Research Article | Open Access

Wei-Dong Zhou, Cheng-Yi Liao, Lan Zheng, "Adaptive Backstepping Control for a Class of Uncertain Nonaffine Nonlinear Time-Varying Delay Systems with Unknown Dead-Zone Nonlinearity", *Abstract and Applied Analysis*, vol. 2014, Article ID 758176, 14 pages, 2014. https://doi.org/10.1155/2014/758176

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

**Academic Editor:**Gani Stamov

#### Abstract

An adaptive backstepping controller is constructed for a class of nonaffine nonlinear time-varying delay systems in strict feedback form with unknown dead zone and unknown control directions. To simplify controller design, nonaffine system is first transformed into an affine system by using mean value theorem and the unknown nonsymmetric dead-zone nonlinearity is treated as a combination of a linear term and a bounded disturbance-like term. Owing to the universal approximation property, fuzzy logic systems (FLSs) are employed to approximate the uncertain nonlinear part in controller design process. By introducing Nussbaum-type function, the a priori knowledge of the control gains signs is not required. By constructing appropriate Lyapunov-Krasovskii functionals, the effect of time-varying delay is compensated. Theoretically, it is proved that this scheme can guarantee that all signals in closed-loop system are semiglobally uniformly ultimately bounded (SUUB) and the tracking error converges to a small neighbourhood of the origin. Finally, the simulation results validate the effectiveness of the proposed scheme.

#### 1. 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 [2–4], 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 [8–27]. 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 performance and indeed make the system unstable. Many controller design schemes are developed for systems with unknown dead zone [2, 3, 15–17, 28–36]. 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, 28–31]. 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 dead-zone 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 dead-zone 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, 17–25, 37–39]. In [17–19, 22–24], 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 time-delay 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. 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]. 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. 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 closed-loop system are bounded and the tracking error is asymptotic convergence. Finally, effectiveness of the developed scheme is demonstrated by the simulation examples.

#### 2. Problem Formulation and Preliminaries

Consider the following SISO nonaffine nonlinear time-varying delay system: where and are the system control input and output, respectively, and are system states, , , and are unknown smooth functions, and . is an unknown smooth function with the unknown time-varying delay terms . There exists a positive constant satisfying , and , is an unknown continuous bounded initial function. denotes the unknown external disturbance, . is the unknown dead-zone input.

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 closed-loop system are bounded.

Utilizing the mean value theorem [40], function in (1) can be rewritten as with and ; .

*Assumption 1. *There exist constants and satisfying .

The nonsymmetric dead-zone input is defined as [17] where and are unknown right and left slopes of the dead zone and and are breakpoints of the dead zone. To deal with dead-zone nonlinearity, the following assumptions are put forward.

*Assumption 2. * and are unknown bounded constants. and are unknown functions and there are unknown positive constants , , , and satisfying

Define vectors and as with and .

Based on the above analysis, the dead zone can be expressed as where

According to Assumption 2, it can be concluded that and is an unknown positive constant meeting . Considering the definition of and , we have

It is easy to obtain that ; that is, is a positive discontinuous bounded function.

Using (2) and (6), system (1) can be transformed into the following form:

*Notation. *, , , , , , and denote , , , , , , and , respectively.

*Assumption 3. *The unknown smooth functions satisfy the inequality
where are unknown positive smooth functions.

*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.

*Assumption 5. *The time derivatives of the time-varying delay terms are and satisfy where is an unknown positive constant.

*Assumption 6. *The external disturbances satisfy , where is defined as an unknown positive constant.

*Remark 7. *The constants , , , , and are only required for analytical purposes and their values are not necessarily known in control laws and adaptive laws.

Before we derive our results, the FLSs and Nussbaum-type 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 and are the FLS inputs and output, respectively. , , 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 center-average defuzzifier as follows [41]: where is the point at which the membership function achieves its maximum value and we assume that . Let be a vector grouping all consequent parameters and , 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 . Then according to the universal approximation theorem, any continuous nonlinear function can be approximated by the FLS as where is an optimal parameter satisfying and is the minimum approximation error satisfying ( 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 , and , are commonly used as Nussbaum functions for nonlinear systems with unknown control direction. In this paper, the Nussbaum function is employed.

Lemma 8 (see [17]). * and are smooth functions with . If the inequality holds with , is a suitable constant, , is a positive constant, and is a time-varying parameter which takes values in unknown closed intervals and , then , , and must be bounded on .*

#### 3. 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 is a virtual control which should be designed for the corresponding ()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 are used to estimate with estimation errors defined as .

