Adaptive Fuzzy Iterative Learning Control for Systems with Saturated Inputs and Unknown Control Directions

Abstract

An adaptive fuzzy iterative learning control (ILC) algorithm is designed for the iterative variable reference trajectory problem of nonlinear discrete-time systems with input saturations and unknown control directions. Firstly, an adaptive fuzzy iterative learning controller is constructed by combining with the fuzzy logic system (FLS), which can compensate the loss caused by input saturation. Then, the discrete Nussbaum gain technique is adopted along the iteration axis, which can be embedded to the learning control method to identify the control direction of the system. Finally, based on the nonincreasing Lyapunov-like function, it is proven that the adaptive iterative learning controller can converge asymptotically when the number of iterations tends to infinity, and the system signals always remain bounded in the learning process. A simulation example verifies the feasibility and effectiveness of the learning control method.

1. Introduction

In the practical system of industrial production, the system input always encounters saturation constraints to some extent, leading to an undesirable control effect. For example, due to the physical limitations of the control torque gyroscope, reaction flywheel, and reaction thruster of the spacecraft actuator, an input saturation problem always exists in the spacecraft attitude control system [1], the control rudder surface is constrained by physical devices and the deflection angle is limited when the fighter is performing large maneuvers [2], and so on. Some nonlinear dynamic inversion methods have been developed to deal with the nonaffine pure-feedback nonlinear systems with input saturations [3]. Using neural network to estimate nonlinear function, reference [4] introduced command filtering technology to solve the problem of “computational complexity” in traditional backstepping, and introduced a compensation mechanism to solve the problem of filtering error and reduces the impact of input saturation. A novel input saturated control approach is proposed in [5] based on the Kalman filter framework. For the quadrotor unmanned aerial vehicles, which has bounded uncertainties and input constraints, a novel adaptive barrier function-based nonsingular terminal sliding mode control approach is proposed in [6]. Moreover, some novel adaptive control methods were proposed for the systems with nonlinear input characteristics, such as the riser systems [7,8], the flexible string systems [9], and the nonlinear fractional order multi-agent systems [10]. Additionally, researchers have paid much attention to the actuator failure problem, with fruitful results [11,12,13].

In reality, the control direction of the control system is often affected by various uncertain factors, and the control direction affects the operation direction of the system. Therefore, the control direction has a great impact on the control performance. Many systems cannot know their control direction in advance, such as flexible manipulators in [14] and the hypersonic vehicles in [15], so it is very crucial to design the correct identification method of control direction. For the systems with unknown control directions, a Nussbaum gain technique-based adaptive fuzzy control has been introduced to address the tracking control of multi agent systems in [16]. With the help of the Nussbaum gain technique, a filter-based recursive tracker was developed in [17].

At the same time, many systems in the real process operate repeatedly. For example, the expressway ramp system in [18], the electromechanical servo system in [19], and parametric nonlinear system in [20]. However, the repeatability is not taken into account in the above literatures [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Up to now, many scholars have applied iterative learning contro (ILC) to solve the tracking control of repeated systems with saturation constraints, because of the leaning ability and simple structure of ILC. For linear systems with saturation constraints, as in references [21,22], the problem can be transformed into a linear constrained quadratic programming problem and a convex quadratic programming problem. For nonlinear systems, the authors in [13,14,15,16,17,18,19,20,21,22,23,24,25] designed adaptive ILC schemes by introducing saturation functions. The speed trajectory tracking problem for high-speed trains with input saturations is solved by an adaptive ILC in [26]. An adaptive ILC controller is designed based on the command filter adaptive backstepping method, where an auxiliary system is established to overcome the control difficulty caused by input saturation [27]. Considering the iteration and time dependent constraint, the authors have designed a novel adaptive ILC to achieve the leader-follower formation tracking control of the nonlinear multi agent systems [28]. In addition, a constrained ILC was designed for the high-speed rack feeder, which subject to the saturation characteristic [24]. The ILC methods proposed in [21,22,23,24,25,26,27,28,29] have achieved good control results in different systems, but they require control direction information in controller design or convergence analysis. Therefore, they are not always feasible for systems with unknown control directions.

The Nussbaum gain technique is the main method to solve the ILC problem in systems with unknown control directions. After the concept of continuous-time Nussbaum gain was proposed in [30], scholars have designed several adaptive ILC algorithms based on Nussbaum gain technique for parameterized nonlinear systems with unknown control directions [31,32,33,34,35]. After the Nussbaum gain concept was extended to discrete-time systems in [36], Yu et al. used the n-step prediction method to estimate the future states of the system, then established the corresponding adaptive ILC method to settle the unknown control direction [37]. For nonparametric nonlinear systems, in addition to transforming the systems to compact structures, the adaptive ILC developed to the systems with unknown control directions discussed not only the randomly varying initial errors, but also the randomly varying reference trajectories [38,39]. However, the systems considered in the above adaptive ILC methods have neglected the input saturations of the real systems.

