Stability Analysis of High-Order Iterative Learning Control for a Class of Nonlinear Switched Systems

Abstract

This paper considers the stability of high-order PID-type iterative learning control law for a class of nonlinear switched systems with state delays and arbitrary switched rules, which perform a given task repeatedly. The stability condition for the proposed high-order learning control law is first established, and then the stability is analyzed based on contraction mapping approach in the sense of 𝜆 norm. It is shown that the proposed iterative learning control law can guarantee the asymptotic convergence of the tracking error for the entire time interval through the iterative learning process. Two examples are given to illustrate the effectiveness of the proposed approach.

1. Introduction

A switched system is a hybrid dynamical system, which consists of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching between them. During the past decades, switched systems have been widely studied, and many interesting results have been reported in the literature, for instance [1–3] and the references therein. The motivation to study switched systems is mainly in twofold. First of all, many engineering systems can be represented by switched systems, such as networked control systems (NCS) [4, 5], traffic control [6], automotive engine control, and aircraft control [7]. Secondly, the idea of controller switching is introduced in order to overcome the shortcomings of the single controller and improve system performance [8, 9]. Some methods have been used in the study of switched systems such as the multiple Lyapunov functions [10, 11], the concept of average dwell time [12], and piecewise quadratic Lyapunov functions [13].

Recent researches in switched system typically focus on the analysis of dynamic behaviors, such as stability [14], controllability, reachability [15, 16], and observability [17] aiming to design controllers with guaranteed stability and performance [18, 19]. Besides the aforementioned problem, designing a controller to achieve tracking for switched systems is a challenging problem [20–22]. In [20], the tracking control problem for switched linear systems with time-varying delays is investigated, and the average dwell time approach and piecewise Lyapunov functional methods are utilized to the stability analysis and controller design. In [21], an observer-based tracking control approach is proposed for switched linear time-varying delay systems with unavailable states. The single Lyapunov-Krasovskii functional method is utilized to the stability analysis and controller design. In [22], the tracking control problem for switched nonlinear systems subject to an output constraint is considered. It is worth pointing out that most of these approaches are given based on the accurate model of nonlinear switched systems, and thus their control performances depend on the accuracy of the models. In addition, various unmodelled dynamics and uncertainties always exist in practical systems. The above-mentioned model-based control approaches for switched systems may lead to bad performance or cause closed-loop system unstable in practice. Therefore, it is necessary and practical to design tracking control methods for switched systems requiring less knowledge about the system dynamics.

Fortunately, for repetitive systems, iterative learning control (ILC) offers a systematic design that can improve the tracking performance by iterations in a fixed time interval. The key feature of this technique is to use information from the previous operation in order to enable the controlled system to perform better progressively from operation to operation. It seems that the main advantage of the iterative learning control strategy is to require less a priori knowledge about the system dynamics and less computational effort than many other types of control strategies. Hence, iterative learning control for repetitive dynamical systems has received considerable attention, and it has made significant progresses over the past two decades (see [23–30] and references therein). However, to the best of our knowledge, no one has been studied the iterative learning control for switched systems. This observation motivates the present study.

In practice, some switched systems are in general repeated, such as traffic system and batch process with multiprocedure. The traffic system can be viewed as a switched system [6]. We may easily find that traffic flow patterns in two consecutive days, or the same weekday of two consecutive weeks, are very close. Ruling out the occasional occurrence of accidents, the routine traffic flow on freeway in the macroscopic level will show inherent repeatability every day. Likely we can find the similarities on a monthly basis, or even a yearly basis [26]. For the batch process with multiprocedure, the system can also be considered as a switched system, and the different procedures represent the different subsystems. If the product is batch processing, then the switched system is operated repetitively. In this paper, the problem of iterative learning control for a class of nonlinear switched systems with arbitrary switched rules is considered. Despite much progress, significant research remains to be done in the direction of linear switched systems, especially since the majority of practical switched systems exhibit inherently nonlinear dynamics and delays [31–33]. Hence, we are focusing on ILC for nonlinear switched systems with time delay.

Most of the existing ILC schemes are based on the first-order updating laws; that is, only the information of one previous iteration is employed. [34] used a high-order ILC law for tracking control of nonlinear systems, where, to form the control in current iteration, the information of several previous learning iterations, including control functions, tracking errors as well as their derivatives, is used. It is demonstrated that high-order ILC schemes have potential to give a better convergence performance than the first-order ILC schemes. Therefore, it is beneficial to investigate high-order ILC schemes for the tracking control of nonlinear switched systems.

The rest of this paper is organized as follows. In Section 2, the problem formulation is described. In Section 3, a sufficient condition which guarantees the stability of high-order PID-type ILC for the nonlinear switched ILC system is given. In Section 4, two examples are presented to validate the theoretical results. Finally, some conclusions are given in Section 5.

2. Problem Formulation

Consider a nonlinear switched system with time delay, which performs a given task repeatedly, as follows:𝑥̇_𝑘(𝑡)=𝑓_𝜎(𝑡)𝑥_𝑘(𝑡),𝑥_𝑑(𝑡−ℎ),𝑡+𝐵_𝜎(𝑡)𝑥_𝑘(𝑡),𝑥_𝑘()𝑢(𝑡),𝑦𝑡−𝑑,𝑡_𝑘(𝑡)=𝑔_𝜎(𝑡)𝑥_𝑘(𝑡),𝑡, (1) where 𝑘 denotes the 𝑘th repetitive operation of the system. 𝑥_𝑘(𝑡)∈𝑅^𝑛 is the state vector, 𝑢_𝑘(𝑡)∈𝑅^𝑝 is the input vector, and 𝑦_𝑘(𝑡)∈𝑅^𝑚 is the output vector. 𝑡∈[0,𝑇] is the finite time interval, and ℎ,𝑑<𝑇 are known time delay. 𝜎(𝑡)∶{1,2,…}→Ψ={1,2,…𝑚} is a switching signal, that is, a piecewise constant function. 𝑚 is the number of models (called subsystems) of the switched system. The vector nonlinear function 𝑓_𝜎(𝑡)(⋅), 𝑔_𝜎(𝑡)(⋅), and matrix function 𝐵_𝜎(𝑡)(⋅) have appropriate dimensions. In this paper, we assume 𝜎(𝑡) is an arbitrary switched rule on the time domain, and it is invariable on the iteration domain, that means the functions (𝑓_𝜎(𝑡),𝑔_𝜎(𝑡), and 𝐵_𝜎(𝑡)) are allowed to take values, at an arbitrary discrete time, in the finite set 𝑓₁,𝑔₁,𝐵₁,…,𝑓_𝑚,𝑔_𝑚,𝐵_𝑚. (2) In this case, the nonlinear switched system (1) can be described as 𝑥̇_𝑘(𝑡)=𝑓_𝑖𝑥_𝑘(𝑡),𝑥_𝑘(𝑡−ℎ),𝑡+𝐵_𝑖𝑥_𝑘(𝑡),𝑥_𝑘(𝑡−𝑑),𝑡𝑢_𝑘(𝑡),𝑦_𝑘(𝑡)=𝑔_𝑖𝑥_𝑘(𝑡),𝑡,𝑖∈{1,2,…,𝑚}. (3)

Basic assumptions for the nonlinear switched system are given as follows.

