Iterative learning control realized using an iteration-varying forgetting factor based on optimal gains

Abstract

Iterative learning control with forgetting factor (ILCFF) is widely used in control engineering. However, choosing the optimal parameters of ILCFF to improve system-output characteristics has been a challenging issue for controller designers. This paper proposes an iterative learning control (ILC) algorithm that involves a variable forgetting factor based on optimal gains for a class of discrete linear time-invariant systems with aperiodic disturbances. The convergence of the algorithm is analyzed, and the necessary and sufficient condition for its convergence is derived in terms of proportional–integral–derivative coefficients. A design method based on optimal gains is established to determine the algorithm coefficients and to accelerate system convergence. Furthermore, the influence of the forgetting factor on both the system-output error and the scope of the proposed algorithm is analyzed. Additionally, the most suitable system type for the application of the forgetting factor is determined. The effectiveness of the algorithm is verified by performing a theoretical analysis and a case-based simulation. The proposed iteration-varying optimal forgetting-factor-based ILC algorithm undergoes fast convergence with a small system-output error. The findings disrupt the conventional view that the use of the forgetting factor increases system-output error. In fact, in a system with small trajectory and increased disturbances, the error induced by the forgetting factor may be smaller than that of the traditional optimal ILC algorithm.

Keywords

Iterative learning control forgetting factor optimal gains

Introduction

As a feedforward control technique, iterative learning controls (ILC) provide tracking-control capabilities for systems that work in a repetitive mode. The approach is used to help realize high-precision tracking for a given trajectory within a specified time range (Liu, 2014) and has been widely used in numerous repetitive-motion systems (e.g., robots (Jin, 2018), aerial vehicles (He et al., 2019), and motors (Hui et al., 2021; Song et al., 2018)) involving stringent requirements for trajectory tracking.

ILC applies an application-oriented control algorithm that emphasizes excellent output characteristics rather than convergence. There are various ILC design strategies that are used to improve system output (Bouakrif and Zasadzinski, 2018; Chi et al., 2017; Liu et al., 2020; Meng and Moore, 2016). Optimization theory has been used to maximize convergence rates and minimize errors (Harte et al., 2005; Liu, 2014). Ideally, learning performance can be predicted if the system Markov parameters are available and utilizable (Liu et al., 2019). In this regard, earlier studies have proposed optimal learning-control schemes that applied numerical techniques (e.g., gradient descent, Newton-Raphson, and Gauss-Newton) to minimize tracking errors (Furuta and Yamakita, 1987; Togai and Yamano, 1985). Amann et al. (1996) presented a norm-optimal ILC (NOILC) solution that demonstrated well-defined convergence properties and design guidelines and provided experimental results using an electromechanical test facility. The NOILC framework was designed to minimize the quadratic optimization problem by selecting weighting matrices to target convergence and robustness criteria (Memon and Shao, 2020). It deals with linear and nonlinear systems in the time domain and has been applied to a variety of systems (Owens et al., 2013; Rafajowicz and Rafajowicz, 2018; Sun and Alleyne, 2014; Yu et al., 2018, 2020). For the sake of deducing computation complexity, Owens and Feng (2003) proposed a parameter-optimal ILC (POILC) scheme for discrete-time systems. The POILC was successfully applied to several different repetitive environments (Owens et al., 2012; Yahyazadeh and Noei, 2015). Based on the above studies, a novel method is suggested to find the optimal gains of the proportional–integral–derivative (PID) type ILC using the optimization techniques in the discrete time domain. Madady (2008, 2013) and Madady et al. (2018) developed a method of finding optimal gains for PID type, extended PID types, and N-parametric ILC controllers to achieve monotonic convergence to generate meaningful results. Memon and Shao (2020, 2021) presented new approaches using the quadratic performance index to optimize the gains of PID-type and data-driven ILCs. However, the system presented in Madady (2008), Memon and Shao(2020, 2021) was ideal and did not include disturbances. The system presented in Madady (2013) and Madady et al. (2018) had repetitive interferences; however, did not improve ILC performance. Although various optimal ILC approaches have been proposed, several widely used algorithms are not optimized yet. By applying optimization method, the algorithm efficiency and control performance can be further improved.

ILC with forgetting factor (ILCFF) has been studied and applied to industrial applications, because of their inherent advantages such as fluctuation suppression, smoothing of control curves, and effective attenuation of unstable high-frequency signals during the iterative process (Arimote et al., 1990; Wang and Peng, 2014). Dai et al. (2014) discussed the convergence problem of ILCFF for a class of parabolic linear distributed parameter systems with uncertainty. Wang et al. (2015) and Wang et al. (2016) proposed high-order feedback and feedback–feedforward ILCFFs for a class of nonlinear systems considering uncertain and non-repetitive disturbances. Liu and Wang (2017) proposed a new ILCFF comprising an output equation with nonlinear input to deal with various tracking problems in the finite time intervals of fractional order systems. In Zheng et al. (2017), a memory ILC controller with a variable forgetting factor was constructed for a nonlinear system. However, they focused only on algorithm convergence, and an in-depth analysis of the forgetting factor effect on system output characteristics was not performed. In recent years, ILCs with forgetting factor, without enough supporting theory, have been applied to several engineering domains (Cao et al., 2015; Dong et al., 2019; Lan et al., 2017; Lin et al., 2019). Most reports indicated that the forgetting factor prevented the accumulation of initial resetting errors and interference. However, these conclusions were reached at the expense of increases in the number of tracking-output errors (Cao et al., 2015; Dai et al., 2014; Xu, 2011). Thus, it is vital to continue to examine the effects of the forgetting factor on system-output characteristics. The optimality of ILCFF has not been thoroughly discussed, and there has not been an in-depth analysis of the impact of forgetting factor and its optimal gains on system-output characteristics. This continues to restrict its application to a certain extent and motivates our research. In Lin et al. (2019), a forgetting-factor based data-driven optimal terminal ILC (FF-DDOTILC) was proposed for the product concentration control of an ethanol fermentation process (EFP). Although an index function of control input was designed with weighting and forgetting factors, the selection principles were not provided. The authors of the present work have discussed the convergence condition of ILCFF and proposed an optimal design method in terms of the PID coefficients (Dai et al., 2019). However, the influences of forgetting factor and optimal gains on the system output have not been examined. Therefore, it is of great significance for the optimization and application of ILCFF to study the calculation method of optimal control gains in ILCFF and its specific influence on system output.

This paper extends the previous work and proposes design of an optimal ILCFF by considering the non-repetitive disturbances existed in the industrial processes. Rigorous analysis demonstrates that the proposed optimal ILCFF can achieve a monotonic convergence and favorable output characteristics. The comparative results in the simulation section show the advantages of the designed controller. The reason lies in the fact that an excellent control performance can be obtained by the proposed optimal methods. Further, this work significantly improves traditional optimal ILC performance for existing non-ideal applications.