In this study, the iterative variable reference trajectory problem for nonlinear discrete-time systems with input saturation and unknown control direction is discussed, analyzed, and verified. Firstly, an adaptive fuzzy iterative learning controller is designed, which consist of the adaptive item and fuzzy logic system (FLS) to compensate the loss caused by input saturation. Then, the Nussbaum gain technique is adopted in the iteration domain and embedded to the adaptive learning algorithm to identify the control direction of the system. Finally, using the nonincreasing Lyapunov-like function, we show that the adaptive iterative learning controller can make the tracking error converge asymptotically when the number of iterations tends to infinity, and the system signals always remain bounded in the learning process. To the best of our knowledge, this is the first time to discuss the adaptive ILC design for the nonlinear systems with saturated inputs and unknown control directions. Moreover, several uncertain varying factors, including random iterative initial errors, iteration-varying reference trajectories, and external disturbances, are investigated. Therefore, the adaptive fuzzy ILC design proposed in this study has a wide range of practical scenarios. A simulation example validates the feasibility and effectiveness of the learning control method.

2. Problem Description

2.1. System Model and Input Saturation Model

Consider the following ILC problem for nonlinear discrete-time systems with input saturation:

$$\left\{\begin{array}{c}{x}_i{(t+1,k)=p}_i\left({\overline{x}}_i{(t,k))+q}_i\right({\overline{x}}_i{(t,k))x}_{i+1}(t,k),(i=1,2,\dots ,m-1)\\ x_m(t+1,k)=p_m\left({\overline{x}}_m{(t,k))+q}_m\right({\overline{x}}_m(t,k)\left)u\right(t,k)+d(t,k)\\ {y(t,k)=x}_1(t,k)\end{array}\right.$$

(1)

where $k=0,1,2, \dots$ and $t\in \{0,1, \dots ,T\}$ refer to the number of iterations and discrete-time instants, respectively. For $i=1,2,\dots ,m, (m>2)$, ${\overline{x}}_i(t,k)={\left[x_{\mathit{1}}{(t,k) x}_{\mathit{2}}{(t,k) \dots x}_i(t,k)\right]}^T\in R^i$ correspond to the system state vectors, $p_i({\overline{x}}_i(t,k))$ and $q_i({\overline{x}}_i(t,k))$ are unknown nonlinear functions, and $u(t,k)\in R$, $y(t,k)\in R$, $\mathrm{and} d(t,k)\in R$ represent the system input, output, and external disturbance, respectively. In addition, there exists an unknown constant $\overline{d} > 0$ that makes $\left|d\left(t,k\right)\right|\le \overline{d}$. That is to say, the iteration-varying $d(t,k)$ is a bounded exteral disturbance. The saturated input model $u(t,k)=U\left(\tau \right(t,k\left)\right), (t\in \{0,1, \dots ,T-m\})$ is given as:

$$u(t,k)=U\left(\tau \right(t,k\left)\right)=\left\{\begin{array}{c}\mu , \mu \le \tau (t,k) \\ \tau (t,k), -\mu \le \tau (t,k)\le \mu \\ -\mu , \tau (t,k)\le -\mu \end{array}\right.$$

(2)

where $\tau (t,k)\in R$ is the input to the saturated nonlinearity and μ is a specific boundary value that determines the upper limit of the control signal. It comes from physical constraints or artificial limiters.

It is assumed that the nonlinear SISO system (1) with input saturation (2) needs to track the iterative variable trajectory $y_o(t,k)$ on $t\in \{m, m+1, \dots , T\}.$ The purpose of this study is to propose an adaptive fuzzy ILC method to determine the control signal $\tau (t,k)$, so that the plant (1) under randomly initial error $y(t,k)-y_o(t,k)$ can still track the randomly varying trajectory $y_o(t,k)$ on $t\in \{m, m+1, \dots , T\}$ when the number of iteration approaches to infinity.

Remark 1.

It can be seen from the nonlinear SISO system (1) that the system input $u(t,k)$ cannot determine or influence the output $y(t,k)$ at the initial time until t ≥ m. Therefore, the tracking on instants $t\in \{\mathit{0}, \mathit{1}, \dots , m-\mathit{1}\}$ is not included in our control objective.

Assumption 1.

Let $q(t,k)=\partial y(t+m,k)/\partial u(t,k)={\mathsf{\prod }}_{i=1}^m{q}_i{(\overline{x}}_i(t,k))$; for each fixed t∈{0, 1, …, T$-$m}, not only do we not know the control gain $q(t,k)$, but we also do not know the sign of its value. However, for all k, the sign is iteratively invariant and there are two constants $\overline{q} > \underset{¯}{q} >0$, such that:

$$0 < \underset{¯}{q} \le |q(t,k)|< \overline{q}$$

(3)

Remark 2.

