Adaptive NN Backstepping With Considering Integral of Tracking Error

In order to simplify the design procedure of traditional neural network backstepping and improve the robustness and precision of control, an improved scheme is studied for a class of nonlinear systems. To avoid reconstructing virtual control inputs in each recursive step, RBF neural networks are utilized as approximators to estimate the desired feedback control of the whole system only. Meanwhile, the integral action of tracking error is introduced into the backstepping design procedure, which not only participates in updating the neural network weight, but also serves as a component part of the control input. This design may benefit the parameter tuning and make controller perform better sometimes. Based on the Lyapunov synthesis approach, theoretical analysis and simulation results are provided to show the feasibility of the improved scheme.


I. INTRODUCTION
Backstepping technique can date back to [1]. As a systematic design approach, its recursive design procedure is suitable for complex strict-feedback systems. Owe to some significant works such as [2], backstepping technique has been further developed and perfected in theory. However, ''explosion of complexity'' arises with the growth of system order. So the application in practice may be limited for high order systems until dynamic surface control is proposed [3]. However, because of filtering technique, dynamic surface control doesn't have the form of exponential convergence. In recent years, backstepping and techniques derived from it are increasingly common in various engineering fields. For air-breathing hypersonic vehicle control, dynamic surface control technique is involved in conjunction with the backstepping control approach [4]. A tracking controller is designed for autonomous helicopter based on a backstepping procedure [5]. Besides, backstepping tracking control law is applied to an autonomous underwater vehicle under a novel framework [6]. And [7] gives distributed controllers to solve flocking problem of multiple mobile robots with the aid of backstepping techniques.
The associate editor coordinating the review of this manuscript and approving it for publication was Juntao Fei .
In terms of engineering applications, precise modeling may be hard to establish for most systems. In practice, uncertainty is one of the important factors that affect the control performance and the closed-loop stability of the whole system [8]. To deal with the parametric uncertainties and unknown nonlinearities, much progress has been made by combining backstepping technique with robust control strategy. At the same time, increasing research interests focus on neural networks (NNs). For example, radial basis function (RBF) networks have been proposed [9]. For RBF networks, F. Girosi and T. Poggio have proved existence and uniqueness of best approximation [10]. This kind of neural networks have recently drawn much attention due to their good generalization ability and a simple network structure that avoids unnecessary and lengthy calculation [11]. In recent years, stability, dissipativity and extended dissipativity analysis problems have been investigated for NNs with time delay [12], [13]. With the development of NNs, neural network control of unknown nonlinear dynamic systems has caused wide public concern [14]. One of the great advantages is that exact values of the system base parameters are not required to be known a priori, such as [15]. Based on Lyapunov's stability theory, stable adaptive NN controller can be designed, so that the neural network weight can achieve on-line adjustment. In this field, many significant VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ works [14], [16]- [19] have been made by combining adaptive neural design with backstepping methodology. Different from them, a systematic design of adaptive NN backstepping in this paper is developed inspired by [1], [11]. Instead of reconstructing virtual control inputs in each recursive step, the design process of RBF neural network is carried out until we step back to the whole system. By utilizing some properties of the system, the desired feedback control of the whole system has the form in which the unknown nonlinearities and known quantities are separated. It is not necessary for each subsystem to acquire a virtual control input estimated by neural network. In this scheme, the number of neural networks can be cut down and will probably no longer be the same as the order of the system. Meanwhile, we introduce the integral action of tracking error into the backstepping design procedure, which not only participates in updating the neural network weight, but also serves as a component part of the control input. Appropriate integral operation can improve the control effect sometimes. So a parameter is provided to adjust the intensity of integral action, which allows for more flexibility in parameter tuning. It helps to enhance the disturbance suppression performance and meet higher precision requirement, but may not hold good for all cases.
This paper uses fundamental and simple mathematical knowledge, overall adaptive scheme is shown to guarantee the stability of the closed-loop system. The remaining part of the paper is organized as follows. The system under consideration is described in section II. In section III, the desired feedback control is presented without repeating the complex design procedure for system with arbitrary order. In section IV, an adaptive neural network controller is provided for controlling uncertain nonlinear systems. Section V demonstrates the simulation results to verify the feasibility.

