FLS-Based Nonuniform Trajectory Tracking AILC for Uncertain Nonlinear Systems with Nonsymmetric Dead-Zone Input and Initial State Error

This paper proposes an AILC method for uncertain nonlinear system to solve diﬀerent target tracking problems. The method uses fuzzy logic systems (FLS) to approximate every uncertain term in systems. All closed-loop signals are bounded on [ 0 , T ] according to the Lyapunov theory. A time-varying boundary layer and a typical convergent series are introduced to handle initial state error, unknown bounds of errors, and nonuniform target tracking, respectively. The result is that the tracking error’s norm can converge to a small neighborhood along iteration increasing asymptotically. Finally, the simulation results of mass-spring mechanical system show the correctness of the theory and validity of the method.


Introduction
e research of the nonuniform trajectory is an interesting problem. e paper [1] proposed a new ILC law for firstorder hybrid parametric system and the paper [2] proposed a novel AILC method for nonlinear hybrid parameter systems. Recently, AILC is presented; a nonuniform target tracking AILC method was proposed in the paper [3]. e paper [4] proposed a fault-tolerant ILC technique for mobile robots' nonrepetitive target tracking with output constraints. From the above literature analysis, solving the nonuniform target tracking problem for uncertain mechanical nonlinear system is an important problem.
Adaptive control is used to handle system's control problem about uncertainties which is a challenging problem. Adaptive control schemes learn uncertainties by adaptive laws. NN and FLS are used in the method as function approximators, such as the papers [5][6][7][8]. e literature [9] could complete the varying control tasks by designing an adaptive fuzzy ILC for uncertain nonlinear system. Based on RBF neural network approximation, the literature [10] 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 paper [11]; by Lyapunov theory, it can prove the convergence. It should be noted that Lyapunov function-based AILC plays an important role in dealing with the timevarying parameter in the literatures [9,11,12]. But initial state error is a challenging one as they need to converge to zero for keeping stability. Only the papers [9,10,13] considered this problem recently it is an important problem for AILC.
Due to the physical limitations of actuators, in real systems, control inputs are often constrained, such as dead-zone inputs. However, these constraints may damage the performance of the system. In the papers [14][15][16][17][18][19][20][21][22][23], control performance could be changed by using different techniques recently. e question of adaptive stabilization for time-delay system had been solved in the literature [16]. e paper [17] 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 nonsymmetric dead-zone 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 nonsymmetric dead-zone 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 nonsymmetric dead-zone input and initial state error issues. (ii) e AILC method is used to uncertain nonlinear systems. FLS 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 mass-spring mechanical system are given to verify the validity of the designed controller. e remainder of this paper is organized as follows: in Section 2, the system description and related concepts are given in detail.
e controller design process and 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 , x � x n is the state that is measured; Γ(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.
Here, the function Γ(u k ) represents actuator output with nonsymmetric dead-zone and can be expressed as where m r and m l present the right and left slopes, respectively; b r and b l present the breakpoint of dead-zone. e nonsymmetric dead-zone can be rewritten as a combination of a line and a disturbance-like term [18]. where Assumption 1. Parameters m l , m r , b l , and b r are uncertain constants. ere exists an unknown ] satisfing 0 < ] ≤ min m l , m r , which is small enough. e unknown constant d is the upper bound of d(t).
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 our control objective, that is to say, lim k⟶∞ ‖y k (t) − y r,k (t)‖ ≤ ϱ, where ϱ is the small positive error. Guarantee that the closed-loop signals are bounded. y r,k (t) is the smooth desired target. k is the iteration index.

Convergent Series Sequence.
e following definition and lemma can be used in the design process.

