Noise-Tolerant ZNN-Based Data-Driven Iterative Learning Control for Discrete Nonaffine Nonlinear MIMO Repetitive Systems

Yunfeng Hu , Chong Zhang , Bo Wang , Jing Zhao , Xun Gong ,Jinwu Gao , and Hong Chen ,,

Abstract—Aiming at the tracking problem of a class of discrete nonaffine nonlinear multi-input multi-output (MIMO) repetitive systems subjected to separable and nonseparable disturbances, a novel data-driven iterative learning control (ILC) scheme based on the zeroing neural networks (ZNNs) is proposed.First, the equivalent dynamic linearization data model is obtained by means of dynamic linearization technology, which exists theoretically in the iteration domain.Then, the iterative extended state observer(IESO) is developed to estimate the disturbance and the coupling between systems, and the decoupled dynamic linearization model is obtained for the purpose of controller synthesis.To solve the zero-seeking tracking problem with inherent tolerance of noise,an ILC based on noise-tolerant modified ZNN is proposed.The strict assumptions imposed on the initialization conditions of each iteration in the existing ILC methods can be absolutely removed with our method.In addition, theoretical analysis indicates that the modified ZNN can converge to the exact solution of the zeroseeking tracking problem.Finally, a generalized example and an application-oriented example are presented to verify the effectiveness and superiority of the proposed process.

I.INTRODUCTION

A. Motivation

MANY applications, such as mechatronic motion system[1], manipulator [2], train operation systems [3] and magnetic memory alloy [4], require executing the same control tasks repeatedly over a finite-time interval.Iterative learning control manages the control purpose of systems with repetitive traits, where “l(fā)earning” means that when constructing the current control algorithm, the control input and tracking error information of the previous iterations stored by the memory device can be utilized to learn and correct the current control input to enhance the control performance of the current iteration [5], [6].Data-driven control means that the controller design does not explicitly contain the mathematical model information of the controlled process, but only uses the online or offline input-output (I/O) data of the controlled system and the knowledge obtained through data processing to design the controller.Therefore, data-driven control methods provide a new method for the precise control and optimization of complex systems [7]-[9].Furthermore, the noise is widespread and inescapable in practical engineering applications, which may lead some algorithms to fail to solve tasks.Consequently, this paper proposes a novel data-driven ILC for a class of discrete-time nonaffine nonlinear MIMO systems polluted by various noises.

B. Related Work

According to the stability analysis tools, the ILC methods can be divided into ILCs based on the contraction mapping[10], [11], composite energy function [12], [13] and normoptimal [14], [15].Traditional ILCs, such as P-type ILC, are analyzed and designed on the basis of the contraction mapping theory.Its main drawbacks come from two aspects: first,the system is required to meet the global Lipschitz continuous condition, which is strict and harsh for certain systems; The second is that the system converges in the sense of artificialλnorm rather than in the sense of reliable two-norm, which may produce undesirable transient behavior due to ignoring the dynamic characteristics of the system.Compared with ILC based on compression mapping, ILCs based on composite energy function [12], [13] and norm-optimal ILCs [14], [15]can improve undesirable transient problems.However, most of them are based on the availability of the affine dynamic model of the system, which can be classified as model-based ILCs.In addition, the ILC methods based on contraction mapping and composite energy functions need to make strict and harsh assumptions about the initialization conditions.The identical initial condition must be fulfilled or the initial state deviation of each iteration is tolerated in a small range, which is inconsistent with some practical engineering problems.

In practical engineering, systems are strongly nonlinear and subject to uncertainties, including external disturbances and parameter perturbations, so establishing a perfect model for the system is extremely difficult and even infeasible.Thus, the model-based ILC methods may lead to some safety and robustness concerns.Motivated by this problem, a data-driven ILC is useful for systems with complex dynamics [16], [17].Neural networks are considered a promising means for approximating nonlinear systems [18], [19].Two feedforward neural networks (FNNs) in [20] and [21] are utilized for the purpose of system output prediction and controller synthesis,however, they converge in the sense of theλ-norm and do not eliminate the problem of poor transient performance.Simultaneously, the gradient information of the objective function of the two neural networks with respect to the variables to be optimized needs to be obtained, which increases the burden of online calculation and reduces the feasibility.In addition,ILCs have been developed and successfully applied to various systems with specific attributes.For high-order uncertain nonlinear fractional order systems, an adaptive ILC method is proposed in [22].An open-loop PD type ILC scheme is developed in [23] for distributed parameter systems, and the convergence is proved theoretically.An ILC algorithm using quantized error information is designed in [24] and successfully applied to stochastic linear and nonlinear systems.In this paper, an ILC of discrete-time nonaffine nonlinear MIMO systems with complex dynamic characteristics polluted by various noises is developed.

