# Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- An adaptive fuzzy output-feedback control-strategy-based disturbance-observer for strict-feedback FO nonlinear systems with unknown external disturbances is achieved for the first time. It should be noted that the authors in [25,26,27,28] have considered a related topic. However, the references [25,26,27,28] are based on the complete measurement of system state information.
- (2)
- A novel FO nonlinear filter based on an auxiliary function is constructed to approximately replace the virtual control functions together with the corresponding fractional derivative, which not only erases the issue of complexity explosion, but also completely compensates for the effects of the boundary errors induced by the constructed filters. Although the authors of [40,41,42] considered adaptive control based on a filter signal for FO nonlinear systems, these results were obtained on the basis of a linear filter, and cannot directly compensate for the aforementioned effects.

**Notations:**In this paper, some specific notations are employed. ${R}^{i}$ means i-dimensional Euclidean space; $\u2225\xb7\u2225$ represents the Euclidean norm of a vector or matrix; ${N}^{+}$ is a positive integer.

## 2. Preliminaries and Problem Formulations

#### 2.1. Preliminaries

**Definition**

**1**

**.**Let $F:[{t}_{0},+\infty )\to R$ be a continuously differentiable function, then the Caputo FO derivative of F with order α satisfies:

**Remark**

**1.**

**Lemma**

**1**

**.**If the α-order derivative of a continuous function $V\left(t\right):[0,\infty )\u27f6R$ satisfies

**Lemma**

**2**

**.**Fractional differential operators of Caputo type satisfy the linear relation, i.e.,

**Lemma**

**3**

**.**For any $x,y\in {R}^{n},\u03f5>0,p>1,q<1$, and $\frac{1}{p}+\frac{1}{q}=1$, one has

**Lemma**

**4**

**.**For all $t\ge {t}_{0}$, a smooth function $\iota \left(t\right)\in R$ satisfies:

#### 2.2. System Descriptions and Control Objective

**Control Objective**: Design an adaptive fuzzy output-feedback controller, such that all the closed-loop signals are bounded and the system output y can track the reference signal ${y}_{d}$ well.

**Assumption**

**A1**

**Assumption**

**A2**

**Remark**

**2.**

## 3. Nonlinear Filter-Based Adaptive Fuzzy Output-Feedback Control Design

#### 3.1. Fractional-Order Fuzzy Observer Design

#### 3.2. Fractional-Order Nonlinear Filter Design

**Remark**

**3.**

#### 3.3. Disturbance Observer-Based Controller Design

**Assumption**

**A3**

**.**There are unknown constants ${\overline{d}}_{i}$ and ${\epsilon}_{i}^{*}$ satisfying ${|}_{0}^{C}{D}_{t}^{\alpha}{d}_{i}\left(t\right)|\le {d}_{i}^{*}$ and $\left|{}_{0}^{C}{D}_{t}^{\alpha}{\epsilon}_{i}\right|\le {\overline{\epsilon}}_{i}$.

**Remark**

**4.**

#### 3.4. Stability Analysis

**Theorem**

**1.**

**Proof.**

- (1)
- Construct the IF-THEN rules, select fuzzy membership functions, and generate the FLS (9).
- (2)
- Choose the observer gains ${r}_{1},{r}_{2},\dots ,{r}_{n}$ such that A is Hurwitz.
- (3)
- Select the matrix $Q>0$, and, by solving (14), the symmetric matrix $P>0$ is acquired.
- (4)
- Choose appropriate parameters to ensure ${\overline{a}}_{1}>0$, ${\overline{a}}_{k}>0,k=2,3,\dots ,n-1$, ${\overline{a}}_{n}>0$, $H>0$, $b{b}_{i}>0,i=1,2,\dots ,n$, and construct the FO state observer (11), the virtual controller and the actual controller (24), the parameter adaptation law (26), the disturbance observer (28), and the FO nonlinear filter (20), respectively.

**Remark**

**5.**

**Remark**

**6.**

## 4. Simulation Study

**Remark**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ma, Z.; Sun, K.
Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance. *Fractal Fract.* **2023**, *7*, 694.
https://doi.org/10.3390/fractalfract7090694

**AMA Style**

Ma Z, Sun K.
Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance. *Fractal and Fractional*. 2023; 7(9):694.
https://doi.org/10.3390/fractalfract7090694

**Chicago/Turabian Style**

Ma, Zhiyao, and Ke Sun.
2023. "Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance" *Fractal and Fractional* 7, no. 9: 694.
https://doi.org/10.3390/fractalfract7090694