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

Authors: Jing He; Lin Mi; Songan Mao; Changfan Zhang; Houguang Chu
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

This paper presents a fault-tolerant control scheme for a class of nonlinear systems with actuator faults and unknown input disturbances. First, the sliding mode control law is designed based on the reaching law method. 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.


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## Body

## 1. Introduction

In recent years, theoretical research in fault-tolerant control has made great progress in the practical applications [1–8]. 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 real-time 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. Section2 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.


## 2. Problem Description

We have an uncertain nonlinear system affected by actuator faults and unknown disturbances:(1)x˙t=Axt+Φx,u+Efat+Ddt+But,yt=Cxt,where x∈Rn is the state variables, u∈Rr is the system inputs, and y∈Rp is the outputs. Assume the nonlinear continuous term Φ(x,u)∈Rn is known. The unknown nonlinear term d(t)∈Rq models the lumped uncertainties and disturbances experienced by the system, which is assumed to be bounded; that is, a positive constant γ1 exists such that d(t)≤γ1. The unknown nonlinear termfa(t)∈Rq represents actuator faults, supposed to be norm bounded; that is, there exists a constant γ2 such that fa(t)≤γ2. A∈Rn×n, B∈Rn×r, C∈Rp×n, D∈Rn×q, and E∈Rn×q are known constant matrices with n>p>q.


## 3. 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, fa(t)=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, fa(t)≠0, an additional control law uf is designed to address and fix the fault. Thus, for a faulty system, the control law can be designed:(2)ut=unt+uft,where unt is the nominal control law for the corresponding fault-free system.If the specified system state command is defined asr(t), r(t)∈Rn, the system error can be represented as(3)ex=rt-xt.Therefore, the sliding mode surface is represented as(4)s0=Gex,where G is a matrix that will be designed.From (4), the following equation can be obtained:(5)s˙0=Ge˙x=Gr˙-x˙=Gr˙-GAxt+Φx,u+Efat+Ddt+But.The control law is designed as(6)u=GB∗Gr˙-GAx-GΦx,u-GEfat-GDdt+ks0+ε⁡sgns0,where parameters k and ε are constant to be designed and GB∗ is the generalized inverse matrix of (GB). If there is no fault, that is, fa(t)=0, the control law of the nominal system is(7)unt=GB∗Gr˙-GAx-GΦx,u-GDdt+ks0+ε⁡sgns0.If there is a fault, that is, fa(t)≠0, the additional control law uf is(8)uft=GB∗-GEfat.Lemma 1.
For the nonlinear system described in (1), the designed fault-tolerant control is shown in (6). When k>0 and ε>0, the system is in the stability condition of a sliding mode.Proof.
Consider the following Lyapunov function:(9)V0=12s02.The time derivative of V0, along with (5), is(10)V˙0=s0s˙0=sGr˙-GAxt+Φx,u+Efat+Ddt+But.The following equation can be obtained by taking the substitution of (6) into (10): (11)V˙0=-ks02-εs0.When k>0 and ε>0, the following equation can be obtained:(12)V˙0≤0.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.


