# Disturbance Observer-Based Event-Triggered Adaptive Command Filtered Backstepping Control for Fractional-Order Nonlinear Systems and Its Application

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Problem Formulation

#### 2.1. Fractional Calculus

**Definition**

**1**

**Definition**

**2**

**Lemma**

**1**

**Lemma**

**2**

#### 2.2. Fuzzy Logic Systems

**Lemma**

**3**

**Lemma**

**4**

#### 2.3. Problem Formulation

**Lemma**

**5**

**Remark**

**1.**

**Assumption**

**1.**

**Assumption**

**2.**

## 3. Main Results

#### NDO-Based Event-Triggered Adaptive Command-Filtered Quantized Control Design

**Step 1**. At first, we introduce the change of coordinates as:

**Step i**($i=2,\dots ,n-1$). Similar to the previous procedure, the FO filter is designed as:

**Remark**

**2.**

**Step n**. In this step, the following saturation function is used to bound the actual control signal. Then, one has

**Theorem**

**1.**

**Proof.**

**Case****(I):**- ${u}_{min}\le {u}_{Max}$: for this case, the following two sub-cases need to be discussed.
**Case****(i):**- $\left|u\left(t\right)\right|\le {u}_{min}$: according to Lemma 4 with ${\theta}^{2}\le {\left(\frac{1-\kappa}{\kappa}{u}_{min}\right)}^{2}$, we can obtain$$\begin{array}{cc}\hfill {D}^{\alpha}{V}_{n}\le & -\sum _{j=1}^{n}{c}_{j}{z}_{j}^{2}-\sum _{j=1}^{n}{\overline{\rho}}_{j}{\tilde{W}}_{j}^{T}{\tilde{W}}_{j}-\sum _{j=1}^{n}{\overline{a}}_{j}{\gamma}_{j}^{2}-\sum _{j=1}^{n}{\overline{l}}_{j}{\tilde{\Lambda}}_{j}^{2}+{\Xi}_{n}.\hfill \end{array}$$
**Case****(ii):**- $\left|u\left(t\right)\right|\ge {u}_{min}$, the following inequality holds in accordance with the relationship ${\theta}^{2}\le {\left(\frac{\kappa +\delta}{\kappa}u\right)}^{2}$ in Lemma 4$$\begin{array}{cc}\hfill {D}^{\alpha}{V}_{n}\le & -\sum _{j=1}^{n}{c}_{j}{z}_{j}^{2}-\sum _{j=1}^{n}{\overline{\rho}}_{j}{\tilde{W}}_{j}^{T}{\tilde{W}}_{j}-\sum _{j=1}^{n}{\overline{a}}_{j}{\gamma}_{j}^{2}-\sum _{j=1}^{n}{\overline{l}}_{j}{\tilde{\Lambda}}_{j}^{2}-\left|{z}_{n}\right|(1-\kappa ){u}_{min}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +|{z}_{n}|\left(\kappa +\delta \right)\left|u\right|+{\Xi}_{n}.\hfill \end{array}$$

**Case****(II):**- ${u}_{min}\ge {u}_{Max}$: for this case, we have $\left|u\right|\le {u}_{min}$. Therefore, a similar result can be obtained by referring to Case (i) in Case (I).

