# Adaptive Robust Fault-Tolerant Synchronization Control for a Dual Redundant Hydraulic Actuation System with Common-Mode Fault

Authors: Ting Li; Ting Yang; Yuyan Cao; Rong Xie; Xinmin Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper investigates the fault-tolerant synchronization control (FTSC) problem for a dual redundant hydraulic actuation system (DRHAS), which works on active/active (A/A) mode and suffers from a kind of common-mode fault (CMF), i.e., internal leakage faults occurring in both hydraulic actuator (HA) channels simultaneously due to a common cause. Firstly, in order to follow the position command and synchronize the force outputs of the two channels, a desired trajectory generator derived from the dynamics of the control surface is employed. Then, considering model uncertainties and nonlinear dynamics of the plant, an FTSC controller is designed based on adaptive robust control (ARC) theory and backstepping technology. The controller parameters, closely related to the fault parameters, are updated online to make the controller adapt to the fault condition only when the system performance degradation exceeds a prescribed tolerable level. It has been verified that the proposed FTSC scheme can guarantee the bounded stability of output tracking error system under common-mode fault. Finally, simulation results under two scenarios demonstrate the effectiveness of the proposed FTSC scheme.


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## Body

## 1. Introduction

For safety and reliability consideration, the primary flight control systems for modern airplanes have universally adopted redundant hydraulic actuation systems (RHAS) to drive important control surfaces. One typical example is the Airbus A320, in which both ailerons are driven by a dual redundant hydraulic actuation system (DRHAS), the rudder is driven by a triply redundant hydraulic actuation system (TRHAS), and the elevators are driven by DRHASs [1].In a typical RHAS configuration, several similar hydraulic actuators (HA), which are supplied from three redundant hydraulic power sources, connect to a common control surface in parallel. Each HA includes an electro-hydraulic servo valve (EHSV), a hydraulic cylinder, and other accessories. There are two distinct operating modes for a RHAS, i.e., active/active (A/A) mode and active/standby (A/S) mode. In A/A mode, all HA channels are working together at the same point and their outputs can be summed in three ways, i.e., parallel force summing, parallel velocity summing, and parallel position summing [2]. Among them, the parallel force summing is the most common choice for the primary flight control system of airliners, such as A320, A380, B747, and B787. In A/S mode, the HA channels operate independently and only one channel is working at one time while others are isolated by bypass-valves.The RHAS on A/A mode has a potential risk of suffering from common-mode fault (CMF), which is a coincidence of fault states of components in separate channels caused by the same event(s) [3]. As stated in [4], the CMF is difficult to predict or to avoid by system design as its nature is unexpected or unrecognized. Reference [5] shows that design faults which are not found and removed at stages prior to the operational stage constitute a major part of CMF in redundant system. In addition, CMF can also occur due to external causes such as environmental disturbance and power supply disturbance. Considering the critical functions of the RHAS in primary flight control systems, CMF tolerance is very important.More recently, theoretical research on fault-tolerant control (FTC) problems, especially on active fault-tolerant control (AFTC) problems for hydraulic actuation system, is receiving more and more attention. However, most research focuses on the FTC problems for single-channel hydraulic actuator (SHA). The existing design methods include quantitative feedback theory (QFT) [6, 7], adaptive robust control (ARC) [8–10], and intelligent control [11]. Among them, [7–11] belong to the AFTC method, which needs to employ real-time fault detection and diagnosis (FDD) module to provide up to date information about the true status of a system [12]. The FDD methods in [7–11] include neural network [7], adaptive robust observer [8], parameter adaptive methodology [9], and stochastic filtering theory [10, 11].Compared with SHA, to the best of the authors’ knowledge, the studies on FTC strategies for the RHAS, especially for the RHAS suffering from CMF, are very limited. In [13, 14], a disturbance-decoupled robust adaptive observer was designed and applied to a RHAS-driven elevator for faulty parameters estimation. A set of local fuzzy PI controllers with respect to different operating conditions of the system were designed. Then, a tolerant control output was generated by synthesizing the output of each local controller according to the obtained fault information. However, the FTC controllers are designed on the basis of an approximate linear model, dynamics of the control surface, and model uncertainties of the plant are not considered in the existing results.To deal with the control problem for nonlinear system with model uncertainties, a great number of nonlinear control schemes have been developed by employing backstepping technology. For example, in [15–18], several backstepping-based adaptive neural (or fuzzy) control schemes were presented and neural networks (or fuzzy logic systems) were employed to approximate the unmodeled dynamics. However, the application of these schemes is limited by the large computation load. In [19], a FTC scheme based on adaptive sliding mode backstepping was proposed for a nonlinear system under external disturbances and faults. In [20, 21], with the aid of the backstepping technology, several nonlinear ARC controllers were constructed, in which the adaptive part was employed to handle parametric uncertainties and the robust part was mainly used to accommodate the unmodeled dynamics. To the authors’ knowledge, a critical issue for FTC design is the limited amount of time for the control system reconfiguration [12]. Therefore, due to its simple structure and fast execution speed, the backstepping-based ARC scheme is more suitable for the FTC design for RHAS with model uncertainties.The force fighting between HA channels is an inherent problem for the RHAS which works on A/A mode and has a parallel force summing configuration, namely, the HAs fight against each other to find an equilibrium point [22].The primary cause of the force fighting problem lies in the unsynchronized force outputs of HAs. In normal situation, manufacturing tolerances and individual nonlinear property of each HA can lead to unsynchronized force outputs. Moreover, the faults occurring in the RHAS will aggravate the unsynchronization of the force outputs and finally enlarge the force fighting between HA channels. Since a serious force fighting may slow down the system response, reduce the position tracking accuracy, or even damage the control surface, solving the problem of unsynchronized force outputs under fault conditions is very important for the FTC controller design for RHAS. Recently, several control strategies have been developed for force outputs synchronization, such as pressure differential equalization control [23], decoupling control [24], and motion state synchronization control [25, 26]. However, to the authors’ knowledge, most of the results are based on linear RHAS models without consideration of nonlinear dynamics and model uncertainties. Besides, no FTC scheme for RHAS can achieve force outputs synchronization control for HAs.In this paper, considering the nonlinear dynamics of the hydraulic system, model uncertainties, and dynamics of the control surface, an adaptive robust fault-tolerant synchronization control (FTSC) scheme is proposed for a DRHAS suffering from an internal leakage CMF. The contributions of the paper are twofold:(1) Based on the introduction of two reference trajectories, a general nonlinear model for the DRHAS with model uncertainties and an internal leakage CMF is constructed to facilitate the controller design. (2) A novel FTSC controller based on ARC control and backstepping technology is designed, which can handle the CMF tolerant control problem and the force outputs synchronization control problem simultaneously. Simulation results further demonstrate the effectiveness of the proposed scheme.The rest of the paper is organized as follows. The model for a DRHAS working on A/A mode is given in Section2. Then, the FTSC scheme which includes desired trajectory generation, model transformation, and FTSC controller design is presented in Section 3. Simulation results are provided in Section 4. Finally, a conclusion is given in Section 5.


## 2. Model of a Dual Redundant Hydraulic Actuation System

The structure of a typical DRHAS is shown in Figure1. Two hydraulic actuators, powered by different hydraulic power sources, connect to a common control surface and operate on A/A mode. Each HA mainly consists of an EHSV and a hydraulic cylinder.Figure 1
Structure diagram of a DRHAS operating on A/A mode.


