RBFNN-Based Nonuniform Trajectory Tracking Adaptive Iterative Learning Control for Uncertain Nonlinear System with Continuous Nonlinearly Input

This paper proposes an adaptive iterative learning control (AILC) method for uncertain nonlinear system with continuous nonlinearly input to solve diﬀerent target tracking problem. The method uses the radial basis function neural network (RBFNN) to approximate every uncertain term in systems. A time-varying boundary layer, a typical convergent series are introduced to deal with initial state error and unknown bounds of errors, respectively. The conclusion is that the tracking error can converge to a very small area with the number of iterations increasing. All closed-loop signals are bounded on ﬁnite-time interval [ 0 ,T ] . Finally, the simulation result of mass-spring mechanical system shows the correctness of the theory and validity of the method.


Introduction
e research of the nonuniform trajectory is an interesting problem. Two new AILC methods for first-order hybrid parametric systems and high-order nonlinear hybrid parameter systems were proposed in the literature studies [1,2], respectively. Recently, AILC was presented, the literature [3] proposed a nonuniform target tracking AILC method, and the literature [4] proposed a fault-tolerant ILC technique for mobile robot nonrepetitive target tracking with output constraints. An ILC for a flapping wing micro aerial vehicle under distributed disturbances was proposed in the literature [5]. It can be seen from the above literature studies that solving the nonuniform target tracking problem for uncertain nonlinear systems is an important problem.
Adaptive control is used to handle system control problem about uncertainties. Adaptive control schemes learn uncertainties by adaptive laws. NN and FLS are used in the method as function approximators, for example, the paper [6,7]. e literature [8] could complete the varying control tasks by designing an adaptive fuzzy ILC for the uncertain nonlinear system. Based on RBF neural network approximation, the paper [9] proposed AILC for nonlinear pure-feedback systems to solve the nonuniform target tracking problem. e uniform AILC frame for uncertain nonlinear system was proposed in the literature [10], by Lyapunov theory, and it could prove the convergence. It should be noted that Lyapunov function-based AILC played an important role in dealing with the time-varying parameter [8,10,11]. However, initial state error problem is a challenging one as they need to converge to zero for keeping stability.
e literature studies [8,9,12] considered this problem recently. It is an important problem for AILC.
Because the actuator's physical is limited, the control input widely exists continuous nonlinearly such that system performance can be deteriorated. In the literature studies [13][14][15][16][17][18][19][20][21][22], control performance could change by using different technique recently. e literature [15] solved the question of adaptive stabilization for the time-delay system. e literature [16] developed an adaptive backstepping method of uncertain nonlinear systems about nonsymmetric dead-zone. As yet, there is no report from the literature for the AILC of nonlinear systems with continuous nonlinearly input and initial state error. is is a problem that needs to be solved urgently.
In this paper, the nonuniform trajectory tracking issue is discussed for the uncertain nonlinear systems with continuous nonlinearly input and initial state error. e contributions of the proposed control method are presented as follows: (i) e nonuniform trajectory tracking issue is studied for uncertain nonlinear systems under continuous nonlinearly input and initial state error issues. (ii) e AILC method is used to uncertain nonlinear systems. e RNFNN is introduced to learn unknown dynamic. A convergence order is introduced to solve the unknown bound and nonuniform target tracking problem.
Finally, simulation results of the mass-spring mechanical system are given to verify the validity of the designed controller.
is paper is organized as follows: in Section 2, the system description and related concepts are given in detail.
e main results are presented in Section 3. A simulation is shown in Section 4. Section 5 is the conclusion.

System Model.
e following nonlinear systems are considered: where x j,k � [x 1,k , . . . , x j,k ] T ∈ R j , and x � x n is the state that is measured. N(u k ) ∈ R represents the actuator characteristics, and y k ∈ R is the system output. f j (x j,k ), j � 1, 2, . . . , n, are smooth unknown nonlinear functions. N(u k ) represents continuous nonlinearly input with N(0) � 0 which belongs to [g 1 , g 2 ], i.e., (2) Assumption 1. g 1 and g 2 are unknown nonzero positive parameters.

Remark 1.
is is a reasonable assumption, because many system constraints can satisfy this condition.
Designing an AILC law u k (t) on [0, T] to make the output y k (t) following the target trajectory y r,k (t) is the control objective of this paper; that is to say, lim k⟶∞ ‖y k (t) − y r,k (t)‖ ≤ ϱ, where ϱ is a very small positive number. All closed-loop signals are guaranteed to be bounded. y r,k (t) is the smooth desired target. k is the iteration index.

Convergent Series Sequence.
e following definition and lemma are used in the controller design process.

Description of RBFNN Approximation
(see [23]) where , μ j ∈ Ω, and η > 0 are the center and the width of s j (x j,k ), respectively. e truth weight W � [w 1 , . . . , w l ] T is given as follows: In this section, we will discuss the detailed controller design process, main conclusion, and specific proof process.

Adaptive Iterative Learning Controller Design.
e following Lemma 2 is used in the controller design process.

Lemma 2.
For the controller u k and the error z nϕ,k , the following inequality holds: z nϕ,k N u k ≤ gz nϕ,k u k , g ∈ g 1 , g 2 .
Proof. If both sides of equation (2) multiply u k , we obtain If both sides of equation (10) multiply z 2 nϕ,k , we obtain i.e., If u k satisfies z nϕ,k u k ≤ 0, then If u k satisfies z nϕ,k u k ≥ 0, then So, the result is obtained. e specific process about designing controller is given as follows.

