Event-triggered adaptive fuzzy tracking control for stochastic nonlinear systems

doi:10.1016/j.jfranklin.2020.07.023

Journal of the Franklin Institute

Volume 357, Issue 14, September 2020, Pages 9505-9522

https://doi.org/10.1016/j.jfranklin.2020.07.023 Get rights and content

Abstract

In this paper, the problem of event-triggered mechanism based adaptive fuzzy tracking control is studied for strict-feedback stochastic nonlinear systems. The fuzzy logic systems are introduced to approximate the unknown nonlinear terms. A novel event-triggered tracking controller is designed by incorporating the backstepping design approach with the method of fuzzy control. The newly designed controller not only ensures that the output signal tracks the given reference signal within a sufficiently small neighborhood of the origin, but also ensures that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in the sense of the four-moment. Meanwhile, an event-triggered control mechanism is introduced to reduce the communication burden from the controller to the actuator. Finally, simulation results demonstrate the effectiveness of the designed controller.

Introduction

In the past several decades, backstepping design has become a powerful tool for nonlinear systems control design, and a lot of outstanding research results have been obtained [1], [2]. In [1], by employing backstepping design and sliding mode observer, an adaptive multiple-input multiple-output backstepping controller is designed for the induction machine. Meanwhile, stochastic disturbance signals inevitably appear in practical systems, therefore, many scholars have paid more and more research on the stochastic nonlinear systems. And some outstanding results have been developed on the stochastic nonlinear systems in the past decades [3], [4], [5], [6], [7]. Specifically, by using Lyapunov-Krasovskii functional and backstepping recursive design, an adaptive output-feedback controller was designed in [3] for stochastic nonlinear systems with input saturation and time-varying delay. Based on the Barrier Lyapunov Function (BLF) and backstepping design approach, an adaptive finite-time control scheme was proposed in [4] for stochastic nonlinear systems subject to full state constraints. The adaptive tracking control problem for switched stochastic nonlinear systems with unknown actuator dead-zone was investigated in [6]. However, a major limitations in the aforementioned results [3], [4], [5], [6], [7] were that the uncertain nonlinearities in the systems considered are either functions whose parameters are unknown and linear with respect to some known functions, or those bounded by known nonlinear functions. To relax the restrictions, some adaptive fuzzy or neural network controller [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] have been designed. For example, in [9], based on the fuzzy logic system (FLS) and backstepping design, an adaptive fuzzy controller was designed for stochastic non-strict feedback nonlinear systems with output constrained. In [11], the problem of adaptive fuzzy tracking control for stochastic nonlinear systems with input constraints was researched. In view of neural network and backstepping control approach, an adaptive control scheme was proposed in [13] for uncertain nonstrict-feedback stochastic nonlinear systems with output constraints and unknown dead zone. With the help of neural network and dynamic surface control approach, an adaptive control algorithm was proposed in [14] for nonstrict-feedback stochastic interconnected nonlinear systems with dead zone.

On the other hand, the classical sample data control scheme needs to transmit control signals to the system without interruption, which leads to the waste of communication resources and the increase of computation burden. This prompted some scholars to study new control schemes to reduce the communication resources and computing burden. Fortunately, an event-triggered control (ETC) schemes were proposed due to the ability in reducing communication resources and computation burden. To some extent, the researches about the event-triggered control of nonlinear systems are relatively perfect and has achieved outstanding results [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. For details, In [23], an adaptive event-triggered control problem was presented for nonlinear system. In [24], the results of [23] were extended to nonlinear systems with full state constraints and actuator fault which greatly reduces the communication burden from controller to actuator. Furthermore, an adaptive event-triggered controller was designed in [27] for stochastic nonlinear systems with actuator fault and unmeasured states via backstepping control approach and BLF approach. To overcome the serious uncertainty and sampling error of the system, a dynamic gain approach is proposed in [29]. However, to the best of our knowledge, until now, the research results about adaptive fuzzy control of stochastic nonlinear systems based on ETC are few. This motivates our current research.

In this paper, an adaptive event-triggered fuzzy control scheme is proposed for stochastic nonlinear systems. Compared with previous results, the contributions of this paper are summered as follows

(1)
The adaptive event-triggered tracking controller is designed for stochastic nonlinear systems via backstepping design approach, where the proposed controller could effectively saves network communication resources and computation burden.
(2)
Compared to existing results, in this paper, only one adaptive parameter is needed to estimate online the bound of unknown parameters for an $n -$ order stochastic nonlinear systems. This leads to the significantly reduction in the computational burden, and the designed control law is simpler and easier to implement in practical applications.

This paper consists of the following chapters. In Section 2, the state model of the system and the theory of fuzzification are described. In Section 3, an event-triggered controller based on fuzzy approximation nonlinear term is proposed firstly. Then, the stability of event-triggering is studied under the conditions of fixed threshold and relative threshold, respectively. Finally, the simulation result is given.