The dynamic linearization (DL) technology proposed in recent years is an effective tool to solve the control task of systems with complex dynamic characteristics [25], [26].Datadriven control technology based on DL was first proposed by Hou in [27], and has been successfully applied in various fields [28], [29].DL has also been extended to nonlinear repetitive systems to develop iterative dynamic linearization (IDL)technology.The dynamic linearization model obtained by DL or IDL is completely equivalent to the original system and does not take advantage of any model information from the controlled plant, so it is data-driven [30], [31].The difference between these two methods is that DL is equivalent to the original system in the time domain, while IDL is equivalent in the iterative domain.Some previous studies have reported that data-driven ILCs based on IDL are suitable for single-input single-output (SISO) systems [16], [26].Limited by the complexity and convergence analysis tools of MIMO systems,there are few data-driven ILC solutions for MIMO systems.The IDL-based ILC for MIMO systems was proposed in [17],but it does not consider the coupling between systems and the uncertainties of the original system.If the system has excessively strong uncertainties and nonlinearities, the dynamic characteristics become extraordinarily complicated and difficult to estimate effectively.Because the extended state observer (ESO) can yield excellent estimations of uncertainties, an ILC method based on ESO is proposed in [32].

The iterative learning tracking controller can be described as solving the iteration-varying zero-finding problem.The effectiveness of recurrent neural networks (RNNs) in dealing with time-varying problems has become a consensus [33].Assuming that evolution along the iteration axis is regarded as a discrete operation, they can also be transplanted to solve iteration-varying issues.Zeroing neural networks (ZNNs), first proposed by Zhanget al.in [34] to solve time-varying problems, originate from RNNs.The outstanding performance of ZNNs in solving time-varying problems has two specific aspects [35], [36].First, the dynamic equation related to the system state is defined in ZNNs, and obtaining gradient information is not required, thus improving the computational efficiency.Also, the convergence of the ZNN can be guaranteed in the form of no residual error.Noise is widespread and inescapable in practical engineering applications.Noise may lead some models to fail to solve tasks.Therefore, a noise-tolerant ZNN model must be developed [37].The solution of various time-varying problems has been excellently verified on ZNNs, for instance, matrix inversion problems [33], reciprocal problems [38], and quadratic programming problems[39], but no relevant research has been reported in which they have been used to solve iterative learning tracking control problems.

The existing ILCs have the following limitations:

1) The ILC methods based on the composite energy function and norm optimization need to achieve an accurate system model in advance.Strong nonlinearities and uncertainties make it difficult or even infeasible to establish a perfect model of the systems in practical engineering.

2) Limited by the complexity and convergence analysis tools of MIMO systems, the existing research on ILC mostly focuses on SISO systems, there are few ILC solutions for MIMO systems.

3) Noise is widespread and inescapable in practical engineering applications, which may lead some algorithms to fail to solve tasks.Existing ILC solutions rarely consider the configuration of noise suppression mechanisms.

4) Some existing ILC methods need to meet harsh assumptions that the identical initial condition must be fulfilled or the initial state deviation of each iteration is tolerated in a small range, which is inconsistent with some practical engineering problems.

C. Main Contribution

The main challenges of this consider are shown in Fig.1(b):nonlinear control, decoupling control of MIMO system and noise suppression.Therefore, this paper aims to design ILCs for discrete nonlinear nonaffine MIMO repetitive systems with noise pollution.Specifically, iterative dynamic linearization technology and iterative extended state observation are used to obtain the decoupling dynamic linearization model(DDLM) equivalent to the original system for the purpose of controller synthesis, as shown in Fig.1(a).Then the ILCs are designed on the basis of DDLM combined with the noise-tolerant ZNNs, the control block diagram is shown in Fig.1(c).

Fig.1.Noise-tolerant ZNN-based data-driven ILC scheme: (a) Decoupling dynamic linearization model; (b) Major considerations; (c) System control scheme.

Compared with the previous research, the principal contributions are summarized as follows:

1) Compared with most existing ILC methods that need an accurate system model, a pure data-driven ILC method is developed for discrete nonlinear nonaffine MIMO repetitive systems.That is, the ILC design and analysis process is independent of the model information of the controlled plant and only uses the measured I/O data of the system.

2) According to the coupling characteristics of MIMO systems, iterative dynamic linearization technology and iterative extended state observation are used to obtain the decoupling dynamic linearization model equivalent to the original system for the purpose of controller synthesis.

3) The convergence of the proposed noise-tolerant ZNNbased data-driven ILC can be guaranteed in the form of no residual error.Aslo, the convergence of the proposed noisetolerant ZNNs is analyzed theoretically in the presence of various noises.

4) The guaranteed convergence of ILC proposed in this paper is not based on any assumptions or restrictions on the initialization conditions, which is more in line with the actual engineering application conditions.Additionally, the proposed ILC is convergent in the sense of 2-norm (rather thanλnorm).