□
Step , α 1,k is the virtual controller. Introduce error function z 1ϕ,k and z 2ϕ,k by Section 2.4 to deal with initial state error as , , Recall that e derivative of z 1ϕ,k is as follows: According to Section 2.3, by RBFNN, F 1 (x 1,k ) is approximated and approximate error δ 1 (x 1,k ) is as follows: where W 1 is optimal weight vector. Denote N 1 � ω 2 M1 , which is needed later, Δ k � a/k l , a > 0, and l ≥ 2. e virtual controller is taken as When (18) and (19) are substituted into (17), we have where estimated W 1 and N 1 are W 1,k and N 1,k , respectively. W 1,k � W 1,k − W 1 and N 1,k � N 1,k − N 1 are the errors of estimated parameters. e last two terms of (20) can be changed to Mathematical Problems in Engineering By (20), (21) can be rewritten as Let where ω M1 is a positive parameter. Take the following nonnegative function: where Γ 11 and Γ 21 are the symmetric positive definite matrixes. e derivative of V 1,k along (23) is as follows: In previous equation, mn ≤ (1/r) Step 2. j: Mj , which is given later. z j+1,k � x j+1,k − α j,k , similar to Step 1, and z jϕ,k and z (j+1)ϕ,k by Section 2.4 are introduced as follows: e derivative of z jϕ,k is as follows: According to Section 2.3, F j (x j,k ) can become by the RBFNN where δ j (x j,k ) are the approximation errors and W j are optimal weight vectors. e virtual controller is chosen as follows: When (29) and (30) are substituted into (28), we have where W j,k and N j,k are the estimated parameters of W j and N j , respectively. W j,k � W j,k − W j and N j,k � N j,k − N j are estimated parameter errors. e last two terms of (31) can be changed as Let Assumption 5. j: ω j satify |ω j | ≤ ω Mj , here ω Mj is unknown. e following positive definite function is chosen: e derivative of V j,k is as follows by substituting (33): Step 3. n, define z n,k � x n,k − α n− 1,k , z nϕ,k by Section 2.4 is introduced as follows: e derivative of z nϕ,k is as follows: where _ α n− 1,k � n− 1 l�1 zα n− 1,k /zx l,k (x l+1,k + f l (x l,k ))+ (zα n− 1,k /zW n− 1,k ) _ W n− 1,k + (zα n− 1,k / zN n− 1,k ) _ N n− 1,k + α n− 1,k / zt.
Denote F n (x n,k ) � f n (x n,k ) − n l�1 zα n− 1,k /zx l,k (x l+1,k + f l (x l,k )), P n− 1,k � (zα n− 1,k /zW n − 1, k) _ W n− 1,k + (zα n− 1,k /zN n− 1,k ) _ N n− 1,k + (α n− 1,k /zt), and then (37) can be rewritten as follows: (38) F n (x n,k ) can be rewritten by RBFNN approximation as follows: F n x n,k � W T n ζ n x n,k + δ n x n,k , where δ n (x n,k ) is the approximation error and W n is an optimal weight vector. en, (38) can be rewritten by substituting (39): Lyapunov function is chosen as follows: where estimated W n , N n and ϱ � 1/g are W n,k , N n,k and ϱ k (t), respectively. W n,k � W n,k − W n , N n,k � N n,k − N n , and ϱ k (t) � ϱ k (t) − ϱ are parameter estimation errors. And Γ 4 is learning gain to be designed. Using the derivative of V n,k along (40), then we have (42)

Mathematical Problems in Engineering
Choose Take the actual controller as follows: where u 2,k is needed to design for dealing with g. en, (42) becomes When ω n � δ n (x n,k ), the following equation holds: − η n z n,k − sgn z nϕ,k _ ϕ n,k (t) � − η n z nϕ,k − η n ϕ n,k (t)sat z n,k ϕ n,k (t) − sgn z nϕ,k _ ϕ n,k (t) and then we have Take u 2,k � − sat(z n,k /ϕ n,k (t))(ϱ k (t) + 1)|u 1,k |, and then (46) becomes Assumption 6. n: bounded ω n satisfies |ω n | ≤ ω Mn , here ω Mn is an unknown parameter. Choose the AILC laws as follows: and then we obtain 3.2. Convergence Analysis. According to the above design process, we can get the following conclusion expressed by eorem 1. e following assumption is needed.

Remark 4.
is is the general setting of initial state in ILC. Under Assumptions 1-7, by controller (43) and adaptive parameter estimation laws (48)-(50) for system (1), we get all signals are bounded, and

Simulation
In this section, a mass-spring mechanical system is considered to show the effectiveness of the proposed controller. m is a mass, and assume that resistive force caused by friction is zero. e external force u k drives the mass, which is a control variable. y k is the displacement from a reference position, and the motion equation of the system with continuous nonlinearly input is as follows: where t ∈ [0, π] and F ms (·) is the restoring force of the spring. k denotes the iteration index.
We define x 1,k � y k , x 2,k � _ y k , and m � 1 which transform (58) into the state-space form (59) e restoring force of the spring can be modeled as In the system, we have k � 1, a 0 � 0, a 1 � a 2 � a 3 � a 4 � 1, and q � 4.
Continuous nonlinearly input N(u) is shown as N(u) � (0.5 + 0.1 sin(u))u. System objective is that the output of (59) can follow the reference trajectory y r,k on [0, π] when k ⟶ ∞. In the different target case, y r,k � g k sin(2t) as k is even, and y r,k � g k cos(t) as k is odd, where g k � rand(0, 1).

Conclusions
e different target tracking problem for unknown nonlinear systems with continuous nonlinearly input is solved in this paper. We introduce the RBFNN to deal with the uncertain dynamics. e problems of approximation error and initial state error can be efficiently solved by suitable means. is paper can keep all signals being bounded on [0, T], and errors can converge with the number of iterations increasing. Finally, the effectiveness of the proposed control method is verified via a simulation example.

Data Availability
No data were used to support this study.   Mathematical Problems in Engineering 9