In this paper, a new ILCFF approach for PID gains with optimization is proposed to expedite convergence speed and reduce tracking error. The influence of the forgetting factor on the convergence speed and output characteristics of a class of discrete linear single-input single-output (SISO) systems with aperiodic interferences is examined, and optimal control gains of the algorithm are evaluated. First, a mathematical model of discrete linear systems having aperiodic interferences is established based on the ILCFF. Then, necessary and sufficient conditions for system convergence are derived. Subsequently, considering the optimal control theory, the sufficient conditions for the monotonic convergence of the linear-time-invariant system with time-varying disturbances are derived. Then, a method of calculating the optimal control gains is introduced. In addition, the influence of the forgetting factor on both optimal control gains and output error of the system is clarified, and an optimal iteration-varying ILCFF algorithm design is provided. Finally, the control effect of the proposed method is verified using a case-based simulation.

Theory analysis

Problem description

We consider the standard state-space equation to represent the discrete-time, linear, and time-invariant SISO system

{\begin{matrix} x_{j} (i + 1) = A x_{j} (i) + B u_{j} (i) + w_{j} (i) \\ y_{j} (i) = C x_{j} (i) + v_{j} (i) \\ x_{j} (0) = x_{0} \\ i = 0, 1, \dots, M, j = 0, 1, \dots \end{matrix},

(1)

where $x_{j} \in R^{n}$ , $u_{j} \in R^{n}$ , and $y_{j} \in R^{n}$ denote state vector, input, and output of the system, respectively. $w_{j} (i) \in R^{n}$ and $v_{j} (i) \in R^{n}$ represent unknown time-varying disturbances. $A$ , $B$ , and $C$ are real matrices with appropriate dimensions. $x_{0}$ is system initial condition, $j$ is iteration or trial number, and $M$ is total number of discrete samples for an iteration. $i \in [0, M]$ is discrete sample-point number in an iteration. It is assumed that $CB \neq 0$ is a real number, and desired output trajectory, $y_{d} (i)$ , is known.

By increasing the number of iterations, the error between the desired output $y_{d} (i)$ and the resultant output $y_{j} (i)$ can be minimized to ensure that the following tracking can be realized

lim_{j \to \infty} (y_{d} (i) - y_{j} (i)) = 0 for i = 1, 2, \dots, M .

(2)

The forgetting factor is introduced into the ILC algorithm to suppress system disturbances. The open-loop PID ILCFF is used to realize system control. The updating law is as follows

\begin{matrix} u_{j + 1} (i) = λ_{j} u_{j} (i) + s_{P} e_{j} (i + 1) + \\ s_{I} \sum_{m = 1}^{i + 1} e_{j} (m) + s_{D} (e_{j} (i + 1) - e_{j} (i)), \end{matrix}

(3)

where $e_{j} (i) = y_{d} (i) - y_{j} (i)$ , and $1 \leq i \leq M$ ; $e_{j} (0) \approx 0$ . $s_{P}$ , $s_{I}$ , and $s_{D}$ denote real constant control gains (i.e., the proportional, integration and derivative learning gains, respectively). $λ_{j}$ represents the forgetting factor, which varies with the iteration number, $j$ , and $0 < λ_{j} < 1$ .

The following reasonable assumptions are made considering equation (1):

Assumption 1: The state and output disturbances are completely suppressed under ideal conditions, and the following relations hold

{\begin{matrix} x_{d} (i + 1) = A x_{d} (i) + B u_{d} (i) \\ y_{d} (i) = C x_{d} (i) \end{matrix},

(4)

where $x_{d} (i)$ and $u_{d} (i)$ denote desired state and input, respectively.

Assumption 2: The state and output disturbances of the system are bounded. In addition, $max_{1 \leq i \leq M} ‖ w_{j} (i) ‖ \leq b_{w}$ and $max_{1 \leq i \leq M} ‖ v_{j} (i) ‖ \leq b_{v}$ , where $b_{w} > 0$ and $b_{v} > 0$ are constants.

Convergence analysis

Theorem 1: The introduced ILCFF is convergent if and only if the control gains, $s_{P}$ , $s_{I}$ , and $s_{D}$ , lie in the following interval

| λ_{j} - CB (s_{P} + s_{I} + s_{D}) | < 1 .

(5)

Proof: Let vectors $U (j)$ , $Y (j)$ , $W (j)$ , $V (j)$ , $Y_{d}$ , and $E (j)$ be defined as follows

\begin{matrix} U (j) = [u_{j} (0) u_{j} (1) \dots u_{j} (M - 1)]^{T} \\ Y (j) = [y_{j} (1) y_{j} (2) \dots y_{j} (M)]^{T} \\ W (j) = [w_{j} (0) w_{j} (1) \dots w_{j} (M - 1)]^{T} \\ V (j) = [v_{j} (1) v_{j} (2) \dots v_{j} (M)]^{T} \\ Y_{d} = [y_{d} (1) y_{d} (2) \dots y_{d} (M)]^{T} \\ E (j) = Y_{d} - Y (j) \\ = [e_{j} (1) e_{j} (2) \dots e_{j} (M)]^{T}, \end{matrix}

(6)

Where, $T$ denotes transpose.

We define $a \in R^{M}$ and $ψ \in R^{M \times M}$ as follows

\begin{matrix} \forall a = [\begin{matrix} a_{1} & a_{2} & \dots & a_{M} \end{matrix}]^{T}, \\ ψ (a) = [\begin{matrix} a_{1} & 0 & 0 & \dots & 0 \\ a_{2} & a_{1} & 0 & \dots & 0 \\ a_{3} & a_{2} & a_{1} & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ a_{M} & a_{M - 1} & a_{M - 2} & \dots & a_{1} \end{matrix}] . \end{matrix}

(7)

In this case, operator $ψ$ produces a low triangular Toeplitz matrix from vector $a$ .

Consequently

\begin{matrix} I = ψ (α), F_{1} = ψ (β), F_{2} = ψ (γ), \\ G = ψ (g), and G_{w} = ψ (g_{w}), \end{matrix}

(8)

where $α = {[\begin{matrix} 1 & 0 & \dots & 0 \end{matrix}]}^{T}$ , $β = {[\begin{matrix} 1 & 1 & \dots & 1 \end{matrix}]}^{T}$ , $γ = {[\begin{matrix} 0 & 1 & 0 & \dots & 0 \end{matrix}]}^{T}$ , $g = {[\begin{matrix} g_{1} & g_{2} & \dots & g_{M} \end{matrix}]}^{T}$ , and $g_{w} = [\begin{matrix} C & CA & \dots & C A^{M - 1} \end{matrix}]^{T}$ .