D. Outline

The remainder of this paper is organized as follows.Section II introduces the problem formulation.Section III describes the dynamic linearization in the iterative domain based on IESO.In Section IV, the ILC based on ZNN is proposed.In Section V, the theoretical analyses and results are reported.Then, an illustrative example is presented in Section VI.Finally, conclusions are given in Section VII.For convenience, all variables involved are listed in Table I.

II.PROBLEM FORMULATION

A discrete nonaffine nonlinear MIMO system that operates repeatedly over a finite-time duration is as follows:

The motivation for considering the problem formulation (1)mainly comes from two aspects: i) The problem formulation(1) is a more general description of discrete nonlinear systems.Some specific discrete nonlinear systems, such as Hammerstein model, Bilinear model, Wiener model and nonlinear auto-regressive moving average with external input (NARMAX) model, etc., can be expressed as special cases of problem formulation (1); ii) The problem formulation (1) can be completely equivalent to a dynamic linearized data model for controller design theoretically.

We give the following assumptions about the standard properties of system (1).

whereL＞0 is a Lipschitz constant;ΔY(ε,n+1)=Y(ε,n+1)-Y(ε-1,n+1); ΔU(ε,n) =U(ε,n) -U(ε-1,n); ΔΨ(ε,n)=Ψ(ε,n)-Ψ(ε-1,n).

TABLE I NOMENCLATURE

Remark 1: Several standard property assumptions required on system (1) are reasonable for many control systems.Assumptions 1 and 2 are typical constraints for general nonlinear systems in control system design.From an energy point of view, when the change in control input energy and generalized disturbance energy is at a finite height, the change in output energy is also bounded, so many practical systems have the property of Assumption 3.

Remark 2: In the study of ILC theory [11], [13], [20], [21],the identical initial conditions need to be fulfilled, that is,yd j(ε,0)=yj(ε,0)is imposed on the controlled system, or the initial state deviation in each trial is tolerated in a small range,which are somewhat strong and harsh.However, the change of the initial state at each trial in practical applications is unavoidable, i.e.,yd j(ε,0)≠yj(ε,0).In this study, the above initialization conditions can be absolutely removed, which is more in line with the actual engineering application conditions.The details are provided below.

Based on the assumptions and remarks above, the control tasks can be defined as below.DLT and IESO are employed to obtain a decoupled dynamic linearized data model equivalent to system (1) for control synthesis purposes.Then, a modified ZNN is designed to achieve the ILC inputuj(ε,n) so that the system outputyj(ε,n+1) tracks the desired trajectoryyd j(ε,n+1) with the evolution of iteration,j∈{1,2,...,m}.Meanwhile, the convergence and anti-noise analysis of the entire system is carried out on the iterative domain.

III.DYNAMIC LINEARIZATION IN ITERATIVE DOMAIN BASED ON IESO

We take into account the case where the accurate mathematical model of system (1) is not available.Therefore, this paper simply exploits the I/O data of system (1), any structural information and parameter information of the controlled plant model need not be available and the DLT is utilized to obtain the dynamic linearization data model equivalent to system (1)in the iterative domain.Aslo, IESO is implemented to compensate for the unmodeled dynamics and the coupling between the loops of the system to further realize the decoupling of the MIMO system.

A. Decoupling Dynamic Linearization for Nonaffine Nonlinear MIMO Systems

For general nonaffine nonlinear MIMO systems (1), the following theorem proves that it can be completely equivalent to a dynamic linearized data model in the iterative domain.

Theorem 1: For system (1), if Assumptions 1-3 are satisfied, when ‖ ΔU(ε,n)‖≠0, there must be Φ (ε,n), pseudo Jacobian matrix (PJM), enabling (1) to be transformed into an equivalent dynamic linearization model in the iterative domain by using the DLT,

Proof: According to system (1), the following formula holds:

Define:

If the partial derivatives ofQ(·) with respect to Ψ(ε,n) are differentiable.In light of the differential mean value theorem and Assumption 2, (4) can be reorganized as

where

Consider the following equation with matrixM(ε,n):

Since ‖ΔU(ε,n)‖≠0 holds, the above formula has at least one solutionM?(ε,n).The PJM is defined as follows:

A new disturbance term is defined as follows:

Then, system (1) is able to equivalently transform to the DL model (3) in view of (4)-(9).■

Theorem 2: For system (1), if Assumptions 1-3 are satisfied, when ‖ ΔU(ε,n)‖≠0, the dynamic linearization model (3)can realize the decoupling of each control loop in the iteration domain by IESO, that is,

Proof: Define the following new diagonal matrix:

Then the dynamic linearization model (3) is further equivalent to the following form:

Then, (12) is further transformed as follows:

Δyj(ε,n+1)=φjj(ε,n)Δuj(ε,n)+?ψj(ε,n).(13)

Consequently, a decoupled dynamic linearization model(12) is available drawing support from DLT and IESO.■

B. Practical Decoupling Dynamic Linearization Model

For controller synthesis purposes, the two unknown terms Π(ε,n) and Ψ?(ε,n) in the decoupled dynamic linearization model (12) need to be estimated online with the I/O data of the controlled plant.

