Citation: Yang, P.; Shen, Z.; Ding, Y.;
Feng, K. Fast Terminal Sliding Mode
Fault-Tolerant Control for Markov
Jump Nonlinear Systems Based on an
Adaptive Observer. Drones 2022, 6,
233. https://doi.org/10.3390/
drones6090233
Academic Editors: Andrzej
Łukaszewicz, Wojciech Giernacki,
Zbigniew Kulesza, Jaroslaw Pytka
and Andriy Holovatyy
Received: 25 July 2022
Accepted: 30 August 2022
Published: 2 September 2022
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Article
Fast Terminal Sliding Mode Fault-Tolerant Control for Markov
Jump Nonlinear Systems Based on an Adaptive Observer
Pu Yang *, Ziwei Shen, Yu Ding and Kejia Feng
Department of Automation, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
* Correspondence: ppyang@nuaa.edu.cn
Abstract:
In this paper, a new adaptive observer is proposed to estimate the actuator fault and distur-
bance of a quadrotor UAV system with actuator failure and disturbance. Based on this, a nonsingular
fast terminal sliding mode controller is designed. Firstly, according to the randomness of faults and
disturbances, the UAV system under faults and disturbances is regarded as one of the Markov jump
nonlinear systems (MJNSs). Secondly, an adaptive observer is designed to simultaneously observe
the system state, fault, and disturbance. In order to improve the precision, the fast adaptive fault
estimation (FAFE) algorithm is adopted in the adaptive observer. In addition, a quasi-one-sided
Lipschitz condition is used to deal with the nonlinear term, which relaxes the condition and contains
more nonlinear information. Finally, a nonsingular fast terminal sliding mode controller is designed
for fault-tolerant control of the system. The simulation results show that the faults and disturbances
can be observed successfully, and that the system is stochastic stable.
Keywords:
fault-tolerant control (FTC); nonsingular fast terminal sliding mode control (NFTSMC);
UAV; FAFE; Markov jump nonlinear systems (MJNSs)
1. Introduction
Markov jump systems (MJSs) were firstly proposed by N. M. Krasovskii and E. A.
Lidskii in 1961 [
1
]. Over the years, relevant theories have been continuously improved.
Since they can better describe the system of stochastic mode jump, MJSs have been gradually
proven valid in the practical [
2
–
4
], and have had considerable research and application
in the fields of economics, physics, unmanned systems, machine learning, and so on.
On the other hand, unmanned aerial vehicles (UAVs) first appeared in the 1920s. Due
to their outstanding performance on the battlefield, Western countries ushered in an
upsurge of UAV research in the 1990s. In recent years, more and more cases of UAVs
being used in rescue and disaster relief have sprung up [
5
–
8
]. Considering the safety and
stability of UAVs, they can replace human beings by going to dangerous disaster relief sites
and accomplishing some dangerous tasks. Due to the particular working environment,
a quadrotor UAV system is easily affected by the wind environment, carried objects, human
intervention, and other factors in the process of performing missions. Therefore, the output
of the UAV system could be unstable. This kind of fault can be regarded as a Markov jump
process, and the UAV system can be described by the continuous-time MJSs.
At present, there have been many studies on the stability and control law of Markov
jump systems. Guan studies the stability of T-S fuzzy Markov jump systems based on
sampling control in [
9
]. Wang studies the stochastic stability of the MJSs, which are affected
by parameter uncertainty and actuator saturation [
10
]. Instead of asymptotic stability,
Chen et al. [
11
] paid more attention to the changes in the transient properties. They studied
the finite-time stability of a class of disturbed MJSs with random time delay. In terms of
the reinforcement learning of agents, Jiang creatively combines the reinforcement learning
method with Markov jump nonlinear systems (MJNSs). Based on this, he realizes optimal
tracking control for MJNSs in [
12
], which opened new data-based fields in the studies
Drones 2022, 6, 233. https://doi.org/10.3390/drones6090233 https://www.mdpi.com/journal/drones
