layer functions. The uncertainties with time-varying delays are compensated for with assistance of appropriate Lyapunov-Krasovskii functional and Young’s inequality. The hyperbolic tangent function is employed to avoid the possible singularity problem. According to a property of hyperbolic tangent function, the system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapunov-like composite energy function (CEF) in two cases, while maintaining all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

1. Introduction

Practically, many engineering systems carry out repetitive tasks in fixed finite space, such as manipulators [1–3]. In high precision engineering, perfect tracking for such tasks is highly desirable. Whereas existing control approaches, such as classical PID controllers, feedback linearization, and adaptive control, may guarantee closed-loop stability, they can hardly achieve perfect tracking. Fortunately, learning and repetitive control are the alternatives to address this problem, which enhance the tracking accuracy from operation to operation for systems executing repetitive tasks. By now, iterative learning control (ILC) has become one of the most important learning strategies owing to its implementation simplicity under the repeatable control environment. The basic idea of ILC is to improve the control performance of systems from trial to trial. Traditional iterative learning controllers have been developed for more than two decades for nonlinear plants [1–12]. The control input of traditional ILC is directly updated by a learning mechanism using the information of error and input in the previous iteration, and the contraction mapping theorem is often used to analyze the stability of closed-loop systems. However, the studied systems must satisfy global Lipschitz continuous condition. Thus, there are some difficulties or limitations to apply traditional ILC for certain systems. In order to circumvent this problem, some other new ILC algorithms have been widely studied. One of the most important developments is adaptive ILC (AILC) [13–15], in which the control parameters are adjusted between successive iterations, and the so-called composite energy function (CEF) [16] is usually constructed to derive the stability conclusions. In recent years, the control community has witnessed great progress in AILC of uncertain nonlinear systems [17–27].

In practical control tasks, systems with time delays are frequently encountered. The existence of time delays may make the controllers design more complicated and challenging, especially for systems with unknown time-delays. Stabilization problem of control systems with time delay has drawn much attention [28–35] due to its mathematical challenge and application demand in real-time control. In [28–31], the controller design and stability analysis for state-delayed systems were presented and [32–35] discussed the stabilization of systems with input delays. In the field of ILC, although so many results have been obtained, only few ones were available for time-delay nonlinear systems [36–41], and the majority of these works were traditional iterative learning controllers. In the framework of AILC, Chen and Zhang [40] proposed an AILC scheme for a class of scalar systems with unknown time-varying parameters and unknown time-varying delay. In [41], an adaptive learning control design was developed for a certain class of first-order nonlinearly parameterized systems with unknown periodically time-varying delay and further extended to a class of high-order systems with both time-varying and time-invariant parameters. However, they all required the identical initial conditions on the initial states and the reference trajectory for the AILC design, which is necessary for the stability and convergence analysis but can hardly be satisfied in practical systems.

In practice, nonsmooth and nonlinear characteristics such as dead-zone, hysteresis, saturation, and backlash are common in actuator and sensors. Dead-zone is one of the most important nonsmooth and non-affine-in-input nonlinearities in many industrial processes, which can severely deteriorate system performances and give rise to extra difficulties in the controller design. Therefore, the effect of dead-zone should be taken into consideration and has been drawing much interest in the control community for a long time [42–48]. To handle the problem of unknown dead-zone in control system design, an immediate method is to construct an adaptive dead-zone inverse [42]. Continuous and discrete adaptive dead-zone inverses were built for linear systems with unmeasurable dead-zone outputs [43, 44]. Based on the assumption of the consistent dead-zone slopes in the positive and negative regions, a robust adaptive control approach was given for a class of special nonlinear systems without using the dead-zone inverse [45]. In [46, 47], the dead-zone is reconstructed into the form of a linear system with a static time-varying gain and bounded disturbances by introducing characteristic function. In [48], input dead-zone is taken into account and it is proved that the simplest ILC scheme retains its ability of achieving the satisfactory performance in tracking control. To the best of our knowledge, there is little work from the viewpoint of AILC to deal with nonlinear systems with time-delay and dead-zone nonlinearity in the literature at present stage.

In this paper, we present a novel AILC scheme for a class of nonlinear time-varying systems with unknown time-varying delays and unknown input dead-zone. To the best of our knowledge, up to now, few works have been reported in the field of AILC to deal with such kinds of systems. The main design difficulty comes from how to deal with dead-zone nonlinearity and delay-dependent uncertainty. In our work, the dead-zone output is represented as a novel simple nonlinear system with a time-varying gain, which is more general than the linear form in [36]. The approach removes the assumption of linear function outside the dead-band without necessarily constructing a dead-zone inverse. An appropriate Lyapunov-Krasovskii functional and Young’s inequality are combined to eliminate the unknown time-varying delays such that the design of the control law is free from these uncertainties. Furthermore, the possible singularity which may be caused by the appearance of the reciprocal of tracking error is avoided by employing the hyperbolic tangent function. By constructing a Lyapunov-like CEF, the stability conclusion is obtained in two cases by exploiting the properties of the hyperbolic tangent function via a rigorous analysis. In addition, the boundary layer function is introduced to remove the requirement for identical initial condition which is required for the majority of ILC schemes.

The rest of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. The AILC design is developed in Section 3. The CEF-based stability analysis is presented in Section 4. A simulation example is presented to verify the validity of the proposed scheme in Section 5, followed by conclusions in Section 6.

2. Problem Formulation and Preliminaries

2.1. Problem Formulation

Consider a class of nonlinear time-varying systems with unknown time-varying time-delays and dead-zone running on a finite time interval [0,𝑇] repeatedly which is given by 𝑥̇_𝑖,𝑘(𝑡)=𝑥_𝑖+1,𝑘(𝑡)𝑥̇,𝑖=1,…,𝑛−1,_𝑛,𝑘(𝑡)=𝑓𝑋_𝑘(𝑡),𝑋_𝜏,𝑘(𝑡),𝜃(𝑡)+𝑏(𝑡)𝑢_𝑘(𝑡)(𝑡),𝑦+𝑑_𝑘(𝑡)=𝑥_1,𝑘(𝑡),𝑢_𝑘(𝑡)=𝐷𝑣_𝑘(𝑡)[],𝑥,𝑡∈0,𝑇_𝑖,𝑘(𝑡)=𝜛_𝑖(𝑡),𝑡∈−𝜏_max,0,𝑖=1,…,𝑛, (1) where 𝑡 is the time, 𝑘∈𝑁 denotes the times of iteration, 𝑁 is the integer set and denotes the sets of iteration times, 𝑦_𝑘(𝑡)∈𝑅 and 𝑥_𝑖,𝑘(𝑡)∈𝑅,𝑖=1,…,𝑛 are the system output and states, respectively, 𝑋_𝑘(𝑡)≜[𝑥_1,𝑘(𝑡),…,𝑥_𝑛,𝑘(𝑡)]^𝑇 is the state vector, 𝜏(𝑡) is unknown time-varying delay of states and 𝑥^𝜏_𝑖,𝑘≜𝑥_𝑖,𝑘(𝑡−𝜏(𝑡)), 𝑖=2,…,𝑛, and 𝑋_𝜏,𝑘(𝑡)=[𝑥^𝜏_1,𝑘(𝑡),…,𝑥^𝜏_𝑛,𝑘(𝑡)]^𝑇,𝑓(⋅,⋅,⋅) are unknown smooth functions, and 𝑏(𝑡) is the unknown continuous time-varying gain of the system input. 𝜃(𝑡) is unknown continuous time-varying parameter vector; 𝑑(𝑡) is unknown bounded external disturbance. 𝜛_𝑖(𝑡) denote the initial functions for delayed states, 𝑖=1,…,𝑛. Consider 𝑣_𝑘(𝑡)∈𝑅 is the control input and the actuator nonlinearity 𝐷(𝑣_𝑘(𝑡)) is described as a dead-zone characteristic.

