Fault-Tolerant Control of a Nonlinear System Actuator Fault Based on Sliding Mode Control

This paper presents a fault-tolerant control scheme for a class of nonlinear systems with actuator faults and unknown input disturbances. First, the slidingmode control law is designed based on the reaching lawmethod.Then, in view of unpredictable state variables and unknown information in the control law, the original system is transformed into two subsystems through a coordinate transformation. One subsystem only has actuator faults, and the other subsystem has both actuator faults and disturbances. A sliding mode observer is designed for the two subsystems, respectively, and the equivalence principle of the sliding mode variable structure is used to realize the accurate reconstruction of the actuator faults and disturbances. Finally, the observation value and the reconstruction value are used to carry out an online adjustment to the designed sliding mode control law, and fault-tolerant control of the system is realized. The simulation results are presented to demonstrate the approach.


Introduction
In recent years, theoretical research in fault-tolerant control has made great progress in the practical applications [1][2][3][4][5][6][7][8].Fault-tolerant control scheme is widely studied in linear systems.Tao and Xu studied the fault-tolerant control of known and unknown parameters of high-speed train dynamics model [9,10].Yu and Jiang proposed an innovative strategy for compensating the actuator faults to optimize system performance [11].Shen et al. presented an integrated design method of adaptive robust control for a linear system with adaptive fault identification [12].And Zhao et al. studied an adaptive sliding mode control for damage problems [13].However, most of the actual objects are nonlinear, and some of the working points in the linear systems will enter the nonlinear region when it has a fault.For systems of actuator with random failures and uncertain parameters, Fan et al. studied its stabilization and tracking problem, but they did not consider the disturbance of unknown inputs [14].Yin et al. presented a fault-tolerant control system scheme for realtime performance optimization, but they did not consider the effect of actuator faults [15].In view of uncertain overdrive systems, Zhang et al. proposed a robust control allocation algorithm based on pseudoinverse, which is compensated for the negative influence of the failure and stuck fault [16].Hu et al. presented an adaptive terminal sliding mode control method, found the finite time control of the attitude tracking, and solved the problem of actuator control input saturation [17].However, these two methods did not obtain accurate fault values.
The concept of sliding mode variable structure control is to design the switching hyperplane of the system according to the expected dynamic characteristics of the system.The variable structure controller is used to drive the system state from the initial state to the switching hyperplane in a finite time and then maintain its state at the switching hyperplane.
Once the system state reaches the switching hyperplane, the control function will ensure that the system travels along the switching hyperplane to reach the origin of the system.The system characteristics and parameters are entirely dependent on the designed switching hyperplane but are unrelated to external disturbances.Hence, the sliding mode variable structure control is extremely robust and has been widely applied in studies of nonlinear systems.
Based on the work of Zhao et al. about the adaptive sliding mode control in the literature [13], considering a class of nonlinear systems with actuator faults and unknown input disturbances, this paper innovatively presents precisely a fault-tolerant control method with disturbance and fault reconstruction.To fulfill the above scheme, a fault-tolerant control law with sliding mode control is first proposed here.A fault diagnosis and reconstruction method is used to carry out accurate reconstruction of unknown information, and finally the observation value and the reconstruction value are used to carry out the corresponding adjustment of the control law.The simulation results show that the proposed control method can meet the requirements of control accurately and reliably.
The remainder of the paper is organized as follows.Section 2 describes the mathematical model of nonlinear systems with actuator faults and unknown input disturbances.Section 3 presents a fault-tolerant control law based on the sliding mode control.The design of observer is discussed in Section 4. Then the fault reconstruction and the disturbance estimation are discussed in Section 5. Section 6 presents an online adjustment of the fault-tolerant control law.Simulation results are presented in Section 7, and conclusion is in Section 8.

Problem Description
We have an uncertain nonlinear system affected by actuator faults and unknown disturbances: where  ∈   is the state variables,  ∈   is the system inputs, and  ∈   is the outputs.Assume the nonlinear continuous term Φ(, ) ∈   is known.The unknown nonlinear term () ∈   models the lumped uncertainties and disturbances experienced by the system, which is assumed to be bounded; that is, a positive constant  1 exists such that ‖()‖ ≤  1 .