## 4. Observer Design

Equation (6) contains the system state variable x, the disturbance d, and the actuator fault fa, where these variables are often unknown in control law. Therefore, we carry out the system state observations x through the method of “designing observer,” where the disturbance d and actuator fault fa are reconstructed accurately again on this basis to obtain the corresponding state observation value x^ and reconstruction values d^ and f^a of the disturbance and fault, which are substituted into (6) to determine the control.Assumption 2.
D is a column full rank matrix and rank(CD)=rank(D).Remark 3.
If the disturbance distribution matrixD is not a column full rank matrix, for example, rank(D)=q1<q, then a rank decomposition D=D1′D2′ could be applied, where D1′ is a column full rank matrix and d1′(t)=D2′d2′(t) could be considered as a new unknown disturbance [18].Assumption 4.
The matrix pairA,C is observable.
The matrix is partitioned for (1) to obtain the following equation:(13)x˙t=x˙1tx˙2t=A11A12A21A22x1tx2t+Φ1x,uΦ2x,u+E1E2ft+D1D2dt+B1B2ut,where x1(t)∈Rn-q, x2(t)∈Rq, A11∈R(n-q)×(n-q), A12∈R(n-q)×q, A21∈Rq×(n-q), A22∈Rq×q, Φ1(x,u)∈Rn-q, Φ2(x,u)∈Rq, E1∈Rn-q, E2∈Rq, D1∈R(n-q)×q, and D2∈Rq×q is a nonsingular matrix.Based on Assumption2, two transformation matrices, namely, T and S, exist [19] such that(14)xt=T-1zt=T-1z1tz2tyt=S-1v1tv2t.Therefore, (1) could be converted as follows:(15)z˙t=z˙1tz˙2t=TAT-1zt+TΦz,u+TEfat+TDdt+TBut,vt=v1tv2t=SCT-1zt,where(16)SCT-1=C1100C22and C22 is a nonsingular matrix.The following nonsingular transformation matrixT is then constructed [19]:(17)T=In-q-D1D2-10Iq.Therefore, the coefficient matrices in (15) are as follows:(18)TAT-1=A-=A-11A-12A-21A-22,TB=B-=B-1B-2,TD=D-=0D-2,TE=E-=E-1E-2,TΦx,u=Φ-x,u=Φ-1x,uΦ-2x,u,where A-11=A11-D1D2-1A21, A-12=(A11-D1D2-1A21)D1D2-1+A12-D1D2-1A22, A-21=A21, A-22=A21D1D2-1+A22, Φ-1(x,u)=Φ1(x,u)-D1D2-1Φ2(x,u), Φ-2(x,u)=Φ2(x,u), E-1=E1-D1D2-1E2, E-2=E2, D-2=D2, B-1=B1-D1D2-1B2, B-2=B2, z1(t)∈Rn-q, z2(t)∈Rq, A-11∈R(n-q)×(n-q), A-12∈R(n-q)×q, A-21∈Rq×(n-q), A-22∈Rq×q, Φ-1(x,u)∈Rn-q, Φ-2(x,u)∈Rq, E-1∈Rn-q, E-2∈Rq, D-2∈Rq, C11∈R(p-q)×(n-q), C22∈Rq×q, v1∈Rp-q, and v2∈Rq.System (15) is then transformed into the following subsystems:(19)z˙1t=A-11z1t+A-12z2t+Φ-1x,u+E-1fat+B-1ut,v1t=C11z1t,(20)z˙2t=A-21z1t+A-22z2t+Φ-2x,u+E-2fat+D-2dt+B-2ut,(21)v2t=C22z2t.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.
(
A
-
11
,
C
11
) and (A-22,C22) are observable.Assumption 6.