In fact, in many real dynamic systems, the control gains are bounded. Except for $q(t,k)$, the unknown nonlinear functions appearing in the system dynamics are not affected by a restriction such as the one in (3). Therefore, it is generally assumed that the control gain with the iteration-invariant feature of the control direction is bounded in adaptive ILC literatures [23,24,25,26,27,28,29]. Different from this kind of literature [23,24,25,26,27,28,29], the control direction of this study is unknown, so the key to completing the tracking task is to correctly identify the control direction.

2.2. Nussbaum Gain Function

Let $\{w(t,k)\}$ is a discrete sequence, then:

w(0) = 0, w(k) ≥ 0, |Δw(k)| = |w(k+1) − w(k)| ≤ T₀, (k = 0, 1, …)

(4)

where the constant T₀ is positive. Based on the sequence {w(k)}, the discrete Nussbaum gain function F(w(k)) is defined as:

$$F\left(w\right(k\left)\right)={ w}_{\Gamma }\left(k\right)\mathsf{\Gamma }\left(w\right(k\left)\right)$$

(5)

where $w_{\mathsf{\Gamma }}\left(k\right)=\underset{j\le k}{\sup }\left\{w\right(j\left)\right\}$. Γ(w(k)) is a sign function, either take 1 or take −1, and the default initial value is 1. Then, the value of Γ(w(k)) is determined by the following algorithm described in Figure 1.

Figure 1. The selection law of Γ(w(k)).

When k→+∞, it can be found from the definition (5) of discrete Nussbaum gain F(w(k)) and the selection law of Γ(w(k)) that if the sequence {w(k)} is convergent, the discrete Nussbaum gain F(w(k)) converges to a constant [38]. Therefore, the control direction can be recognized by the sign of the discrete Nussbaum gain F(w(k)).

2.3. Approximation Based on Fuzzy Systems

The knowledge base constructed by the fuzzy reasoning rule set is as follows [40]:

$$\begin{gathered} R_j{: \mathrm{If} z}_{1 }{\mathrm{is} B}_{1j}{ \mathrm{and} z}_{2 }{\mathrm{is} B}_{2j}{ \mathrm{and} \dots \mathrm{and} z}_q{ \mathrm{is} B}_{\mathrm{qj}}, \\ \mathrm{then} \varphi { \mathrm{is} D}_j, j=1,2, \dots ,M, \end{gathered}$$

where the input of FLS is z = [z₁ z₂ … z_q]^T ∈ Ω_z ⊂ R^q, and the output is $\varphi$∈ R. Fuzzy sets B_ij and D_j (i = 1,2, …,q, j = 1,2, …, M) correspond to fuzzy membership functions $s_{B_{\mathrm{ij}}}{(z}_i)$ and $s_{D_j}(\varphi ),$ respectively. M refers to the fuzzy rules number.

FLS with single valued fuzzification, product inference engine, and central average defuzzifier can be expressed as:

$$\mathsf{\varphi }(z,W)=\frac{{\sum }_{j=1}^M{w}_j({\prod }_{i=1}^q{s}_{B_{\mathrm{ij}}}{(z}_i\left)\right)}{{\sum }_{j=1}^M{\prod }_{i=1}^q{s}_{B_{\mathrm{ij}}}{(z}_i)}=\sum _{j=1}^M{w}_j{s}_j{\left(z\right)=W}^{T}S\left(z\right)$$

(6)

where $w_j{=\max }_{\varphi \in R}{\{s}_D(\varphi \left)\right\}$ and W = [w₁ w₂ … w_M]^T represents the parameter vector; $s_j\left(z\right)=\frac{{\prod }_{i=1}^q{s}_{B_{\mathrm{ij}}}{(z}_i)}{{\sum }_{j=1}^M{\prod }_{i=1}^q{s}_{B_{\mathrm{ij}}}{(z}_i)}, and$S(z) = [s₁(z) s₂(z) … s_M(z)]^T denotes the fuzzy basis function vector.

Obviously, based on the definition in (6), we obtain:

$$\sum _{j=1}^M{s}_j\left(z\right){=1,\left|\right|S\left(z\right)\left|\right|}^2\le 1$$

(7)

where ${\left|\right|S\left(z\right)\left|\right|}^2{=S}^T\left(z\right)S\left(z\right)$.

Reference [40] proved that for any given compact set ${\mathsf{\Omega }}_z\in R^q$ and any $\overline{\epsilon }$ > 0, continuous nonlinear function p(z), there exists the following FLS form:

$$p\left(z\right)=\varphi {\left(z\right)+\epsilon }^{*}\left(z\right)$$

2.4. The System Transformation

In order to facilitate the design of adaptive fuzzy ILC controllers for systems with input saturation and unknown control direction, the nonlinear SISO system (1) is rewritten and deformed as:

$${y(t+m,k)=x}_1{(t+m,k)=K(\overline{x}}_m{(t,k))+q(t,k\left)u\right(t,k)+d}_1(t,k)$$

