Adaptive neural network control for nonlinear non-strict feedback time-delay systems

This paper focuses on adaptive neural control for a class of non-strict feedback nonlinear systems with state delays and input delay. By combining integral transformation with adaptive neural control approach, a backstepping-based adaptive neural control scheme is proposed. The suggested control schemes guarantees that the tracking error converges to a small neighbourhood of the origin, meanwhile, all the closed-loop signals remain bounded. Simulation examples are used to verify the effectiveness of the proposed method.


Introduction
In the past three decades, backstepping has been developed an useful method for nonlinear system controller design (see Kanellakopoulos et al., 1991;Schwartz et al., 1999;Yao & Tomizuka, 1997). Particularly, adaptive neural or fuzzy control approach is combined with backstepping to cope with control design of nonlinear systems. In  and , the problem of adaptive fuzzy output feedback control is discussed for a class of strict-feedback nonlinear systems. The designed adaptive fuzzy controllers guarantee achievement of output tracking issue and boundedness of all the closed-loop signals. For more results on approximation-based adaptive control of nonlinear systems, please see Rubio and Yu (2007), M. Wang et al. (2010), Ge et al. (2004) and Ho et al. (2005).
Note that delays are usually the source of system instability or performance degradation. Recently, the control systems with input delay have also received more and more attention. In Liu et al. (2018), the stabilization of linear interconnected systems with input delays is studied. Decentralized controllers are designed based on the solvability of linear matrix inequalities. In Lin et al. (2020), the problem of finite-time stabilization is addressed for linear switched systems with input delays. By converting the time continuous system into a discrete system, a digital state feedback and a digital output feedback controllers are proposed respectively. In Zhu et al. (2012), the authors introduce the integral transformation in the control design to deal with the input delay. Furthermore, CONTACT Bing Chen chenbing1958@126.com an adaptive neural controller is constructed for strictfeedback nonlinear systems with input delay. This way is applied to multi-agent systems with input delays in Y. M. Li et al. (2020). However, the existing results on adaptive control for nonlinear input delay systems are mainly presented for strict-feedback systems. Theoretically, these control strategies cannot be directly applied to the systems with non-strict feedback form, which include the strictfeedback form as a special case. Even though there are some backstepping-based adaptive neural or fuzzy control schemes of non-strict feedback systems to be reported, in these existing control designs the proposed virtual control signals involve the information on the state variables of the subsequent subsystems. That generates the algebraic loop phenomenon and does not meet the backstepping design rule.
In the current paper, we will consider the adaptive neural control for non-strict feedback systems with state and input delays. In controller design, the integral transformation is utilized to cope with the input delay in order to ensure the feasibility of controller design. The structural feature of the basis vector functions of RBF NNs is used to design the virtual control signals. By this way, the algebraic loop problem is successfully avoided. The suggested adaptive neural controller is shown to ensure good tracking performance and the boundedness of all the closed-loop signals. At last, two examples are used to illustrate the effectiveness of the developed control strategy and give a comparison to the existing results.

Preliminaries and problem formulation
Consider a nonlinear non-strict feedback system, which is represented by the following differential equations: . . x n ] T , u ∈ R and y ∈ R are system's state, input and output variables, respectively, f i (·), g i (·) and q i (·) are unknown smooth nonlinear functions, with f i (0) = 0 and q i (0) = 0, and input delay τ is known positive constant, state delay τ i is unknown. In addition, the system function ∈ R denotes the control input subject to saturation nonlinearity described by where u max is known upper bound of u(t). According to H. Wang et al. (2013), introduce the function g(u) as follows Then D(u) can be expressed in the following form: where d(u) = D(u) − g(u) is a bounded function, and its bound can be obtained as In addition, by the mean value theorem, there exists μ(t)(0 < μ < 1) such that where g u μ = ∂g(u) ∂u | u=u μ , u μ = μu + (1 − μ)u 0 . By choosing u 0 = 0, (5) can be written as In order to facilitate the control design, the following assumptions and lemmas are given.
Assumptions 2.1: The control gain function g i (1 ≤ i ≤ n) is bounded and meets the inequality: where g s and g t are unknown constant.
Assumptions 2.2: For the function g u μ in (6), there is unknown constant g l > 0 such that Assumptions 2.3: The reference signal y d and its nth order time derivatives are continuous and bounded. Therefore, we assume that there exist positive constant d such that |y d (t)| ≤ d and |y (k) Assumptions 2.4: There exist strictly increasing smooth functions ϕ i (·) : R + −→ R + , with ϕ i (0) = 0, such that for i = 1, 2, . . . , n Remark 2.1: From the above assumption, we know that there are smooth functions h i (s) make ϕ i (s) = sh i (s), and the following inequality holds Definition 2.1: To construct the controller by using backstepping method, the following transformation is defined where α i is the virtual control signal of the ith subsystem and has the form where α 0 = y d , U i = z i φ i (x i ) δ i , a i and δ i are common positive constants,θ i is the estimation of θ i , an unknown constant that will be shown below. The estimation error is defined asθ

