# Robust Adaptive Sliding Mode Fault Tolerant Control for Nonlinear System with Actuator Fault and External Disturbance

Authors: Liang Zheng; Xuelian Dong; Qian Luo; Menglan Zeng; Xinping Yang; Rong Zhou
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

In this paper, an adaptive sliding mode fault tolerant control (ASMFTC) approach is proposed for a class of nonlinear systems with actuator fault, uncertainty, and external disturbance. Specifically, first, a novel observer is proposed to estimate the state, actuator fault, and external disturbance. Then, by utilising the observed information, a novel output sliding mode observer is constructed. In the control method, an adaptive law and two compensators are designed to attenuate the unknown estimation errors, actuator fault, and disturbance. Furthermore, the reaching ability of the sliding motion is analysed and the H-infinite performance is introduced to ensure the robustness of the system. Finally, a flexible single joint manipulator system and a two-cart system are used to verify the effectiveness of the proposed method.


---

## Body

## 1. Introduction

Industrial control system is inevitable to suffer from faults caused by the actuators, sensors, and other components of the system because of long time operation or unexpected operation. When fault occurs, the system performance will be certainly reduced and even the stability of the system cannot be guaranteed [1, 2]. With the growing requirement of the reliability and maintainability of the system, fault tolerant control (FTC) becomes an import and indispensable way when designing the controller [3, 4]. Generally, fault tolerant control (FTC) can be divided into two parts, i.e., passive FTC and active FTC. For passive FTC, the controller is designed based on the prior knowledge. On the contrary, the active FTC is constructed and integrated with the fault diagnosis, and the fault is detected online. Many literatures have been achieved about the FTC for the manipulator system; see [5–7] and the references therein.It is worth noting that disturbance exists widely in many industry processes [8], which will directly lead to the instability of the control system. There are many reasons for disturbance for mechanical systems, such as load fluctuation, noise in the working environment, and parameters vibration. Disturbance rejection control is also of great importance when designing the controller. Many control methods have been proposed in the last decades to attenuate, such as H∞ control [8], robust control [9], and sliding mode control [4]. In [8], the problem of reliable H∞ control is investigated for discrete-time T-S fuzzy systems with infinite distributed delay disturbance and actuator faults; the stability of the system and the H∞ performance index were guaranteed. In [9], the disturbance rejection control strategy for attitude tracking of an aircraft with both internal uncertainties and external disturbances was investigated; a robust feedback controller was designed to achieve the desired tracking performance. In [4], an adaptive sliding controller was proposed for a class of single-link flexible-joint robot with mismatched disturbance; a back stepping method was used to deal with the disturbance such that the tracking performance can be guaranteed.In most practical industry control systems, fault and external disturbance are unknown or are too expensive to measure, which drives people regard the fault or disturbance signal as an unknown input of the system. Aiming at this problem, the observer based method is of great importance in order for the accommodate fault and disturbance. Actually, the observer based methods have achieved abundant research results since the last 1970s. For example, in [10], a class of linear time-invariant systems with unknown inputs was discussed, and the design procedure of the observer to estimate the maximum subspace of the state was presented. In [11], a class of Lipschitz system was considered; an asymptotic observer with respect to estimate both the state and input with the features of considering additive disturbance and uncertainty was designed. In [12], an adaptive unknown input observer was constructed and the robustness performance was analysed. Reference [13] addressed the problem of the disturbances observer constructing; the disturbance rejection was considered simultaneously; and the method was applied to a robot manipulator regulation.Sliding mode control (SMC), which has potential advantages of robustness to uncertainties and disturbances and low sensitivity to the system parameter variations, has been widely investigated in nonlinear system [14]. However, the inherent robustness properties of conventional sliding mode control approaches are guaranteed only after the occurrence of a sliding mode [15]. In order to circumvent this problem, the idea of integral sliding mode control (ISM) control was introduced. In the ISMC scheme, the reaching phase is eliminated; i.e., the sliding motion occurs from the initial time. In [16], an adaptive robust controller for robot manipulators system with parameter variations and disturbances was developed. By using adaptive integral sliding mode control and time-delay estimation, the robustness and stability of the system were achieved. In addition, the observer based sliding mode control has an advantage of the disturbance attenuation control, because the estimated information is concluded in the controller scheme. In [17], the disturbance observer based SMC was considered for a special second order system with mismatched uncertainties. In [18], a disturbance observer based integral sliding mode control approach for continuous system was investigated; by incorporating the estimated information in the controller to counteract the disturbance, the chattering problem was eased and the stability was guaranteed.Motived by [16, 17], this paper focuses on the robust fault tolerant control for a class of nonlinear system with actuator fault, uncertainty, and external disturbance. The main works and contributions are as follows: (1) a novel observer is presented to estimate the states, fault signal, and disturbance simultaneously; (2) a new sliding mode robust control scheme is constructed, in which two compensators are included to accommodate the fault signal and to attenuate the disturbance; (3) the reaching ability of the sliding motion is proved. Finally, the proposed method is applied to the flexible single joint manipulator system; the simulation results illustrate the effectiveness of the proposed method.The remaining parts of this paper are organized as follows. In Section2, the system description and some assumptions are given. In Section 3, the construction of observer and the design of the controller are presented. In Section 4, a flexible single joint manipulator system and a two-cart system are presented and some conclusions are given in Section 5.