II. SYSTEM DESCRIPTION
Consider the SISO nonlinear system with a simplified triangular structure [20]: x n ] T ∈ R n , u ∈ R, y ∈ R are the state variables, system input and output respectively; g and f i , i = 1, 2, · · · , n can be known or unknown smooth functions. Disturbance is included in f n . The objective is to force the output x 1 follow a desired trajectory x 1d . Note that many control problem (e.g., single inverted pendulum control, flight-path angle control [21], and manipulator control [22]) can be formulated in such structure.
Besides, systems in a more general form may be transformed into (1). A case study of three order system is provided here and similar procedure may be extended to n order systems. Consider the following nonlinear dynamic system: , we obtain thaṫ , we obtain thaṫ Define , then we arrive aṫ All signals in the closed-loop system must remain bounded, so the derivation above is not available for all cases.

III. DESIRED FEEDBACK CONTROL
For the sake of brevity, the following step-by-step procedure only shows Step 1, 2, i, n, in detail with redundant equations and repetitive steps being omitted.
Step 1: Define s 1 = e 1 + ξ e 1 dt, whose derivative iṡ View x 2 as a virtual control input, the desired value can be expressed as where, e 2 = x 2 − x 2d .
Taking V 1 = 1 2 s 2 1 as a Lyapunov function candidate, its derivative iṡ Step 2: Define s 2 = e 2 + ξ e 2 dt, whose derivative iṡ View x 3 as a virtual control input, the desired value can be expressed as (10) where, e 3 = x 3 − x 3d . Taking V 2 = V 1 + 1 2 s 2 2 as a Lyapunov function candidate, its derivative iṡ Step View x i+1 as a virtual control input, the desired value can be expressed as where, Step n : Define s n = e n + ξ e n dt, whose derivative iṡ Desired feedback control input is chosen as Choose overall Lyapunov function V n = V n−1 + 1 2 s 2 n . As a positive definite function, it is differentiable with a negative time derivative, that iṡ The desired feedback control input u * is given in the last step. By suitably choosing value of ξ, k 1 , k 2 , · · · , k n , stability of the whole close-loop system can be obtained with good performance. Then, supposing that nonlinear function g and f i , i = 1, 2, · · · , n are known exactly, the realizable and practicable form of u * is derived as follows.
Differentiating (7) and (13) yields It follows that The above equation can be rewritten as Substituting the above equation into (16) to have the form System (1) can be transformed into the following equations: The above equation can be rewritten as Combining (7) and (13) with the define of e i+1 and s i+1 , we obtain .
Substituting (19) and (22) into (24) leads to Remarks: 1. It should be noticed that x i , i = 2, 3, · · · , n is forced to follow the desired trajectory x id , i = 2, 3, · · · , n witch contains an integral term. In this way, singularity-free desired feedback controller can be designed by taking advantage of the integral operation.
2. Actually, (7) and (13) are first order differential equations for x id , i = 2, 3, · · · , n, respectively. It is not necessary to solve them, though analytical solutions are available.
3. Considering (25), s i , i = 1, 2 · · · n depends on constant coefficients, s 1 and its derivative. 4. There exists η = 2 min(k 1 , k 2 , · · · , k n ) such thatV n ≤ −ηV n , then we have V n (t) ≤ V n (0) · e −ηt , that is For high order systems, improved computing power nowadays may be qualified for the task. However, because of filtering technique, dynamic surface control doesn't have the form of exponential convergence.

