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

doi:10.1016/j.fss.2015.07.012

Fuzzy Sets and Systems

Volume 290, 1 May 2016, Pages 56-78

https://doi.org/10.1016/j.fss.2015.07.012 Get rights and content

Abstract

In this paper, the problem of adaptive fuzzy quantized output-feedback control is investigated for a class of uncertain switched nonlinear systems in strict feedback form. The considered switched systems contain unknown nonlinearities, hysteretic quantized input and without requiring the system states being available for measurement. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, and a switched fuzzy state observer is designed to estimate the unmeasured states. The hysteretic quantized input is implemented to avoid the oscillation caused by logarithmic quantizer and decomposed into two bounded nonlinear functions. Based on the estimated states and using the backstepping design principle, an adaptive fuzzy quantized output feedback control scheme is developed. It is proved that whole adaptive fuzzy quantized control scheme can guarantee that all the variables in the closed-loop system are bounded under a class of switching signals with average dwell time, and also that the system output can track a given reference signal as closely as possible. The simulation results are given to check the effectiveness of the proposed approach.

Introduction

In several control engineering problems, the system to be controlled is characterized by a finite set of possible control actions. Such systems are referred to as systems with quantized control input and the possible values of the input represent the levels of quantization. For example, hydraulic systems using on/off valves are systems with quantized input, digital control, hybrid systems, automotive powertrain systems, networked control systems (for information processing of networked systems, all signals must be quantized before data transmission) [3], [4], [5], [41], [42], [43], [44]. Therefore, the control design for quantized control systems is important.

Recently quantized feedback control has attracted a great deal of attention. An important aspect is that utilizing quantization schemes can not only have sufficient precision, but also require low communication rate [1], [2]. Compared to the traditional logarithmic quantizer [1], [2], Hayakawaa et al. [3] extended the results from uncertain linear systems to uncertain nonlinear systems, and first developed a hysteretic quantizer, which can avoid the oscillation caused by logarithmic quantizer. The work in [4] investigated the adaptive quantized backstepping feedback control problems for SISO strict-feedback uncertain nonlinear systems with hysteretic quantized input. However, the choice of quantization parameters depends on controller design parameters and certain system parameters, it is difficult to choose appropriate quantizer parameters for a complex nonlinear system. It is easy to see that the results in [3], [4] include only the nonlinear uncertainties in strict-feedback form. In addition, Hayakawaa et al. [3] and Zhou et al. [4] required that the nonlinear functions included in the controlled systems are known or can be linearly parameterized. To overcome this difficult, Liu et al. [5] proposed an adaptive fuzzy quantized control for a class of nonlinear stochastic systems, which don't need assume the controlled systems are completely known. Although, the results of the above have made some achievement, they all required the states of the controlled systems are measured directly, and did not consider the control design problem for uncertain switched nonlinear systems in strict-feedback form, in which the states are unavailable for measurement.

In the past decade, adaptive backstepping fuzzy or neural control design schemes have got great development. The authors in [6], [7], [8], [9], [10] studied the adaptive fuzzy or neural control design problems for non-strict-feedback nonlinear systems by combing backstepping technique. The characteristic of the non-strict-feedback nonlinear system is that system functions contain the whole state variables. In [11], [12], the adaptive fuzzy state feedback control approaches were proposed for a class of SISO nonlinear systems with unknown dead zone output and unknown virtual control coefficients, respectively. The authors in [13], [14], [15] proposed adaptive fuzzy state feedback control design methods for a class of MIMO nonlinear systems with unknown directions and time-varying delays, respectively. Different from the above state feedback control design, the authors in [16], [17], [18], [19], [20], [21], [22], [23] proposed adaptive fuzzy output feedback control approaches for a class of uncertain nonlinear systems with input nonlinearities. It should be mentioned that the adaptive fuzzy control design methods on uncertain nonlinear systems in strict-feedback form with input quantization has not been reported yet. And also, although many fuzzy or neural adaptive control design problems have been investigated for uncertain nonlinear systems by combining the adaptive backstepping design technique with fuzzy-logic-systems (FLSs) or neural-networks (NNs) [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [45], [46], these adaptive fuzzy or neural control schemes are all focused on the non-switched nonlinear systems.