## 2. System Descriptions

The following nonlinear system with actuator fault and mismatched disturbance is(1)x˙t=Axt+But+Ewt+Bfft+Δξx,t+Mgx,tyt=Cxtwhere x(t)∈Rn is the state vector, u(t)∈Rm is the control input of the system, and yt∈Rp is the output. w(t)∈Rk is the external disturbance of the system and satisfies w(t)≤ϖ, w˙(t)≤λ, where ϖ and λ are known constants. f(t)∈Rp is the actuator fault signal, which satisfies f(t)≤f∗, where f∗ is known constant. Δξ(x,t) represents the system uncertainty with Δξ(x,t)≤ϖκ. g(x,t)∈Rn is a nonlinear Lipschitz function representing the unmodeled dynamics, i.e., g(xt,t)-g(x^t,t)≤ηxt-x^t, where η is the Lipschitz constant. A, B, C, E, Bf, and M are known as system parameter matrices with proper dimensions.


## 3. Main Results

### 3.1. Observer Design

Motived by [12, 13], in this part, a novel composite observer is designed as follows:(2)x^˙t=Ax^t+But+Bfυ+Lyt-y^t+Mgx^,tp˙=-HCEp-HCEHyt-HCAx^t-HCBut+Mgx^,t-usy^t=Cx^tw^=p+Hytwhere x^, w^, y^, Δξ(x^,t), and g(x^,t) are estimates of x, w, y, Δξ(x,t), and g(x,t); L and H are the observer gains. p is an intermediate variable of the observer, us is a compensator, and υ is the estimation of ft. us and υ are designed as follows:(3)us=P1ewP1ew2αt(4)υ=BfTP2exexTP2Bff∗+δwhere ew=w-w^, ex=x-x^, αt=-2ewTP1HCϖκ+2ewTP1λ+2exTP2ϖκ, and δ is a positive scalar; P1 and P2 are two positive definite symmetric matrices. From (2), one can deduce that(5)w^˙=-HCEw^-HCAx^t-HCBut+HCAxt+But+Ewt+Bfft+Mgx,t+Δξx,t-us-HCMgx^,tThrough further simplification, one can obtain that(6)w^˙=HCEew+HCΔξx,t+HCBfft+HCMg~x,t+HCAex-uswhere g~(x,t)=g(x,t)-g(x^,t).Then, the error dynamic systems can be obtained as follows:(7)e˙w=-HCEew-HCMg~x,t-HCBfft-HCAex-HCΔξx,t+us+w˙(8)e˙x=Axt+But+Ewt+Bfft+Δξx,t+Mgx,t-Ax^t-But-Gυ-Lyt-y^t-Mgx^,t=A-LCex+Δξx,t+Ewt+Mg~x,t+Bfft-υDefine e=ewexT, then the augmented system can be written as(9)e˙=A-e+M-g~x,t+G-ζw+w˙-+G∗ft-υ+H∗Δξx,t+u-swhere(10)A-=-HCE-HCA0A-LC,M-=-HCMM,w˙-=w˙0,M-=-HCMM,u-s=us0,G-=-HCG00E,H∗=-HCI,G∗=0Bf,ζw=ftwtT.ζw is the equivalent disturbance, and the H∞ performance index is defined as(11)J=∫0∞esTesds-∫0∞ζwsTζwsdsTheorem 1.
The dynamic error systems (7) and (8) are stable and satisfy the H∞ performance index, if there exist two positive definite symmetry matrices P1 and P2, such that the following (12) holds:(12)ψmP1HCA-P1H-P1HCM0∗ψnP2E-0P2M∗∗-γ200∗∗∗-σ1-10∗∗∗∗-σ-1<0where ψm=-P1HCE-P1HCET+I, H-=HCBf0, E-=E0, and ψn=P2A-LC+A-LCTP2+I.Proof.
The Lyapunov candidate function is chosen as(13)V=eTPewhere P=P100P2.
From (13), the following (14) can be obtained(14)V˙=2ewTP1e˙w+2exTP2e˙xAccording to (7) and (8), it can be deduced that(15)V˙=2ewTP1e˙w+2exTP2e˙x=2ewTP1-HCEew-HCMg~x,t+us-HCBfft-HCAex-HCΔξx,t+w˙+2exTP2A-LCex+Δξx,t+Ewt+Mg~x,t+Bfft-υFrom (4), one can check that(16)exTP2Bfft-υ=exTP2Bfft-exTP2Bfυ≤exTP2Bff∗-exTP2BfBfTP2exexTP2Gf∗+δ≤-δexTP2Bf<0Then (15) can be written as(17)V˙=2ewTP1e˙w+2exTP2e˙x≤2ewTP1-HCEew-HCMg~x,t+us-H-ζw-HCAex-HCΔξx,t+w˙+2exTP2A-LCex+Δξx,t+E-ζw+Mg~x,twhere E-=0E.
