Finite-time adaptive neural dynamic surface control for non-linear systems with unknown dead zone

This paper proposes a new adaptive neural network ﬁnite-time dynamic surface control scheme for inaccurate non-linear systems subjected to unknown dead zones. The ‘explosion of differentiation’ is eliminated by the ﬁrst-order ﬁlter in backstepping design. The parameter normalization scheme updates the coefﬁcients of activation functions in the radial basis function neural networks. The dead zone inverse method estimates the dead zone parameters. It is proved that the proposed controller achieves the faster tracking performance and the boundedness of all signals in ﬁnite time. The contribution of this paper is that the presented controller not only has the advantages of transient performance and program execution time in comparison with traditional backstepping methods, but also its computational cost is lower than command ﬁltered methods. Simulation experiment results are included to illustrate the effectiveness of the proposed scheme.


INTRODUCTION
At present, fuzzy logic systems (FLSs) [1,2] and neural networks [3][4][5] are regarded as two kinds of intelligent approximators with different structures, because they exist excellent characteristics which can approximate the unknown non-linear smooth functions with great accuracy [6,7]. For example, an intelligent fuzzy approximator was presented for a class of singleinput-single-output (SISO) strict-feedback systems with a triangular structure to solve the singularity problem of the control boundary [1,8]. Liu et al. [9] utilized universal approximators to approximate the differential functions and greatly simplified the derivation process of the control law for a class of non-affine systems. Radial basis function neural networks (RBFNNs) were employed to model the unavailable smooth functions in nonlinear quantized systems [10]. In conclusion, the main role of an intelligent approximator in a control system is to approximate the unknown dynamic function [11] or estimate the uncertain parameters in real time [12]. This paper will employ RBFNNs to approximate uncertain or unavailable non-linear functions during the design procedure of controller.
Dead zones may be an unknown and unmeasurable physical phenomenon in practical industrial environments including hydraulic servo control [13], flexible joint industrial robots [14] and pneumatic servo driving systems [15]. Therefore, it This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Control Theory & Applications published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology is of great significance to improve the robustness and antiinterference ability of the control system when considering the influence of the dead zone in the controller design process. Shahriari and Rahmani [16] established a type-2 network system to identify the key parameters (the left and right breakpoint positions) of dead zone models. In addition to using neural networks to deal with dead zone phenomenon, there also exist other effective methods. Tao and Kokotovic [17] proposed a dead zone inverse to model the dead zone phenomenon for the first time. Then, the vibration problem of an actual control signal was tackled by introducing a deformed Sigmoid function to smooth the dead zone inverse [18]. Furthermore, time-varying integral functions [19], indirect adaptive updating parameters [20] and other schemes which are utilized to overcome the dead zones are not listed here. It can be concluded that combining different control algorithms with the dead zone model can improve the response performance of the control system and suppress the vibration of the tracking trajectory under the guidance of the asymptotic Lyapunov stability theory. On the basis of the proposed theories, above-mentioned research results [8,[13][14][15][16][17][18][19][20] cannot quantitatively estimate when the control system can enter the desired steady state [21].