### 2.1. Model of the Hydraulic Actuator

The two HAs have the same structure, and the mathematical model for HA1 is presented below.The model of the EHSV can be described by a proportional function [27, 28] as(1)xv1=ki1u1where xv1 is the servo valve displacement, u1 is the control input, and ki1 is the servo valve gain.The cylinder dynamic can be represented by(2)x¨h1=1mh1A1Ph11-A2Ph12-Bh1x˙h1-Fh1-d1where xh1 is the piston rod displacement, mh1 is the piston mass, A1 is the piston area of the active chamber, A2 is the piston area of the passive chamber, Bh1 is the damping coefficient, Fh1 is the external load of the cylinder, and d1 is the model uncertainty which includes unmodeled nonlinear dynamics and lumped parameter variations.The chamber pressures of the cylinder, i.e.,Ph11 and Ph12, are calculated by(3)P˙h11=βeVh11-A1x˙h1-qh1+Qh11,(4)P˙h12=βeVh12A2x˙h1+qh1-Qh12where βe is the effective oil bulk modulus, Vh11=V10+A1xh1 is the volume of the active chamber, Vh12=V20-A2xh1 is the volume of the passive chamber, and V10 and V20 are the initial volumes of two chambers, respectively. Qh11 is the supply flow rate of the active chamber and Qh12 is the return flow rate of the passive chamber.When the DRHAS suffers from an internal leakage CMF, i.e., the internal leakage faults occurring in both HAs simultaneously due to a common cause, the internal leakage flow rateqh1 of HA1 can be modeled by [9](5)qh1=Ci1Ph11-Ph12+β1t-tfCt1Ph11-Ph12sgnPh11-Ph12where Ci1 is the normal internal leakage coefficient, Ct1 represents the fault internal leakage coefficient, sgn(·) is the sign function, tf is the fault occurrence time, β1(t-tf) is the time profile of the fault which can be described by (6)β1t-tf=0t<tf1-e-μ1t-tft≥tf,and μ1 is a positive constant which represents the fault evolution speed.The flow ratesQh11 and Qh12 can be calculated by(7)Qh11=kqki1u1su1Ps1-Ph11+s-u1Ph11-Pr1,(8)Qh12=kqki1u1su1Ph12-Pr1+s-u1Ps1-Ph12where kq=Cdω2/ρ is the flow rate gain of the servo valve, Ps1 is the supply oil pressure, Pr1 is the return oil pressure, and s(x) is defined as (9)sx≜1x≥00x<0.For simplicity, we define(10)Rh11=su1Ps1-Ph11+s-u1Ph11-Pr1,(11)Rh12=su1Ph12-Pr1+s-u1Ps1-Ph12.Therefore, (7) and (8) can be rewritten as (12)Qh11=kqki1Rh11u1,(13)Qh12=kqki1Rh12u1.


### 2.2. Model of the Control Surface

Supposing both HAs are connected rigidly to the control surface (see Figure1), the driving forces Fdh1 and Fdh2 which are generated by the two HAs can be described by (14)Fdh1=Fh1=Kh1xh1-xd,(15)Fdh2=Fh2=Kh2xh2-xdwhere Kh1, Kh2 are the connection stiffness and xd is the linear displacement of the control surface. The angular displacement θd can be approximately represented as a linear function of xd, then we have (16)θd=xdrd.Assuming that the outputs of the HAs are summed by force, the control surface dynamics can be represented as [24](17)Fdh1+Fdh2rd=Jdθ¨d+βdθ˙d+TLwhere rd represents the radial distance of the control surface, Jd is the moment of inertia, βd is the damping coefficient, and TL is the load torque.


### 2.3. State-Space Equation of the Dual Redundant Hydraulic Actuation System

Define the state variables as follows.(18)x=x11,x12,x13x21,x22,x23T≜xh1,x˙h1,A1Ph11-A2Ph12xh2,x˙h2,A1Ph21-A2Ph22TAccording to (3)-(13), we have(19)x˙13=fh11xu1-fh12x-fh13x,(20)fh11x=A1βeVh11kqki1Rh11+A2βeVh12kqki1Rh12,(21)fh12x=A1βeVh11A1x˙h1+Ci1Ph11-Ph12+A2βeVh12A2x˙h1+Ci1Ph11-Ph12,(22)fh13x=β1t-tfCt1A1βeVh11+A2βeVh12Ph11-Ph12sgnPh11-Ph12,and x˙23 can be described in a similar manner. Then, a state-space equation for the DRHAS can be written as (23)x˙11x˙21=x12x22x˙12x˙22=-H1x11x21-H2x12x22+H3x13x23+H4xcoupling-H3dx˙13x˙23=-F2x-βt-TfF3xθ+F1xuwhere H1=diag(Kh1/mh1,Kh2/mh2), H2=diag(Bh1/mh1,Bh2/mh2), H3=diag(1/mh1,1/mh2), H4=diag(Kh1rd/mh1,Kh2rd/mh2), d=d1,d2T, F1(x)=diag(fh11(x),fh21(x)), F2(x)=fh12(x),fh22(x)T, F3(x)=diag(fh13(x),fh23(x)), β(t-tf)=diag(β1(t-tf),β2(t-tf)), θ=Ct1,Ct2T is the fault parameter vector, u=u1,u2T is the control input vector, and xcoupling=θd,θdT is the coupling part between two HA channels [24].Defining the output variables asy1=θd,y2=θ˙d, then according to (14)-(17), the output equation can be written as (24)y˙1=y2y˙2=rdJdKh1xh1+Kh2xh2-rd2JdKh1+Kh2y1-βdJdy2-1JdTL.Assumption 1.
The chamber pressurePh11, Ph12 are bounded and satisfy Pr1<Ph11<Ps1, Pr1<Ph12<Ps1. Similarly, the chamber pressure Ph21, Ph22 are bounded and satisfy Pr2<Ph21<Ps2, Pr2<Ph22<Ps2. Meanwhile, all the state variables in (18) are bounded no matter under fault-free or faulty conditions.Remark 2.
From Assumption1 and (10)-(11), it can be concluded that Rh11>0 and Rh12>0. Similarly, it can be derived that Rh21>0 and Rh22>0. Therefore, the inequalities fh11(x)>0 and fh21(x)>0 always hold and the matrix F1(x) is reversible.


## 2.1. Model of the Hydraulic Actuator

The two HAs have the same structure, and the mathematical model for HA1 is presented below.The model of the EHSV can be described by a proportional function [27, 28] as(1)xv1=ki1u1where xv1 is the servo valve displacement, u1 is the control input, and ki1 is the servo valve gain.The cylinder dynamic can be represented by(2)x¨h1=1mh1A1Ph11-A2Ph12-Bh1x˙h1-Fh1-d1where xh1 is the piston rod displacement, mh1 is the piston mass, A1 is the piston area of the active chamber, A2 is the piston area of the passive chamber, Bh1 is the damping coefficient, Fh1 is the external load of the cylinder, and d1 is the model uncertainty which includes unmodeled nonlinear dynamics and lumped parameter variations.The chamber pressures of the cylinder, i.e.,Ph11 and Ph12, are calculated by(3)P˙h11=βeVh11-A1x˙h1-qh1+Qh11,(4)P˙h12=βeVh12A2x˙h1+qh1-Qh12where βe is the effective oil bulk modulus, Vh11=V10+A1xh1 is the volume of the active chamber, Vh12=V20-A2xh1 is the volume of the passive chamber, and V10 and V20 are the initial volumes of two chambers, respectively. Qh11 is the supply flow rate of the active chamber and Qh12 is the return flow rate of the passive chamber.When the DRHAS suffers from an internal leakage CMF, i.e., the internal leakage faults occurring in both HAs simultaneously due to a common cause, the internal leakage flow rateqh1 of HA1 can be modeled by [9](5)qh1=Ci1Ph11-Ph12+β1t-tfCt1Ph11-Ph12sgnPh11-Ph12where Ci1 is the normal internal leakage coefficient, Ct1 represents the fault internal leakage coefficient, sgn(·) is the sign function, tf is the fault occurrence time, β1(t-tf) is the time profile of the fault which can be described by (6)β1t-tf=0t<tf1-e-μ1t-tft≥tf,and μ1 is a positive constant which represents the fault evolution speed.The flow ratesQh11 and Qh12 can be calculated by(7)Qh11=kqki1u1su1Ps1-Ph11+s-u1Ph11-Pr1,(8)Qh12=kqki1u1su1Ph12-Pr1+s-u1Ps1-Ph12where kq=Cdω2/ρ is the flow rate gain of the servo valve, Ps1 is the supply oil pressure, Pr1 is the return oil pressure, and s(x) is defined as (9)sx≜1x≥00x<0.For simplicity, we define(10)Rh11=su1Ps1-Ph11+s-u1Ph11-Pr1,(11)Rh12=su1Ph12-Pr1+s-u1Ps1-Ph12.Therefore, (7) and (8) can be rewritten as (12)Qh11=kqki1Rh11u1,(13)Qh12=kqki1Rh12u1.