Furthermore

g_{k} = C A^{k - 1} B, k = 1, 2, \dots, M .

(9)

From equation (1), the following relation can be obtained

\begin{matrix} Y (j) = GU (j) + G_{x} x_{0} + \\ G_{w} W (j) + V (j), j = 0, 1, \dots, \end{matrix}

(10)

where $G_{x} = [CA C A^{2} C A^{3} \dots C A^{M}]^{T}$ .

By subtracting the two consecutive iterations, $j$ and $j + 1$ , from equation (10), one can obtain

\begin{matrix} Y (j + 1) - Y (j) = GZ (j) + \\ G_{w} (W_{x} (j + 1) - W_{x} (j)) + V (j + 1) - V (j), \end{matrix}

(11)

where $Z (j) = U (j + 1) - U (j)$ .

Consequently

\begin{matrix} E (j + 1) = E (j) - GZ (j) - \\ G_{w} (W (j + 1) - W (j)) - V (j + 1) + V (j) . \end{matrix}

(12)

According to equation (3)

\begin{matrix} Z (j) = (λ_{j} - 1) U (j) + [(s_{P} + s_{D}) I + \\ s_{I} F_{1} - s_{D} F_{2}] E (j), j = 0, 1, \dots . \end{matrix}

(13)

Substituting $Z (j)$ from equation (13) into equation (12) yields

\begin{matrix} E (j + 1) = G_{c} E (j) - (λ_{j} - 1) GU (j) - \\ G_{w} (W (j + 1) - W (j)) - V (j + 1) + V (j), \end{matrix}

(14)

where $G_{c} = I - G [(s_{P} + s_{D}) I + s_{I} F_{1} - s_{D} F_{2}]$ .

From equations (6) and (14), one can obtain

E (j + 1) = G_{c}' E (j) + M',

(15)

where $G_{c}' = λ_{j} I - (s_{P} + s_{D}) G - s_{I} G F_{1} + s_{D} G F_{2}$ , and

\begin{matrix} M' = (1 - λ_{j}) (Y_{d} - G_{x} x_{0}) + \\ G_{w} (λ_{j} W (j) - W (j + 1)) + (λ_{j} V (j) - V (j + 1)) . \end{matrix}

(16)

From equation (7), it can be noted that $G_{c}' = ψ (g_{c})$ , and

\begin{matrix} g_{c} = [\begin{matrix} g_{c 1} & g_{c 2} & g_{c 3} & \dots & g_{cM} \end{matrix}]^{T} \\ = [\begin{matrix} λ_{j} - g_{1} s_{P} - g_{1} s_{I} - g_{1} s_{D} \\ - g_{2} s_{P} - (\sum_{i = 1}^{2} g_{i}) s_{I} - (g_{2} - g_{1}) s_{D} \\ - g_{3} s_{P} - (\sum_{i = 1}^{3} g_{i}) s_{I} - (g_{3} - g_{2}) s_{D} \\ ⋮ \\ - g_{M} s_{P} - (\sum_{i = 1}^{M} g_{i}) s_{I} - (g_{M} - g_{M - 1}) s_{D} \end{matrix}] . \end{matrix}

(17)

Remark 1: In equations (15) and (16), the presence of the forgetting factor considerably influences the system-output tracking error. If $W_{x}$ and $W_{y}$ occur repeatedly under the condition where $λ_{j} \to 1$ , the system output tracking error can be efficiently reduced. However, the system matrix $G_{w}$ notably influences the disturbances that do not occur repeatedly with the number of iterations.

In contrast to the approaches employed by Madady (2008, 2013), vector $E (j + 1)$ in equation (15) changes. Subsequently, the iterative theorem derived from matrix theory (Gene and Van Loan, 1983) is used for further analysis.

Theorem 2: Suppose $A = L - N \in R^{n \times n}$ indicates the splitting of a non-singular matrix, and $b \in R^{n}$ . Assuming that $L$ is non-singular, the iteration, $L x^{(k + 1)} = N x^{(k)} + b$ , converges to $x = A^{- 1} b$ for all starting n-vectors, $x^{(0)}$ , if and only if $ρ (L^{- 1} N) < 1$ .

According to Theorem 2 and equation (15), when $M' \neq 0$ and $ρ (G_{c}') < 1$ , the convergence solution for equation (15) is as follows

E^{*} = (I - G_{c}')^{- 1} M', j = 0, 1, \dots

(18)

Remark 2: According to Theorem 2 and equations (15)–(18), the convergence of $E (j)$ is considerably influenced by $G_{c}'$ . However, the convergent solution depends on $G_{c}'$ and $M'$ . Therefore, decrease in the disturbance value and the values of $G_{w}$ and $λ_{j}$ influence the output-tracking error. Generally, the system-state parameters have small values. Thus, if the forgetting factor approaches one, and the system disturbances decrease with the number of iterations, the output-tracking error can be effectively reduced.

As a low-triangular Toeplitz matrix, necessary and sufficient condition for $G_{c}'$ is that all eigenvalues lie within the unit circle. From equation (15), the following necessary and sufficient condition can be derived for the convergence of the considered ILC

| λ_{j} - g_{1} (s_{P} + s_{I} + s_{D}) | < 1 .

(19)

Note that this condition is equivalent to equation (5).

Remark 3: Equation (19) represents the relationship between the forgetting factor and the control gains when the system converges. The values of the control gains depend on the system-state parameter and forgetting factor, and the value range varies with the change in the forgetting factor. When the forgetting factor and control gains satisfy equation (19), the system converges. However, the monotonic convergence of the system cannot be ensured. Thus, it requires further analysis, as discussed in the following section.

Optimal control gain

To achieve monotonic convergence, the following conditions should be satisfied (Madady, 2008, 2013)

{\begin{matrix} {‖ E (j + 1) ‖}_{σ} < {‖ E (j) ‖}_{σ}, (E (j) \neq 0) \\ {‖ E (j + 1) ‖}_{σ} = {‖ E (j) ‖}_{σ}, (E (j) = 0) \end{matrix},

(20)

where, $σ = 1, 2, \infty, j = 0, 1, \dots$

Lemma 1: The presented ILC system exhibits monotonic convergence if

‖ g_{c} ‖_{1} < 1 - \frac{{‖ M' ‖}_{σ}}{‖ E (j) ‖_{σ}} .

(21)

Proof: This proof is similar to that of the lemma presented by Madady (2008).