Assumption 1. For a desired trajectory 𝑦_𝑑(𝑡), it exists 𝑢_𝑑(𝑡) and 𝑥_𝑑(𝑡) satisfying 𝑥̇_𝑑(𝑡)=𝑓_𝑖𝑥_𝑑(𝑡),𝑥_𝑑(𝑡−ℎ),𝑡+𝐵_𝑖𝑥_𝑑(𝑡),𝑥_𝑑(𝑡−𝑑),𝑡𝑢_𝑑(𝑡),𝑦_𝑑(𝑡)=𝑔_𝑖𝑥_𝑑(𝑡),𝑡, (4) where 𝑢_𝑑(𝑡) is the desired input and 𝑥_𝑑(𝑡) is the desired state.

Assumption 2. The nonlinear function 𝑓_𝑖,𝐵_𝑖,𝑔_𝑖,𝑔_𝑖𝑥=𝜕𝑔_𝑖/𝜕𝑥, 𝑔_𝑖𝑡=𝜕𝑔_𝑖/𝜕𝑡, 𝑖∈{1,2,…,𝑚} is uniformly globally Lipschitz in 𝑥(𝑡) on interval [0,𝑇], that is, for all 𝑡∈[0,𝑇], ∃ constants 𝑘_ℎ,𝑘_𝑓,𝑘_𝐵 such that ℎ𝑥₁(𝑡),𝑡−ℎ𝑥₂(𝑡),𝑡≤𝑘_ℎ𝑥₁(𝑡)−𝑥₂(𝑡,𝑓_𝑖)𝑥₁(𝑡),𝑥₁(𝑡−ℎ),𝑡−𝑓_𝑖𝑥₂(𝑡),𝑥₂(𝑡−ℎ),𝑡≤𝑘_𝑓𝑥₁(𝑡)−𝑥₂(𝑡)+𝑥₁(𝑡−ℎ)−𝑥₂(,𝐵𝑡−ℎ)_𝑖𝑥₁(𝑡),𝑥₁(𝑡−𝑑),𝑡−𝐵_𝑖𝑥₂(𝑡),𝑥₂(𝑡−𝑑),𝑡≤𝑘_𝐵𝑥₁(𝑡)−𝑥₂(𝑡)+𝑥₁(𝑡−𝑑)−𝑥₂(𝑡−𝑑), (5) for any pair (𝑥₁(𝑡),𝑥₂(𝑡)), where ℎ∈{𝑔_𝑖,𝑔_𝑖𝑥,𝑔_𝑖𝑡}.

Assumption 3. The resetting condition is satisfied for all the iteration; that is, 𝑥_𝑘(𝑡)=𝑥_𝑑(𝑡),𝑡∈−𝜇,0, (6) where 𝜇=max{ℎ,𝑑} and 𝑥_𝑑(𝑡) is the desired initial function.

Remark 4. From Assumption 1, since 𝑢_𝑑(𝑡) exists uniquely, the uniform convergence of the control profile 𝑢(𝑡) to 𝑢_𝑑(𝑡) implies that the state and output tracking errors will vanish. Assumption 2 is a basic condition for switched systems which means that system states are continuous, even for structure switches.

Remark 5. Assumption 3 means 𝑥_𝑘(0)=𝑥_𝑑(0) for all 𝑘, which is the identical initial condition for ILC system. For a switched system, even though the dynamic behavior is changing between different subsystems, the initial condition reset is often satisfied in many practical systems. For instance, when a product is processed by several different procedures, then the system can be considered as a switched system, and the different procedures represent the different subsystems. If the product is batch processing, then the switched system is operated repetitively. Even though the processing is changing between different procedures, the initial state of the product is identical.

Remark 6. The switched rule 𝜎(𝑡) in system (1) is an arbitrary switched rule on the time domain, and it is invariable on the iteration domain. One may argue that the problem considered in this paper is similar to the ILC for time varying nonlinear system. However, there are some crucial differences. The time varying system admits a family of solutions that can be parameterized solely by the initial condition, whereas the switched system admits a family of solutions that is parameterized both by the initial condition and the switching signal. Besides, the switched system has sudden transient at some time instants.

The control target is to find a control input sequence 𝑢_𝑘(𝑡)=𝑢_𝑑(𝑡), such that 𝑦_𝑘(𝑡) converges to 𝑦_𝑑(𝑡) as 𝑘→∞; that is, as the learning iteration repeats, the system output converges to the desired trajectory.

Now, the following high-order ILC updating law for the system (3) is proposed which uses the 𝑃, 𝐼, and 𝐷 information of tracking errors: 𝑢_𝑘+1(𝑡)=^𝑁_𝑗=1𝑃_𝑗𝑢_{𝑘−𝑗+1}(𝑡)+^𝑁_𝑗=1𝑄_𝑗𝑒_{𝑘−𝑗+1}(𝑡)+𝑅_𝑗𝑒̇_{𝑘−𝑗+1}(𝑡)+𝑆_𝑗^𝑡₀𝑒_{𝑘−𝑗+1}(𝑡)𝑑𝑡, (7) where 𝑒_𝑘(𝑡)=𝑦_𝑑(𝑡)−𝑦_𝑘(𝑡) is the tracking error, integer 𝑁≥1 is the order of the ILC law. 𝑃_𝑗, 𝑄_𝑗, 𝑅_𝑗, and 𝑆_𝑗 are learning gain matrices. The learning operators 𝑄_𝑗, 𝑅_𝑗, and 𝑆_𝑗 are chosen to be bounded, and their upper bounds denoted by 𝑏_𝑄, 𝑏_𝑅, and 𝑏_𝑆, respectively, are defined by 𝑏_𝑄=max_{1≤𝑗≤𝑁}𝑄_𝑗,𝑏_𝑅=max_{1≤𝑗≤𝑁}𝑅_𝑗,𝑏_𝑆=max_{1≤𝑗≤𝑁}𝑆_𝑗. (8) As usual, it is assumed that 𝑢_𝑘(𝑡)=0 and 𝑒_𝑘(𝑡)=0 for 𝑘<0.