The nonlinear functionsΦ-1 and Φ-2 are assumed to be known and satisfied the Lipschitz conditions, such that(22)Φ-1x,u-Φ-1x^,u≤γ-3x-x^=γ-3T-1z-z^=γ3z-z^,Φ-2x,u-Φ-2x^,u≤γ-4x-x^=γ-4T-1z-z^=γ4z-z^,where γ3 and γ4 are known Lipschitz positive constants.
Based on the transformed systems (19) and (20), the present study proposes the following two sliding mode observers:(23)z^˙1t=A-11z^1t+A-12z^2t+Φ-1x^,u+E-1r1t+B-1ut+L1v1t-v^1t,v^1t=C11z^1t,z^˙2t=A-21z^1t+A-22z^2t+Φ-2x^,u+E-2r2t+B-2ut+L2v2t-v^2t,v^2t=C22z^2t,where superscript “ ^ ” indicates an estimated value and r1(t) and r2(t) represent the input signals of the sliding mode observers:(24)r1t=-ρ1F1v^1t-v1tF1v^1t-v1tifv^1t-v1t≠00ifv^1t-v1t=0,r2t=-ρ2F2v^2t-v2tF2v^2t-v2tifv^2t-v2t≠00ifv^2t-v2t=0,where F1 and F2 are the observer gains and ρ1 and ρ2 are positive scalars that will be designed.Assumption 7.
The existing arbitrary matricesF1 and F2 and symmetric positive definite matrices P1 and P2 will satisfy the following equations:(25)P1E-1=C11TF1T,P2E-2=C22TF2T.
From Assumption5 we know that the existing matrices L1 and L2 will make A10 and A20 become stable matrices:(26)A-11-L1C11=A10,A-22-L2C22=A20.If the state estimation errors are defined as e1=z1(t)-z^1(t), e2=z2(t)-z^2(t), and e=e1e2T and the output estimation errors are defined as ev1=v1(t)-v^1(t)=C11e1(t) and ev2=v2(t)-v^2(t)=C22e2(t), then, from (19), (20), and (23), the state estimation errors dynamical systems are described by(27)e˙1t=A-11-L1C11e1t+A-12e2t+Φ-1z,u-Φ-2z^,u+E-1fat-r1t,(28)e˙2t=A-22-L2C22e1t+A-21e1t+Φ-2z,u-Φ-2z^,u+E-2fat-r2t+D-2dt.Lemma 8.
Considering the error dynamics system (27) and (28) and Assumptions 6 and 7, if the following LMI is satisfied(29)A10TP1+P1A10+ξγ2In-qA-21TP2+P1A-12P10A-12TP1+P2A-21A20TP2+P2A20+ξγ2Iq0P2P10-ξIn-q00P20-ξIq<0and the parameters ρ1 and ρ2 satisfy(30)ρ1>γ2,ρ2>γ4+D-2E-2γ1then e1 and e2 are asymptotically convergent; that is,(31)limt→∞e1t=0,limt→∞e2t=0,where ξ is a positive constant and Iq is aq-dimensional identity matrix.Proof.
Consider the following Lyapunov function:(32)V1=e1TP1e1+e2TP2e2.The derivative of V along with the error dynamic systems (27) and (28) is(33)V˙1=e1TA10TP1+P1A10e1+2e1TP1A-12e2+Φ-1z,u-Φ-1z^,u+E-1fat-E-1r1t+e2TA20TP2+P2A20e2+2e2TP2A-21e1+Φ-2z,u-Φ-2z^,u+E-2fat+D-2dt-E-2r2t.Let(34)A-=A-11-L1C11A-12A-21A-22-L2C22,P=P100P2;then(35)V˙1=eTA-TP+PA-e+2eTPΦ-z,u-Φ-z^,u+2e1TP1E-1fat-E-1r1t+2e2TP2E-2fat+D-2dt-E-2r2t.Since the inequality 2XTY≤(1/ξ)XTX+ξYTY is true for any scalar ξ>0, then(36)2eTPΦ-z,u-Φ-z^,u≤1ξeTP2e+ξΦ-z,u-Φ-z^,uTΦ-z,u-Φ-z^,u≤1ξeTP2e+ξγ2e2.From Assumption 7, the following equation can be