Section snippets

Problem statement and preliminaries

The following uncertain stochastic nonlinear systems are considered $\begin{matrix} {\begin{matrix} d x_{i} = (g_{i} ({\bar{x}}_{i}) x_{i + 1} + f_{i} ({\bar{x}}_{i})) d t + ψ_{i}^{T} ({\bar{x}}_{i}) d w, i = 1, \dots, n - 1, \\ d x_{n} = (g_{n} (x) u + f_{n} (x)) d t + ψ_{n}^{T} (x) d w, \\ y = x_{1}, \end{matrix} \end{matrix}$ where $x = {[x_{1}, \dots, x_{n}]}^{T} \in R^{n},$ u(t) ∈ R and y ∈ R are the system states, input and output, respectively; ${\bar{x}}_{i} = {[x_{1}, \dots, x_{i}]}^{T} \in R^{i}, i = 1, 2, \dots, n$ ; $f_{i} ({\bar{x}}_{i})$ and $ψ_{i} ({\bar{x}}_{i})$ are unknown smooth nonlinear functions. $g_{i} ({\bar{x}}_{i})$ is known smooth nonlinear function. For simplicity, we denote f_i(·) as f_i, g_i(·) as g_i and ψ_i(·) as ψ_i.

Consider the following stochastic nonlinear

Event-triggered control design scheme

To design an adaptive event-triggered controller, the following coordinate transformations are introduced: $ξ_{1} = x_{1} - y_{d},$ $ξ_{i} = x_{i} - α_{i - 1}, i = 2, \dots, n,$ where ξ_i denotes the error variable, $α_{i - 1}$ denotes the virtual control signal.

Step 1:From Eqs. (1) and (5), it follows: $d ξ_{1} = (g_{1} x_{2} + f_{1} - {\dot{y}}_{d}) d t + ψ_{1}^{T} d w .$ Select the following Lyapunov candidate function: $V_{1} = \frac{1}{4} ξ_{1}^{4} + \frac{1}{2 γ} {\tilde{θ}}^{2},$ where $\tilde{θ} = θ - \hat{θ}$ being the parameter estimation error, $θ = \max_{1 \leq i \leq n} {∥ Φ_{i} ∥^{2}}$ . By using It $\hat{o}$ formula, there holds $L V_{1} = ξ_{1}^{3} (g_{1} (ξ_{2} + α_{1}) + f_{1} - {\dot{y}}_{d}) + \frac{3}{2} ξ_{1}^{2} ψ_{1}^{T} ψ_{1} - \frac{1}{γ} \tilde{θ} \dot{\hat{θ}} .$ By

Stability analysis

Theorem 1

Consider the stochastic nonlinear system (1) with the event-triggering rules (18) and (20), the designed virtual controller (11), (16) and (28), the actual controller (19), the adaptive law $\dot{\hat{θ}} = ν_{n} - γ \hat{θ}$ . Suppose Assumption 1, Assumption 2, Lemma 1, Lemma 2 hold. Then, the closed-loop system satisfies the following properties

(i) All the signals in the closed-loop system are four-moment SGUUB.

(ii) The tracking error converges to a small neighborhood of the origin in probability.

(iii)The

Simulation example

In this part, an example is given to illustrate the effectiveness of the proposed control scheme. Consider the following systems: $\begin{matrix} {\begin{matrix} d x_{1} = (x_{1}^{2} + x_{2}) d t, \\ d x_{2} = (x_{3} + x_{1} x_{2}) d t, \\ d x_{3} = (u + x_{1} x_{2} x_{3}) d t + d w, \\ y = x_{1} . \end{matrix} \end{matrix}$ The reference signal is $y_{d} (t) = 0.5 \sin t$ . The fuzzy membership functions are denoted as follows $\begin{matrix} μ_{F_{i}^{1}} = e^{- \frac{1}{2} {(x_{i} + 1.5)}^{2}}, μ_{F_{i}^{2}} = e^{- \frac{1}{2} {(x_{i} + 1)}^{2}}, \\ μ_{F_{i}^{3}} = e^{- \frac{1}{2} {(x_{i} + 0.5)}^{2}}, μ_{F_{i}^{4}} = e^{- \frac{1}{2} {(x_{i})}^{2}}, \\ μ_{F_{i}^{5}} = e^{- \frac{1}{2} {(x_{i} - 0.5)}^{2}}, μ_{F_{i}^{6}} = e^{- \frac{1}{2} {(x_{i} - 1)}^{2}}, \\ μ_{F_{i}^{7}} = e^{- \frac{1}{2} {(x_{i} - 1.5)}^{2}} . \end{matrix}$ The fixed threshold strategy event-triggered controller is designed as $\begin{matrix} {\begin{matrix} α_{1} = \frac{1}{g_{1}} (- (c_{1} + 1) ξ_{1} - \frac{ξ_{1}^{3}}{2} - ξ_{1}^{3} θ \end{matrix} \end{matrix}$

Conclusions

In this paper, an adaptive fuzzy event-triggered tracking control scheme has been invested for a class of stochastic nonlinear systems. Based on the fuzzy logic systems and backstepping design technique approach, an adaptive fuzzy event-triggered tracking controller is designed. The proposed control scheme can not only guarantees the output signal tracks the given reference signal under the bounded error, but also all the signals in the closed-loop system are SGUUB in the sense of the

Declaration of Competing Interest

None.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61973148, 61773191, the Natural Science Foundation of Shandong Province under Grant ZR2018MF028, and the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant ZR2016JL025.