## 4. Simulation Verification

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Tejado, I.; Pérez, E.; Valério, D. Fractional calculus in economic growth modeling of the group of seven. Fract. Calc. Appl. Anal.
**2019**, 22, 139–157. [Google Scholar] [CrossRef] - Zhang, X.; Huang, W. Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties. Fractal Fract.
**2020**, 4, 50. [Google Scholar] [CrossRef] - Zouari, F.; Ibeas, A.; Boulkroune, A.; Cao, J.; Arefi, M.M. Adaptive neural output-feedback control for nonstrict-feedback time-delay fractional-order systems with output constraints and actuator nonlinearities. Neural Netw.
**2018**, 105, 256–276. [Google Scholar] [CrossRef] [PubMed] - Bao, H.; Park, J.H.; Cao, J. Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn.
**2015**, 82, 1343–1354. [Google Scholar] [CrossRef] - Lin, C.; Chen, B.; Shi, P.; Yu, J. Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems. Syst. Control. Lett.
**2018**, 112, 31–35. [Google Scholar] [CrossRef] - Coronel-Escamilla, A.; Gomez-Aguilar, J.F.; Stamova, I.; Santamaria, F. Fractional order controllers increase the robustness of closed-loop deep brain stimulation systems. Chaos Solut. Fractals
**2020**, 140, 110149. [Google Scholar] [CrossRef] - Luo, S.; Lewis, F.L.; Song, Y.; Ouakad, H.M.J. Accelerated adaptive fuzzy optimal control of three coupled fractional-order chaotic electromechanical transducers. IEEE Trans. Fuzzy Syst.
**2021**, 29, 1701–1714. [Google Scholar] [CrossRef] - Zouari, F.; Ibeas, A.; Boulkroune, A.; Cao, J.; Arefi, M.M. Neural network controller design for fractional-order systems with input nonlinearities and asymmetric time-varying Pseudo-state constraints. Chaos Solut. Fractals
**2021**, 144, 110742. [Google Scholar] [CrossRef] - Wei, Y.; Tse, P.W.; Yao, Z.; Wang, Y. Adaptive backstepping output feedback control for a class of nonlinear fractional order systems. Nonlinear Dyn.
**2016**, 86, 1047–1056. [Google Scholar] [CrossRef] - Liu, H.; Pan, Y.; Li, S.; Chen, Y. Adaptive fuzzy backstepping control of fractional-order nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst.
**2017**, 47, 2209–2217. [Google Scholar] [CrossRef] - Li, Y.; Wang, Q.; Tong, S. Fuzzy adaptive fault-tolerant control of fractional-order nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst.
**2021**, 51, 1372–1379. [Google Scholar] [CrossRef] - Yu, J.; Zhao, L.; Yu, H.; Lin, C.; Dong, W. Fuzzy finite-time command filtered control of nonlinear systems with input saturation. IEEE Trans. Cybern.
**2018**, 48, 2378–2387. [Google Scholar] [PubMed] - Li, Y.; Li, K.; Tong, S. Finite-time adaptive fuzzy output feedback dynamic surface control for MIMO nonstrict feedback systems. IEEE Trans. Fuzzy Syst.
**2019**, 27, 96–110. [Google Scholar] [CrossRef] - Qiu, J.; Sun, K.; Rudas, I.J.; Gao, H. Command filter-based adaptive NN control for MIMO nonlinear systems with full-state constraints and actuator hysteresis. IEEE Trans. Cybern.
**2020**, 50, 2905–2915. [Google Scholar] [CrossRef] [PubMed] - Niu, B.; Li, H.; Karimi, H.R. Adaptive NN dynamic surface controller design for nonlinear pure-feedback switched systems with time-delays and quantized input. IEEE Trans. Syst. Man Cybern. Syst.
**2018**, 48, 1676–1688. [Google Scholar] [CrossRef] - Liu, Y. Adaptive dynamic surface asymptotic tracking for a class of uncertain nonlinear systems. Int. J. Robust Nonlinear Control.
**2018**, 28, 1233–1245. [Google Scholar] [CrossRef] - Ma, Z.; Ma, H. Adaptive fuzzy backstepping dynamic surface control of strict-feedback fractional-order uncertain nonlinear systems. IEEE Trans. Fuzzy Syst.
**2020**, 28, 122–133. [Google Scholar] [CrossRef] - Song, S.; Zhang, B.; Song, X.; Zhang, Z. Neuro-fuzzy-based adaptive dynamic surface control for fractional-order nonlinear strict-feedback systems with input constraint. IEEE Trans. Syst. Man Cybern. Syst.
**2021**, 50, 3575–3586. [Google Scholar] [CrossRef] - Song, S.; Park, J.H.; Zhang, B.; Song, X. Observer-based adaptive hybrid fuzzy resilient control for fractional-order nonlinear systems with time-varying delays and actuator failures. IEEE Trans. Fuzzy Syst.
**2021**, 29, 471–485. [Google Scholar] [CrossRef] - Liu, H.; Pan, Y.; Cao, J. Composite learning adaptive dynamic surface control of fractional-order nonlinear systems. IEEE Trans. Cybern.
