#### Abstract

For the nonlinear discrete-time system, higher-order iterative learning control (HOILC) with optimal control gains based on evolutionary algorithm (EA) is developed in this paper. Since the updating actions are constituted by the tracking information from several previous iterations, the suitably designed HOILC schemes with appropriate control gains usually achieve fast convergence speed. To optimize the control gains in HOILC approach, EA is introduced. The encoding strategy, population initialization, and fitness function in EA are designed according to the HOILC characteristics. With the global optimization of EA, the optimal control gains of HOILC are selected adaptively so that the number of convergence iteration is reduced in ILC process. It is shown in simulation that the sum absolute error, total square error, and maximum absolute error of tracking in the proposed HOILC based on EA are convergent faster than those in conventional HOILC.

#### 1. Introduction

In real applications such as robot manipulator systems [1–5] and flexible systems [6–8], there are many unmanned autonomous systems in complex environments. The exact mathematical model is hard to construct. For these systems, iterative learning control (ILC) is proposed. It is an effective intelligent control approach applied in dynamical systems that perform repetitive tasks to track a specific trajectory in a certain time interval. By using the control input and tracking information of previous iterations, the control input signal can be gradually updated from iteration to iteration such that the tracking performance can be improved. Less previous knowledge about the controlled systems makes ILC popular in theoretical fields [9–14] as well as applicable fields [15–19].

First-order ILC, which generates the control input from tracking information at last iteration, is widely applied to dynamical systems for perfect tracking in a finite time interval [20–26]. However, only the tracking information of last iteration is utilized to update the current control input in first-order ILC, and thus it is difficult to obtain a satisfactory convergence speed. To achieve faster convergence speed, higher-order ILC (HOILC) adopting the tracking information of many previous iterations to generate the current control input signal was proposed [27–31]. Since the updating actions are constituted by the tracking information from several previous iterations, the tracking performance of suitably designed HOILC is better than that of first-order ILC. Specifically, the appropriate control gains can accelerate the convergence process of HOILC. Thus, how to select optimal control gains is a significant issue in HOILC designs.

Motivated by the above observation, in this paper, the evolutionary algorithm (EA) originating from biological evolutionism is adopted to choose the optimal control gains in HOILC scheme adaptively. EA is a heuristic optimizing algorithm which simulates the reproduction, selection, crossover, and mutation in biological evolution process. It has been widely introduced to deal with various optimal issues [32–34]. In this paper, the encoding strategy, population initialization, and fitness function of EA are designed according to the HOILC characteristics such that the generations in EA are reduced. Then, the designed EA is involved in HOILC to optimize the control gains. After that, the optimal control gains and the control inputs are generated simultaneously. Comparing with traditional HOILC, the number of convergence iteration is reduced in the proposed EA scheme based HOILC (EA-HOILC). The EA with global optimization is introduced to optimize the control gains of HOILC in this paper.

The rest of paper is organized as follows. The problem formulation is given in Section 2. The HOILC with its convergence analysis is provided in Section 3. Section 4 presents the designed EA-HOILC scheme with optimal control gains. In Section 5, an example is provided to illustrate the effectiveness of the proposed EA-HOILC. Section 6 concludes this paper.

#### 2. Problem Formulation

Consider the following nonlinear discrete-time system which performs repetitive operation:where and represent the iteration index and the time point, respectively. , , and denote the state, control input, and output of system (1), respectively. , , and . for is the reference output, where is the corresponding reference state. is the ILC tracking error at th iteration for . The following assumptions are required for the technical analysis. represents the required norm in this paper.

*Assumption 1. *For all , the initial state satisfiesAs the identical initial condition considered in Assumption 1 cannot be satisfied, the techniques proposed in [29–31] can be introduced to deal with the vibration of initial state.

*Assumption 2. *The nonlinear function in system (1) is assumed to be differentiable to and to be globally Lipschitz in the first variable, that is, ,where is the Lipschitz constant.

*Assumption 3. *The number .

*Remark 1. *It is noted that Assumption 3 implies that the relative degree of system (1) is one. For the nonlinear discrete-time systems with higher relative degree, the ILC law can be modified according to the order of system relative degree as discussed in [31].

Suppose that the reference output is realizable, there exists a unique control input such thatThe objective of this paper is to develop an EA-HOILC method, which generates the control input from the tracking information of several previous iterations. The control gains are optimized by EA to reduce the number of convergence iteration. For HOILC convergence analysis, the following lemma is adopted.

