Volume 13 Number 3
June 2016
Article Contents
Xiao-Qiang Li, Dan Wang and Zhu-Mu Fu. Adaptive NN Dynamic Surface Control for a Class of Uncertain Non-affine Pure-feedback Systems with Unknown Time-delay. International Journal of Automation and Computing, vol. 13, no. 3, pp. 268-276, 2016. doi: 10.1007/s11633-015-0924-8
Cite as: Xiao-Qiang Li, Dan Wang and Zhu-Mu Fu. Adaptive NN Dynamic Surface Control for a Class of Uncertain Non-affine Pure-feedback Systems with Unknown Time-delay. International Journal of Automation and Computing, vol. 13, no. 3, pp. 268-276, 2016. doi: 10.1007/s11633-015-0924-8

Adaptive NN Dynamic Surface Control for a Class of Uncertain Non-affine Pure-feedback Systems with Unknown Time-delay

Author Biography:
  • Dan Wang graduated in automation engineering from Dalian University of Technology, China in 1982. He received the M. Sc. degree in marine automation engineering from Dalian Maritime University, China in 1987, and the Ph.D. degree in automation and computer-aided engineering from Chinese University of Hong Kong, China in 2001. He is currently a professor at Dalian Maritime University, China. His research interests include nonlinear control theory and applications, neural networks, adaptive control, robust control, fault detection and isolation, and system identification. E-mail: dwangdl@gmail.com

    Zhu-Mu Fu graduated in mechanical design and manufacturing from Luoyang Institute of Technology, China in 1998. He received the M. Sc. degree in vehicle engineering from Henan University of Science and Technology, China in 2003. He received the Ph. D. degree in control theory and control engineering from Southeast University, China in 2007. He is currently an associate professor at Henan University of Science and Technology, China. His research interests include nonlinear control theory and applications, H-infinity control, energy management strategy for hybrid electric vehicle. E-mail: fzm1974@163.com

  • Corresponding author: Xiao-Qiang Li graduated in mathematics from Datong University, China in 2004. He received the M. Sc. degree in mathematics from Dalian Maritime University, China in 2007 and the Ph.D. degree in control theory and control engineering from Dalian Maritime University, China in 2011. He is currently an associate professor at Henan of University Science and Technology, China. His research interests include nonlinear control theory and applications, neural networks, adaptive control. E-mail: sxxqli@126.com (Corresponding author) ORCID iD: 0000-0001-7118-6816
  • Received: 2014-05-06
Fund Project:

the Science and Technology Innovative Foundation for Distinguished Young Scholar of Henan Province No. 144100510004

the Science and Technology Programme Foundation for the Innovative Talents of Henan Province University No. 13HASTIT038

the Key Program of Henan Provincial Department of Education No. 13A470254

National Natural Science Foundation of China Nos. 61273137 and 51375145

  • Adaptive neural network (NN) dynamic surface control (DSC) is developed for a class of non-affine pure-feedback systems with unknown time-delay. The problems of "explosion of complexity" and circular construction of the practical controller in the traditional backstepping algorithm are avoided by using this controller design method. For removing the requirements on the sign of the derivative of function fi, Nussbaum control gain technique is used in control design procedure. The effects of unknown time-delays are eliminated by using appropriate Lyapunov-Krasovskii functionals. Proposed control scheme guarantees that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. Two simulation examples are presented to demonstrate the method.
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  • [1] D. Wang, Z. H. Peng, T. S. Li, X. Q. Li, G. Sun. Adaptive dynamic surface control for a class of uncertain nonlinear systems in Pure-feedback form. In Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, IEEE, Shanghai, China, pp. 1956- 1961, 2009.
    [2] R. Su, L. R. Hunt. A canonical expansion for nonlinear systems. IEEE Transactions on Automatic Control, vol. 31, no. 7, pp. 670-673, 1986.
    [3] K. Nam, A. Arapostathis. A model reference adaptive control scheme for pure-feedback nonlinear systems. IEEE Transactions on Automatic Control, vol. 33, no. 9, pp. 803- 811, 1988.
    [4] I. Kanellakopoulos, P, V. Kokotovic, A. S. Morse. Systematic design of adaptive controllers for feedback linearizable systems. IEEE Transactions on Automatic Control, vol. 36, no. 11, pp. 1241-1253, 1991.
    [5] D. Wang, J. Huang. Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form. Automatica, vol. 38, no. 8, pp. 1365-1372, 2002.
    [6] A. M. Zou, Z. G. Hou, M. Tan. Adaptive control of a class of nonlinear pure-feedback systems using fuzzy backstepping approach. IEEE Transactions on Fuzzy Systems, vol. 16, no. 4, pp. 886-896, 2008.
    [7] C. Wang, D. J. Hill, S. S. Ge, G. R. Chen. An ISS-modular approach for adaptive neural control of pure-feedback systems. Automatica, vol. 42, no. 5, pp. 723-731, 2006.
    [8] T. P. Zhang, S. S. Ge. Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form. Automatica, vol. 44, no. 7, pp. 1895-1903, 2008.
    [9] Z. G. Hou, A.M. Zou, F. X. Wu, L. Cheng, M. Tan. Adaptive dynamic surface control of a class of uncertain nonlinear systems in pure-feedback form using fuzzy backstepping approach. In Proceedings of International Conference on Automation Science and Engineering, IEEE, Washington DC, USA, pp. 821-826, 2008.
    [10] S. S. Ge, C. Wang. Adaptive NN control of uncertain nonlinear pure-feedback systems. Automatica, vol. 38, no. 4, pp. 671-682, 2002.
    [11] B. B. Ren, S. S. Ge, C. Y. Su, T. H. Lee. Adaptive neural control for a class of uncertain nonlinear systems in Pure-feedback form with hysteresis input. IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, vol. 39, no. 2, pp. 431-443, 2009.
    [12] Y. J. Liu, S. C. Tong, Y. M. Li. Adaptive fuzzy control for a class of nonlinear systems with unknown time-delays. In Proceedings of the 7th World Congress on Intelligent Control and Automation, IEEE, Chongqing, China, pp. 4361- 4364, 2008.
    [13] Z. X. Yu, S. G. Li, H. B. Du. Razumikhin-Nussbaum- Lemma-based adaptive neural control for uncertain stochastic pure-feedback nonlinear systems with time-varying delays. International Journal of Robust and Nonlinear Control, vol. 23, no. 11, pp. 1214-1239, 2013.
    [14] M.Wang, S. S. Ge, K. S. Hong. Approximation-based adaptive tracking control of pure-feedback nonlinear systems with multiple unknown time-varying delays. IEEE Transactions on Neural Networks, vol. 21, no. 11, pp. 1804-1816, 2010.
    [15] M.Wang, X. P. Liu, P. Shi. Adaptive neural control of purefeedback nonlinear time-delay systems via dynamic surface technique. IEEE Transactions on Systems, Man, and Cybernetics, vol. 41, no. 6, pp. 1681-1692, 2011.
    [16] D. Wang, J. Huang. Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form. IEEE Transactions on Neural Networks, vol. 16, no. 1, pp. 195-202, 2005.
    [17] D. Swaroop, J. C. Gerdes, P. P. Yip, J. K. Hedeick. Dynamic surface control of nonlinear systems. In Proceeding of the American Control Conference, IEEE, Albuquerque, USA, pp. 3028-3034, 1997.
    [18] D. Swaroop, J. K. Hedrick, P. P. Yip, J. C. Gerdes. Dynamic surface control for a class of nonlinear systems. IEEE Transactions on Automatic Control, vol. 45, no. 10, pp. 1893- 1899, 2000.
    [19] P. P. Yip, J. K. Hedrick. Adaptive dynamic surface control: A simplified algorithm for adaptive backstepping control of nonlinear systems. International Journal of Control, vol. 71, no. 5, pp. 959-979, 1998.
    [20] S. J. Yoo, J. B. Park, Y. H. Choi. Adaptive dynamic surface control for stabilization of parametric strict-feedback nonlinear systems with unknown time delays. IEEE Transactions on Automatic Control, vol. 52, no. 12, pp. 2360-2365, 2007.
    [21] S. J. Yoo, J. B. Park, Y. H. Choi. Adaptive neural dynamic surface control of nonlinear time-delay systems with model uncertainties. In Proceedings of the American Control Conference, IEEE, Minneapolis, USA, pp. 3140-3145, 2006.
    [22] T. S. Li, X. F. Wang, X. Y. Yang. DSC design of a robust adaptive NN control for a class of nonlinear MIMO systems. Journal of Harbin Engineering University, vol. 30, no. 2, pp. 121-125, 2009. (in Chinese)
    [23] Z. G. Hou, A. M. Zou, F. X. Wu, L. Cheng, M. Tan. Adaptive dynamic surface control of a class of uncertain nonlinear systems in pure-feedback form using fuzzy backstepping approach. In Proceedings of the 4th IEEE Conference on Automation Science and Engineering, IEEE, Washington DC, USA, pp. 821-826, 2008.
    [24] S. J. Yoo, J. B. Park, Y. H. Choi. Adaptive dynamic surface control of flexible-joint robots using self-recurrent wavelet neural networks. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 36, no. 6, pp. 1342- 1355, 2006.
    [25] S. J. Yoo, J. B. Park, Y. H. Choi. Adaptive output feedback control of Flexible-joint robots using neural networks dynamic surface design approach. IEEE Transactions on Neural Networks, vol. 19, no. 10, pp. 1712-1726, 2008.
    [26] G. Z. Zhang, J. Chen, Z. P. Lee. Adaptive robust control for servo mechanisms with partially unknown states via dynamic surface control approach. IEEE Transactions on Control Technology, vol. 18, no. 3, pp. 723-731, 2010.
    [27] D. Q.Wei, X. S. Luo, B. H.Wang, J. Q. Fang. Robust adaptive dynamic surface control of chaos in permanent magnet synchronous motor. Physics Letters A, vol. 363, no. 1, pp. 71-77, 2007.
    [28] Z. J. Yang, K. Miyazaki, S. Kanae, K. Wada. Robust position control of a magnetic levitation system via dynamic surface control technique. IEEE Transactions on Industrial Electronics, vol. 51, no. 1, pp. 26-34, 2004.
    [29] C. L. Wang, Y. Lin. Adaptive dynamic surface control for linear multivariable systems. Automatica, vol. 46, no. 10, pp. 1703-1711, 2010.
    [30] H. Y. Yue, J. M. Li. Adaptive fuzzy dynamic surface control for a class of perturbed nonlinear time-varying delay systems with unknown dead-zone. International Journal of Automation and Computing, vol. 9, no. 5, pp. 545-554, 2012.
    [31] P. L. Liu, T. J. Su. Robust stability of interval time-delay systems with delay-dependence. Systems & Control Letters, vol. 33, no. 4, pp. 231-239, 1998.
    [32] S. L. Niculescu. Delay Effects on Stability: A Robust Control Approach, New York, USA: Springer-Verlag, 2001.
    [33] D. F. Coutinho, C. E. de Souza. Delay-dependent robust stability and L2-gain analysis of a class of nonlinear timedelay systems. Automatica, vol. 44, no. 8, pp. 2006-2018, 2008.
    [34] M. Jankovic. Control Lyapunov-razumikhin functions and robust stabilization of time delay systems. IEEE Transactions on Automatic Control, vol. 46, no. 7, pp. 1048-1050, 2001.
    [35] Z. D. Tian, X. W. Gao, K. Li. A hybrid time-delay prediction method for networked control system. International Journal of Automation and Computing, vol. 11, no. 1, pp. 19-25, 2014.
    [36] T. M. Apostol. Mathematical Analysis, 2nd ed., Reading, MA: Addison-Wesley, 1974.
    [37] T. P. Zhang, S. S. Ge. Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs. Automatica, vol. 43, no. 6, pp. 1021-1033, 2007.
    [38] E. P. Ryan. A universal adaptive stabilizer for a class of nonlinear systems. Systems & Control Letters, vol. 16, no. 3, pp. 209-218, 1991.
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Adaptive NN Dynamic Surface Control for a Class of Uncertain Non-affine Pure-feedback Systems with Unknown Time-delay