## 2.2. Model of the Control Surface

Supposing both HAs are connected rigidly to the control surface (see Figure1), the driving forces Fdh1 and Fdh2 which are generated by the two HAs can be described by (14)Fdh1=Fh1=Kh1xh1-xd,(15)Fdh2=Fh2=Kh2xh2-xdwhere Kh1, Kh2 are the connection stiffness and xd is the linear displacement of the control surface. The angular displacement θd can be approximately represented as a linear function of xd, then we have (16)θd=xdrd.Assuming that the outputs of the HAs are summed by force, the control surface dynamics can be represented as [24](17)Fdh1+Fdh2rd=Jdθ¨d+βdθ˙d+TLwhere rd represents the radial distance of the control surface, Jd is the moment of inertia, βd is the damping coefficient, and TL is the load torque.


## 2.3. State-Space Equation of the Dual Redundant Hydraulic Actuation System

Define the state variables as follows.(18)x=x11,x12,x13x21,x22,x23T≜xh1,x˙h1,A1Ph11-A2Ph12xh2,x˙h2,A1Ph21-A2Ph22TAccording to (3)-(13), we have(19)x˙13=fh11xu1-fh12x-fh13x,(20)fh11x=A1βeVh11kqki1Rh11+A2βeVh12kqki1Rh12,(21)fh12x=A1βeVh11A1x˙h1+Ci1Ph11-Ph12+A2βeVh12A2x˙h1+Ci1Ph11-Ph12,(22)fh13x=β1t-tfCt1A1βeVh11+A2βeVh12Ph11-Ph12sgnPh11-Ph12,and x˙23 can be described in a similar manner. Then, a state-space equation for the DRHAS can be written as (23)x˙11x˙21=x12x22x˙12x˙22=-H1x11x21-H2x12x22+H3x13x23+H4xcoupling-H3dx˙13x˙23=-F2x-βt-TfF3xθ+F1xuwhere H1=diag(Kh1/mh1,Kh2/mh2), H2=diag(Bh1/mh1,Bh2/mh2), H3=diag(1/mh1,1/mh2), H4=diag(Kh1rd/mh1,Kh2rd/mh2), d=d1,d2T, F1(x)=diag(fh11(x),fh21(x)), F2(x)=fh12(x),fh22(x)T, F3(x)=diag(fh13(x),fh23(x)), β(t-tf)=diag(β1(t-tf),β2(t-tf)), θ=Ct1,Ct2T is the fault parameter vector, u=u1,u2T is the control input vector, and xcoupling=θd,θdT is the coupling part between two HA channels [24].Defining the output variables asy1=θd,y2=θ˙d, then according to (14)-(17), the output equation can be written as (24)y˙1=y2y˙2=rdJdKh1xh1+Kh2xh2-rd2JdKh1+Kh2y1-βdJdy2-1JdTL.Assumption 1.
The chamber pressurePh11, Ph12 are bounded and satisfy Pr1<Ph11<Ps1, Pr1<Ph12<Ps1. Similarly, the chamber pressure Ph21, Ph22 are bounded and satisfy Pr2<Ph21<Ps2, Pr2<Ph22<Ps2. Meanwhile, all the state variables in (18) are bounded no matter under fault-free or faulty conditions.Remark 2.
From Assumption1 and (10)-(11), it can be concluded that Rh11>0 and Rh12>0. Similarly, it can be derived that Rh21>0 and Rh22>0. Therefore, the inequalities fh11(x)>0 and fh21(x)>0 always hold and the matrix F1(x) is reversible.


## 3. Fault-Tolerant Synchronization Control Scheme

For system (18)-(24), the purpose of this section is to design a fault-tolerant synchronization control scheme such that the output tracking error system is bounded stable under the internal leakage CMF condition and the following control performance can be guaranteed:(a)
The output deflection angle of the control surface can follow the system deflection command.(b)
The force outputs synchronization for two HAs can be maintained.The structure of the proposed FTSC scheme is illustrated in Figure2. To serve the control objectives, a desired trajectory generator is employed to adjust command inputs. Two reference trajectories xr1 and xr2 are generated for position tracking and for force outputs synchronization, respectively. Based on a transformed nonlinear model, an adaptive robust FTSC controller is designed by using backstepping technology. A Lyapunov function based method is used to evaluate the control performance of the closed-loop system. The controller reconfigures its control action through parameters adaptation online as the degraded control performance induced by the CMF exceeds a tolerable level.Figure 2
Structure diagram for the proposed FTSC scheme.


### 3.1. Desired Trajectory Generation

To realize the position control of the closed-loop system, a reference trajectoryxr1 is generated.Theorem 3.
The actual deflection angley1 can trace the deflection command θr accurately if the function f1(xh1,xh2)=Kh1xh1+Kh2xh2 can trace the reference trajectory xr1 which satisfies(25)xr1=Jdrdrd2JdKh1+Kh2y1+βdJdy2+1JdTL+α˙r1-k2z2-z1.Proof.
The reference trajectoryxr1 can be derived by using backstepping technique. Define two error variables as(26)z1=y1-θr,z2=y2-αr1.
Defining a Lyapunov function asV1=0.5z12, its time derivative is (27)V˙1=z1z˙1=z1z2+αr1-θ˙r.
Choosing the virtue control inputαr1 as (28)αr1=-k1z1+θ˙r,and substituting it into (27) yields(29)V˙1=-k1z12+z1z2where k1 is a positive constant.
Then choosing a Lyapunov function asV2=V1+0.5z22 and based on (24), (26), and (29), the time derivative of V2 can be written as (30)V˙2=V˙1+z2z˙2=-k1z12+z1z2+z2rdJdKh1xh1+Kh2xh2-rd2JdKh1+Kh2y1-βdJdy2-1JdTL-α˙r1.
Defining a function asf1(xh1,xh2)=Kh1xh1+Kh2xh2 and from (30), the function f1(xh1,xh2) can be seen as a control input, and when it satisfies(31)f1xh1,xh2=Jdrdrd2JdKh1+Kh2y1+βdJdy2+1JdTL+α˙r1-k2z2-z1, (30) can be rewritten as (32)V˙2=-k1z12-k2z22<0where k2 is a positive constant.
From (32), an asymptotic tracking can be achieved; i.e., z1→0 and y1→θr as t→∞. Therefore, Theorem 3 is proved.In order to achieve the synchronous force outputs of two HAs and eliminate the fighting force as much as possible, a reference trajectoryxr2 is generated.According to (14) and (15), the fighting force between two HAs can be represented as (33)ΔF=Fh1-Fh2=Kh1xh1-Kh2xh2-Kh1-Kh2rdθd.Defining a function asf2(xh1,xh2)=Kh1xh1-Kh2xh2 and setting xr2=(Kh1-Kh2)rdθr. From (33), it is evident that the fighting force ΔF will approach zero as the function f2(xh1,xh2) traces the reference trajectory xr2 accurately.Assumption 4.
The deflection commandθr is C5continuous and bounded.Remark 5.
Noting (24), (26), (28), (31), and Assumption 4, z1, z2, αr1, α˙r1, y1, and y2 are all bounded. Then, from (25), it is obvious that xr1 is bounded. According to the definition of xr2 and following Assumption 4, it is evident that xr2 is bounded. Then, by similar analysis, it can be derived that xr1 and xr2 are C3 continuous and bounded.