(8)

where $q(t,k)$, ${K(\overline{x}}_m(t,k))$ and $d_1(t,k)=d(t,k){\prod }_{i=1}^{m-1}{Q}_i{(\overline{x}}_m(t,k)),(i=1,2,\dots ,m-1)$ can refer to Appendix A of [39]. $Q_i{(\overline{x}}_m(t,k))$ and $q_i{(\overline{x}}_i(t,k))$ have the same range for i = 1,2, …, 𝑚 − 1. Therefore, the boundedness of $Q_i{(\overline{x}}_m(t,k))$ can be achieved from Assumption 1. Moreover, due to the boundedness of the external disturbance $d(t,k)$, the boundedness of $d_1(t,k)$ is guaranteed. That is to say, there exists a constant ${\overline{d}}_1>0$ such that:

$${|d}_1(t,k)|=|d(t,k)\prod _{i=1}^{m-1}{Q}_i{(\overline{x}}_m(t,k)\left)\right|\le {\overline{d}}_1$$

(9)

Substitute (2) into (8), we obtain:

$$\begin{gathered} {y(t+m,k)=K(\overline{x}}_m{(t,k)+q(t,k)\left[U\right(\tau (t,k)\left)\tau \right(t,k)+h(\tau (t,k),t\left)\right]+d}_1(t,k) \\ {=E(\overline{x}}_m(t,k),\tau (t,k)+q\left(t,k\right){h\left(\tau \right(t,k),t)+d}_1(t,k) \end{gathered}$$

(10)

where:

$${E(\overline{x}}_m{(t,k),\tau (t,k)=K(\overline{x}}_m\left(t,k\right)+q\left(t,k\right)U\left(\tau \right(t,k),t)\tau (t,k).$$

(11)

Define:

$${H(\overline{x}}_m(t,k),\tau (t,k))=\frac{\partial {E(\overline{x}}_m(t,k),\tau (t,k))}{\partial \tau (t,k)}, t\in \{0,1,\dots ,T-m\}$$

From (11) and (2):

$${\mathrm{H}(\overline{x}}_m(t,k),\tau (t,k))=\left\{\begin{array}{c}\mu q(t,k), \mu \le \tau \left(t,k\right) \\ \tau \left(t,k\right)q(t,k), -\mu \le \tau \left(t,k\right)\le \mu \\ -\mu q(t,k), \tau \left(t,k\right)\le -\mu \end{array}\right.$$

(12)

Clearly, it can be known that μ in (2) and 0 < $\underset{¯}{q}$≤ |q(t,k)| ≤ $\overline{q}$ in (3) are bounded, so that, when $\tau (t,k) \ne 0$, ${H(\overline{x}}_m(t,k),\tau (t,k))$ is nonzero and bounded. Moreover, according to (12), ${H(\overline{x}}_m(t,k),\tau (t,k))$ and $q(t,k)$ have the same sign.

3. Adaptive Fuzzy ILC Design and Analysis

In order to carry out the design of the adaptive fuzzy ILC controller, first, we define the tracking error as $e(t,k)=y(t,k)-y_o(t,k)$, according to (8):

$$\begin{gathered} e(t+m,k)=y(t+m,k-y_o(t+m,k) \\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =E(\overline{x}}_m(t,k),\tau (t,k))+q\left(t,k\right){h\left(\tau \right(t,k),t)+d}_1(t,k)-y_o(t+m,k) \end{gathered}$$

(13)

If $\mu \to \propto$ in input saturation (2) and no external disturbance exist in system (1) with $d(t,k)=0$. As a result, we obtain $h\left(\tau \right(t,k),t)=$0. In addition, on the basis of (12), we have:

$$\frac{\partial {\left(E\right(\overline{x}}_m(t,k),\tau (t,k))-y_o(t+m,k))}{\partial \tau (t,k)} \ne 0$$

(14)

Using the implicit function theorem, there exists the ideal control signal ${\tau }^{*}\left(\overline{z}\right(t,k\left)\right)$ as follows:

$${E(\overline{x}}_m{(t,k),\tau }^{*}\left(\overline{z}\right(t,k\left)\right))-y_o(t+m,k)=0$$

(15)

where $\overline{z} (t,k)=[\begin{array}{cc}{\overline{x}}_m^T(t,k)& y_o(t+m,k)\end{array}{]}^T\in R^{m+1}$. According to (13) and (15), when $h\left(\tau \right(t,k),t)=0$ and $d_1(t,k)=0$, we can obtain:

$$\begin{array}{c}{e(t+m,k)=E(\overline{x}}_m{(t,k),\tau }^{*}(\overline{z} (t,k\left)\right))-y_o(t+m,k)=0,t\in \{0,1,\dots ,T-m\}.\end{array}$$

However, due to the unknown characteristics of $p_i({\overline{x}}_i(t,k))$, $q_i({\overline{x}}_i(t,k))$, and μ related in the input saturation (2), the ideal control signal ${\tau }^{*}(\overline{z} (t,k\left)\right)$ cannot be obtained, so FLS is used to approximate it.

$${\tau }^{*}{(\overline{z} (t,k\left)\right)=W}^{*T}{\left(t\right)S(\overline{z} (t,k\left)\right)+\epsilon }^{*}(\overline{z} (t,k\left)\right)$$