In this paper, a reference trajectory vector is given by 𝑋_𝑑(𝑡)=[𝑦_𝑑(𝑡),𝑦̇_𝑑(𝑡),…,𝑦_𝑑^(𝑛−1)(𝑡)]^𝑇. The tracking error vector is 𝑒_𝑘(𝑡)=[𝑒_1,𝑘,𝑒_2,𝑘,…,𝑒_𝑛,𝑘]^𝑇=𝑋_𝑘(𝑡)−𝑋_𝑑(𝑡). The control objective of this paper is to design an adaptive iterative learning controller 𝑢_𝑘(𝑡), such that the tracking error 𝑒_𝑘(𝑡) converges to a small neighborhood of the origin as 𝑘→∞; that is, lim_𝑘→∞‖𝑒_𝑘(𝑡)‖≤𝜀_𝑒∞, while all the signals in the closed-loop system remain bounded, where 𝜀_𝑒∞ is a small positive error tolerance which will be given in the subsequent context and ‖⋅‖ denotes the Euclidian norm. Define the filtered tracking error as 𝑒_𝑠𝑘(𝑡)=[Λ^𝑇1]𝑒_𝑘(𝑡), where Λ=[𝜆₁,𝜆₂,…,𝜆_𝑛−1]^𝑇 and 𝜆₁,…,𝜆_𝑛−1 are the coefficients of Hurwitz polynomial 𝐻(𝑠)=𝑠^𝑛−1+𝜆_𝑛−1𝑠^𝑛−2+⋯+𝜆₁.

To facilitate control system design, we make the following reasonable assumptions for the system functions, unknown time delays, and reference signals.

Assumption 1. The unknown state time-varying delay 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤𝜏_max, 𝜏̇(𝑡)≤𝜅<1, where 𝜏_max and 𝜅 are unknown positive constants.

Assumption 2. The unknown smooth functions 𝑓(⋅,⋅,⋅) satisfy inequality 𝑓𝑋_𝑘,𝑋_𝜏,𝑘,𝜃(𝑡)−𝑓𝑋_𝑑,𝑋_𝑑,𝜏(𝑡)≤𝑋,𝜃_𝑘−𝑋_𝑑ℎ₁𝑋_𝑘,𝑋_𝑑𝜉₁(𝜃)+𝑋_𝜏,𝑘−𝑋_𝑑,𝜏ℎ₂𝑋_𝜏,𝑘,𝑋_𝑑,𝜏𝜉₂(𝜃), (2) where 𝑋_𝑑,𝜏≜𝑋_𝑑(𝑡−𝜏(𝑡)), ℎ₁(⋅,⋅) and ℎ₂(⋅,⋅) are known positive smooth functions, and 𝜉₁(𝜃) and 𝜉₂(𝜃) are unknown smooth functions of 𝜃(𝑡).

Assumption 3. The sign of 𝑏(𝑡) is known; without loss of generality, we always assume 𝑏(𝑡)>0.

Assumption 4 (see [23]). The initial state errors 𝑒_𝑖,𝑘(0) at each iteration are not necessarily zero small and fixed, but they are assumed to be bounded.

Assumption 5. The reference state trajectory 𝑋_𝑑(𝑡) is continuous, bounded, and available.

Assumption 6. The unknown external 𝑑(𝑡) is bounded; that is, |𝑑(𝑡)|≤𝑑_max with an unknown constant 𝑑_max.

Remark 7. Assumption 1 is common in the control problem of time-varying delay systems, which guarantees that the time delay terms can be eliminated by using Lyapunov-Krasovskii functional. Moreover, Assumption 1 is milder than that in [39–41] as it does not require the true value of 𝜏_max and 𝜅.

Remark 8. As 𝑏(𝑡) is continuous on [0,𝑇], there exist constants 0<𝑏_min≤𝑏_max such that 𝑏_min≤𝑏(𝑡)≤𝑏_max. However, the control gain bounds 𝑏_min and 𝑏_max are only required for analytical purposes; their true values are not necessarily known in the sense that they are not used for controller design.

2.2. Dead-Zone Characteristic

The dead-zone characteristic can be described as 𝑢_𝑘(𝑡)=𝐷𝑣_𝑘(𝑡)=𝑚(𝑡)𝑣_𝑘(𝑡)−𝑏_𝑟for𝑣_𝑘(𝑡)≥𝑏_𝑟,0for𝑏_𝑙<𝑣_𝑘(𝑡)<𝑏_𝑟,𝑚(𝑡)𝑣_𝑘(𝑡)−𝑏_𝑙for𝑣_𝑘(𝑡)≤𝑏_𝑙, (3) where 𝑏_𝑟≥0 and 𝑏_𝑙≤0 are unknown constants, 𝑚(𝑡)>0 is unknown time-varying slopes, and 𝑣_𝑘(𝑡) is the input and 𝑢_𝑘(𝑡) is the output of dead-zone. A graphical representation of the dead-zone in this paper is shown in Figure 1.

Figure 1 

Dead-zone model.

The dead-zone output 𝑢_𝑘(𝑡) is not available for measurement. We make the following assumption on the dead-zone parameters.

Assumption 9. The dead-zone parameters 𝑏_𝑟, 𝑏_𝑙, and 𝑚(𝑡) are bounded. That is, there exist unknown constants 𝑏_𝑟min, 𝑏_𝑟max, 𝑏_𝑙min, 𝑏_𝑙max, 𝑚_min, and 𝑚_max, such that 𝑏_𝑟min≤𝑏_𝑟≤𝑏_𝑟max, 𝑏_𝑙min≤𝑏_𝑙≤𝑏_𝑙max, and 𝑚_min≤𝑚(𝑡)≤𝑚_max.

From a practical point of view, we can redefine the dead-zone nonlinearity as 𝑢_𝑘(𝑡)=𝐷𝑣_𝑘=𝑚(𝑡)𝑣_𝑘(𝑡)−𝑑₁𝑣_𝑘(𝑡) (4) with 𝑑₁𝑣_𝑘(𝑡)=𝑚(𝑡)𝑏_𝑟for𝑣_𝑘(𝑡)≥𝑏_𝑟,𝑚(𝑡)𝑣_𝑘(𝑡)for𝑏_𝑙<𝑣_𝑘(𝑡)<𝑏_𝑟,𝑚(𝑡)𝑏_𝑙for𝑣_𝑘(𝑡)≤𝑏_𝑙. (5) It is obvious that 𝑑₁(𝑣_𝑘(𝑡)) is bounded.

Remark 10. Obviously, the dead-zone characteristic is nonlinear. And the form in [36] is the special case of (3) when 𝑚(𝑡) is invariant. Therefore, the presentation of dead-zone in our work is more general than the earlier results.

2.3. A Motivating Example

In order to clarify the main idea of AILC, we show the design procedure briefly by a simple scalar system running on [0,𝑇] as follows: 𝑧̇_𝑘(𝑡)=𝜃(𝑡)𝜉𝑧_𝑘,𝑡+𝑢^𝑧_𝑘(𝑡), (6) where 𝑧_𝑘(𝑡) and 𝑢^𝑧_𝑘(𝑡) are the system state and the control input in the 𝑘th iteration, respectively, 𝜃(𝑡) is an unknown time-varying parameter, and 𝜉(𝑧_𝑘,𝑡) is a known time-varying function. The reference trajectory is 𝑧_𝑟(𝑡), 𝑡∈[0,𝑇]. Define the tracking error as 𝑒^𝑧_𝑘(𝑡)=𝑧_𝑘(𝑡)−𝑧_𝑟(𝑡) and design the control law and adaptive learning law for the unknown time-varying parameter for the 𝑘th iteration as follows: 𝑢^𝑧_𝑘(𝑡)=−𝑘₁𝑒^𝑧_𝑘(𝑡)+𝑧̇_𝑟(𝑡)−𝜃_𝑘(𝑡)𝜉𝑧_𝑘,𝜃,𝑡_𝑘(𝑡)=𝜃_𝑘−1(𝑡)+𝑞𝜉𝑧_𝑘,𝑡𝑒^𝑧_𝑘(𝑡),𝜃₀(𝑡)=0,𝑡∈[0,𝑇], (7) where 𝑘₁,𝑞>0 are design parameters. Define the estimate error as 𝜃_𝑘(𝑡)=𝜃_𝑘(𝑡)−𝜃(𝑡). Choose a Lyapunov-like CEF as 𝐸^𝑧_𝑘(𝑡)=12𝑒^𝑧_𝑘(𝑡)+12𝑞^𝑡₀𝜃_𝑘(𝜎)d𝜎. (8)

Throughout this paper, 𝜎 denotes the integral variable. Then it can be derived that Δ𝐸^𝑧_𝑘(𝑡)=𝐸^𝑧_𝑘(𝑡)−𝐸^𝑧_𝑘−1(𝑡)≤−^𝑡₀𝑒^𝑧_𝑘(𝜎)²d𝜎. (9) We can further derive that lim_𝑘→∞^𝑇₀𝑒^𝑧_𝑘(𝜎)²d𝜎=0. (10) Therefore, the system state 𝑧_𝑘(𝑡) converges to the reference trajectory 𝑧_𝑟(𝑡) on [0,𝑇] as 𝑘→∞.