### 3.2. Model Transformation

A linear transformationT for the first two equations of (23) is introduced [24] and a new state-space model can be derived as (34)f˙x11,x21,t=fx12,x22,tf˙x12,x22,t=-M1fx11,x21,t-M2fx12,x22,t+M3x3+M4xcoupling-d~x˙3=-F2x-βt-tfF3xθ+F1xuwhere T=Kh1Kh2Kh1-Kh2, f(x11,x21,t)=Tx11x21=f1(x11,x21,t)f2(x11,x21,t), x3=x13x23, M1=TH1T-1, M2=TH2T-1, M3=TH3, M4=TH4 and d~=M3d=d~1,d~2T.To facilitate using backstepping method [29] to design the FTSC controller, a new state variable x-3=M3x3 is defined and the state-space model (34) can be rewritten into the following semi-strict-feedback form [30].(35)f˙x11,x21,t=fx12,x22,tf˙x12,x22,t=-M1fx11,x21,t-M2fx12,x22,t+x-3+M4xcoupling-d~x-˙3=-M3F2x-βt-tfM3F3xθ+M3F1xuAssumption 6.
The components of the model uncertainty vectord~ after transformation satisfy(36)d~1≤d-1,d~2≤d-2where d-1 and d-2 are known constants.Assumption 7.
The components of the fault parameter vectorθ=Ct1,Ct2Tsatisfy(37)θ1min≤Ct1≤θ1max,θ2min≤Ct2≤θ2maxwhere θ1min, θ1max, θ2min, and θ2max are known constants.
The effects of the internal leakage CMF can be represented by(38)qf=βt-tfM3F3xθwhere qf=qf1,qf2T.


### 3.3. Fault-Tolerant Synchronization Controller Design

Based on the transformed nonlinear model (35), an adaptive robust FTSC controller is developed and two control objectives should be achieved:(1)
The state variablef1(x11,x21,t) can track the reference trajectory xr1.(2)
The state variablef2(x11,x21,t) can track the reference trajectory xr2.The specific design procedure is described as follows.Step 1.
Defining a vector of reference trajectory asxr=xr1,xr2T, the tracking error vector z1 is calculated by (39)z1=z11,z12T=fx11,x21,t-xr.
Introducing a virtue control vectorα1=α11,α21T to stabilize f(x12,x22,t), the error vector z2 is calculated by (40)z2=z21,z22T=fx12,x22,t-α1.
Defining a Lyapunov function asV1=0.5z1Tz1, its time derivative is(41)V˙1=z1Tz˙1=z1Tfx12,x22,t-x˙r=z1Tz2+α1-x˙r.
Settingα1=-k1z1+x˙r and substituting it into (41) yields(42)V˙1=z1Tz2-z1Tk1z1where k1=diag(k11,k12) is a diagonal positive definite matrix.Step 2.
Introducing a virtual control vectorα2 to stabilize x-3, the tracking error z3 can be calculated by(43)z3=z31,z32T=x-3-α2.
Constructing a Lyapunov functionV2=V1+0.5z2Tz2, then according to (35), (40), (42), and (43), the time derivative of V2 can be written as (44)V˙2=z1Tz2-z1Tk1z1+z2T-M1fx11,x21,t-M2fx12,x22,t+z3+α2+M4xcoupling-d~-α˙1.
Choose the virtual control vectorα2in (43) as (45)α2=α2a+α2sα2a=-z1+M1fx11,x21,t+M2fx12,x22,tα2a=-M4xcoupling+α˙1,α2s=α2s1+α2s2α2s1=-k2s1z2where α2a is an adjustable model compensation term, α2s1 is a nominal stabilizing feedback, α2s2 is an additional feedback term to tackle the model uncertainties, and k2s1=diag(k2s11,k2s12) is a diagonal positive definite matrix.
The itemα2s2=[α2s21,α2s22]T in (45) should satisfy the following conditions:(46)iz21α2s21-d~1≤ε1andz22α2s22-d~2≤ε2,iiz21α2s21≤0andz22α2s22≤0where ε1 and ε2 are positive design parameters. In condition (i), α2s21 and α2s22 are used to tackle the model uncertainties d~1 and d~2 in two HA channels, respectively. Condition (ii) is to ensure that the control items α2s21 and α2s22 are dissipating in nature without affecting the functionality of the control item α2a.Remark 8 (see [30]).
To meet the conditions of (46), we can choose (47)α2s2=α2s21,α2s22T=-h214ε1z21,-h224ε2z22Twhere h21 and h22 are positive design parameters and satisfy (48)h21≥d-12,h22≥d-22.Remark 9.
From condition (i) of (46), it can be derived that(49)z2Tα2s2-d~≤εwhere ε=ε1+ε2. Note that the smaller the value of ε, the better the robust control performance can be achieved.
Substituting (45) into (44) yields(50)V˙2=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z2Tz3.Step 3.
Similarly, constructing a Lyapunov functionV3=V2+0.5z3Tz3, then according to (35), (43), and (50), the time derivative of V3 can be calculated by(51)V˙3=V˙2+z3Tz˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z2Tz3+z3T-M3F2x-βt-tfM3F3xθ+M3F1xu-α˙2.
The control lawu can be described as follows:(52)u=ua+usua=F1-1xF2x+F3xθ^+M3-1α˙2-z2us=F1-1xM3-1us1+us2us1=-k3s1z3where θ^ is an estimation of θ, ua is an adjustable model compensation term, us1 is a nominal stabilizing feedback, us2 is a nonlinear robust control law to tolerate the effects of the slight internal leakage CMF, and k3s1=diag(k3s11,k3s12) is a diagonal positive definite matrix.Remark 10.
Under no leakage or slight internal leakage CMF condition, the proposed FTSC control law (52) without parameters adaptation is actually a nonlinear robust synchronization control (NRSC) law.
From (38), the effect of the slight internal leakage CMF for each HA channel, which can be tolerated by the robust control law us2, should satisfy (53)qf1≤q-f1,qf2≤q-f2where q-f1, q-f2 are predefined constants and represent the tolerable level of the slight internal leakage for HA1 and HA2, respectively.
Subsequently, the robust control lawus2=[us21,us22]T in (52) should satisfy the following conditions.(54)iz31us21-qf1≤ξ1andz32us22-qf2≤ξ2,iiz31us21≤0andz32us22≤0
In condition (i) of (54), us21 and us22 are robust control laws to tolerate the effects of the slight internal leakage qf1 and qf2 in two HA channels, respectively, and the control accuracy for them is quantified by the positive design parameters ξ1 and ξ2, respectively. The condition (ii) of (54) is to ensure that the control law us2 is damping without affecting the functionality of ua.Remark 11.
The control lawus2 is chosen as (55)us2=us21,us22T=-h314ξ1z31,-h324ξ2z32Twhere h31 and h32 are positive design parameters which satisfy (56)h31≥q-f12,h32≥q-f22,and it is easy to verify that the control law (55) satisfies both the conditions of (54).Remark 12.
According to the condition (i) of (54), it can be derived that(57)z3Tus2-qf≤ξ,z3Tus2≤0with ξ=ξ1+ξ2. Similarly, the smaller the value of ξ, the better the robust control performance that can be achieved.
Then substituting (52) into (51) yields(58)V˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z2Tz3+z3T-βt-tfM3F3xθ+M3F3xθ^-k3s1z3+us2-z2.Theorem 13.
For the model uncertainties satisfying inequality (36) or/and the slight internal leakage CMF occurring in the DRHAS fulfilling inequality (53), the robust control law (52) without parameters adaptation (i.e., θ^=0) can guarantee the following:
(a) All signals are bounded and the Lyapunov functionV3 is bounded by(59)V3t≤Vrtwhere(60)Vrt=exp-λtV30+μλ1-exp⁡-λt,and λ=-2min(k11,k12,k2s11,k2s12,k3s11,k3s12), μ=ε+ξ, and V3(0) is the initial value of V3.
(b) If there are no model uncertainties and internal leakage CMF after a finite timet0 (i.e., d~=0,qf=0,∀t≥t0), then, except for the results in (a) of Theorem 13, zero final tracking error can also be achieved (i.e., z1→0 as t→∞).Proof.
Substituting (38), (49), and (57) into (58) yields(61)V˙3=-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+z2Tα2s2-d~+z3Tus2-qf≤-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+ε+ξ≤-2min⁡k11,k22,k2s11,k2s12,k3s11,k3s12V3+μ,which leads to inequality (59) and (60). Therefore, z1, z2, and z3 are bounded. Since xr is C3continous and bounded, from (39)-(41), f(x11,x21,t), f(x12,x22,t), α1, z˙1, and α˙1 are bounded. As y1 and y2 are bounded, xcoupling and x˙coupling are bounded. Then, from (45) and (43), α2 and x-3 are bounded. Noting Assumption 6, (35), and (39), f˙(x12,x22,t), z¨1, and α¨1 are bounded. Hence, from (45), α˙2 is bounded. Then, noting Assumption 1 and (52), the control input u is bounded. Therefore, the (a) of Theorem 13 is proved.
Furthermore, whend~=0,qf=0 after a finite time t0, (61) can be rewritten as(62)V˙3=-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+z2Tα2s2+z3Tus2,and then following condition (ii) of (46) and (54), we have(63)V˙3≤-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3≤-λV3.Hence, the (b) of Theorem 13 is proved.Remark 14.
If a moderate or severe internal leakage CMF occurs, inequality (53) will be violated which means that the control law (52) without fault parameter adaptation cannot tolerant the CMF any more. Meanwhile, the performance of the closed-loop system will deteriorate immediately, followed by a violation of inequality (59). Therefore, the quadratic Lyapunov function V3(t) is adopted as a performance indicator.Remark 15.
The functionVr(t) is used as a tolerable performance degradation level. As shown in (60), the maximum value of Vr(t) can be freely chosen by adjusting the controller parameters μ and λ. Note that the smaller the value of μ or the larger the value of λ, the better the transient performance that can be guaranteed.Remark 16.
Through online monitoring of the value ofV3(t) and comparing it with the value of Vr(t), a moderate or severe internal leakage CMF can be indicated as inequality (59) is violated over a period of time, and a condition-based maintenance operation can be performed to take the place of the scheduled maintenance, which will effectively reduce the maintenance costs.
If the performance indicator exceeds a tolerable performance degradation level, the controller reconfigures its control actions through controller parameter adaptation.
Ast-tf→∞, β(t-tf)M3F3(x)θ→M3F3(x)θ, (58) can be rewritten as (64)V˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z3Tus2-z3Tk3s1z3-z3TM3F3xθ~where θ~=θ-θ^.
Furthermore, construct a Lyapunov function shown as follows:(65)V=V3+0.5θ~TΓ-1θ~where the learning rate matrix Γ=diag(Γ1,Γ2) is a diagonal positive definite matrix and the time derivative of V is equivalent to(66)V˙=V˙3+θ~TΓ-1θ~˙=-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+ε+z3Tus2-θ~TΓ-1ΓF3TxM3Tz3+θ^˙.
From the above equation, an online updating law for estimation of internal leakage CMF can be derived by using the discontinuous projection mapping [30]. (67)θ^˙=0V3t≤VrtProjθ^σV3t>Vrtwhere σ=-ΓF3T(x)M3Tz3 and the discontinuous projection mapping is defined as follows:(68)Projθ^σ=0ifθ^=θmaxandσ>00ifθ^=θminandσ<0σotherwise,which guarantees that the parameters adaptation process has the following properties:(69)P1.θ^∈Ωθ^=θ^:θ^min≤θ^≤θ^max,P2.θ~TΓ-1Projθ^σ-σ≤0,∀σ.Theorem 17.
For the DRHAS suffering from moderate or severe internal leakage CMF which violates inequality (53), the control law (52) combined with the parameter adaptation law (67) can guarantee that the closed-loop control system gradually recover its performance if the following persistent excitation (PE) condition is satisfied:(70)∃T,δ>0s.t.∫tt+TφυφυTdυ≥δI,∀twith φ(υ)=F3T(x)M3T.Proof.
In this situation, the third equation of (35) can be rewritten as (71)x-˙3=-M3F2x-M3F3xθ+M3F1xu.
Then, the time derivative ofz3 is equal to(72)z˙3=x-˙3-α˙2=-z2-k3s1z3+us2-φTθ~.
Note that there is no unknown disturbance in (72) and the PE condition (70) can be easily satisfied as the integral term is a diagonal positive definite matrix. Therefore, the CMF estimation θ^ will converge to its real value (i.e., θ~→0 as t→∞). Thus, (64) can be rewritten as(73)V˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z3Tus2-z3Tk3s1z3≤-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+ε+z3Tus2≤λV3+ε,which leads to the same result as (59). If there are no model uncertainties after a finite time t1(i.e., d~=0,qf=0,∀t≥t1), the same result as (63) can be obtained. Thus, Theorem 17 is proved.