Remark 4: To ensure that the system has the highest convergence rate, the value of $‖ G_{c}' ‖_{σ}$ should be minimized. The approximate value of $‖ G_{c}' ‖_{σ}$ can be obtained by determining the minimum $‖ g_{c} ‖_{1}$ value.

Therefore, the optimal control gains of the algorithm can be determined by performing the following theoretical analysis:

From equation (7)

λ_{j} I = ψ (μ),

(22)

where

μ = {[\begin{matrix} λ_{j} & 0 & 0 & \dots & 0 \end{matrix}]}^{T} .

(23)

Furthermore, $g_{c} = μ - QS$ , where $S \in R^{3}$ , $Q \in R^{M \times 3}$ , $S = [\begin{matrix} s_{P} & s_{I} & s_{D} \end{matrix}]^{T}$ , and

Q = [\begin{matrix} g_{1} & g_{1} & g_{1} \\ g_{2} & g_{1} + g_{2} & g_{2} - g_{1} \\ g_{3} & g_{1} + g_{2} + g_{3} & g_{3} - g_{2} \\ ⋮ & ⋮ & ⋮ \\ g_{M} & g_{1} + g_{2} + \dots + g_{M} & g_{M} - g_{M - 1} \end{matrix}] .

(24)

Considering the findings of Madady (2008), it can be stated that

{\begin{matrix} {‖ g_{c} ‖}_{1} < \sqrt{M} {‖ g_{c} ‖}_{2} \\ ρ = ‖ g_{c} ‖_{2}^{2} = {g_{c}}^{T} g_{c} \end{matrix},

(25)

where

ρ = λ_{j}^{2} - 2 μ^{T} QS + S^{T} Q^{T} QS .

(26)

By deriving $ρ$ with respect to $S$ , the optimal $S^{*}$ can be obtained as

\begin{matrix} S^{*} = (Q^{T} Q)^{- 1} Q^{T} μ \\ = g_{1} (Q^{T} Q)^{- 1} [\begin{matrix} λ_{j} & λ_{j} & λ_{j} \end{matrix}]^{T} . \end{matrix}

(27)

Subsequently, it is assumed that

(Q^{T} Q)^{- 1} = [\begin{matrix} q_{1} & q_{2} & q_{3} \\ q_{2} & q_{4} & q_{5} \\ q_{3} & q_{5} & q_{6} \end{matrix}] .

(28)

The optimal control gains can thus be obtained as follows

{\begin{matrix} s_{P}^{*} = λ_{j} g_{1} (q_{1} + q_{2} + q_{3}) \\ s_{I}^{*} = λ_{j} g_{1} (q_{2} + q_{4} + q_{5}) \\ s_{D}^{*} = λ_{j} g_{1} (q_{3} + q_{5} + q_{6}) \end{matrix} .

(29)

This perspective shows that the optimal control gains of the ILCFF vary with the change in the forgetting factor, ensuring that the system exhibits monotonic convergence. Therefore, implementing a variable forgetting factor can effectively improve system-convergence performance, accelerate system convergence, and suppress the system interferences.

Equation (26) thus yields

\begin{matrix} ρ^{*} = {λ_{j}}^{2} - μ^{T} Q (Q^{T} Q)^{- 1} Q^{T} μ \\ = {λ_{j}}^{2} - {λ_{j}}^{2} g_{1}^{2} (q_{1} + 2 q_{2} + 2 q_{3} + q_{4} + 2 q_{5} + q_{6}) . \end{matrix}

(30)

If $ρ_{1}$ is the index function for the traditional ILC algorithm, the optimal $ρ_{1}^{*}$ can be obtained as follows

ρ_{1}^{*} = 1 - g_{1}^{2} (q_{1} + 2 q_{2} + 2 q_{3} + q_{4} + 2 q_{5} + q_{6}) .

(31)

Consequently

ρ^{*} < ρ_{1}^{*} .

(32)

And the sufficient condition for monotonic convergence can be defined as follows

ρ^{*} < M^{- 1} or ρ_{1}^{*} < M^{- 1} .

(33)

According to the findings of Madady (2013), Theorem 3 can be described as follows:

Theorem 3: Because $ρ^{*} < ρ_{1}^{*}$ , the performance of the proposed algorithm is better than that of the traditional PID-type ILC.

Proof: This proof is similar to that of the Theorem 3 reported by Madady (2013).

Thus, it can be concluded that the convergence rate of the ILCFF is higher than that of the traditional ILC algorithm.

Influence of forgetting factor on the convergence error

An ILCFF with optimal control gains can effectively accelerate the convergence of the system. However, its influence on the system output-tracking error, which is the basis of system control, must be considered. It is conventionally assumed that the use of the forgetting factor increases the system-output error. Consequently, it is necessary to analyze the influence of the optimal forgetting factor on the convergence error. In this study, a variable forgetting factor that is more suitable for practical systems is implemented.

Equation (18) indicates that different ILC control laws will lead to different convergence errors. Let the optimal gain of the traditional ILC algorithm be $S_{1}^{*} = [s_{P 1}^{*} s_{I 1}^{*} s_{D 1}^{*}]^{T}$ and the convergence error be $E_{1}^{*}$ . As shown in equation (3), the optimal gain of the ILCFF is $S_{2}^{*} = [s_{P 2}^{*} s_{I 2}^{*} s_{D 2}^{*}]^{T}$ , and the convergence error is $E_{2}^{*}$ . Consequently

{\begin{matrix} E_{1}^{*} = {(I - G_{c 1}')}^{- 1} M_{1}' = N_{1}^{- 1} M_{1}' \\ E_{2}^{*} = {(I - G_{c 2}')}^{- 1} M_{2}' = N_{2}^{- 1} M_{2}' \end{matrix},

(34)

where $N_{1} = ψ (g_{m})$ , and $N_{2} = ψ (g_{n})$ .

\begin{matrix} g_{m} = {[\begin{matrix} g_{m 1} & g_{m 2} & \dots & g_{mM} \end{matrix}]}^{T} \\ = [\begin{matrix} g_{1} (s_{P 1}^{*} + s_{I 1}^{*} + s_{D 1}^{*}) \\ g_{2} s_{P 1}^{*} + (\sum_{i = 1}^{2} g_{i}) s_{I 1}^{*} + (g_{2} - g_{1}) s_{D 1}^{*} \\ g_{3} s_{P 1}^{*} + (\sum_{i = 1}^{3} g_{i}) s_{I 1}^{*} + (g_{3} - g_{2}) s_{D 1}^{*} \\ ⋮ \\ g_{M} s_{P 1}^{*} + (\sum_{i = 1}^{M} g_{i}) s_{I 1}^{*} + (g_{M} - g_{M - 1}) s_{D 1}^{*} \end{matrix}] \end{matrix}