A Fault-Tolerant Control Law
The active fault-tolerant control for faulty system (1) can be described as follows: If there is no fault, that is,   () = 0, it is considered to be a nominal system; any appropriate control law is designed to achieve the gradual stability of the nominal system.If there is a fault, that is,   () ̸ = 0, an additional control law   is designed to address and fix the fault.Thus, for a faulty system, the control law can be designed: where   () is the nominal control law for the corresponding fault-free system.
If the specified system state command is defined as (), () ∈   , the system error can be represented as Therefore, the sliding mode surface is represented as where  is a matrix that will be designed.From (4), the following equation can be obtained: −  ( () + Φ (, ) +   () +  () +  ()) . ( The control law is designed as where parameters  and  are constant to be designed and () * is the generalized inverse matrix of ().If there is no fault, that is,   () = 0, the control law of the nominal system is   () = () * ( ṙ −  − Φ (, ) −  () +  0 If there is a fault, that is,   () ̸ = 0, the additional control law   is   () = () * (−  ()) . ( Lemma 1.For the nonlinear system described in (1), the designed fault-tolerant control is shown in (6).When  > 0 and  > 0, the system is in the stability condition of a sliding mode.
Proof.Consider the following Lyapunov function: The time derivative of  0 , along with (5), is The following equation can be obtained by taking the substitution of ( 6) into (10): When  > 0 and  > 0, the following equation can be obtained: This completes the proof.
It can be seen from the above analysis that the system can satisfy the asymptotic stability requirement; namely, it will be driven to the corresponding sliding mode surface in a finite period of time.

Observer Design
Equation (6) contains the system state variable , the disturbance , and the actuator fault   , where these variables are often unknown in control law.Therefore, we carry out the system state observations  through the method of "designing observer," where the disturbance  and actuator fault   are reconstructed accurately again on this basis to obtain the corresponding state observation value x and reconstruction values d and f of the disturbance and fault, which are substituted into (6) to determine the control.
Assumption 2.  is a column full rank matrix and rank() = rank().
Remark 3. If the disturbance distribution matrix  is not a column full rank matrix, for example, rank() =  1 < , then a rank decomposition  =   1   2 could be applied, where   1 is a column full rank matrix and   1 () =   2   2 () could be considered as a new unknown disturbance [18].Assumption 4. The matrix pair (, ) is observable.
The matrix is partitioned for (1) to obtain the following equation: where Based on Assumption 2, two transformation matrices, namely,  and , exist [19] such that Therefore, (1) could be converted as follows: where and  22 is a nonsingular matrix.
The following nonsingular transformation matrix  is then constructed [19]: Therefore, the coefficient matrices in (15) are as follows: where System (15) is then transformed into the following subsystems: Two subsystems, (19) and (20), could be obtained from system (1) by matrix transformation.Subsystem ( 19) is free from any uncertainties but is subjected to system actuator faults; subsystem (20) has actuator faults and uncertainties.The detail design of two sliding mode observers corresponding to the above subsystems will be presented as follows.
Assumption 5. ( 11 ,  11 ) and ( 22 ,  22 ) are observable.Assumption 6.The nonlinear functions Φ 1 and Φ 2 are assumed to be known and satisfied the Lipschitz conditions, such that where  3 and  4 are known Lipschitz positive constants.
Based on the transformed systems ( 19) and ( 20), the present study proposes the following two sliding mode observers: where superscript "̂" indicates an estimated value and  1 () and  2 () represent the input signals of the sliding mode observers: where  1 and  2 are the observer gains and  1 and  2 are positive scalars that will be designed.
Assumption 7. The existing arbitrary matrices  1 and  2 and symmetric positive definite matrices  1 and  2 will satisfy the following equations: (25) From Assumption 5 we know that the existing matrices  1 and  2 will make  10 and  20 become stable matrices: If the state estimation errors are defined as and the output estimation errors are defined as , (20), and (23), the state estimation errors dynamical systems are described by Lemma 8. Considering the error dynamics system ( 27) and ( 28) and Assumptions 6 and 7, if the following LMI is satisfied and the parameters  1 and  2 satisfy where  is a positive constant and   is a q-dimensional identity matrix.
Proof.Consider the following Lyapunov function: The derivative of  along with the error dynamic systems ( 27) and ( 28) is then Since the inequality 2   ≤ (1/)  +   is true for any scalar  > 0, then From Assumption 7, the following equation can be obtained: Equations ( 36) to (37) are substituted into (33) to obtain V1 turns out to be negative definite by imposing The linear matrix inequality is satisfied so that () will make a global asymptotic convergence to zero; that is, lim This completes the proof. . ( Then the error system ( 27) and ( 28) will be driven to the corresponding sliding mode surface   = 0 ( = 1, 2) in finite time. 1 and  2 are positive constants.
Proof.(1) We select the Lyapunov function as follows: The time derivative of  2 along the trajectories of system ( 27) is given by From  min ( then (2) Consider a Lyapunov function candidate The time derivative of  3 along the trajectories of system ( 28) is given by This completes the proof.
This result shows that the sliding mode reachability condition is satisfied.As a consequence, based on the sliding mode equivalent principle, an ideal sliding motion will take place on the surface   = 0 ( = 1, 2) and after some finite time.