SSD算法利用多層特征圖來預(yù)測目標(biāo),但并未充分利用各個網(wǎng)絡(luò)層之間信息的結(jié)合。對于小目標(biāo)的檢測,主要利用淺層的網(wǎng)絡(luò)層去匹配,然而淺層網(wǎng)絡(luò)的特征圖表征能力較弱,對于小目標(biāo)容易出現(xiàn)誤檢和漏檢現(xiàn)象。

We consider the following PJM estimation criterion function:

where Π ?(ε,n) indicates the estimated value of Π (ε,n) andωis the positive penalty coefficient used to penalize excessive varieties in PJM estimation.Ψ??(ε-1,n) is the estimated value of Ψ?(ε-1,n), which is implemented by IESO as described below.The criterion function (14) is minimized to obtain the PJM estimation algorithm as follows:

where ξ ∈(0,1] is the added step factor.In the decoupled model, only the estimation of diagonal matrix Π is required,which reduces the computational burden compared with previous studies [17].

whereY?(ε,n) is the estimated value ofY(ε,n); Ψ??(ε,n) is the estimated value of Ψ?(ε,n); andLi(i=1,2) are the observer gains, which are set by the pole placement method.

The learning attribute of the algorithm proposed in this paper is also embodied in the learnable parameters that can be estimated and updated online by the parameter estimation laws (15) and (16), and the proposed parameter estimation laws have the ability to learn from the information in the previous iteration.

Remark 4: The “adaptability” of the proposed scheme comes from the parameter estimation laws (15) and (16).References [40] and [41] introduce the adaptive mechanism into the control architecture to improve the quality of the control system.Then, this paper is different from them in that the estimation and update of parameters are carried out in the iterative domain rather than the time domain.That is, the proposed parameter adaptive laws have the intelligence of learning in repetition.In addition, the implementation of parameter adaptive laws only uses online I/O data, so the proposed scheme is purely data-driven.

Proof: The system output estimation error vector is defined asE?(ε,n)=Y(ε,n)-Y?(ε,n).The IESO estimation error equation can be obtained as follows:

The pole placement method can be used to configure the poles of the estimation error equation at θi(i=1,2) to obtain the gainLi(i=1,2), namely,L1=2-θ1-θ2andL2=θ1θ2-θ1-θ2+1, |θi|≤1.Therefore, as ε →∞, the estimation error Ψ??(ε,n)converges to 0.

The following relationship can be obtained by subtracting Π(ε,n)from both ends of (15) and combining model (12):

According to Theorem 1, ‖Φ(ε,n)‖≤L, it can be seen that‖Π(ε-1,n)-Π(ε,n)‖≤2L.Applying the norm to(18) gives

Consequently, the above formula signifies that the estimation error Π ?(ε,n) of PJM is also bounded.■

Finally, the practical decoupling dynamic linearization model for controller synthesis can be expressed as

Remark 5: As stated in Theorem 3, the estimation error of φj j(ε,n) converges to zero, and the estimation error ofψ?j(ε,n)is bounded.Therefore, the reconstruction error of the unknown nonlinear system obtained by the IDL technology and parameter estimation laws is also bounded.

IV.ILC DESIGN BASED ON ZNN

The ILC based on the decoupled dynamic linearization model (22) is developed so that the output of each control loopyj(ε,n) can track its expected valueyd j(ε,n).

A. ZNN Dynamic Model

Consider thejth outputyjtrajectory tracking problem,which is formulated as follows:

whereyj(ε,n+1) is thejth output at timen+1 in theε-th iteration andyd j(n+1) represents the expected value of thej-th output at timen+1.Then the purpose is to design the unknown control inputuj(ε,n) to make (23) hold.Suppose that there is always an ideal control inputud j(ε,n) in each iteration.The following error function is defined:

Theej(ε,n+1) can be regarded as the gap between the control inputuj(ε,n) and the ideal control inputud j(ε,n).In other words, ifuj(ε,n) is sufficiently close toud j(ε,n) when ε →∞,e(ε,n+1)can converge to zero.Therefore, for the sake of solving the tracking problem (23), theej(ε,n+1) is defined as

The principle of the ZNN dynamic model is applied to solve tracking problem (23), and the evolution form of the defined error function is as follows:

where α ＞0 can adjust the convergence speed of the error function ande˙j(ε,n+1) is the derivative ofej(ε,n+1).The ZNN dynamic equation (26) has been proven to be globally convergent, that is, the control inputuj(ε,n) can globally converge asymptotically to the ideal inputud j(ε,n).However,ILC is essentially a discrete process in the iterative domain, so a discrete ZNN dynamic model needs to construct for solving the tracking problem (23).Therefore, the discrete ZNN can be obtained by adopting the backward difference for the continuous ZNN dynamic model (26) in the iterative domain

ej(ε,n+1)-ej(ε-1,n+1)=-αej(ε,n+1).(27)

Combining (27) with (22) and (25), the tracking problem(23) is solved as follows:

For the convenience of notation, (27) is recorded as a traditional zeroing neural network (TZNN), and (28) is written as an ILC based on TZNN (TZNN-ILC).