The 𝜆-norm will be used in this paper. It is defined by 𝑓(⋅)_𝜆=sup_{0≤𝑡≤𝑇}e^−𝜆𝑡𝑓(𝑡),𝜆>0, (9) for a vector function 𝑓∶[0,𝑇]→𝑅^𝑛.

For the sake of brevity, the following notations will be used: ℎ_𝑘=ℎ𝑥_𝑘(𝑡),𝑡,ℎ_𝑑=ℎ𝑥_𝑑(𝑡),𝑡,𝛿ℎ_𝑘=ℎ_𝑑−ℎ_𝑘,𝑏_ℎ=sup‖ℎ(𝑥(𝑡),𝑡)‖, (10) where ℎ represents a function concerned. The partial derivatives of ℎ(𝑥(𝑡),𝑡) are denoted by ℎ_𝑥𝑘=𝜕ℎ(𝑥,𝑡)𝜕𝑥_{𝑥=𝑥𝑘(𝑡)},ℎ_𝑡𝑘=𝜕ℎ(𝑥,𝑡)𝜕𝑡_{𝑥=𝑥𝑘(𝑡)},ℎ_𝑥𝑑=𝜕ℎ(𝑥,𝑡)𝜕𝑥_{𝑥=𝑥𝑑(𝑡)},ℎ_𝑡𝑑=𝜕ℎ(𝑥,𝑡)𝜕𝑡_{𝑥=𝑥𝑑(𝑡)}. (11) Let 𝑘_ℎ be the Lipschitz constant of the function ℎ with respect to 𝑥 in [0,𝑇]. Then it is easy to see that 𝛿ℎ_𝑘≤𝑘_ℎ𝛿𝑥_𝑘. (12)

3. Main Result

Note that the 𝜎(𝑡) is an arbitrary switching rule during the finite time interval [0,𝑇], which is different from the aforementioned studies [14–19]. The switching rule 𝜎(𝑡) can be described as 𝜎=𝑖₀,𝑡₀,𝑖₁,𝑡₁,𝑖₂,𝑡₂,…,𝑖_𝑛,𝑡_𝑛, (13) where 𝑡₀ is the initial time instant, by default 𝑡₀=0. 𝑡_𝑛 is the terminal time instant with 𝑡_𝑛=𝑇. 𝑡₁,𝑡₂,⋯,𝑡_𝑛 denote the switching instants, and the pair (𝑖_𝑛,𝑡_𝑛) which represents subsystem 𝑖_𝑛 is active during the interval 𝑡_𝑛−1≤𝑡≤𝑡_𝑛. 𝑡_𝑛−𝑡_𝑛−1 is the dwell time of the subsystem 𝑖_𝑛. Clearly, the control input of overall systems is a piecewise function, and the discontinuity points are those switching instants.

Without loss of generality, we can assume that the arbitrary switching rule 𝜎(𝑡) is given as𝜎(𝑡)=𝑖=1,𝑡∈0,𝑡₁,2,𝑡∈𝑡₁,𝑡₂,⋮𝑚,𝑡∈𝑡_𝑚−1,𝑇. (14) Switched sequence (14) implies each subsystem only operated once during the whole interval [0,𝑇].

To proof the main result, we first give the following lemmas.

Lemma 7. Suppose that a real positive series {𝑎_𝑛}^∞₁ satisfies 𝑎_𝑛≤𝜌₁𝑎_𝑛−1+𝜌₂𝑎_𝑛−2+⋯+𝜌_𝑁𝑎_𝑛−𝑁(),+𝜀,𝑛=𝑁+1,𝑁+2,… (15) where 𝜌_𝑖≥0(𝑖=1,2,…,𝑁), 𝜀≥0 and 𝜌=^𝑁_𝑖=1𝜌_𝑖<1. (16) Then the following holds: lim_𝑛→∞𝑎_𝑛≤𝜀.1−𝜌 (17)

Proof. Let 𝑛₁∈{𝑛−1,𝑛−2,…,𝑛−𝑁} be an index number such that 𝑎_𝑛1=max{𝑎_𝑛−1,𝑎_𝑛−2,…,𝑎_𝑛−𝑁}. Then, from (15), we have 𝑎_𝑛≤𝜌₁𝑎_𝑛−1+𝜌₂𝑎_𝑛−2+⋯+𝜌_𝑁𝑎_𝑛−𝑁+𝜀,≤𝜌𝑎_𝑛1+𝜀. (18) Similarly, let 𝑛₂∈{𝑛₁−1,𝑛₁−2,…,𝑛₁−𝑁} such that 𝑎_𝑛2=max𝑎_𝑛1−1,𝑎_𝑛1−2,…,𝑎_𝑛1−𝑁. (19) Then 𝑎_𝑛1≤𝜌₁𝑎_𝑛1−1+𝜌₂𝑎_𝑛1−2+⋯+𝜌_𝑁𝑎_𝑛1−𝑁+𝜀,≤𝜌𝑎_𝑛2+𝜀. (20) Therefore 𝑎_𝑛≤𝜌²𝑎_𝑛2+𝜌𝜀+𝜀. (21) In general, 𝑎_𝑛≤𝜌^𝑚𝑎_𝑛𝑚+𝜌^𝑚−1𝜀+𝜌^𝑚−2𝜀+⋯+𝜌𝜀+𝜀, (22) where 𝑚 and 𝑛_𝑚 are positive integers. If 𝑚 is chosen such that 𝑛_𝑚≤𝑁, then (𝑛/𝑁)−1≤𝑚≤𝑛−𝑁, and therefore 𝑚→∞ when 𝑛→∞. Let 𝑀=max{𝑎₁,𝑎₂,…,𝑎_𝑁}, and then 𝑎_𝑛≤𝜌^𝑚𝑀+1−𝜌^𝑚1−𝜌𝜀, (23) which implies lim_𝑛→∞𝑎_𝑛≤𝜀.1−𝜌 (24)
This completes the proof of Lemma 7.

Lemma 8 ((Bellman-Gronwall lemma) [35]). Assume that functions 𝜀(𝑡),𝑐(𝑡),𝑎(𝑡)∶𝑅^𝑛×[0,𝑇]→𝑅^𝑛 are continuous and nonnegative function in 𝑡. If the following holds: 𝜀(𝑡)≤𝑐(𝑡)+^𝑡₀𝑎(𝜏)𝜀(𝜏)𝑑𝜏,𝑡∈[0,𝑇], (25) then 𝜀(𝑡)≤𝑐(𝑡)𝑒^{∫𝑡0𝑎(𝜏)𝑑𝜏},𝑡∈[0,𝑇]. (26) Now, we can give the following result.