(16)

For the established FLS, $S(\overline{z} (t,k\left)\right)\in R^M$, $W^{*}\left(t\right)\in R^M$, and ${\epsilon }^{*}(\overline{z} (t,k\left)\right)\in R^M$ denote fuzzy basic function vector, the optimal fuzzy parameter vector, and approximation error, respectively. Since the optimal fuzzy parameter vector $W^{*}\left(t\right)$ in (16) is unknown, $\hat{W} (t,k)$, which denotes the estimation of $W^{*}\left(t\right)$ at the k-th iteration, is used to estimate $W^{*}\left(t\right)$ iteratively. Based on the framework of (16), the adaptive fuzzy ILC law is appropriately designed as:

$${\tau (t,k)=\hat{W}}^T(t,k)S(\overline{z} (t,k\left)\right)+\hat{\omega } (t,k)$$

(17)

where $\hat{\omega } (t,k)$ represents the estimation of ${\omega }^{*}$, which is set in (20). Next, $\tilde{w} (t,k)=\hat{\omega } (t,k)-{\omega }^{*}$ and $\tilde{W} (t,k)=\hat{W} (t,k)-W^{*}\left(t\right)$ are used to represent the estimation error of the k-th iteration of ${\omega }^{*}$ and $W^{*}\left(t\right)$.

According to (15) and the mean difference value theorem, (13) is transformed to:

$$\begin{gathered} {e(t+m,k)=E(\overline{x}}_m{(t,k),\tau (t,k))+q(t,k\left)h\right(\tau (t,k),t)+d}_1(t,k)-{E(\overline{x}}_m{(t,k),\tau }^{*}(\overline{z} (t,k\left)\right)) \\ {=H(\overline{x}}_m{(t,k),\tau }_c(t,k)\left)\right[\tau (t,k)-{\tau }^{*}{(\overline{z} (t,k\left)\right)]+q(t,k\left)h\right(\tau (t,k),t)+d}_1(t,k) \end{gathered}$$

(18)

${\mathrm{where} H(\overline{x}}_m(t,k),\tau (t,k))$ is defined in (12) and ${H(\overline{x}}_m{(t,k),\tau }_c{(t,k))=H(\overline{x}}_m{(t,k),\tau (t,k)\left)\right|}_{{\tau (t,k)=\tau }_c(t,k)}$, ${\tau }_c(t,k)\in {\left[\min \right\{\tau (t,k),\tau }^{*}(\overline{z} (t,k\left)\right)\},$ $max\{{\tau (t,k),\tau }^{*}(\overline{z} (t,k\left)\right)\left\}\right]$.

Define $H_c{(t,k)=H(\overline{x}}_m{(t,k),\tau }_c(t,k))$; thus, with the help of the boundedness of ${H(\overline{x}}_m{(t,k),\tau }_c(t,k))$, we have two unknown constants $\underset{¯}{H}$ and $\overline{H}$ that meet $0 < \underset{¯}{H}\le { |H}_c(t,k)| \le \overline{H}.$

Remark 3.

In allusion to the description in Section 2.4, although the signs of ${H(\overline{x}}_m{(t,k),\tau }_c(t,k))$in (16) and $H_c(t,k)$are unknown, they have the same constant sign for every$t\in \{\mathit{0},\mathit{1}, \dots ,T-m\}$ as $q(t,k)$. Therefore, in the proposed adaptive fuzzy ILC algorithm, the discrete Nussbaum gain technique can be applied to recognize the unknown sign of $H_c(t,k)$, so as to determine the sign of $q(t,k)$.

Substituting (16) and (17) into (18), we obtain:

$$\begin{gathered} {e(t+m,k)=H}_c{(t,k)[\hat{W}}^T(t,k)S(\overline{z} (t,k\left)\right)+\hat{\omega } (t,k) -W^{*T}\left(t\right)S(\overline{z} (t,k\left)\right)-(\overline{z} (t,k\left)\right)] \\ {+q(t,k)h\left(\tau \right(t,k),t)+d}_1(t,k)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ {=H}_c{(t,k)[\hat{W}}^T{(t,k)S(\overline{z} (t,k\left)\right)+\hat{\omega } (t,k)]+d}_2(t,k) \end{gathered}$$

(19)

where $d_2{(t,k)=H}_c{(t,k)[\rho }^{*}{-\epsilon }^{*}{(\overline{z} (t,k\left)\right)]+q(t,k\left)h\right(\tau (t,k),t)+d}_1(t,k)$, due to $\underset{\_}{H}\le { |H}_c(t,k)| \le \overline{H}$, ${|\omega }^{*}(\overline{z} (t,k\left)\right)| \le \overline{\omega }$, $\underset{\_}{q}\le \left|q\right(t,k\left)\right| \le \overline{q}$ in (3), $\left|h\right(\tau (t,k),t\left)\right| \le \overline{h}$ in (2) and ${|d}_1(t,k)| \le {\overline{d}}_1$ in (9), we can obtain:

$${\omega }^{*}{=\overline{H}}^{-1}\overline{q} \overline{h}$$

(20)