References (36)

J. Zhang et al.
Finite-time tracking control for stochastic nonlinear systems with full state constraints
Appl. Math. Comput.
(2018)
X.D. Zhao et al.
Adaptive tracking control for switched stochastic nonlinear systems with unknown actuator dead-zone
Automatica
(2015)
Y.J. Liu et al.
Adaptive control-based barrier Lyapunov functions for a class of stochastic nonlinear systems with full state constraints
Automatica
(2018)
B. Chen et al.
Direct adaptive fuzzy control of nonlinear strict-feedback systems
Automatica
(2009)
W.H. Zhang et al.
Fuzzy adaptive control of nonlinear MIMO systems with unknown dead zone outputs
J. Frankl. Inst.
(2018)
X. Huo et al.
Observer-based fuzzy adaptive stabilization of uncertain switched stochastic nonlinear systems with input quantizatio
J. Frankl. Inst.
(2019)
X.Y. Liang et al.
Dissipativity-based sampled-data control for fuzzy Markovian jump systems
Appl. Math. Comput.
(2019)
S. Yan et al.
Reliable h_∞ output control of nonlinear systems with dynamic event-triggered scheme
J. Frankl. Inst.
(2019)
T. Wang et al.
Robust adaptive fuzzy output feedback control for stochastic nonlinear systems with unknown control direction
Neurocomputing
(2013)
C.B. Regaya et al.
An adaptive sliding-mode speed observer for induction motor under backstepping control
ICIC Express Lett.
(2017)

C.B. Regaya et al.

A novel adaptive control method for induction motor based on backstepping approach using dspace DS 1104 control board

Mech. Syst. Signal Process.

(2018)

H.F. Min et al.

Output-feedback control for stochastic nonlinear systems subject to input saturation and time-varying delay

IEEE Trans. Autom. Control

(2019)

Z.Y. Sun et al.

Global output-feedback stabilization for stochastic nonlinear systems: a double-domination approach

Int. J. Robust Nonlinear Control

(2018)

W. Sun, S.F. Su, Y.Q. Wu, J.W. Xia, A novel adaptive fuzzy control for output constrained stochastic non-strict...

J.W. Xia et al.

Finite-time adaptive fuzzy control for nonlinear systems with full state constraints

IEEE Trans. Syst. Man Cybern.: Syst.

(2019)

H.Y. Li et al.

Adaptive neural control of uncertain nonstrict-feedback stochastic nonlinear systems with output constraint and unknown dead zone

IEEE Trans. Syst. Man Cybern.: Syst.

(2017)

B. Niu et al.

Adaptive neural-network-based dynamic surface control for stochastic interconnected nonlinear nonstrict-feedback systems with dead zone

IEEE Trans. Syst. Man Cybern. Syst.

(2019)

S. Sui et al.

Neural network filtering control design for nontriangular structure switched nonlinear systems in finite time

IEEE Trans. Neural Netw. Learn. Syst.

(2019)

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Event-triggered adaptive fuzzy tracking control for stochastic nonlinear systems

Abstract

Introduction

Section snippets

Problem statement and preliminaries

Event-triggered control design scheme

Stability analysis

Simulation example

Conclusions

Declaration of Competing Interest

Acknowledgments

Appl. Math. Comput.

Automatica

Automatica

Automatica

J. Frankl. Inst.

J. Frankl. Inst.

Appl. Math. Comput.

J. Frankl. Inst.

Neurocomputing

An adaptive sliding-mode speed observer for induction motor under backstepping control

ICIC Express Lett.

A novel adaptive control method for induction motor based on backstepping approach using dspace DS 1104 control board

Mech. Syst. Signal Process.

Output-feedback control for stochastic nonlinear systems subject to input saturation and time-varying delay

IEEE Trans. Autom. Control

Global output-feedback stabilization for stochastic nonlinear systems: a double-domination approach

Int. J. Robust Nonlinear Control

Finite-time adaptive fuzzy control for nonlinear systems with full state constraints

IEEE Trans. Syst. Man Cybern.: Syst.

Adaptive neural control of uncertain nonstrict-feedback stochastic nonlinear systems with output constraint and unknown dead zone

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