Theorem 9. Consider the nonlinear switched system (3) with switching rule (14), and Assumptions 1–3 are satisfied. If ^𝑁_𝑗=1𝑃_𝑗=𝐼_𝑚, (27) and there exist positive numbers 𝜌_𝑗 satisfying 𝑃_𝑗−𝑅_𝑗𝑔_𝑖𝑥𝐵_𝑖≤𝜌_𝑗,_𝑁𝑗=1𝜌_𝑗=𝜌<1, (28) then the system output converges to the desired output; that is, 𝑦_𝑑(𝑡)→𝑦_𝑘(𝑡) is ensured as 𝑘→∞ for all 𝑡∈[0,𝑇].

Proof. Denote 𝑙=𝑘−𝑗+1, and the ILC law (7) can be rewritten as 𝑢_𝑘+1(𝑡)=^𝑁_𝑗=1𝑃_𝑗𝑢_𝑙(𝑡)+^𝑁_𝑗=1𝑄_𝑗𝑒_𝑙(𝑡)+𝑅_𝑗𝑒_𝑙̇(𝑡)+𝑆_𝑗^𝑡₀𝑒_𝑙(𝑡)𝑑𝑡. (29)
From (3), we have 𝑒_𝑙(𝑡)=𝑦_𝑑(𝑡)−𝑦_𝑙(𝑡)=𝑔_𝑖𝑥_𝑑(𝑡),𝑡−𝑔_𝑖𝑥_𝑙(𝑡),𝑡=𝛿𝑔_𝑖,𝑙,𝑒_𝑙̇(𝑡)=𝑦̇_𝑑(𝑡)−𝑦̇_𝑙(𝑡)=𝑔_{𝑖,𝑥𝑑}𝑥̇_𝑑+𝑔_{𝑖,𝑡𝑑}−𝑔_{𝑖,𝑥𝑙}𝑥̇_𝑙−𝑔_{𝑖,𝑡𝑙}=𝛿𝑔_{𝑖,𝑥𝑙}𝑥̇_𝑑+𝑔_{𝑖,𝑥𝑙}𝛿𝑥̇_𝑙+𝛿𝑔_{𝑖,𝑡𝑙}, (30) where 𝑙=𝑘,𝑘−1,…,𝑘−𝑁+1, and 𝑥̇_𝑑=𝑓_𝑖,𝑑+𝐵_𝑖,𝑑𝑢_𝑑,𝛿𝑥̇_𝑙=𝑓_𝑖,𝑑+𝐵_𝑖,𝑑𝑢_𝑑−𝑓_𝑖,𝑙−𝐵_𝑖,𝑙𝑢_𝑙=𝛿𝑓_𝑖,𝑙+𝛿𝐵_𝑖,𝑙𝑢_𝑑+𝐵_𝑖,𝑙𝛿𝑢_𝑙, (31) where 𝛿𝑓_𝑖,𝑙=𝑓_𝑖,𝑑−𝑓_𝑖,𝑙=𝑓_𝑖𝑥_𝑑(𝑡),𝑥_𝑑(𝑡−ℎ),𝑡−𝑓_𝑖𝑥_𝑙(𝑡),𝑥_𝑙(𝑡−ℎ),𝑡,𝛿𝐵_𝑖,𝑙=𝐵_𝑖,𝑑−𝐵_𝑖,𝑙=𝐵_𝑖𝑥_𝑑(𝑡),𝑥_𝑑(𝑡−𝑑),𝑡−𝐵_𝑖𝑥_𝑙(𝑡),𝑥_𝑙(𝑡−𝑑),𝑡. (32) Now, by using ILC updating law (29) and condition (27), we have 𝛿𝑢_𝑘+1=𝑢_𝑑−𝑢_𝑘+1=^𝑁_𝑗=1𝑃_𝑗𝛿𝑢_𝑙−^𝑁_𝑗=1𝑄_𝑗𝛿𝑔_𝑖,𝑙−^𝑁_𝑗=1𝑆_𝑗^𝑡₀𝛿𝑔_𝑖,𝑙−𝑑𝜏_𝑁𝑗=1𝑅_𝑗𝛿𝑔_{𝑖,𝑥𝑙}𝑥̇_𝑑+𝑔_{𝑖,𝑥𝑙}𝛿𝑥̇_𝑙+𝛿𝑔_{𝑖,𝑡𝑙}=^𝑁_𝑗=1𝑃_𝑗𝛿𝑢_𝑙−^𝑁_𝑗=1𝑄_𝑗𝛿𝑔_𝑖,𝑙−^𝑁_𝑗=1𝑆_𝑗^𝑡₀𝛿𝑔_𝑖,𝑙−𝑑𝜏_𝑁𝑗=1𝑅_𝑗𝛿𝑔_{𝑖,𝑥𝑙}𝑥̇_𝑑+𝛿𝑔_{𝑖,𝑡𝑙}+𝑔_{𝑖,𝑥𝑙}𝛿𝑓_𝑖,𝑙+𝛿𝐵_𝑖,𝑙𝑢_𝑑+𝐵_𝑖,𝑙𝛿𝑢_𝑙=^𝑁_𝑗=1𝑃_𝑗−𝑅_𝑗𝑔_{𝑖,𝑥𝑙}𝐵_𝑖,𝑙𝛿𝑢_𝑙−^𝑁_𝑗=1𝑄_𝑗𝛿𝑔_𝑖,𝑙−^𝑁_𝑗=1𝑆_𝑗^𝑡₀𝛿𝑔_𝑖,𝑙−𝑑𝜏_𝑁𝑗=1𝑅_𝑗𝛿𝑔_{𝑖,𝑥𝑙}𝑥̇_𝑑+𝛿𝑔_{𝑖,𝑡𝑙}+𝑔_{𝑖,𝑥𝑙}𝛿𝑓_𝑖,𝑙+𝛿𝐵_𝑖,𝑙𝑢_𝑑. (33) Taking norms yields 𝛿𝑢_𝑘+1≤^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_𝑙+^𝑁_𝑗=1𝑏_𝑄𝑘_𝑔𝛿𝑥_𝑙+^𝑁_𝑗=1𝑏_𝑆^𝑡₀𝑘_𝑔𝛿𝑥_𝑙+𝑑𝜏_𝑁𝑗=1𝑏_𝑅𝑘_𝑔𝑥𝑏_𝑥𝑑𝛿𝑥_𝑙+𝑘_𝑔𝑡𝛿𝑥_𝑙+𝑏_𝑔𝑥𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙+𝑘_𝑓𝛿𝑥_𝑙(𝑡−ℎ)+𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙(≤𝑡−𝑑)_𝑁𝑗=1𝜌_𝑗𝛿𝑢_𝑙+𝑎₀^𝑁_𝑗=1𝛿𝑥_𝑙+𝑎₁^𝑁_𝑗=1^𝑡₀𝛿𝑥_𝑙𝑑𝜏+𝑏_𝑅^𝑁_𝑗=1𝑏_𝑔𝑥𝑘_𝑓𝛿𝑥_𝑙(𝑡−ℎ)+𝑏_𝑔𝑥𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙(𝑡−𝑑), (34) where 𝑎₀=𝑏_𝑄𝑘_𝑔+𝑏_𝑅𝑘_𝑔𝑥𝑏_𝑥𝑑+𝑘_𝑔𝑡+𝑏_𝑔𝑥𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵,𝑎₁=𝑏_𝑆𝑘_𝑔. (35) Denoting 𝜃∈{ℎ,𝑑}, from Assumption 3, we have ^𝑡₀𝛿𝑥_𝑘(𝜏−𝜃)𝑑𝜏=⁰_−𝜃𝑥_𝑑(𝜏)−𝑥_𝑘(𝜏)𝑑𝜏+₀^𝑡−𝜃𝑥_𝑑(𝜏)−𝑥_𝑘(𝜏=)𝑑𝜏_0𝑡−𝜃𝛿𝑥_𝑘(𝜏)𝑑𝜏≤^𝑡₀𝛿𝑥_𝑘(𝜏)𝑑𝜏, (36) which implies ‖𝛿𝑥_𝑙(𝑡−𝜃)‖≤‖𝛿𝑥_𝑙(𝑡)‖.
