High-order model-free adaptive iterative learning control

Abstract

In this paper, a high-order model-free adaptive iterative learning control (HOMFAILC) scheme is proposed for nonlinear non-affine discrete-time systems. The main feature of the proposed HOMFAILC is that not only the control input iterative learning law is designed with higher-order algorithms but also the parameter iterative adaptive updating law has the similar forms. Using additional control knowledge in both the iterative learning control law and the parameter estimation law, the control performance of the proposed HOMFAILC can be improved consequently. Furthermore, the similar structure in the control law and parameter updating law can also help to achieve a better control performance. The convergence is proved with rigorous mathematical analysis. Simulation study shows the effectiveness of the proposed HOMFAILC.

Keywords

Iterative learning control higher-order algorithms model-free adaptive ILC nonlinear non-affine systems

Introduction

In practice, many processes perform the same tasks repeatedly, such as high-speed trains (Sun et al., 2013), industrial robots (Chien and Tayebi, 2008), and so on. Iterative learning control (ILC) (Arimoto et al., 1984) is suitable for systems characterized by repeated operation in a finite time interval. It improves the control effect by learning from previous experience. After nearly 40 years of development, ILC has obtained rich theoretical results (Amann et al., 1996; Freeman, 2017; Jonnalagadda and Elumalai, 2020; Volckaert et al., 2013; Xu and Li, 2018) and many successful applications (Li et al., 2016; Sampson et al., 2016; Won et al., 2017) in practice.

High-order ILC (Bien and Huh, 1989; Chen et al., 1997a, 1998; Gunnarsson and NorrlöF, 2006; HäTöNen et al., 2006) is an important branch of ILC which uses more information to further improve control performance. At present, scholars have done a lot of work on high-order ILC. Li et al. (2012) proposed a high-order ILC with initial value learning for linear systems. For the possible output measurement data dropout, Bu et al. (2011) designed a high-order ILC using the super-vector approach. In addition, Chen et al. (1997b) and Boudria and Gauthier (2012) investigated high-order terminal ILC using only the terminal state.

Note that some of the high-order ILCs (Bien and Huh, 1989; Boudria and Gauthier, 2012; Bu et al., 2011; Chen et al., 1997a, 1998; Gunnarsson and NorrlöF, 2006; Li et al., 2012) focus on proportional–integral–derivative (PID)-type control laws, and the control performance is limited by the fixed learning gains that cannot be adapted to change. Beyond that, most of the controller design and analysis are for linear systems (Boudria and Gauthier, 2012; Bu et al., 2011; Chen et al., 1997b; Gunnarsson and NorrlöF, 2006; HäTöNen et al., 2006; Li et al., 2012) or affine nonlinear systems (Bien and Huh, 1989; Chen et al., 1997a, 1998). It is worth pointing out that these methods (Bien and Huh, 1989; Boudria and Gauthier, 2012; Bu et al., 2011; Chen et al., 1997a, 1997b, 1998; Gunnarsson and NorrlöF, 2006; HäTöNen et al., 2006; Li et al., 2012) need to know the model information of the system in actual application. However, in practice, it is difficult to establish an accurate system model, especially for increasingly complex industrial processes.

In response to such problems, data-driven high-order ILC (Ai et al., 2020; Chi et al., 2008b) has been developed, which not only retains the advantages of the original high-order ILC but also combines the data-driven feature, that is, requiring only I/O data instead of explicit model knowledge. In addition, Chi et al. (2018) apply the same idea to terminal ILC. However, the above data-driven high-order ILCs (Ai et al., 2020; Chi et al.,2008b, 2018) only adopt a high-order algorithm in either the control law or parameter estimation law. It is clear that the control law and parameter updating law are not symmetric in the structure form although they are two complementary parts of the whole control strategy.

In fact, the symmetrical similarity is common in nature and comes from “self-optimization” to have a comparatively superior behavior. Therefore, the study of symmetric similarity in control systems becomes interesting but challenging due to few mathematical tools to be employed. Hou (1994) and Hou and Huang (1997) introduced the basic idea of adaptive control system design with symmetric similarity structure. Similar results can also be found in studies by Alessandri et al. (2003), Goodwin et al. (2005), and Gopaluni et al. (2004). Recently, model-free adaptive ILC (MFAILC), which is similar to the existing model-free adaptive control, has been developed by Chi and Hou (2007). Some adaptive ILC schemes based on the similarity analysis between adaptive control and ILC have been proposed by Chi et al. (2008a).