**2020**, 50, 2557–2567. [Google Scholar] [CrossRef] - Song, X.; Sun, P.; Song, S.; Stojanovic, V. Finite-time adaptive neural resilient DSC for fractional-order nonlinear large-scale systems against sensor-actuator faults. Nonlinear Dyn.
**2023**, 111, 12181–12196. [Google Scholar] [CrossRef] - Xu, B.; Sun, F.; Pan, Y.; Chen, B. Disturbance observer-based composite learning fuzzy control of nonlinear systems with unknown dead zones. IEEE Trans. Syst. Man Cybern. Syst.
**2017**, 47, 1854–1862. [Google Scholar] [CrossRef] - Chen, M.; Shao, S.; Shi, P. Disturbance-observer-based robust synchronization control for a class of fractional-order chaotic systems. IEEE Trans. Circuits Syst. II Express Briefs
**2017**, 64, 417–421. [Google Scholar] [CrossRef] - Min, H.; Xu, S.; Ma, Q.; Zhang, B.; Zhang, Z. Composite-observer-based output-feedback control for nonlinear time-delay systems with input saturation and its application. IEEE Trans. Ind. Electron.
**2018**, 65, 5856–5863. [Google Scholar] [CrossRef] - Hou, C.; Liu, X.; Wang, H. Adaptive fault tolerant control for a class of uncertain fractional-order systems based on disturbance observer. Int. J. Robust Nonlinear Control.
**2020**, 30, 3436–3450. [Google Scholar] [CrossRef] - Han, S.I. Fuzzy supertwisting dynamic surface control for MIMO strict-feedback nonlinear dynamic systems with supertwisting nonlinear disturbance observer and a new partial tracking error constraint. IEEE Trans. Fuzzy Syst.
**2019**, 27, 2101–2114. [Google Scholar] [CrossRef] - Wei, X.; Dong, L.; Zhang, H.; Hu, X.; Han, J. Adaptive disturbance observer-based control for stochastic systems with multiple heterogeneous disturbances. Int. J. Robust Nonlinear Control.
**2019**, 29, 5533–5549. [Google Scholar] [CrossRef] - Gao, Z.; Guo, G. Command-filtered fixed-time trajectory tracking control of surface vehicles based on a disturbance observer. Int. J. Robust Nonlinear Control.
**2019**, 29, 4348–4365. [Google Scholar] [CrossRef] - Liu, Z.; Wang, F.; Zhang, Y.; Chen, C.L.P. Fuzzy adaptive quantized control for a class of stochastic nonlinear uncertain systems. IEEE Trans. Cybern.
**2016**, 46, 524–534. [Google Scholar] [CrossRef] - Liu, W.; Lim, C.C.; Shi, P.; Xu, S. Backstepping fuzzy adaptive control for a class of quantized nonlinear systems. IEEE Trans. Fuzzy Syst.
**2017**, 25, 1090–1101. [Google Scholar] [CrossRef] - Song, S.; Park, J.H.; Zhang, B.; Song, X.; Zhang, Z. Adaptive command filtered neuro-fuzzy control design for fractional-order nonlinear systems with unknown control directions and input quantization. IEEE Trans. Syst. Man Cybern. Syst.
**2021**, 51, 7238–7249. [Google Scholar] [CrossRef] - Sui, S.; Chen, C.L.P.; Tong, S.; Feng, S. Finite-time adaptive quantized control of stochastic nonlinear systems with input quantization: A broad learning system based identification method. Int. J. Robust Nonlinear Control.
**2020**, 67, 8555–8565. [Google Scholar] [CrossRef] - Xing, L.; Wen, C.; Liu, Z.; Su, H.; Cai, J. Event-triggered adaptive control for a class of uncertain nonlinear system. IEEE Trans. Autom. Control.
**2017**, 62, 2071–2076. [Google Scholar] [CrossRef] - Song, X.; Wu, C.; Stojanovic, V.; Song, S. 1 bit encoding–decoding-based event-triggered fixed-time adaptive control for unmanned surface vehicle with guaranteed tracking performance. Control. Eng. Pract.
**2023**, 135, 105513. [Google Scholar] [CrossRef] - Hayakawaa, T.; Ishii, H.; Tsumurac, K. Adaptive quantized control for nonlinear uncertain systems. Syst. Control. Lett.
**2009**, 58, 625–632. [Google Scholar] [CrossRef] - Song, X.; Wang, M.; Ahn, C.K.; Song, S. Finite-time fuzzy bounded control for semilinear PDE systems with quantized measurements and Markov jump actuator failures. IEEE Trans. Cybern.
**2009**, 52, 5732–5743. [Google Scholar] [CrossRef] [PubMed] - Li, Y.; Chen, Y.; Podlubny, I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica
**2009**, 45, 1965–1969. [Google Scholar] [CrossRef] - Li, Y.; Yang, G. Adaptive asymptotic tracking control of uncertain nonlinear systems with input quantization and actuator faults. Automatica
**2016**, 96, 23–29. [Google Scholar] [CrossRef] - Ge, Z.M.; Yu, T.C.; Chen, Y.S. Chaos synchronization of a horizontal platform system. J. Sound Vib.
**2003**, 268, 731–749. [Google Scholar] [CrossRef] - Aghababa, M.P. Chaotic behavior in fractional-order horizontal platform systems and its suppression using a fractional finite-time control strategy. J. Mech. Sci. Technol.
**2014**, 28, 1875–1880. [Google Scholar] [CrossRef]