3. AILC Design

According to Assumption 4, we know that there exist known constants 𝜀_𝑖 such that |𝑒_𝑖,𝑘(0)|≤𝜀_𝑖, 𝑖=1,2,…𝑛 for any 𝑘∈𝑁. In order to relax the identical initial condition in ILC, we employ a boundary layer function [23] as follows: 𝑠_𝑘(𝑡)=𝑒_𝑠𝑘(𝑡)(𝑡)𝑒−𝜂sat^𝑠𝑘(𝑡)𝜂(𝑡),𝜂(𝑡)=𝜀𝑒^−𝐾𝑡,𝐾>0, (11)(12) where 𝜀=[Λ^𝑇1][𝜀₁,𝜀₂,…,𝜀_𝑛]^𝑇 and 𝐾 is a design parameter. The saturation function sat(⋅) is given by 𝑒sat^𝑠𝑘(𝑡)𝜂(𝑡)=1,if𝑒_𝑠𝑘(𝑡)(𝑡),𝑒>𝜂_𝑠𝑘(𝑡)𝜂(𝑡),if𝑒_𝑠𝑘(𝑡)≤𝜂(𝑡),−1,if𝑒_𝑠𝑘(𝑡)<−𝜂(𝑡). (13)

Remark 11. Note that 𝜂(𝑡) decreases along time axis with initial condition 𝜂(0)=𝜀 and 0<𝜂(𝑇)≤𝜂(𝑡)≤𝜀, ∀𝑡∈[0,𝑇], and then if 𝑠_𝑘(𝑡) can be derived to zero ∀𝑡∈[0,𝑇], the states will asymptotically converge to the reference trajectory for all 𝑡∈[0,𝑇].

It can be easily shown that 𝑒_𝑠𝑘(0)=𝜆₁𝑒_1,𝑘(0)+𝜆₂𝑒_2,𝑘(0)+⋯+𝑒_𝑛,𝑘(0)≤𝜆₁𝑒_1,𝑘(0)+𝜆₂𝑒_2,𝑘(0)+⋯+𝑒_𝑛,𝑘(0)≤𝜆₁𝜀₁+𝜆₂𝜀₂+⋯+𝜀_𝑛=𝜀=𝜂(0) (14) which implies that 𝑠_𝑘(0)=𝑒_𝑠𝑘(0)−𝜂(0)(𝑒_𝑠𝑘(0)/𝜂(0))=0 is satisfied forall 𝑘∈𝑁. For the subsequent controller design, we firstly give the dynamic of 𝑒_𝑛,𝑘(𝑡) as follows: 𝑒̇_𝑛,𝑘(𝑡)=𝑓𝑋_𝑘(𝑡),𝑋_𝜏,𝑘,𝜃(𝑡)+𝑏(𝑡)𝑢_𝑘+𝑑(𝑡)−𝑦_𝑑^(𝑛)(𝑡)=𝑓𝑋_𝑘,𝑋_𝜏,𝑘,𝜃(𝑡)−𝑓𝑋_𝑑,𝑋_𝑑,𝜏(𝑡)𝑋,𝜃+𝑓_𝑑,𝑋_𝑑,𝜏(𝑡)(𝑡)𝑚(𝑡)𝑣,𝜃+𝑏_𝑘(𝑡)−𝑑₁𝑣_𝑘(𝑡)+𝑑(𝑡)−𝑦_𝑑^(𝑛)(𝑡)=𝑓𝑋_𝑘,𝑋_𝜏,𝑘,𝜃(𝑡)−𝑓𝑋_𝑑,𝑋_𝑑,𝜏(𝑡)𝑋,𝜃+𝑓_𝑑,𝑋_𝑑,𝜏,𝜃(𝑡)+𝑏(𝑡)𝑚(𝑡)𝑣_𝑘(𝑡)+𝑑₂(𝑡)−𝑦_𝑑^(𝑛)(𝑡), (15) where 𝑑₂(𝑡)=−𝑏(𝑡)𝑑₁(𝑣_𝑘(𝑡))+𝑑(𝑡). By Assumptions 3 and 6, we know that 𝑑₂(𝑡) is bounded; that is, there exists an unknown smooth positive function 𝑑(𝑡) such that |𝑑₂(𝑡)|≤𝑑(𝑡). For the simplicity of expression, we define 𝑏_𝑚(𝑡)=𝑏(𝑡)𝑚(𝑡), Θ(𝑡)=𝑓(𝑋_𝑑,𝑋_𝑑,𝜏,𝜃(𝑡)), and Δ_𝑘(𝑡)=𝑓(𝑋_𝑘,𝑋_𝜏,𝑘,𝜃(𝑡))−𝑓(𝑋_𝑑,𝑋_𝑑,𝜏,𝜃(𝑡)). It is clear that Θ(𝑡) is an unknown time-varying function which is invariant in the iteration domain and 𝑏_𝑚=𝑚_min𝑏_min≤𝑏_𝑚(𝑡)≤𝑚_max𝑏_max=𝑏_𝑚. Define a smooth scalar function as 𝑉_𝑠𝑘(𝑡)=12𝑠²_𝑘(𝑡). (16) Differentiating 𝑉_𝑠𝑘(𝑡) with respect to time, we can obtain 𝑉̇_𝑠𝑘(𝑡)=𝑠_𝑘(𝑡)𝑠_𝑘̇(𝑡)=𝑠_𝑘(𝑡)𝑒̇_𝑠𝑘(𝑡)−𝜂̇(𝑡),if𝑒_𝑠𝑘(𝑡)(𝑡)𝑒>𝜂0,if_𝑠𝑘(𝑡(𝑡)𝑠)≤𝜂_𝑘(𝑡)𝑒̇_𝑠𝑘(𝑡)+𝜂̇(𝑡),if𝑒_𝑠𝑘(𝑡)<−𝜂(𝑡)=𝑠_𝑘(𝑡)𝑒̇_𝑠𝑘(𝑡)−𝜂̇(𝑡)sgn𝑠_𝑘(𝑡)=𝑠_𝑘(𝑡)^𝑛−1_𝑗=1𝜆_𝑗𝑒_𝑗+1,𝑘(𝑡)−𝜂̇(𝑡)sgn𝑠_𝑘(𝑡)+Θ(𝑡)+Δ_𝑘(𝑡)+𝑏_𝑚(𝑡)𝑣_𝑘(𝑡)+𝑑₂(𝑡)−𝑦_𝑑^(𝑛)(𝑡)=𝑠_𝑘(𝑡)^𝑛−1_𝑗=1𝜆_𝑗𝑒_𝑗+1,𝑘(𝑡)+𝐾𝜂(𝑡)sgn𝑠_𝑘(𝑡)+𝐾𝑒_𝑠𝑘(𝑡)−𝐾𝑒_𝑠𝑘(𝑡)+Θ(𝑡)+Δ_𝑘(𝑡)+𝑏_𝑚(𝑡)𝑣_𝑘(𝑡)+𝑑₂(𝑡)−𝑦_𝑑^(𝑛)(𝑡)=𝑠_𝑘(𝑡)Θ(𝑡)+Δ_𝑘(𝑡)+𝑏_𝑚(𝑡)𝑣_𝑘(𝑡)+𝜇_𝑘(𝑡)+𝑑₂(𝑡)−𝐾𝑠²_𝑘(𝑡), (17) where 𝜇_𝑘(𝑡)=∑^𝑛−1_𝑗=1𝜆_𝑗𝑒_𝑗+1,𝑘(𝑡)+𝐾𝑒_𝑠𝑘(𝑡)−𝑦_𝑑^(𝑛)(𝑡), and we use the relation 𝑠_𝑘(𝑡)−𝐾𝑒_𝑠𝑘(𝑡)+𝐾𝜂(𝑡)sgn𝑠_𝑘(𝑡)=𝑠_𝑘(𝑡)−𝐾𝑠_𝑘(𝑡)(𝑡)𝑒−𝐾𝜂sat^𝑠𝑘(𝑡)𝜂(𝑡)+𝐾𝜂(𝑡)sgn𝑠_𝑘(𝑡)=−𝐾𝑠²_𝑘(𝑡)−𝐾𝜂(𝑡)𝑠_𝑘(𝑡)+𝐾𝜂(𝑡)𝑠_𝑘(𝑡)=−𝐾𝑠²_𝑘(𝑡). (18) Utilizing Young’s inequality and noting Assumption 2, it follows that 𝑠_𝑘(𝑡)Δ_𝑘(𝑡)≤𝑠_𝑘(𝑡×𝑋)_𝑘−𝑋_𝑑ℎ₁𝑋_𝑘,𝑋_𝑑𝜉₁(𝜃)+𝑋_𝜏,𝑘−𝑋_𝑑,𝜏ℎ₂𝑋_𝜏,𝑘,𝑋_𝑑,𝜏𝜉₂(𝜃≤1)2𝑠²_𝑘(𝑡)𝜉²₁(𝜃)+12𝑒_𝑘²ℎ²₁𝑋_𝑘,𝑋_𝑑+12𝑠²_𝑘(𝑡)𝜉²₂(𝜃)+12𝑒_𝜏,𝑘²ℎ²₂𝑋_𝜏,𝑘,𝑋_𝑑,𝜏. (19) Substituting (19) into (17) leads to 𝑉̇_𝑠𝑘(𝑡)≤𝑠_𝑘(𝑡)×Θ(𝑡)+𝑏_𝑚(𝑡)𝑣_𝑘(𝑡)+𝜇_𝑘(𝑡)+𝑑₂(𝑡)+12𝑠_𝑘(𝑡)𝜉²₁(𝜃)+12𝑠_𝑘(𝑡)𝜉²₂(𝜃)−𝐾𝑠²_𝑘(𝑡)+12𝑒_𝑘²ℎ²₁𝑋_𝑘,𝑋_𝑑+12𝑒_𝜏,𝑘²ℎ²₂𝑋_𝜏,𝑘,𝑋_𝑑,𝜏. (20)