  • Corresponding author: Xiao-Qiang Li graduated in mathematics from Datong University, China in 2004. He received the M. Sc. degree in mathematics from Dalian Maritime University, China in 2007 and the Ph.D. degree in control theory and control engineering from Dalian Maritime University, China in 2011. He is currently an associate professor at Henan of University Science and Technology, China. His research interests include nonlinear control theory and applications, neural networks, adaptive control. E-mail: sxxqli@126.com (Corresponding author) ORCID iD: 0000-0001-7118-6816
Fund Project:

the Science and Technology Innovative Foundation for Distinguished Young Scholar of Henan Province No. 144100510004

the Science and Technology Programme Foundation for the Innovative Talents of Henan Province University No. 13HASTIT038

the Key Program of Henan Provincial Department of Education No. 13A470254

National Natural Science Foundation of China Nos. 61273137 and 51375145

Abstract: Adaptive neural network (NN) dynamic surface control (DSC) is developed for a class of non-affine pure-feedback systems with unknown time-delay. The problems of "explosion of complexity" and circular construction of the practical controller in the traditional backstepping algorithm are avoided by using this controller design method. For removing the requirements on the sign of the derivative of function fi, Nussbaum control gain technique is used in control design procedure. The effects of unknown time-delays are eliminated by using appropriate Lyapunov-Krasovskii functionals. Proposed control scheme guarantees that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. Two simulation examples are presented to demonstrate the method.

Xiao-Qiang Li, Dan Wang and Zhu-Mu Fu. Adaptive NN Dynamic Surface Control for a Class of Uncertain Non-affine Pure-feedback Systems with Unknown Time-delay. International Journal of Automation and Computing, vol. 13, no. 3, pp. 268-276, 2016. doi: 10.1007/s11633-015-0924-8
Citation: Xiao-Qiang Li, Dan Wang and Zhu-Mu Fu. Adaptive NN Dynamic Surface Control for a Class of Uncertain Non-affine Pure-feedback Systems with Unknown Time-delay. International Journal of Automation and Computing, vol. 13, no. 3, pp. 268-276, 2016. doi: 10.1007/s11633-015-0924-8
  • The control problem of the systems with lower triangular structure, such as strict-feedback system and pure-feedback system, is a focus in the field of nonlinear control during past decades. It is due to the structural characteristics of pure-feedback system that it has a more representative form than strict-feedback system and the controller design procedure of pure-feedback system is more complicated than that of strict-feedback system.

    While the strict-feedback nonlinear systems have been much studied, relatively fewer results are available in the literature for the control of a class of uncertain nonlinear systems in the pure-feedback form[1]. Next are some works about pure-feedback system. The definition of pure-feedback system is given[2]. A model reference adaptive control scheme is presented for nonlinear systems in a pure-feedback canonical form with unknown parameters in [3]. Based on the backstepping, a controller design procedure for a class of pure-feedback systems with unknown parameters is given in [4]. The control problem of pure-feedback systems with unknown nonlinearities were researched later and the approximators such as neural networks or fuzzy logic systems are used to parameterize the unknown nonlinearities. For examples, the control problem of affine pure-feedback system with unknown nonlinearities is considered in [1, 4-5]. Zou et al.[6-15] consider the control problem of non-affine pure-feedback system and the system considered in [8] is uncertain non-affine pure-feedback system with unknown dead zone, whereas in [11], it is an uncertain non-affine pure-feedback system with hysteresis input, and in [12-15], it is an uncertain non-affine pure-feedback system with time-delay.

    The main design technique for the systems with lower triangular structure is backstepping. In the past decades, backstepping based adaptive control has been extensively investigated for uncertain nonlinear systems. In [4], adaptive backstepping, as a breakthrough in nonlinear control area, was introduced. In the paper, Kanellakopoulos et al. proposed stable controllers for strict-feedback systems and pure-feedback systems. After that, adaptive backstepping approach, a recursive design procedure, has been found to be particularly useful for nonlinear systems with triangular structures and there are lots of representative works for designing controller using backstepping, to just name a few [4-16]. In particular, references [4, 16] design controllers for strict-feedback systems and references [1, 5-15] design controllers for pure-feedback system using backstepping technique.