Inspired by the above analysis, a novel high-order model-free adaptive iterative learning control (HOMFAILC) approach is presented. Two high-order laws are contained in the proposed HOMFAILC: a high-order ILC law and a high-order adaptive iterative learning parameter estimation law. At first, a linear data model is obtained by dynamic linearization technique, which is equal to the original nonlinear system. On this basis, a high-order control law is obtained by designing an objective function containing multiple control input signals in historical batches. Similarly, an objective function containing multiple parameters in the previous batches is developed and a high-order parameter iterative updating law is designed by solving the optimal problem of the objective function. The main innovations of the paper include:

Compared with first-order method (Chi and Hou, 2007), the parameter adaptive estimation law and control law are both designed with high-order algorithms, which can enhance the control performance by utilizing extra historical data.

Compared with high-order ILC methods (Ai et al., 2020; Chi et al., 2008b, 2018), the proposed HOMFAILC has a dual high-order symmetric structure, which can help improve control performance and to facilitate the design and analysis of the control system.

Convergence analysis becomes more difficult because more information from previous batches is used in the HOMFAILC. To solve this problem, clamping criterion and lifting technology are introduced as basic mathematical tools to obtain theoretical analysis and proof.

The proposed HOMFAILC is data-driven and does not use any process modle information except for the I/O data, which is more suitable for practical systems.

The remainder of this paper is constructed as follows. Section 2 gives the problem formulation. Section 3 presents the HOMFAILC method. In section 4 the convergence analysis is shown. Section 5 gives the simulation results. Conclusions are in section 6.

Problem formulation

Consider the following repeatable non-affine nonlinear discrete-time system

\begin{matrix} z (s + 1, v) = f (z (s, v), \dots, z (s - n_{z}, v), u (s, v), \dots, u (s - n_{u}, v)) \end{matrix}

(1)

where $z (s, v) \in R$ and $u (s, v) \in R$ are output and input at the $s$ sampling moment of iteration $v$ , respectively; $s \in Z^{T}$ ; $v \in Z^{+}$ ; $n_{z}$ , $n_{u} \in Z^{+}$ ; $Z^{T} = {0, 1, \dots, T}$ ; $Z^{+}$ means positive integer, and $f (\cdot)$ is an unknow function.

Two assumptions are made for system (1).

Assumption 1 (Chi and Hou, 2007): The partial derivative $\partial f (\cdot) / \partial u (s, v)$ is continuous.

Assumption 2 (Chi and Hou, 2007): System (1) is generalized Lipschitz, $\forall s \in Z^{T}$ and $\forall v \in ℤ^{+}$ , that is

| Δ z (s + 1, v) | \leq b | Δ u (s, v) |

for $| Δ u (s, v) | \neq 0$ , where $Δ$ is an iterative difference operator, and $b > 0$ .

Remark 1: By nature, Assumption 1 means that the process is smooth without an abrupt change, which is a most common assumption for practical physical processes. Assumption 2 means that a finite input can produce a finite output, which is easy to understand in terms of the conservation of energy law.

Lemma 1 (Chi and Hou, 2007): System (1) satisfies assumptions 1–2. If $| Δ u (s, v) | \neq 0$ , then the nonlinear system can be transformed into a data model along the direction of the iteration axis

Δ z (s + 1, v) = ψ (s, v) Δ u (s, v)

(2)

where $ψ (s, v)$ is an unknown parameter which satisfies $| ψ (s, v) | \leq b$ according to Assumption 2.

In this work, a control input is intended to be established so that the tracking error, $ϵ (s + 1, v) = z_{d} (s + 1) - z (s + 1, v)$ , iteratively asymptotically tends to 0, where $z_{d} (s + 1)$ is the desired output.

High-Order Model Free Adaptive ILC

The following objective function is considered

\begin{matrix} J (u (s, v), α) = {| z_{d} (s + 1) - z (s + 1, v) |}^{2} \\ + λ {| u (s, v) - \sum_{m = 1}^{l} α_{m} u (s, v - m) |}^{2} \end{matrix}

(3)

where $λ > 0$ is a penalty factor; $α = [α_{1}, α_{2}, \dots, α_{l}]$ represents the weights of the parameters to input information at the previous batches, $\sum_{m = 1}^{l} α_{m} = 1$ , and $l$ is an artificial order length.