To overcome the design difficulty arising from the unknown time-varying delay term, consider the following Lyapunov-Krasovskii functional: 𝑉_𝑈𝑘(𝑡)=12(1−𝜅)^𝑡_{(𝑡)𝑡−𝜏}𝑒_𝑘(𝜎)²ℎ²₂𝑋_𝑘(𝜎),𝑋_𝑑(𝜎)d𝜎. (21) Recalling Assumption 1, taking the time derivative of 𝑉_𝑈𝑘(𝑡) leads to 𝑉̇_𝑈𝑘(𝑡)=12()𝑒1−𝜅^𝑘2ℎ²₂𝑋_𝑘,𝑋_𝑑−1−𝜏̇(𝑡)2()𝑒1−𝜅^𝜏,𝑘2ℎ²₂𝑋_𝜏,𝑘,𝑋_𝑑,𝜏≤12()𝑒1−𝜅^𝑘2ℎ²₂𝑋_𝑘,𝑋_𝑑−12𝑒_𝜏,𝑘²ℎ²₂𝑋_𝜏,𝑘,𝑋_𝑑,𝜏. (22) Define a Lyapunov functional as 𝑉_𝑘(𝑡)=𝑉_𝑠𝑘(𝑡)+𝑉_𝑈𝑘(𝑡); combining (20) and (22), we can obtain 𝑉̇_𝑘(𝑡)≤𝑠_𝑘(𝑡)×Θ(𝑡)+𝑏_𝑚(𝑡)𝑣_𝑘(𝑡)+𝜇_𝑘(𝑡)+𝑑₂(𝑡)+12𝑠_𝑘(𝑡)𝜉²₁(𝜃)+12𝑠_𝑘(𝑡)𝜉²₂(𝜃)−𝐾𝑠²_𝑘(𝑡)+12𝑒_𝑘²ℎ²₁𝑋_𝑘,𝑋_𝑑+12()𝑒1−𝜅^𝑘2ℎ²₂𝑋_𝑘,𝑋_𝑑. (23) For the convenience of expression, denote 𝜁_𝑘(𝑡)=(1/2)‖𝑒_𝑘‖²ℎ²₁(𝑋_𝑘,𝑋_𝑑)+(1/2(1−𝜅))‖𝑒_𝑘‖²ℎ²₂(𝑋_𝑘,𝑋_𝑑) then (23) can be simplified as 𝑉̇_𝑘(𝑡)≤𝑠_𝑘(𝑡)×Θ(𝑡)+𝑏_𝑚(𝑡)𝑣_𝑘(𝑡)+𝜇_𝑘(𝑡)+𝑑₂(𝑡)+12𝑠_𝑘(𝑡)𝜉²₁(𝜃)+12𝑠_𝑘(𝑡)𝜉²₂(𝜃)+𝜁_𝑘(𝑡)𝑠_𝑘(𝑡)−𝐾𝑠²_𝑘(𝑡). (24)

Here, we note that singularity problem may occur in (24) due to the term 𝜁_𝑘(𝑡)/𝑠_𝑘(𝑡) which approaches ∞ as 𝑠_𝑘(𝑡) approaches zero. In order to tackle this problem, we exploit the following characteristic of hyperbolic tangent function.

Lemma 12 (see [49]). For any constant 𝜂>0 and any variable 𝑝∈𝑅, lim_𝑝→0tanh²𝑝/𝜂𝑝=0. (25)