In order to satisfy the requirements of real-time control, some specific industrial occasions must limit the total time of transient response [22]. Therefore, a finite-time stability theory which defines the specific solution formula for the time required to enter the steady state came into being [23]. Lee et al. [24] combined finite-time theory with a sliding mode technique to establish the precise control of piezoelectric actuators. In [25], state observers and FLSs have been utilized to design a finitetime controller for multiple-input-multiple-output (MIMO) systems as state variables of the control system are not measurable. The backstepping technology, as a popular control method, also has wide application [26][27][28][29][30][31][32][33]. For example, the backstepping technique was used to control linear systems or time-varying strict feedback systems. The shortcoming of the backstepping approach is 'explosion of differentiation' due to the repeated differentiations of immediate control functions, which has received considerable attention. To overcome the above drawback, Wang et al. [34] introduced a class of FLSs to approximate continuous functions which contain immediate control signals, so a finite-time adaptive fuzzy backstepping controller was designed for non-linear systems. Then, the backstepping controller was further successfully extended to non-affine systems with dead zones [35], which expands its application field. There also exist other methods to deal with the 'explosion of differentiation' in the process of backstepping design, for example, the first-order filters replace the virtual control signals; therefore the dynamic surface control (DSC) scheme appeared [36][37][38][39][40][41][42][43]. Ling et al. [44] normalized the adaptive parameters by using the maximum value of norm such that the running efficiency of the standard DSC algorithm is improved for flexible manipulators. Some command filtered control technology mainly utilizes Levant differentiators to approximate and compensate the intermediate control signals [45][46][47][48][49][50], which can also deal with the above 'explosion of differentiation' problem. Yu et al. [51] developed a finite-time command filtered method (the control scheme of [51] is abbreviated FTCF) to achieve the high-precision tracking of reference trajectory in finite time. In summary, finite-time controller design for different dynamic systems mainly combines finite-time stability theory with sliding mode, state observer or backstepping techniques. However, within our knowledge, there is not a finitetime DSC which has been applied to non-linear systems with unknown dead zones in the previous references.
Based on the improved DSC algorithm of [44], we turn to the finite-time tracking control issues for a class of inaccurate nonlinear systems with unknown dead zones. First, the RBFNNs model the unknown functions of the dynamic system. Then the first-order filter is employed to overcome the 'explosion of differentiation'. Next, the key parameters of the dead zone are estimated by the dead zone inverse approach [17]. Finally, a new finite-time dynamic surface controller is constructed in this paper. Compared with the existing research results, our work has the following novelties: 1. The finite-time stability theory and the smooth inverse dead zone model are introduced into the existing control algorithm [44] such that a new finite-time DSC scheme that considers the dead zone is proposed. Compared with traditional backstepping methods, this proposed control scheme has the advantages of lower computational burden and faster transient response. 2. The first derivative of a reference signal is only needed in the process of control signal design; therefore, the described algorithm has obvious strengths when the reference signal exists no high-order derivative. 3. The first-order filter accurately approximates the derivative of the intermediate virtual control signal so that the 'differential explosion' problem in backstepping design is solved. 4. In the control design process, since the first-order filter and the adaptive parameter normalization scheme are introduced such that the number of differential equations greatly reduced. Thus compared with FTCF [51], the proposed algorithm significantly degrades the computational burden and accelerates the efficiency of online tuning parameters.
Note that the organization of this paper is constructed as follows. Section 2 represents the problem formulation and preliminaries. The detailed design and stability proof of the finite-time dynamic surface control scheme (the proposed control scheme is abbreviated as FTDSC) considering dead zones are given in Section 3. Simulation comparison examples are presented to verify the effectiveness of the FTDSC in Section 4. Finally, the conclusions are given in Section 5.

Description of the system with unknown dead zone
In the subsequent derivation, let estimation error( ⋅) = (⋅) −( ⋅), where( ⋅) is the estimation value, (⋅) denotes the true value.
Throughout the brief, ‖ ‖= is a positive real number, ℝ n is the real space and its superscript n represents dimension. Consider a class of n order non-linear systems whose general model is given as following form: is the state vector, y denotes the measured output signal, u T is the actual control signal (the output of dead zone). Due to the inaccuracy of physical system models, F i (x i ) and F n (x n ) are difficult to obtain, so we only assume that they are bounded unknown continuous differentiable functions. Notice that P i (x i ), P n (x n ) are continuously bounded and often used in strict-feedback systems [51], for example. The dead zone model is described by the following equation [18]: where u C is plant signal (the input of dead zone), and D(u C ) is the dead zone function. d r > 0, d l > 0, d − < 0, d + > 0 denote right slope, left slope, right breakpoint and left breakpoint of the dead zone, respectively, which are assumed unknown. Symbol D −1 (u C ) represents the inverse function of D(u C ); then one will obtain with where S r (t ) and S l (t ) are membership functions of the true dead zone model. Now, the parameter vector T is defined to estimate the key parameters of the dead zone model, S r (t ) and S l (t ) belong to the elements of S S S that will cause chattering of the system output [18]; therefore two kinds of sigmoid functions are employed to replace S r (t ) and S l (t ) respectively, and thenŜ SŜ is introduced to replace S S S . Note that increasing the approximation betweenŜ SŜ and S S S , which can be achieved by reducing the design parameter e .
