Adaptive Backstepping Control for a Class of Uncertain Nonaffine Nonlinear Time-Varying Delay Systems with Unknown Dead-Zone Nonlinearity

and Applied Analysis 3 the unknown external disturbance, i = 1, . . . , n. D(u(t)) is the unknown dead-zone input. The control objective is to design an adaptive backstepping controller for system (1) such that the system output y(t) tracks the desired trajectory y d (t) and all signals in closedloop system are bounded. Utilizing the mean value theorem [40], function f(⋅) in (1) can be rewritten as f (x (t) , D (u (t))) = f (x (t) , 0) + ∂f (x (t) , D (u (t))) ∂D(u(t)) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨D(u)=λ D (u (t)) = f n (x (t)) + gn (x (t)) D (u (t)) , (2) with f n (x(t)) = f(x(t), 0) and g n (x(t)) = ∂f(x(t), D(u(t)))/∂D(u(t))| D(u(t))=λ ; λ ∈ (0, D(u)). Assumption 1. There exist constants g and g satisfying 0 < g ≤ |g i (x i (t))| ≤ g. The nonsymmetric dead-zone input is defined as [17] D (u (t)) = {{ {{ { φ r (t) (u (t) − mr) , u (t) > mr, 0, −m l ≤ u (t) ≤ mr, φ l (t) (u (t) + ml) , u (t) < −ml, (3) where φ r (t) and φ l (t) are unknown right and left slopes of the dead zone andm r andm l are breakpoints of the dead zone. To deal with dead-zone nonlinearity, the following assumptions are put forward. Assumption 2. m r and m l are unknown bounded constants. φ r (t) and φ l (t) are unknown functions and there are unknown positive constants φ r0 , φ r1 , φ l0 , and φ l1 satisfying 0 < φ r0 ≤ φ r (t) ≤ φr1, 0 < φ l0 ≤ φ l (t) ≤ φl1. (4) Define vectors η(t) and κ(t) as η (t) = [ηr (t) , ηl (t)] T , κ (t) = [φr (t) , φl (t)] T , (5) with η r (t) = { 1, u(t)≥−ml 0, u(t)<−ml and η l (t) = { 1, u(t)≤mr 0, u(t)>mr . Based on the above analysis, the dead zone can be expressed as D(u) = η T (t) κ (t) u (t) + d (u (t)) , (6) where


