Abstract—This paper investigates an active fuzzy fault tolerant tracking control (AFFTTC) for a class of multi-input, multi-output (MIMO) nonlinear systems with completely unknown dynamic, sensor faults, actuator faults and external disturbances. The proposed (AFFTTC) based on backstepping technique deals with three additive (bias, drift and loss of accuracy) and one multiplicative (loss of effectiveness) sensor and actuator faults. With the help of fuzzy logic systems (FLS’s), two adaptive fuzzy control laws are proposed. Lyapunov approach and Barbalat’s lemma are using to prove that the designed control strategy can stabilize the nonlinear system, which is subjected to the unknown nonlinear dynamic, actuator faults, sensor faults and disturbances, simultaneously. Numerical simulations on a quadrotor helicopter system are presented to show the effectiveness of the investigated control scheme
Index Terms— Adaptive fault-tolerant control; Fuzzy systems; MIMO unknown nonlinear systems; Backstepping method; Sensor fault; Actuator fault.
In the last decade universal approximation systems have been widely used in the area of the control theory, such as fuzzy systems and neuronal networks 1-4. Most of these works treat the case of uncertain nonlinear multivariable systems, varying in both structures and parameters, due to external disturbances, harsh operational environment or plant aging, however the aforementioned papers did not include the case when the systems are subject to various faults caused by actuators and sensors. These faults may provide undesirable effects and even can drive systems to become unstable or damaged 5-7. As control of systems becomes more complex, the security remains a key point and the development of new control theory integrating the faults that can occur on a system is of a great interest. Faults Diagnosis 5,6, and fault tolerant control 7 have become challenging in the area of modern control theory. For the fault tolerant control problem, one has to find a control law such that when a fault occurs from sensor and/or actuator, the system has to keep his own properties in terms of stability and tracking performances. The main idea to ensure these properties is to make the control law immune against these faults by canceling its bad effects.
Generally speaking, fault tolerant control (FTC) techniques can be classified into two categories 8,9. In passive FTC (PFTC) systems a single controller, with fixed structure or parameters, is used to deal with all possible failure scenarios which are assumed to be known a priori. Furthermore, the passive technique does not require online fault detection, diagnosis and control reconfiguration. Besides the implementation of this technique is easy and it is more conservative 10-12. In the literature, many suitable methods have been used and combined with each other for giving reliable controllers. In 13-16, authors proposed a lot of techniques specifically for linear system. In 13 suitable guaranteed cost control technique is proposed based on the LMI technique. Moreover, a reliable controller was developed in 14,15 based on LMI approach and static output feedback control and in 16, a reliable non-fragile compensation filter has been investigated. In 17-19, authors proposed fault tolerant control techniques specifically for non-linear systems. In 17 a mixed fuzzy static output feedback control was proposed using iterative LMI approach. A reliable filter was designed in 18 for a class of nonlinear networked control system using T-S fuzzy model and LMI technique. A static output feedback fuzzy controller has been employed in 19 for T-S fuzzy systems with sensor faults, from which the stability of these T-S fuzzy systems is guaranteed thanks to a sufficient condition based on LMIS.