IV. ADAPTIVE NN CONTROL
There have been several studies concerning neural network approximation, using hidden units described by so-called ''radial basis functions,'' h ( x − c ), where h is some smooth real function of the distance x − c from the ''center'' vector c in the input space [23]. Consider RBF neural networks where b j notes a positive scalar called a width, positive integer l denotes the neural network node number, W is an adjustable network weight vector. An assumption should be made that the reconstruction errors of RBF neural networks are bounded in the following discussions. Supposing that nonlinear function g and f n are unknown, letŴ be an estimate of the ideal NN weight W * , and definẽ W =Ŵ − W * . It can be proved that the following function approximation holds: where, z = [x 1 , x 2 , · · · , x n ] T is the input of RBF neural network.
In the above section, a desired feedback control input is given Based on it, an adaptive NN controller is designed as follows Considering (20), the derivative of s n can be expressed aṡ Choose the following Lyapunov function candidate for the design of the adaptive and control laws Using (17) where f n , g > 0 are diagonal adaptation rate matrixes. The parameter of robust term can be set as M ≥ µ f n + µ g u , so that asymptotic tracking can be retained. Similarly, when nonlinear function g and f i , i = 1, 2, · · · n are unknown, adaptive NN controller can be designed. Let where, z = x 1 ,ẋ 1 , · · · , x (n−1) 1 T is the input of RBF neural network.
The design of adaptive and control laws and Lyapunov function candidate are given directly gẆ g −s n uh g (z) +s n µ f +µ g u −M |s n | (52) M ≥ µ f n + µ g u It's worth mentioning that the chattering phenomena in sliding modes have been studied for many years. Various methods of chattering suppression is introduced in [24].
After RISE-based control [25], [26] came out, the method has been improved and creatively applied in industry [27].
In this paper, by rising k n and cutting down M , chattering can be reduced and the stability can be guaranteed in the presence of great disturbance. It's uniformly ultimate bounded result in theory, but good control effect can be achieved. Simulation results show the feasibility when M = 0.
Besides, another approach is provided as follows to solve the chattering problem. In (37) and (49), robust term may cause chattering, so a low pass filter can be designed to deal with the drawback brought by the sign function.
The desired continuous control input is given by (16). Let e n+1 = u − u * denotes the difference between actual and desired control input, then (15) can be rewritten aṡ The low pass filter can be expressed aṡ where r is the input of the filter, and T is time constant. Desired input of the filter is chosen as Let p = −u/T −u * ,the following function approximation holds: By utilizing the reconstruction of NN, input of the filter is where e n+1 is accessible. According to (15) and (16) e n+1 = u − u * = 1 g (ṡ n + k n s n + s n−1 ) Stability analysis is given as followṡ Choose Lyapunov function candidate Then, the updating algorithm for NN weight is chosen aṡ If g is known, it can be achieved. We can see that input of the filter contains the sign function, but output of the filter is a relatively smooth signal, which is the actual control input. Remarks: 1. If the unknown nonlinearities are only g and f n , the system is observable. So the input of RBF neural network is chosen as z = [x 1 , x 2 , · · · , x n ] T . However, if g and f i , i = 1, 2, · · · n are unknown, z = x 1 ,ẋ 1 , · · · , x (n−1) 1 T is chosen because every state variable is a function of x 1 and its derivatives according to system (1).
2. For systems having local precise model, substituting the accurate parts into f directly makes it easy for RBF design. For high order systems, the estimation of f using one RBF neural network may not reconstruct the complex dynamic characteristics well. Number of the RBF neural network depends on the specific system and control effect, though approximation term by term is theoretically possible.

