Adaptive Fuzzy Fault-Tolerant Control for a Class of Nonlinear Systems under Actuator Faults: Application to an Inverted Pendulum

Recently, the Adaptive control technique has witnessed much attention in control theory society due to its ability to deal with system uncertain or unknown dynamics [1]-[9]. Generally speaking, universal approximator such as neural networks (NNs) and fuzzy logic systems (FLSs) tools was a good solution to overcome system uncertainty [10] [11], or fuzzy systems as universal approximator [1]-[6]. Various Fuzzy adaptive techniques have been developed in the literature classified from SISO to MIMO linear and nonlinear systems. In the design stage of fuzzy adaptive control law, direct and indirect approaches have been studied. In the direct, one controller consists to approximate the ideal control law with the help of a fuzzy system (see Refs. [12] [13] [14] [1]-[9]). However, the indirect resides on the approximation of the uncertain nonlinear system using fuzzy systems and based on these approximations. A general adaptive controller is built [1-5] [12] [13] [16]-[19]. On the other side, the adaptive technique was integrated with fault-tolerant control approach to handle actuator and sensor failures. Practically, sensor and/or actuator faults seem unavoidable separately or collectively due to their importance. If an actuator or sensor faults occur during the system operation, this can lead to a catastrophic behavior and drive the system to instability. Authors in [1] have investigated an adaptive fuzzy fault-tolerant control scheme for a class of nonlinear systems ARTICLE INFO ABSTRACT

with simultaneous actuator and sensor failures. A combination method based on fuzzy systems (FSs) and backstepping approach allowed the online estimation of the adaptive parameters and guaranteed the boundedness of all signals in the closed-loop system, while in [4], an active fault-tolerant control technique has been proposed for a class of second-order nonlinear system subjected to state-dependent actuator faults with the presence of unknown control gain sign and external disturbances. In [20], adaptive fault-tolerant control is applied on a flexible spacecraft with state-dependent actuator failures using simple linear sets of system states and errors combination. In [21], the authors proposed a dynamic surface-based control approach using the Nussbaum-type function for attitude stabilization of a spacecraft under actuator saturation. More results can be found in [22], where an active fault-tolerant control scheme has been developed for a class of MIMO nonlinear systems with sensor failures based on dynamic surface control (DSC).
Based on the aforementioned works, a fuzzy adaptive fault-tolerant control strategy is proposed for a class on the nonlinear system with actuator faults, exogenous disturbance, and uncertainties. A modified controller with new adaptive algorithms are designed and the upper and lower bounds of the control gain sign (CGS). An additional robust control term is added to circumvent the problem of approximation errors and mollify the tracking curves.
The main contributions of the proposed controller are summarized below: i. The proposed controller, along with a robust term, is superior to the controller performance in [14].
ii. The actuator faults model is time-varying parameters with bias, drift, loss of accuracy, and loss of effectiveness, which make the controller affordable against large faults scale.
iii. The exogenous disturbance is handled theoretically instead of approximation.
The rest of this paper is designed as follows: Problem formulation along with the studied class is first described, followed by a brief description of the universal approximation, i.e., fuzzy logic systems. Then, the proposed direct adaptive fuzzy fault-tolerant control scheme is presented with the corresponding adaptive laws and the stability analysis using Lyapunov methodology. A simulation example on the dynamic model of an inverted pendulum is performed to evaluate the accuracy of the proposed technique. Finally, some conclusions and general comments are given.