For any two positive scalarsσ and σ1, the following inequalities are satisfied:(18)2exTP2Mg~x,t≤σexTP2MMTP2ex+σ-1g~Tx,tg~x,t≤exTσP2MMTP2+σ-1η2ex(19)-2ewTP1HCMg~x,t≤σ1ewTP1HCMMTCTHTP1ew+exTσ1-1η2exFrom (3), one can check that(20)2ewTP1us+2ewTP1-HCΔξx,t+w˙+2exTP2Δξx,t≤2ewTP1P1ewP1ew2αt+2ewTP1HCΔξx,t+2ewTP1w˙+2exTP2Δξx,t≤0Combining (18) and (19), (17) can be written as(21)V˙t≤ξtTΞξtwhere ξt=ewtTextTζwT, Ξ=ψ1P1HCA-P1H-∗ψ2P2E-∗∗0,(22)ψ1=-P1HCE-P1HCET+σ1P1HCMMTCTHTP1,ψ2=P2A-LC+A-LCTP2+η2σ+σP2MMTP2+η2σ1.Then the H∞ performance index (11) is changed to the following (21):(23)J=∫0∞esTesds-∫0∞γ2ζwsTζwsds≤∫0∞esTesds-∫0∞γ2ζwsTζwsds+∫0∞V˙≤ξtTΞ1ξtwhere Ξ1=ψaP1HCA-P1H-∗ψbP2E-∗∗-γ2, ψa=ψ1+I, ψb=ψ2+I.
Thus, according to (12) and the Schur complement, the dynamic error system is stable and the H∞ performance index is satisfied. This completes the proof.From Theorem1, the estimation errors ex, ew, and ef=ft-υt will converge to zero asymptotically; i.e., there exist two positive scalars κ-1, κ-2, such that ef≤κ-1, ew≤κ-2. In the following part, we will reconstruct the fault signal.Define the following sliding mode surface(24)S=St∣S=exTheorem 2.
With the observer (2) and the condition (24), if (12) has a feasible solution, then the state of the system will be driven to the sliding surface S in finite time, if the following condition is satisfied(25)δ>Bf-1Δξx,t+Ewt+Mg~x,tand the fault signal can be approximated by(26)ft≈BfTP2exexTP2Bff∗+δProof.
The Lyapunov candidate function is defined as follows:(27)V1=exTP2exThe derivation of (27) can be obtained as (28)V˙1=2exTP2e˙x=2exTP2A-LCex+Δξx,t+Ewt+Mg~x,t+Bfft-υFrom Theorem 1, if (12) holds, then ψn=P2A-LC+A-LCTP2+I<0; one can check that 2exTP2A-LCex<0; then (28) can be rewritten as follows (29):(29)V˙1≤2exTP2Δξx,t+Ewt+Mg~x,t+Bfft-υ≤2exTP2Δξx,t+2exTP2Ewt+2exTP2Mg~x,t+2exTP2Bfft-υ≤2exTP2BfBf-1Δξx,t+Bf-1Ewt+Bf-1Mg~x,t-2exTP2Bfδ=-2exTP2Bfδ-Bf-1Δξx,t+Ewt+Mg~x,t≤-2exTP2Bf≤-2BfλminP2V1Thus, we can conclude that the state of the system will reach the sliding mode surface in finite time. When the state of the system reaches the sliding surface, one can obtain that(30)S=ex=0From (30), it can be obtained that(31)0=Δξx,t+Ewt+Mg~x,t+Bfft-υFrom Theorem 1, there exist a normal scalar μa which satisfies g~(x,t)≤μa; it can be obtained by (31) that(32)ft-υ=-Bf†Δξx,t+Ewt+Mg~x,t≤λminBf†Δξx,t+λminBf†Ewt+λminBf†Mg~x,t≤λminxBf†ϖκ+λminBf†Eϖ+λminBf†Mμa≤σϖwhere Bf†=BfTBf-1BfT and(33)σϖ≥λminxBf†ϖκ+λminBf†Eϖ+λminBf†MμaIt can be seen from (33) that if σϖ is small enough, (26) is established. It can be seen that (26) is a discontinuous functions; (34) formula can be used to replace (26)(34)ft≈BfTP2exexTP2Bf+αmf∗+δwhere αm is a positive constant.