According to equation (2), one can rewrite equation (3) as

\begin{matrix} J (u (s, v), α) = {| ϵ (s + 1, v - 1) - ψ (s, v) (u (s, v) - u (s, v - 1)) |}^{2} \\ + λ {| u (s, v) - \sum_{m = 1}^{l} α_{m} u (s, v - m) |}^{2} \end{matrix}

Using optimal technique, one can have

\begin{matrix} u (s, v) = \frac{ψ {(s, v)}^{2}}{λ + ψ {(s, v)}^{2}} u (s, v - 1) + \frac{λ}{λ + ψ {(s, v)}^{2}} \\ \sum_{m = 1}^{l} α_{m} u (s, v - m) \\ + \frac{ψ (s, v)}{λ + ψ {(s, v)}^{2}} ϵ (s + 1, v - 1) \end{matrix}

(5)

Note that, parameter $ψ (s, v)$ is unknown in equation (5). Define $\hat{ψ} (s, v)$ as the estimate of $ψ (s, v)$ . The following objective function is considered

\begin{matrix} J (\hat{ψ} (s, v), β) = {| Δ z (s + 1, v - 1) - \hat{ψ} (s, v) Δ u (s, v - 1) |}^{2} \\ + μ {| \hat{ψ} (s, v) - \sum_{m = 1}^{l} β_{m} \hat{ψ} (s, v - m) |}^{2} \end{matrix}

(6)

where $μ > 0$ is an adjustable factor to influence the rate at which parameter estimates change, $β = [β_{1}, β_{2}, \dots, β_{l}]$ represents the weights of the parameters estimation at the previous batches, and $\sum_{m = 1}^{l} | β_{m} | = 1$

Minimizing equation (6) with respect to $\hat{ψ} (s, v)$ , we get

\begin{matrix} \hat{ψ} (s, v) = \sum_{m = 1}^{l} β_{m} \hat{ψ} (s, v - m) + \frac{Δ u (s, v - 1)}{μ + Δ u {(s, v - 1)}^{2}} \\ \times (Δ z (s + 1, v - 1) - Δ u (s, v - 1) \sum_{m = 1}^{l} β_{m} \hat{ψ} (s, v - m)) \end{matrix}

(7)

Therefore, we obtain the following high-order model free adaptive ILC (HOMFAILC)

\begin{matrix} \hat{ψ} (s, v) = \sum_{m = 1}^{l} β_{m} \hat{ψ} (s, v - m) + \frac{η Δ u (s, v - 1)}{μ + Δ u {(s, v - 1)}^{2}} \\ \times (Δ z (s + 1, v - 1) - Δ u (s, v - 1) \sum_{m = 1}^{l} β_{m} \hat{ψ} (s, v - m)) \end{matrix}

(8)

\begin{matrix} \hat{ψ} (s, v) = \hat{ψ} (s, 0), if | \hat{ψ} (s, v) | \leq ς or \\ sign (\hat{ψ} (s, v)) \neq sign (\hat{ψ} (s, 0)) or | Δ u (s, v - 1) | \leq ς \end{matrix}

(9)

\begin{matrix} u (s, v) = \frac{\hat{ψ} {(s, v)}^{2}}{λ + \hat{ψ} {(s, v)}^{2}} u (s, v - 1) \\ + \frac{λ}{λ + \hat{ψ} {(s, v)}^{2}} \sum_{m = 1}^{l} α_{m} u (s, v - m) \\ + \frac{ρ \hat{ψ} (s, v)}{λ + \hat{ψ} {(s, v)}^{2}} ϵ (s + 1, v - 1) \end{matrix}

(10)

where equation (9) is a reset algorithm which is added to improve the parameter estimation ability of equation (8). $ς > 0$ is a small constant. $\hat{ψ} (s, 0)$ is the initial value of $\hat{ψ} (s, v)$ . $0 < η \leq 1$ and $ρ > 0$ are step size factors.

In order to be more intuitive, a block diagram of the proposed HOMFAILC is given in Figure 1.

Figure 1.

Block diagram of the proposed HOMFAILC.

Remark 2: Compared with the first-order method (Chi and Hou, 2007), the parameter adaptive estimation law and control law in the proposed HOMFAILC are both high-order algorithms, which can enhance the control performance by utilizing extra historical data. Compared with high-order ILC methods (Ai et al., 2020; Chi et al., 2008b, 2018), the proposed HOMFAILC is a dual high-order symmetric similar design, which can help to improve control performance and facilitate the design and analysis of the control system.