## 3.1. Desired Trajectory Generation

To realize the position control of the closed-loop system, a reference trajectoryxr1 is generated.Theorem 3.
The actual deflection angley1 can trace the deflection command θr accurately if the function f1(xh1,xh2)=Kh1xh1+Kh2xh2 can trace the reference trajectory xr1 which satisfies(25)xr1=Jdrdrd2JdKh1+Kh2y1+βdJdy2+1JdTL+α˙r1-k2z2-z1.Proof.
The reference trajectoryxr1 can be derived by using backstepping technique. Define two error variables as(26)z1=y1-θr,z2=y2-αr1.
Defining a Lyapunov function asV1=0.5z12, its time derivative is (27)V˙1=z1z˙1=z1z2+αr1-θ˙r.
Choosing the virtue control inputαr1 as (28)αr1=-k1z1+θ˙r,and substituting it into (27) yields(29)V˙1=-k1z12+z1z2where k1 is a positive constant.
Then choosing a Lyapunov function asV2=V1+0.5z22 and based on (24), (26), and (29), the time derivative of V2 can be written as (30)V˙2=V˙1+z2z˙2=-k1z12+z1z2+z2rdJdKh1xh1+Kh2xh2-rd2JdKh1+Kh2y1-βdJdy2-1JdTL-α˙r1.
Defining a function asf1(xh1,xh2)=Kh1xh1+Kh2xh2 and from (30), the function f1(xh1,xh2) can be seen as a control input, and when it satisfies(31)f1xh1,xh2=Jdrdrd2JdKh1+Kh2y1+βdJdy2+1JdTL+α˙r1-k2z2-z1, (30) can be rewritten as (32)V˙2=-k1z12-k2z22<0where k2 is a positive constant.
From (32), an asymptotic tracking can be achieved; i.e., z1→0 and y1→θr as t→∞. Therefore, Theorem 3 is proved.In order to achieve the synchronous force outputs of two HAs and eliminate the fighting force as much as possible, a reference trajectoryxr2 is generated.According to (14) and (15), the fighting force between two HAs can be represented as (33)ΔF=Fh1-Fh2=Kh1xh1-Kh2xh2-Kh1-Kh2rdθd.Defining a function asf2(xh1,xh2)=Kh1xh1-Kh2xh2 and setting xr2=(Kh1-Kh2)rdθr. From (33), it is evident that the fighting force ΔF will approach zero as the function f2(xh1,xh2) traces the reference trajectory xr2 accurately.Assumption 4.
The deflection commandθr is C5continuous and bounded.Remark 5.
Noting (24), (26), (28), (31), and Assumption 4, z1, z2, αr1, α˙r1, y1, and y2 are all bounded. Then, from (25), it is obvious that xr1 is bounded. According to the definition of xr2 and following Assumption 4, it is evident that xr2 is bounded. Then, by similar analysis, it can be derived that xr1 and xr2 are C3 continuous and bounded.