(35)

\begin{matrix} g_{n} = {[\begin{matrix} g_{n 1} & g_{n 2} & \dots & g_{nM} \end{matrix}]}^{T} \\ = [\begin{matrix} (1 - λ_{j}) + g_{1} (s_{P 2}^{*} + s_{I 2}^{*} + s_{D 2}^{*}) \\ g_{2} s_{P 2}^{*} + (\sum_{i = 1}^{2} g_{i}) s_{I 2}^{*} + (g_{2} - g_{1}) s_{D 2}^{*} \\ g_{3} s_{P 2}^{*} + (\sum_{i = 1}^{3} g_{i}) s_{I 2}^{*} + (g_{3} - g_{2}) s_{D 2}^{*} \\ ⋮ \\ g_{M} s_{P 2}^{*} + (\sum_{i = 1}^{M} g_{i}) s_{I 2}^{*} + (g_{M} - g_{M - 1}) s_{D 2}^{*} \end{matrix}] . \end{matrix}

(36)

According to equations (29) and (35)–(36)

N_{2} = λ_{j} N_{1} + (1 - λ_{j}) I and S_{2}^{*} = λ_{j} S_{1}^{*} .

(37)

For a system, the value of $g_{k}$ is fixed. However, the presence of the forgetting factor directly influences the values of $N$ and $M'$ . Furthermore, because of the uncertainty in the value of $M'$ , the influence of the forgetting factor on the output error, $E^{*}$ , of the system can be determined only by analyzing $N$ and $M'$ separately.

Therefore, we first analyze $N_{1}^{- 1}$ and $N_{2}^{- 1}$ as follows:

According to the definition of the induced matrix norm (Gene and Van Loan, 1983)

‖ N_{1}^{- 1} ‖ ‖ N_{2}^{- 1} ‖^{- 1} = {(min_{g_{m} \neq 0} \frac{‖ N_{1} g_{m} ‖}{‖ g_{m} ‖})}^{- 1} (min_{g_{n} \neq 0} \frac{‖ N_{2} g_{n} ‖}{‖ g_{n} ‖}) .

(38)

The 1-norm and infinite norm of the matrix can be defined according to the properties of the Toeplitz matrix and the norm of its inverse, as follows

{\begin{matrix} min_{g_{m} \neq 0} \frac{{‖ N_{1} g_{m} ‖}_{1, \infty}}{{‖ g_{m} ‖}_{1, \infty}} = g_{1} (s_{P 1} + s_{I 1} + s_{D 1}) \\ min_{g_{n} \neq 0} \frac{{‖ N_{2} g_{n} ‖}_{1, \infty}}{{‖ g_{n} ‖}_{1, \infty}} = (1 - λ_{j}) + g_{1} (s_{P 2} + s_{I 2} + s_{D 2}) \end{matrix} .

(39)

When the optimal control gains of the algorithm are determined using equation (29)

\frac{{‖ N_{1}^{- 1} ‖}_{1, \infty}}{{‖ N_{2}^{- 1} ‖}_{1, \infty}} = \frac{(1 - λ_{j}) + λ_{j} g_{1} (s_{P 1}^{*} + s_{I 1}^{*} + s_{D 1}^{*})}{g_{1} (s_{P 1}^{*} + s_{I 1}^{*} + s_{D 1}^{*})} .

(40)

Equation (30) indicates that

ρ^{*} > 0 and 0 < g_{1} (s_{P 1}^{*} + s_{I 1}^{*} + s_{D 1}^{*}) < 1 .

(41)

Consequently

\begin{matrix} \frac{{‖ N_{1}^{- 1} ‖}_{1, \infty}}{{‖ N_{2}^{- 1} ‖}_{1, \infty}} = \frac{(1 - λ_{j}) + λ_{j} g_{1} (s_{P 1}^{*} + s_{I 1}^{*} + s_{D 1}^{*})}{g_{1} (s_{P 1}^{*} + s_{I 1}^{*} + s_{D 1}^{*})} \\ = λ_{j} + \frac{1 - λ_{j}}{g_{1} (s_{P 1}^{*} + s_{I 1}^{*} + s_{D 1}^{*})} \\ > λ_{j} + 1 - λ_{j} = 1 . \end{matrix}

(42)

The 2-norms of $N_{1}^{- 1}$ and $N_{2}^{- 1}$ must be analyzed separately. $N_{1}$ and $N_{2}$ are lower triangular Toeplitz matrices, and the same is true of $N_{1}^{- 1}$ and $N_{2}^{- 1}$ by the definition and properties of the matrix. If the elements of $N_{1}$ and $N_{2}$ are positive, and

N_{1}^{- 1} = ψ (n_{m}) = ψ [\begin{matrix} n_{m 1} \\ n_{m 2} \\ ⋮ \\ n_{mM} \end{matrix}]

(43)

N_{2}^{- 1} = ψ (n_{n}) = ψ [\begin{matrix} n_{n 1} \\ n_{n 2} \\ ⋮ \\ n_{nM} \end{matrix}],

(44)

the following expressions can be obtained

{\begin{matrix} n_{m 1} = \frac{1}{g_{m 1}} \\ n_{m 2} = - \frac{g_{m 2} n_{m 1}}{g_{m 1}} = - \frac{g_{m 2}}{g_{m 1}^{2}} \\ n_{m 3} = - \frac{g_{m 3} n_{m 1} + g_{m 2} n_{m 2}}{g_{m 1}} = - \frac{g_{m 3}}{g_{m 1}^{2}} + \frac{g_{m 2}^{2}}{g_{m 1}^{3}} \\ ⋮ \\ n_{mM} = - \frac{\sum_{k = 1}^{M - 1} n_{mk} g_{mM - k + 1}}{g_{m 1}} = \sum_{k = 1}^{M - 1} \frac{{(- 1)}^{k} f (g_{mh}^{(k)})}{g_{m 1}^{k + 1}} \end{matrix}

(45)

{\begin{matrix} n_{n 1} = \frac{1}{g_{n 1}} \\ n_{n 2} = - \frac{g_{n 2} n_{n 1}}{g_{n 1}} = - \frac{g_{n 2}}{g_{n 1}^{2}} \\ n_{n 3} = - \frac{g_{n 3} n_{n 1} + g_{n 2} n_{n 2}}{g_{n 1}} = - \frac{g_{n 3}}{g_{n 1}^{2}} + \frac{g_{n 2}^{2}}{g_{n 1}^{3}} \\ ⋮ \\ n_{nM} = - \frac{\sum_{k = 1}^{M - 1} n_{nk} g_{nM - k + 1}}{g_{n 1}} = \sum_{k = 1}^{M - 1} \frac{{(- 1)}^{k} f (g_{nh}^{(k)})}{g_{n 1}^{k + 1}} \end{matrix},

(46)

where $f (g_{mh}^{(k)})$ is the product of $g_{mh}, h \in [2, M]$ , and number of multiplicands is $k$ . The matrices do not differ in terms of the coefficient $f (g_{mh}^{(k)})$ . Thus, this aspect is not discussed.

