Global output feedback stabilization control for non-strict feedback nonlinear systems

doi:10.1016/j.ejcon.2021.09.002

European Journal of Control

Volume 63, January 2022, Pages 126-132

https://doi.org/10.1016/j.ejcon.2021.09.002 Get rights and content

Abstract

This paper studies global output feedback stabilization (OFS) control for non-strict feedback nonlinear systems with unknown functions (UFs). In order to deal with the UFs problem, a Lemma is proposed. This Lemma not only avoids the disadvantages of approximation methods, but also avoids the Assumptions of UFs. Therefore, the algorithm in the paper reduces the conservatism. Then, a control algorithm is proposed to solve the global OFS control problem of closed-loop systems. At the same time, the “explosion of terms” problem of backstepping is also solved. Finally, the algorithm is applied to Duffing system and Chua’s oscillator system to verify the effectiveness of the algorithm.

Introduction

Practical systems such as aeroelastic systems and thermodynamic systems are nonlinear and have many uncertainties [12], [30]. Therefore, it is very meaningful to study the control problem of uncertain nonlinear systems [2], [13], [23], [25], [29], [31], [33], [34]. Among them, the OFS control based on part of the measurement information of the system has been widely concerned [34], and a large number of important results have been obtained [5], [9], [10], [36], [39]. Among the results of many OFS control methods, the controlled system has not only unmeasurable system states, but also UFs [5], [9], [10], [22], [36], [39]. Zhao et al. [39] proposed a solution to adaptive OFS control for a class of nonsmooth nonlinear systems with UFs based on fuzzy control. An adaptive neural OFS compensation control scheme based on command-filtered backstepping was developed in Nai et al. [22]. It is worth mentioning that the above methods are based on fuzzy approximation method and neural network approximation method [22], [39].

According to the characteristic of fuzzy control and neural network control, any smooth function in compact sets can be approximated by the fuzzy logic function or the neural network [8]. Based on this characteristic, the UFs problem of nonlinear systems are transformed into the unknown parameters problem. Then adaptive control is used to design the controller [1], [14], [16], [20], [21], [24], [26], [28], [35], [37]. Therefore, it can be seen from many similar references that the problem of OFS control for nonlinear systems with UFs has been successfully solved. However, the approximation method also has some defects, as described in reference [8]: In most of approximation methods, controllers and adaptive laws are derived under Assumptions that systems states remain in some compact sets for all time. If systems states leaves the region due to the bad transient behavior, then the modeling errors may be large, which could force systems states even further out from compact sets, eventually causing instability of the systems. However, in the current similar results, many results have not given how to obtain this compact sets, so this method is defective. At the same time, if approximation methods are introduced to approximate UFs, then the approximation errors are also introduced [27]. This makes the algorithm only achieve semi-globally uniformly ultimately bounded (SGUUB) results [39]. Then, the global OFS control of nonlinear systems with UFs has not been solved.

In fact, in many results, the global OFS control algorithm of nonlinear systems has been well solved [4], [6], [7], [15], [18], [19], [38]. Liu [19] was concerned with the global regulation by output feedback for a class of uncertain nonlinear systems with unmeasurable states and UFs. The problem of OFS was investigated for a class of nonlinear systems with UFs in the [18]. In the above results [4], [6], [7], [15], [18], [19], [38], UFs in nonlinear systems need to satisfy certain Assumptions. Then the controller can be designed to stabilize the nonlinear system globally. According to the [4], [6], [7], [15], [18], [19], [38], we need to calculate the corresponding parameters and functions to satisfy these Assumptions. In practical application, it is not easy to obtain these parameters and functions. Therefore, these Assumptions increase the conservatism of the control algorithm and reduce the applicability. Therefore, it is very meaningful to avoid these Assumptions to design a global OFS control algorithm for nonlinear systems with UFs.

In this paper, a class of non-strict feedback systems with UFs are studied based on backstepping. In order to solve the global OFS problem, the contributions of this paper are summarized as follows

1)
Compared with the approximation methods [1], [14], [16], [20], [21], [24], [26], [28], [35], [37], the algorithm in the paper does not need to introduce approximation errors. Therefore, this algorithm can solve the global control problem. And the algorithm does not use adaptive control, so the algorithm is relatively simple and easy to implement.
2)
A Lemma is proposed such that Assumptions of UFs are avoided [4], [6], [7], [15], [18], [19], [38]. Therefore, the algorithm in the paper reduces the conservatism and increases the applicability.
3)
This Lemma also solves the “explosion of terms” of backstepping. Compared with the dynamic surface control (DSC) method [32], the proposed control algorithm can not only avoid the introduction of filters, but also solve the global OFS control problem of the controlled system.

In summary, a global OFS control algorithm is proposed for nonlinear non-strict feedback systems with UFs by using backstepping. Finally, the algorithm is applied to Duffing systems and Chua’s oscillator control [17], [39], and the simulation results verify the effectiveness of the algorithm.

The remaining parts of this paper are as follow: Systems and preliminaries are described in Section 2, the controller design of nonlinear systems is designed in Section 3, the stability analysis is written in Section 4, two examples are given in Section 5, and the conclusions of this paper is described in Section 6.