Introduction
In the past decade, adaptive backstepping design technique has received a great deal of attention since it was pioneered by Kanellakopoulos et al. in 1991 [1].In [2][3][4], adaptive backstepping is utilized to construct robust adaptive backstepping controller.The main feature of this approach is that it can handle nonlinear systems without satisfying the matching conditions, but the backstepping design procedure has a shortcoming named explosion of complexity because of the repeated differentiations of virtual controllers.By using dynamic surface control technique, the explosion of complexity shortcoming is overcome [5].References [6,7] develop a command filtered backstepping approach which is feasible even when the number of iterations of the backstepping method is large.However, it should be noted that the nonlinear functions are all assumed to be known in the abovementioned methods.Recently, many adaptive backstepping controllers with FLSs or neural networks (NNs) have been developed for nonlinear systems in strict feedback form [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Owing to the universal approximation property of FLSs or NNs, these control approaches do not require the precise knowledge of system nonlinearities.Nevertheless, the introduced FLSs or NNs may lead to a burdensome computation when the number of the parameters which need to be tuned by online learning laws increases significantly.
To handle the inevitable weakness meeting when increasing the number of fuzzy rules or neural network nodes, the optimal weighting vector in FLSs is used as the estimation parameter [8,9].In [10,14,19,21,25,27], FLSs are utilized to directly approximate the desired control signals instead of the unknown nonlinearities in each backstepping design step.Consequently, the number of parameters needed to be adapted is significantly reduced for only one parameter needed to be estimated online no matter how many fuzzy rules are selected.On the basis of the work in [10], a novel adaptive fuzzy backstepping controller construct method without requirement of the fuzzy basis functions is exploited [22,23].
Dead-zone characteristic is one of the most common actuator nonsmooth nonlinearities encountered in many industrial processes, which can seriously affect the system 2 Abstract and Applied Analysis performance and indeed make the system unstable.Many controller design schemes are developed for systems with unknown dead zone [2,3,[15][16][17][28][29][30][31][32][33][34][35][36].Generally, the dead zone is first treated as a combination of a linear and a bounded disturbance-like term, and then the controller that can achieve a good control performance is designed by adopting robust control technique [16,17,[28][29][30][31].In [32], a novel two-layered fuzzy logic controller which consists of a fuzzy logic-based precompensator and a usual fuzzy PD controller are developed for controlling systems with dead zone.In [33,34], by introducing a fuzzy logic deadzone compensator two fuzzy controllers are constructed for motion control system and a DC motor system, respectively.Nevertheless, when there are no suitable rules for the deadzone nonlinearity, this method may be unfeasible for it depends much on operators or experts experience.In [2,3,15,35,36], the inverse function of dead zone is utilized to compensate the effect of the dead zone.Using this method, an effective control has been achieved, but the shortcoming that the dead-zone parameters are required to be constants is inevitable.Regrettably, although much progress has been made in the fields of controller design for nonlinear systems with unknown dead zone, nonaffine nonlinear systems with unknown dead zone are seldomly investigated.
Time delays frequently occur in practical control systems, such as electrical networks and hydraulic systems.Considering that the existing time delays often cause system instability and performance deterioration, to handle the control problem for systems with time delays is an unavoidable issue.Two main tools Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions are usually applied to nonlinear time-delay systems [4,[17][18][19][20][21][22][23][24][25][37][38][39].In [17][18][19][22][23][24], Lyapunov-Krasovskii functionals are constructed to compensate the unknown time delays.Within these schemes, the condition that the unknown time delays are assumed to be unknown constants is too strict.To solve time-varying delays problem, a novel Lyapunov-Krasovskii functionals are designed on condition that the derivative of time delay functions is less than one [20,25,30,37].In [4,21,38], Lyapunov-Razumikhin lemma-based adaptive backstepping control approaches are proposed for nonlinear systems in which the limitation condition on the derivative of time delay is cancelled.In [17,24,25], adaptive fuzzy or neural backstepping controllers are designed for a class of nonlinear time-delay systems with unknown control directions.As control direction, that is, the sign of control gain, decide the direction along which the controller parameters are updated, designing adaptive controllers for these unknown systems with the control direction becoming much more difficult.Nussbaum-type function is utilized to deal with the unknown control direction [17,24,25].A robust adaptive NNs controller is first proposed for a class of nonlinear timedelay systems with unknown dead-zone nonlinearity and unknown control direction [17].However, in this method, the time delay is supposed to be unknown constants and the NNs introduced to approximate the uncertain nonlinear term may result in complexity computation when the dimension of system increases.