    However, a drawback with the backstepping technique is the problem of "explosion of complexity"[1, 8-9, 15-30]. In the design procedure of the controller using the backstepping technique, certain nonlinear functions such as virtual controls are differentiated repeatedly. The increasing of the order n of systems and the number of states will result in the increasing of complexity of the virtual controllers drastically. In [17], dynamic surface control (DSC) technique was proposed to overcome this problem by introducing a first-order filter of the synthetic input at each step of the traditional backstepping approach. DSC for the tracking problem of non-Lipschitz systems was considered in [18]. In [19], a point controller (i.e., the reference input is a constant) was designed using DSC which drives the system to achieve semi-global exponential stability. The design procedure of the NN DSC controller for a class of strict-feedback nonlinear systems was given in [16] and the controller guarantees that the solution of the closed-loop is uniformly ultimately bounded. After this work, DSC technique has been widely used in solving the problem of "explosion of complexity" for the controller design of all kinds of uncertain systems with cascade structures such as strict-feedback system[16, 20-22], pure-feedback system[1, 8, 15, 23] and so on[24-30].

    Another drawback of traditional backstepping in the control design for pure-feedback systems is circular construction of the practical controller[6, 7]. In [16], a controller design procedure is developed for a class of pure-feedback systems and this drawback, circular construction of the practical controller, was avoided skillfully. But, to avoid this problem, the method requires an assumption that the control gain of the transformed system should not be equal to zero. And the problem of circular construction of the practical controller in the control design was not pointed out in [16]. In [7], circular construction of the practical controller was mentioned for the first time and a control design process is developed for a class of non-affine pure-feedback system and input-to-state stability (ISS)-modular approach is adopted to overcome this difficulty. In [6], to circumvent this problem, filtered signals are used at every step in the control design procedure using backstepping and the given controller can guarantee that all the signals in the closed-loop system are uniformly ultimately bounded. In our paper[1], combining the DSC technique with backstepping, a controller design procedure was presented for a class of pure-feedback systems with constant control gain, the DSC technique was used to solve the problem of "explosion of complexity" and the problem of circular construction simultaneously.

    One of the challenging problems in control of nonlinear systems is the nonlinear systems with time-delay. Time-delay is often encountered in various systems, such as in recycled reactors, recycled storage tanks, nuclear reactors, rolling mills and chemical processes[31]. The existence of time-delays can destroy the stability or debase the performance of control systems[32]. Therefore, the stability analysis and controller design of time-delay systems are very important both in theory and in practice. However, the control problem and the stability analysis of the systems with time-delay are difficult. The main methods to solve the control problem of time-delay systems are Lyapunov-Krasovskii method[33], Lyapuov-Razumikhin method[34] and prediction method[35].

    In this paper, neural network based adaptive control is investigated for a class of nonaffine pure-feedback nonlinear systems with time-delay by combining DSC technique with backstepping technique. The difficulties of the control problem of the augmented system in this paper are the problem of circular construction of controller, singularity problem during controller design procedure and unknown time-delay. The DSC technique is used to overcome the problem of "explosion of complexity" and the problem of circular construction of controller in the traditional backstepping algorithm is avoided by choosing xi*+1=0. For solving the singularity problem, unit step functions are defined and different controllers are given according to the value of error. The given appropriate Lyapunov-Krasovskii function eliminates the effects of the unknown time-delay in the system. The analysis based on the Lyapunov stability theorem shows that under the appropriate assumptions, the solution of the closed-loop system is globally uniformly ultimately bounded. In addition, the output of the system is proven to converge to a small neighborhood of the origin.

    The rest of the paper is organized as follows: The problem formulation and some preliminary results are presented in Section 2. Section 3 describes the adaptive dynamic surface controller by using backstepping technique and Nussbaum control gain technique for considered systems of our work. The stability of the closed-loop system consisting of considered systems, control law and update laws is analyzed in Section 4. Two examples are given in Section 5 to illustrate the effectiveness of our method. Finally, Section 6 concludes this paper.

  • We consider a single-input single-output pure-feedback nonlinear system with unknown time-delay described by

    $\left\{ \begin{matrix} {{{\dot{x}}}_{i}}={{f}_{i}}({{{\bar{x}}}_{i}}, {{x}_{i+1}})+{{h}_{i}}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}})), 1\le i\le n-1 \\ \begin{align} & {{{\dot{x}}}_{n}}={{f}_{n}}({{{\bar{x}}}_{n}}, u)+{{h}_{n}}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}})) \\ & y={{x}_{1}} \\ \end{align} \\ \end{matrix} \right.$

    (1)

    where $\bar x_i=[x_1, x_2, \cdots, x_i]\in {\bf R}^i, i=1, \cdots, n, u\in {\bf R}, y\in {\bf R}$ are state variables, system input and output, respectively. $f_i(\cdot), h_i(\cdot)(i=1, \cdots, n)$ are unknown smooth functions; $\tau_i(i=1, \cdots, n)$ is the unknown time-delay of state.

    Remark 1. The system in [12] is given as

    $\left\{ \begin{matrix} {{{\dot{x}}}_{i}}={{f}_{i}}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}}), {{x}_{i+1}}), 1\le i\le n-1 \\ \begin{align} & {{{\dot{x}}}_{n}}={{f}_{i}}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}}), u) \\ & y={{x}_{1}}. \\ \end{align} \\ \end{matrix} \right.$

    (2)

    Comparing this system with our system in this paper, we can find that there are some differences in their structure. Those differences cause the differences in the controller design method. The control problem of non-affine pure-feedback systems free of time-delay has been solved by [7-11].

    Our objective is to design a robust adaptive controller for (1) such that the closed-loop system is stable and the output y(t) of the system tracks the reference signal yr(t).

    According to the mean value theorem[36], function fi(·)(i=1, …, n) in (1) can be rewritten as

    ${{f}_{i}}({{\bar{x}}_{i}}, {{x}_{i+1}})={{f}_{i}}({{\bar{x}}_{i}}, x_{i+1}^{*})+{{g}_{{{\lambda }_{i}}}}({{x}_{i+1}}-x_{i+1}^{*})$

    (3)

    where $\displaystyle g_{\lambda_i}=g_i(\bar x_i, \lambda_i x_{i+1}+(1-\lambda_i)x_{i+1}^*)=\frac{\partial f_i(\bar x_i, x_{i+1})}{\partial x_{i+1}}, 0<\lambda_i< 1$ $(1\leq i\leq n)$ and $x_{n+1}=u$ and $x^*_{i+1}\in {\bf R}$ . By choosing xi*+1=0[21-23], (3) can be expressed as $f_i(\bar x_i, x_{i+1})=f_i(\bar x_i, 0)+g_{\lambda_i}x_{i+1}$ and $g_{\lambda_i}=g_i(\bar x_i, \lambda_i x_{i+1})$ .

    Then, the system (1) can be transformed into the form as

    $\left\{ \begin{matrix} {{{\dot{x}}}_{i}}={{f}_{i}}({{{\bar{x}}}_{i}}, 0)+{{g}_{{{\lambda }_{i}}}}{{x}_{i+1}}+{{h}_{i}}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}})), 1\le i\le n-1 \\ \begin{align} & {{{\dot{x}}}_{n}}={{f}_{n}}({{{\bar{x}}}_{n}}, 0)+{{g}_{{{\lambda }_{i}}}}u+{{h}_{n}}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}})) \\ & y={{x}_{1}}. \\ \end{align} \\ \end{matrix} \right.$

    (4)

    For the design of the controller, we give some assumptions as follows:

    Assumption 1. The state $\bar x_n$ of the system (1) is available for feedback.

    Assumption 2. There exist known constants 0<gmingmax such that gmin≤|i|≤gmax.

    Assumption 3. There exist function him(·) and known constant τm such that ${{h}_{i}}(\cdot )\le {{h}_{im}}(\cdot ), {{\tau }_{i}}\le {{\tau }_{m}}(1\le i\le n)$ , i.e., function ${{h}_{i}}(\cdot )(1\le i\le n)$ and τi are bounded.

    Assumption 4. The reference signal yr(t) is a sufficiently smooth function of t. $y_r, \dot y_r$ and yr(2) are bounded.

    Before introducing our control design method, we first recall the technique of Nussbaum control gain[37] and the approximation property of the radial basis function (RBF) NN[16].

    An even differentiable function is called Nussbaum-type function if it has the following properties:

    $ \begin{array}{rcl}\displaystyle \lim_{s\to +\infty}\sup \frac {1}{s}\int_0^s N(\zeta){\rm d} \zeta=+ \infty \end{array} {\nonumber} $ $ \begin{array}{rcl}\displaystyle \lim_{s\to +\infty}\inf \frac {1}{s}\int_0^s N(\zeta){\rm d} \zeta=- \infty. \end{array} {\nonumber} $

    The continuous functions ζ2 cos ζ and eζ2 cos $(\frac{\pi}{2})$ have those two properties and they are Nussbaum functions. In this paper, Nussbaum function ζ2 cos ζ is exploited.