The real-time estimation functionû T (t ) can be expressed aŝ where (û T (t ) and̂̂̂T d will be designed in Section 3) Based on the definitions of (6) and (7), (3) can be rewritten as According to the designedû T (t ) and̂̂̂T d , u C can be obtained. Now, subtract (6) from (5) to get the estimation errorũ T (t ) = T d S S S −̂̂̂T dŜ SŜ , and thenũ T (t ) can be further described as Then, by means of deforming (9), u T (t ) can be expressed as Substituting (10) into (1), the n order system is further described as The control issue of this paper is to establish an actual controllerû T (t ) for system (1) and guarantee the output signal y tracking the reference signal y r in a finite period of time.

Lemmas and assumptions
To cope with the design and stability analysis of controller, the following Lemmas and Assumptions should be included.
Lemma 1 see [52]. Assume that there exists a smooth positive definite quadratic function L(x) for a dynamic system, whose time derivative can be expressed asL If three constants satisfy 0 > 0, 0 < a < 1, c 0 > 0, then the system is semi-global practical finite-time stable (SGPFS), and the finite time T R of which the output signal enters the steady state can be described as with 1 ∈ (0, 1).
Assumption 2 see [36]. In the dead zone model, the specific values of the breakpoint and slope parameters are unknown. However, they should not only satisfy d − < 0, d + > 0, but also the slope of the dead zone at the breakpoint has the same sign.
Assumption 3 see [55]. In order to ensure the continuity and boundedness of the intermediate and final control signals, one assumes that 0 < |P i | < P max , P max ∈ ℝ + , i = 1, 2, … , n, and the sign of P i is not allowed to change.

Radial basis function neural networks
In this research, we choose RBFNNs as a class of intelligent approximators [52], the output function y out (x, W ) can be gained as where E (x) m is the Gaussian activation function and bounded input vector, weight vector of the RBFNN, respectively. c ∈ ℝ K is the centre position vector, b denotes the Gaussian width. According to the universal approximation theorem [56], the ideal output y out (x,W ) satisfies wherēis an arbitrarily small positive real number,W ∈ ℝ K × ℝ K denotes the optimal approximation weight matrix. An adaptive constant parameter will be defined to solve the optimal weight vector of n RBFNNs:

CONTROLLER DESIGN AND ANALYSIS
In this section, the design process of the FTDSC controller for the n order non-linear system (1) with unknown dead zone are divided into three steps. The proof of SGPFS for this system is given after the above steps are completed.
Define a general formula for the first-order filter: where Q j , j denote the output of first-order filter and the filter constant, respectively. j −1 represents the intermediate control signal that needs to be designed. Then, the transformations of coordinates can be described as follows: where e 1 , e j are the errors of state variable, and e j denotes the filter error.
Remark 1. The derivative of intermediate control signal j −1 is approximated by the output Q j , j = 2, 3, … , n of the firstorder filter to deal with the 'explosion of differentiation' problem in the backstepping design. In contrast, above signals are modelled by Levant differentiators in [51]. Compared with the FTCF, the FTDSC diminishes significantly the number of differential equations that need to be solved; therefore it has the advantage of low time complexity.