Inspired by the preceding discussion, in this paper, a class of nonaffine nonlinear time-varying delay systems with both unknown dead-zone input and completely unknown control direction is investigated and an adaptive fuzzy backstepping control scheme is exploited.The main contributions of this paper can be summarized as follows.(1) Few papers consider nonaffine systems with unknown dead-zone nonlinearity.The difficulty of design controller for nonaffine systems is that the control input appears nonlinear in unknown nonlinear systems.Mean value theorem is used to transform the nonaffine form into an affine form, and then the existing approaches for affine systems can be directly applied [40].(2) Similar to [10], FLSs are directly employed to approximate the unknown nonlinearities.Considering the norm of the ideal weighting vector in FLSs as the estimation parameter instead of the elements of weighting vector, there is only one parameter that needs to be estimated online in each step.Meanwhile, it should be noted that in this control approach, the basic functions of FLSs do not occur in the control laws and adaptive laws.This improvement can overcome the explosion of complexity caused by repeated differentiations of virtual controllers and the increase of system dimension.
(3) The other encountered trouble is how to cope with the unknown time delay terms in system.Compared with [17], the time delay term considered in this paper is time varying and the novel Lyapunov-Krasovskii functionals are employed to stability analysis and synthesis.In particular, here, the reason we use Lyapunov-Krasovskii functionals to construct controller is that this method can provide less conservative and delay-independent results.Using the Lyapunov stability theorem, it is proved that the proposed control schemes can guarantee that all the signals in closedloop system are bounded and the tracking error is asymptotic convergence.Finally, effectiveness of the developed scheme is demonstrated by the simulation examples.
The control objective is to design an adaptive backstepping controller for system (1) such that the system output () tracks the desired trajectory   () and all signals in closedloop system are bounded.
Remark 4. Compared with [17] in which the bounding functions are required to be known, it should be emphasized that the nonlinear functions  , (⋅) are unknown in this paper.Before we derive our results, the FLSs and Nussbaumtype function should be introduced.
The FLS has a basic configuration which contains fuzzifier, fuzzy rule base, fuzzy reference engine, and defuzzifier, such four components.The fuzzy rule base is composed of a series If-Then inference rules in the following form [41]: where x = [ 1 ,  2 , . . .,   ]  and  are the FLS inputs and output, respectively.   ,  = 1, . . ., , and   are fuzzy sets characterized by fuzzy membership functions     (  ) and    (), respectively, and  is the number of fuzzy rule.The final output of the fuzzy system can be expressed by using the singleton fuzzifier, product inference engine, and centeraverage defuzzifier as follows [41]: where   is the point at which the membership function    () achieves its maximum value and we assume that . .,   ]  be a vector grouping all consequent parameters and (x) = [ 1 (x), . . .,   (x)]  , where . .,  is the vector of fuzzy basis function.Then, using the conception of fuzzy basis functions [41], the output of the fuzzy logic system can be formulated as (x) =   (x).Then according to the universal approximation theorem, any continuous nonlinear function (x) can be approximated by the FLS as where  *  is an optimal parameter satisfying  *  = arg min   (sup ∈x | f(x) − (x)|) and (x) is the minimum approximation error satisfying |(x)| ≤  0 ( 0 is a positive constant).Nussbaum-type function is successfully applied to cope with the problem caused by unknown control direction [17,24,25,27].A function which has the following properties is called Nussbaum function [42]: Functions, such as  2 cos(),  2 sin() and exp( 2 ) cos((/2)), are commonly used as Nussbaum functions for nonlinear systems with unknown control direction.In this paper, the Nussbaum function () =  2 cos() is employed.