    The following lemma regarding the property of Nussbaum function is used in the controller design and stability analysis in the next section.

    Lemma 1[37]. Let V and κ be smooth functions defined on [0, tf) with V(t) ≥ 0, ∀ t ∈ [0, tf) and be an even smooth Nussbaum-type function. The following inequality holds:

    $ \begin{array}{rcl}\displaystyle 0 \leq V(t)\leq c_0+{\rm e}^{-c_1t}\int^t_0(G(x(\tau))N(\kappa)+1)\dot \kappa {\rm e}^{c_1\tau}{\rm d}\tau \end{array} {\nonumber} $

    where c1>0 and t∈0, tf), G(x(t)) is a time-varying parameter which takes values in the unknown closed intervals I:=[l-, l+] with 0∗ I, and c0 represents some suitable constant, V(t), κ(t) and $\displaystyle\int_0^tg(x(\tau))N(\kappa)\dot\kappa {\rm d}\tau$ must be bounded on [0, tf).

    According to Proposition 2 in [38], if the solution of the resulting closed-loop system is bounded, then tf=∞.

    The RBF neural networks take the form $\theta^{\rm T} \xi$ , where $\theta=[\theta_1, \theta_2, \cdots, \theta_N]^{\rm T}$ is called weight vector, ξ is a vector valued function defined in RN. Denote the components of ξ by ρi (i=1, …, N), then ρi(x) (i=1, …, N) is called a basis function. A commonly used basis function is the so-called Gaussian function of the following form:

    $ \begin{array}{rcl}\displaystyle \rho_i(x)=\frac{1}{\sqrt{2\pi}\vartheta } {\rm e}^{-\frac{\parallel x-\zeta_j\parallel^2}{2\vartheta^2}}, \quad \vartheta > 0, j=1, \cdots, N \end{array} {\nonumber} $

    where ζ (j=1, …, N)∈ Rn are the constant vectors called the center of the basis function, and $\vartheta$ is a real number called the width of the basis function. According to the approximation property of the RBF network, given a continuous real valued function f:Ω → R with Ω∈ Rn, a compact set, and any δm>0 by appropriately choosing $\sigma$ and $\zeta_j (j=1, \cdots, N)\in {\bf R}^n$ for some sufficiently large integer N, there exists an ideal $\theta^*=[\theta^*_1, \theta^*_2, \cdots, \theta_N^*]^{\rm T}$ such that the RBF network $\theta^{\rm *T}\xi$ can approximate the given function $f(x):{\bf R}^m\mapsto {\bf R}$ with the approximation error bounded by δm, i.e.,

    $ \begin{array}{rcl}\displaystyle f(x)=\theta^{\rm *T}\xi(x)+\delta^* \end{array} {\nonumber} $

    with $\mid \delta^* \mid \leq \delta_m$, where $\delta^*$ represents the reconstruction error and $x\in \Omega_x$ .

    The ideal weigh vector θ* is an artificial quantity required for analytical purposes. θ* is defined as the value of θ that minimizes |δ| for all $x\in \Omega_x \subset {\bf R}^m$ , i.e.,

    $ \begin{array}{rcl}\displaystyle \theta^*\cong \arg \min{\sup_{x\in {\bf R}^N}\mid f(x)-\theta^{\rm T}\xi(x)}\mid. \end{array} {\nonumber} $

    Assumption 5. θ* is bounded parameter, i.e., there exists a constant vector θM such that $\mid \theta^*\mid \leq \theta_M$ .

  • In this section, we will incorporate the DSC technique into a neural network based adaptive control design scheme for the n-th-order system described by (4). Similar to traditional backstepping, the design of adaptive control laws is based on the adjusting of the errors $s_1=x_1-y_r, \cdots, s_i=x_i-z_i (i=2, \cdots, n)$ , where zi is the output of a first-order filter with virtual controller αi-1 as the input. Finally, an overall control law u is constructed at step n.

    In the controller design procedure, we define $\tilde \theta_i=\hat \theta_i-\theta^*_i (i=1, \cdots, n)$ , where $\hat \theta_i$ is the estimation of $\theta^*_i, \Gamma_i (i=1, \cdots, n)$ is a constant matrix satisfying $\Gamma_i=\Gamma^{\rm T}_italic>0, k_{i1}, k_{i2}(i=1, \cdots, n) \gamma$ are positive constants, $\lambda_{\max}(A)$ denotes the largest eigenvalue of a square matrix A. Given any compact set Ωi, the set $\Omega_{s_i}(i=1, \cdots, n)$ is defined as $\Omega_{s_i}=\{s_i\mid \mid s_i\mid\geq c_{s_i}, s_i+y_r\in \Omega_i\}$ and $\Omega_i-\Omega_{s_i}=\{s_i\mid s_i+y_r\in \Omega_i, s_i+y_r\notin \Omega_{s_i}\}$ . Function $q(s_i, c_{s_i})(1\leq i\leq n)$ is defined as follows and this function will be used in the control design later

    $q({{s}_{i}}, {{c}_{{{s}_{i}}}})=\left\{ \begin{matrix} \begin{align} & 1, {{s}_{i}}\in {{\Omega }_{{{s}_{i}}}} \\ & 0, {{s}_{i}}\in {{\Omega }_{i}}-{{\Omega }_{{{s}_{i}}}} \\ \end{align} \\ \end{matrix} \right.$

    (5)

    where csi is a positive design constant that can be chosen arbitrarily small. Next is the controller design procedure.

    Step i. At this step, we consider the first equation in (4). We define si=xi-zi which is called the error surface with zi as the desired trajectory and z1=yr. Then,

    $ \begin{array}{rcl} \dot s_i=f_i(\bar x_i, 0)+g_{\lambda_i}x_{i+1}+h_i(\bar x_i(t-\tau_i))-\dot z_i. \end{array} $

    (6)

    In order to design the control law, we define a smooth scalar function as

    $ \begin{array}{rcl}\displaystyle V_i=\frac {1}{2}s^2_i+\frac {1}{2} \tilde \theta^{\rm T}_i \Gamma^{-1}_i \tilde \theta_i+\int_{t-\tau_i}^th^2_{im}(\bar x_i(\sigma)){\rm d}\sigma. \end{array} $

    (7)

    Differentiating Vi with respect to time t, we obtain

    $\begin{align} & {{{\dot{V}}}_{i}}={{s}_{i}}{{{\dot{s}}}_{i}}+\tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}}+h_{im}^{2}({{{\bar{x}}}_{i}}(t))-h_{im}^{2}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}}))= \\ & {{s}_{i}}[{{f}_{i}}({{{\bar{x}}}_{i}}, 0)+{{g}_{{{\lambda }_{i}}}}{{x}_{i+1}}+{{h}_{i}}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}}))-{{{\dot{z}}}_{i}}]+ \\ & \tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}}+h_{im}^{2}({{{\bar{x}}}_{i}}(t))-h_{im}^{2}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}}))= \\ & {{s}_{i}}{{f}_{i}}({{{\bar{x}}}_{i}}, 0)+{{s}_{i}}{{h}_{i}}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}}))-{{s}_{i}}{{{\dot{z}}}_{i}}+{{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{x}_{i+1}}+ \\ & \tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}}+h_{im}^{2}({{{\bar{x}}}_{i}}(t))-h_{im}^{2}({{{\bar{x}}}_{i}}(t-{{\tau }_{i}}))\le \\ & {{s}_{i}}{{F}_{i}}+\frac{s_{i}^{2}}{4}+{{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{x}_{i+1}}-{{s}_{i}}{{{\dot{z}}}_{i}}+\tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}} \\ \end{align}$

    (8)

    where

    $ \begin{array}{rcl}\displaystyle F_i=f_i(\bar x_i, 0)+\frac{h^2_{im}(\bar x_i(t))}{s_i}. \end{array} $

    (9)

    Define a compact set $\Omega_i\subset {\bf R}$ , let $\theta^{\rm *T}_i$ and $\varepsilon^*_i$ be such that for any $x_i\in \Omega_i$ ,

    $ \begin{array}{rcl} F_i=q(s_i, c_{s_i})(\theta^{\rm *T}_i\xi_i+\varepsilon^*_i) \end{array} $

    (10)

    where $ \mid \varepsilon^*_i\mid\leq \varepsilon$ .