To ensure that the closed-loop control system converges in finite time, we introduce the following control signal i for i = 1, 2, … , n and adaptive functions: where adjustable parameters i ∈ ℝ + , i ∈ ℝ + , 1 ∈ ℝ + , ∈ ℝ + ,̄∈ ℝ + ,̄∈ ℝ + , and the input vectors of RBFNNs are It can be inferred from (21) that e 2 −1 i will be infinite as 2 − 1 < 0, in order to avoid singularity [35], we assume that ∈ (0.5, 1). Notice that sometimes let E i (Z i ) = E i , for the simplicity of presentation.
Remark 2. On the one hand, the robust compensator̂̂̂d is employed to deal with the dead zone, which improves the disturbance-rejection ability of controller, on the other hand, the control signal i whose design procedure is based on the framework of the SGPFS, therefore the transient performance of the FTDSC is better than traditional backstepping schemes.
Step 1 (i = 1): Taking the derivative of e 1 = x 1 − y r , and according to the transformation of coordinates (20), one haṡ Substituting (21) into (23), and multiplying by e 1 , one can obtain The input function of the first RBFNN is selected as Note that the input functions of the RBFNN of ith and nth subsystem areF It can be inferred from (17) that there is an infinitesimal con-stant̄1 that satisfies the following equilibrium: where 1 (Z 1 ) which is simplified as 1 represents the approximation error of first RBFNN. Now, substituting (27) into (26), e 1̇e1 can be expressed as To facilitate controller design, (28) is introduced a tuning parameter 1 , so e 1̇e1 can be rewritten as Based on the Young's inequality and (18), the following inequalities hold: Substituting (30) into (29), one can get According to (19) and (20), we havė The designed intermediate control signal 1 is a bounded function; thus −̇1 is also a bounded function, namely, there is a positive number Θ 1 which satisfies the inequality | −̇1| ≤ Θ 1 . Substituting this inequality into (32); then multiplying both sides by e 2 , and according to the Young's inequality, the following inequality holds: Choose a positive definite Lyapunov function L 1 = 1 2 (e 2 1 + e 2 2 ), and its derivative follows: Substituting (31) and (33) into (34) results iṅ Step 2 (i = 2, … , n − 1): With the help of (20) and (21), the tracking error of the ith order subsystem can be described as Similar to the derivation process of (27)-(35), we havė Step 3 (i = n): Taking the time derivative of e n = x n − Q n ; then substitutingẋ n of (11) into above equation, we geṫe n =ẋ n −Q n = F n + P n (û The n is viewed asû T (t ), then we can obtain the error dynamic of the nth-order subsystem: Similar to the derivation procedure from (27) Introduce a smooth Lyapunov function as follows: where is a 4 × 4 positive definite diagonal matrix, −1 represents the inverse of . The derivative of (41) iṡ wherẽ= -̂. It can be inferred from (18) that is a constant value and its derivative is 0, so the following equation holds:̇̃=̃(̇−̇̂) Similarly,̃̃̃d = d −̂̂̂d , d is a key constant parameter vector of the dead zone model, and its derivative is also 0; thus̃̃T Substituting (40), (43) and (44)  . (45) Now, define the final smooth quadratic Lyapunov function for entire non-linear control system as follows: 2̄. (46) The derivative of L all with respect to time can be described aṡ . (47) The summation of (45), (37) and (35) into (47) produceṡ Substituting (22) into (48) results iṅ Remark 3. We choose only one adaptive law to update the coefficients of activation functions in RBFNNs. Compared with the traditional backstepping algorithm [32], the computational burden of FTDSC is less than the backstepping scheme, based on the parameter normalization algorithm (18). It can be inferred from Lemma 3 that the following inequalities are satisfied (the detailed proof is shown in the Appendix.):1 where q w = e ( ln )∕(1− ) ∈ ℝ + , substituting (50) into (49) yields Let min( , j = 2, 3, … , n and 0 = min{ ,̄}, where = min{2 1 , … , 2 n }. According to the Lemma 2, the following inequalities can be obtained: with According to above-mentioned derivation process, 0.5 < < 1, c 0 > 0, 0 ≥ 0, and L all is a positive