## 3.2. Model Transformation

A linear transformationT for the first two equations of (23) is introduced [24] and a new state-space model can be derived as (34)f˙x11,x21,t=fx12,x22,tf˙x12,x22,t=-M1fx11,x21,t-M2fx12,x22,t+M3x3+M4xcoupling-d~x˙3=-F2x-βt-tfF3xθ+F1xuwhere T=Kh1Kh2Kh1-Kh2, f(x11,x21,t)=Tx11x21=f1(x11,x21,t)f2(x11,x21,t), x3=x13x23, M1=TH1T-1, M2=TH2T-1, M3=TH3, M4=TH4 and d~=M3d=d~1,d~2T.To facilitate using backstepping method [29] to design the FTSC controller, a new state variable x-3=M3x3 is defined and the state-space model (34) can be rewritten into the following semi-strict-feedback form [30].(35)f˙x11,x21,t=fx12,x22,tf˙x12,x22,t=-M1fx11,x21,t-M2fx12,x22,t+x-3+M4xcoupling-d~x-˙3=-M3F2x-βt-tfM3F3xθ+M3F1xuAssumption 6.
The components of the model uncertainty vectord~ after transformation satisfy(36)d~1≤d-1,d~2≤d-2where d-1 and d-2 are known constants.Assumption 7.
The components of the fault parameter vectorθ=Ct1,Ct2Tsatisfy(37)θ1min≤Ct1≤θ1max,θ2min≤Ct2≤θ2maxwhere θ1min, θ1max, θ2min, and θ2max are known constants.
The effects of the internal leakage CMF can be represented by(38)qf=βt-tfM3F3xθwhere qf=qf1,qf2T.


## 3.3. Fault-Tolerant Synchronization Controller Design

Based on the transformed nonlinear model (35), an adaptive robust FTSC controller is developed and two control objectives should be achieved:(1)
The state variablef1(x11,x21,t) can track the reference trajectory xr1.(2)
The state variablef2(x11,x21,t) can track the reference trajectory xr2.The specific design procedure is described as follows.Step 1.
Defining a vector of reference trajectory asxr=xr1,xr2T, the tracking error vector z1 is calculated by (39)z1=z11,z12T=fx11,x21,t-xr.
Introducing a virtue control vectorα1=α11,α21T to stabilize f(x12,x22,t), the error vector z2 is calculated by (40)z2=z21,z22T=fx12,x22,t-α1.
Defining a Lyapunov function asV1=0.5z1Tz1, its time derivative is(41)V˙1=z1Tz˙1=z1Tfx12,x22,t-x˙r=z1Tz2+α1-x˙r.
Settingα1=-k1z1+x˙r and substituting it into (41) yields(42)V˙1=z1Tz2-z1Tk1z1where k1=diag(k11,k12) is a diagonal positive definite matrix.Step 2.
Introducing a virtual control vectorα2 to stabilize x-3, the tracking error z3 can be calculated by(43)z3=z31,z32T=x-3-α2.
Constructing a Lyapunov functionV2=V1+0.5z2Tz2, then according to (35), (40), (42), and (43), the time derivative of V2 can be written as (44)V˙2=z1Tz2-z1Tk1z1+z2T-M1fx11,x21,t-M2fx12,x22,t+z3+α2+M4xcoupling-d~-α˙1.
Choose the virtual control vectorα2in (43) as (45)α2=α2a+α2sα2a=-z1+M1fx11,x21,t+M2fx12,x22,tα2a=-M4xcoupling+α˙1,α2s=α2s1+α2s2α2s1=-k2s1z2where α2a is an adjustable model compensation term, α2s1 is a nominal stabilizing feedback, α2s2 is an additional feedback term to tackle the model uncertainties, and k2s1=diag(k2s11,k2s12) is a diagonal positive definite matrix.
The itemα2s2=[α2s21,α2s22]T in (45) should satisfy the following conditions:(46)iz21α2s21-d~1≤ε1andz22α2s22-d~2≤ε2,iiz21α2s21≤0andz22α2s22≤0where ε1 and ε2 are positive design parameters. In condition (i), α2s21 and α2s22 are used to tackle the model uncertainties d~1 and d~2 in two HA channels, respectively. Condition (ii) is to ensure that the control items α2s21 and α2s22 are dissipating in nature without affecting the functionality of the control item α2a.Remark 8 (see [30]).
To meet the conditions of (46), we can choose (47)α2s2=α2s21,α2s22T=-h214ε1z21,-h224ε2z22Twhere h21 and h22 are positive design parameters and satisfy (48)h21≥d-12,h22≥d-22.Remark 9.
From condition (i) of (46), it can be derived that(49)z2Tα2s2-d~≤εwhere ε=ε1+ε2. Note that the smaller the value of ε, the better the robust control performance can be achieved.
Substituting (45) into (44) yields(50)V˙2=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z2Tz3.Step 3.
Similarly, constructing a Lyapunov functionV3=V2+0.5z3Tz3, then according to (35), (43), and (50), the time derivative of V3 can be calculated by(51)V˙3=V˙2+z3Tz˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z2Tz3+z3T-M3F2x-βt-tfM3F3xθ+M3F1xu-α˙2.
The control lawu can be described as follows:(52)u=ua+usua=F1-1xF2x+F3xθ^+M3-1α˙2-z2us=F1-1xM3-1us1+us2us1=-k3s1z3where θ^ is an estimation of θ, ua is an adjustable model compensation term, us1 is a nominal stabilizing feedback, us2 is a nonlinear robust control law to tolerate the effects of the slight internal leakage CMF, and k3s1=diag(k3s11,k3s12) is a diagonal positive definite matrix.Remark 10.
Under no leakage or slight internal leakage CMF condition, the proposed FTSC control law (52) without parameters adaptation is actually a nonlinear robust synchronization control (NRSC) law.
From (38), the effect of the slight internal leakage CMF for each HA channel, which can be tolerated by the robust control law us2, should satisfy (53)qf1≤q-f1,qf2≤q-f2where q-f1, q-f2 are predefined constants and represent the tolerable level of the slight internal leakage for HA1 and HA2, respectively.
Subsequently, the robust control lawus2=[us21,us22]T in (52) should satisfy the following conditions.(54)iz31us21-qf1≤ξ1andz32us22-qf2≤ξ2,iiz31us21≤0andz32us22≤0
In condition (i) of (54), us21 and us22 are robust control laws to tolerate the effects of the slight internal leakage qf1 and qf2 in two HA channels, respectively, and the control accuracy for them is quantified by the positive design parameters ξ1 and ξ2, respectively. The condition (ii) of (54) is to ensure that the control law us2 is damping without affecting the functionality of ua.Remark 11.
The control lawus2 is chosen as (55)us2=us21,us22T=-h314ξ1z31,-h324ξ2z32Twhere h31 and h32 are positive design parameters which satisfy (56)h31≥q-f12,h32≥q-f22,and it is easy to verify that the control law (55) satisfies both the conditions of (54).Remark 12.
