Adaptive output feedback dynamic surface control of nonlinear systems with unmodeled dynamics and unknown high-frequency gain sign

doi:10.1016/j.neucom.2014.05.061

Neurocomputing

Volume 143, 2 November 2014, Pages 312-321

https://doi.org/10.1016/j.neucom.2014.05.061 Get rights and content

Abstract

In this paper, two adaptive output feedback control schemes are proposed for a class of nonlinear systems with unmodeled dynamics and unmeasured states as well as unknown high-frequency gain. Radial basis function (RBF) neural networks (NNs) are used to approximate the unknown nonlinear functions. K-filters are designed to estimate the unmeasured states. An available dynamic signal is introduced to dominate the unmodeled dynamics. By introducing the dynamic surface control (DSC) method, the bounded condition of the approximation error is removed, and the tracking control is achieved. Moreover, the number of adjustable parameters and the complexity of the design are both reduced. By theoretical analysis, the closed-loop system is shown to be semi-globally uniformly ultimately bounded (SGUUB). Simulation results are provided to illustrate the effectiveness of the proposed approach.

Introduction

Since adaptive tracking control was proposed based on backstepping for a class of strict-feedback nonlinear systems in [1], backstepping has become one of the most popular design methods and has been widely used to design adaptive control for a large class of nonlinear systems with a triangular structure in [1], [2], [3], [4]. As we know, unmodeled dynamics exists in many practical nonlinear systems, due to some factors, such as measurement noises, modeling errors, external disturbances, and modeling simplifications, and they can severely degrade the closed-loop system performance. Therefore, some different methods were discussed to deal with such systems with unmodeled dynamics using backstepping in [5], [6], [7]. By introducing an available dynamic signal and using backstepping design method, robust adaptive control was proposed for a class of nonlinear systems with unknown parameters and unmodeled dynamics as well as uncertain nonlinearities in [5]. Furthermore, a novel adaptive control was developed for similar systems in [6]. In contrast to [5], the adaptive control laws in [6] did not require any dynamic dominating signal to guarantee the robustness property of global stability. In [7], robust adaptive backstepping control was presented for a class of nonlinear systems with unmodeled dynamics and unit virtual gains by applying input-to-state stability property. In [8], robust adaptive neural tracking control was proposed for a class of pure-feedback nonlinear systems with unmodeled dynamics and unknown gain signs.

An obvious drawback in the traditional backstepping is the problem of “explosion of complexity”, which is caused by the repeated differentiations of certain nonlinear functions such as virtual control in [2], [3], [4]. To overcome the ‘explosion of complexity’, several dynamic surface control schemes were developed for a class of strict-feedback or pure-feedback nonlinear systems in [9], [10], [11], [12], [13]. Using implicit function theorem and DSC method as well as integral type Lyapunov function, adaptive neural dynamic surface control was proposed for a class of pure-feedback nonlinear systems with unknown dead-zone in [11], [12]. Two adaptive tracking DSC schemes were developed for a class of strict-feedback and non-affine pure-feedback nonlinear systems using radial basis function neural networks (RBFNNs). Using mean value theorem and Young׳s inequality, only one learning parameter needs to be tuned online in the whole controller design, and the computational burden was effectively alleviated in [13]. Combining backstepping with dynamic surface control, an adaptive control scheme was presented for a class of nonlinear systems in pure feedback form with unmodeled dynamics in [14].