obtained:(37)2e1TP1E-1fat-2e1TP1E-1r1t=2e1TP1E-1fat-2ρ1e1TP1E-1F1C11e1F1C11e1≤2F1C11e1γ2-ρ1≤0,2e2TP2E-2fat+2e2TP2D-2dt-2e2TP2E-2r2t=2e2TP2E-2fat+2e2TP2D-2dt-2ρ2e2TP2E-2F2C22e2F2C22e2≤2F2C22e2γ2+D-2E-2γ1-ρ2≤0.Equations (36) to (37) are substituted into (33) to obtain(38)V˙1≤eTA-TP+PA-+1ξP2+ξγ2Ine.V˙1 turns out to be negative definite by imposing(39)A-TP+PA-+1ξP2+ξγ2In<0.The linear matrix inequality is satisfied(40)A-TP+PA-+ξγ2InPP-ξIn<0such that(41)A10TP1+P1A10+ξγ2In-qA-21TP2+P1A-12P10A-12TP1+P2A-21A20TP2+P2A20+ξγ2Iq0P2P10-ξIn-q00P20-ξIq<0,so that e(t) will make a global asymptotic convergence to zero; that is,(42)limt→∞e1t=0,limt→∞e2t=0.This completes the proof.Remark 9.
Lemma8 implies that e1 and e2 are bounded; that is, t0 will exist if t>t0(43)e1≤δ1,e2≤δ2,e≤δ,where δ1, δ2, and δ are three finite positive scalars.Lemma 10.
Choose the sliding mode surfacess1=F1ev1 and s2=F2ev2. Suppose Assumptions 6 and 7 and inequality (43) hold; besides, ρi  (i=1,2) is chosen sufficiently large to satisfy (44)ρ1≥A-11-L1C11+γ3δ1+A-12+γ3δ2+E-1γ2+K1λminE-1TP1E-1,ρ2≥F2C22A-22-L2C22+γ4δ2+A-21+γ4δ1+E-2γ2+D2γ1+K2λminE-2TP2E-2.Then the error system (27) and (28) will be driven to the corresponding sliding mode surface si=0  (i=1,2) in finite time. K1 and K2 are positive constants.Proof.
(1) We select the Lyapunov function as follows:(45)V2=12s1Ts1.The time derivative of V2 along the trajectories of system (27) is given by(46)V˙2=s1TF1C11e˙1=s1TF1C11A-11-L1C11e1t+A-12e2t+Φ-1z,u-Φ-1z^,u+E-1fat-E-1r1t=s1TF1C11e˙1=s1TF1C11A-11-L1C11e1t+A-12e2t+Φ-1z,u-Φ-1z^,u+E-1fat-F1C11E-1r1t=s1TF1C11A-11-L1C11e1t+A-12e2t+Φ-1z,u-Φ-1z^,u+E-1fat-ρ1s1TE-1TP1E-1s1s1.From λminE-1TP1E-1s12≤s1TE-1TP1E-1s1 and (43), it follows that(47)V˙2≤s1TF1C11A-11-L1C11e1t+A-12e2t+Φ-1z,u-Φ-1z^,u+E-1fat-ρ1λminE-1TP1E-1s1≤s1F1C11A-11-L1C11+γ3δ1+A-12+γ3δ2+E-1γ2-ρ1λminE-1TP1E-1. We design ρ1 which satisfies(48)ρ1≥A-11-L1C11+γ3δ1+A-12+γ3δ2+E-1γ2+K1λminE-1TP1E-1;then(49)V˙2≤-K1s1.
(2) Consider a Lyapunov function candidate(50)V3=12s2Ts2.The time derivative of V3 along the trajectories of system (28) is given by(51)V˙3=F2C22e2TF2C22e˙2=F2C22e2TF2C22A-22-L2C22e2t+A-21e1t+Φ-2z,u-Φ-2z^,u+E-2fat+D2dt-E-2r2t=s2TF2C22A-22-L2C22e2t+A-21e1t+Φ-2z,u-Φ-2z^,u+E-2fat+D2dt-ρ2s2TE-2TP2E-2s2s2.
By the same reasoning as in the proof of the previous steps, it follows that(52)V˙3≤s2TF2C22A-22-L2C22e2t+A-21e1t+Φ-2z,u-Φ-2z^,u+E-2fat+D2dt-ρ2λminE-2TP2E-2s2≤s2F2C22A-22-L2C22+γ4δ2+A-21+γ4δ1+E-2γ2+D2γ1-ρ2λminE-2TP2E-2.
We designρ2 which satisfies(53)ρ2≥F2C22A-22-L2C22+γ4δ2+A-21+γ4δ1+E-2γ2+D2γ1+K2λminE-2TP2E-2;then(54)V˙3≤-K2s2.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 surfacesi=0  (i=1,2) and after some finite time.