Description of Fuzzy Logic System (FLS)
. e FLS has a good approximation property [24]. For a smooth f(x) ∈ R can be approximated by where δ(x) is the fuzzy approximated error, (W * ) T is the optimal weight vector satisfing where D x represents the set of x and f(x|W) that is estimated f, shown by with an adjustable W, and Ω W * is given as follows: with W * M being a positive constant.
Assumption 2. In this paper, the following inherent approximation error δ j (x j,k ) is assumed to be bounded with |δ j (x j,k )| ≤ θ j , where the unknown parameters θ j (1 ≤ j ≤ n) denote the smallest upper bounds of |δ j (x j,k )| with θ j ≥ 0.
where δ 1 (x 1,k ) is the approximation error and W 1 is the optimal weight vector. Define 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 Substitute equations (16) and (17) into equation (15), then where the 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 equation (18) can be changed as Equations (18) and (19) can be rewritten as Let Assumption 4. e bounded term ω 1 satisfies |ω 1 | ≤ ω M1 , where ω M1 is a positive parameter.
Take the following nonnegative function: where Γ 11 and Γ 21 are symmetric matrices and are positive. Derivate V 1,k according to equation (21), then In the previous equation Step j.
Mj , which is given later. z j+1,k � x j+1,k − α j,k , the same to Step 1; z jϕ,k and z (j+1)ϕ,k from Section 2.4 are introduced as Derivate z jϕ,k as follows: where (25) can be rewritten as According to Section 2.3, F j (x j,k ) by FLS can become where δ j (x j,k ) is the approximation error and W j is the optimal weight vector. e virtual controller is taken as Equations (27) and (28) are substituted into equation (26), then 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 equation (30) can be changed as Let ω j � δ j (x j,k ) + ϕ (j+1)ϕ,k (t)sat(z (j+1)ϕ,k /ϕ (j+1),k (t)), then equation (29) becomes Assumption j. ω j satisfies |ω j | ≤ ω Mj ; here, ω Mj is unknown. Positive definite function is chosen: Derivate V j,k according to equation (31), Step n. Define z n,k � x n,k − α n−1,k ; z nϕ,k from Section 2.4 is introduced as Derivate z nϕ,k as follows: (zα n−1,k /zN n−1,k ) _ N n−1,k + (α n−1,k /zt), and by equation (3), then equation (35) can be rewritten as from which F n (x n,k ) can be rewritten as follows: F n x n,k � W T n S 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. Let u 1,k � −z (n−1)ϕ,k − W T n,k S n (x n,k ) − N n,k (1/Δ k )z n,k + P n−1,k − η n z n,k . Take the actual controller as where u 2,k is designed to compensate for unknown input gain m(t). en, according to equations (37) and (38), equation (36) can be rewritten as _ z nϕ,k � u k +(m(t) − 1)u k + W T n S n x n,k + δ n x n,k − P n−1,k − sgn z nϕ,k _ ϕ n,k (t) + d(t) � −z (n−1)ϕ,k − W T n,k S n x n,k − N n,k 1 Δ k z n,k + m(t)u 2,k +(m(t) − 1)u 1,k where W n,k and N n,k are the estimated parameters of W n and N n , respectively. W n,k � W n,k − W n and N n,k � N n,k − N n are the estimated parameter errors. e last two terms of equation (39) can be changed as −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) � −η n z nϕ,k .

Simulation
In this section, a mass-spring mechanical system is considered to show the effectiveness of the proposed controller. m is a mass; 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; the motion equation of the system with nonsymmetric deadzone input is as follows: where t ∈ [0, π], F ms (·) is the spring's restoring force; k denotes the iteration index.
We define x 1,k � y k , x 2,k � _ y k , and m � 1, so equation (54) can be transformed into the state-space form (55) e spring's restoring force 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.

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Mathematical Problems in Engineering   Mathematical Problems in Engineering e nonsymmetric dead-zone is shown as System objective is that the output of system (54) 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 nonsymmetric dead-zone input is solved. Introduce FLS to deal with the uncertain dynamics, and the problem 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]; errors can converge to a small set along iteration increasing. Simulation study proves the correctness of AILC method of this paper.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.