Fault Reconstruction and Disturbance Estimation
When the system reaches the sliding mode surface,   = ṡ  = 0 ( = 1, 2) according to the sliding mode equivalent principle [20].The following equations are obtained: 5.1.The Reconstruction of Actuator Fault   ().From (55), we obtain the following: From (42), it follows that The sigmoid function was used to replace sgn() to weaken the chattering problem of the sliding mode, where the reconstructed value of the actuator fault is where  1 is a positive constant to be designed.

The Estimation of the Unknown Input Disturbance 𝑑(𝑡).
From (56), we obtain the following: From (42), it follows that The sigmoid function was used to replace sgn() to weaken the chattering problem of the sliding mode, where the reconstruction value of the disturbance is where  2 is a positive constant to be designed.

Adjustment of the Fault-Tolerant Control Law
In view of the control law  designed for the faulted system in (1), the observed value x of the system state, the reconstructed values d from (62), and f from (59) of the disturbance and fault, respectively, were substituted into (6).The adjusted control law becomes At this point, the control law in (7) adjusted to a nominal system is The additional control law in ( 8) is adjusted to Stability analysis is as follows.
Consider the following Lyapunov function: The time derivative of  4 along with ( 5) and ( 63) is Finally, if the positive constant  is chosen such that we obtain It can be seen from the above analysis that the system can satisfy the asymptotic stability.
From (1), the parameter matrixes are The transformational matrix  is The transformational matrix  is Set the matrices as In the control law of (63), the constants are taken to be  = 5 and  = 2.The robotic arm system (71) is a dual input system, where the input signal 1 is  1 = 2 cos() and the input  signal 2 is  2 = 5 sin().Two square signals were chosen as the "unknown" input disturbance, where one had a period of 1 s and an amplitude of 1 and the other had a period of 2 s with an amplitude of 2. The initial values of the system state variable () were taken as 0.6, 0.3, −1.5, and 0.5, respectively.For faultless systems, the simulated experimental results are as follows.
Figures 1 and 3 show two states of  1 and  3 used to separately track the two input signals in the case of different initial values; the given input values were reached quickly, and they remained stable.The errors,  1 and  2 , shown in Figures 2  and 4, prove the point that the tracking effect is good.Figures 5 and 6 are two output signals of the control law, respectively.
Whena system contains an actuator fault, a sinusoidal signal and a cosine signal were chosen to, respectively, simulate the fault: for example,  1 () = sin(5) and  2 () = cos(5).Thus the fault-tolerant control was not carried out; namely, the additional control law   was not added at this point.Figures 7 and 9

Conclusions
In this paper, we proposed a fault-tolerant control scheme based on fault reconstruction for a class of nonlinear systems with actuator faults and unknown input disturbances.It can obtain information from the unknown system state and can restrain the influence of the fault and disturbance.Therefore, this method has very strong engineering practicability.From the simulated experimental results applied on a mechanical  are, our method can maintain the system asymptotic stability and can achieve a precise control purpose when it contains an actuator fault and an unknown input disturbance, which demonstrates that our proposed method provides a new way to control nonlinear systems with actuator faults and unknown disturbances in practical engineering applications.Natural Science Foundation of China (nos.2016JJ5007 and 2015JJ5011).

Figure 1 :
Figure 1: Actual value  1 and tracking state of input signal 1.

Figure 3 :
Figure 3: Actual value  3 and tracking state of input signal 2.

Figure 7 :Figure 8 :
Figure 7: Actual value  1 and tracking state of input signal 1.

Figure 9 :
Figure 9: Actual value  3 and tracking state of input signal 2.
show the result of two input signals tracking, and Figures 8 and 10 show the result of two tracking errors.It can be seen from Figures 7, 8, 9, and 10 that the input signals cannot be accurately tracked, and the tracking errors converge to 0. When the additional control law   is added for fault-tolerant control, the input signal tracking results are shown in Figures15 and 17 , and the results of two tracking errors are shown in Figures16 and 18 . The reconstruction results are shown in Figures 11, 12, 13, and 14.Figures 11 and 12 display the unknown disturbances  1 and  2 and their estimated values, respectively.Meanwhile, Figures 13 and 14 show the actuator faults  1 and  2 and their reconstruction values separately.The simulation

Figure 11 :
Figure 11: Unknown input disturbance  1 and its estimation value.

Figure 12 :
Figure 12: Unknown input disturbance  2 and its estimation value.

Figure 15 :
Figure 15: Actual value  1 and tracking state of input signal 1.