## 5. Fault Reconstruction and Disturbance Estimation

When the system reaches the sliding mode surface,si=s˙i=0(i=1,2) according to the sliding mode equivalent principle [20]. The following equations are obtained:(55)F1ev1=F1e˙v1=F1C11e˙1=0,(56)F2ev2=F2e˙v2=F2C22e˙2=0.


### 5.1. The Reconstruction of Actuator Faultfa(t)

From (55), we obtain the following:(57)F1C11A-11-L1C11e1t+A-12e2t+Φ-1z,u-Φ-1z^,u+E-1fat-E-1r1t=0.From (42), it follows that(58)fat≈r1t=ρ1sgnF1ev1.The sigmoid function was used to replace sgn(s) to weaken the chattering problem of the sliding mode, where the reconstructed value of the actuator fault is(59)f^a=ρ121+e-a1F1ev1-1,where a1 is a positive constant to be designed.


### 5.2. The Estimation of the Unknown Input Disturbanced(t)

From (56), we obtain the following:(60)F2C22A-22-L2C-22e2t+A-21e1t+Φ-2x,u-Φ-2x^,u+E-2fat-E-2r2t+D-2dt=0.From (42), it follows that(61)dt≈D-2-1E-2r2t-E-2fat=D-2-1E-2ρ2sgnF2ev2-ρ1sgnF1ev1.The sigmoid function was used to replace sgn(s) to weaken the chattering problem of the sliding mode, where the reconstruction value of the disturbance is(62)d^t=D-2-1E-2ρ221+e-a2F2ev2-1-ρ121+e-a1F1ev1-1,where a2 is a positive constant to be designed.


## 5.1. The Reconstruction of Actuator Faultfa(t)

From (55), we obtain the following:(57)F1C11A-11-L1C11e1t+A-12e2t+Φ-1z,u-Φ-1z^,u+E-1fat-E-1r1t=0.From (42), it follows that(58)fat≈r1t=ρ1sgnF1ev1.The sigmoid function was used to replace sgn(s) to weaken the chattering problem of the sliding mode, where the reconstructed value of the actuator fault is(59)f^a=ρ121+e-a1F1ev1-1,where a1 is a positive constant to be designed.


## 5.2. The Estimation of the Unknown Input Disturbanced(t)

From (56), we obtain the following:(60)F2C22A-22-L2C-22e2t+A-21e1t+Φ-2x,u-Φ-2x^,u+E-2fat-E-2r2t+D-2dt=0.From (42), it follows that(61)dt≈D-2-1E-2r2t-E-2fat=D-2-1E-2ρ2sgnF2ev2-ρ1sgnF1ev1.The sigmoid function was used to replace sgn(s) to weaken the chattering problem of the sliding mode, where the reconstruction value of the disturbance is(62)d^t=D-2-1E-2ρ221+e-a2F2ev2-1-ρ121+e-a1F1ev1-1,where a2 is a positive constant to be designed.


## 6. Adjustment of the Fault-Tolerant Control Law

In view of the control lawu designed for the faulted system in (1), the observed value x^ of the system state, the reconstructed values d^ from (62), and f^a from (59) of the disturbance and fault, respectively, were substituted into (6). The adjusted control law becomes(63)u=GB∗Gr˙-GAx^-GΦx^,u-GEf^at-GDd^t+ks0+ε⁡sgns0.At this point, the control law in (7) adjusted to a nominal system is(64)unt=GB∗Gr˙-GAx^-GΦx^,u+ks0+ε⁡sgns0-GDd^t.The additional control law in (8) is adjusted to(65)uft=GB-GEf^at.Stability analysis is as follows.Consider the following Lyapunov function:(66)V4=12s02.The time derivative of V4 along with (5) and (63) is(67)V˙4=s0s˙0=s0Gr˙-GAx-GDd-GEfa-GΦx,u-GBGB∗Gr˙-GAx^-GΦx^,u-GEf^at-GDd^t+ks0+εsgn⁡s0=s0-ks0-εsgn⁡s0-GAx-x^-GΦx,u-Φx^,u-GEfa-f^a-GDd-d^=s0-ks0-εsgn⁡s0-GAT-1e-GΦx,u-Φx^,u-GEfa-f^a-GDd-d^.It follows from (43) that(68)V˙4≤-ks02-εs0+s0GAT-1δ+GDD-2-1E-2ρ2+ρ1+GEρ1+Gγδ.Finally, if the positive constant ε is chosen such that(69)ε≥δGAT-1+ρ2+ρ1GDD-2-1E-2+ρ1GE+γδGwe obtain(70)V˙4≤-ks02.It can be seen from the above analysis that the system can satisfy the asymptotic stability.