According to equations (32) and (35)–(36)

1 - M^{- 1} < g_{m 1} < g_{n 1} < 1 .

(47)

If $ξ = \frac{g_{n 1}}{g_{m 1}} = \frac{1 - λ + λ g_{m 1}}{g_{m 1}}$ ,

1 < ξ < 1 + \frac{1 - λ}{M - 1} .

(48)

According to equations (35)–(36) and (45)–(46)

\frac{{(- 1)}^{k} f (g_{nh}^{(k)})}{g_{n 1}^{k + 1}} / \frac{{(- 1)}^{k} f (g_{mh}^{(k)})}{g_{m 1}^{k + 1}} = \frac{λ^{k}}{ξ^{k + 1}} < 1 .

(49)

If $λ \to 1$ , then

| n_{mi} | > | n_{ni} |, 1 \leq i \leq M .

(50)

Equations (35)–(36) and (48)–(50) indicate that the matrix elements at the same positions in $N_{1}^{- 1}$ and $N_{2}^{- 1}$ have the same signs, and the absolute values for the $N_{1}^{- 1}$ elements are greater than those of the $N_{2}^{- 1}$ elements. Furthermore, the absolute values of the $(N_{1}^{- 1})^{T} N_{1}^{- 1}$ elements are greater than those of the $(N_{2}^{- 1})^{T} N_{2}^{- 1}$ elements in the same positions.

The sum method in the analytic hierarchy process (Zhang, 2014) can be employed to estimate that the maximum eigenvalue of $(N_{1}^{- 1})^{T} N_{1}^{- 1}$ is greater than that of $(N_{2}^{- 1})^{T} N_{2}^{- 1}$ .

Consequently

‖ N_{1}^{- 1} ‖_{σ} > ‖ N_{2}^{- 1} ‖_{σ}, (σ = 1, 2, \infty) .

(51)

Subsequently, the system error caused by time-varying nonlinear disturbances is analyzed.

According to equation (16)

{\begin{matrix} M_{1}' = G_{w} (W (j) - W (j + 1)) + (V (j) - V (j + 1)) \\ \begin{matrix} M_{2}' = (1 - λ_{j}) (Y_{d} - G_{x} x_{0}) + \\ G_{w} (λ_{j} W (j) - W (j + 1)) + (λ_{j} V (j) - V (j + 1)) \end{matrix} \end{matrix}

(52)

M_{2}' = M_{1}' + Δ,

(53)

$where Δ = (1 - λ_{j}) (Y_{d} - G_{x} x_{0} - G_{w} W (j) - V (j)) .$

Then

\frac{‖ E_{2}^{*} ‖}{‖ E_{1}^{*} ‖} = \frac{‖ N_{2}^{- 1} M_{2}' ‖}{‖ N_{1}^{- 1} M_{1}' ‖} = \frac{‖ N_{2}^{- 1} (M_{1}' + Δ) ‖}{‖ N_{1}^{- 1} M_{1}' ‖} .

(54)

Remark 5: If matrices $M_{1}'$ and $Δ$ in equation (54) have the same sign, it will directly affect the error-comparison result. According to the definition of $Δ$ , signs of its elements depend mainly on $Y_{d}$ , which does not exist in $M_{1}'$ ; however, it is introduced by the forgetting factor to $M_{2}'$ . According to equation (52), value of $M_{1}'$ depends on the system parameters and changes with the values of $W (j)$ and $V (j)$ . Meanwhile, several other influencing factors, such as $Y_{d}$ , $x_{0}$ , pertain to $M_{2}'$ . Additionally, the value of $Y_{d}$ is usually far greater than the others. Thus, $‖ M_{2}' ‖ >> ‖ M_{1}' ‖$ frequently occurs, implying $‖ E_{1}^{*} ‖ < ‖ E_{2}^{*} ‖$ . Consequently, it is widely believed that the forgetting factor increases the system-output error. However, $‖ M_{2}' ‖ >> ‖ M_{1}' ‖$ is not fixed, and the prerequisite condition of $‖ M_{2}' ‖ < ‖ M_{1}' ‖$ should be discussed.

Equations (52)–(54) indicate that the condition $‖ E_{2}^{*} ‖ < ‖ E_{1}^{*} ‖$ can occur if $Y_{d}$ is sufficiently small, and the value of sequence $M_{2}'$ is less than that of sequence $M_{1}'$ . This indicates that the implementation of the forgetting factor with the optimal gain can lead to faster system convergence and fewer tracking errors. This finding provides a theoretical basis for the application of the forgetting factor. Specifically, in the case of systems with a small expected trajectory, the output error of the ILCFF with optimal control gains would be close to or even better than that of the traditional algorithm. Combined with Theorem 3, it can be concluded that the output characteristics of the ILCFF with optimal gains are superior to those of the traditional algorithm.

Designed controller

The variable forgetting-factor-based ILC algorithm is primarily applied to complex systems involving system-state interferences and measurement noise. However, this algorithm is limited, because it is difficult to simultaneously realize both adaptive adjustment and algorithm simplicity. Moreover, the variable forgetting factor should conform to the following three basic principles:

As the number of iterations increases, the forgetting factor should approach one to ensure that the system-output tracking error is minimized. Specifically, $0 < λ_{j} < 1$ , and $lim_{j \to \infty} λ_{j} \to 1$ .

When the system output error decreases, $λ_{j} \to 1$ to ensure that the system output converges to the minimum error value. However, when the output error exhibits notable fluctuations, $λ_{j}$ should change accordingly to enhance the system robustness and suppress the interferences.

The use of $λ_{j}$ with optimal control gains can enhance system convergence. However, the system-output error increases in several cases. To reduce the system-output error and enhance the convergence speed, $λ_{j}$ should be used in systems with a small desired trajectory and large disturbances.