Lemma 2.2 (Chen et al., 2014):
, the following holds: withā i = 1.5 + a 2 i + 2 z iθ i , and d is the upper bound of y d .

Controller design
This section will propose an adaptive neural network control scheme based on backstepping. The design process is divided into n steps. For convenience, we substitute Step 1. For the first subsystem, define the Lyapunov function as by (10), we havė The derivative of V 1 is given by, By Lemma 2.2 and Assumption 2.4, for z 1 q 1 (x τ 1 ) of (15), one has: where C 1 = n j=1 (n + 1) 2ā2 j + 1 β 1 (n + 1) 2ā2 n h 2 1 , β 1 are given parameters. To simplify control design, introduce functions h t 1 and H t τ 1 . h t 1 will be defined below, H t τ 1 is defined as: Substituting (16) and (17) into (15) yieldṡ wheref 1 is:f The functionf 1 can not be used to design virtual control signal because it contains unknown functions. By the approximation ability of RBF NNs, the following formula holdsf where δ(Z 1 ) is estimation error, ε 1 is known accuracy of estimation. Because the virtual control α 1 can not contain the state after the first subsystem, the state vector in the neural network is changed to the current state vector in the first system by Lemma 2.1. By the above discussion and Lemma 2.3, we can get: where The parameter δ 1 is positive constant.
Substituting (19) and (21) into (18) produces: We can get virtual control: where , then, the virtual control is substituted into the inequality (22) and expressed as: And it turns out that the adaptive lawθ 1 iṡ then substituting adaptive law into (24), we have: Step i. For the nth subsystem, define the Lyapunov function as: For z i by transformation we can have: Combine (28) and (29), take the derivative of the V i For the delay term notational simplicity, make ∂α i−1 ∂x i = −1, by Assumption 2.4 and Lemma 2.2, one has: where C i = n j=1 (n + 1) 2ā2 j + 1 β k (n + 1) 2ā2 n h 2 i , β k are the given parameters. For convenience, substitute h 2 j , h 2 i for h 2 j ((n + 1)ā j |z j (τ k )|), h 2 i [(n + 1)ā n d] respectively. Similar to the first step, introduce functions h t i and H t τ i . h t i will be defined below, H t τ i is defined as: Substituting (31) and (32) into (30) yieldṡ wherē By neural networks approachf i , where Substituting (34) into (33) and then using (36) result iṅ we can get virtual control α i as: where U i = z i φ i (x i ) δ i , a i > 0 and δ i > 0 are design parameters,θ i is the estimation of θ i . Thus, replacing α i into (37) giveṡ We getθ Then, substituting (40) into (39) produces: Step n−1. For the (n − 1)th subsystem, consider a Lyapunov function V n−1 as For z n−1 , by transformation, it can be obtained thaṫ The time derivative of V n−1 is given bẏ The integral term and time delay terms are dealt with as following: For the integral term −z n−1 g n−1 g n t t−τ D(u(θ )) dθ , by mean value theorem since the control signal has a saturation-limited nature, there is a positive constant D n that makes τ |u|(ξ ) ≤ τ D n , we get: − z n−1 g n−1 g n t t−τ D(u (θ)) dθ ≤ z n−1 g n−1 g n τ D n tanh z n−1 g n−1 g n τ D n δ n−1,1 + kδ n−1,1 .
In order to eliminate the influence of the time delay term, for all of these systems, define Lyapunov function where the function W i is defined as: The derivative of the V showṡ − a 2 n z 2 n g s g l + g s 2γ 2 n σ 2 nθ 2 n + n−1 i=1 kδ i θ i + kδ n−1,2 θ n−1 g s + kδ n θ n g l g s + n i=1 g s 2γ 2 i σ 2 i θ 2 i + kδ n,1 + 1 2 g l g n D n 2 + 1 2 By rearranging the term n i=1 H t τ i we have with c(n, k) = n−k+1 2 . From the (80), we select By substituting (80) and (81) into (79), one haṡ − a 2 n z 2 n g s g l + g s 2γ 2 n σ 2 nθ 2 n + n−1 i=1 kδ i θ i + kδ n−1,2 θ n−1 g s + kδ n θ n g l g s where a 0 = min 2a 2 i g s , σ 2 i : 1 i n − 1, 2a 2 n g s g l , σ 2 n , 1 , kδ i θ i θ n−1 g s + kδ n−1,2 + kδ n θ n g l g s Further, DefineV = max{V 0 , b 0 a 0 }. Therefore, we have z = √ 2V, that means that all the signals are bounded in the closedloop system. In addition where k * = 2 a 0 , δ * = √ b 0 . It is shown that the tracking error is bounded and the limit tends to a small enough set.