However, the above results were only suitable for the nonlinear systems with measurable states. When the states of system are not measured, several adaptive output feedback control approaches were proposed in [l5–18]. Adaptive output feedback control was proposed for a class of nonlinear systems with unmodeled dynamics using a reduced-order partial-state observer and small gain approach in [15]. K-filters were introduced in [16]. Using the nonlinear small-gain approach and backstepping, adaptive fuzzy output feedback control was developed for a class of SISO nonlinear systems with the unstructured uncertainties, dynamic disturbances, and unmodeled dynamics in [17]. An adaptive dynamic surface control was developed in [18] for a class of output-feedback nonlinear systems discussed in [16]. In [19], an adaptive output feedback control approach is developed using K-filers and DSC for a class of uncertain nonlinear systems in the parametric output feedback form. In [20], globally stable adaptive output-feedback tracking control was proposed by using backstepping and Nussbaum function. In [21], adaptive fuzzy output feedback DSC was proposed by using K-filters and Nussbaum function for a class of nonlinear systems with unknown dead-zone and control direction. In [22], adaptive fuzzy output feedback control approach was proposed based on backstepping for a class of SISO nonlinear strict-feedback systems with unknown high-frequency gain sign and unmodeled dynamics. In [20], [21], [22], K-filters were designed to estimate unmeasured states. In [23], decentralized adaptive fuzzy output feedback control approach was proposed for a class of large-scale strict-feedback nonlinear systems with the unmeasured states. K-filters were designed to estimate the unmeasured states, and a dynamical signal was introduced to cope with dynamic uncertainties. Using Ito differential formula and fuzzy observer, two adaptive fuzzy output feedback control were proposed for a class of uncertain stochastic nonlinear systems with the known or unknown control direction in [24], [25]. Typically, the above methods used fuzzy systems or neural networks as approximation models for unknown system nonlinearities. A key condition in most of these methods was that the approximation error or the observer error was assumed to be bounded before the stability analysis of the closed-loop system was implemented.

Motivated by the previous works in [5], [16], [20], [21], [22], in this paper, adaptive neural network output feedback control is developed by combining K-filters with dynamic surface control. The main contributions and some comparisons of the paper are summarized as follows:

(i)
Two adaptive output feedback neural tracking control schemes are proposed for a class of nonlinear systems with unmodeled dynamics and unmeasured states by combining K-filters with DSC techniques in this paper while the considered systems do not include unmodeled dynamics in [18], [19], [20], [21], and all the states need to be measurable in [7], [8], [9], [10], [11], [12], [13], [14]. The first control scheme deals with unknown high-frequency gain by constructing appropriate virtual control at the first step of backstepping without using a Nussbaum gain compared with [20], [22], and the second scheme tackles unknown high-frequency gain by using a Nussbaum gain, whereas the control gain sign needs to be known in [16].
(ii)
The upper bound of the dynamic uncertain term is assumed to be the sum of two unknown continuous functions in unmodeled dynamics and output in this paper while it is supposed to be a polynomial in unmodeled dynamics and output with unknown coefficients in [22].
(iii)
The extra term $Q (y, v)$ which the approximation errors and the dynamic uncertainties bring about is effectively dealt with using the defined compact sets of DSC in the final step of the stability analysis without discussing two cases in [8].
(iv)
The circular arguments and the tuning function are avoided using DSC approach in this paper while the approximation error is assumed to be bounded before the closed-loop system is shown to be stable compared with the existing adaptive fuzzy /neural control results in [2], [3], [4], [7], [8], [14], [19], [22], [23], [24], [25], and backstepping method and the tuning function are employed in [20], [22]. Moreover, the tracking control is carried out in this paper while the tuning objective is implemented in [22].

The rest of the paper is organized as follows. The problem formulation and preliminaries are given in Section 2. The filter design is proposed based on radial basis function neural networks in Section 3. Adaptive output feedback dynamic surface control is developed without using Nussbaum gain, and the stability of the closed-loop system is analyzed in Section 4. In Section 5, adaptive output feedback dynamic surface control is discussed using a Nussbaum gain. Simulation results are performed to demonstrate the effectiveness of the approach in Section 6. Section 7 contains the conclusions.