Performing the 𝜆-norm operation for (34) and considering ‖𝛿𝑥_𝑙(𝑡−𝜃)‖≤‖𝛿𝑥_𝑙(𝑡)‖, we can obtain 𝛿𝑢_𝑘+1𝜆≤^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_𝑙𝜆+𝑎₂^𝑁_𝑗=1𝛿𝑥_𝑙𝜆, (37) where 𝑎₂=𝑎₀+𝑎₁+𝑏_𝑅𝑏_𝑔𝑥(𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵).
(1) When 𝑡∈[0,𝑡₁]. In this case, the subsystem 1 is active. From (3) and Assumption 1, we can obtain 𝛿𝑥_𝑙=𝛿𝑥_𝑙(0)+^𝑡₀𝛿𝑥̇_𝑙𝑑𝜏=𝛿𝑥_𝑙(0)+^𝑡₀𝛿𝑓_1,𝑙+𝛿𝐵_1,𝑙𝑢_𝑑+𝐵_1,𝑙𝛿𝑢_𝑙𝑑𝜏. (38)
Taking norms yields 𝛿𝑥_𝑙=𝛿𝑥_𝑙(0)+^𝑡₀𝛿𝑥̇_𝑙≤𝑑𝜏𝛿𝑥_𝑙(0+_𝑡0)𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙+𝑘_𝑓𝛿𝑥_𝑙(+𝜏−ℎ)𝑑𝜏_𝑡0𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙(𝜏−𝑑)+𝑏_𝐵𝛿𝑢_𝑙𝑑𝜏. (39) Since 𝑥_𝑘(0)=𝑥_𝑑(0), then 𝛿𝑥_𝑙(0)=0. From (36) and (39), we have 𝛿𝑥_𝑙≤^𝑡₀𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵+𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙+𝑏_𝐵𝛿𝑢_𝑙𝑑𝜏. (40) Defining 𝑎₃=𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵+𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵, using Lemma 8 for (40), we have 𝛿𝑥_𝑙≤^𝑡₀𝑒^{𝑎3(𝑡−𝜏)}𝑏_𝐵𝛿𝑢_𝑙𝑑𝜏,𝑡∈0,𝑡₁. (41) Taking 𝜆-norm yields 𝛿𝑥_𝑙𝜆≤𝑏_𝐵𝑂₁𝜆⁻¹1−𝑎₃𝑂₁𝜆⁻¹𝛿𝑢_𝑙𝜆, (42) where 𝑂₁(𝜆⁻¹)=(1−𝑒^−𝜆𝑡1)/𝜆. Combing (37) and (42) yields 𝛿𝑢_𝑘+1𝜆≤^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_𝑙𝜆+𝑎₂𝑏_𝐵𝑂₁𝜆⁻¹1−𝑎₃𝑂₁𝜆^−1𝑁_𝑗=1𝛿𝑢_𝑙𝜆,=^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_𝑙𝜆, (43) where 𝜌_𝑗=𝜌_𝑗+𝑎₂𝑏_𝐵𝑂₁𝜆⁻¹1−𝑎₃𝑂₁𝜆⁻¹. (44) By condition (28) in Theorem 9, one can find a sufficiently large 𝜆 such that 𝜌_𝑗<1 and ∑^𝑁_𝑗=1𝜌_𝑗=𝜌<1. Then, according to Lemma 7, it can be concluded that lim_𝑘→∞𝛿𝑢_𝑘𝜆=0,𝑡∈0,𝑡₁. (45) From (42) and ‖𝑒_𝑘‖_𝜆≤𝑘_𝑔‖𝛿𝑥_𝑘‖_𝜆, we can observe that the tracking errors ‖𝑒_𝑘‖_𝜆 and ‖𝛿𝑥_𝑘‖_𝜆 both tend to zero for 𝑡∈[0,𝑡₁] when 𝑘→∞.
(2) When 𝑡∈[𝑡₁,𝑡₂]. In this case, the subsystem 2 is active. The state error has the following expression: 𝛿𝑥_𝑙=𝛿𝑥_𝑙𝑡₁+^𝑡_𝑡1𝛿𝑥̇_𝑙𝑑𝜏=𝛿𝑥_𝑙𝑡₁+^𝑡_𝑡1𝛿𝑓_2,𝑙+𝛿𝐵_2,𝑙𝑢_𝑑+𝐵_2,𝑙𝛿𝑢_𝑙𝑑𝜏. (46) Taking norms yields 𝛿𝑥_𝑙=𝛿𝑥_𝑙𝑡₁+^𝑡_𝑡1𝛿𝑥̇_𝑙≤𝑑𝜏𝛿𝑥_𝑙𝑡₁+^𝑡_𝑡1𝑘_𝑓+𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙+𝑘_𝑓𝛿𝑥_𝑙(+𝑡−ℎ)𝑑𝜏_𝑡𝑡1𝑏_𝑢𝑑𝑘_𝐵𝛿𝑥_𝑙(𝑡−𝑑)+𝑏_𝐵𝛿𝑢_𝑙𝑑𝜏. (47) Due to the fact that ‖𝛿𝑥_𝑙(𝑡−ℎ)‖≤‖𝛿𝑥_𝑙(𝑡)‖, ‖𝛿𝑥_𝑙(𝑡−𝑑)‖≤‖𝛿𝑥_𝑙(𝑡)‖, we have 𝛿𝑥_𝑙≤𝛿𝑥_𝑙𝑡₁+^𝑡_𝑡1𝑎₃𝛿𝑥_𝑙+𝑏_𝐵𝛿𝑢_𝑙𝑑𝜏,𝑡∈𝑡₁,𝑡₂. (48) Using Lemma 8 for (40) and taking 𝜆-norm yield 𝛿𝑥_𝑙𝜆≤𝛿𝑥_𝑙𝑡₁+𝑏_𝐵𝑂₂𝜆⁻¹1−𝑎₃𝑂₂𝜆⁻¹𝛿𝑢_𝑙𝜆, (49) where 𝑂₂(𝜆⁻¹)=(1−𝑒^−𝜆𝑡2)/𝜆. Combing (37) and (49) yields 𝛿𝑢_𝑘+1𝜆≤^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_𝑙𝜆+𝑎₂^𝑁_𝑗=1𝛿𝑥_𝑙𝑡_1𝜆+𝑎₂𝑏_𝐵𝑂₂𝜆⁻¹1−𝑎₃𝑂₂𝜆^−1𝑁_𝑗=1𝛿𝑢_𝑙𝜆=^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_𝑙𝜆+𝑎₂^𝑁_𝑗=1𝛿𝑥_𝑙(𝑡₁)_𝜆, (50) where 𝜌_𝑗=𝜌_𝑗+𝑎₂𝑏_𝐵𝑂₂𝜆⁻¹1−𝑎₃𝑂₂𝜆⁻¹. (51) By condition (28) in Theorem 9, one can find a sufficiently large 𝜆 such that 𝜌_𝑗<1 and ∑^𝑁_𝑗=1𝜌_𝑗=𝜌<1.
Note that (50) can be rewritten as 𝛿𝑢_𝑘𝜆≤^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_{𝑘−𝑗𝜆}+𝜉_𝑘, (52) where 𝜉_𝑘=𝑎₂∑^𝑁_𝑗=1‖𝛿𝑥_𝑘−𝑗(𝑡₁)‖_𝜆.
Let 𝑘₁∈{𝑘−1,…,𝑘−𝑁} be an iteration number such that 𝑢_𝑘1=max𝑢_𝑘−1,𝑢_𝑘−2,…,𝑢_𝑘−𝑁. (53) Then, from (50), it is easy to see that 𝛿𝑢_𝑘𝜆≤𝜌𝛿𝑢_𝑘1𝜆+𝜉_𝑘. (54) Similarly, let 𝑘₂∈{𝑘₁−1,𝑘₁−2,…,𝑘₁−𝑁} such that 𝑢_𝑘2=max𝑢_𝑘1−1,𝑢_𝑘1−2,…,𝑢_𝑘1−𝑁. (55) then 𝛿𝑢_𝑘1𝜆≤^𝑁_𝑗=1𝜌_𝑗𝛿𝑢_{𝑘1−𝑗𝜆}+𝜉_𝑘1≤𝜌𝛿𝑢_𝑘2𝜆+𝜉_𝑘1, (56) therefore 𝛿𝑢_𝑘𝜆≤𝜌²𝛿𝑢_𝑘2𝜆+𝜌𝜉_𝑘1+𝜉_𝑘. (57) In general, 𝛿𝑢_𝑘𝜆≤𝜌^𝑚′𝛿𝑢_𝑘𝑚𝜆+𝜌^𝑚′−1𝜉_{𝑘𝑚′−1}+𝜌^𝑚′−2𝜉_{𝑘𝑚′−2}+⋯+𝜌𝜉_𝑘1+𝜉_𝑘, (58) where 𝑚^′ and 𝑘_𝑚′ are positive integers. If 𝑚^′ is chosen such that 𝑘_𝑚′≤𝑁, then 𝑘𝑁−1≤𝑚^′≤𝑘−𝑁, (59) therefore 𝑚^′→∞ when 𝑘→∞. Let 𝑀=max{𝛿𝑢₁,𝛿𝑢₂,…,𝛿𝑢_𝑁}, then 𝛿𝑢_𝑘𝜆≤𝜌^𝑚′𝑀+^𝑚′−1_𝑗=0𝜌^𝑗𝜉_𝑘𝑗, (60) where 𝜉_𝑘0=𝜉_𝑘. Note that 𝑘→∞ means 𝑚^′→∞, 𝑘_𝑗→∞,𝑗∈{0,1,2,…,𝑚^′}. From (45), we know that lim_𝑘→∞‖𝛿𝑥_𝑘(𝑡₁)‖_𝜆=0, it also means lim_𝑘→∞𝜉_𝑘𝑗=0. Taking 𝑘→∞ for (60) gives lim_𝑘→∞𝛿𝑢_𝑘𝜆≤𝑀lim_𝑘→∞𝜌^𝑚′+lim^𝑚_𝑘→∞^′−1_𝑗=0𝜌^𝑗𝜉_𝑘𝑗. (61) Considering lim_𝑘→∞𝜌^𝑚′=0 and lim_𝑘→∞∑^𝑚′−1_𝑗=0𝜌^𝑗𝜉_𝑘𝑗=0, (61) implies lim_𝑘→∞𝛿𝑢_𝑘𝜆=0. (62) From (49) and ‖𝑒_𝑘‖_𝜆≤𝑘_𝑔‖𝛿𝑥_𝑘‖_𝜆, we can also observe that the tracking errors ‖𝑒_𝑘‖_𝜆 and ‖𝛿𝑥_𝑘‖_𝜆 both tend to zero for 𝑡∈[𝑡₁,𝑡₂] when 𝑘→∞.
In this analogy, lim_𝑘→∞‖𝑒_𝑘‖_𝜆=0, lim_𝑘→∞‖𝛿𝑥_𝑘‖_𝜆=0, and lim_𝑘→∞‖𝛿𝑢_𝑘‖_𝜆=0 can also be obtained for 𝑡∈[𝑡₂,𝑡₃],𝑡∈[𝑡₃,𝑡₄],…,[𝑡_𝑚−1,𝑇]. Hence, for the whole time interval 𝑡∈[0,𝑇], we have lim_𝑘→∞[𝑦_𝑑(𝑡)−𝑦_𝑘(𝑡)]=0.
This completes the proof.

