Observer-based adaptive fuzzy quantized control of uncertain nonlinear systems with unknown control directions

doi:10.1016/j.fss.2018.10.006

Fuzzy Sets and Systems

Volume 371, 15 September 2019, Pages 61-77

https://doi.org/10.1016/j.fss.2018.10.006 Get rights and content

Abstract

This paper studies the output-feedback tracking control problem for a class of strict-feedback systems with input quantization and unknown control directions. A coordinate transformation is first introduced such that the unknown control coefficients are lumped together and the studied system is transformed into an equivalent system. Subsequently, a high-gain fuzzy state observer is designed to estimate unmeasurable states. Based on the high-gain fuzzy state observer, the adaptive backstepping design procedure is proposed for the new defined system. Then, by introducing a Nussbaum function and a smooth function, the obstacle caused by unknown control direction and input quantization is successfully circumvented, and a coarse quantization can be achieved. Moreover, the proposed adaptive controller can guarantee that the closed-loop system is bounded, and the tracking error converge to a predefined small residue set. The main advantages of our method are that: first, the restrictions on system control gains are relaxed. Second, the controller design does not depend on the quantization parameter. Finally, simulation results are provided to illustrate the effectiveness and benefits of the proposed results.

Introduction

During the past decades, the study on uncertain nonlinear systems has attracted growing attention due to the widespread applications, such as one-link manipulators, hydraulic equipments, mass-spring-damper systems, and chemical reactors [1]. The main difficulty for controlling such complex systems is the presence of large parametric uncertainties or unknown nonlinear functions. To solve this problem, several methodologies, such as robust control, adaptive control [2], [3], [4], and fuzzy or neural control [5] have been proposed based on backstepping techniques. Among all, fuzzy or neural control is certainly the most popular one and involves an approximation-based procedure to compute the controller [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. In general, these methods of analysis have been proven to be instrumental for the design of several different controllers during the last years.

Although excellent results have been proposed for nonlinear systems, several challenging issues still remain open and need to be addressed. In particular when the nonsmooth nonlinearity, input quantization, is taken into account, which exists commonly in the modern control systems [28]. On the other hand, as we know, the quantization error cannot be avoided, which can severely limit the system performance and even make the system unstable [28]. Therefore, it is important and necessary to develop algorithms to compensate the quantization errors and maintain the desired performance.

Note that some effective approaches have been presented for the linear or nonlinear systems [29], [30], [31], [32], [33], [34], [35], [36] based on robust quantized control schemes. Especially, an adaptive quantized control approach was given in [37] for a discrete time linear uncertain system. Additionally, a direct adaptive quantized control framework was established in [38] for nonlinear system. In [39], a stabilization problem was considered for strict-feedback nonlinear systems based on the assumption that all the nonlinear functions of the system must be known and satisfy the global Lipschitz continuity condition. This result was later extended in [40], [41], [42] for a class of parameter nonlinear systems. The common feature of [39], [40], [41], [42] is that the state variables are all required to be available. In fact, such a requirement is strict for some practical control systems.

Recently, an output-feedback adaptive quantized control method was developed in [43] for a class of nonlinear systems, where a combination of a logarithmic (or a hysteresis) quantizer and a uniform quantize is further introduced to reduce communication expenses and minimize the effects of quantization error. However, a main limitation in [43] is that the nonlinear uncertainties are assumed to be linear with the unknown parameters. Considering the more general nonlinear systems, [44] investigated the fuzzy adaptive state-feedback tracking control problem for stochastic strict-feedback nonlinear systems via backstepping technique. Then, the fuzzy quantized output-feedback control problem is considered for the same stochastic strict-feedback nonlinear systems with unmodeled dynamics [45]. Recently, the fuzzy quantized output-feedback control problem is extended for a class of uncertain switched nonlinear systems in [46]. Although the above works investigated the hysteretic quantized control problem of the strict-feedback nonlinear systems from several different angles, there are few works published for nonlinear systems with unknown control direction. If the control direction is considered, the observer and controller designs in [43], [44], [45], [46] fail to work. Therefore, it is necessary to develop a new quantized control technique to address these problems, which motivates the present investigation.

In this paper, the problem of the adaptive output-feedback tracking control problem for a class of uncertain nonlinear system with input quantization and unknown control direction is studied. The main contributions of this paper are summarized as follows:

1) The restrictive assumptions on quantization parameter are removed [39], a novel quantizer control design method, which is independent of the quantization parameters, is introduced to compensate for the effect of input quantization. Besides, this paper also extends the state-feedback control in [39], [40], [41], [42] to the output-feedback case.

2) The restriction on the known control direction has been removed [43], [44], [45]. A coordinate transformation and the Nussbaum gain technique are employed to overcome the obstacle caused by the unknown control coefficients.

3) In contrast to the traditional fuzzy state observer adopted in existing output feedback adaptive control schemes [6], [7], we propose a novel high-gain fuzzy state observer to estimate the unmeasured states, which can make the tracking error $z_{1}$ converge to an arbitrarily small residual set by increasing the extra design parameter μ.

In what follows, the problem formulation and preliminaries are presented in Section 2 and high-gain fuzzy state observer is proposed in Section 3. The output feedback backstepping design method is given in Section 3. A simulation example is presented in Section 4. Finally, a conclusion of this paper is presented within Section 5.