According to the condition (i) of (54), it can be derived that(57)z3Tus2-qf≤ξ,z3Tus2≤0with ξ=ξ1+ξ2. Similarly, the smaller the value of ξ, the better the robust control performance that can be achieved.
Then substituting (52) into (51) yields(58)V˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z2Tz3+z3T-βt-tfM3F3xθ+M3F3xθ^-k3s1z3+us2-z2.Theorem 13.
For the model uncertainties satisfying inequality (36) or/and the slight internal leakage CMF occurring in the DRHAS fulfilling inequality (53), the robust control law (52) without parameters adaptation (i.e., θ^=0) can guarantee the following:
(a) All signals are bounded and the Lyapunov functionV3 is bounded by(59)V3t≤Vrtwhere(60)Vrt=exp-λtV30+μλ1-exp⁡-λt,and λ=-2min(k11,k12,k2s11,k2s12,k3s11,k3s12), μ=ε+ξ, and V3(0) is the initial value of V3.
(b) If there are no model uncertainties and internal leakage CMF after a finite timet0 (i.e., d~=0,qf=0,∀t≥t0), then, except for the results in (a) of Theorem 13, zero final tracking error can also be achieved (i.e., z1→0 as t→∞).Proof.
Substituting (38), (49), and (57) into (58) yields(61)V˙3=-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+z2Tα2s2-d~+z3Tus2-qf≤-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+ε+ξ≤-2min⁡k11,k22,k2s11,k2s12,k3s11,k3s12V3+μ,which leads to inequality (59) and (60). Therefore, z1, z2, and z3 are bounded. Since xr is C3continous and bounded, from (39)-(41), f(x11,x21,t), f(x12,x22,t), α1, z˙1, and α˙1 are bounded. As y1 and y2 are bounded, xcoupling and x˙coupling are bounded. Then, from (45) and (43), α2 and x-3 are bounded. Noting Assumption 6, (35), and (39), f˙(x12,x22,t), z¨1, and α¨1 are bounded. Hence, from (45), α˙2 is bounded. Then, noting Assumption 1 and (52), the control input u is bounded. Therefore, the (a) of Theorem 13 is proved.
Furthermore, whend~=0,qf=0 after a finite time t0, (61) can be rewritten as(62)V˙3=-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+z2Tα2s2+z3Tus2,and then following condition (ii) of (46) and (54), we have(63)V˙3≤-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3≤-λV3.Hence, the (b) of Theorem 13 is proved.Remark 14.
If a moderate or severe internal leakage CMF occurs, inequality (53) will be violated which means that the control law (52) without fault parameter adaptation cannot tolerant the CMF any more. Meanwhile, the performance of the closed-loop system will deteriorate immediately, followed by a violation of inequality (59). Therefore, the quadratic Lyapunov function V3(t) is adopted as a performance indicator.Remark 15.
The functionVr(t) is used as a tolerable performance degradation level. As shown in (60), the maximum value of Vr(t) can be freely chosen by adjusting the controller parameters μ and λ. Note that the smaller the value of μ or the larger the value of λ, the better the transient performance that can be guaranteed.Remark 16.
Through online monitoring of the value ofV3(t) and comparing it with the value of Vr(t), a moderate or severe internal leakage CMF can be indicated as inequality (59) is violated over a period of time, and a condition-based maintenance operation can be performed to take the place of the scheduled maintenance, which will effectively reduce the maintenance costs.
If the performance indicator exceeds a tolerable performance degradation level, the controller reconfigures its control actions through controller parameter adaptation.
Ast-tf→∞, β(t-tf)M3F3(x)θ→M3F3(x)θ, (58) can be rewritten as (64)V˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z3Tus2-z3Tk3s1z3-z3TM3F3xθ~where θ~=θ-θ^.
Furthermore, construct a Lyapunov function shown as follows:(65)V=V3+0.5θ~TΓ-1θ~where the learning rate matrix Γ=diag(Γ1,Γ2) is a diagonal positive definite matrix and the time derivative of V is equivalent to(66)V˙=V˙3+θ~TΓ-1θ~˙=-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+ε+z3Tus2-θ~TΓ-1ΓF3TxM3Tz3+θ^˙.
From the above equation, an online updating law for estimation of internal leakage CMF can be derived by using the discontinuous projection mapping [30]. (67)θ^˙=0V3t≤VrtProjθ^σV3t>Vrtwhere σ=-ΓF3T(x)M3Tz3 and the discontinuous projection mapping is defined as follows:(68)Projθ^σ=0ifθ^=θmaxandσ>00ifθ^=θminandσ<0σotherwise,which guarantees that the parameters adaptation process has the following properties:(69)P1.θ^∈Ωθ^=θ^:θ^min≤θ^≤θ^max,P2.θ~TΓ-1Projθ^σ-σ≤0,∀σ.Theorem 17.
For the DRHAS suffering from moderate or severe internal leakage CMF which violates inequality (53), the control law (52) combined with the parameter adaptation law (67) can guarantee that the closed-loop control system gradually recover its performance if the following persistent excitation (PE) condition is satisfied:(70)∃T,δ>0s.t.∫tt+TφυφυTdυ≥δI,∀twith φ(υ)=F3T(x)M3T.Proof.
In this situation, the third equation of (35) can be rewritten as (71)x-˙3=-M3F2x-M3F3xθ+M3F1xu.
Then, the time derivative ofz3 is equal to(72)z˙3=x-˙3-α˙2=-z2-k3s1z3+us2-φTθ~.
Note that there is no unknown disturbance in (72) and the PE condition (70) can be easily satisfied as the integral term is a diagonal positive definite matrix. Therefore, the CMF estimation θ^ will converge to its real value (i.e., θ~→0 as t→∞). Thus, (64) can be rewritten as(73)V˙3=-z1Tk1z1-z2Tk2s1z2+z2Tα2s2-d~+z3Tus2-z3Tk3s1z3≤-z1Tk1z1-z2Tk2s1z2-z3Tk3s1z3+ε+z3Tus2≤λV3+ε,which leads to the same result as (59). If there are no model uncertainties after a finite time t1(i.e., d~=0,qf=0,∀t≥t1), the same result as (63) can be obtained. Thus, Theorem 17 is proved.


