Adaptive Fuzzy Tracking Control for a Class of Uncertain Nonlinear Time-Delayed Systems with Saturation Constrains

1Software School, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China 2School of Science, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China 3School of Material and Metallurgy, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China 4College of International of Finance and Banking, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China


Introduction
Over the past decades, the modeling and control design problem of nonlinear systems have attracted considerable attentions because of the extensively practical applications.Consequently, a large number of successful control schemes for uncertain nonlinear systems with dead-zone, time delays, and actuator failures have been developed in this area; see [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein.More specifically, Tong and Li [1] studied the adaptive fuzzy tracking control problem for a class of nonlinear systems with dead-zone nonlinearities.In [9], by employing a nonlinear fault estimator, the output fault-tolerant tracking control was developed using the adaptive backstepping technique.Zhou et al. [13] considered the adaptive output tracking control problem for a class of nonlinear systems with stochastic disturbances and time delays.Recently, the authors in [14] proposed the adaptive tracking control approaches for a class of nonlinear timedelayed systems with dead-zone nonlinearities.The global stabilization problem for a class of nonlinear time-delayed systems [15] was considered by using multiswitching-based adaptive neural network control method.Besides, by combining fuzzy approximation and adaptive backstepping technique, a novel robust fault-tolerant control scheme [16] was developed for a class of non-lower-triangular nonlinear systems with actuator failures.
It should be pointed out that all the abovementioned control strategies are merely suitable for the considered nonlinear systems in affine form rather than nonaffine form.It is well known that the nonaffine nonlinear systems represent more general cases which can describe many practical processes.Li and Tong [17] proposed an adaptive fuzzy output control approach for a class of pure-feedback nonlinear systems with dead-zone constrains.In [18], an adaptive fuzzy asymptotic tracking controller was designed for a class of uncertain nonaffine nonlinear systems with dead-zone inputs.Meanwhile, the actuator saturation constrain control implies that the input signals are always bounded for most of practical systems.The saturation problem is very important for the actuator control design, and the performance and stability of the closed-loop systems will be severely effective if the input constrain in the design process is ignored.Thus, the adaptive control design for nonlinear systems with saturation constrains is a challenging topic.Wen et al. [19] studied the adaptive control problem for a class of uncertain nonlinear systems with saturation constrains.Based on the disturbance observer, direct adaptive NNs control strategies in [20] were developed for a class of uncertain nonaffine nonlinear systems with saturation inputs.Additionally, the adaptive fuzzy tracking control scheme for a class of nonaffine nonlinear systems with saturation constrains and stochastic disturbances in [21] was proposed.
Motivated by the above considerations, this paper studies the problem of adaptive fuzzy tracking control for a class of uncertain nonaffine nonlinear systems with input saturations.Compared with the existing results, the main contributions of this paper are as follows: (1) The approximation-based adaptive tracking control scheme is extended to nonaffine nonlinear systems with multiple time delays and saturation constrains.(2) Different from the design methods proposed in [20,21], it is obtained that the tracking errors can asymptotically converge to zero rather than being uniformly ultimately bounded.(3) The mean value theorem is used to deal with the nonaffine term with input saturation, and thus the desired asymptotic tracking performance of the closedloop systems can be achieved by using Lyapunov stability analysis.
The rest of the paper is organized as follows.Section 2 gives the problem statement and preliminaries.A novel adaptive fuzzy asymptotic tracking controller is designed in Section 3. Simulation studies are then provided in Section 4 to verify the effectiveness of the proposed control method, and Section 5 draws the conclusions.
Assumption 3 (see [18,20]).For all  ∈ R  and  ∈ R in system (5), there always exist positive constants  1 and  2 such that the following inequality holds: Assumption 4. Given the practical system described by (1) satisfying the input saturation (2), there exists feasible actual control input , which can achieve the desired control objective.
Remark 5. Clearly, Assumption 1 is quite standard and means that the external disturbance, the reference output signal, and its time derivatives are bounded, respectively.It follows from Assumption 2 that the change rate of the input gain is bounded.Particularly, different from [20,21], the tracking error of this paper can asymptotically converge to zero rather than to a desired compact set.
Similar to [10,11,[16][17][18], the following fuzzy approximation lemma is given by the following lemma.Lemma 6.Let () be a continuous function that is defined on a compact set Ω  .For any given positive constant , there always exists a fuzzy logic system () in the form of (7) Consequently, the optimal parameter vectors  * of fuzzy logic system (FLS) is defined as where Ω  and Ω  are compact regions for  and , respectively.In addition, the fuzzy approximation error  * () is defined as