Section snippets

Problem statement and preliminaries

Consider the following uncertain nonlinear systems with unmodeled dynamics in the output feedback form: ${\begin{matrix} \dot{z} = q (z, y) \\ \dot{x} = Ax + f (y) + G σ (y) u + Δ (z, y, t) \\ y = e_{1}^{T} x \end{matrix}$ where $A = [\begin{matrix} 0 & I_{n - 1} \\ 0 & 0 \end{matrix}], f (y) = [\begin{matrix} f_{1} (y) \\ ⋮ \\ f_{n} (y) \end{matrix}], Δ (z, y, t) = [\begin{matrix} Δ_{1} (z, y, t) \\ ⋮ \\ Δ_{n} (z, y, t) \end{matrix}]$ $x = [\begin{matrix} x_{1} \\ ⋮ \\ x_{n} \end{matrix}], e_{1} = [\begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}], G = [\begin{matrix} 0_{(n - m - 1) \times 1} \\ b \end{matrix}]$ $b = {[b_{m}, \dots, b_{1}, b_{0}]}^{T} \in R^{m + 1}$ , x is the unmeasured state; $u \in R$ is the input, and $y \in R$ is the output; $σ (y) \neq 0$ is a known positive continuous function; $f_{i} (y)$ is the unknown smooth function; $z \in R^{n_{0}}$ is the unmodeled dynamics, and $Δ_{i} (z, y, t)$ is the unknown smooth

Filter design based on neural networks

We employ the following filters: ${\begin{matrix} \dot{ξ} = A_{0} ξ + Ly, ξ \in R^{n} \\ {\dot{Ω}}^{T} = A_{0} Ω^{T} + F^{T} (y, u), Ω^{T} \in R^{n \times (m + 1 + N)} \end{matrix}$ where $A_{0} = A - {Le}_{1}^{T}, L = {[l_{1}, \dots, l_{n}]}^{T}, A_{0}$ is a Hurwitz matrix, that is, ${PA}_{0} + A_{0}^{T} P = - hI, P = P^{T} > 0$ where $h > 0$ is a design constant.

Define the state estimate as follows: $\hat{x} = ξ + Ω^{T} θ$ The observer error is defined as $ε = x - \hat{x}$ . Thus $x = ξ + Ω^{T} θ + ε$ $\dot{ε} = A_{0} ε + δ (y) + Δ (z, y, t)$ Let $Ω^{T} = [v_{m}, \dots, v_{1}, v_{0}, Ξ], Ω^{T} \in R^{n \times (m + 1 + N)}$ Inspired by [16], the novel filters are described as follows: ${\begin{matrix} \dot{ξ} = A_{0} ξ + Ly, ξ \in R^{n} \\ \dot{Ξ} = A_{0} Ξ + Φ^{T} (y), Ξ \in R^{n \times N} \\ \dot{λ} = A_{0} λ + e_{n} σ (y) u, λ \in R^{n} \end{matrix}$ According to the discussion in [16] and (10)

Adaptive DSC design without using Nussbaum gain

In this section, based on dynamic surface control method, we propose an adaptive output feedback tracking control scheme. The whole design needs ρ steps.

For convenience, some notations are presented below. ${\bar{s}}_{i} = {[s_{1}, s_{2}, \dots, s_{i}]}^{T}$ , ${\bar{y}}_{j} = {[y_{2}, \dots, y_{j}]}^{T}$ , where $s_{i}, y_{j}$ will be given in the controller design later, $i = 1, 2, \dots, ρ, j = 2, \dots, ρ . y_{j} = ω_{j} - α_{j - 1}, j = 2, \dots, ρ, ω_{j}$ is the output of a first-order filter with α_i−1 as the input, and α_i−1 is an intermediate control which shall be developed for the corresponding (i −1)th

Adaptive DSC design using Nussbaum gain

In this section, we will use Nussbaum gain to deal with unknown high-frequency gain sign. All notations are the same as Section 4.

In order to deal with the unknown control gain sign, the Nussbaum gain technique is employed in this section. A function $N (ζ)$ is called a Nussbaum-type function if it has the following properties: $(i) \lim_{s \to \infty} \sup \frac{1}{s} \int_{0}^{s} N (ζ) d ζ =+ \infty (ii) \lim_{s \to \infty} \inf \frac{1}{s} \int_{0}^{s} N (ζ) d ζ = - \infty$ Commonly used Nussbaum functions include: $ζ^{2} \cos (ζ)$ , $ζ^{2} \sin (ζ)$ , and $\exp (ζ^{2}) \cos ((π / 2) ζ)$ in [29], [30].