Remark 10. Theorem 9 is given for the nonlinear switched system with the switched sequence (14), which assumes each subsystem only operated once during the whole interval [0,𝑇], and the active sequence for subsystems is {1,2,…,𝑚}. From the proof process of Theorem 9, we know that the result can also be extended to the arbitrary switch sequence with more times active for each subsystem.

Remark 11. High-order ILC updating law utilizes the past experiences comprehensively and has more flexibilities in choosing learning operators and parameters. Hence, the better ILC performance can be expected. If the system dynamics is totally unknown, like the selection of learning parameters in traditional ILC laws, the order 𝑁 selection is also based on a trial-and-error method, which should be also in an iterative learning way. In practice, 𝑁 should normally be chosen to be less than or equal to 3.

Remark 12. In many practical systems, stochastic disturbances are unavoidable [36–38]. In this case, the nonlinear switched system (1) can be described by 𝑥̇_𝑘(𝑡)=𝑓_𝜎(𝑡)𝑥_𝑘(𝑡),𝑥_𝑑(𝑡−ℎ),𝑡+𝐵_𝜎(𝑡)𝑥_𝑘(𝑡),𝑥_𝑘(𝑡−𝑑),𝑡𝑢(𝑡)+𝑤_𝑘(𝑡),𝑦_𝑘(𝑡)=𝑔_𝜎(𝑡)𝑥_𝑘(𝑡),𝑡+𝑣_𝑘(𝑡), (63) where 𝑤_𝑘(𝑡)∈𝑅^𝑛 and 𝑣_𝑘(𝑡)∈𝑅^𝑚 are state disturbances and output disturbances, respectively. If the disturbances are bounded in the sense of ‖𝑤_𝑘(𝑡)‖≤𝑏_𝑤 and ‖𝑣_𝑘(𝑡)‖≤𝑏_𝑣 for all 𝑘≥0, where 𝑏_𝑤, and 𝑏_𝑣 are positive constants, from the proof of Theorem 9, it is easy obtained that, when the PID ILC law (7) is used for system (63), if the conditions in Theorem 9 are guaranteed, then the system output converges to the desired output with a bounded error. The error bound depends on the 𝑏_𝑤 and 𝑏_𝑣. The similar results can also be founded in ILC for general nonlinear systems with disturbances [39, 40].

4. Simulation Illustrations

In this section, we consider two illustrative examples.

Example 1 (SISO nonlinear switched system). In this simulation test, let us consider the following SISO nonlinear switched system with two subsystems: 𝑆₁∶𝑥̇_1,𝑘(𝑡)𝑥̇_2,𝑘(𝑡)=0.8sin𝑥_1,𝑘(𝑡)0.3𝑒^−𝑡cos𝑥_1,𝑘(𝑡)−0.4sin𝑥_2,𝑘(𝑡)+10𝑢_𝑘(𝑡),𝑦_𝑘(𝑡)=𝑥_1,𝑘(𝑡)+0.5𝑥_2,𝑘(𝑡),𝑆₂∶𝑥̇_1,𝑘(𝑡)𝑥̇_2,𝑘(𝑡)=−0.6sin𝑥_1,𝑘(𝑡)+0.3sin𝑥_2,𝑘(𝑡)0.5𝑒^−𝑡sin𝑥_2,𝑘(𝑡)+𝑢0.50.7^𝑘(𝑡),𝑦_𝑘(𝑡)=0.2𝑥_1,𝑘(𝑡)+𝑥_2,𝑘(𝑡). (64)
The nonlinear switched system is operated during the two subsystems with arbitrary switched law in time domain, and the desired output is given as 𝑦_𝑑(𝑡)=sin(30𝑡)+𝑡𝑒^2𝑡, and 𝑡∈[0,1]. For the initial state, it is assumed that 𝑥_𝑘(0)=𝑥_𝑑(0)=0 for all 𝑘. The ILC law (7) is applied by adopting the zero initial control input 𝑢₀(𝑡)=0 for all 𝑡∈[0,1]. In simulation, we produces a random sequence 𝜎(𝑡),𝑡∈[0,1] with the value is 1 and 2, as shown in Figure 1. If 𝜎(𝑡)=1, the system (64) is 𝑆₁; otherwise, if 𝜎(𝑡)=2, the system (64) is 𝑆₂.
To check the tracking performance, we consider the following second-order ILC law 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+𝑄₁𝑒_𝑘−1(𝑡)+𝑄₂𝑒_𝑘(𝑡)+𝑅₁𝑒̇_𝑘−1(𝑡)+𝑅₂𝑒_𝑘̇(𝑡), (65) where 𝑄₁=0.3, 𝑄₂=0.5, 𝑅₁=0.9, and 𝑅₂=1.2. Note that the nonlinear functions sin(𝑥),cos(𝑥) are both uniformly globally Lipschitz in 𝑥 and uniformly bounded for all 𝑡∈[0,1] and 𝑥∈𝑅. It is obvious that 𝑔_𝑖𝑥 and 𝐵_𝑖 are also bounded. Checking the condition in Theorem 9, we have 𝑔_1𝑥𝐵₁=1010.5=1,𝑔_2𝑥𝐵₂=𝜌0.210.50.7=0.8,₁=max𝐼−𝑅₁𝑔_1𝑥(𝑥(𝑡))𝐵₁(𝑥(𝑡)),𝐼−𝑅₁𝑔_2𝑥(𝑥(𝑡))𝐵₂(𝑥(𝑡𝜌))=0.28,₂=max𝐼−𝑅₂𝑔_1𝑥(𝑥(𝑡))𝐵₁(𝑥(𝑡)),𝐼−𝑅₂𝑔_2𝑥(𝑥(𝑡))𝐵₂(𝑥(𝑡))=0.2, (66) then, 𝜌₁+𝜌₂=0.48<1. From Theorem 9, we know that the nonlinear switched ILC system is asymptotically stable. Figures 2, 3, and 4 give simulation results for system output trajectories at 5th, 10th, and 30th iterations. In these figures, we plot the system output in solid line and the desired output trajectory in dashed line. It can be observed that the better tracking performance can be obtained after 30th iteration. Intuitively, this result can be understood in the following way: although the switched system has an arbitrary switched rule on the time domain, the arbitrary switched rule is invariable on the iteration domain. If we update the control signals on the repetitive iteration domain, the perfect tracking performance can be obtained eventually.
To further check the effectiveness of high-order ILC law, we also did simulation tests using first-order ILC given as 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+0.6𝑒_𝑘(𝑡)+0.5𝑒_𝑘̇(𝑡), and the control input at the first iteration is also given as 𝑢₀(𝑡)=0 for all 𝑡∈[0,1]. The tracking errors for second-order ILC and first order ILC are given in Figure 5. Clearly, the high-order ILC scheme can be faster convergence than the first-order one.

Figure 1 

The random switched rule 𝜎(𝑡) for system (64).