By employing the hyperbolic tangent function, (24) can be rewritten as 𝑉̇_𝑘(𝑡)≤𝑠_𝑘(𝑡)×Θ(𝑡)+𝑑₂(𝑡)+𝑏_𝑚(𝑡)𝑣_𝑘(𝑡)+𝜇_𝑘(𝑡)+12𝑠_𝑘(𝑡)𝜉²₁(𝜃)+12𝑠_𝑘(𝑡)𝜉²₂(𝜃)+𝑎2𝑠_𝑘(𝑡)tanh²𝑠_𝑘(𝑡)𝜂(𝑡)𝑒_𝑘²ℎ²₁𝑋_𝑘,𝑋_𝑑+𝑎2(1−𝜅)𝑠_𝑘(𝑡)tanh²𝑠_𝑘(𝑡)𝜂(𝑡)×𝑒_𝑘²ℎ²₂𝑋_𝑘,𝑋_𝑑−𝐾𝑠²_𝑘(𝑡)+1−𝑎tanh²𝑠_𝑘(𝑡)𝜂(𝑡)𝜁_𝑘(𝑡), (26) where 𝑎>1 is a constant. From Lemma 12, we know that lim_{𝑠𝑘(𝑡)→0}(𝑎/𝑠_𝑘(𝑡))tanh²(𝑠_𝑘(𝑡)/𝜂(𝑡))𝜁_𝑘(𝑡)=0. Hence, (𝑎/𝑠_𝑘(𝑡))tanh²(𝑠_𝑘(𝑡)/𝜂(𝑡))𝜁_𝑘(𝑡) is defined at 𝑠_𝑘(𝑡)=0 and the possible singularity problem has been avoided. Upon multiplication of (26) by 1/𝑏_𝑚(𝑡), it becomes 𝑉̇_𝑘(𝑡)𝑏_𝑚(𝑡)≤𝑠_𝑘(𝑡)×1𝑏_𝑚(𝑡)Θ(𝑡)+𝑑₂(𝑡)+𝑣_𝑘(𝑡)+1𝑏_𝑚(𝑡)𝜇_𝑘(𝑡)+12𝑏_𝑚(𝑡)𝑠_𝑘(𝑡)𝜉²₁(𝜃)+𝜉²₂(𝜃+𝑎)2𝑏_𝑚(𝑡)𝑠_𝑘(𝑡)tanh²𝑠_𝑘(𝑡)𝜂(𝑡)×𝑒_𝑘²ℎ²₁𝑋_𝑘,𝑋_𝑑+𝑎2𝑏_𝑚(𝑡)(1−𝜅)𝑠_𝑘(𝑡)tanh²𝑠_𝑘(𝑡)𝜂(𝑡)×𝑒_𝑘²ℎ²₂𝑋_𝑘,𝑋_𝑑−𝐾𝑏_𝑚(𝑡)𝑠²_𝑘(𝑡)+1𝑏_𝑚(𝑡)×1−𝑎tanh²𝑠_𝑘(𝑡)𝜂(𝑡)𝜁_𝑘(𝑡)=𝑠_𝑘(𝑡)𝛽^𝑇(𝑡)Φ_𝑘(𝑡)+𝑣_𝑘(𝑡)−𝐾𝑏_𝑚(𝑡)𝑠²_𝑘(𝑡)+1𝑏_𝑚(𝑡)1−𝑎tanh²𝑠_𝑘(𝑡)𝜂(𝑡)𝜁_𝑘(𝑡), (27) where 𝛽(𝑡)=[(1/𝑏_𝑚(𝑡))(Θ(𝑡)+𝑑₂(𝑡)), 1/𝑏_𝑚(𝑡), (1/𝑏_𝑚(𝑡))(𝜉²₁(𝜃)+𝜉²₂(𝜃)),   (1/(1−𝜅)𝑏_𝑚(𝑡))]^𝑇 denote the unknown time-varying parameter vector that is invariant along the iteration axis and Φ_𝑘(𝑡)=[1, 𝜇_𝑘(𝑡)+(𝑎/2𝑠_𝑘(𝑡))tanh²(𝑠_𝑘(𝑡)/𝜂(𝑡))‖𝑒_𝑘‖²ℎ²₁(𝑋_𝑘,𝑋_𝑑),𝑠_𝑘(𝑡), (𝑎/2𝑠_𝑘(𝑡))tanh²(𝑠_𝑘(𝑡)/𝜂(𝑡))‖𝑒_𝑘‖²ℎ²₂(𝑋_𝑘,𝑋_𝑑)]^𝑇. Based on (27), we can design the adaptive iterative learning controller as follows: 𝑣_𝑘(𝑡)=−𝛽_𝑘(𝑡)Φ_𝑘(𝑡)−𝐾₁𝑠_𝑘(𝑡), (28) where 𝐾₁>0 is design parameters and 𝛽_𝑘(𝑡) is the estimate of 𝛽(𝑡) in the 𝑘th iteration. The adaptive learning algorithms for unknown parameter are given by 𝛽_𝑘(𝑡)=𝛽_𝑘−1(𝑡)+𝑞𝑠_𝑘(𝑡)Φ_𝑘(𝑡),𝛽₀(𝑡)=0,𝑡∈[0,𝑇], (29) where 𝑞>0 is the learning gain. Define the estimation error as 𝛽_𝑘(𝑡)=𝛽_𝑘(𝑡)−𝛽(𝑡). Hence, substituting the controller (28) back into (27) yields 𝑉̇_𝑘(𝑡)𝑏_𝑚(𝑡)≤−𝑠_𝑘(𝑡)𝛽^𝑇_𝑘(𝑡)Φ_𝑘(𝑡)−𝐾𝑏_𝑚(𝑡)+𝐾₁𝑠²_𝑘(𝑡)+1𝑏_𝑚(𝑡)1−𝑎tanh²𝑠_𝑘(𝑡)𝜂(𝑡)𝜁_𝑘(𝑡)≤−𝑠_𝑘(𝑡)𝛽^𝑇_𝑘(𝑡)Φ_𝑘(𝑡)−𝐾𝑏_𝑚+𝐾₁𝑠²_𝑘(𝑡)+1𝑏_𝑚(𝑡)1−𝑎tanh²𝑠_𝑘(𝑡)𝜂(𝑡)𝜁_𝑘(𝑡). (30) For simplicity in expression, we denote (𝐾/𝑏_𝑚+𝐾₁) by 𝐾₂=(𝐾/𝑏_𝑚+𝐾₁). Then, (30) can be continued as 𝑠_𝑘(𝑡)𝛽^𝑇_𝑘(𝑡)Φ_𝑘(𝑡)𝑉̇≤−^𝑘(𝑡)𝑏_𝑚(𝑡)−𝐾₂𝑠²_𝑘(𝑡)+1𝑏_𝑚(𝑡)1−𝑎tanh²𝑠_𝑘(𝑡)𝜂(𝑡)𝜁_𝑘(𝑡). (31)

4. Stability and Convergence Analysis

In this section, we will check the stability of the closed-loop system and the convergence of tracking errors by CEF-based analysis. First of all, we give the following property of the tangent hyperbolic function.

Lemma 13. Define a compact set Ω_𝑠𝑘 as Ω_𝑠𝑘∶={𝑠_𝑘(𝑡)∣|𝑠_𝑘(𝑡)|≤𝑚𝜂(𝑡)}. Then, for any 𝑠_𝑘(𝑡)∉Ω_𝑠𝑘, the following inequality holds: 1−𝑎tanh²𝑠_𝑘(𝑡)𝜂(𝑡)<0, (32) where 𝑚=ln(𝑎/(𝑎−1)+1/(𝑎−1)).

Proof. See the appendix.

The stability and convergence property of the proposed AILC scheme is summarized as follows.

Theorem 14. Considering closed-loop system (1), if Assumptions 1–6 and 9 hold, designing the control laws (28) with adaptive updating laws (29), the following properties can be guaranteed: (i) all the signals of the closed-loop system are bounded; (ii) the filtered tracking error 𝑒_𝑠𝑘(𝑡) converges to a small neighborhood of zero as 𝑘→∞ in 𝐿₂ norm; that is, lim_𝑘→∞∫^𝑇₀(𝑒_𝑠𝑘(𝜎))²d𝜎≤𝜀_𝑒𝑠𝑘, 𝜀_𝑒𝑠𝑘=(1/2𝐾)(1+𝑚)²𝜀²; and (iii) the tracking error vector satisfies lim_𝑘→∞‖𝑒_𝑘(𝑡)‖≤𝜀_𝑒∞, 𝜀_𝑒∞=(1+‖Λ‖)(𝑘₀∑^𝑛−1_𝑖=1𝜀_𝑖+(1/(𝜆₀−𝐾))(1+𝑚)𝜀𝑘₀)+(1+𝑚)𝜀, where 𝜆₀ and 𝑘₀ are positive constants and will be given later.