B. Noise-Tolerant Modified ZNN Dynamic Models

To alleviate the sensitivity of the ZNN dynamic model polluted by different noises, in this section, several noise-tolerant modified ZNN dynamic models are designed.Inspired by the fact that integral control technology can effectively deal with noise, the integral of the error function is introduced into the TZNN to obtain the following noise-tolerant modified ZNN:

where β ＞0 is the parameter to be designed.Similarly, combining (29) with (22) and (25), the tracking problem (23) is solved as follows:

Equations (29) and (30) are, respectively, recorded as a type I modified zeroing neural network (MZNNI) and an ILC based on MZNNI (MZNNI-ILC).Furthermore, combining TZNN with two integral terms will further enhance its ability to suppress the effect of noise.Then another noise-tolerant modified ZNN dynamic model is designed as follows:

The tracking problem (23) based on (31) is solved as follows:

Equations (31) and (32) are recorded as a type II modified zeroing neural network (MZNNII) and an ILC based on MZNNII (MZNNII-ILC), respectively.The algorithm steps of MZNNII-ILC are presented in Table II.

TABLE II ALGORITHM STEPS OF THE MZNNII-ILC

It is worth emphasizing that the dynamic change information of the proposed ZNN model is reflected on the iteration axis rather than the time axis.Therefore, the physical meaning of iteration-derivation information in noise-tolerant modified ZNN models refers to the tracking error change between two adjacent iterations atn-th time instance.The system structure of the ZNN model (32) for solving the trajectory tracking optimization problem is shown in Fig.2, which utilizes both the iteration-derivation information and iteration-integration information.

To facilitate the subsequent robustness analysis of the MZNN dynamic models, the noisewj(ε) is injected into the dynamic models (27), (29) and (31) to derive the following noise-polluted ZNN dynamic models:

Fig.2.The system structure of the ZNN model (32).

Next, the stability and robustness of the MZNNII-ILC will be investigated.In addition, the analyses on stability and robustness for TZNN-ILC and MZNNI-ILC are similar to MZNNII-ILC, and thus, are omitted.

Remark 6: Noise is widespread and inescapable in practical engineering.Therefore, it is necessary to consider the noise when solving the trajectory tracking problem (23).Note thatwj(ε)may be bounded noise (constant noise), unbounded noise (velocity or acceleration noise), random noise and even combinations of the three.Therefore, the noise-tolerant ZNN models (29) and (31) are developed by introducing the integral term.The noise pollution is effectively suppressed when solving the problem (23).

V.THEORETICAL ANALYSES AND RESULTS

In this section, the proof of the convergence and robustness of MZNNII-ILC for suppressing noise pollution is provided.The corresponding theoretical analysis results are also presented.

The following theorem illustrates that MZNNII-ILC has the propertyofglobalconvergence.Lettheoutput trackingerror vectorbeE(ε,n)=[e1(ε,n),...,em(ε,n)]T∈Rm, where thejth output tracking error is marked asej(ε,n).

Theorem 4: The tracking errorof MZNNIIILC for solving the tracking problem (23) is 0, and ‖·‖2is the Euclidean norm of the vector.

Proof: The MZNNII dynamic model (31) can be evolved as follows:

Then, (34) is subtracted from (31) to obtain

Furthermore, (35) can be rewritten as follows:

Then, (36) is subtracted from (35) to obtain

The linear dynamic equation of thejth output tracking error is obtained through a series of mathematical processings, so the properties of MZNNII-ILC can be analyzed under the framework of linear control theory.Define a new error vector as.Then equation(37) can be rearranged into the following form:

where

By performing norm operation on (38) and according to its recurrence property, we can obtain

It can be ensured that the real part of the eigenvalues of matrixAis less than 1 by selecting appropriate parametersαandβto be designed.Thenholds, so the following formula is also true:

Consequently, according to the relationship between Πεjandej(ε,n), l imε→∞‖E(ε,n)‖2=0 holds.

Remark 7:It can be concluded from (40) that, as stated in Remark 2, this paper does not need to impose on the controlled system to fulfill the same initial conditions, and any restrictions on the initial conditions are completely removed.This is more conducive to practical engineering applications.In addition, Theorem 4 illustrates that the proposed MZNNIIILC can globally converge to the ideal control inputud j(?,n)of tracking problem (23).

To analyze the robustness of MZNNII-ILC to inhibit noise pollution, the results of a series of theorems and theoretical analysis processes are given below.

Theorem 5: The tracking error limε→∞‖E(ε,n)‖2of constant noise-polluted MZNNII (33c) for solving the tracking problem (23) is 0 and is independent of the amplitude of unknown constant noise.