Remark 3: In theory, the bigger values of $λ$ and $μ$ mean a faster convergence because they are present in the numerator, but an overshoot may occur as a trade-off. While the smaller values of $ρ$ and $η$ mean a faster convergence because they are present in the denominator. In addition, if the control knowledge is close to the current iteration, its weight coefficient may be large, and vice versa. Therefore, one can generally select $α_{1} \geq α_{2} \geq \dots \geq α_{l}$ and $β_{1} \geq β_{2} \geq \dots \geq β_{l}$ for the $l$ th-order algorithm.

Remark 4: In practice, the initial estimation of $\hat{ψ} (s, v)$ is selected such that its sign is the same as that of $ψ (s, v)$ , whose sign can be determined using the I/O data by conducting a closed-loop experiment.

Remark 5: Although high-order algorithms mean more storage units and more computing resource requirements (Liu et al., 2021), one can make a trade-off between control performance improvement and control resource use, which is a significant topic in the future research.

Convergence analysis

The following lemmas are useful in the convergence analysis.

Lemma 2 (Jury, 1964): Let

A = {[\begin{matrix} a_{1} & a_{2} & \dots & a_{L - 1} & a_{L} \\ 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 & 0 \end{matrix}]}_{L \times L .}

If $\sum_{j = 1}^{L} | a_{j} | < 1$ , we have $r (A) < 1$ , where $r (\cdot)$ represents the spectral radius.

Lemma 3 (Huang, 1984): For $Γ \in R^{n \times n}$ and $ε > 0$ , there always exists a proper matrix norm $| | \cdot | |_{w}$ on the normed vector space $w$ , such that $| | Γ | |_{w} \leq r (Γ) + ε .$

Theorem 1: For system (1) under assumptions 1–2, applying the presented HOMFAILC (8)–(10), the following two conclusions can be obtained:

1. $\hat{ψ} (s, v)$ is bounded, $\forall s \in Z^{T}$ and $\forall v \in Z^{+}$ ;

2. The tracking error converges to 0 iteratively, that is, $lim_{v \to \infty} ϵ (s + 1, v) = 0$ .

Proof: The following two parts prove Theorem 1 in detail.

(i) The boundedness of $\hat{ψ} (s, v)$

If condition $| \hat{ψ} (s, v) | \leq ς or | Δ u (s, v - 1) | \leq ς$ is satisfied, then $\hat{ψ} (s, v)$ is clearly bounded.

In other case, define $\tilde{ψ} (s, v) = \hat{ψ} (s, v) - ψ (s, v)$ as the parameter estimation error. From equation (8), one has

\begin{matrix} \tilde{ψ} (s, v) = \sum_{m = 1}^{l} β_{m} \hat{ψ} (s, v - m) - ψ (s, v) + \frac{η Δ u (s, v - 1)}{μ + Δ u {(s, v - 1)}^{2}} \\ \times (Δ z (s, v) - Δ u (s, v - 1) \sum_{m = 1}^{l} β_{m} \hat{ψ} (s, v - m)) \end{matrix}

(11)

Furthermore, equation (11) can be rewritten

\begin{matrix} \tilde{ψ} (s, v) = (1 - \frac{η Δ u {(s, v - 1)}^{2}}{μ + Δ u {(s, v - 1)}^{2}}) \sum_{m = 1}^{l} β_{m} \tilde{ψ} (s, v - m) \\ + (1 - \frac{η Δ u {(s, v - 1)}^{2}}{μ + Δ u {(s, v - 1)}^{2}}) \sum_{m = 1}^{l} β_{m} ψ (s, v - m) \\ + \frac{η Δ u {(s, v - 1)}^{2}}{μ + Δ u {(s, v - 1)}^{2}} ψ (s, v - 1) - ψ (s, v) \end{matrix}

(12)

And then we convert this formula to a lifted form

\begin{matrix} {\tilde{ψ}}_{l} (s, v) = Θ (s, v) {\tilde{ψ}}_{l} (s, v - 1) + Θ (s, v) ψ_{l} (s, v - 1) \\ - θ (s, v) ψ_{l} (s, v - 1) \\ - ψ_{l} (s, v) + ψ_{l} (s, v - 1) \end{matrix}