Therefore:

$${|d}_2(t,k)| \le \overline{ H}{(\omega }^{*}{+\overline{\epsilon } )+\overline{q}\overline{h}+\overline{d}}_1{=\overline{H} \overline{\epsilon }+2 \overline{q} \overline{h}+\overline{d}}_1{:=\overline{d}}_2$$

(21)

where ${ \overline{H} \overline{\epsilon }+2 \overline{q} \overline{h}+\overline{d}}_1{:=\overline{d}}_2$ refers to the definition of ${ \overline{H} \overline{\epsilon }+2 \overline{q} \overline{h}+\overline{d}}_1$ as ${ \overline{d}}_2$, and ${ \overline{d}}_2>0$ is an unknown constant. Here, we can easily find from the definition that ${\omega }^{*}$ is an unknown constant related to saturated input (2). An additional adaptive term $\hat{\omega } (t,k)$ in the control signal $\tau (t,k)$ in the fuzzy controller (17) is used to estimate ${\omega }^{*}$, thus compensating for the influence of the saturated input (2).

For the fuzzy controller (17), when k = 0,1,2, …, the parameter adaptive law is designed as:

$$\hat{W} (t,k)=\hat{W} (t,k-1)-\delta F\left(w\right(t,k\left)\right)S( \overline{z} (t,k-1\left)\right)\times \frac{\theta (t,k)\rho (t,k)}{N(t,k)}$$

(22)

where $\hat{W} (t,-{1)=0}_{M\times 1}$, ${ \overline{z} (t,-1)=0}_{M\times 1}$.

$$\hat{\omega } (t,k)=\hat{\omega } (t,k-1)-\delta F\left(w\right(t,k\left)\right)\times \frac{\theta (t,k)\rho (t,k)}{N(t,k)}$$

(23)

where $\hat{\omega } (t,-1)=0$.

$$w(t,k+1)=w(t,k)+\frac{{\theta (t,k)V(t,k)\rho }^2(t,k)}{N(t,k)}$$

(24)

where $w(t,0)=0$.

$$\rho (t,k+1)=\frac{\delta e(t+m,k)}{V(t,k+1)}, \rho (t,0)=2\beta$$

(25)

$$V(t,k)=1+\left|F\right(w(t,k)\left)\right|$$

(26)

$$N\left(t,k\right)=[1+|F\left(w\right(t,k\left)\right)\left|\right][1+\left|\right|S(\overline{z}{(t,k-1)\left)\right||}^2{+\rho }^2\left(t,k\right)]$$

(27)

$$\theta (t,k+1)=\left\{\begin{array}{ll}1,\hfill & \mathrm{if} \left|\rho \right(t,k+1\left)\right|\ge \beta \hfill \\ 0,\hfill & \mathrm{others} \hfill \end{array}\right., \theta (t,0)=1$$

(28)

Among them, the design parameter $\delta > 0$ is a constant. The discrete Nussbaum gain $F\left(w\right(t,k\left)\right)$ is updated in the iterative direction at time $t\in \{0,1, \dots ,T-m\}$. $\beta$ > 0 is the threshold of $\rho (t,k)$ in (28). That is, only if $\left|\rho \right(t,k\left)\right| \ge \beta$, then the adaptive update laws (22)–(24) will be carried out. M is the number of fuzzy rules in (16). With (24), (26) and (27), there is:

$$\begin{gathered} \mathrm{\Delta }w(t,k)=w(t,k+1)-w(t,k)=\frac{{\theta (t,k)V(t,k)\rho }^2(t,k)}{N(t,k)} \\ =\frac{{\theta (t,k)\rho }^2(t,k)}{1+\left|\right|S( \overline{z} (t,k-{1\left)\right)\left|\right|}^2{+\rho }^2(t,k)} \end{gathered}$$

(29)

From (28) and (27), we can obtain:

$$0\le \mathrm{\Delta }w\left(t,k\right)\le 1$$

(30)

Therefore, $F\left(w\right(t,k\left)\right)$ is a discrete Nussbaum gain relative to k according to (4).

Theorem 1.

For the nonparametric nonlinear SISO system (1) with input saturation (2) and unknown control direction, design the adaptive fuzzy ILC controller (17) and the adaptive update laws (22)–(28), then, under assumption 1, we can ensure that:

(1) When$t\in \{m,m+\mathit{1}, \dots ,T\}$, the ILC tracking error$e(t,k)$can converge to a small region around zero as$k\to \infty$, that is:

$${\mathit{lim}}_{k\to +\infty }\mathit{sup}\left|e\right(t,k\left)\right|<\eta \beta /\delta , t\in \{m,m+1,\dots ,T\}$$

where the design parameter$\delta$> 0 and the threshold$\beta$>0 are defined in the adaptive update laws (22)–(28), and${\eta =\mathit{lim}}_{k\to +\infty }\mathit{sup}\left|V\right(t,k\left)\right|$.