As we know, switched systems are an important class of hybrid systems, which can be described by a family of subsystems and a rule that orchestrates the switching between them [26], [27], [28], [29], [30]. The control design and stability on the switched systems have attracted many researchers a great interest. Recently, some control design methods have been proposed via the backstepping design technique for several classes of switched nonlinear systems [31], [32], [33], [34], [35], [36], [37], [38]. The works [31], [32], [33], [34] have investigated for a class of switched nonlinear systems based on the common Lyapunov function method. Two adaptive neural control schemes have been proposed for a class of switched nonlinear systems in [35], [36] based on average dwell-time technique. The work [37] proposed an adaptive neural network feedback control scheme for nonlinear switched impulsive systems under all admissible switched strategy. And also, Han et al. [38] proposed an adaptive neural network control method for a class of switched nonlinear systems with switching jumps and uncertainties in both system models and switching signals based on dwell-time property. However, all the above-mentioned adaptive control approaches required the states are measured directly, which limits the applicability of these control schemes in practical industrial systems. In addition, in the control design, the aforementioned adaptive control schemes do not consider the quantized input effect on the control performance. To the best of our knowledge, to date now, there are no results on adaptive fuzzy output feedback control available for uncertain switched nonlinear systems with immeasurable states and input quantization.

In this paper, the problem of adaptive fuzzy quantized output-feedback control is investigated for a class of uncertain switched nonlinear systems in strict-feedback form. The considered switched systems contain unknown nonlinearities, hysteretic quantized input and without requiring the system states being available for measurement. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, and a switched fuzzy state observer is designed to estimate the unmeasured states. By using the nonlinear decomposition of input quantization and in the framework of adaptive backstepping technique, a robust adaptive fuzzy output-feedback tracking control approach is developed. The stability of the closed-loop system is proved based on Lyapunov function method and the average dwell time method. Compared with the existing literature, the main contributions of this paper are summarized as follows.

(i) This paper proposed an adaptive fuzzy tracking output feedback control method for a class of switched nonlinear systems in strict-feedback form. The proposed adaptive control method has solved the state unmeasured problem via designing fuzzy switched state observer. Although the previous results in [31], [32], [33], [34], [35], [36], [37], [38] also studied the design problem for switched nonlinear systems, they all require that the states must be available for measurement. In addition, the control methods in [31], [32], [33], [34], [35], [36], [37], [38] do not consider the problem of quantized input.

(ii) This paper first investigated the adaptive fuzzy output-feedback quantized control design problem for uncertain switched nonlinear system with hysteretic quantized input. The proposed control scheme can not only guarantee the stability of the whole switched control system, but also can attenuate the effect of the hysteretic quantizer on the control performance. Note that recent results in [3], [4], [5] only focus on the non-switched nonlinear systems. In addition, the controlled systems in [3], [4], [5] all require that the states must be available for measurement. Thus they cannot be applied to the switched nonlinear system under consideration in this study.

Section snippets

System descriptions and assumptions

Consider the following uncertain switched nonlinear system in strict-feedback form: ${\begin{matrix} {\dot{x}}_{i} = x_{i + 1} + f_{i, σ (t)} ({\underline{x}}_{i}) + d_{i, σ (t)} (t), \\ i = 1, \dots, n - 1, \\ {\dot{x}}_{n} = q_{σ (t)} (u_{σ (t)}) + f_{n, σ (t)} (x) + d_{n, σ (t)} (t), \\ y = x_{1} \end{matrix}$ where ${\underline{x}}_{i} = {[x_{1}, x_{2}, \dots, x_{i}]}^{T} \in ℜ^{i}$ , $i = 1, 2, \dots, n ()$ are the states, $y \in ℜ$ is the output of system. The function $σ (t) : [0, \infty) \to M = {1, 2, \dots, m}$ is a switching signal which is assumed to be a piecewise continuous (from the right) function of time. Moreover, $σ (t) = k$ implies that the kth subsystem is active. $f_{i, σ (t)} (x), (i = 1, 2, \dots, n,)$ are unknown smooth

Switched fuzzy state observer design

In this section, a switched fuzzy state observer is proposed for estimating the immeasurable states $x_{i}$ , $i = 2, \dots, n$ in the switched strict-feedback nonlinear system (1).