V. SIMULATION STUDY
In this section, two second order plants are considered to verify the effectiveness of the improved scheme.
In the first example, simulation results of our improved scheme are provided in contrast to a traditional method. Consider the following strict-feedback system [17]: The desired trajectory y d is generated from the following van der Pol oscillator system: The initial conditions [x 1 (0), x 2 (0)] T = [1.2, 1.0] T and [x d1 (0), x d2 (0)] T = [1.5, 0.8] T . In [17], an ingenious adaptive NN controller is designed using two neural networks containing 25 and 135 nodes respectively. Fig. 1 shows the control effect of applying the controller to the system for tracking desired signal y d with β = 0.2.
Based on section II, define x 1 = x 1 , x 2 = 1 + 0.1x 2 1 x 2 . The above system can be transformed into a simplified triangular structure: As the plant is a second-order system, according to (49), the adaptive NN backstepping shall be chosen as where,  Fixed-step size is chosen as 0.01s. Fig. 2 shows the simulation results for our designed adaptive NN controller with two neural networks containing 7 and 7 nodes respectively.
Through comparing Fig. 1 with Fig. 2(a), it can be seen that RBF neural networks while simpler in structure, offer a better tracking performance under the improved scheme. The system output y tracks the desired trajectory y d more quickly and precisely.
The second example is used to show the function of integral terms. For a variable length pendulum [28], the plant dynamics can be expressed as followṡ The reference signal is sin t, t > 4πs.
As the plant is a second order system, according to (21) and (49), the desired feedback control input and the adaptive NN control input shall be chosen as where, s 2 = k 1 s 1 +ṡ 1 , z = [x 1 ,ẋ 1 ] T . Take the physical meaning into consideration, x 1 ∈ [− π 2 , π 2 ]. If circular frequency of y d is 1rad/s, then Change ξ by setting ξ = 0 and 15 respectively with other parameters being fixed at D = 5, k 1 = k 2 = 20, M = 0. Fig. 3 and 4 present the simulation results for the desired feedback control. The tracking error given in Fig. 4(a) is restricted in a smaller region than that given in Fig. 3(a). The control input and system states are shown in Fig. 3(b)-(d) and 4(b)-(d).
Remarks: 1. In the first example, choose ξ = 0 because the adding of integral term will go against the fast convergence and less steady residual error. The difference between desired and practical initial conditions may be one of the main causes. Use simpler neural networks with fewer nodes, fairly good tracking performance can be obtained under the scheme in this paper.
2. Due to many factors, it is hard to set optimal control parameters for controller in engineering practice. When the design of control parameters is inappropriate, adding integral action of tracking error may significantly improve the control effect. When suitably choosing the control parameters, integral terms can still contribute to the precision of control. The second example belong to the latter. Note that simulation results are sensitive to the parameters of neural network. It is hard to just simply compare the adaptive NN backstepping control effects with different ξ . Desired feedback control effects with different ξ can be convincing.
3. The output of RBF may be far from the exact value of f affected by large disturbance in the second example. Actually it is the estimate of f + d(t). Neural network control makes breakthrough in control of nonlinear uncertain systems. But it's difficult to acquire accurate estimation of unknown dynamics in many cases [11]. Reasons for this are manifold. Fundamentally, the adaptive law cannot guarantee the convergence of approximation error. And parameters determination of the neural network mainly depends on experience in system debugging. Moreover, actual operation may not always run on nominal condition due to modeling error as well as occurrence of external disturbance. So the deviation of estimation and tracking error that followed are hard to avoid. Increase of tracking error will further cause deterioration of the approximation effect. Such positive feedback behavior could lead to divergence of the system. Integral action of tracking error introduced into each step of backstepping may help to solve the problem mentioned above sometimes.
4. The reason for the jumps in Fig. 3-7 is the switching of the command signal. So there is a jump for tracking the abruptly altered desired trajectory.

VI. CONCLUSION
In this paper, for a class of nonlinear systems with a simplified triangular structure, a systematic design of adaptive neural network controllers has been developed. We incorporate backstepping technique into neural network based adaptive control design framework in a different way. During this process, integral action of tracking error takes effect and the Lyapunov function with integral term contained in it can still converge to zero exponentially. Two simulation examples indicate that our improved scheme can reduce the design complexity of neural network controller, and improve the precision of control. System stability and better asymptotic tracking are shown to be maintained in the improved scheme. YA YANG was born in Shandong, China, in 1992. He received the B.S. degree in reliability engineering from Harbin Engineering University, where he is currently pursuing the Ph.D. degree in mechanics with the College of Aerospace and Civil Engineering. His research interests include aerospace dynamics and control, and aeroelasticity of vehicles. VOLUME 8, 2020