(13)

where

\begin{matrix} {\tilde{ψ}}_{l} (s, v) = {[\tilde{ψ} (s, v), \tilde{ψ} (s, v - 1), \dots, \tilde{ψ} (s, v - l + 1)]}^{T} \\ {\tilde{ψ}}_{l} (s, v) = {[ψ (s, v), ψ (s, v - 1), \dots, ψ (s, v - l + 1)]}^{T} \\ θ (s, v) = diag [1 - \frac{η Δ u {(s, v - 1)}^{2}}{μ + Δ u {(s, v - 1)}^{2}}, 1, \dots, 1] \end{matrix}

Θ (s, v) = {[\begin{matrix} Θ_{1} (s, v) & Θ_{2} (s, v) & \dots & Θ_{l - 1} (s, v) & Θ_{l} (s, v) \\ 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 & 0 \end{matrix}]}_{l \times l}

Θ_{m} (s, v) = (1 - \frac{η Δ u {(s, v - 1)}^{2}}{μ + Δ u {(s, v - 1)}^{2}}) β_{m}, m = 1, 2, \dots, l

Taking norms on both sides of equation (13), one can guarantee that

‖ {\tilde{ψ}}_{l} (s, v) ‖_{w} \leq ‖ Θ (s, v) ‖_{w} ‖ {\tilde{ψ}}_{l} (s, v - 1) ‖_{w} + q (s, v)

(14)

where

\begin{matrix} q (s, v) = ‖ Θ (s, v) ‖_{w} ‖ ψ_{l} (s, v - 1) ‖_{w} + ‖ θ (s, v) ‖_{w} ‖ ψ_{l} (s, v - 1) ‖_{w} \\ + ‖ ψ_{l} (s, v - 1) - ψ_{l} (s, v) ‖_{w} \end{matrix}

Owing to $μ > 0$ , $0 < η \leq 1$ , and $| Δ u (s, v - 1) | > ς$ , we have $\frac{η ς^{2}}{(μ + ς^{2})} \leq \frac{η Δ u {(s, v - 1)}^{2}}{(μ + Δ u {(s, v - 1)}^{2})}$ . Then, there exists a constant $d_{0}$ such that

0 \leq | 1 - \frac{η Δ u {(s, v - 1)}^{2}}{μ + Δ u {(s, v - 1)}^{2}} | \leq | 1 - \frac{η ς^{2}}{μ + ς^{2}} | \leq d_{0} < 1

(15)

By setting $\sum_{m = 1}^{l} | β_{m} | = 1$ , one can guarantee that

\sum_{m = 1}^{l} | Θ_{m} (s, v) | = | 1 - \frac{η Δ u {(s, v - 1)}^{2}}{μ + Δ u {(s, v - 1)}^{2}} | \sum_{m = 1}^{l} | β_{m} | < 1

It means that $r (Θ (s, v)) < 1$ by the Lemma 2. According to Lemma 3, we have

‖ Θ (s, v) ‖_{w} \leq r (Θ (s, v)) + ε \leq d_{1} < 1

(16)

where $0 < d_{1} < 1$ is a constant.

Since $| ψ (s, v) | \leq b$ , and according to the definition of $ψ_{l} (s, v)$ , the boundedness of $ψ_{l} (s, v)$ is obtained. By virtue of equation (15), we can get $θ (s, v)$ is bounded. So there is a positive constant $d_{2}$ that makes

0 < q (s, v) \leq d_{2}

(17)

Combining equations (14), (15), and (17), it results

\begin{matrix} ‖ {\tilde{ψ}}_{l} (s, v) ‖_{w} \leq d_{1} ‖ {\tilde{ψ}}_{l} (s, v - 1) ‖_{w} + d_{2} \leq d_{1}^{2} ‖ {\tilde{ψ}}_{l} (s, v - 2) ‖_{w} \\ + d_{1} d_{2} + d_{2} \\ \leq \dots \leq d_{1}^{v} ‖ {\tilde{ψ}}_{l} (s, 0) ‖_{w} + \frac{d_{2} (1 - d_{1}^{v})}{1 - d_{1}} \end{matrix}

(18)

which means that $\tilde{ψ} (s, v)$ is bounded. Since $ψ_{l} (s, v)$ is bounded and $\hat{ψ} (s, 0)$ is given bounded, the boundedness of $\hat{ψ} (s, v)$ is obtained directly.

(ii) The convergence of $ϵ (s + 1, v)$

Suppose $u (s, v)$ is the optimal input that minimizes $J (u (s, v), α)$ in the $v th$ iteration, and the corresponding optimal weight is $α$ . We can clearly know that the following inequality is satisfied

J (u (s, v), α) \leq J (u (s, v - 1), Λ_{0})

(19)

where $Λ_{0} = [1, 0, \dots, 0]^{T}$ .