Lemma 1 (see [31]). *Let be a real sequence defined asfor , where is a specific real sequence. If are nonnegative numbers satisfyingthen implies that .*

#### 3. HOILC Design and Convergence Analysis

In this section, for nonlinear discrete-time system (1) under Assumptions 1–3, the following HOILC law is designed for and :where is the order of HOILC law (7), and and () for are the control gains.

*Remark 2. *In the existing HOILC schemes [30, 31], the initial control inputs are normally set as zero vectors. In this paper, since the control inputs can be obtained by EA along with the optimal control gains, we can set the initial control inputs same as the generated control inputs. It means that the initial control inputs are optimized by EA, which can also accelerate the convergence speed.

Theorem 1. *For nonlinear discrete-time system (1) under Assumptions 1–3, the HOILC law (7) is applied. If the control gains and () for are selected to makethen for .*

*Proof. *Let and . Subtracting both sides of (7) with and considering (1), (4), and (8), we obtainThen, noting convergence condition (9) and Assumption 2, we can further deduce thatwhere and for .

On the other hand, it follows from (1) and (4) thatTaking norm on both sides of (13) and considering Assumptions 1-2, it yieldswhere . Substituting (14) into (12),As , considering (2) of Assumption 1, it is derived from (12) thatApplying Lemma 1 to (16) with convergence condition (10), we haveAs , from (16), there isApplying Lemma 1 to (18) with convergence condition (10) and considering (17), we obtainAssume that for , there isAs , it follows from (15) that Applying Lemma 1 to (21) with convergence condition (10) and considering (17) and (20), we can deriveFinally, based on the mathematical induction, the following result can be deduced:Noting (2) in Assumption 1, then it can be obtained from (15) and (24) thatFurthermore, for , it follows from (1) and (4) thatThen, we have for . The proof is completed.

#### 4. EA-HOILC Scheme with Optimal Control Gains

Theorem 1 provides the asymptotic convergence of the proposed HOILC. It is well known that the control gains can affect the convergence performance significantly. In this section, the control gains of the HOILC developed in Section 3 are optimized by EA to reduce the number of convergence iteration.

EA is an intelligent optimization algorithm which simulates the process of biological evolution to gain the optimal solution. The main idea of EA-HOILC is presented as follows.

##### 4.1. Encoding Strategy

In this paper, the control gains of HOILC are real numbers, so it is appropriate to choose the real encoding strategy. The control gains to be optimized in HOILC law (7) are and (). Since the convergence condition (8) holds, it is easily obtained that . As a result, we can assume the variable vector in EA to be , and the encoding strategy is represented as

##### 4.2. Population Initialization and Individual Evaluation

Based on the convergence conditions (8)–(10), the value range of control gains and for could be determined. Thus, the initial population can be produced according to the convergent conditions. Let be the population size, without loss of generality, assume that is even. The variable vector of th individual in the population is represented as which are initialized to for . On the other hand, for variable , let the system output for th individual at th time point be . To evaluate the individual superiority, the following fitness function of th individual is established:where is a constant large enough and is the sum of absolute value of tracking error represented as

From the fitness function (27) and the initial variable , the initial fitness value of th individual is obtained. Then, we have the following initial fitness vector of population:

Hence, the initial population is constructed aswhere the initial variable of population is

From (31), the th () variable of the th () individual is represented as the th row of th column. The last column of shown in (30) is the fitness value of the corresponding initial variable in population.

##### 4.3. Selection Strategy

The individuals are selected into next generation by roulette strategy and elitism strategy. The individual with bigger fitness value is selected at higher probability by the roulette strategy. However, one shortcoming of the roulette strategy is that the best individual in old population might be missed. So, we adopt elitism strategy to ensure that the best individual of last generation can be retained. Due to these two strategies, the number of convergence generation of EA can be reduced.

##### 4.4. Crossover Operator

The crossover probability depends whether an individual needs to cross. For th individual, where , a random number is produced between 0 and 1, which is represented as . If , the crossover operation occurs. Otherwise, the crossover operation does not occur. Due to the real encoding strategy, arithmetical crossover operator is adopted. Assume the variable vectors of th and th parent individuals to be and , respectively, which are selected to cross. After crossover, they generate two new individuals, of which the variable vectors are represented as and . For , the crossover operation is expressed aswhere is the crossover weighting for the th and th parent individuals.