Proof. Define a Lyapunov-like CEF as follows: 𝐸_𝑘(𝑡)=12𝑞^𝑡₀𝛽^𝑇_𝑘(𝜎)𝛽_𝑘(𝜎)d𝜎. (33) The difference of 𝐸_𝑘(𝑡) is Δ𝐸_𝑘(𝑡)=𝐸_𝑘(𝑡)−𝐸_𝑘−1(𝑡)=12𝑞^𝑡₀𝛽^𝑇_𝑘(𝜎)𝛽_𝑘(𝜎)−𝛽^𝑇_𝑘−1(𝜎)𝛽_𝑘−1(𝜎)d𝜎. (34) Utilizing the algebraic relation (𝑎−𝑏)^𝑇(𝑎−𝑏)−(𝑎−𝑐)^𝑇(𝑎−𝑐)=(𝑐−𝑏)^𝑇[2(𝑎−𝑏)+(𝑏−𝑐)] and taking adaptive learning law (29) into account, we have the following inequality: Δ𝐸_𝑘(𝑡)=^𝑡₀𝑠_𝑘(𝜎)𝛽^𝑇_𝑘(𝜎)Φ_𝑘(𝜎)−𝑞d𝜎2^𝑡₀𝑠²_𝑘(𝜎)Φ_𝑘(𝜎)²≤d𝜎_𝑡0𝑠_𝑘(𝜎)𝛽^𝑇_𝑘(𝜎)Φ_𝑘(𝜎)d𝜎. (35) Substituting (31) into (35), it follows that Δ𝐸_𝑘(𝑡)≤−^𝑡₀𝑉̇_𝑘(𝜎)𝑏_𝑚(𝜎)+d𝜎_𝑡01𝑏_𝑚(𝜎)1−𝑎tanh²𝑠_𝑘(𝜎)𝜂(𝜎)𝜁_𝑘(𝜎)−d𝜎_𝑡0𝐾₂𝑠²_𝑘(𝜎)1d𝜎≤−𝑏_𝑚𝑉_𝑘(𝑡)−𝐾₂^𝑡₀𝑠²_𝑘(𝜎)+d𝜎_𝑡01𝑏_𝑚(𝜎)1−𝑎tanh²𝑠_𝑘(𝜎)𝜂(𝜎)𝜁_𝑘(𝜎)d𝜎. (36)
For analysis of stability, we consider two cases.
Case  1. Consider (𝑠_𝑘(𝑡)∈Ω_𝑠𝑘). When 𝑠_𝑘(𝑡)∈Ω_𝑠𝑘, |𝑠_𝑘(𝑡)|≤𝑚𝜂(𝑡) holds. If 𝑠_𝑘(𝑡)=0, we know 𝑒_𝑠𝑘(𝑡) is bounded by 𝜂(𝑡); that is, |𝑒_𝑠𝑘(𝑡)|≤𝜂(𝑡). If 𝑠_𝑘(𝑡)>0, we have 𝑠_𝑘(𝑡)=𝑒_𝑠𝑘(𝑡)−𝜂(𝑡), and from |𝑠_𝑘(𝑡)|≤𝑚𝜂(𝑡) we can obtain 𝑠_𝑘(𝑡)=𝑒_𝑠𝑘(𝑡)−𝜂(𝑡)≤𝑚𝜂(𝑡) which further implies 0<𝑒_𝑠𝑘≤(1+𝑚)𝜂(𝑡). Similarly, if 𝑠_𝑘(𝑡)<0, we have 𝑠_𝑘(𝑡)=𝑒_𝑠𝑘(𝑡)+𝜂(𝑡)≥−𝑚𝜂(𝑡) which means 0>𝑒_𝑠𝑘(𝑡)≥−(1+𝑚)𝜂(𝑡). Synthesizing the above analysis, we know that |𝑒_𝑠𝑘(𝑡)|≤(1+𝑚)𝜂(𝑡) holds. Obviously, 𝑥_𝑖,𝑘(𝑡) are bounded since 𝑋_𝑑(𝑡) is bounded. According to the smoothness of ℎ₁(⋅,⋅) and ℎ₂(⋅,⋅), we know that Φ_𝑘(𝑡) is a bounded vector. Recalling updating law (29), we know 𝛽₀(𝑡)=0, 𝑡∈[0,𝑇]; then, when 𝑠_𝑘(𝑡)∈Ω_𝑠𝑘, 𝛽_𝑘(𝑡) is bounded as well, 𝑘∈𝑁. Following this chain of reasoning, the boundedness of 𝑣_𝑘(𝑡) can be deduced. As such, all closed-loop signals are bounded.
Remark  15. Theoretically, 𝑚 can be made arbitrarily small by choosing 𝑎, for example, when we choose 𝑎=100 and 𝑚=0.099. This leads to 𝑠_𝑘 arbitrarily small. However, large 𝑎 may give rise to high gain control which can deteriorate the transient performance of closed-loop system. Consequently, in practical applications, the designers should choose appropriate design parameters to gain satisfactory transient performance and the ideal tracking error.
Case  2. Consider (𝑠_𝑘(𝑡)∉Ω_𝑠𝑘). According to Lemma 13, we know that the last term of Δ𝐸_𝑘(𝑡) can be removed from the analysis. Therefore, (36) can be simplified as Δ𝐸_𝑘(𝑡)1≤−𝑏_𝑚𝑉_𝑘(𝑡)−𝐾₂^𝑡₀𝑠²_𝑘(𝜎)d𝜎<0. (37)
Inequality (37) shows that 𝐸_𝑘(𝑡) is decreasing along iteration axis. Thus, the boundedness of 𝐸_𝑘(𝑡) can be guaranteed as long as 𝐸₁(𝑡) is finite. According to the definition 𝐸_𝑘, 𝐸₁(𝑡) is given by 𝐸₁(𝑡)=12𝑞^𝑡₀𝛽^𝑇₁(𝜎)𝛽₁(𝜎)d𝜎. (38) Taking the time derivative of 𝐸₁(𝑡) results in 𝐸̇₁(𝑡)=1𝛽2𝑞^𝑇1(𝑡)𝛽₁(𝑡). (39) Recalling parameter adaptive laws (29), we have 𝛽₁(𝑡)=𝑞𝑠₁(𝑡)Φ₁(𝑡), and then we obtain 𝐸̇₁(𝑡)=1𝛽2𝑞^𝑇1(𝑡)𝛽₁(𝑡)=1𝛽2𝑞^𝑇1(𝑡)𝛽₁(𝑡)−2𝛽^𝑇₁(𝑡)𝛽₁(𝑡)+1𝑞𝛽^𝑇₁(𝑡)𝛽₁(𝑡)=1𝛽2𝑞¹(𝑡)−𝛽(𝑡)^𝑇𝛽₁(𝑡)(𝑡)𝛽−𝛽−2₁(𝑡)−𝛽(𝑡)^𝑇𝛽₁(𝑡)+𝑠₁(𝑡)𝛽^𝑇₁(𝑡)Φ₁(𝑡)=1−𝛽2𝑞^𝑇1(𝑡)𝛽₁+𝛽^𝑇(𝑡)𝛽(𝑡)+𝑠₁(𝑡)𝛽^𝑇₁(𝑡)Φ₁(𝑡). (40) Substituting (31) into (40) yields 𝐸̇₁(𝑡)𝑉̇≤−¹(𝑡)𝑏_𝑚(𝑡)−𝐾₂𝑠²₁(𝑡)+1𝛽2𝑞^𝑇(𝑡)𝛽(𝑡). (41) Denote 𝛽_max=max_{𝑡∈[0,𝑇]}{(1/2𝑞)𝛽^𝑇(𝑡)𝛽(𝑡)}. Integrating the above