    Note that if Fi is utilized to construct the controller, controller singularity may occur since $\displaystyle \frac{h^2_{im}(\bar x_i(t))}{s_i}$ is not well-defined at si=0. Therefore, care must be taken to guarantee the boundedness of the control as discussed in [33]. So, we choose the virtual control αi+1 as

    $ \begin{array}{lcr} \alpha_{i+1} =q(s_i, c_{s_i})N(\kappa_i)[K_is_i+\hat\theta_i^{\rm T}\xi_i-\dot z_i] \end{array} $

    (11)

    with

    $ \begin{array}{lcr} \dot\kappa_i=q(s_i, c_{s_i})(K_is_i^2+\hat\theta_i^{\rm T}\xi_is_i-\dot z_is_i) \end{array} $

    (12)

    $ \begin{array}{lcr} \dot {\hat\theta}_i=q(s_i, c_{s_i})\Gamma_i(\xi_is_i-\gamma\hat\theta_i). \end{array} $

    (13)

    Define a new state variable zi+1 and let αi+1 pass through a first-order filter with time constant βi+1 to obtain zi+1:

    $ \begin{array}{lcr} \beta_{i+1}\dot z_{i+1}+z_{i+1}=\alpha_{i+1}, \quad z_{i+1}(0)=\alpha_{i+1}(0). \end{array} $

    (14)

    Step n. At this step, we consider the n-th equation in (4), i.e.,

    $ \begin{array}{lcr} \dot x_n=f_n(\bar x_n, 0)+g_{\lambda_n}u+h_n(\bar x_n(t-\tau_n)). \end{array} $

    (15)

    We define $s_n=x_n-z_n$ , then

    $ \begin{array}{lcr} \dot s_n=f_n(\bar x_n, 0)+g_{\lambda_n}u+h_n(\bar x_n(t-\tau_n))-\dot z_n. \end{array} $

    (16)

    In order to design the control law, define a smooth scalar function as

    $ \begin{array}{lcr}\displaystyle V_n=\frac {1}{2}s^2_n+\frac {1}{2} \tilde \theta^{\rm T}_n \Gamma^{-1}_n \tilde \theta_n+\int_{t-\tau_n}^th^2_{nm}(\bar x_n(\sigma)){\rm d}\sigma. \end{array} $

    (17)

    Differentiating Vn with respect to time t, we obtain

    $\begin{align} & {{{\dot{V}}}_{n}}={{s}_{n}}{{{\dot{s}}}_{n}}+\tilde{\theta }_{n}^{\text{T}}\Gamma _{n}^{-1}{{{\dot{\hat{\theta }}}}_{n}}+ \\ & h_{nm}^{2}({{{\bar{x}}}_{n}}(t))-h_{nm}^{2}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}}))= \\ & {{s}_{n}}[{{f}_{n}}({{{\bar{x}}}_{n}}, 0)+{{g}_{{{\lambda }_{n}}}}u+{{h}_{n}}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}}))-{{{\dot{z}}}_{n}}]+ \\ & \tilde{\theta }_{n}^{\text{T}}\Gamma _{n}^{-1}{{{\dot{\hat{\theta }}}}_{n}}+h_{nm}^{2}({{{\bar{x}}}_{n}}(t))-h_{nm}^{2}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}}))= \\ & {{s}_{n}}{{f}_{n}}({{{\bar{x}}}_{n}}, 0)+{{s}_{n}}{{h}_{n}}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}}))- \\ & {{s}_{n}}{{{\dot{z}}}_{n}}+{{s}_{n}}{{g}_{{{\lambda }_{n}}}}u+\tilde{\theta }_{n}^{T}\Gamma _{n}^{-1}{{{\dot{\hat{\theta }}}}_{n}}+ \\ & h_{nm}^{2}({{{\bar{x}}}_{n}}(t))-h_{nm}^{2}({{{\bar{x}}}_{n}}(t-{{\tau }_{n}}))\le \\ & {{s}_{n}}{{F}_{n}}+\frac{s_{n}^{2}}{4}+{{s}_{n}}{{g}_{{{\lambda }_{n}}}}u-{{s}_{n}}{{{\dot{z}}}_{n}}+\tilde{\theta }_{n}^{\text{T}}\Gamma _{n}^{-1}{{{\dot{\hat{\theta }}}}_{n}} \\ \end{align}$

    (18)

    where

    $ \begin{array}{rcl}\displaystyle F_n=f_n(\bar x_n, 0)+\frac{h^2_{nm}(\bar x_n(t))}{s_n}. \end{array} {\nonumber} $

    Define a compact set $\Omega_n\subset {\bf R}^n$ , let $\theta^*_n$ and $\varepsilon^*_n$ be such that for any $\bar x_n\in \Omega_n$ ,

    $ \begin{array}{lcr} F_n=q(s_i, c_{s_i})(\theta^{\rm *T}_n\xi_n+\varepsilon^*_n) \end{array} $

    (19)

    where $\mid \varepsilon^*_n\mid\leq \varepsilon$ .

    Similarly, we choose the control input u as

    $ \begin{array}{lcr} u=q(s_n, c_{s_n})N(\kappa_n)[K_ns_n+\hat\theta_n^{\rm T}\xi_n-\dot z_n] \end{array} $

    (20)

    with

    $ \begin{array}{lcr} \dot\kappa_n=q(s_n, c_{s_n})(K_ns_n^2+\hat\theta_n^{\rm T}\xi_ns_n-\dot z_ns_n) \end{array} $

    (21)

    $ \begin{array}{lcr} \dot {\hat\theta}_n=q(s_n, c_{s_n})\Gamma_n(\xi_ns_n-\gamma\hat\theta_n). \end{array} $

    (22)
  • In this section, we show that the control law and update law introduced in the above design procedure guarantee the uniform ultimate boundedness of the solution of the closed-loop system.

    Theorem 1. Under Assumptions 1-5, consider the closed-loop system consisting of the plant, the proposed robust adaptive state feedback control law (20) and the adaptive laws (13) and (22), it is guaranteed that all of the signals of the closed-loop system are semi-globally uniformly ultimately bounded and the output y(t) of the given system (1) converges to the reference signal yr(t).

    Proof. This theorem's proof is separated n steps to complete.

    Step i. First of all, we consider the Lyapunov candidate function

    $\begin{align} & {{V}_{{{s}_{i}}}}={{V}_{i}}+\frac{1}{2}y_{i+1}^{2}= \\ & \frac{1}{2}s_{i}^{2}+\frac{1}{2}\tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\tilde{\theta }}}_{i}}+ \\ & \int\limits_{t-{{\tau }_{i}}}^{t}{h_{im}^{2}}({{{\bar{x}}}_{i}}(\sigma ))\text{d}\sigma +\frac{1}{2}y_{i+1}^{2} \\ \end{align}$

    (23)

    where $y_{i+1}=z_{i+1}-\alpha_{i+1}$ .