Proof: Suppose the constant noise pollution iswj(ε)=bj.ThenZ-transform is performed in the iteration domain for the constant noise-polluted MZNNII (33c),

The transfer function ofej(z,n) is

According to the properties of the terminal value theorem ofZ-transformation, we can obtain

The transfer functions ofej(z,n) are

According to the properties of the terminal value theorem ofZ-transformation, (45a) can be converted to

Similarly, (45b) can be converted to

TABLE III THE COMPARISON OF CONVERGENCE PERFORMANCE OF DIFFERENT ZNN DYNAMIC MODELS

Furthermore, the following theorem gives a proof of the robustness of MZNNII-ILC to deal with more general random noise pollution.Assume that the random noise-pollution of (33c) iswj(ε)=δj(ε).

Theorem 7: The random noise-polluted MZNNII (33c) for solving the tracking problem (23) is of boundary input and boundary output (BIBO) stability, and limε→∞‖E(ε,n)‖2＜

Proof: Adopting the same mathematical approach as in(34)-(37) for the noise-pollution MZNNII dynamic model(33c), we have

Then, (48) is rearranged into the following state space expression:

Similarly, by performing norm operation on (50) and according to its recurrence property, we have

Remark 8: As stated in Theorems 5-7, although the proposed MZNNII-ILC is contaminated by noise, the convergence guarantee of the iterative learning controller does not require the controlled system to meet the harsh same initial conditions.In addition, the noise in (33c) may be a combination of several types of noises.Since the dynamic model (33c)is essentially a linear system, the combined noise-pollution MZNNII-ILC can be analyzed by the superposition principle.

VI.ILLUSTRATIVE EXAMPLES

In this section, two trajectory tracking examples of the coupled nonlinear MIMO system are analyzed and studied.The solution accuracy of trajectory tracking problem (23) with different noise-pollutions are compared, and the effectiveness and superiority of the MZNNII-ILC and MZNNI-ILC proposed in this paper compared with the TZNN-ILC are verified.Simultaneously, the proposed ILC schemes are analyzed and compared with existing ILC schemes.

A. Generalized Example

A discrete-time MIMO system is considered as

1)Noise-Tolerant Performance Verification: The parameters of the PJM estimation algorithm (PJMEA), IESO and MZNNII-ILC are designed as shown in Table IV.The mean absolute errors (MAEs) ofy1andy2at theith iteration are defined as MAE1 and MAE2, respectively, and a new vector MAE = [MAE1, MAE2] is defined.E=[e1(ε,n),e2(ε,n)] is defined as the error index vector.Next, the effectiveness and advantage of the proposed ILCs will be verified in four different noise-pollution cases.

TABLE IV THE PARAMETERS INVOLVED IN SIMULATION

Constant noise: Fig.3 shows the simulation results of three different ILCs in the presence of constant noise.Figs.3(a) and 3(b) are the tracking curves ofy1andy2at the 50th iteration,respectively.Figs.3(c) and 3(d) show the maximum tracking errors ofy1andy2at each iteration, respectively.Fig.3(e)shows the evolution curve of ||MAE||2.Figs.3(f), 3(g) and 3(h) indicate the tracking error ||E||2curves under different ILCs.Fig.3 shows that MZNNII-ILC and MZNNI-ILC proposed can make the tracking error converge to near zero without being affected by the constant noise, while tracking error under the TZNN-ILC converges to a certain bound, which is affected by the constant noise amplitude.

Velocity noise: Fig.4 shows the simulation results of three different ILCs in the presence of velocity noise.Since the system output under TZNN-ILC tends to diverge and is not in the observable field of vision, Figs.4(a) and 4(b) only show the tracking performances ofy1andy2under MZNNII-ILC and MZNNI-ILC at the 50th iteration.Fig.4 shows that MZNNIIILC can accurately solve the tracking problem (23), and the tracking error is driven to a certain limit under the MZNNIILC.However, TZNN-ILC cannot solve tracking problem(23) with velocity noise pollution, and its noise-tolerant performance is the worst.

Fig.3.Solving the trajectory tracking problem (23) with constant noise w j(ε)=10.(a) y1 tracking performance at 50th iteration; (b) y2 tracking performance at 50th iteration; (c) The learning performance of y1 under different ILCs; (d) The learning performance of y2 under different ILCs; (e) The evolution curve of||MAE||2; (f) The tracking error curve under MZNNII-ILC; (g) The tracking error curve under MZNNI-ILC; (h) The tracking error curve under TZNN-ILC.

Fig.4.Solving the trajectory tracking problem (23) with velocity noise w j(ε)=2ε+10.(a) y1 tracking performance at 50th iteration; (b) y2 tracking performance at 50th iteration; (c) The learning performance of y1 under different ILCs; (d) The learning performance of y2 under different ILCs; (e) The evolution curve of | |MAE||2; (f) The tracking error curve under MZNNII-ILC; (g) The tracking error curve under MZNNI-ILC; (h) the tracking error curve under TZNNILC.