## 7. Simulation Example

Consider a robot arm whose motion equation is as follows [21]:(71)Mqq¨+Bq,q˙q˙+Gq=u,Mq=J100Jm,Bq,q˙=Fn00Fm,Gq=mglsin⁡q1+aq1-q2-aq1-q2,where q1 and q2 are the link displacement and the rotor displacement, respectively. The link inertia J1, motor rotor inertia Jm, elastic constant a, link mass m, gravity g, connecting rod lengthl, and viscous friction coefficients Fn and Fm are all positive constant parameters. The control u is the torque delivered by the motor. The robot parameters are Fn=0.5Nm/(rad/s), Jm=1Nm2, k=2Nm/rad,Fm=1, J1=2Nm2, m=0.15kg, g=9.8m/s2, and l=0.3m.From (1), the parameter matrixes are(72)A=0100-kJ1-F1J1kJ100001kJm0-kJm-FmJm=0100-1-0.2510000120-2-1,fx,u,t=0-0.2205sin⁡x100,E=00240021,D=10001002,B=00100001,C=1000001001010001.The transformational matrix T is(73)T=10-10010000100001.The transformational matrix S is(74)S=1-100001-101000001.Set the matrices as (75)P1=2001,P2=1001,F1=0204,F2=0201,L1=3-123.25,L2=4-12-1,G=51000051.In the control law of (63), the constants are taken to be k=5 and ε=2. The robotic arm system (71) is a dual input system, where the input signal 1 is r1=2cos⁡πt and the input signal 2 is r2=5sin⁡(πt). 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 x(t) were taken as 0.6, 0.3, −1.5, and 0.5, respectively. For faultless systems, the simulated experimental results are as follows.Figures1 and 3 show two states of x1 and x3 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, e1 and e2, 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.Figure 1
Actual valuex1 and tracking state of input signal 1.Figure 2
Tracking errore1 of input signal 1.Figure 3
Actual valuex3 and tracking state of input signal 2.Figure 4
Tracking errore2 of input signal 2.Figure 5
Control input signalu1.Figure 6
Control input signalu2.When a system contains an actuator fault, a sinusoidal signal and a cosine signal were chosen to, respectively, simulate the fault: for example,fa1t=sin⁡5t and fa2(t)=cos⁡(5t). Thus the fault-tolerant control was not carried out; namely, the additional control law uf was not added at this point. Figures 7 and 9 show the result of two input signals tracking, and Figures 8 and 10 show the result of two tracking errors.Figure 7
Actual valuex1 and tracking state of input signal 1.Figure 8
Tracking errore1 of input signal 1.Figure 9
Actual valuex3 and tracking state of input signal 2.Figure 10
Tracking errore2 of input signal 2.It can be seen from Figures7, 8, 9, and 10 that the input signals cannot be accurately tracked, and the tracking errors converge to 0. When the additional control law uf is added for fault-tolerant control, the input signal tracking results are shown in Figures 15 and 17, and the results of two tracking errors are shown in Figures 16 and 18. The reconstruction results are shown in Figures 11, 12, 13, and 14.Figure 11
Unknown input disturbanced1 and its estimation value.Figure 12
Unknown input disturbanced2 and its estimation value.Figure 13
Actuator faultfa1 and its reconstruction value.Figure 14
Actuator faultfa2 and its reconstruction value.Figure 15
Actual valuex1 and tracking state of input signal 1.Figure 16
Tracking errore1 of input signal 1.Figure 17
Actual valuex3 and tracking state of input signal 2.Figure 18
Tracking errore2 of input signal 2.Figures11 and 12 display the unknown disturbances d1 and d2 and their estimated values, respectively. Meanwhile, Figures 13 and 14 show the actuator faults fa1 and fa2 and their reconstruction values separately. The simulation results indicate that the method can estimate the unknown disturbances and reconstruct the actuator faults accurately. And the tracking effect is good.Compared with Figures7, 8, 9, and 10, it can be seen from Figures 15, 16, 17, and 18 that the input signals can be accurately tracked after the additional control law uf is added, and the tracking errors can converge to 0. Figures 19 and 20 are two output signals of the control law, respectively.Figure 19
Control input signalu1.Figure 20
Control input signalu2.


## 8. 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.



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*Source: 8595960-2017-03-09.xml*