According to equations (2) and (3) can be re-expressed as

\begin{matrix} J (u (s, v), α) = {| ϵ (s + 1, v - 1) - ψ (s, v) Δ u (s, v) |}^{2} \\ + λ {| u (s, v) - \sum_{m = 1}^{l} α_{m} u (s, v - m) |}^{2} \end{matrix}

(20)

Combining equation (20), we get

\begin{matrix} J (u (s, v - 1), Λ_{0}) = J (u (s, v - 1), {[1, 0, \dots, 0]}^{T}) \\ = {| ϵ (s + 1, v - 1) |}^{2} \end{matrix}

(21)

According to equation (3), yields

{| ϵ (s + 1, v) |}^{2} \leq J (u (s, v), α)

(22)

As a result, from equations (19)–(22) we can get

{| ϵ (s + 1, v) |}^{2} \leq J (u (s, v), α) \leq {| ϵ (s + 1, v - 1) |}^{2}

(23)

Obviously, $ϵ (s + 1, v)$ exists as $v$ approaches infinity, which means that

lim_{v \to \infty} {| ϵ (s + 1, v) |}^{2} = lim_{v \to \infty} J (u (s, v), α)

(24)

According to equations (3) and (24), one has

\begin{matrix} lim_{v \to \infty} λ {| u (s, v) - \sum_{m = 1}^{l} α_{m} u (s, v - m) |}^{2} \\ = lim_{v \to \infty} J (u (s, v), α) - lim_{v \to \infty} {| ϵ (s + 1, v) |}^{2} \end{matrix}

(25)

Since $λ$ is a positive constant, we have

lim_{v \to \infty} (u (s, v) - \sum_{m = 1}^{l} α_{m} u (s, v - m)) = 0

(26)

According to equation (10), it obtains

\begin{matrix} u (s, v) - \sum_{m = 1}^{l} α_{m} u (s, v - m) = \frac{ρ \hat{ψ} (s, v)}{λ + \hat{ψ} {(s, v)}^{2}} ϵ (s + 1, v) \\ + \frac{\hat{ψ} {(s, v)}^{2}}{λ + \hat{ψ} {(s, v)}^{2}} (u (s, v - 1) - \sum_{m = 1}^{l} α_{m} u (s, v - m)) \end{matrix}

(27)

Combining equations (26) and (27), we obtain

lim_{v \to \infty} \frac{ρ \hat{ψ} (s, v)}{λ + \hat{ψ} {(s, v)}^{2}} ϵ (s + 1, v) = 0

(28)

According to (i), the boundeness of $\hat{ψ} (s, v)$ is proved. Moreover, $\hat{ψ} (s, v) \neq 0$ can be guaranteed by the reset algorithm (9). Consequently, conclusion (ii) of Theorem 1 is the direct result of equation (28), that is, $lim_{v \to \infty} ϵ (s + 1, v) = 0$ .

Simulation

Example 1: numerical simulation

The following nonlinear system is considered

\begin{matrix} z (s + 1, v) = \\ {\begin{matrix} \frac{z (s, v)}{1 + z^{2} (s, v)} + u^{3} (s, v) + σ (s, v), 0 \leq s \leq 50 \\ (z (s, v) z (s - 1, v) z (s - 2, v) u (s - 1, v) \\ \times (z (s - 2, v) - 1) + a (s) u (s, v)) / \\ (1 + z^{2} (s - 1, v) + z^{2} (s - 2, v)) + σ (s, v) 50 < s \leq 100 \end{matrix} \end{matrix}

where $a (s) = round (s / 50)$ , $s \in {0, 1, \dots, 100}$ , and $σ (s, v)$ is a random disturbance that varies with time and iteration, as shown in Figure 2.

Figure 2.

The random disturbance in Example 1.

The desired output signal is

z_{d} (s) = {\begin{matrix} 0.5 \times {(- 1)}^{round (s / 10)}, & 0 \leq s \leq 30 \\ 0.5 \sin (s π / 10) + 0.3 \cos (s π / 10), & 30 < s \leq 70 \\ 0.5 \times {(- 1)}^{round (s / 10)}, & 70 < s \leq 100 \end{matrix}

In the simulation, the system runs 100 iterations. The proposed HOMFAILC algorithm (8)–(10) is applied. The controller parameters are $μ = 0.1$ , $η = λ = ρ = 0.6$ , $α = [1.25, - 0.25]$ , and $β = [0.55, 0.45]$ . The initial parameter estimate is chosen as $\hat{ψ} (s, 0) = 0.4$ , and the initial output varies randomly in interval $[- 0.05, 0.05]$ .