Acceleration noise: Fig.5 shows the simulation results of three different ILCs in the presence of acceleration noise.Figs.5(a) and 5(b) show the tracking performances ofy1andy2under MZNNII-ILC for the 5th, 20th and 50th iterations.Fig.5 illustrates that neither MZNNI-ILC nor TZNN-ILC can essentially deal with acceleration noise pollution, while the tracking error is driven to a certain bound under the MZNNIILC, which is affected by the acceleration noise amplitude.

Random noise: Many analyses of the controller noise tolerance assume that the noise type or the noise distribution is known.However, the uncertainty of the actual system makes it more meaningful to study random noise tolerance.Fig.6 shows the simulation results of three different ILCs under random noises.The proposed MZNNII-ILC and MZNNI-ILC exhibit remarkable robustness against random noise pollutions compared to TZNN-ILC.

Fig.5.Solving the trajectory tracking problem (23) with acceleration noise w j(ε)=ε?ε+10.(a) y1 tracking performance at different iterations under MZNNII-ILC; (b) y2 tracking performance at different iterations under MZNNII-ILC; (c) The learning performance of y1 under different ILCs; (d) The learning performance of y2 under different ILCs; (e) The evolution curve of ||MAE||2; (f) The tracking error curve under MZNNII-ILC; (g) The tracking error curve under MZNNI-ILC; (h) the tracking error curve under TZNN-ILC.

Fig.6.Solving the trajectory tracking problem (23) with random noise [-5,5]+10.(a) y1 tracking performance at 50th iteration; (b) y2 tracking performance at 50th iteration; (c) The learning performance of y1 under different ILCs; (d) The learning performance of y2 under different ILCs; (e) The evolution curve of||MAE||2; (f) The tracking error curve under MZNNII-ILC; (g) The tracking error curve under MZNNI-ILC; (h) The tracking error curve under TZNN-ILC.

Step trajectory: Fig.7 shows the simulation results of three different ILCs in the presence of velocity noise.Since the system output under TZNN-ILC tends to diverge and is not in the observable field of vision, Figs.7(a) and 7(b) only show the tracking performances ofy1andy2under MZNNII-ILC and MZNNI-ILC at the 50th iteration.Figs.7(c) and 7(d) are the evolution curves of the mean absolute errors ofy1andy2under three different ILCs, respectively.Fig.7 shows that MZNNII-ILC can accurately solve the random step trajectory tracking problem, and the tracking error is driven to a certain limit under the MZNNI-ILC.However, TZNN-ILC cannot solve the tracking problem with velocity noise pollution, and its noise-tolerant performance is the worst.Therefore, this set of simulation results further verified the theoretical analyses stated in Theorems 5 and 6.

2)Comparisons: For the sake of verifying the superiority of the developed ILC methods based on the modified ZNN in the presence of noise pollution, three existing ILC schemes were applied for comparative analysis.The first scheme is the classical P-type ILC method, called P-type-ILC.The second scheme is the model-free adaptive ILC proposed in [42],labeled MFAILC, which is designed according to the dynamic linearized data model (22).The third control scheme is the ILC method on account of feedforward neural network proposed in [20], labeled FNN-ILC, which takes advantage of RBFNN to learn ILC according to the (22), and the network parameters are updated by the gradient descent method.The parameters of the comparison controllers are designed as shown in Table IV.The ZNN based ILCs proposed in this paper are (28), (30), and (32), respectively.The control laws under different algorithms are summarized in Table V.

Fig.7.Solving the problem of random step trajectory tracking with velocity noise w j(ε)=2ε+10.(a) y1 tracking performance at 50th iteration; (b) y2 tracking performance at 50th iteration; (c) The evolution curve of MAE1; (d) The evolution curve of MAE2.

The system outputs are considered to be polluted by noise

wherec(ε,n) andd(ε,n) are the noises imposed ony1(ε,n) andy2(ε,n), respectively, considering that they have the following time-iteration-varying form:

Record the noise pollution shown in (56) as Case I.Figs.8(a)and 8(b) show time-iteration-varying noisesc(ε,n) andd(ε,n)in Case I, respectively.Figs.8(c) and 8(d) are the evolution curves of the mean absolute errors ofy1andy2under different algorithms, respectively.Fig.8 illustrates that when the system outputs are polluted by (56), the tracking errors of the six algorithms can converge to the residual error within a certain range.However, the MZNNI-ILC and MZNNII-ILC proposed in this paper exhibit excellent accuracy in solving the tracking problem (23) due to their essentially noise-tolerant characteristics.

Next, the system outputs are subject to more severe noise pollution, as follows:

Record the noise pollution shown in (57) as Case II.Figs.9(a) and 9(b) show time-iteration-varying noisesc1(ε,n)andd1(ε,n) in Case II, respectively.Figs.9(c) and 9(d) are the evolution curves of the mean absolute errors ofy1andy2under different algorithms, respectively.Fig.9 shows that when the system is more strongly polluted by noise, the tracking errors under P-type-ILC, MFAILC, FNN-ILC and TZNNILC will diverge.The proposed MZNNII-ILC combines two integral terms and can effectively deal with noise, so it shows good convergence despite noise pollution.