inequality over [0,𝑡] leads to 𝐸₁(𝑡)−𝐸₁(0)1≤−𝑏_𝑚𝑉₁(𝑡)−𝐾₂^𝑡₀𝑠²₁(𝜎)d𝜎+𝑡⋅𝛽_max. (42) Obviously, 𝐸₁(0)=0, and then it follows from (42) that 𝐸₁(𝑡)≤𝑡⋅𝛽_max<∞ (43) which indicates the boundedness of 𝐸₁(𝑡), so 𝐸_𝑘(𝑡) is finite for any 𝑘∈𝑁. Using (37) repeatedly, we have 𝐸_𝑘(𝑡)=𝐸₁(𝑡)+^𝑘_𝑙=2Δ𝐸_𝑙(𝑡)<𝐸₁(𝑡)−1𝑏_𝑚^𝑘_𝑙=2𝑉_𝑙(𝑡)−^𝑘_𝑙=2𝐾₂^𝑡₀𝑠²_𝑙(𝜎)d𝜎≤𝐸₁(𝑡)−^𝑘_𝑙=2𝐾₂^𝑡₀𝑠²_𝑙(𝜎)d𝜎. (44) We rewrite inequality (44) as ^𝑘_𝑙=2𝐾₂^𝑡₀𝑠²_𝑙(𝜎)d𝜎≤𝐸₁(𝑡)−𝐸_𝑘(𝑡)≤𝐸₁(𝑡). (45) Let 𝑡=𝑇, and, taking the limitation of (45), it follows that lim^𝑘_{𝑘→∞𝑙=2}^𝑇₀𝑠²_𝑙(𝜎)1d𝜎≤𝐾₂𝐸₁(𝑇). (46)
Since 𝐸₁(𝑇) is bounded, with the aid of the convergence theorem of the sum of series, lim_𝑘→∞∫^𝑇₀𝑠²_𝑘(𝜎)d𝜎=0, which implies that lim_𝑘→∞𝑠_𝑘(𝑡)=𝑠_∞(𝑡)=0, ∀𝑡∈[0,𝑇]. Moreover, from definition (11), we can know that, when |𝑒_𝑠𝑘(𝑡)|≤𝜂(𝑡), 𝑠_𝑘(𝑡)=0, then lim_𝑘→∞∫^𝑇₀𝑠²_𝑘(𝜎)d𝜎=0 is equivalent to lim_𝑘→∞|𝑒_𝑠𝑘(𝑡)|≤𝜂(𝑡), which furthermore implies that lim_𝑘→∞∫^𝑇₀(𝑒_𝑠𝑘(𝜎))²d𝜎≤∫^𝑇₀(𝜂(𝜎))²d𝜎.
According to the boundedness of 𝐸_𝑘(𝑡), we can obtain the boundedness of 𝛽_𝑘(𝑡). From ∫^𝑡₀𝑠²_𝑘(𝜎)d𝜎≤∫^𝑇₀𝑠²_𝑘(𝜎)d𝜎, we can get the boundedness of 𝑠_𝑘(𝑡). Considering the finiteness of reference trajectory 𝑋_𝑑(𝑡), we further obtain that 𝑥_𝑖,𝑘(𝑡) are bounded. Based on the above reasoning, we can obtain the boundedness of 𝑣_𝑘(𝑡) by similar analysis in Case 1.
Synthesizing the derivations in two cases, we can conclude that the proposed control algorithm is able to guarantee that all closed-loop signals are bounded and lim_𝑘→∞|𝑒_𝑠𝑘(𝑡)|≤(1+𝑚)𝜂(𝑡). Therefore, we can obtain lim_𝑘→∞∫^𝑇₀(𝑒_𝑠𝑘(𝜎))²d𝜎≤𝜀_𝑒,  𝜀_𝑒=∫^𝑇₀((1+𝑚)𝜂(𝜎))²d𝜎=(1/2𝐾)(1+𝑚)²𝜀²(1−𝑒^−2𝐾𝑇) ≤ 1/2𝐾)(1+𝑚)²𝜀²=𝜀_𝑒𝑠𝑘; thus, the control objective is achieved. Furthermore, the bound of 𝑒_𝑠∞(𝑡) will satisfy lim_𝑘→∞|𝑒_𝑠𝑘(𝑡)|=𝑒_𝑠∞(𝑡)=(1+𝑚)𝜀𝑒^−𝐾𝑡, ∀𝑡∈[0,𝑇].
Define the vector 𝜓_𝑘(𝑡)=[𝑒_1,𝑘(𝑡),𝑒_2,𝑘(𝑡),…,𝑒_{𝑛−1,𝑘}(𝑡)]^𝑇, and then a state representation of 𝑒_𝑠𝑘(𝑡)=Λ^𝑇1𝑒_𝑘(𝑡) can be expressed as 𝜓̇_𝑘(𝑡)=𝐴_𝑠𝜓_𝑘(𝑡)+𝑏_𝑠𝑒_𝑠𝑘(𝑡), (47) where 𝐴_𝑠=01⋯0⋮⋮⋱⋮00⋯1−𝜆₁−𝜆₂⋯−𝜆_𝑛−1∈𝑅^{(𝑛−1)×(𝑛−1)},𝑏_𝑠=0⋮01∈𝑅^𝑛−1 (48) with 𝐴_𝑠 as a stable matrix. In addition, there are two constants 𝑘₀>0 and 𝜆₀>0 such that ‖𝑒^𝐴𝑠𝑡‖≤𝑘₀𝑒^−𝜆0𝑡 [50]. The solution for 𝜓̇_𝑘(𝑡) is 𝜓_𝑘(𝑡)=𝑒^𝐴𝑠𝑡𝜓_𝑘(0)+^𝑡₀𝑒^{𝐴𝑠(𝑡−𝜎)}𝑏_𝑠𝑒_𝑠𝑘(𝜎)d𝜎. (49) Accordingly, it follows from (49) that 𝜓_𝑘(𝑡)=𝑘₀𝜓_𝑘(0)𝑒^−𝜆0𝑡+𝑘₀^𝑡₀𝑒^{−𝜆0(𝑡−𝜎)}𝑒_𝑠𝑘(𝜎)d𝜎. (50) When we choose suitable parameters such that 𝜆₀>𝐾, from lim_𝑘→∞|𝑒_𝑠𝑘(𝑡)|≤(1+𝑚)𝜂(𝑡), we can have 𝜓_∞(𝑡)=𝑘₀𝜓_∞(0)𝑒^−𝜆0𝑡+𝑘₀^𝑡₀𝑒^{−𝜆0(𝑡−𝜎)}𝑒_𝑠∞(𝜎)d𝜎≤𝑘₀𝜓_∞(0)+(1+𝑚)𝜀𝑘₀^𝑡₀𝑒^{−𝜆0(𝑡−𝜎)}𝑒^−𝐾𝜎d𝜎=𝑘₀𝜓_∞(0)+(1+𝑚)𝜀𝑘₀1𝜆₀𝑒−𝐾^−𝐾𝑡−𝑒^−𝜆0𝑡≤𝑘₀𝜓_∞(0+1)𝜆₀()−𝐾1+𝑚𝜀𝑘₀. (51) Noting 𝑒_𝑠𝑘(𝑡)=[Λ^𝑇1]𝑒_𝑘(𝑡) and 𝑒_𝑘(𝑡)=𝜓^𝑇_𝑘(𝑡)𝑒_𝑛,𝑘(𝑡)^𝑇, we have 𝑒_𝑘(𝑡)≤𝜓_𝑘(𝑡)+𝑒_𝑛,𝑘(𝑡=𝜓_𝑘)(𝑡)+𝑒_𝑠𝑘(𝑡)−Λ^𝑇𝜓_𝑘(𝑡≤(‖Λ𝜓)1+‖)_𝑘(𝑡)+𝑒_𝑠𝑘(𝑡). (52) Combining the previous two inequalities, we can obtain 𝑒_∞(𝑡)≤(1+‖Λ‖)𝜓_∞(𝑡)+𝑒_𝑠∞(𝑡≤(‖Λ𝑘)1+‖)₀𝜓_∞(0+1)𝜆₀()−𝐾1+𝑚𝜀𝑘₀+()𝜂(𝑡)≤(‖Λ𝑘1+𝑚1+‖)₀^𝑛−1_𝑖=1𝜀_𝑖+1𝜆₀()−𝐾1+𝑚𝜀𝑘₀+()𝜂(𝑡)≤(‖Λ𝑘1+𝑚1+‖)₀^𝑛−1_𝑖=1𝜀_𝑖+1𝜆₀()−𝐾1+𝑚𝜀𝑘₀+(1+𝑚)𝜀=𝜀_𝑒∞. (53) This concludes the proof.