Simulation example
To illustrate the validity of the control method proposed in this paper and further compare with other control methods, consider the following two simulation examples: Example 4.1: Consider the following second-order nonstrict feedback nonlinear system: where x 1 , x 2 denote the state variables, y is the system output, and the input saturation limits are chosen as u max = 5 and u min = −5. The system satisfies Assumptions 2.1-2.4, by selecting parameters a 1 = 2, a 2 = 5, δ 1 = δ 2 = 2, σ 1 = σ 2 = 1, γ 1 = γ 2 = 1, g l = 5. The simulation is carried out with the initial conditions x 1 (0) = 0.5, x 2 (0) = 0.5,θ 1 =θ 2 = 0, the reference signal is assumed to be y d = sin(x), state delay τ 1 = τ 2 = 2, input delay τ = 0.1. ψ i (t) = 0, (i = 1, 2), ϑ(t) = 0 when t < 0. Each control signal is as follows Figures 1-4 illustrate the simulation results. Figure 1 shows the input signal of the system can well track the given reference signal. Figures 2 and 3 show that all the closed-loop signals are bounded. Figure 4 displays the saturation control input signal u. Figures 1-4 show that the proposed control method can be applied to a class of non-strict feedback systems.
Example 4.2: Consider the following nonlinear system, which is taken from H. Li et al. (2017),  The work of the original literature is to use the Pade approximation method to deal with the input delay. The virtual control, real control law and adaptive law are designed as  According to the original article, the given tracking signal is y d = sin(t), the initial conditions of the system are [x 1 (0), x 2 (0)] T = [0.07, 0.2] T , [θ 1 (0), θ 2 (0)] T = [0, 0] T , select control parameters as a 1 = 2, a 2 = 3, c 1 = 50, c 2 = 54, b 1 = 0.08, b 2 = 0.09, σ 1 = 1, σ 2 = 2, r 1 = 1, r 2 = 2, k d1 = 0.12, η 1 = 0.4, η 2 = 0.2. When the input delay is τ = 0.0043, system output can track the given reference signal well. The tracking effect is shown in Figure 5. When the input delay is selected as 0.1, the system output can no longer track the given reference signal, as shown in Figure 6. Under the same reference signal and initial conditions, the method proposed in this paper is used in the control system (85). The control parameters are a 1 = a 2 = 12, δ 1 = δ 2 = 0.1, γ 1 = γ 2 = 5, σ 1 = σ 2 = 0.03, g l = 0.3. When the input delay is τ = 0.0043, the simulation is  shown in Figure 7, when τ = 0.1, the simulation effect is shown in Figure 8. The simulation results show that the method proposed in this paper is more effective for the control of the system (85).

Conclusion
This paper addresses the problem of adaptive tracking control for a class of SISO nonlinear systems with state and input delays. The controller is constructed by backstepping adaptive neural control approach, which ensures that all signals in the closed-loop system are bounded and the tracking error converges to a small neighbourhood around the origin. In the future research, we hope further develop some new approaches to nonlinear systems with input delay. In addition, finite-time control issue is also considered for nonlinear systems with state and input delays.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by the National Natural Science Foundation of China [grant numbers 61873137, 61673227].