definite function, which satisfies the four conditions of the proposed Lemma 1, in addition, if the (1) also satisfies Assumptions 1-3; then all signals will be SGPFS, i.e. e 1 , e 2 , … , e n , e 2 , e 3 , … , e n , and d are semi-global practical finite-time bounded. Furthermore, we can define the finite time T R that the control system enters the steady state Furthermore, the tracking error of the dynamic system satisfies with ∈ (0, 1). For ∀t ≥ T R , under the virtual control signal 1 , 2 … n−1 , actual control signal n , adaptive updating laŵ and dead zone compensator̂̂̂d , and then the tracking error e 1 approaches an infinitesimal domain of the origin. It can be inferred from the Assumption 1 that the reference trajectory y r is a bounded signal. Because e 1 = x 1 − y r , the boundedness of state x 1 is proven. The dynamic model (1) implies that P 1 (x 1 ) is a bounded continuous function. Since 1 is a function about P 1 (x 1 ) and e 1 , the 1 is a bounded virtual control law. It can be observed from e 2 = Q 2 − 1 that Q 2 is also a bounded signal. Because e 2 = x 2 − Q 2 , the boundedness of state x 2 is proven.
Following the similar analysis process, the boundedness of all the signals in the closed-loop systems will be proven.

SIMULATION EXAMPLS
In order to show that the described algorithm can overcome the dead zone phenomenon and exhibit excellent tracking performances, we assume that the plant signal u C passes the following dead zone model: To reduce the number of parameters that needs to be debugged in the simulation [18], the breakpoint parameters d 2 and d 4 of the dead zone model are only estimated. Furthermore, other constant parameters of dead zone are chosen as: (12,12),̄= 20,̄= 0.5. Simulation 1. Consider a simulation example of a third-order electromechanical system as follows: Comparing the state equations of the electromechanical system with the general formula of non-linear system (1), we can conclude: P 1 =1, P 3 =1∕15 , P 2 =1∕0.0642 , F 1 =0, The correlation coefficient of each variable in (57) refers to the simulation example given in [57], but the difference is that we will use RBFNNs to tackle with the F i (x x x i ), i = 1, 2, 3. The initial conditions are defined as x 1 (0) = 0.5, x 2 (0) = 0.5, Taking approximation accuracy and program running efficiency into account, the centre position vector of the Gaussian function is designed as c = [−3.5, −2.5, −1.5, −0.5, 0.5, 1.5, 2.5, 3.5, 4.5] , and the Gaussian width b m = 3, m = 1, 2, … , 9 for every RBFNN [35]. Notice that above RBFNNs will be used in subsequent backstepping scheme.
Remark 4. It can be known from the definitions of (54) and (55) that there exists a serious coupling relationship between adjustable parameters ( , i , i ) and integrated system performances (T R and e 1 ) such that the specific values of above parameters can be comprehensively weighed by simulations. On the one hand, the tracking error e 1 is mainly impacted by i , i = 1, 2, … , n, therefore these design parameters can be determined first. On the other hand, the design parameter affects the time T R for the controller to enter a steady state. Specifically, when a larger is selected, the controller can enter a steady state more quickly. Therefore, on the basis of ensuring the convergence of the tracking error, we can fine-tune to make the controller exhibit more excellent transient performance.
Simulation 2. The traditional backstepping algorithm [32] is applied to the same actual electromechanical system (57), we can obtain the following control laws and adaptive parameter update functions (in order to save space, we have omitted the specific design process of backstepping controller here.): According to backstepping algorithm, the input vectors of all RBFNNs can be, respectively, defined as Z 1 = [x 1 , y r ,̇y r ] T , , it contains the derivative information of the reference signal, and its derivative order is the same as the system order. It is worth mentioning that the proposed FTDSC only needs the first derivative of the reference signal, which may be more suitable in real applications.