5. Simulation Studies

In this section, a simulation example is presented to verify the effectiveness of the AILC scheme. Consider the following second-order nonlinear system with unknown time-varying delays and unknown dead-zone running on [0,10], repetitively: 𝑥̇_1,𝑘(𝑡)=𝑥_2,𝑘(𝑡),𝑥̇_2,𝑘(𝑡)=𝑓𝑋_𝑘(𝑡),𝑋_𝜏,𝑘(𝑡),𝜃(𝑡)+𝑏(𝑡)𝑢_𝑘(𝑡)(𝑡),𝑦+𝑑_𝑘(𝑡)=𝑥_1,𝑘(𝑡),𝑢_𝑘(𝑡)=𝐷𝑣_𝑘(𝑡), (54) where  𝑓(𝑋_𝑘(𝑡),𝑋_𝜏,𝑘(𝑡),𝜃(𝑡))=−(𝑥_1,𝑘(𝑡)+𝑥_2,𝑘(𝑡))𝜃(𝑡)+exp(−𝜃(𝑡)((𝑥^𝜏_1,𝑘(𝑡))²+(𝑥^𝜏_2,𝑘(𝑡))²)), 𝑏(𝑡)=2+0.5sin𝑡, 𝑑(𝑡)=0.1sin𝑡, and the time delays are 𝜏(𝑡)=0.5(1+sin𝑡) with 𝜏_max=1, 𝜃(𝑡)=|cos(𝑡)|. It can be easily verified that exp−𝜃(𝑡)𝑋_𝑘²−exp−𝜃(𝑡)𝑋_𝑑²≤𝑋_𝑘−𝑋_𝑑2|𝜃(𝑡)|𝑒^−0.5. (55) Obviously, Assumptions 1–3 and Assumptions 5, 6, and 9 are satisfied. Moreover, we can know that ℎ₁=1 and ℎ₂=1. We give the simulation study in the following three cases.

Case 1. The reference trajectory to be tracked by the state vector is given by 𝑋_𝑑(𝑡)=[sin𝑡,cos𝑡]^𝑇. The design parameters are chosen as 𝜀₁=𝜀₂=1, 𝜆=2, 𝐾=3, 𝛾=0.5, 𝐾₁=2, 𝑞=1, 𝑎=5, 𝜀=𝜆𝜀₁+𝜀₂=3. The parameters for dead-zone are specified by 𝑚=1+0.2sin𝑡, 𝑏_𝑟=0.25, 𝑏_𝑙=−0.25. The initial conditions 𝑥_1,𝑘(0) and 𝑥_2,𝑘(0) are randomly taken in the intervals [−0.5,0.5] and [0.5,1.5], respectively. Parts of the simulation results are shown in Figures 2, 3, 4, 5, and 6. From the simulation results, we can see that the proposed AILC is effective in the sense that it can drive the tracking errors converge to zero along the iteration axis.

Figure 2 

System output 𝑦 on 𝑦𝑑(𝑘=1) in Case 1.

Figure 3 

System output 𝑦 on 𝑦𝑑(𝑘=10) in Case 1.

Figure 4 

Control input 𝑢𝑘 and 𝑣𝑘(𝑘=1) in Case 1.

Figure 5 

Control input 𝑢𝑘 and 𝑣𝑘(𝑘=10) in Case 1.

Figure 6 

∫𝑇0𝑠2𝑘(𝑡)d𝑡 versus the number of iterations in Case 1.

Case 2. To show the control performance for more complicated reference trajectory, we choose the reference trajectory as 𝑋_𝑑(𝑡)=[sin𝑡+sin(1.5𝑡),cos𝑡+1.5cos(1.5𝑡)]^𝑇. The design parameters are chosen the same as those in case 1. The initial conditions 𝑥_1,𝑘(0) and 𝑥_2,𝑘(0) are randomly taken in the intervals [−0.5,0.5] and [2,3], respectively. Parts of the simulation results are shown in Figures 7, 8, 9, 10, and 11. It shows that for more complicated reference trajectory the proposed approach is also able to achieve excellent tracking performance.

Figure 7 

System output 𝑦 on 𝑦𝑑(𝑘=1) in Case 2.

Figure 8 

System output 𝑦 on 𝑦𝑑(𝑘=10) in Case 2.

Figure 9 

Control input 𝑢𝑘 and 𝑣𝑘(𝑘=1) in Case 2.

Figure 10 

Control input 𝑢𝑘 and 𝑣𝑘(𝑘=10) in Case 2.

Figure 11 

∫𝑇0𝑠2𝑘(𝑡)d𝑡 versus the number of iterations in Case 2.

Case 3. Finally, the contribution of this paper is shown by comparing the proposed controller with traditional adaptive controller. The controller is the same, but the adaptive laws using 𝜎-modification for parameters are given by ̇𝛽(𝑡)=−ΓΦ_𝑘(𝑡)𝑠_𝑘+𝜎𝛽(𝑡),𝛽(0)=0. (56) The design parameters are given by Γ=diag{0.01} and 𝜎=0.5. Since traditional adaptive controller does not run repeatedly, the notation 𝑘 in this case does not have any practical meaning. Figure 12, 13, and 14 provide simulation results. From the simulation results shown below, it is obvious that the adaptive controller cannot achieve perfect tracking performance of the system output and reference trajectory.

Figure 12 

System output 𝑦 on 𝑦𝑑 in Case 3.

Figure 13 

Control input 𝑢𝑘 and 𝑣𝑘 in Case 3.

Figure 14 

Tracking error 𝑠(𝑡) versus time.

As observed in simulation results above, the proposed AILC can achieve a good tracking performance and tracking errors decrease along the iteration axis, which demonstrates the validity of the proposed AILC approach in this paper.

6. Conclusions

In this paper, a new AILC scheme is proposed for a class of nonlinear time-varying systems with both unknown time-varying time-delay and unknown input dead-zone nonlinearity in the presence of disturbance running on a finite time interval repetitively. A novel representation of the dead-zone output is given. Using appropriate Lyapunov-Krasovskii functional in the Lyapunov function candidate, the uncertainties from unknown time-varying delays are removed such that control law is delay-independent. The identical initial condition for ILC is relaxed by introducing the boundary layer. The hyperbolic tangent function is employed to avoid the possible singularity problem. Theoretical analysis by constructing Lyapunov-like CEF has shown that the tracking errors converge to a small residual domain around the origin as iteration goes to infinity. At the same time, all the closed-loop signals remain bounded. Simulation results have been provided to demonstrate the effectiveness the proposed control scheme.

Appendix

Proof of Lemma 13. For convenience in expression, denote 𝑥=𝑠_𝑘(𝑡)/𝜂(𝑡). We rewrite inequality (32) as 1𝑏<tanh²(𝑥)=𝑒^𝑥−𝑒^−𝑥𝑒^𝑥+𝑒^−𝑥22=1−𝑒^𝑥+𝑒^−𝑥2. (A.1) Noting the fact that 𝑒^𝑥 and 𝑒^−𝑥 are positive, it follows from (A.1) that 𝑒^𝑥+𝑒^−𝑥>2𝑏.𝑏−1 (A.2) Then, we can obtain 𝑒^2𝑥−2𝑏𝑒𝑏−1^𝑥+1>0. (A.3) Solving the quadratic inequality (A.3), we can have 0<𝑒^𝑥<𝑏()−𝑏−11(𝑏−1)or𝑒^𝑥>𝑏()+𝑏−11().𝑏−1 (A.4)
On the other hand, from |𝑠_𝑘(𝑡)|>𝑚𝜂(𝑡), we know that 𝑥<−𝑚or𝑥>𝑚 (A.5) which implies 0<𝑒^𝑥<1𝑏/(𝑏−1)+()=1/𝑏−1𝑏()−𝑏−11(𝑏−1)or𝑒^𝑥>𝑏()+𝑏−11().𝑏−1 (A.6)
Obviously, from the homology of (A.4) and (A.6), we know that Lemma 13 holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.