Controller Design and Stability Analysis
In this section, an adaptive fuzzy control scheme is presented by using backstepping technique combined with Lyapunov-Krasovskii functionals and Nussbaum type functions.The backstepping design is based on the following change of coordinates: where  −1 is a virtual control which should be designed for the corresponding ( − 1)th subsystem.In general, the design procedure contains  steps.FLSs are employed to approximate the unknown nonlinear term.Then, let us define unknown constants satisfying For   are unknown and θ are used to estimate   with estimation errors θ defined as θ =   − θ .
The detailed design procedure is described in the following steps.
Step 1.Consider the Lyapunov-Krasovskii function as where ),  1 and  are design positive parameters.

Giving a compact set
with  1 , a positive design parameter, then a function defined as follows will be employed to design controller We choose the virtual control law  1 and adaptive laws as where  1 and  1 are design positive parameters.
Choose the following Lyapunov-Krasovskii function: with   being a design positive parameter, Similar to Step 1, the virtual control law   and adaptive laws are designed as with   and   being design positive parameters; the function   (  ) is defined as where Ω   := {  | |  | < 0.2554  } stands for a compact set and   is a design positive parameter which decides the size of convergence region.
The time derivative of  ,0 is where with Abstract and Applied Analysis 7 Owing to Assumption 3, we get Utilizing Young's inequality (42) yields Substituting ( 39) into (44) results in with Similarly,   can be approximated by FLSs to an arbitrary given accuracy as where represents approximation error, and   is an unknown positive constant.
Remark 11.The discussion of ( 53) is similar to the analysis of (31).If  +1 can be regulated as bounded, by utilizing Lemma 8, the boundedness of signals   (),   (), and ∫  0   (  ) ς   is achieved.Thus, we can guarantee that signals   , (  ), θ , and θ are all bounded on [0,   ).The effect of the extra term  −   ∫  0  2 +1      will be handled in the next step.
Step n.Consider Lyapunov-Krasovskii function as follows: where   is a design positive parameter, and We choose the following actual control input  and adaptive laws: where   and   are design positive parameters.The function   (  ) is defined as with From the definition of  −1 , we obtain Applying Young's inequality and ( 43), ( 60) can be rewritten as The time derivative of  ,0 is Abstract and Applied Analysis 9 where By utilizing (63), (62) yields with By using FLSs, function   can be approximated as where Z  = [x  ,   , ẏ  ,  −1 , θ−1 ]  with  −1 = [ 1 , . . .,  −1 ]  and θ−1 = [ θ1 , . . ., θ−1 ]  ,   (Z  ) expresses the approximation error, and   is an unknown positive constant.
Remark 13.According to the above analysis, we know that tracking error depends on   ,   ,   ,   ,   ,   , ,   , and   .As   ,   ,   , and   are unknown, a concrete estimation of the tracking error is impossible.From inequality (75), it is clear that by reducing   and   , meanwhile increasing   ,   , and , the tracking error will be diminished.Simultaneously, it is worth pointing out that the parameters   ,   ,   , and   are not used in the control law and adaptive laws design, which are employed for stability analysis.
From Figure 1, it can be seen that good tracking performance is achieved.The response curve of state variable is shown in Figure 2. Figure 3 depicts the trajectory of the control input.We can conclude that the control input is bounded.Figures 4 and 5 display the adaptive parameters θ1 , θ2 , and  1 ,  2 , respectively.Example 2. To further demonstrate the feasibility of the controller, we present the following nonlinear system: where  1 () = 0.2(1 + cos()),  2 () = 0.3(1 + sin()), and  3 () = 0.1(0.5 + 0.2 cos()).Similar to Example 1, the control laws and the adaptive laws are chosen as  From Figure 6, it can be concluded that a good tracking performance is obtained.Figures 7 and 8 show the trajectory of state variables  2 and  3 , respectively.Figure 9 depicts the curve of the control input signal.Figures 10 and 11 display the adaptive parameters θ1 , θ2 , θ3 , and  1 ,  2 ,  3 , respectively.
From the simulation results, it is seen that fairly good tracking performances are achieved; meanwhile, all the other signals in closed-loop system are bounded.

Conclusions
In this paper, an adaptive fuzzy backstepping control scheme is presented for a class of nonaffine nonlinear time-delay systems with unknown control direction and unknown deadzone input nonlinearity.By choosing appropriate Lyapunov-Krasovskii functionals, the adaptive fuzzy controller is designed based on backstepping technique and FLSs.The proposed controller guarantees that all the signals in the closed-loop system are bounded and the tracking error eventually converges to a small neighbourhood of the origin.In addition, the number of the parameters which need to be tuned online is significantly reduced.This makes our scheme easily realized in practice.The simulation results illustrate the effectiveness and feasibility of the proposed approach.

Assumption 5 .Assumption 6 .Remark 7 .
The time derivatives of the time-varying delay terms   are τ  and satisfy τ  ≤  * < 1 where  * is an unknown positive constant.The external disturbances   satisfy |  | ≤   , where   is defined as an unknown positive constant.The constants  0 ,  1 ,  0 ,  * , and   are only required for analytical purposes and their values are not necessarily known in control laws and adaptive laws.

Figure 1 : 2 Figure 2 :
Figure 1: Trajectories of system output  and reference signal   .