### 3.2. Controller Design

The main purpose of this part is to design a controller to guarantee the stability of the system. In this paper, a novel integral sliding mode method based on the observer is presented, and the reaching ability is proved; the controller is designed as follows:(35)ut=u1t+u2tThe sliding mode surface is designed as(36)st=Gyt-y0-C∫0tAx^θ+Bu2θ+Ew^θ+Bfυθ+Mgx^,tdθwhere G is a parameter matrix, which need to be chosen to satisfy that GCB is inevitable.From (36), (37) can be obtained(37)s˙t=GCx˙t-Ax^t-Bu2t-Ew^t-Bfυt-Mgx^,t=GCBu1t+GCAex+Eewt+Bfeft+Mg~x,t+Δξx,tFrom Theorems 1 and 2, the estimation errors ex, ew, ef, and g~x,t are bounded; in accordance with (1), one can define the following equivalent uncertain term:(38)ϕ=Aex+Eewt+Geft+Mg~x,t+Δξx,tThen, one can check that ϕ is bounded.Then the controlleru1t is given by(39)u1t=-ψtGCBTstsTGCBwhere ψt=ϕ^+δk. δk is a positive scalar and ϕ^ is the estimation of ϕ with the following adaptive rules:(40)ϕ^˙=βmsTGCB+δmwhere βm and δm are two positive scalars.Lemma 3 (see [19]).
With the sliding mode surface (40) and controller (39), ϕ^ is bounded; i.e., there exist a positive constant such that the following inequality is satisfied: (41)ϕ^≤ϕ-,∀t>0Theorem 4.
With the controller (39), the system trajectory will converge to the sliding mode surface s in finite time.Proof.
The Lyapunov function is given as(42)V2t=12sTtst+12βmϕ^-ϕ-2The derivation of (42) can be obtained as (43)V˙2t=sTts˙t+ϕ^-ϕ-ϕ^˙=sTtGCBu1t+sTtGCBAex+Eewt+Geft+Mg~x,t+Δξx,t+ϕ^-ϕ-ϕ^˙≤-ψtsTtGCGCBTstsTGCB+sTtGCBϕ+ϕ^-ϕ-ϕ^˙=-sTGCBϕ-ϕ^-δksTGCB+ϕ^-ϕ-sTGCB+δm≤-δksTGCB-δmϕ^-ϕ-<0Then the trajectory of the system will reach the sliding mode surface in finite time. This completes the proof.When the system reaches the sliding surface, i.e.,s˙t=0, then the equivalent control of the system can be obtained(44)ueqt=-GCB-1GCAex+Eewt+Bfeft+Mg~x,t+Δξx,tIn this paper, the idea of equivalent control is applied to design the controller. In the next part, the design of u2t and the stability analysis of the system are given. u2t is designed as (45)u2t=-GCB-1GCAx^t+Ex^t+Bfνt+Mg^x,t-K1x^t-K2w^t-K3f^t+uktwhere ukt is given as follows:(46)ukt=-BTN-TQxtxTtQN-B2xTtQN-ϖκwhere N-=I-BGCB-1GC and Q is a positive symmetric matrix.Substituting (45) and (46) into (1), the closed loop system can be obtained as(47)x˙t=Axt+Bu2t+ueqt+Ewt+Bfft+Δξx,t+Mgx,t=A--BK1xt+E--BK2wt+B-f-BK3ft+M-x,t+N-Δξx,t+BK1ex+BK2ew+BK3ef+Buktwhere A-=A-BGCB-1GCA, E-=E-BGCB-1GCE, B-f=Bf-BGCB-1GCBf, and M-=M-BGCB-1GCM.TakeHe=E--BK2B-f-BK3M-BK1BK2BK3; we=w(t)Tf(t)TgT(x,t)exTewTefTT; then the upper (48) can be written as (49)(48)x˙t=A--BK1xt+Hewet+N-Δξx,t+BuktThe following theorem gives the stability analysis of the system.Theorem 5.
With (44)-(46), system (48) is asymptotically stable and the H∞ performance
J = ∫ 0 t y τ T y τ d τ ≤ γ 2 ∫ 0 t H e τ T H e τ d τ is satisfied, if there exist a positive symmetric matrix Q, such that the following conditions holds:(49)QA--BK1+A--BK1TQQHeCT∗-γ20∗∗-I<0Proof.
The Lyapunov function is designed as(50)Vt=xtQxtThe derivation of (51) can be obtained as(51)V˙t=2xTtQx˙t=2xTtQA--BK1xt+2xTtQHewet+2xTtQN-Δξx,t+BuktFrom (46), one can obtain that(52)xTtQN-Δξx,t+Bukt≤xTtQN-Δξx,t-xTtQN-BBTN-TQxtxTtQN-BxTtQN-ϖκ≤0Then, (51) can be rewritten as(53)V˙t≤2xTtQA--BK1xt+2xTtQHewetThen, on can obtain that(54)J=∫0tyτTyτdτ-γ2∫0tweτTweτdτ≤∫0tyτTyτdτ-γ2∫0tweτTweτdτ+V˙t=2xTtQA--BK1xt+2xTtQHewet=ζTtΞaζtwhere ζTt=xTtHeT, Ξa=QA--BK1+A--BK1TQ+CTCQHe∗-γ2.
According to (49), J<0, which implies that the system is stable and the H∞ performance is satisfied. The proof is completed.