Figure 2 

System output trajectory at 5th iteration for Example 1.

Figure 3 

System output trajectory at 10th iteration for Example 1.

Figure 4 

System output trajectory at 30th iteration for Example 1.

Figure 5 

The max tracking error of Example 1 for different ILC laws.

Example 2 (MIMO nonlinear switched system with state delay). In this example, let us consider the following MIMO nonlinear switched system with time delay: 𝑆₁∶𝑥̇_1,𝑘(𝑡)𝑥̇_2,𝑘(𝑡)=0.5𝑥_1,𝑘(𝑡−ℎ)+𝑥_2,𝑘(𝑡)0.3𝑥_1,𝑘()+1𝑡−ℎ1+𝑥_2,𝑘(𝑡)+()𝑥10sin𝑡−𝑑_1,𝑘(𝑡−𝑑)cos(𝑡−𝑑)𝑥_2,𝑘()×𝑢𝑡−𝑑_1,𝑘(𝑡)𝑢_2,𝑘(𝑡),𝑦_1,𝑘(𝑡)𝑦_2,𝑘(𝑡)=sin𝑥_1,𝑘(𝑡)+0.1𝑥_2,𝑘(𝑡)cos𝑥_2,𝑘(𝑡),𝑆₂∶𝑥̇_1,𝑘(𝑡)𝑥̇_2,𝑘(𝑡)=0.2𝑥_1,𝑘(𝑡)+𝑥_2,𝑘(𝑡−ℎ)0.6𝑥_1,𝑘(𝑡)+10.5+𝑥_2,𝑘()+()𝑥𝑡−ℎ1cos𝑡−𝑑_1,𝑘()()𝑥𝑡−𝑑0sin𝑡−𝑑_2,𝑘()𝑢𝑡−𝑑^1,𝑘(𝑡)𝑢_2,𝑘(𝑡),𝑦_1,𝑘(𝑡)𝑦_2,𝑘(𝑡)=0.3𝑥_1,𝑘(𝑡)+sin𝑥_2,𝑘(𝑡)cos𝑥_1,𝑘(𝑡), (67) where ℎ=𝑑=0.1. The switched system is also operated during the two subsystems with arbitrary switched law in time domain, and the desired trajectories are given as 𝑦_1,𝑑(𝑡)𝑦_2,𝑑(𝑡)=[]sin3𝜋𝑡cos3𝜋𝑡,𝑡∈0,1. (68)
For the initial state, it is assumed that 𝑥_1,𝑘(0)=𝑥_2,𝑘(0)=0 for all 𝑘. The ILC law is applied by adopting the zero initial control input 𝑢_1,0(𝑡)=0,𝑢_2,0(𝑡)=0, for all 𝑡∈[0,1]. In simulation, we produces a random sequence 𝜎(𝑡), and 𝑡∈[0,1] with the value is 1 and 2, as shown in Figure 6, If 𝜎(𝑡)=1, the system (67) is 𝑆₁; otherwise, if 𝜎(𝑡)=2, the system (67) is 𝑆₂. We consider the following second order ILC law: 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+𝑄₁𝑒_𝑘−1(𝑡)+𝑄₂𝑒_𝑘(𝑡)+𝑅₁𝑒̇_𝑘−1(𝑡)+𝑅₂𝑒_𝑘̇(𝑡), (69) where 𝑄₁=0.5, 𝑄₂=0.4, 𝑅₁=0.8, and 𝑅₂=1.1. Note that the nonlinear system only contains nonlinear argument sin(𝑥), cos(𝑥), and they are both uniformly globally Lipschitz in 𝑥 and uniformly bounded for all 𝑡∈[0,1]. To guarantee the ILC convergence, 𝜌 should be less than one. It is easy to obtain that the previous choices of 𝑅₁, 𝑅₂ clearly satisfy the convergence condition. The simulation results for system output trajectories at 5th, 10th, and 30th iterations are shown in Figures 7, 8, and 9. In these figures, we plot the system output in solid line and the desired output trajectory in dashed line. It can be observed that the switched ILC system is asymptotically stable, and the better tracking performance can be obtained after 30th iteration for the whole time interval. We also did simulation tests using first order ILC given as 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+0.6𝑒_𝑘(𝑡)+0.5𝑒_𝑘̇(𝑡), and the control input at the first iteration is also given as 𝑢_1,0(𝑡)=0,𝑢_2,0(𝑡)=0, for all 𝑡∈[0,1]. The simulation results of tracking errors for second order ILC and first order ILC are shown in Figure 10. It is presented to illustrate the effectiveness of high-order ILC law over the conventional first-order one.

Figure 6 

The random switched rule 𝜎(𝑡) for system (67).

(a)

(b)

Figure 7 

System output trajectory at 5th iteration for Example 2: (a) 𝑦1,𝑘(𝑡), (b) 𝑦2,𝑘(𝑡).

(a)

(b)

Figure 8 

System output trajectory at 10th iteration for Example 2: (a) 𝑦1,𝑘(𝑡), (b) 𝑦2,𝑘(𝑡).

(a)

(b)

Figure 9 

System output trajectory at 30th iteration for Example 2: (a) 𝑦1,𝑘(𝑡), (b) 𝑦2,𝑘(𝑡).

(a)

(b)

Figure 10 

The max tracking error of Example 2 for different ILC laws: (a) 𝑒1,𝑘(𝑡), (b) 𝑒2,𝑘(𝑡).

5. Conclusions

The stability of high-order ILC law for nonlinear switched systems with state delays is analyzed. It is shown that the asymptotic convergence of the tracking error can be guaranteed under some conditions, and the perfect tracking performance can be obtained during the entire time interval. This result can be understood in the following way: although the switched system has an arbitrary switched rule on the time domain, the arbitrary switched rule is invariable on the iteration domain. If we update the control signals on the repetitive iteration domain, the perfect tracking performance can be obtained eventually.

In the previous works [20–22], the asymptotic convergence of tracking control is only guaranteed in the time domain; that is, the system output tracking errors converge to 0 when 𝑡→∞. When the ILC is considered for switched systems, the perfect tracking performance can be obtained during the entire time interval. ILC can use the previous operation information of the switched systems, and then the tracking can be performed better progressively from operation to operation. It also requires less a priori knowledge about the system dynamics and less computational effort than many other kinds of control approaches. In future work, we will consider the stability of nonlinear switched systems with arbitrary switched rule on the iteration domain.

Acknowledgments

This work is supported by the Program of NSFC (no. 61203065, no. 61240049), the Program of Open Laboratory Foundation of Control Engineering Key Discipline of Henan Provincial High Education (KG 2011-10), and the Program of Key Young Teacher of Henan Polytechnic University.