    The derivative of Vsi along the subsystem (6) is

    $\begin{align} & {{{\dot{V}}}_{{{s}_{i}}}}={{{\dot{V}}}_{i}}+{{y}_{i+1}}{{{\dot{y}}}_{i+1}}\le \\ & {{s}_{i}}{{F}_{i}}+\frac{s_{i}^{2}}{4}+{{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{x}_{i+1}}- \\ & {{s}_{i}}{{{\dot{z}}}_{i}}+\tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}}+{{y}_{i+1}}{{{\dot{y}}}_{i+1}}. \\ \end{align}$

    (24)

    Let $x_{i+1}=x_{i+1}-z_{i+1}+z_{i+1}-\alpha_{i+1}+\alpha_{i+1}=s_{i+1}+y_{i+1}+\alpha_{i+1}$ , we have

    $\begin{align} & {{{\dot{V}}}_{{{s}_{i}}}}\le {{s}_{i}}{{F}_{i}}+\frac{s_{i}^{2}}{4}+{{s}_{1}}{{g}_{{{\lambda }_{i}}}}{{s}_{i+1}}+{{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{y}_{i+1}}+ \\ & {{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{\alpha }_{i+1}}-{{s}_{i}}{{{\dot{z}}}_{i}}+\tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}}+{{y}_{i+1}}{{{\dot{y}}}_{i+1}}. \\ \end{align}$

    (25)

    Substituting the virtual control law (9) and using the Young's inequality $\displaystyle ab\leq a^2+\frac{b^2}{4}$ , we have

    $\begin{align} & {{{\dot{V}}}_{{{s}_{i}}}}\le {{s}_{i}}{{F}_{i}}+\frac{s_{i}^{2}}{4}+{{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{s}_{i+1}}+{{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{y}_{i+1}}+ \\ & {{s}_{i}}{{g}_{{{\lambda }_{i}}}}{{\alpha }_{i+1}}-{{s}_{i}}{{{\dot{z}}}_{i}}+\tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}}+{{y}_{i+1}}{{{\dot{y}}}_{i+1}}\le \\ & {{s}_{i}}{{F}_{i}}+\frac{3s_{i}^{2}}{4}+{{g}_{{{\max }^{2}}}}s_{i+1}^{2}+{{g}_{{{\max }^{2}}}}y_{i+1}^{2}+ \\ & [{{g}_{{{\lambda }_{i}}}}q({{s}_{i}}, {{c}_{{{s}_{i}}}})N({{\kappa }_{i}})+1]{{{\dot{\kappa }}}_{i}}- \\ & q({{s}_{i}}, {{c}_{{{s}_{i}}}})[{{K}_{i}}s_{i}^{2}+{{{\hat{\theta }}}_{i}}{{\xi }_{i}}{{s}_{i}}-{{{\dot{z}}}_{i}}{{s}_{i}}]+ \\ & \tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}{{{\dot{\hat{\theta }}}}_{i}}+{{y}_{i+1}}{{{\dot{y}}}_{i+1}}. \\ \end{align}$

    (26)

    Case 1. When $\mid s_i\mid\geq c_{s_i}$ , then q(si, csi)=1. Noticing that $\displaystyle\dot y_{i+1}=\dot z_{i+1}-\dot\alpha_{i+1}=\frac{\alpha_{i+1}-z_{i+1}}{\beta_{i+1}}-\dot\alpha_{i+1}=-\frac{y_{i+1}}{\beta_{i+1}}-\dot\alpha_{i+1}, \mid\varepsilon_i^*\mid\leq \varepsilon$ and substituting (10), $\dot\kappa_i$ , and the update law (13) into (26), we obtain

    $\begin{align} & {{{\dot{V}}}_{{{s}_{i}}}}\le -({{K}_{i}}-1)s_{i}^{2}+{{g}_{{{\max }^{2}}}}s_{i+1}^{2}+ \\ & {{g}_{{{\max }^{2}}}}y_{i+1}^{2}+[{{g}_{{{\lambda }_{i}}}}N({{\kappa }_{i}})+1]{{{\dot{\kappa }}}_{i}}+ \\ & {{\varepsilon }^{2}}-\gamma \tilde{\theta }_{i}^{\text{T}}{{{\hat{\theta }}}_{i}}+{{y}_{i+1}}\left( -\frac{{{y}_{i+1}}}{{{\beta }_{i+1}}}-{{{\dot{\alpha }}}_{i+1}} \right). \\ \end{align}$

    Since N(κ) is continuous, $\displaystyle\dot\alpha_{i+1}=\frac{\partial N(\kappa_i)}{\partial \kappa_i}\dot\kappa_i[K_is_i+\hat\theta_i^{\rm T}\xi_i-\dot z_i]+N(\kappa_i)[K_i\dot s_i$ $+\dot{\hat\theta}_i\xi_i+\hat\theta_i\dot\xi_i]$ is a continuous function. All of variables of the function $\dot\alpha_{i+1}$ are from compact sets, so the variables of this function belong to a compact set $\Omega_{\alpha_{i+1}}$ . Then $\mid\dot\alpha_{i+1}\mid$ has a maximum Bi+1 on a compact set $\Omega_{\alpha_{i+1}}$ , i.e., $\mid\dot\alpha_{i+1}\mid\leq B_{i+1}$ [11].

    Using $2\tilde\theta^{\rm T}_i\hat \theta _i\geq \parallel \tilde\theta_i\parallel^2-\parallel\theta^*_i\parallel^2$ , Assumption 5 and $\displaystyle y_{i+1}B_{i+1}\leq \frac{y^2_{i+1}B_{i+1}^2}{2\omega}+\frac{\omega}{2}, \omega>0$ , we have

    $\begin{align} & {{{\dot{V}}}_{{{s}_{i}}}}\le -({{K}_{i}}-1)s_{i}^{2}+{{g}_{{{\max }^{2}}}}s_{i+1}^{2}+{{g}_{{{\max }^{2}}}}y_{i+1}^{2}+ \\ & [{{g}_{{{\lambda }_{i}}}}N({{\kappa }_{i}})+1]{{{\dot{\kappa }}}_{i}}+{{\varepsilon }^{2}}-\frac{\gamma }{2}(\parallel {{{\tilde{\theta }}}_{i}}{{\parallel }^{2}}-\parallel {{\theta }_{M}}{{\parallel }^{2}})- \\ & \frac{y_{i+1}^{2}}{{{\beta }_{i+1}}}+\frac{y_{i+1}^{2}B_{i+1}^{2}}{2\omega }+\frac{\omega }{2}. \\ \end{align}$

    Let $\displaystyle K_i-1=k_{i1}+k_{i2}\geq k_{i1}+b_i\int^t_{t-\tau_m}h_{im}^2(\bar x_i(\sigma)){\rm d}\sigma$ , where $b_is^2_i\geq a_1, a_1\geq 0$ , since $\tau_i\leq \tau_m$ , we can get

    $ \begin{array}{rcl}\displaystyle -\int^t_{t-\tau_m}h_{im}^2(\bar x_i(\sigma)){\rm d}\sigma\leq -\int^t_{t-\tau_i}h_{1m}^2(\bar x_i(\sigma)){\rm d}\sigma. \end{array} {\nonumber} $

    So we have

    $\begin{align} & {{{\dot{V}}}_{{{s}_{i}}}}\le -{{k}_{i1}}s_{i}^{2}-\frac{\gamma }{2{{\lambda }_{\max }}(\Gamma _{i}^{-1})}\tilde{\theta }_{i}^{\text{T}}\Gamma _{i}^{-1}\tilde{\theta }_{i}^{\text{T}}+\frac{\omega }{2}- \\ & {{a}_{i}}\int\limits_{t-{{\tau }_{i}}}^{t}{h_{im}^{2}}({{{\bar{x}}}_{i}}(\sigma ))\text{d}\sigma -(\frac{1}{{{\beta }_{i+1}}}- \\ & {{g}_{{{\max }^{2}}}}-\frac{{{B}_{i+1}}}{2\omega })y_{i+1}^{2}+[{{g}_{{{\lambda }_{i}}}}N({{\kappa }_{i}})+1]{{{\dot{\kappa }}}_{i}}+ \\ & {{\varepsilon }^{2}}+\frac{\gamma }{2}\parallel {{\theta }_{M}}{{\parallel }^{2}}+{{g}_{{{\max }^{2}}}}{{s}_{i+1}}. \\ \end{align}$

    Define the following constants:

    $\begin{align} & {{\delta }_{i}}=\min \left\{ {{k}_{i1}}, {{a}_{i}}, \frac{\gamma }{{{\lambda }_{\max }}(\Gamma _{i}^{-1})}, \frac{1}{{{\beta }_{i+1}}}-{{g}_{{{\max }^{2}}}}-\frac{{{B}_{i+1}}}{2\omega } \right\} \\ & \begin{matrix} {{\eta }_{i}}={{\varepsilon }^{2}}+\frac{\gamma }{2}\parallel {{\theta }_{M}}{{\parallel }^{2}}+\frac{\omega }{2}. \\ \end{matrix} \\ \end{align}$

    Thus, we have

    $ \begin{array}{lcr} \dot V_{s_i}\leq \delta_iV_{s_i}+\eta_i+[g_{\lambda_i}N(\kappa_i)+1]\dot\kappa_i+g_{\max}^2s_{i+1}^2. \end{array} $

    (27)