## 3.1. Observer Design

Motived by [12, 13], in this part, a novel composite observer is designed as follows:(2)x^˙t=Ax^t+But+Bfυ+Lyt-y^t+Mgx^,tp˙=-HCEp-HCEHyt-HCAx^t-HCBut+Mgx^,t-usy^t=Cx^tw^=p+Hytwhere x^, w^, y^, Δξ(x^,t), and g(x^,t) are estimates of x, w, y, Δξ(x,t), and g(x,t); L and H are the observer gains. p is an intermediate variable of the observer, us is a compensator, and υ is the estimation of ft. us and υ are designed as follows:(3)us=P1ewP1ew2αt(4)υ=BfTP2exexTP2Bff∗+δwhere ew=w-w^, ex=x-x^, αt=-2ewTP1HCϖκ+2ewTP1λ+2exTP2ϖκ, and δ is a positive scalar; P1 and P2 are two positive definite symmetric matrices. From (2), one can deduce that(5)w^˙=-HCEw^-HCAx^t-HCBut+HCAxt+But+Ewt+Bfft+Mgx,t+Δξx,t-us-HCMgx^,tThrough further simplification, one can obtain that(6)w^˙=HCEew+HCΔξx,t+HCBfft+HCMg~x,t+HCAex-uswhere g~(x,t)=g(x,t)-g(x^,t).Then, the error dynamic systems can be obtained as follows:(7)e˙w=-HCEew-HCMg~x,t-HCBfft-HCAex-HCΔξx,t+us+w˙(8)e˙x=Axt+But+Ewt+Bfft+Δξx,t+Mgx,t-Ax^t-But-Gυ-Lyt-y^t-Mgx^,t=A-LCex+Δξx,t+Ewt+Mg~x,t+Bfft-υDefine e=ewexT, then the augmented system can be written as(9)e˙=A-e+M-g~x,t+G-ζw+w˙-+G∗ft-υ+H∗Δξx,t+u-swhere(10)A-=-HCE-HCA0A-LC,M-=-HCMM,w˙-=w˙0,M-=-HCMM,u-s=us0,G-=-HCG00E,H∗=-HCI,G∗=0Bf,ζw=ftwtT.ζw is the equivalent disturbance, and the H∞ performance index is defined as(11)J=∫0∞esTesds-∫0∞ζwsTζwsdsTheorem 1.
The dynamic error systems (7) and (8) are stable and satisfy the H∞ performance index, if there exist two positive definite symmetry matrices P1 and P2, such that the following (12) holds:(12)ψmP1HCA-P1H-P1HCM0∗ψnP2E-0P2M∗∗-γ200∗∗∗-σ1-10∗∗∗∗-σ-1<0where ψm=-P1HCE-P1HCET+I, H-=HCBf0, E-=E0, and ψn=P2A-LC+A-LCTP2+I.Proof.
The Lyapunov candidate function is chosen as(13)V=eTPewhere P=P100P2.
From (13), the following (14) can be obtained(14)V˙=2ewTP1e˙w+2exTP2e˙xAccording to (7) and (8), it can be deduced that(15)V˙=2ewTP1e˙w+2exTP2e˙x=2ewTP1-HCEew-HCMg~x,t+us-HCBfft-HCAex-HCΔξx,t+w˙+2exTP2A-LCex+Δξx,t+Ewt+Mg~x,t+Bfft-υFrom (4), one can check that(16)exTP2Bfft-υ=exTP2Bfft-exTP2Bfυ≤exTP2Bff∗-exTP2BfBfTP2exexTP2Gf∗+δ≤-δexTP2Bf<0Then (15) can be written as(17)V˙=2ewTP1e˙w+2exTP2e˙x≤2ewTP1-HCEew-HCMg~x,t+us-H-ζw-HCAex-HCΔξx,t+w˙+2exTP2A-LCex+Δξx,t+E-ζw+Mg~x,twhere E-=0E.