According to these principles, the variable forgetting factor is designed as an iteration-varying factor, whose initial value is lower than one and increases with the increase of the iteration number and approaches one, as follows

λ_{j} = {\begin{matrix} λ_{1} + (j - 1) Λ, (0 < τ \leq L_{1}, j \leq j_{L}) \\ m_{L}, (τ > L_{1}, j > j_{L}, ‖ E_{j - 1} ‖ > ‖ E_{j_{L}} ‖) \\ m_{T}, (otherwise) \end{matrix},

(55)

where

τ = \frac{‖ E_{j - 1} ‖}{‖ E_{j - 2} ‖},

(56)

and $λ_{1}$ is the first iterative value of $λ$ . $Λ$ is change value when the iterations commence. $L_{1}$ is allowed maximum ratio of the two preceding output-error norms. $j_{L}$ is minimum iteration number when the output error tends to stabilize. $m_{L}$ is the fixed value of the forgetting factor when the error increases considerably. $m_{T}$ is the fixed value of the forgetting factor when the value of $τ$ is in the normal range, and the output error is small. Furthermore, the optimal control gains of the ILCFF change with the change of the forgetting factor according to equation (29).

Remark 6: Variable $τ$ indicates change trend of the last two system-output errors, which are used to prevent sudden wide-range fluctuations. If the system-output error is convergent, and the error fluctuation by disturbances is small, the system output will have better performance when the value of $λ_{j}$ rapidly increases and approaches $m_{T}$ . However, because of the uncontrollability of the system disturbances, it is necessary to set $m_{L}$ to accelerate the convergence when the system-output error increases suddenly.

Simulation

An example similar to the case described by Madady (2013) pertaining to a position servo-control system was considered. The discrete dynamic model of the motor was as follows

\begin{matrix} {\begin{matrix} x_{j} (i + 1) = A_{D} x_{j} (i) + B_{D} u_{j} (i) + w_{j} (i) \\ y_{j} (i) = C_{D} x_{j} (i) + v_{j} (i) \end{matrix} \\ i = 0, 1, \dots, 100, j = 0, 1, \dots, \end{matrix}

(57)

where $A_{D} = [\begin{matrix} 0.8869 & 0 & 0 \\ 0.1507 & 0.9989 & 0 \\ 0.0008 & 0.0100 & 1 \end{matrix}]$ , $B_{D} = [\begin{matrix} 0 \\ 0.0097 \\ 0.0093 \end{matrix}]$ , $C_{D} = [\begin{matrix} 0 & 0 & 1 \end{matrix}]$ , and $w_{j} (i) = [\begin{matrix} 20 \sin (\frac{i}{M}) \cos (\frac{6 i}{M}) \\ 10 \sin (\frac{i}{M}) \cos (\frac{6 i}{M}) \\ 0.1 j (\sin (\frac{2 j π i}{M} + \frac{π}{3}) \cos (\frac{3 j π i}{M})) \end{matrix}]$ , with $v_{j} (i) = [\begin{matrix} 0 & 0 & 0.05 j \sin (\frac{i}{M}) \cos (\frac{2 i}{M}) \end{matrix}]^{T}$ .

The desired output trajectory is defined as follows

y_{d} (i) = \frac{1}{8000} i (M - i), 1 \leq i \leq M, M = 100,

(58)

where the sampling period, $T_{1} = 0.01 s$ , and initial state of the system is $x_{0} = [\begin{matrix} 0 & 0 & 0 \end{matrix}]^{T}$ .

According to Madady (2013), the optimal control gains of the conventional system can be defined as follows:

\begin{matrix} s_{p} = g_{1} (q_{1} + q_{2} + q_{3}) = - 1.0585 \\ s_{i} = g_{1} (q_{2} + q_{4} + q_{5}) = 0.0076 \\ s_{d} = g_{1} (q_{3} + q_{5} + q_{6}) = 108.5768 . \end{matrix}

(59)

To analyze the influence of a time-varying nonlinear disturbance on the convergence characteristics of the ILC algorithm under different forgetting factors, different forgetting factors and control gains were combined, and the following four ILCFF algorithm configurations were obtained:

A1: traditional algorithm with $λ = 1$ and fixed control gains, according to equation (59).

A2: variable forgetting factor $λ_{j} = 1 - 0 . 1^{j}$ (Wang et al., 2015) and fixed control gains, according to equation (59).

A3: variable forgetting factor $λ_{j} = 1 - 0 . 1^{j}$ and variable control gains, according to equations (29) and (59).

A4: variable forgetting factor according to equation (55) and variable control gains according to equations (29) and (59), where $λ_{1} = 0.85$ , $Λ = 0.03$ , $j_{L} = 3$ , $L_{1} = 2$ , $m_{L} = 0.95$ , and $m_{T} = 0.98$ .

Remark 7: An important role of the forgetting factor is accelerating the convergence speeds. According to equations (30) and (31) and Theorem 3, it can be observed that the value of the forgetting factor directly affects the convergence speed. Thus, selection of an initial forgetting factor value, $λ_{1}$ , is highly important. Although a smaller $λ_{1}$ can provide a faster convergence, early convergence can lead to excessive output errors. Thus, a relatively small initial forgetting factor that approaches one quickly with the number of iterations is expected to result in improvements to both convergence speed and error reduction.

Prior to the simulation, different forgetting factors in the aforementioned four algorithms were introduced, as shown in Figure 1.

Case 1: The disturbances mentioned in equation (57) are non-repeating with respect to iterations. Performances of the Algorithms A1–A4 are shown in Figure 2. Furthermore, the error norm per iteration is given in Table 1, wherein the tracking performances of the A1–A4 approaches are compared, given $m_{T} = 0.98$ . The variance of the error norm is calculated from the fourth-to-tenth iterations.

Figure 1.

Curves of different forgetting factors.

Figure 2.

(Case 1) Comparison of output tracking-error norm of the system with non-repeatable interference.

Table 1.

(Case 1) Comparison of error norm and variance.

Algorithm	Initial	First iteration	Fourth iteration	Tenth iteration	Variance of norm
A1	2.291	1.907	0.293	0.231	1.32×10⁻³
A2	2.291	1.907	0.299	0.231	1.46×10⁻³
A3	2.291	1.552	0.281	0.231	1.06×10⁻³
A4( $λ_{1} = 0.85$ )	2.291	1.380	0.238	0.226	4.50×10⁻⁴
A4( $λ_{1} = 0.9$ )	2.291	1.552	0.266	0.226	9.21×10⁻⁴
A4( $λ_{1} = 0.92$ )	2.291	1.622	0.281	0.226	1.24×10⁻³

As shown in Table 1 and Figure 2, convergence rates of the Algorithms A1 and A2 are low, and Algorithm A2 cannot exploit the suppression effect of the forgetting factor. When the control gains of A2 are calculated using the proposed optimal ILCFF methods, the convergence speed is improved in A3. The proposed algorithm (Algorithm A4) adopts a smaller forgetting factor to accelerate convergence at the beginning of the iterative process, and a fixed forgetting factor is selected to suppress the output error when the tracking error tends to be stable. Therefore, convergence speed and output characteristics of the Algorithm A4 are better than those of the other algorithms. Furthermore, it can be observed that $λ_{1}$ and $Λ$ in the proposed technique obviously affect the convergence speed. However, $m_{T}$ determines the output convergence error.