##### 4.5. Mutation Operator

In this paper, we adopt the stochastic mutation strategy. Let be the mutation probability. For th individual, a number is produced randomly between 0 and 1 represented as , . If , the mutation operation occurs. Otherwise, the mutation operation does not occur. Let be the variable vector of th individual which is selected to mutate. After mutation, a new variable vector is produced. The mutation operator is defined aswhere is the mutation weighting for th individual.

##### 4.6. Terminative Conditions

The terminative conditions can be determined by the fitness value or by the tracking error. In simulation, the number of generation in EA is set to be 100. Finally, we can obtain the optimal control gains () and () from the best individual produced by EA. According to the convergence condition (8), the last control gain is derived by .

##### 4.7. Overview of the Proposed EA-HOILC

The flowchart of the proposed EA-HOILC is depicted in Figure 1. First of all, according to the control gains characteristics and convergence conditions (9) and (10), the initial variable () is obtained. Then, we apply the traditional HOILC at with initial control gains in each , where convergence condition (8) is considered. By using the tracking error with as shown in (29) produced by HOILC, the corresponding fitness () is derived from (27). Combining and for , the initial population as shown in (31) with (30) and (32) is produced. Secondly, selection, crossover, and mutation are performed by the selection strategy, crossover operator, and mutation operator, respectively. After that, a new input , , is obtained along with the optimal control gains. Set the initial control inputs of the EA-HOILC , . Then, the HOILC with optimal control gains process begins.

#### 5. Simulation

To verify the effectiveness of the proposed EA-HOILC, a two-link robotic fish is employed. The system dynamic of the two-link robotic fish is described as follows [22]:where is the mass of robotic fish, kg/m is the water resistance coefficient, denotes the velocity, and is the forward thrust produced by the tail motion. Let the velocity and the forward thrust be the system state and the control input , respectively, where is the sampling time. We can discretize system (34) by using . Thus, the discrete-time system iswhere and .

The reference output trajectory is represented aswith . For the proposed HOILC algorithm, set the order . The control gains , , and are selected by EA. Another control gain is obtained from convergence condition (8). The crossover probability , and the mutation probability . To evaluate the tracking performance, three tracking indexes on sum absolute error , total square error , and maximum absolute error are defined as follows:

In the simulation, the EA-HOILC is run 10 times, and the optimized control gains are shown in Table 1.

Figure 2 exhibits the system output performance at iterations , , and by using EA-HOILC with the average values of optimal control gains in 10 times. To compare the convergence speed between EA-HOILC and conventional HOILC with different parameters, the control gains in HOILC proposed in [30] with 2-order are, respectively, chosen as following two cases. Case 1: , , , and and Case 2: , , , and . The corresponding sum absolute error , total square error , and maximum absolute error of tracking are shown in Figure 3. From Figure 3, one can observe that the case with lager control gains in , , and can achieve faster convergent speed in conventional HOILC. Moreover, it is clearly revealed that the proposed EA-HOILC can make the convergence iterations less than the conventional HOILC with the same order.

#### 6. Conclusions

In this paper, an HOILC law utilizing the tracking information of several previous iterations is proposed for the nonlinear discrete-time system. The convergence is rigorously analyzed based on the mathematical induction. In order to improve the convergence performance of the developed HOILC, the EA with global optimization is introduced to optimize the control gains. With the optimal control gains, the proposed EA-HOILC can achieve faster convergence speed. In simulation, it is shown that the sum absolute error, total square error, and maximum absolute error of tracking in EA-HOILC are convergent faster than those in the conventional HOILC with same order. However, it is worth noting that because EA is adopted to select the control gains, the offline computing time of EA-HOILC is longer. It is very suitable for the cases in which fewer iterations are required only. For instance, to destroy a target with bombs, the proposed method can reduce the number of bombs at the cost of computing time. Future research will extend the EA-HOILC developed in this paper to the dynamical systems with uncertainties in real application [35–37].

#### Data Availability

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

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

This research was funded in part by National Natural Science Foundation of China with grant nos. 61903096 and 62173101, Science and Technology Program of Guangzhou with grant no. 201904010475, and Zhijiang Laboratory's Open Project with grant no. 2021KF0AB06.