For any two positive scalarsσ and σ1, the following inequalities are satisfied:(18)2exTP2Mg~x,t≤σexTP2MMTP2ex+σ-1g~Tx,tg~x,t≤exTσP2MMTP2+σ-1η2ex(19)-2ewTP1HCMg~x,t≤σ1ewTP1HCMMTCTHTP1ew+exTσ1-1η2exFrom (3), one can check that(20)2ewTP1us+2ewTP1-HCΔξx,t+w˙+2exTP2Δξx,t≤2ewTP1P1ewP1ew2αt+2ewTP1HCΔξx,t+2ewTP1w˙+2exTP2Δξx,t≤0Combining (18) and (19), (17) can be written as(21)V˙t≤ξtTΞξtwhere ξt=ewtTextTζwT, Ξ=ψ1P1HCA-P1H-∗ψ2P2E-∗∗0,(22)ψ1=-P1HCE-P1HCET+σ1P1HCMMTCTHTP1,ψ2=P2A-LC+A-LCTP2+η2σ+σP2MMTP2+η2σ1.Then the H∞ performance index (11) is changed to the following (21):(23)J=∫0∞esTesds-∫0∞γ2ζwsTζwsds≤∫0∞esTesds-∫0∞γ2ζwsTζwsds+∫0∞V˙≤ξtTΞ1ξtwhere Ξ1=ψaP1HCA-P1H-∗ψbP2E-∗∗-γ2, ψa=ψ1+I, ψb=ψ2+I.
Thus, according to (12) and the Schur complement, the dynamic error system is stable and the H∞ performance index is satisfied. This completes the proof.From Theorem1, the estimation errors ex, ew, and ef=ft-υt will converge to zero asymptotically; i.e., there exist two positive scalars κ-1, κ-2, such that ef≤κ-1, ew≤κ-2. In the following part, we will reconstruct the fault signal.Define the following sliding mode surface(24)S=St∣S=exTheorem 2.
With the observer (2) and the condition (24), if (12) has a feasible solution, then the state of the system will be driven to the sliding surface S in finite time, if the following condition is satisfied(25)δ>Bf-1Δξx,t+Ewt+Mg~x,tand the fault signal can be approximated by(26)ft≈BfTP2exexTP2Bff∗+δProof.
The Lyapunov candidate function is defined as follows:(27)V1=exTP2exThe derivation of (27) can be obtained as (28)V˙1=2exTP2e˙x=2exTP2A-LCex+Δξx,t+Ewt+Mg~x,t+Bfft-υFrom Theorem 1, if (12) holds, then ψn=P2A-LC+A-LCTP2+I<0; one can check that 2exTP2A-LCex<0; then (28) can be rewritten as follows (29):(29)V˙1≤2exTP2Δξx,t+Ewt+Mg~x,t+Bfft-υ≤2exTP2Δξx,t+2exTP2Ewt+2exTP2Mg~x,t+2exTP2Bfft-υ≤2exTP2BfBf-1Δξx,t+Bf-1Ewt+Bf-1Mg~x,t-2exTP2Bfδ=-2exTP2Bfδ-Bf-1Δξx,t+Ewt+Mg~x,t≤-2exTP2Bf≤-2BfλminP2V1Thus, we can conclude that the state of the system will reach the sliding mode surface in finite time. When the state of the system reaches the sliding surface, one can obtain that(30)S=ex=0From (30), it can be obtained that(31)0=Δξx,t+Ewt+Mg~x,t+Bfft-υFrom Theorem 1, there exist a normal scalar μa which satisfies g~(x,t)≤μa; it can be obtained by (31) that(32)ft-υ=-Bf†Δξx,t+Ewt+Mg~x,t≤λminBf†Δξx,t+λminBf†Ewt+λminBf†Mg~x,t≤λminxBf†ϖκ+λminBf†Eϖ+λminBf†Mμa≤σϖwhere Bf†=BfTBf-1BfT and(33)σϖ≥λminxBf†ϖκ+λminBf†Eϖ+λminBf†MμaIt can be seen from (33) that if σϖ is small enough, (26) is established. It can be seen that (26) is a discontinuous functions; (34) formula can be used to replace (26)(34)ft≈BfTP2exexTP2Bf+αmf∗+δwhere αm is a positive constant.