Section snippets

Systems and preliminaries

Consider a class of $n$ th order nonlinear uncertain non-strict feedback systems described by $\begin{matrix} {\begin{matrix} {\dot{x}}_{i} = x_{i + 1} + f_{i} (x), i = 1, \dots, n - 1, \\ {\dot{x}}_{n} = u + f_{n} (x), \\ y = x_{1}, \end{matrix} \end{matrix}$ where ${\bar{x}}_{i} = {[x_{1}, \dots, x_{i}]}^{T} \in R^{i}$ and $x = {[x_{1}, \dots, x_{n}]}^{T} \in R^{n}$ are the system state vectors, and the only output variable $y = x_{1} \in R$ is measurable. $u \in R$ is the system input. $f_{i} (x)$ ( $i = 1, \dots, n$ ) is a smooth nonlinear UFs with $f_{i} (0, \dots, 0) = 0$ , respectively.

The control objectives of this paper are as follows: In order to solve global OFS problem of uncertain nonlinear systems (1) with UFs, control

Controller design

In this section, the backstepping method is utilized to designed control law.

Step 1: Define the coordinate transformation $z_{1} = x_{1}$ and $z_{2} = {\hat{x}}_{2} - α_{1} (z_{1})$ , where $α_{1} (z_{1})$ is a virtual controller. From (3), the dynamic of $z_{1}$ is shown as follows ${\dot{z}}_{1} = x_{2} + f_{1} (x) = {\hat{x}}_{2} + E_{2} + f_{1} (x) = α_{1} (z_{1}) + z_{2} + E_{2} + F_{1} (x),$ where $F_{1} (x) = f_{1} (x)$ .

Define Lyapunov function $V_{1}$ as $V_{1} = V_{0} + \frac{1}{2} z_{1}^{2} .$

From (8) and (9), the dynamic of $V_{1}$ is given by $\begin{matrix} {\dot{V}}_{1} & \leq & - {ℓ (E) ∥ E ∥}^{2} + 2 E^{T} Φ_{0} F (x) \\ + z_{1} (α_{1} (z_{1}) + z_{2} + E_{2} + F_{1} (x)) . \end{matrix}$

Based on Young’s inequality, the following formulas are given by $2 E^{T} Φ_{0}$

Stability analysis

Based on the control law above, the following Theorem is summarized.

Theorem 1

Consider global OFS problem of uncertain nonlinear systems (1) with UFs. Construct the controller (29) and virtual laws (16) and (22). Thus, all signals of the closed-loop systems are globally bounded, and the state ${\hat{x}}_{i}$ ( $i = 2, \dots, n$ ) of observer systems (3) can observe the state $x_{i}$ of the controlled systems (1). At the same time, $\lim_{t \to \infty} x_{i} (t) = 0$ ( $i = 1, \dots, n$ ).

Proof 2

According to inequality (30), since ${∥ F (x) ∥}^{2} \geq 0$ and $\sum_{j = 1}^{n} \frac{1}{τ_{j 2}} F_{j}^{2} (x) \geq 0$ , it is

Simulation example

In this section, two simulation examples are given to elaborate the effectiveness of the control method in the paper.

Example 1

In order to better illustrate the effectiveness of the algorithm in this paper, consider the following control Duffing system [11] ${\ddot{y}}_{1} + p_{1} y_{1} + p_{2} y_{1}^{3} + p_{3} {\dot{y}}_{1} = u,$ where $p_{1}$ , $p_{2}$ and $p_{3}$ are constant parameters. $u$ is the control input.

According to [11], we can know that the control problem of nonlinear system (45) can be described as follows: A controller $u$ is designed so that all signals of

Conclusions

Global OFS control for nonlinear systems with UFs and unmeasurable states has been studied. Compared with the existing results, the proposed Lemma 1 makes the algorithm in this paper have the following advantages: (1) Compared with the approximation algorithms, the algorithm in the paper can guarantee that all signals of the closed-loop systems are globally bounded; (2) compared with the existing global algorithms, the algorithm in this paper can avoid Assumptions of UFs. This can reduce the

Declaration of Competing Interest

Authors declare that they have no conflict of interest.

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^☆: This work was supported by the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant KYCX21_0305.

View full text

Global output feedback stabilization control for non-strict feedback nonlinear systems☆

Abstract

Introduction

Section snippets

Systems and preliminaries

Controller design

Stability analysis

Simulation example

Conclusions

Declaration of Competing Interest

Automatica

J. Frankl. Inst.

Eur. J. Control

Automatica

J. Frankl. Inst.

Automatica

Syst. Control Lett.

Eur. J. Control

Adaptive decentralized fuzzy dynamic surface control scheme for a class of nonlinear large-scale systems with input and interconnection delays

Eur. J. Control

Static state feedback linearization of nonlinear control systems on homogeneous time scales

Math. Control Signals Syst.

Global exponential stabilization of a class of nonlinear systems by output feedback

IEEE Trans. Autom. Control

Cooperative output regulation problem of nonlinear multiagent systems with proximity graph via output feedback control

IEEE Trans. Cybern.

Norm estimators and global output feedback stabilization of nonlinear systems with ISS inverse dynamics

IEEE Trans. Autom. Control

Global fixed-time output feedback stabilization of perturbed planar nonlinear systems

IEEE Trans. Circuits Syst. II

Stable Adaptive Neural Network Control

On an output feedback finite-time stabilization problem

IEEE Trans. Autom. Control

Nonlinear Output Regulation: Theory and Applications

Robust adaptive prescribed performance control for a class of nonlinear pure-feedback systems

Int. J. Robust Nonlinear Control

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

IEEE Trans. Syst. Man Cybern.