Case 2: The disturbances and algorithm parameters remain the same as in Case 1, except when $m_{T} = 0.99$ . Also, error norms and variance of A4 are changed. The results are given in Figure 3 and Table 2.

Figure 3.

(Case 2) Comparison of output tracking-error norm of the system with non-repeatable interference.

Table 2.

(Case 2) Comparison of error norm and variance.

Algorithm	Initial	First iteration	Fourth iteration	Tenth iteration	Variance of norm
A4( $λ_{1} = 0.85$ )	2.291	1.380	0.244	0.228	4.53×10⁻⁴
A4( $λ_{1} = 0.9$ )	2.291	1.552	0.268	0.228	8.69×10⁻⁴
A4( $λ_{1} = 0.92$ )	2.291	1.622	0.281	0.228	1.16×10⁻³

As shown in Table 2 and Figure 3, the convergence speed of A4 did not change when $λ_{1}$ and $Λ$ were the same as in Case 1. Additionally, the convergence error is increased when $m_{T} = 0.99$ , which means that $m_{T}$ approaching one is equal to smallest convergence error is not suitable for every system.

Case 3: Assuming that the disturbances and algorithm parameters remain the same as in Case 1, except when $Λ = 0$ , error norms and variances of Algorithm A4 change. The results are given in Figure 4 and Table 3. To enhance the comparison, Algorithm A4 with a forgetting factor value of 0.7 (black dotted line) is added to Figure 3.

Figure 4.

(Case 3) Comparison of output tracking-error norm of the system with non-repeatable interference.

Table 3.

(Case 3) Comparison of error norm and variance.

Algorithm	Initial	First iteration	Fourth iteration	Tenth iteration	Variance of norm
A4( $λ_{1} = 0.7$ )	2.291	0.913	0.327	0.226	2.28×10⁻³
A4( $λ_{1} = 0.85$ )	2.291	1.380	0.227	0.226	2.64×10⁻⁴
A4( $λ_{1} = 0.9$ )	2.291	1.552	0.232	0.226	3.71×10⁻⁴
A4( $λ_{1} = 0.92$ )	2.291	1.622	0.239	0.226	4.72×10⁻⁴

As shown in Table 3 and Figure 4, when the forgetting factor is $λ_{1} = 0.7$ , the output tracking error is larger after it gains stability. There is an obvious re-convergence process when $λ_{j} = m_{T}$ . Therefore, using increment-factor $Λ$ to accelerate the convergence is adequate. Although a small forgetting factor implies rapid convergence, it causes large errors. Therefore, it is necessary to ensure that the forgetting factor approaches one prior to the convergence error stabilizing, to reduce the system-output error. The proposed algorithm, which uses the increment factor, $Λ$ , and fixes the forgetting factor from the fourth iteration ensures minimum output error and a rapid convergence rate.

Case 4: The disturbances mentioned in equation (57) constitute both non-repeating noise and random disturbances. The performances of the error norms in the Algorithms A1–A4 are shown in Figure 5. Furthermore, the error norms per iteration and variance are given in Table 4, when $m_{T} = 0.98$ . A random disturbance, $f_{rand}$ , with a maximum amplitude of 0.01 was added to $v_{j} (i)$ to yield the following system disturbance

w_{1 j} (i) = w_{j} (i); v_{1 j} (i) = v_{j} (i) + f_{rand} {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T} .

(60)

Figure 5.

(Case 4) Comparison of output tracking-error norm of the system with non-repeatable interference and random disturbances.

Table 4.

(Case 4) Comparison of error norm and variance.

Algorithm	Initial	First iteration	Fourth iteration	Tenth iteration	Variance of norm
A1	2.246	1.874	0.278	0.233	6.24×10⁻⁴
A2	2.246	1.874	0.283	0.233	6.95×10⁻⁴
A3	2.246	1.526	0.268	0.233	5.01×10⁻⁴
A4( $λ_{1} = 0.85$ )	2.246	1.358	0.237	0.226	3.54×10⁻⁴
A4( $λ_{1} = 0.9$ )	2.246	1.526	0.254	0.226	5.17×10⁻⁴
A4( $λ_{1} = 0.92$ )	2.246	1.595	0.265	0.226	6.47×10⁻³

According to Figure 5 and Table 4, Algorithm A4 exhibits a fast convergence speed with small convergence errors when the system has time-varying nonlinear interference and random disturbances. The forgetting factor demonstrates better performance when suppressing the random disturbance than those of the traditional ILC algorithms.

Remark 8: 2-norm is used to compare system-output errors in these simulations, giving the square root of the sum of the squares of the elements in the vector. As a vector-comparison tool, 1-norm and infinite norm can achieve the same comparative conclusions.

Remark 9: Simulation of the Algorithms A1–A4 is actually a comparison of output characteristics of different algorithms with various optimal control gains. Algorithms A1 and A2 represent application of the traditional optimal control gains in the ILC and the ILCFF, where Algorithms A3 and A4 represent application of the proposed optimal control gains in the different ILCFFs. Algorithm A4 proposes the direct possibility of ILCFF improvement. Our target is to propose a new algorithm that can be optimized or improved in some system-output characteristic. The simulation results demonstrate the effectiveness of the proposed optimal gains and prove the correction of the influence analysis of the forgetting factor on the convergence error.

Conclusion

System disturbances are inevitable in practical applications. By taking a class of linear discrete-time systems with time-varying non-repetitive disturbances into consideration, this research work proposed an optimal ILC algorithm with an iteration-varying forgetting factor to accelerate system convergence and reduce convergence error. Mathematical proof was provided for the convergence of the proposed ILCFF, and a method for calculating the optimal control gains using the forgetting factor was established. Additionally, the influence of the forgetting factor on the system output was examined. Conventional wisdom suggests that introducing a forgetting factor to an ILC will increase system-output error. However, the traditional argument fails when the desired system trajectory is small and disturbances are increased over iterations. The proposed optimal ILCFF demonstrated a better performance in terms of convergence rate and output-tracking error in both theoretical analyses and simulations. In addition, handling the problem when in the system model (1) and updating law (3), initial condition $x_{0}$ , initial control $u_{0}$ , and parameters of updating law are output-related is a subject of another research work.

Footnotes

Acknowledgements

We would like to thank Editage () for English-language editing.

Data availability

The data used to support the findings of this study are included within the article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [grant number 51565033] and the Gansu Provincial Natural Science Foundation of China [grant number 18JR3RA139].

ORCID iDs

Baolin Dai

Jun Gong

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