(2) The discrete Nussbaum gain$F\left(w\right(t,k\left)\right)$ will eventually converge to a function relative to t, meanwhile, all system signals remain bounded.

The Proof of Theorem 1 is presented in the Appendix A, and the control block diagram is given in Figure 2.

Figure 2. Control block diagram.

Remark 4.

Theorem 1 indicates that the bounded convergence of the tracking error $e(t,k)$: ${\mathit{lim}}_{k\to +\infty }\mathit{sup}|e\left(t,k\right)|<\eta \beta /\delta , t\in \{m,m+\mathit{1},\dots ,T\}$can be achieved. Obviously, the size of the convergence boundary of $e(t,k)$is determined by the design parameter, $\delta$, the given threshold $\beta$, and ${\eta =\mathit{lim}}_{k\to +\infty }\mathit{sup}\left|V\right(t,k\left)\right|$. Therefore, for the adaptive ILC scheme in practical application, the choice of control parameters $\delta$and $\beta$is balanced between the control parameters $\delta$, $\beta$, and FLSto achieve better tracking profiles. In addition, the size of the convergence boundary of tracking error $e(t,k)$is also influenced by the external disturbance $d(t,k)$and the established FLS.

4. Simulation Results

In this section, one example is applied to illustrate the applicability of the obtained adaptive fuzzy ILC. Considering the existence of input saturation and unknown control direction in system (1), we have:

$$\left\{\begin{array}{c}{x}_1(t+1,k)=\frac{1{.15x}_1(t,k)}{{1+(x}_1{(t,k))}^2}+(0.35+0.05\sin (x_1{(t,k)\left)\right)x}_2(t,k)\\ x_2(t+1,k)=\frac{2{.5x}_1(t,k)}{{1+(x}_1{(t,k))}^2{+(x}_2{(t,k))}^2}-u(t,k)+d(t,k)\\ {y(t,k)=x}_1(t,k)\end{array}\right.$$

(31)

where $t\in \{0,1, \dots ,100\}$, $d(t,k)=0.1\cos (0{.05t\left)\cos \right(x}_1(t,k))$ is the iteration variable external disturbance. The saturation input $u(t,k)$ is constructed as:

$$u(t,k)=U\left(\tau \right(t,k\left)\right)=\left\{\begin{array}{c}3.6, 3.6\le \tau (t,k) \\ \tau (t,k), -3.6\le \tau (t,k)\le 3.6\\ -3.6, \tau (t,k)\le -3.6\end{array}\right.$$

(32)

where $t\in \{0,1,\dots ,98\}$. The saturation model (32) is only used for simulation, and will be unknown for the designer. The iterative variable reference trajectory of system (31) is $y_o(t,k)=\sin (0.01\pi t)+m(k)$, $\mathrm{where} m\left(k\right)\in [0.15,0.5] \mathrm{varied} \mathrm{randomly} \mathrm{along} \mathrm{iteration} \mathrm{axis}.$ For more details, we can see from Figure 3. The initial states ${ \overline{x}}_2{(0,k)=[x}_1{(0,k) x}_2{(0,k)]}^T=[\begin{array}{cc}0.05& 0.05\end{array}{]}^T$. It should be checked that the initial errors $e(0,k)=y(0,k)-y_o{(0,k)=x}_1(0,k)-y_o(0,k)$ will change iteratively when the reference trajectories $y_o(t,k)$ vary randomly.

Figure 3. The randomly varying factor m(k) in $y_o(t,k)$.

One can conclude with the nonlinear discrete-time system (31) that $q_1{( \overline{x}}_1(t,k))=0.35+0.05\sin (x_1(t,k))>0$ and $q_2{( \overline{x}}_2(t,k))=-1$. Accordingly, we have $q(t,k)=\partial y(t+2,k)/\partial u(t,k)={\prod }_{i=1}^2{q}_i{( \overline{x}}_i(t,k))<0$. While from the definition of the Nussbaum gain function in (5), we can obtain that the default initial value of the sign function $\mathsf{\Gamma }\left(w\right(t,k\left)\right)$ in $F\left(w\right(t,k\left)\right)$ is $\mathsf{\Gamma }\left(w\right(t,0\left)\right)=+1$, $t\in \{0,1,\dots ,98\}$, although $F\left(w\right(t,0\left)\right)=0$ according to $w(t,0)=0$ in (24). Therefore, we can find that the initial sign of the Nussbaum gain function of $F\left(w\right(t,k\left)\right)$ is opposite to the real control direction of the controlled system (31).