    Multiplying (27) by $\displaystyle {\rm e}^{\delta_it}$ yields

    $ \begin{array}{lcr}\displaystyle (V_{s_i}{\rm e}^{\delta_it})'\leq [\eta_i+[g_{\lambda_i}N(\kappa_i)+1]\dot\kappa_i+g_{\max}^2s_{i+1}^2]{\rm e}^{\delta_it} \end{array} $

    (28)

    Integrating (28) over [0, t] yields

    $\begin{align} & {{V}_{{{s}_{i}}}}(t)\le \frac{{{\eta }_{i}}}{{{\delta }_{i}}}+{{V}_{{{s}_{i}}}}(0)+ \\ & {{e}^{-{{\delta }_{i}}t}}\int_{0}^{t}{\left[ g{{\lambda }_{t}}N\left( {{\kappa }_{i}} \right)+1 \right]}{{{\dot{\kappa }}}_{i}}{{e}^{{{\delta }_{i}}\sigma }}d\sigma + \\ & {{e}^{-{{\delta }_{i}}t}}\int_{0}^{t}{{}}g_{\max }^{2}s_{i+1}^{2}{{e}^{{{\delta }_{i}}\sigma }}d\sigma . \\ \end{align}$

    If $\displaystyle {\rm e}^{-\delta_it}\int^t_0g_{\max}^2s_{i+1}^2{\rm e}^{\delta_i\sigma}{\rm d}\sigma$ is bounded, we can replace it with a constant. And applying Lemma 1, we can conclude that $\displaystyle V_{s_i}, \kappa_i, \int_0^t[g_{\lambda_i}N(\kappa_i)+1]\dot\kappa_i{\rm d}\sigma$ are bounded in $[0, t_f]$ .

    According to (18), if we know the upper bound of si+1, we can know $\displaystyle {\rm e}^{-\delta_it}\int^t_0g_{\max}^2s_{i+1}^2{\rm e}^{\delta_i\sigma}{\rm d}\sigma$ is bounded.

    $\begin{align} & {{\text{e}}^{-{{\delta }_{i}}t}}\int\limits_{0}^{t}{{{g}_{{{\max }^{2}}}}}s_{i+1}^{2}{{\text{e}}^{{{\delta }_{i}}\sigma }}\text{d}\sigma \le \\ & {{g}_{{{\max }^{2}}}}{{\text{e}}^{-{{\delta }_{i}}t}}su{{p}_{{{t}_{{{s}_{i+1}}}}\in [0, t]}}s_{i+1}^{2}({{t}_{{{s}_{i+1}}}})\int\limits_{0}^{t}{{{\text{e}}^{{{\delta }_{i}}\sigma }}}\text{d}\sigma \le \\ & \frac{{{g}_{{{\max }^{2}}}}}{{{\delta }_{i}}}su{{p}_{{{t}_{{{s}_{i+1}}}}\in [0, t]}}s_{i+1}^{2}({{t}_{{{s}_{i+1}}}}). \\ \end{align}$

    (29)

    Case 2. When si <csi, it is a bounded variable. The system does not need to be controlled, so we have αi+1=0.

    Step n. We consider the following Lyapunov candidate function:

    $ \begin{array}{lcr}\displaystyle V_n=V_{s_n}=\frac {1}{2}s^2_n+\frac {1}{2} \tilde \theta^{\rm T}_n \Gamma^{-1}_n \tilde \theta_n+\int_{t-\tau_n}^th^2_{nm}(\bar x_n(\sigma)){\rm d}\sigma. \end{array} $

    (30)

    According to (18), we know that

    $ \begin{array}{rcl}\displaystyle \dot V_n\leq s_n F_n+\frac{s_n^2}{4}+s_ng_{\lambda_n}u-s_n\dot z_n+\tilde\theta^{\rm T}_n\Gamma^{-1}_{n}\dot{\hat\theta}_n. \end{array} $

    (31)

    Substituting (19) and the control law (20) into (31), we have

    $\begin{align} & {{{\dot{V}}}_{n}}\le {{s}_{n}}(\theta _{n}^{\text{*T}}+\varepsilon _{n}^{*})+\frac{s_{n}^{2}}{4}+q({{s}_{n}}, {{c}_{{{s}_{n}}}}){{g}_{{{\lambda }_{n}}}}N({{\zeta }_{n}}){{{\dot{\kappa }}}_{n}}+ \\ & {{{\dot{\kappa }}}_{n}}-{{{\dot{\kappa }}}_{n}}-{{s}_{n}}{{{\dot{z}}}_{n}}+\tilde{\theta }_{n}^{\text{T}}\Gamma _{n}^{-1}{{{\dot{\hat{\theta }}}}_{n}}. \\ \end{align}$

    (32)

    Case 1. When |sn|≥ csn, then q(sn, csn)=1. Using the Young's inequality $\displaystyle ab\leq a^2+\frac{b^2}{4}$ , noticing that $ \mid\varepsilon_n^*\mid\leq \varepsilon$ and substituting the update law (22) into (32), we obtain

    $ \begin{array}{rcl}\displaystyle \dot V_n\leq -(K_n-\frac{1}{2})s_n^2+(g_{\lambda_n}N(\zeta_n)+1)\dot\kappa_n-\gamma \tilde\theta_n^{\rm T}\hat\theta_n+\varepsilon^2. \end{array} $

    (33)

    Let $\displaystyle K_n-1=k_{n1}+k_{n2}\geq k_n+b_n\int^t_{t-\tau_m}h_{nm}^2(\bar x_n(\sigma)){\rm d}\sigma$ where $b_ns_n^2\geq a_n$ . Since $\tau_n\leq \tau_m$ , we can get

    $ \begin{array}{rcl}\displaystyle -\int^t_{t-\tau_m}h_{nm}^2(\bar x_n(\sigma)){\rm d}\sigma\leq -\int^t_{t-\tau_n}h_{nm}^2(\bar x_n(\sigma)){\rm d}\sigma. \end{array} $

    Noticing that $2\tilde\theta^{\rm T}_n\hat \theta _n\geq \parallel \tilde\theta_n\parallel^2-\parallel\theta^*_n\parallel^2$ and the Assumption 5, substituting (34) into (33), we can get

    $\begin{align} & {{{\dot{V}}}_{{{s}_{n}}}}\le -{{k}_{n1}}s_{n}^{2}-\frac{\gamma }{2{{\lambda }_{\max }}(\Gamma _{n}^{-1})}\tilde{\theta }_{n}^{\text{T}}{{\Gamma }^{-1}}n{{{\tilde{\theta }}}_{n}}- \\ & {{a}_{n}}\int\limits_{t-{{\tau }_{n}}}^{t}{h_{nm}^{2}}({{{\bar{x}}}_{n}}(\sigma ))\text{d}\sigma + \\ & [{{g}_{{{\lambda }_{n}}}}N({{\kappa }_{n}})+1]{{{\dot{\kappa }}}_{n}}+{{\varepsilon }^{2}}+\frac{\gamma }{2}\parallel {{\theta }_{M}}{{\parallel }^{2}}. \\ \end{align}$

    Define the following constants:

    $\begin{align} & {{\delta }_{n}}=\min \left\{ {{k}_{n}}, {{a}_{n}}, \frac{\gamma }{{{\lambda }_{\max }}(\Gamma _{n}^{-1})} \right\} \\ & \begin{matrix} {{\eta }_{n}}={{\varepsilon }^{2}}+\frac{\gamma }{2}\parallel {{\theta }_{M}}{{\parallel }^{2}}. \\ \end{matrix} \\ \end{align}$

    Thus, we have

    $ \begin{array}{lcr} \dot V_{s_n}\leq \delta_nV_{s_n}+\eta_n+[g_{\lambda_n}N(\kappa_n)+1]\dot\kappa_n. \end{array} $

    (35)

    Multiplying (35) by $\displaystyle {\rm e}^{\delta_nt}$ yields

    $\begin{matrix} ({{V}_{{{s}_{n}}}}{{\text{e}}^{{{\delta }_{n}}t}}{)}'\le [{{\eta }_{n}}+({{g}_{{{\lambda }_{n}}}}N({{\kappa }_{n}})+1){{{\dot{\kappa }}}_{n}}]{{\text{e}}^{{{\delta }_{n}}t}}. \\ \end{matrix}$

    (36)

    Integrating (36) over [0, t] yields

    $\begin{align} & {{V}_{{{s}_{n}}}}(t)\le \frac{{{\eta }_{n}}}{{{\delta }_{n}}}+{{V}_{{{s}_{n}}}}(0)+ \\ & {{e}^{-{{\delta }_{n}}t}}\int_{0}^{t}{{}}[{{g}_{{{\lambda }_{n}}}}N({{\zeta }_{n}})+1]{{{\dot{\kappa }}}_{n}}{{\text{e}}^{{{\delta }_{n}}\sigma }}\text{d}\sigma . \\ \end{align}$

    (37)

    Case 2. When $\mid s_n\mid <c_{s_n}$ , it is a bounded variable. There is no need of control, so u=0.