The control parameters are chosen as: 1 = 20, 2 = 3 = 35, 1 = 2 = 3 = 15, 1 = 2 = 3 = 20, 1 = 2 = 3 = 0.2. The initial values of adaptive parameters are chosen as: Remark 5. It should be pointed out that the backstepping approach requires three adaptive parameters. However, the presented controller needs only an adaptive parameter, which has the advantage of the lower computational burden. The above backstepping approach does not use finite-time theory such that its response speed is slower than FTDSC, which can be verified by Figure 2. Figure 1 shows that both the backstepping approach and FTDSC have fascinating tracking performance. The tracking error curves of the two algorithms are displayed in Figure 2. The tracking error of FTDSC has stopped oscillating at time t = 0.5086 s, namely, has entered a steady state, while the tracking error of the backstepping approach is still unstable at this moment, and the tracking error stop oscillating as t = 1.06 s. If above time which the tracking error stop oscillating is regarded as a transient performance index, then, the transient tracking performance of developed controller increase 200% in comparison with the traditional backstepping approach. Figures 3-6 show the time histories of other state variables, adaptive laws, plant signal and actual control signal. It can be inferred that all curves are bounded and approach a neighbourhood of the origin in finite time, which further validates the correctness of the stability analysis process in Section 3. Figure 5 implies that the output amplitude of the proposed controller at the initial moment is reduced by 3.21% in comparison with the control output of backstepping method, which is more suitable in industrial application. It is worth pointing out that the dead  zone results in the roughness (inflection point) of the curve, and its detailed description is shown in Figure 6.
Based on Remark 1, the first-order filter in FTDSC only needs one differential equation to describe, while a Levant differentiator in FTCF needs two differential equations to describe; therefore, we can further infer that for the n-order systems the number of differential equations involved in FTDSC  is reduced by (n − 1)∕2 , compared with FTCF. It is worth noting that in order to save space, the design process of the FTCF is detailed in [51] and will not be repeated here. To ensure the single variable principle, the parameters of FTCF are debugged, and the FTCF has the same tracking performance as the FTDSC. Under the above conditions, a simulation of time complexity is performed to prove the correctness of the conclusion mentioned in Remark 1.
Simulation 3. Set the same total sampling time in the MAT-LAB2016a software environment. Then run programs corresponding to two different algorithms and record the average running time. Next replace the total sampling time and repeat the above experimental steps. Finally , the simulation results are shown in Table 1.
Hardware configuration of the computer on which MAT-LAB2016a depends: operating system: Windows 8.1; CPU: Intel Core i5-6500@3.2 GHz; RAM: 8 GB. To avoid relative errors, the data in Table 1 are obtained by recording the results of five replicate experiments and calculating their average values under the same simulation environment.
Remark 6. It can be inferred from the data in Table 1 that the average solution time of FTDSC is reduced by 47% with respect to FTCF, and above-mentioned strengths are more obvious, as the total sampling time increase.

CONCLUSION
This paper proposes a new adaptive neural network finite-time DSC approach to solve the tracking problem for uncertain nonlinear systems with unknown dead zones. The first-order filters are employed to approximate the derivatives of virtual control signals, the RBFNNs model the uncertain system functions, and the unknown dead zone is replaced by the adaptive dead zone inverse function. Simulations 1 and 2 imply that the transient tracking performance of the developed controller increases 200% in comparison with the traditional backstepping approach. The results of Simulation 3 show that the average solution time of FTDSC reduces 47%, compared with FTCF. It is worth noting that the proposed FTDSC needs to utilize some knowledge of dynamics systems; therefore future work will mainly account for how to extend FTDSC to a class of totally unknown non-affine non-linear systems. .

ACKNOWLEDGEMENT
(A.1) The auxiliary terms is introduced in the right of above inequality, we have:̃1