## 4. Simulation Results

To illustrate the effectiveness of the proposed FTSC scheme, a model of a DRHAS is established in the MATLAB/Simulink environment. Simulation parameters of the model are given in Table1.Table 1
Simulation parameters of the DRHAS [13, 24].
Component Parameter Value (Unit) Parameter Value (Unit) EHSV EHSV gainki1, ki2 3.04 × 1 0 - 3 ( m / A ) Flow rate gainkq 7.5 × 1 0 - 5 ( m 2 / s / p a 1 / 2 )  Cylinder Piston massmh1, mh1 55 ( k g ) Piston area (active chamber)A1 1.47 × 1 0 - 3 ( m 2 ) Piston area (passive chamber)A2 1.47 × 1 0 - 3 ( m 2 ) Effective oil bulk modulusβe 800 ( M P a ) Damping coefficientBh1, Bh2 1 × 1 0 4 ( N s / m ) Initial volume (active chamber)V10 0.735 × 1 0 - 4 ( m 3 ) Initial volume (passive chamber)V20 0.735 × 1 0 - 4 ( m 3 ) Normal internal leakage coefficientCi1, Ci2 1 × 1 0 - 12 ( m 3 / s / p a )  Control Surface Connection stiffnessKh1, Kh2 1 × 1 0 8 ( N / m ) Radial distancerd 0.15 ( m ) Moment of inertiaJd 13.5 ( k g m 2 ) Damping coefficientβd 51.75 ( N m s ) Elastic load coefficientKd 9.14 × 1 0 5 ( N / m )  Power Supply Oil supply pressurePs1, Ps2 28 ( M P a ) Oil reservoir pressurePr1,Pr2 0 ( M P a )A comprehensive performance evaluation regarding the proposed FTSC scheme is performed in comparison with the other two schemes, i.e., the classical PID and the NRSC. Moreover, the following two scenarios are considered:(a) Scenario S1: A severe internal leakage CMF which has different effects (i.e., different magnitudes and evolution speeds) on two channels occurs in the DRHAS. Specifically, attf=7s, a severe internal leakage fault with Ct1=5×10-7(m3/s/pa), μ1=10 occurs in HA1 and a severe internal leakage fault with Ct2=1×10-7(m3/s/pa), μ2=0.5 occurs in HA2 due to a common cause.(b) Scenario S2: A slight internal leakage CMF occurs in the DRHAS. Specifically, attf=7s, a slight internal leakage with Ct1=1×10-9(m3/s/pa), μ1=5 occurs in HA1 and a slight internal leakage with Ct2=0.5×10-9(m3/s/pa), μ2=10 occurs in HA2 due to a common cause.The parameters for the control schemes are shown in Table2. In addition, θr=arctan⁡(0.2sin⁡(2πt))[1-exp⁡(-0.1t3)] is used as the system deflection command to fulfill Assumption 4. An elastic load FL=Kdxd is used to simulate the air loads acting on the control surface and the unknown model uncertainties d1 and d2 in two HAs are as follows:(74)d1=100Nt<2s100+400sin⁡2πtNt≥2sand d2=400N.Table 2
Parameters of the control schemes.
Scheme Parameter Value Parameter Value FTSC /NRSC k 1 300 k 2 300 k 1 d i a g ( 1000,1000 ) k 2 s 1 d i a g ( 1000,1000 ) k 3 s 1 d i a g ( 1000,1000 ) h 21 , h 22 3.4 × 1 0 18 ε 1 , ε 2 1.0 × 1 0 14 h 31 , h 32 2.5 × 1 0 23 ξ 1 , ξ 2 1.0 × 1 0 21 d - 1 , d - 2 1.8 × 1 0 9 q - f 1 , q - f 2 4.5 × 1 0 11 θ m a x 5.0 × 1 0 - 6 , 5.0 × 1 0 - 6 T θ m i n 0,0 T  FTSC Γ 1,Γ2 1.0 × 1 0 - 37  PID k p 1 , k p 2 90 k i 1 , k i 2 0 k d 1 , k d 2 0For scenario S1, Figure3 shows the position tracking error for three control schemes, and Table 3 summarizes the results of the maximum position tracking error for each scheme under different conditions. It is evident that the proposed FTSC scheme has the best position tracking performance among the three schemes.Table 3
Maximum tracking error for three control schemes under different scenarios.
Maximum tracking error (rad) Fault-freet∈[0s,7s) S1 S2 Fault transientt∈[7s,8s] Steady statet∈[15s,20s] Fault transientt∈[7s,8s] Steady statet∈[15s,20s] PID 0.0250 0.0343 0.0555 0.0250 0.0251 NRSC 2.4546e-004 0.0014 0.0018 2.4585e-004 2.4687e-004 FTSC 2.4546e-004 7.8270e-004 2.4635e-004 2.4585e-004 2.4687e-004Figure 3
Position tracking error under scenario S1.In normal condition, the FTSC and NRSC are the same. As listed in Table3, the maximum position tracking error for FTSC/NRSC is only 2.4546e-004 rad, much less than that for PID. This can be attributed to the robust control design in the FTSC/NRSC which dominates the model uncertainties better than PID.When the severe internal leakage CMF occurs attf=7s, the dynamic performance of the closed-loop system begins to deteriorate rapidly, and this can be reflected from the variation of the performance indicator V3 (see Figure 4). It is observed that the function value of V3 experiences an abrupt variation at tf=7s and exceeds its upper bound Vr immediately, due to the occurrence of the CMF. Then, according to (67), the fault parameter vector and also the control parameter vector θ^ for FTSC begin to be updated online, and more control input is generated to compensate for the CMF effects. Therefore, the dynamic performance of the system under the control of the FTSC is gradually restored. As summarized in Table 3, after experiencing a small jump about 7.8270e-004 rad, the position tracking error for the FTSC finally converges to 2.4635e-004 rad in steady state, much less than that for PID and for NRSC.Figure 4
Indication of performance degradation under scenario S1.Figure5 compares the output fighting forces for three control schemes. Table 4 summarizes the results of the maximum fighting force for each scheme under different conditions. Results indicate that the proposed FTSC scheme has the smallest fighting force among all the schemes no matter during fault-free or faulty time.Table 4
Maximum fighting force for three control schemes under different scenarios.
Maximum fighting force (N) Fault-freet∈[0s,7s) S1 S2 Fault transientt∈[7s,8s] Steady statet∈[15s,20s] Fault transientt∈[7s,8s] Steady statet∈[15s,20s] PID 560.7653 2.5401e+004 1.7943e+004 1.2109e+003 1.2450e+003 NRSC 133.9697 9.7126e+003 8.1759e+003 146.9094 147.5323 FTSC 133.9697 4.9884e+003 133.0907 146.9094 147.5323Figure 5
Force fighting between two HA channels under scenario S1.In normal condition, due to the introduction of the force synchronization strategy, the NRSC and FTSC can reduce the fighting force more effectively than PID. As shown in the second column of Table4, the maximum fighting force is about 133N for FTSC, smaller than that for PID.Under fault condition, note that the CMF which has different effects on HAs may lead to response difference between two HAs and further enlarges the fighting force. Therefore, since the proposed FTSC scheme can accommodate the CMF better than the other two schemes, a more significant elimination of the fighting force between two HAs can be achieved naturally. As shown in Figure5 and Table 4, the fighting force of the FTSC reaches its maximum value 4.9884e+003N, along with the occurrence of the severe internal leakage CMF, and gradually reduces to 133N after entering steady state, far less than that of PID and NRSC.Figure6 shows the control inputs of the DRHAS for three schemes. Note that the fault effect in HA1 is more serious than that in HA2. Therefore, more control input is allocated to HA1 to compensate for the fault effect. As depicted in the upper plot of Figure 6, the control input u1 for the proposed scheme is slightly larger than that for PID and for NRSC, but still maintains a relatively small value. Figure 7 presents the estimation result of the CMF and its real value. It can be found that the fault effect for each channel can be estimated correctly.Figure 6
Control inputs of the DRHAS under scenario S1.Figure 7
Estimate for severe internal leakage CMF under scenario S1.Under scenario S2, the DRHAS suffers from a slight internal leakage CMF. As illustrated in Figure8, neither the maximum fault effect in HA1 nor that in HA2 exceeds the threshold. In this situation, according to inequalities (53) and (54), the leakage effects can be compensated by the robust control law (52). In Figure 9, the variation curve of the performance indicator V3 further confirms that the system performance is still acceptable as the maximum value of V3 does not exceed its upper bound. Therefore, based on (67), the parameters online adaptation in (52) will not be activated and the schemes FTSC and NRSC are the same.Figure 8
The internal leakage effects for two HA channels under scenario S2.Figure 9
Indication of performance degradation under scenario S2.Figure10 compares the position tracking error for three schemes. It is easy to see that the FTSC/NRSC has a much higher tracking accuracy than PID. As summarized in Table 3, by comparing the maximum tracking errors before and after the fault occurrence for each scheme, it seems that the slight internal leakage CMF has little effect on the position tracking accuracy of the system. Figure 11 shows the output fighting forces for three control schemes. Obviously, the FTSC/NRSC can reduce the fighting force more effectively in comparison with the PID, especially after the fault occurrence. As listed in Table 4, the maximum fighting force for PID during fault transient time is about 1.2109e+003N, almost ten times larger than that for FTSC/NRSC. This can be attributed to the robust control design as well as the force synchronization design for the FTSC/NRSC controller.Figure 10
Position tracking error under scenario S2.Figure 11
Force fighting between two HA channels under scenario S2.In conclusion, based on the simulation results, the proposed FTSC scheme can ensure the following:(a)
The closed-loop control system is robust to model uncertainties and slight internal leakage CMF.(b)
Fast and correct CMF estimation can be obtained when the performance degradation exceeds the predesigned tolerable level.(c)
A high position tracking accuracy and an effective reduction of fighting force between channels can be achieved no matter during fault-free or faulty time.


## 5. Conclusions

This paper proposes a fault-tolerant synchronization control scheme for a DRHAS, which operates on A/A mode and suffers from an internal leakage CMF, i.e., internal leakage faults occurring in both HAs simultaneously due to a common cause. Firstly, a desired trajectory generator is employed and two reference trajectories are generated for position tracking and for force outputs synchronization of HA channels, respectively. Secondly, a nonlinear adaptive robust FTSC controller is designed by employing backstepping method. A Lyapunov function based method is used to evaluate the control performance of the closed-loop system. As the degraded control performance induced by the CMF exceeds a tolerable level, the FTSC controller reconfigures its control action through parameters adaptation online to accommodate the fault. It has been verified that the proposed FTSC scheme can guarantee the bounded stability of output tracking error system under CMF. Finally, simulation results under two scenarios demonstrate the effectiveness of the proposed FTSC scheme.



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*Source: 6570104-2018-10-04.xml*