## 3.2. Controller Design

The main purpose of this part is to design a controller to guarantee the stability of the system. In this paper, a novel integral sliding mode method based on the observer is presented, and the reaching ability is proved; the controller is designed as follows:(35)ut=u1t+u2tThe sliding mode surface is designed as(36)st=Gyt-y0-C∫0tAx^θ+Bu2θ+Ew^θ+Bfυθ+Mgx^,tdθwhere G is a parameter matrix, which need to be chosen to satisfy that GCB is inevitable.From (36), (37) can be obtained(37)s˙t=GCx˙t-Ax^t-Bu2t-Ew^t-Bfυt-Mgx^,t=GCBu1t+GCAex+Eewt+Bfeft+Mg~x,t+Δξx,tFrom Theorems 1 and 2, the estimation errors ex, ew, ef, and g~x,t are bounded; in accordance with (1), one can define the following equivalent uncertain term:(38)ϕ=Aex+Eewt+Geft+Mg~x,t+Δξx,tThen, one can check that ϕ is bounded.Then the controlleru1t is given by(39)u1t=-ψtGCBTstsTGCBwhere ψt=ϕ^+δk. δk is a positive scalar and ϕ^ is the estimation of ϕ with the following adaptive rules:(40)ϕ^˙=βmsTGCB+δmwhere βm and δm are two positive scalars.Lemma 3 (see [19]).
With the sliding mode surface (40) and controller (39), ϕ^ is bounded; i.e., there exist a positive constant such that the following inequality is satisfied: (41)ϕ^≤ϕ-,∀t>0Theorem 4.
With the controller (39), the system trajectory will converge to the sliding mode surface s in finite time.Proof.
The Lyapunov function is given as(42)V2t=12sTtst+12βmϕ^-ϕ-2The derivation of (42) can be obtained as (43)V˙2t=sTts˙t+ϕ^-ϕ-ϕ^˙=sTtGCBu1t+sTtGCBAex+Eewt+Geft+Mg~x,t+Δξx,t+ϕ^-ϕ-ϕ^˙≤-ψtsTtGCGCBTstsTGCB+sTtGCBϕ+ϕ^-ϕ-ϕ^˙=-sTGCBϕ-ϕ^-δksTGCB+ϕ^-ϕ-sTGCB+δm≤-δksTGCB-δmϕ^-ϕ-<0Then the trajectory of the system will reach the sliding mode surface in finite time. This completes the proof.When the system reaches the sliding surface, i.e.,s˙t=0, then the equivalent control of the system can be obtained(44)ueqt=-GCB-1GCAex+Eewt+Bfeft+Mg~x,t+Δξx,tIn this paper, the idea of equivalent control is applied to design the controller. In the next part, the design of u2t and the stability analysis of the system are given. u2t is designed as (45)u2t=-GCB-1GCAx^t+Ex^t+Bfνt+Mg^x,t-K1x^t-K2w^t-K3f^t+uktwhere ukt is given as follows:(46)ukt=-BTN-TQxtxTtQN-B2xTtQN-ϖκwhere N-=I-BGCB-1GC and Q is a positive symmetric matrix.Substituting (45) and (46) into (1), the closed loop system can be obtained as(47)x˙t=Axt+Bu2t+ueqt+Ewt+Bfft+Δξx,t+Mgx,t=A--BK1xt+E--BK2wt+B-f-BK3ft+M-x,t+N-Δξx,t+BK1ex+BK2ew+BK3ef+Buktwhere A-=A-BGCB-1GCA, E-=E-BGCB-1GCE, B-f=Bf-BGCB-1GCBf, and M-=M-BGCB-1GCM.TakeHe=E--BK2B-f-BK3M-BK1BK2BK3; we=w(t)Tf(t)TgT(x,t)exTewTefTT; then the upper (48) can be written as (49)(48)x˙t=A--BK1xt+Hewet+N-Δξx,t+BuktThe following theorem gives the stability analysis of the system.Theorem 5.
With (44)-(46), system (48) is asymptotically stable and the H∞ performance
J = ∫ 0 t y τ T y τ d τ ≤ γ 2 ∫ 0 t H e τ T H e τ d τ is satisfied, if there exist a positive symmetric matrix Q, such that the following conditions holds:(49)QA--BK1+A--BK1TQQHeCT∗-γ20∗∗-I<0Proof.
The Lyapunov function is designed as(50)Vt=xtQxtThe derivation of (51) can be obtained as(51)V˙t=2xTtQx˙t=2xTtQA--BK1xt+2xTtQHewet+2xTtQN-Δξx,t+BuktFrom (46), one can obtain that(52)xTtQN-Δξx,t+Bukt≤xTtQN-Δξx,t-xTtQN-BBTN-TQxtxTtQN-BxTtQN-ϖκ≤0Then, (51) can be rewritten as(53)V˙t≤2xTtQA--BK1xt+2xTtQHewetThen, on can obtain that(54)J=∫0tyτTyτdτ-γ2∫0tweτTweτdτ≤∫0tyτTyτdτ-γ2∫0tweτTweτdτ+V˙t=2xTtQA--BK1xt+2xTtQHewet=ζTtΞaζtwhere ζTt=xTtHeT, Ξa=QA--BK1+A--BK1TQ+CTCQHe∗-γ2.
According to (49), J<0, which implies that the system is stable and the H∞ performance is satisfied. The proof is completed.