The detailed design procedure is described in the following steps.

*Step 1. *Consider the Lyapunov-Krasovskii function as
where , and are design positive parameters.

Giving a compact set as with , a positive design parameter, then a function defined as follows will be employed to design controller

We choose the virtual control law and adaptive laws as where and are design positive parameters.

*Case 1.* In this case, we have . Apparently, the tracking error is bounded. According to (18)–(20), we can conclude that when we select bounded initial values, is bounded and ; that is, is bounded. Integrating (21) over we get that signal is bounded.

*Case 2.* When , the following process is needed.

The time derivative of is with .

When Assumption 3 holds, we get

By using Young’s inequality and combining with (23), (22) yields where and are positive constants

*Remark 9. *Note that the function () is not defined at which leads to the fact that it cannot be approximated by FLSs. To cope with this difficulty, we introduce function instead of according to the effective approach in [19]. The design parameter can be adjusted to achieve better performance.

As consists of unknown nonlinear functions , , and , cannot be directly used to construct controller. According to the universal approximation property of FLSs, can be rewritten as where stands for approximation error and is an unknown constant, .

Considering the inequality we obtain

As , we have

Substituting (27) and (28) into (24) results in

Considering and using (19)–(21), (29) yields with and .

Multiplying (30) by and then integrating over , we get where .

Noting that there is an extra term within (31), we suppose that can be regulated as bounded; then we have

As the boundedness of the extra term is obtained from (32), directly applying Lemma 8, we get the conclusion that signals , , and hence , , , and are all bounded on . Consequently, if is bounded, we can get the conclusion that all signals in Step 1 are bounded. In addition, the boundedness of will be proved in step 2 (see Step* k*).

*Step k (). *Considering that steps have a similar procedure, Step* k* is presented as follows.

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 stands for a compact set and is a design positive parameter which decides the size of convergence region.

*Case 1.* In this case, we suppose that . It is obvious that the tracking error is bounded. From (35)–(37), when selecting bounded initial values, we achieve the boundedness of , , and . After integrating (37) over , we conclude that signal is bounded.

*Case 2.* In case 2, the tracking error satisfies .

The time derivative of is where .

Then the time derivative of is with

*Remark 10. *Here is the function of . When , we have ; that is, is the function of , , , and , where and . Then we derive the time derivative of in (41).

Using (40) and (41), the derivative of is

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.

As the fuzzy basis function satisfies , we get the following inequality:

Substituting (48) into (45), we obtain

As , we can derive

Applying (35)–(37) and (50), (49) produces where and .

Multiplying (51) by results in

Integrating (52) over [0,* t*], we obtain
with .

*Remark 11. *The discussion of (53) is similar to the analysis of (31). If can be regulated as bounded, by utilizing Lemma 8, the boundedness of signals , , and is achieved. Thus, we can guarantee that signals , , , and are all bounded on . The effect of the extra term 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 denoting a compact set and is a design positive parameter.

Similarly, we analyze the th-subsystem from two cases.

*Case 1.* In Case 1, satisfies . As is a positive design parameter; we obtain that is bounded. In addition, we can conclude that the signals , , , , and are bounded.

*Case 2.* We suppose that in this case.

The time derivative of is

From the definition of , we obtain

Applying Young’s inequality and (43), (60) can be rewritten as

The time derivative of is where .

By utilizing (63), (62) yields with

By using FLSs, function can be approximated as where with and , expresses the approximation error, and is an unknown positive constant.

Similarly, we can derive the following inequality:

Considering , we get

Applying (56)–(58), (67), and (68), (64) results in where and .

Multiplying (69) by and then integrating it over , we have with .

Considering function satisfies , and Assumption 2 holds, we can derive . Noting (70), applying Lemma 8, we can conclude that signals , , and are bounded. Hence, , , , , and are SUUB on .

The main result is summarized in the following theorem.

Theorem 12. *Consider nonaffine nonlinear time-varying delay system (1), when Assumptions 1–6 hold, by applying the control law (56), virtual control laws (19), and (35) and adaptive laws (20), (21), (36), (37), (57), and (58); then with bounded initial conditions, it is guaranteed that all the signals in closed-loop system are SUUB and the tracking error eventually converges to a small neighbourhood of the origin.*

*Proof. *Owing to the previous analysis, we get the conclusion that the term is bounded.

Noting (70), we suppose that the upper bound satisfies

From (54), (70), and (71), we have

Thus, we can conclude the boundedness of the signals , , and .

In the rest of the steps from to 1, we acquire