Problem Formulation
A class of SISO nonlinear systems without faults (faults free) can be written under the following equations [3] [6] Which can be concise and written as where = [ 1 , … , ] ∈ ℝ , is the vector of the system; ∈ ℝ is the scalar control input; ∈ ℝ is the scalar system output; ( ) and ( ) are unknown smooth nonlinear functions; ( ) is considered as an exogenous disturbance.
In respect to the dynamic of the system (2), the following assumptions will be made:  Regarding the faults given in Table 1, then the faulty actuator can be defined by the following compact form Meanwhile, the system described in (2a) will take the form below The objective is to design an adaptive fuzzy controller for system (2c) under actuator faults, exogenous disturbances, and uncertainties so that the system output ( ) can stably follow a referred trajectory ( ). Its stability can be defined as all signals in the closed-loop system stay bounded.
Nevertheless, the tracking error vector can be defined as If the functions ( ( ) , ( ), ( )and ( )) are known, then the control objective is achieved, and the ideal control law can be considered as: where is the solution of the Lyapunov-like equation, which will be designed later, and, Hence, the dynamic error can be further written as The Lyapunov-like equation is defined as where defines a symmetric positive definite matrix satisfying the Lyapunov-like equation with > 0.
Finally, the time-derivative of the Lyapunov-like equation along with the error dynamic (10) can be rewritten as This can be summarized as followed Based on assumption 3, the above equation can be rewritten as Meanwhile, based on assumption 5, equation 14 can be rewritten as Therefore, it can be concluded that the tracking error and its derivatives asymptotically converge to zero without any compact set ( ) ( ) → 0 as → ∞ for = 0,1, … , − 1 [1], and the system is globally stable. It is tough to implement the ideal control law presented in (7) since the nonlinear functions ( ), ( ), ( ), and the exogenous disturbance are unknown. In this case, our objective is to use fuzzy systems to approach the ideal control law.

Fuzzy Logic Systems
It is shown and proved that any real continuous function could be approximated using fuzzy systems defined on a compact set with arbitrarily high precision [23]. Sugeno et al. [24] have proposed a class of fuzzy systems that allows representing general knowledge to be expressed in analytical form, describing the system's internal behavior.
This class of fuzzy systems is called Takagi-Sugeno (TS) fuzzy systems. The input of the fuzzy system is defined as = [ 1 , … , ] and its output is defined as . As for ∈ , each is associated with , fuzzy sets in . There is at least one non-zero degree of membership ( ) ≠ 0 where = 1,2, … , and = 1,2, … , .
Another notable characteristic of fuzzy systems is associated with their rules. The fuzzy system has = ∏ =1 fuzzy rules, which have the following form where ̆ ∈ { 1 , … , } and ( ) is a numerical function on the output space. In general, ( ) is a polynomial function depending on variable inputs, but it can also be an arbitrary function to adequately describe the studied system's behavior.
First-order and zero-order Takagi-Sugeno fuzzy systems can be distinguished from the ( ). The ( ) of the first-order Takagi-Sugeno fuzzy system is the first-order polynomial form, as in Meanwhile, the ( ) of the zero-order Takagi-Sugeno fuzzy systems (TS-0) is a polynomial of zero-order as in In this work, a zero-order fuzzy system (TS-0) will be considered. A numerical conclusion is subjected to each rule. A weighted average is used to calculate the total output. In this way, the time spent on the defuzzification process can be reduced.
The equation (21) can be simplified and rewritten as follows: Nevertheless, by considering the principle of fuzzy basis functions [25], the output of the TS-0 fuzzy system can be written as:

Adaptive Fuzzy Fault-Tolerant Design
This section discusses approximating the ideal control law to ensure the system can track the desired reference trajectory. A fuzzy system is used to approximate the control law with a direct approach to achieve these goals. Based on the universal approximation theory [23] of fuzzy systems, the ideal control law can be approximated as * = ( ) * + ( ) with ( ) as the approximation error, ( ) is a vector of fuzzy basis functions which was properly assumed and set in advance by the user, and * is the optimal parameters' vector. The * somehow minimizes | ( )| as in * = { | * − ( ) |} The approximation error is assumed to be bounded as follows: with ̅ is an unknown positive constant. It should be noted that the fuzzy basis functions determine the lower bound of the approximation error ̅ . As a result, if these functions are better selected, the ̅ would be smaller.
Since the optimal parameters * are unknown, the synthesis of the controller must be estimated. Hence, the is the estimation of the * which will be calculated from an adaptation algorithm. The fuzzy adaptive approximation of the ideal control law is defined as Then, the following control law can be considered based on the above equation where is a robust control to deal with the approximation errors. The robust control used in this work is defined as The following are the chosen adaption laws that refer to the estimation parameters: when is the time-varying parameter with > 0, > 0, 0 > 0

Remark 1
The general design of the proposed approach can be seen in Fig. 1.