## 4. Simulation

In this part, two examples are used to verify the effectiveness of the proposed method.Example 1.
In this part, the flexible single joint manipulator system borrowed from [7] is used to verify the effectiveness of the proposed method, the diagram for the flexible single joint is shown as Figure 1 [5], the joint is driven by a DC motor, and its dynamic system can be represented as (55)θ˙m=wmw˙m=kJmθ1-θm-BJmwm+kτJmuθ˙1=w1w˙1=kJ1θ1-θm-mgqJ1sin⁡θ1where Jm is the inertia of the DC motor, J1 is the inertia of the link, and θm and θ1 are the angle positions of the motor and the link, respectively. wm and w1 represent the angular velocities of the motor and link. k is the torsional spring constant, kτ is the amplifier gain, m is the mass, B is the viscosity coefficient, g is the gravitational constant, and q is the distance between the rotor center and the center of gravity of the link; u is the control input delivered by the motor. Taking the system state as xt=θmwmθ1w1T, the system can be rewritten in the form of (1), where(56)A=0100-48.6-1.2548.60000101.950-1.950,B=021.600,M=000-0.333,C=100001000010,E=1011.
AssumingBf=B, the system nonlinear term gx,t=sin⁡(x3). The actuator fault and disturbance signal are assumed as(57)ft=00≤t<80.5sin⁡0.5t8≤t<120.212≤t<20,wt=00≤t<50.4sin⁡0.4t5≤t<100.510≤t<20.
In addition, we assume that system uncertainty has the following forms:Δξ(x,t)=ϑmqx+ϑn, in which qx≤qm, qm, ϑn are known parameters.
By solving inequalities (12) and (49), we can obtain that(58)P2=0.68480.0014-0.2221-0.12540.00140.86520.00150.0016-0.22210.00150.6223-1.1022-0.12540.0016-1.10220.4829,L=0.3333-14.8916-0.2074-5.81470.18697.08730.272813.82980.6310-0.65100.00102.7158,H=9.65900.00300.1060,P1=0.1.Q=2.49160.7847-0.71150.87840.78470.7428-0.40840.4250-0.7115-0.40840.5576-0.28030.87840.4250-0.28032.1097.The simulation results are as given in Figures 2–8
Figures2–5 show the states of the system and their estimated response curves, respectively. Figure 6 shows the external disturbance and the estimation curve of the system. Figure 7 expresses the fault and its estimation curves. Figure 8 illustrates the control input signal. From Figures 2–5, it can be observed that the states of the system are stable in a limited time in the presence of and fault signals and external disturbance. These results also show that the system has better convergence with the controller, which means the proposed control method can effectively deal with the above problems.
Seen from Figure6, the accuracy of the observer in this paper is good, which can effectively detect the disturbance signals of the system, thus helping the subsequent controller design. From Figure 7, it can be seen that the fault signal of the system can be reconstructed effectively by adjusting the parameters of the sliding mode observer, which indicates that the observed signal can be used in the design of the controller, and the effectiveness of the method is verified. From Figure 8, one can observe that the controller works when fault occurs and in the presence of disturbance. The simulation results illustrate the effectiveness of the proposed method.Figure 1
Diagram for flexible single joint structure.Figure 2
State ofx1 and its estimation.Figure 3
State ofx2 and its estimation.Figure 4
State ofx3 and its estimation.Figure 5
State ofx4 and its estimation.Figure 6
Disturbance and its estimation.Figure 7
Fault and its estimation.Figure 8
Control inputut.Example 2.
In this section, the two-cart system which borrowed form [20] is provided to illustrate the effectiveness of the proposed method. The geometric of the system is shown as in Figure 9 [20].
As shown in Figure9, the first cart is connected to a rigid wall via a damper, and is connected to a second cart by a spring. The external force is applied to a second cart via an actuator. Both carts have a nominal mass of a=1kg, the damper have a constant of b0=1N/m and the spring constant c0=1N/m. The time constant of the actuator is τ=0.2. The states are the force, velocities and positions of the two carts. The actuator fault and unmatched disturbance are considered. The system parameters are given as follows:(59)A=-1τ00000-b0a0-c0ac0a1a00ca-c0a0100000100,B=1τ0000,BF=1τ0000,C=11000001000001000001,Bd=1100.200.200.100.2.
The nonlinear unmodeled uncertainty and unmatched nonlinearity are assumed to beΔfx,t=sin⁡x4, ξx,t=sin⁡x1. The actuator fault and external disturbance are supposed to be as follows:(60)ft=00≤t<250.5sin⁡0.2t25≤t<40140≤t<80,w1t=00≤t<150.5sin⁡t15≤t<50150≤t<80,w2t=0.5sin⁡3t0≤t<200.5sin⁡3t20≤t<50150≤t<80.
Choosing the matrixG=10000, and one can check that GCB is invertible. By solving (52), one can obtain that(61)P=0.0799-0.000-0.000-0.000-0.000-0.0001.10460.000-0.000-0.000-0.0000.0001.10460.000-0.000-0.000-0.0000.0001.10460.000-0.000-0.000-0.0000.0001.1046.The results of the simulation are as in Figures 10–14.
Figures10 and 11 characterize the trajectories of the system; from Figures 10-11, it can be seen that the stability of the positions and velocities of the first and second cart can be guaranteed. Figure 12 shows the estimation of the actuator fault; one can clearly check that the proposed method performance better than the intermediate method proposed in [20], precisely, the method proposed response faster than the method in [20], and the proposed method has less chattering.
To verify the proposed method, the responses of the system are shown in Figures13-14 without controller. From the two figures, one can conduct that the closed loop system becomes unstable when removing the observer and compensator. The simulation results illustrate the effectiveness of the proposed method.Figure 9
Geometric structure of the two-cart system.Figure 10
Response of statex1t, x2t, and x3t.Figure 11
Response of statex4t and x5t.Figure 12
Estimation of actuator faultft.Figure 13
Response of statex1t, x2t, and x3t without controller.Figure 14
Response of statex4t and x5t without controller.


## 5. Conclusion

In this paper, a class of nonlinear systems with disturbance, fault, and uncertainty is studied. The state and disturbance estimation of the system are realized by designing an observer first, while the sliding mode observer technology is introduced to reconstruct the fault signal. Then, by using the observed information, a novel robust integral sliding mode controller is presented. Furthermore, the reaching ability is proved and the stability of the closed loop system is analysed. Finally, the method is applied to a class of single joint manipulator system and a two-cart system. The simulation results show that the observer has good response characteristics, and the controller has good fault tolerant ability.



---

*Source: 6349510-2019-02-06.xml*