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

This paper presents an adaptive iterative learning control (AILC) scheme for a class of nonlinear systems with unknown timevarying 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.


Introduction
Practically, many engineering systems carry out repetitive tasks in fixed finite space, such as manipulators [1][2][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][2][3][4][5][6][7][8][9][10][11][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][14][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][18][19][20][21][22][23][24][25][26][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][29][30][31][32][33][34][35] due to its mathematical challenge and application demand in real-time control.In [28][29][30][31], the controller design and stability analysis for statedelayed systems were presented and [32][33][34][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][37][38][39][40][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 timevarying delay.In [41], an adaptive learning control design was developed for a certain class of first-order nonlinearly parameterized systems with unknown periodically timevarying delay and further extended to a class of highorder 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][43][44][45][46][47][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 timevarying 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 deadband without necessarily constructing a dead-zone inverse.An appropriate Lyapunov-Krasovskii functional and Young's inequality are combined to eliminate the unknown timevarying 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 Lyapunovlike 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.
To facilitate control system design, we make the following reasonable assumptions for the system functions, unknown time delays, and reference signals.
Assumption 3. The sign of () is known; without loss of generality, we always assume () > 0.
Assumption 4 (see [23]).The initial state errors  , (0) at each iteration are not necessarily zero small and fixed, but they are assumed to be bounded.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][40][41] as it does not require the true value of  max and .
Remark 8.As () is continuous on [0, ], there exist constants 0 <  min ≤  max such that  min ≤ () ≤  max .However, the control gain bounds  min and  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.

Dead-Zone Characteristic.
The dead-zone characteristic can be described as where   ≥ 0 and   ≤ 0 are unknown constants, () > 0 is unknown time-varying slopes, and V  () is the input and   () is the output of dead-zone.A graphical representation of the dead-zone in this paper is shown in Figure 1.
The dead-zone output   () is not available for measurement.We make the following assumption on the dead-zone parameters.From a practical point of view, we can redefine the deadzone nonlinearity as with It is obvious that  1 (V  ()) is bounded.
Remark 10.Obviously, the dead-zone characteristic is nonlinear.And the form in [36] is the special case of (3) when () is invariant.Therefore, the presentation of dead-zone in our work is more general than the earlier results.

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, ] as follows: where   () and    () are the system state and the control input in the th iteration, respectively, () is an unknown time-varying parameter, and (  , ) is a known time-varying function.The reference trajectory is   (),  ∈ [0, ].Define the tracking error as    () =   () −   () and design the control law and adaptive learning law for the unknown timevarying parameter for the th iteration as follows: where  1 ,  > 0 are design parameters.Define the estimate error as θ () = θ () − ().Choose a Lyapunov-like CEF as Throughout this paper,  denotes the integral variable.Then it can be derived that We can further derive that lim Therefore, the system state   () converges to the reference trajectory   () on [0, ] as  → ∞.

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.
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   () converges to a small neighborhood of zero as  → ∞ in  2 norm; that is,

and (iii) the tracking error vector satisfies lim
, where  0 and  0 are positive constants and will be given later.

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: where (  (),  , (), ()) = −( Obviously, Assumptions 1-3 and Assumptions 5, 6, and 9 are satisfied.Moreover, we can know that ℎ 1 = 1 and ℎ 2 = 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   () = [sin , cos ]  .The design  It shows that for more complicated reference trajectory the proposed approach is also able to achieve excellent tracking performance.
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 -modification for parameters are given by  The design parameters are given by Γ = diag{0.01}and  = 0.5.Since traditional adaptive controller does not run repeatedly, the notation  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.
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.

Conclusions
In this paper, a new AILC scheme is proposed for a class of nonlinear time-varying systems with both unknown timevarying 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 deadzone 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 (A.6) Obviously, from the homology of (A.4) and (A.6), we know that Lemma 13 holds.
Assumption 5.The reference state trajectory   () is continuous, bounded, and available.|()| ≤  max with an unknown constant  max .