    According to the Lemma 1, we know that Vsn, κn, $\displaystyle\int^t_0[g_{\lambda_n}N(\kappa_n)+1]\dot\kappa_n{\rm d}\sigma$ and sn are bounded in [0, t]. Letting the bound of $\displaystyle\int^t_0[g_{\lambda_n}N(\kappa_n)+1]\dot\kappa_n{\rm d}\sigma$ is An on [0, tf) for all tf>0, we can get the result as

    $\begin{align} & {{e}^{-{{\delta }_{n}}t}}\int_{0}^{t}{{}}[{{g}_{{{\lambda }_{n}}}}N({{\zeta }_{n}})+1]{{{\dot{\kappa }}}_{n}}{{e}^{{{\delta }_{n}}\sigma }}d\sigma \le \\ & \int_{0}^{t}{{}}\left| [{{g}_{{{\lambda }_{n}}}}N({{\zeta }_{n}})+1]{{{\dot{\kappa }}}_{n}} \right|\left| {{e}^{-{{\delta }_{n}}(t-\sigma )}}\text{d}\sigma \right.\le \\ & \int_{0}^{t}{{}}\left| [{{g}_{{{\lambda }_{n}}}}N({{\zeta }_{n}})+1]{{{\dot{\kappa }}}_{n}} \right|\text{d}\sigma \le {{A}_{n}}. \\ \end{align}$

    Substituting the bound into (37), we can get

    $ \begin{array}{rcl} \displaystyle\frac{1}{2}s_n^2\leq V_{s_n} \leq\frac{\eta_n}{\delta_n}+V_{s_n}(0)+A_n. \end{array} $

    (38)

    So, we have

    $ \begin{array}{rcl} \displaystyle |s_n|\leq \sqrt{2V_{s_n}} \leq\sqrt{2(\frac{\eta_n}{\delta_n}+V_{s_n}(0)+A_n)}. \end{array} $

    (39)

    By adjusting the parameters, we can get that sn is bounded. Using inequality (29) and Lemma 1, we can get that sn-1 is bounded, too. Applying the Lemma 1 (n-1) times backward, we can get that si (1≤in) is bounded, and $\dot s_i(1\leq i\leq n)$ is bounded too.

  • To illustrate and clarify the proposed design procedure, we apply the adaptive neural network controller developed in Section 3 to control a nonlinear system.

    Example 1. Consider a nonlinear system:

    $\left\{ \begin{align} & {{{\dot{x}}}_{1}}={{f}_{1}}({{x}_{1}}, {{x}_{2}})+{{h}_{1}}({{x}_{1}}(t-{{\tau }_{1}})) \\ & {{{\dot{x}}}_{2}}={{f}_{2}}({{x}_{1}}, {{x}_{2}}, u)+{{h}_{2}}({{x}_{1}}(t-{{\tau }_{2}}), {{x}_{2}}(t-{{\tau }_{2}})) \\ & y={{x}_{1}}. \\ \end{align} \right.$

    (40)

    For the purpose of simulation, following plant dynamics are used: $f_1(\cdot)=(1+x_1^2)x_2+x_1{\rm e}^{-x_2}, h_1(\cdot)=2x_1^2$ , $f_2(\cdot)={\rm cos}(x_1x_2)+x_1x_2^2+u+\sin(u), h_2(\cdot)=\sin(x_1x_2)$ and τ1=τ2=1. And we choose $h_1m(\cdot)=2x_1^2, h_2m(\cdot)=1$ and τm=2 for satisfying Assumption 3. The reference trajectory is sin t. The initial values of all the states are 0.4. The initial values of all the time-delay states are 0. Two RBF NN are used in this simulation. The numbers of those NN are chosen as 10 and 103. The centers of NN are chosen as [-0.5, 0.5] and [-0.5, 0.5]3, respectively. The widths of RBF are all chosen as 1. The initial value of parameter θ1 is set to 0.1 and θ2 is set to 0.4. The initial value of parameter κ1 is set to 1.5 and κ2 is set to 0.4. The initial value of parameter z2 is set to 0. The controller parameters chosen for simulation are K1(t)=20, K2(t)=300, Γ1=Γ2=0.000 2I, γ=0.00, and csi=0.001.

    Simulation results are shown in Figs. 1 and 2. Fig. 1 shows that the given control input is bounded. Fig. 2 gives the output of the closed-loop system and the reference signal.

    Figure 1.  Control input u for the augmented system in Example 1

    Figure 2.  Output of the closed-loop system (dash line) and the reference signal (solid line) in Example 1

    Example 2. Consider the inverted pendulum system

    $\left\{ \begin{align} & {{{\dot{x}}}_{1}}={{x}_{2}}+{{h}_{1}}({{x}_{1}}(t-{{\tau }_{1}})) \\ & {{{\dot{x}}}_{2}}={{f}_{2}}({{x}_{1}}, {{x}_{2}}, u)+{{h}_{2}}({{x}_{1}}(t-{{\tau }_{2}}), {{x}_{2}}(t-{{\tau }_{2}})) \\ & y={{x}_{1}}. \\ \end{align} \right.$

    (41)

    For the purpose of simulation, following plant dynamics are used

    $\begin{align} & {{h}_{1}}(\cdot )=2x_{1}^{2} \\ & {{h}_{2}}(\cdot )=\sin ({{x}_{1}}{{x}_{2}}) \\ \end{align}$

    (42)

    $\begin{align} & {{f}_{2}}(\cdot )=\frac{g\sin {{x}_{1}}-\frac{mlx_{2}^{2}\cos {{x}_{1}}\sin {{x}_{1}}}{{{m}_{c}}+m}}{l\left( \frac{4}{3}-\frac{m{{\cos }^{2}}{{x}_{1}}}{{{m}_{c}}+m} \right)}+ \\ & \frac{\frac{m\cos {{x}_{1}}}{{{m}_{c}}+m}}{l\left( \frac{4}{3}-\frac{m{{\cos }^{2}}{{x}_{1}}}{{{m}_{c}}+m} \right)}u \\ \end{align}$

    (43)

    where x1 and x2 denote the angle and angular velocity of the pendulum, respectively. g=9.8 m/s2 is the acceleration due to gravity, mc is the mass of cart, m is the mass of pole, l is the half length of pole, and u is the control input. The values of parameters are given by mc=0.1, m=1, and l=0.5. The reference trajectory is sin t.

    The initial values of states are all 0.1. The initial values of time-delay states are all 0. Two RBF NN are used in this simulation. The numbers of those NN are chosen as 10 and 103. The centers of NN are chosen as [-0.5, 0.5] and [-0.5, 0.5]3, respectively. The widths of RBF are all chosen as 1. The initial value of parameter θ1 and θ2 are set to 0. The initial value of parameter κ1 and κ2 are set to 0. The initial value of parameter z2 is set to 0. The controller parameters chosen for simulation are K1(t)=2, K2(t)=27, Γ1=Γ2=0.000 2I, γ=0.001, and csi=0.001.

    S imulation results are shown in Figs. 3 and 4. Fig. 3 shows that the given control input is bounded. Fig. 4 gives the output of the closed-loop system and the reference signal. The output of the closed-loop system tracks the reference input well.

    Figure 3.  Control input u for the augmented system in Example 2

    Figure 4.  Output of the closed-loop system (dash line) and the reference signal (solid line) in Example 2

  • Adaptive neural control has been proposed for a class of unknown single-input single-output (SISO) nonaffine pure-feedback systems with unknown time-delay. The DSC technique was used in the design procedure for avoiding the problem of explosion of complexity. By using the appropriate Lyapunov-Krasovskii function, the effects of the unknown time-delays are eliminated. Since the transformed system contains the unknown virtual control coefficients, the technique of Nussbaum gain function control scheme is adopted. The proposed control scheme guarantees that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. Simulations are given to show the effectiveness of the presented method.

Reference (38)

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