The system parameter estimation $\hat{ψ} (s, v)$ is shown in Figure 3. One can see from Figure 3 that the parameter estimation is bounded. The maximum tracking error of each batch is denoted as $ϵ_{max} (v) = max_{s \in {1, 2, \dots 100}} | ϵ (s, v) |$ , which is shown in Figure 4. Meanwhile, Figure 5 shows the system output in the $5 th$ , $10 th$ , and $100 th$ iterations. From Figures 4 and 5, we can see that the proposed HOMFAILC method can obtain good control performance even though there exist external disturbances.

Figure 3.

The parameter estimation in Example 1.

Figure 4.

Maximum tracking errors in Example 1.

Figure 5.

Partial batches tracking performance in Example 1.

Compared with the MFAILC (Chi and Hou, 2007) and the high-order ILC proposed by Chi et al. (2008b), the differences are listed in Table 1. For fairness, the same parameters are selected, that is, the initial output varies randomly in interval $[- 0.05, 0.05]$ , $η = λ = ρ = 0.6$ , $μ = 0.1$ , $α = [1.25, - 0.25]$ , and $β = [0.55, 0.45]$ . Figure 6 shows the simulation results. From Figure 6, one can see that the developed HOMFAILC performs better than the other two methods because it utilizes additional control information and has a symmetric similar structure.

Table 1.

Differences of the three methods.

Methods	Orders of the control law	Orders of the parameter estimation law
MFAILC (Chi and Hou, 2007)	One order	One order
High-order ILC (Chi et al., 2008b)	High order	One order
HOMFAILC	High order	High order

MFAILC: model-free adaptive iterative learning control; HOMFAILC: high-order model-free adaptive iterative learning control.

Figure 6.

Maximum tracking errors in Example 1.

Furthermore, to evaluate the control performance of the proposed HOMFAILC numerically, a numerical index is defined as

J = \sum_{v = 1}^{N} ϵ_{max} (v)

(29)

where $N$ is the total number of iterations, and $ϵ_{max} (v)$ is the maximum tracking error defined earlier.

Applying the above three methods, respectively, the index of each method is shown in Table 2, which further confirms the superiority of the proposed HOMFAILC.

Table 2.

The index of each method in Example 1.

Methods	$J$
MFAILC (Chi and Hou, 2007)	11.11
High-order ILC (Chi and Hou, 2007)	9.12
HOMFAILC	7.93

MFAILC: model-free adaptive iterative learning control; HOMFAILC: high-order model-free adaptive iterative learning control.

Example 2: the continuously stirred tank reactor system

In the continuously stirred tank reactor (CSTR; Chang, 2013; Mhaskara et al., 2006), an elementary reaction takes place of the form $A \to B$ , which is a nonlinear process. The model is shown as follows

{\overset{\cdot}{C}}_{A} = \frac{F}{V} (C_{A 0} - C_{A}) - k_{0} e^{- E / R T_{R}} C_{A} + σ_{1}

(30)

{\overset{\cdot}{T}}_{R} = \frac{F}{V} (T_{A 0} - T_{R}) + \frac{- Δ H}{ρ C_{p}} k_{0} e^{- E / R T_{R}} C_{A} + \frac{Q_{σ}}{ρ C_{p} V}

(31)

where $C_{A}$ means the concentration of $A$ . $C_{A 0}$ is the control input. The other parameters are listed in Table 3. $σ_{1}$ is a random disturbance which varies with time and iteration, as shown in Figure 7.

Table 3.

The parameters of a CSTR system.

Symbol	Physical meaning	Value
$V$	The reactor volume	$0.1 m^{3}$
$Δ H$	The reaction enthalpy	$- 4.78 \times 10^{4} kJ / kmol$
$k_{0}$	The pre-exponential factor	$72 \times 10^{9} mi n^{- 1}$
$C_{p}$	The heat capacity	$0.239 kJ / kgK$
$ρ$	The fluid density	$1000 kg / m^{3}$
$E$	The activation energy	$8.314 \times 10^{4} kJ / koml$
$Q_{σ}$	The removed heat	$2.78 \times 10^{4} kJ / \min$
$C_{p}$	/	$0.239 kJ / kgK$
$T_{A 0}$	/	$310.0 K$
$R$	/	$8.314 kJ / kmolK$
$T_{R}$	/	$395.33 K$
$F$	/	$0.1 m^{3} / \min$

CSTR: continuously stirred tank reactor.