Section snippets

Problem statement

Consider the following strict-feedback nonlinear systems $\begin{matrix} {\dot{x}}_{i} & = φ_{i} x_{i + 1} + f_{i} ({\underline{x}}_{i}) + d_{i} (t), 1 \leq i \leq n - 1, \\ {\dot{x}}_{n} & = φ_{n} q (u) + f_{n} ({\underline{x}}_{n}) + d_{n} (t), \\ y & = x_{1} \end{matrix}$ where ${\underline{x}}_{i} = {[x_{1}, x_{2}, \dots, x_{i}]}^{T} \in R^{i}, i = 1, 2, \dots, n$ , $x = {\underline{x}}_{n} \in R^{n}, u \in R$ and $y \in R$ denote the state vector, the control input and the system output, respectively. $f_{i} ({\underline{x}}_{i})$ is an unknown smooth nonlinear function, $φ_{i}$ is an unknown constant, $d_{i} (t)$ denotes bounded external disturbance, and $q (u)$ denotes the quantized input. As in [39], we consider the following hysteresis quantizer (see Fig. 1): $q (u) = {\begin{matrix} u_{i} sgn ( \end{matrix}$

Main results and stabilities

Owing to the presence of immeasurable states, a fuzzy state observer is firstly introduced to estimate them. Further, based on the estimations, an adaptive fuzzy controller is developed.

A simulation example

In this section, the effectiveness and merits of the proposed controller is demonstrated via an electromechanical dynamic system [7]. The dynamics of the electromechanical dynamic system is given by ${\begin{matrix} D \ddot{q} + B \dot{q} + N \sin (q) = I \\ M \dot{I} = - H I - K_{m} \dot{q} + V \end{matrix}$ where the descriptions of $q, \dot{q}, \ddot{q}, I, V$ , D, B, M, H, N, and $K_{m}$ can be found in [7].

Remark 5

Finally, it should be pointed out that since all the nonlinear functions $φ_{i}$ must be known, and only the state-feedback control schemes are studied, the existing adaptive control methods,

Conclusions

In this paper, by using the backstepping technique and a Nussbaum function, the adaptive fuzzy output-feedback tracking controller has been proposed for a class of uncertain nonlinear systems with input quantization and unknown control direction. The considered system is more general than those in most existing results on output-feedback control. The proposed controller guarantees that the closed-loop stability can be achieved, while the error signals converge to a small neighborhood of the

Acknowledgements

This work was supported in part by the Funds of National Science of China (Grant No. 61603166, 61420106016, 61621004), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant No. 2018ZCX03).

References (47)

Z.P. Jiang
A combined backstepping and small-gain approach to adaptive output feedback control
Automatica
(1999)
Y.J. Liu et al.
Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems
Automatica
(2017)
S. Tong et al.
Observer-based fuzzy adaptive control for strict-feedback nonlinear systems
Fuzzy Sets Syst.
(2009)
M. Wang et al.
Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems
Fuzzy Sets Syst.
(2008)
Y.J. Liu et al.
Adaptive fuzzy output feedback control for a class of nonlinear systems with full state constraints
IEEE Trans. Fuzzy Syst.
(2018)
Y. Li et al.
Robust adaptive output feedback control to a class of non-triangular stochastic nonlinear systems
Automatica
(2018)
T.P. Zhang et al.
Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form
Automatica
(2008)
L.J. Long et al.
Adaptive fuzzy output-feedback dynamic surface control of MIMO switched nonlinear systems with unknown gain signs
Fuzzy Sets Syst.
(2016)
A. Boulkroune et al.
Adaptive fuzzy controller for multivariable nonlinear state time-varying delay systems subject to input nonlinearities
Fuzzy Sets Syst.
(2011)
D. Liberzon
Hybrid feedback stabilization of systems with quantized signals
Automatica
(2003)

J. Lunze et al.

Deterministic discrete-event representations of linear continuous variable systems

Automatica

(1999)

T. Liu et al.

A sector bound approach to feedback control of nonlinear systems with state quantization

Automatica

(2012)

T. Hayakawa et al.

Adaptive quantized control for linear uncertain discrete-time systems

Automatica

(2009)

G.Y. lai et al.

Adaptive asymptotic tracking control of uncertain nonlinear system with input quantization

Syst. Control Lett.

(2016)

Y.X. Li et al.

Adaptive asymptotic tracking control of uncertain nonlinear systems with input quantization and actuator faults

Automatica

(2016)

L. Xing et al.

Output feedback control for uncertain nonlinear systems with input quantization

Automatica

(2016)

S. Sui et al.

Fuzzy adaptive quantized output feedback tracking control for switched nonlinear systems with input quantization

Fuzzy Sets Syst.

(2016)

F. Ferrante et al.

Stabilization of continuous-time linear systems subject to input quantization

Automatica

(2015)

H.K. Khalil

Nonlinear Systems

(2002)

C.Y. Wen et al.

Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance

IEEE Trans. Autom. Control

(2011)

L.X. Wang

Stable adaptive fuzzy control of nonlinear systems

IEEE Trans. Fuzzy Syst.

(1993)

S. Tong et al.

Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones

IEEE Trans. Fuzzy Syst.

(2012)

B. Chen et al.

Observer and adaptive fuzzy control design for nonlinear strict-feedback systems with unknown virtual control coefficients

IEEE Trans. Fuzzy Syst.

(2018)

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