However, if a failure that would not be considered in the design occurs, the stability and performance of the closed-loop system cannot be guaranteed. Such potential limitations of passive approaches provide a strong motivation for the development of methods and strategies for Active FTC (AFTC) systems 20. Active methods consist of online reconfiguring or reconstructing the controller to recover the stability and system performance as soon as a diagnostic algorithm has detected the presence of fault 21-32. The recent FTC literature shows a steady increase in interest of developing strategies for nonlinear systems via the use of T-S fuzzy modelling 33-35. T-S models are usually considered for fault estimation in nonlinear systems since the approach provides an opportunity to handle system nonlinearity via well-developed modern linear systems optimization and design tools. However, most of the work on this subject employing T-S fault estimation are focused on the actuator estimation and compensation problem and do not consider simultaneous actuator and sensor faults. In 36,37, robust detection and isolation schemes for nonlinear systems subject to known nonlinear actuator faults is developed with bounded control inputs and state variables. In 38 an active FTC law is designed based on an open-loop model of the system and a function of fault parameters under the assumption that they are immediately identified by an FDI model. In 39,40 a reconfigurable controller is developed based on FDI system represented by nonlinear observers for spacecraft under sensor bias occurrence. In all techniques based on faults detection (FD), there is some time delay between faults occurrence and faults accommodation. In 41 the authors designed active fault-tolerant controller using fault detection observer for T-S fuzzy models via delta operator approach. Some fault-tolerant control methods of T-S fuzzy systems are studied for networked control systems, such as in 42,43. The fault estimation and fault-tolerant control problems for a class of fuzzy stochastic systems with sensor failures are addressed by using fuzzy technique and sliding-mode observer scheme 44. A controller design problem is discussed for a class of nonlinear stochastic active fault-tolerant control system with Makovian parameters 45. In 46 the authors presented an integrated fault diagnosis (FD) and FTC design for polytopic LPV descriptor systems, then based on an adaptive polytopic observer (APO) for providing both states and actuator faults estimation. In 47 a sensor active fault tolerant control is presented for a class of nonlinear systems subjected to input constraints and based on output feedback control using high-gain observers. In 48 a proportional derivative (PD) sliding mode observer-based fault tolerant control scheme for a class of Lipchitz nonlinear systems against sensor faults is synthetized. In 49 a T-S dynamic output feedback control is presented using two T-S fuzzy observers dedicated to provide separate estimates of actuator and sensor faults for purpose of fault compensation. Since, autonomous systems such as aerial, space vehicles have highly nonlinear, interesting and unstable dynamics, fault tolerant control of these MIMO systems is one of the key issues that need to be addressed. In 50-54 the authors addressed the actuator faults case with only attitude measurement. However, the occurrence of sensor faults may affect the dynamics of the controlled system. In 55 the authors combined adaptive disturbance observer with back-stepping technique, under time-varying sensor faults for a class of nonlinear systems. In 56 an active FT tracking controller strategy is proposed for vehicle system to preserve the closed-loop stability in spite of sensor faults, based on T-S fuzzy representation of the system. In 57 an active fault-tolerant control approach is presented for vehicle active suspension systems in finite-frequency domain. In 58, active fault tolerant controller is developed using neural networks combined with sliding mode, and the performance index for spacecraft under one kind of actuator faults (loss of effectiveness). In 59, an active FTC based on back-stepping is developed for a class of MIMO uncertain nonlinear systems subjected to four kinds of velocity sensor faults including bias, drift, loss of accuracy and loss of effectiveness and is applied for attitude stabilizing of quad-rotor. In 60 an active fault tolerant control based on a differential geometry tool, and a fault detection and diagnosis is presented for a class of nonlinear systems subjected to an additive fault and applied on an aircraft. In 61 the authors investigated an adaptive fault tolerant control for a class of nonlinear systems based on Neuronal Networks and implicit function theorem with unknown two kinds of actuator nonlinear faults (bias and loss of effectiveness). In 62 authors proposed an adaptive FTC scheme to solve the tracking problem for a class of uncertain nonlinear MISO discrete-time using RBFNNs to approximate the unknown functions, and studied only the case when actuators are failed with both bias (Lock in place) and loss of effectiveness. In 63, authors based on a quantization strategy to deal with the problem of nonlinear uncertain MIMO systems, in the presence of only two kinds of actuator faults (Lock in place and loss of effectiveness). Two observers based fault estimation (FE) and fault tolerant control (FTC), for fuzzy systems with local nonlinear systems under only loss of accuracy for both sensor and actuator faults simultaneously 64.