Figure 7.

The random disturbance in Example 2.

The control purpose is to stabilize the system at $C_{A} = 0.57 kmol / m^{3}$ . The sampling time is chosen as $0.01$ min, and the simulation lasts for 3 min; thus $s \in {0, 1, \dots, 299}$ . Moreover, the initial concentration fluctuation $C_{A 0}$ is also considered, $C_{A 0} (0, v) = (0.47 + σ_{2} (v)) kmol / m^{3}$ , and $σ_{2} (v)$ iteratively varies in $[- 0.01, 0.01]$ .

In the simulation, the system runs 200 iterations and the HOMFAILC (8)–(10) is applied to CSTR system. The controller parameters are as follows: $\hat{ψ} (s, 0) = 0.5$ , $u (s, 0) = 0.5$ , $ρ = 2$ , $η = 0.001$ , $λ = 0.001$ , and $μ = 0.1$ . The order of the HOMFAILC is set as $l = 4$ . The weight factors of each historical control inputs are $α = [0.6, 0.25, 0.1, 0.05]$ and the weight factors of each historical parameter estimations are $β = [0.6, 0.25, 0.1, 0.05]$ . Figure 8 shows the parameter estimation, which is bounded for all time instants and iterations. Maximum tracking error is plotted in Figure 9 using a red solid line. It can be seen that the tracking error gradually converges with the increase of the number of iterations.

Figure 8.

The parameter estimation in Example 2.

Figure 9.

Maximum tracking errors in Example 2.

For a comparison purpose, the MFAILC (Chi and Hou, 2007) and the high-order ILC (Chi et al., 2008b) are also applied, respectively. For fairness, the parameters are selected to be the same as that of the HOMFAILC method, that is, $λ = 0.001$ , $η = 0.001$ , $ρ = 2$ , $μ = 0.1$ , $α = [0.6, 0.25, 0.1, 0.05]$ , and the intial value $u (s, 0) = 0.5$ , $\hat{ψ} (s, 0) = 0.5$ . The simulation results are shown as the dashed and dotted lines in Figure 9, respectively. Meanwhile, the index $J$ of each method is listed in Table 4. From the comparison results, we can observe that the proposed HOMFAILC performs better than the other two methods due to the use of high-order algorithm and symmetric similarity structure.

Table 4.

The index of each method in Example 2.

Methods	$J$
MFAILC (Chi and Hou, 2007)	3.23
High-order ILC (Chi et al., 2008b)	3.21
HOMFAILC	2.99

MFAILC: model-free adaptive iterative learning control; HOMFAILC: high-order model-free adaptive iterative learning control.

Conclusion

In this work, an HOMFAILC is presented, which includes a high-order ILC law and a high-order adaptive iterative learning parameter estimation law. The design of this dual high-order ILC ensures that the controller can capture more dynamic information of the original system to enhance the control performance. The developed approach is data-driven, which only uses I/O data from previous batches. The convergence of the proposed method is proved by strict mathematical means. The superiority of the presented HOMFAILC approach is confirmed by comparing with the MFAILC and the high-order ILC.

Footnotes

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 in part by the National Science Foundation of China (grant nos. 61873139 and 61833001), in part by the Taishan Scholar Program of Shandong Province of China, in part by the Fundamental Research Funds for the Central Universities, in part by the Independent Innovation Major Project of Qingdao (grant no. 21-1-2-14-zhz), and in part by the Natural Science Foundation of Shandong Province of China (grant no. ZR2019MF036).

ORCID iDs

Na Lin

Ronghu Chi

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High-order model-free adaptive iterative learning control

Abstract

Keywords

Introduction

Problem formulation

High-Order Model Free Adaptive ILC

Convergence analysis

(i) The boundedness of ψ ^ ( s , v )

(ii) The convergence of ϵ ( s + 1 , v )

Simulation

Example 1: numerical simulation

Example 2: the continuously stirred tank reactor system

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

(i) The boundedness of $\hat{ψ} (s, v)$

(ii) The convergence of $ϵ (s + 1, v)$