Theorem:
Consider the faulty system (2c) respecting the assumptions (1-6). The control law defined by (28) and (29)  where * is the ideal control law, considered as an unknown term introduced just for theoretical purpose and its value is unnecessary for the design of the proposed controller.
Then, the dynamics of the error can be written as: Until is stable or (| − |) = ( ) + 1 ( −1) + ⋯ + is stable, it is known that a symmetric positive definite matrix ( , ) that satisfies the Lyapunov equation is existed.
where ( ) is a symmetric positive definite matrix.

Results and Discussion
In this part, an inverted pendulum model is used to evaluate the proposed method's accuracy and prompt. The inverted pendulum mechanism system can be seen in Fig. 2. Tracking control is considered for the system. The dynamic equations of such a system are provided below as in [25].
where 1 is rotational movement, 2 is rotational velocity, = 9.8 / 2 is the gravitational acceleration force, is the cart's mass, is the pole's mass, is the pole's half-length, and is the applied force values. Moreover, = 1 , = 0.1 and = 0.5 are the values of the selected parameter.
The control purpose is to force the system to track the given trajectory ( ) = sin( ). It should be noted that the given reference allows a 1 maximum swing, while it is limited to 0.1 rad in [25]. Meanwhile, the exogenous disturbance is given as ( ) = 0.45sin (3 ).

Fig. 2. The used inverted pendulum
One fuzzy system is used as an approximator for the ideal control law taking the form of (23). Two fuzzy system inputs are selected: 1 ( ) and 2 ( ). Five Gaussian membership functions are subjected to each input, and can be defined as The initial value of (0) was randomly selected in the range of (-2 and 2). The initial value of ̂(0) = 0 and (0) =3. The simulation was conducted with actuator faults instead of sensor faults. The time profile of the faults was chosen to be at the initial time of the simulation. The actuators fault took the following forms and parameters: 1) Bias (0.005 . ); 2) Drift with = 0.07; 3) Loss of accuracy defined by a square waveform with 0.0087 . amplitude and 0.15 frequency; 4) 75% loss of effectiveness.
The simulation results of the angular position = 1 and the angular velocity ̇= 2 are shown in Fig. 3 and Fig. 4, respectively. The control input signal ( ) is shown in Fig. 5. The tracking error signal ( ) is depicted in Fig. 6. We can figure out that the system output converges to the desired trajectory in a short time, even in the presence of actuator faults. So, the proposed control strategy is capable of tracking precisely. Based on the results (Fig. 3-6), we can figure out the proposed approach reaches a good tracking performance against uncertainties, exogenous disturbances, and actuator faults. The position of the inverted pendulum ( ) reaches the desired trajectory ( ) in few seconds (around 2.5 seconds) as shown in Fig. 3 even in the presence of actuator faults, with acceptable angular velocity as depicted in Fig. 4, and the applied effort is smooth without any chattering phenomenon and acceptable power (no saturation) as shown in Fig. 5. Finally, the tracking error is closer to the origin (see Fig. 6), which implies that the control objective is reached.

Conclusion
The effects of time-varying actuator faults and exogenous disturbance on direct adaptive fuzzy fault-tolerant control for a class of unknown nonlinear systems are studied in this paper. Fuzzy logic systems (FLSs) are used to approach the entire adaptive control rule, including the actuator fault and the exogenous disturbance, with one robust controller term to compensate for the FLC approximation errors. The controller does not need any mathematical model of the plant, and no-fault detection and isolation FDI units are needed.