The emphasis of the current work is to design an active fuzzy fault tolerant tracking control (AFFTTC) scheme for a class of huge second-order nonlinear systems subject to four kinds of sensors and actuators faults and external disturbances, where it is assumed that the dynamics of the system are unknown. The controller is updated according to the occurrence of actuator and/or sensor faults. Using the fact that if we introduce actuator and sensor faults models into the system, this change the dynamics of the system into a strict-feedback category, and to deal with this class of nonlinear systems, we use back-stepping technique with a novel fuzzy adaptive fault tolerant strategy. The proposed adaptive control law is composed of two terms and are developed to deal with different kinds of desired performances. The first control term is an adaptive fuzzy control law to deal with the unknown system dynamics and actuator faults, with online updating for all adaption laws for fulfilling the best performances. The second robust control term is designed to overcome the problem of fuzzy approximation errors, sensor faults, and external disturbances, with online updating for all adaption laws. The stability of the closed-loop systems is performed using Lyapunov approach and Barbalat’s Lemma, for ensuring the convergence of the tracking error, taking into consideration the unknown system, the disturbance, sensor and actuator faults.
Compared with other existing works in the same area, the main contributions of this paper lie in the following aspects.
1- Unlike in 50-54, the proposed controller does not need any information about the models of velocity sensor faults. Besides, the controller deals automatically with the sensor fault occurring on the system.
2- unlike in 13-19,39,40,47-49,64 where the authors proposed an FTC for one or two kind of sensor faults which may limits the applicability of the controller since other kinds of sensor faults may occur and then, the system may lose its own performances (tracking, stability). For these reasons, in the proposed work a comfortable controller including four kinds of sensor faults including bias, drift, loss of accuracy, and loss of effectiveness, to keep the proposed controller stronger with many kinds of sensor faults and keep his desired performances (stability, tracking, robustness).
3- Unlike in 46,58,61-64 where the authors presented an AFTC taking into consideration only one or two kind of actuator faults, which considerably limits the range of applicability of these AFTC approaches, while in the proposed controller, we considered four kinds of actuator faults including bias, drift, loss of effectiveness, and loss of accuracy.
4- Unlike in 55,59,64 authors have described the external disturbances as exogenous neuronal stable systems 55, and in 64 authors described the external disturbances into two part, one represents an estimated disturbances and the second is generated by an exogenous systems, while in 59, the authors approximated the external disturbance by free-models, but time-varying with derivable bounds which represents restrictive conditions. In our work, only the boundedness mild condition is made on external perturbations.
5- Unlike in 38-40,46 where fault detection and isolation FDI module is needed, in the proposed work, it is only focused on the on-line estimation, compensation and robust adaptive control scheme.
6- Unlike in 58,61 where the control gain of the system under control is considered as a simple constant, which limits the range of applicability of the approach. In the proposed approach, we assume the system dynamics are unknown with the control gain as an unknown nonlinear function to cover a lot of mechanical systems applications like (inverted pendulum, induction motor drive, single-link robot arm, mass-springer-damper system, flexible spacecraft, DOF helicopter, DOF quadrotor helicopter and many other systems).
7- Unlike in 55,59,63 where in 59, 63 the closed-loop system is (UUB) stable under time-varying with derivable bounds sensor faults and system uncertainties, and it’s globally asymptotically stable under slowly time-varying sensor faults and system uncertainties and the tracking error converge exponentially only to a compact set. While in 55 the closed-loop system is only locally asymptotically stable (UUB). In our work the closed-loop system is globally asymptotically stable under varying both sensor and actuator faults, and also the unknown system dynamics and the tracking error converge exponentially to a neighborhood of the origin.
The remainder of the paper is organized as follows: Section 2 introduces the class of nonlinear systems taking into consideration, the description of the structure for sensor and actuator faults, and some assumptions followed by a brief description of fuzzy systems. In section 3, an active fuzzy fault tolerant tracking control (AFFTTC) is developed and designed. Simulation example is performed on the model of a quadrotor helicopter given in section 4. Finally, the conclusion is given in section 5.