**Figure 3.**The state response of system (87) without control effort.

Parameters | Nomenclature | Value | Unit |
---|---|---|---|

A | Inertia moment of the platform around axis 1 | 0.3 | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

B | Inertia moment of the platform around axis 2 | 0.5 | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

C | Inertia moment of the platform around axis 3 | 0.2 | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

D | Damping coefficient | 0.4 | $\mathrm{kg}\xb7{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ |

F | Amplitude of the harmonic torque | 3.4 | $\mathrm{N}\xb7\mathrm{m}$ |

g | Acceleration constant of gravity | 9.8 | $\mathrm{m}\xb7{\mathrm{s}}^{-2}$ |

k | Proportional constant of the accelerometer | 0.11559633 | $\mathrm{kg}\xb7\mathrm{m}\xb7\mathrm{rad}$ |

R | Radius of the Earth | 6,378,000 | $\mathrm{m}$ |

$\omega $ | Circular frequency of the harmonic torque | 1.8 | $\mathrm{rad}\xb7{\mathrm{s}}^{-1}$ |

Design Parameters | Disturbance Terms |
---|---|

${c}_{1}={c}_{2}=50,{a}_{1}={a}_{2}={\rho}_{2}=2,\iota =0.01$, $\delta =0.2,\kappa =0.15,{u}_{min}=5,{u}_{Max}=8,{l}_{1}=10$, ${l}_{2}=20,{b}_{1}={b}_{2}=1,\chi =0.5,\alpha =0.95$. | ${d}_{1}(x,t)=1.5sin\left(2t\right)+0.5cos\left({x}_{1}{x}_{2}\right)$, ${d}_{2}(x,t)=1.5cos\left(2t\right)+0.5sin\left({x}_{1}{x}_{2}\right).$ |

Initial Conditions | |

${x}_{1}\left(0\right)=0.1$,${x}_{2}\left(0\right)=$ −$0.1,{\gamma}_{1}\left(0\right)={\gamma}_{2}\left(0\right)=0,$ ${\widehat{\Lambda}}_{1}\left(0\right)={\widehat{\Lambda}}_{2}\left(0\right)=0,\widehat{W}\left(0\right)={\underset{9}{\underbrace{[0,0,\dots ,0]}}}^{T}.$ | |

Reference Signal | |

${y}_{r}=0.5sin\left(t\right)+sin\left(0.5t\right)$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Song, S.; Song, X.; Tejado, I.
Disturbance Observer-Based Event-Triggered Adaptive Command Filtered Backstepping Control for Fractional-Order Nonlinear Systems and Its Application. *Fractal Fract.* **2023**, *7*, 810.
https://doi.org/10.3390/fractalfract7110810

**AMA Style**

Song S, Song X, Tejado I.
Disturbance Observer-Based Event-Triggered Adaptive Command Filtered Backstepping Control for Fractional-Order Nonlinear Systems and Its Application. *Fractal and Fractional*. 2023; 7(11):810.
https://doi.org/10.3390/fractalfract7110810

**Chicago/Turabian Style**

Song, Shuai, Xiaona Song, and Inés Tejado.
2023. "Disturbance Observer-Based Event-Triggered Adaptive Command Filtered Backstepping Control for Fractional-Order Nonlinear Systems and Its Application" *Fractal and Fractional* 7, no. 11: 810.
https://doi.org/10.3390/fractalfract7110810