B. Thermal Management System of PEMFC

In this section, a thermal management system of proton exchange membrane fuel cell (PEMFC) is considered, as shown in Fig.10.The main control objective is to control the outlet and inlet temperatures of the cooling water (Tst,outandTst,in) by manipulating the flow rate of the circulating water pump and the air flow rate of the radiator (WclandWair), so as to achieve the purpose of controlling the temperature of the stack.However,Tst,outandTst,indeviate from the expectedvalues due to dynamic load scenarios (varying stack current Ist) and uncertain information.Therefore, the thermal management system of PEMFC is a multi input and multi output coupling nonlinear system.The working mechanism of the thermal management system for PEMFC is described in detail in [43].It is worth emphasizing that we only need it to obtain input and output data, and the proposed method does not need any system model information.

TABLE V CONTROL LAWS UNDER DIFFERENT ALGORITHMS

Fig.8.Comparison results for Case I.(a) Time-iteration-varying c (ε,n) ; (b) Time-iteration-varying d (ε,n); (c) The evolution curves of | MAE1| under different ILCs; (d) The evolution curves of | MAE2| under different ILCs.

The system state is defined as[x1(ε,n),x2(ε,n)]=[Tst,out(ε,n),Tst,in(ε,n)], the control input vector is[u1(ε,n),u2(ε,n)]=[Wcl(ε,n),Wair(ε,n)], the output vector is[y1(ε,n),y2(ε,n)]=[x1(ε,n),x2(ε,n)], and the expected value vector is[yd1(n+1),yd2(n+1)].Consider the stack current shown in Fig.11 to simulate the dynamic load scenario in order to verify the performance of the controllers.The system outputs are assumed to be polluted by noise:

Fig.9.Comparison results for Case II.(a) Time-iteration-varying c (ε,n) ; (b) Time-iteration-varying d (ε,n); (c) The evolution curves of | MAE1| under different ILCs; (d) The evolution curves of | MAE2| under different ILCs.

Fig.10.Schematic diagram of the PEMFC’s thermal management system.

Fig.11.The curve of stack current.

wherec2(ε,n) andd2(ε,n) are the noises imposed ony1(ε,n)andy2(ε,n), considering that they have the following timeiteration-varying form:

Fig.12 shows the simulation results of the thermal management system for PEMFC.Figs.12(a) and 12(b) are the tracking curves ofy1andy2at the 70th iteration, respectively.Figs.9(c) and 9(d) show the evolution curves of the mean absolute errors ofy1andy2under three algorithms based on ZNN,respectively.Figs.12(a)-12(d) show that three ILCs based on ZNN have excellent convergence performances in the orientation of iteration axis.Figs.12(e) and 12(f) show time-iteration-varying noisesc2(ε,n) andc2(ε,n) imposed ony1andy2,respectively.Figs.12(g) and 12(h) are the mean absolute errors polluted by noise under three algorithms based on ZNN,respectively.Figs.12(g) and 12(h) illustrate that the convergence performances of the proposed MZNNII-ILC and MZNNI-ILC in the iterative axis orientation are significantly better than that of TZNN-ILC in the case of noise pollution due to the noise-tolerant mechanism.

VII.CONCLUSIONS

Fig.12.Performance verification of thermal management system for PEMFC: (a) y1 tracking performance at 70th iteration; (b) y2 tracking performance at 70th iteration; (c) The evolution curves of MAE1 under different ILCs; (d) The evolution curves of MAE2 under different ILCs; (e) Time-iteration-varying c2(ε,n); (f) Time-iteration-varying d 2(ε,n); (g) The evolution curves of MAE1 polluted by noise under different ILCs; (h) The evolution curves of MAE2 polluted by noise under different ILCs.

A novel data-driven ILC scheme is proposed for settling the trajectory tracking problem of a class of discrete nonaffine nonlinear MIMO repetitive systems.By using IDL and IESO,a decoupling dynamic linearization model for controller synthesis is obtained.The trajectory tracking problem can be regarded as a zero-seeking optimal problem in the iterative domain.Therefore, an ILC scheme based on ZNN is proposed.Meanwhile, inspired by the fact that the integral control technology can effectively deal with noises, the noise-tolerant ZNN-based ILC is further developed.In addition, several theorems are proposed to investigate the convergence and noise-tolerance of MZNNII-ILC.Simulation results convincingly indicate that the proposed ILCs based on the modified ZNN with noise tolerance have significant advantages over the ILC based on the traditional ZNN.Furthermore, compared with the several existing ILCs, the designed schemes exhibit remarkable tracking accuracy and outstanding convergence in the presence of noise-pollution.