For the kth subsystem in (1), it can be written as ${\begin{matrix} {\dot{x}}_{1} & = & f_{1, k} ({\hat{x}}_{1}) + x_{2} + Δ f_{1, k} + d_{1, k} \\ {\dot{x}}_{2} & = & f_{2, k} ({\underline{\hat{x}}}_{2}) + x_{3} + Δ f_{2, k} + d_{2, k} \\ ⋮ \\ {\dot{x}}_{n - 1} & = & f_{n - 1, k} ({\underline{\hat{x}}}_{n - 1}) + x_{n} + Δ f_{n - 1, k} + d_{n - 1, k} \\ {\dot{x}}_{n} & = & f_{n, k} (\hat{x}) + q_{k} (u_{k}) + Δ f_{n, k} + d_{n, k} \\ y & = & x_{1} \end{matrix}$ where $Δ f_{i, k} = f_{i, k} ({\underline{x}}_{i}) - f_{i, k} ({\underline{\hat{x}}}_{i}), i = 1, \dots, n; k \in {1, 2, \dots, m}$ .

By Lemma 2, we can assume that the nonlinear function $f_{i, k} ({\underline{x}}_{i})$ in (12) $1 \leq i \leq n; k \in {1, 2, \dots, m}$ can be

Adaptive fuzzy control design and stability analysis

To realize the control objective, this section will give the adaptive fuzzy output-feedback control design via backstepping design technique and the designed fuzzy state observer in the last section. The stability of the switched closed-loop system will be proved by using Lyapunov function and average dwell time method.

In the sequel, the n-step adaptive backstepping design procedures will be developed for the kth subsystem.

Let $z_{1} = y - y_{r}, z_{i} = {\hat{x}}_{i} - α_{i - 1, k}, 2 \leq i \leq n; k \in {1, 2, \dots, m}$ where $α_{i - 1, k}$ is a virtual

Simulation studies

In this section, a simulation example is given to illustrate the effectiveness of the proposed adaptive fuzzy output feedback control approach.

Example 1 A numerical example

Consider the following uncertain switched nonlinear system: ${\dot{x}}_{1} = x_{2} + f_{1, σ (t)} (x_{1}) + d_{1, σ (t)} (t)$ ${\dot{x}}_{2} = q_{σ (t)} (u_{σ (t)}) + f_{2, σ (t)} (x_{1}, x_{2}) + d_{2, σ (t)} (t)$ $y = x_{1}$ where $f_{1, 1} (x_{1}, x_{2}) = x_{1} \sin (x_{1}^{2})$ , $f_{2, 1} (x_{1}, x_{2}) = x_{2} / (1 + x_{1}^{2})$ , $d_{1, 1} (t) = 0.1 \sin (t)$ , $d_{2, 1} (t) = 0.1 \cos (t), f_{1, 2} (x_{1}, x_{2}) = 0.2 \cos (x_{1}), f_{2, 2} (x_{1}, x_{2}) = 0.2 x_{1} x_{2}^{2}, d_{1, 2} (t) = \cos (t)$ , $Δ_{1, 2} (t) = 0.1 \sin (t)$ , $d_{2, 2} (t) = 0.1 \cos (t)$ . The output is assumed to

Conclusions

This paper has studied the tracking control design problem for a class of switched nonlinear systems in strict-feedback form. The considered switched systems have completely unknown nonlinear functions, hysteretic quantized input and without direct requirement of the states measurement. The hysteretic quantized input is decomposed by two bounded nonlinear functions, fuzzy logic systems are utilized to model the switched nonlinear systems and a switched fuzzy state observer has been established

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^☆: This work was supported by the National Natural Science Foundation of China (Nos. 61374113, 61203008, 61074014).

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

Abstract

Introduction

Section snippets

System descriptions and assumptions

Switched fuzzy state observer design

Adaptive fuzzy control design and stability analysis

Simulation studies

Conclusions

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Automatica

Neural Netw.

Math. Comput. Simul.

Automatica

Fuzzy Sets Syst.

Automatica

Automatica

Neurocomputing

Neurocomputing

Neurocomputing

Syst. Control Lett.

Automatica

Inf. Sci.

Appl. Math. Comput.

Physica A

Adaptive quantized control for linear uncertain discrete-time systems

Automatica