Step 1: Choose a

Simulation results

To verify the effectiveness of the proposed approach, a numerical example is given. Consider the following nonlinear system with unmodeled dynamics: ${\begin{matrix} \dot{z} = q (z, y) \\ {\dot{x}}_{1} = x_{2} + \frac{x_{1} - x_{1}^{3}}{1 + x_{1}^{4}} + 0.1 \cos t + z \\ {\dot{x}}_{2} = x_{1}^{2} \frac{1 - e^{- x_{1}}}{1 + e^{- x_{1}}} - (x_{1}^{2} + 2 x_{1}) \sin x_{1} + 0.2 σ (y) u + 0.1 \sin t + x_{1} z \\ y = x_{1} \end{matrix}$ where $q (z, y) = - z + y^{2} + 0.5$ , $σ (y) = 15 + y^{2}$ . The desired trajectory $y_{d} = 0.5 \sin (t) + 0.5 \sin (0.5 t)$ .

The filters are selected as follows: ${\begin{matrix} {\dot{ξ}}_{1} = - l_{1} ξ_{1} + ξ_{2} + l_{1} y \\ {\dot{ξ}}_{2} = - l_{2} ξ_{1} + l_{2} y \\ {\dot{Ξ}}_{(1)} = [- l_{1}, 1] Ξ + [ϕ_{1}^{T} (y) 0_{1 \times M_{1}}] \\ {\dot{Ξ}}_{(2)} = [- l_{2}, 0] Ξ + [0_{1 \times M_{2}} ϕ_{2}^{T} (y)] \\ {\dot{λ}}_{1} = - l_{1} λ_{1} + λ_{2} \\ {\dot{λ}}_{2} = - l_{2} λ_{1} + σ (y) u \end{matrix}$

The adaptation laws

Conclusions

Based on dynamic surface control method and using K-filters, adaptive output feedback neural tracking control has been proposed for a class of nonlinear systems with unmodeled dynamics and unmeasured states. Using two different methods, the unknown high-frequency gain sign has been effectively tackled. By introducing an available dynamic signal, unmodeled dynamics and dynamic uncertainties have been dealt with. By constructing appropriate unknown continuous function, the effect which the

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61174046).

Xiaonan Xia received the M.Eng. degree in Control Theory and Control Engineering from Southeast University in 2000. She is currently a Ph.D. degree candidate in Information Computing Science from Yangzhou University. She is now a lecturer at the Department of Automation, College of Information Engineering, Yangzhou University. Her research interests include robust adaptive control, intelligent control, and nonlinear control.

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Tianping Zhang received the B.Sc. degree in Mathematics from Yangzhou Teachers College, Yangzhou, China, in 1986, the M.Sc. degree in Operations Research and Control Theory from East China Normal University, Shanghai, China, in 1992, and the Ph.D. degree in Automatic Control Theory and Applications from Southeast University, Nanjing, China, in 1996. He is now a professor in the College of Information Engineering, Yangzhou University, Yangzhou, China. From October 2005 to October 2006, he was a visiting scientist in the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He has published more than 180 papers on journals and conferences. His current research interests include fuzzy control, adaptive control, intelligent control, and nonlinear control.

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Adaptive output feedback dynamic surface control of nonlinear systems with unmodeled dynamics and unknown high-frequency gain sign

Abstract

Introduction

Section snippets

Problem statement and preliminaries

Filter design based on neural networks

Adaptive DSC design without using Nussbaum gain

Adaptive DSC design using Nussbaum gain

Simulation results

Conclusions

Acknowledgments

Automatica

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Fuzzy Sets Syst.

Neurocomputing

Automatica

Automatica

Fuzzy Sets Syst.

Neurocomputing

J. Frankl. Inst.

Neurocomputing

Syst. Control Lett.

Syst. Control Lett.

Systematic design of adaptive controller for feedback linearizable systems

IEEE Trans. Autom. Control

Direct adaptive NN control of a class of nonlinear systems

IEEE Trans. Neural Netw.

Observer-based adaptive fuzzy backstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems

IEEE Trans. Fuzzy Syst.