Adaptive Fuzzy Tracking Controller Design
In this section, the adaptive fuzzy asymptotic tracking control scheme will be developed for the nonlinear system (1) with external disturbance, multiple time delays, and saturation constrain.For this purpose, taking the time derivative of the tracking error  =  −   with respect to  yields ė =  +  ( (, sat ()) where Then, from (12), + is a stable matrix by properly choosing a gain vector .Moreover, for any given  =   > 0, there exists  =   > 0 such that the Lyapunov equation ( + )   + ( + ) + (/) = − holds, where  is a positive design parameter.
Remark 7. The adaptive fuzzy controller (15) mainly consists of four terms.Concretely,  −1  is the positive design parameter of the adaptive control gain, and the first term  of the right hand side plays a key role for stabilizing system.The second term is used to decouple the nonaffine term with saturation nonlinearity.The third term and the fourth term with adaptation laws ( 16) are used to deal with the effects of multiple time delays and external disturbance, respectively.Now, the stability of the resulting closed-loop system is given in the following theorem.(15) and parameter updated laws (16), the tracking error of the closed-loop system can asymptotically converge to zero; that is, lim →∞ () = 0 for any (, ) ∈ Ω, which is a proper compact set.

Theorem 8. Consider the uncertain nonaffine nonlinear system (1) satisfying Assumptions 1-4. With the application of adaptive fuzzy controller
Proof.For the closed-loop error system (11), choose a Lyapunov function candidate as follows: where θ = θ− * and K = K − *  ,  = 1, 2, are the parameter estimation errors.Then, taking the time derivative of  with respect to  yields By invoking (14), we obtain that Using triangle inequality and according to the definitions of Then, from the adaptive controller (15) and the parameter updated laws ( 16), we can obtain that Using the inequality 0 ≤ /( + ) < 1, ∀ ≥ 0,  > 0, and where  =  * 2 /4 +  * 2 1 /4 +  * 2 2 /4 + 2. Integrating ( 22) from 0 to  yields Thus, it further implies that ∫  0   ()() ≤ (1/  min ())(| =0 + ), ∀ > 0, where  min (⋅) denotes the minimum eigenvalue of a matrix, that is,  ∈  2 .According to Barbalat's lemma [22], it can be concluded that lim →∞ () = 0.The proof is completed.Remark 9.It should be pointed out that the control methods proposed in [20,21] can guarantee that the tracking errors converge to the desired compact sets.The tracking error of the closed-loop system can asymptotically converge to zero by employing the adaptive control scheme in [18]; however, this control scheme cannot deal with nonaffine nonlinear systems with multiple time delays and saturation constrains.In this paper, based on fuzzy approximation technique and the mean value theorem, the proper nonlinear functions, it is proved that the desired asymptotic tracking performance of the closed-loop systems can be achieved via Lyapunov stability analysis.

Conclusion
This paper studies a novel adaptive fuzzy asymptotic tracking control scheme for a class of uncertain nonaffine nonlinear systems with multiple time delays, saturation constrains, and external disturbances.By using the mean value theorem and fuzzy logic system (FLS), the parameter updated laws are constructed to estimate the unknown adaptive controller parameters online.It is also shown that the proposed control method guarantees all the closed-loop system signals to be uniformly bounded and the tracking error can asymptotically converge to zero based on Lyapunov-based analysis.Numerical simulation results are provided to show the effectiveness of the proposed adaptive fuzzy tracking control design approach.