In the following simulations, the developed adaptive fuzzy ILC controller (17) with adaptive updating laws (22)–(28) is used to complete the tracking tasks of system (31) with input saturation (32) and unknown control direction. In terms of Theorem 1, the design parameters $\delta$ and $\beta$ are taken as $\delta =1.35$ and $\beta =0.001$. Meanwhile, the number of FLS rules is M = 5, the corresponding input vector is $\overline{z} (t,k)=[\begin{array}{ccc}{z}_1(t,k)& z_2(t,k)& z_3(t,k)\end{array}{]}^T=\begin{array}{ccc}[x_1(t,k)& x_2(t,k)& y_o(t+2,k)\end{array}{]}^T$. Therefore, the FLS in (17) is constructed as ${ \hat{W}}^T(t,k)S( \overline{z} (t,k\left)\right)=\frac{{\sum }_{j=1}^5{ \hat{W}}_j(t,k){\prod }_{i=1}^3{s}_{B_{\mathrm{ij}}}{(z}_i(t,k))}{{\sum }_{j=1}^5{\prod }_{i=1}^3{s}_{B_{\mathrm{ij}}}{(z}_i(t,k))}$, the membership functions corresponding to the input $z_i(t,k), (i=1,2,3)$ are as follows:

$$\begin{gathered} s_{B_{i1}}{(z}_i(t,k))=\exp \{-\frac{{{(z}_i(t,k)+4)}^2}{5}\} \\ {s}_{B_{i2}}{(z}_i(t,k))=\exp \{-\frac{{{(z}_i(t,k)+2)}^2}{5}\}, \\ {s}_{B_{i3}}{(z}_i(t,k))=\exp \{-\frac{{{(z}_i(t,k))}^2}{5}\}, \\ {s}_{B_{i4}}{(z}_i(t,k))=\exp \{-\frac{{{(z}_i(t,k)-2)}^2}{5}\}, \\ {s}_{B_{i5}}{(z}_i(t,k))=\exp \{-\frac{{{(z}_i(t,k)-4)}^2}{5}\}. \end{gathered}$$

The following tracking error square sum index is used to evaluate the accuracy of ILC tracking:

$$\mathrm{EE}\left(k\right)=\sum _{t=2}^{100}{e}^2(t,k)$$

(33)

Figure 4 shows the discrete sequence $\{w\left(t,k\right)\}$ and discrete Nussbaum gain sequence $\left\{F\right(w(t,k)\left)\right\}$ at t = 70. Obviously, from Figure 4, we can notice that the value of discrete Nussbaum gain $F\left(w\right(70,k)$ turns out to be negative, making it consistent with the correct control direction of system (31) at about 20 iterations.

Figure 4. Discrete sequence $\{w\left(t,k\right)\}$ and discrete Nussbaum gain sequence $\left\{F\right(w(t,k)\left)\right\}$ at t = 70. The iterative tracking error index EE(k) under different iterations is plotted in Figure 5. We can find that at the beginning of the iteration, the error index EE(k) increased significantly due to the control direction of the system (31) has not been correctly judged. After the control direction of the system (31) is correctly identified, the iterative tracking error index EE(k) begins to decrease. Figure 4 and Figure 5 validate that the discrete Nussbaum gain F(w(t,k)) can modify the sign accordingly to accommodate the unknown control direction, and show the effect of the Nussbaum gain function to efficiently identify the right control direction. Figure 6 exhibits the system output $y(t,k)$ and the reference trajectory $y_o(t,k)$ at 100th and 200th iterations, respectively. It can be observed that the system output $y(t,k)$ can track the reference trajectory $y_o(t,k)$ as the iteration increase. Figure 7 shows $u(t,k)$ and $\tau (t,k)$ of the input saturation (32) at t = 70. Figure 8 exhibits the corresponding estimation of $\hat{\omega } (t,k)$ and $\hat{W} (t,k)$ when t = 70.

The simulation results in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 verify the effectiveness of the proposed approach. In the presence of the input saturation and unknown control direction, the adaptive learning method fulfils that the Nussbaum gain function identifies the control direction correctly, the system output tracks the reference trajectory to a small region, and all the system signals are bounded.

Figure 5. Tracking error index EE(k) of different iterations.

Figure 6. The tracking curves of iteration 100 and iteration 200, respectively.

Figure 7. $u(t,k)$ and $\tau (t,k)$ of the input saturation (32) at t = 70.

Figure 8. Estimation of parameters $\hat{\omega }\left(t,k\right)$ and $\hat{W}\left(t,k\right)$ at t = 70.

5. Conclusions

In real life, the performance of a repetitive operating system may be affected by input saturation, and thus, loses stability. In this study, an adaptive fuzzy ILC algorithm is proposed for a class of nonlinear discrete systems with input saturations and unknown control directions under the conditions of randomly varied initial error, reference trajectory and external disturbance. In this algorithm, FLS is constructed to estimate the ideal control signal, while an adaptive fuzzy iterative learning controller is designed to compensate the loss caused by input saturation. Then, the discrete Nussbaum gain technique is applied to identify the unknown control direction along the iteration axis. After theoretical analysis and simulation verification, the conclusions are as follows: when the number of iterations tends to infinity, the ILC tracking error can converge to an adjustable residual set except the initial time, and the system signals can remain bounded in the adaptive ILC process. In addition, some interesting research, such as unknown control direction system combined with actuator failure, will be our future research direction.