# Adaptive Fault-Tolerant Cruise Control for a Class of High-Speed Trains with Unknown Actuator Failure and Control Input Saturation

Authors: Tao Tao; Hongze Xu
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper investigates the position and velocity tracking control of a class of high-speed trains (HST) with unknown actuator failures (AF) and control input saturation (CIS). Firstly, a nonlinear dynamic model for HST at normal operating status is built. The structure of traction system in HST is analyzed and the corresponding model for HST with unknown AF is presented as well. The type of AF under consideration is that some of the plant inputs are influenced by hopping function. An adaptive model-based fault detection and diagnosis (AMFDD) module is proposed based on immersion and invariance (I&I) method to make decisions on whether a fault has occurred. A new framework to design a monotone mapping is proposed in I&I method, that is,P(x)-monotone. Using on-line obtained fault information, an adaptive law is designed to update the controller parameters to handle unknown AF and CIS in HST simultaneously when some of plant parameters are unknown. Closed-loop stability and asymptotic position and velocity tracking are ensured. Numerical simulations of China Railways High-speed 2 (CRH2) train are provided to verify the effectiveness of the presented scheme.


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## Body

## 1. Introduction

In practical applications, nonlinearity caused by imperfections of modelling leads to performance degradation of the closed-loop system. Compensation control of nonlinear systems has been a research topic of wide interest. In [1], unknown continuous and monotone nonlinearities which satisfy some constraints are considered, and an adaptive controller using artificial neural networks is proposed for a class of nonlinear systems with input nonlinearities. In [2], direct adaptive neural network control is presented for uncertain multi-input/multi-output nonlinear systems in block triangular forms. Reference [3] presents two indirect adaptive control schemes which are employed to approximate unknown nonlinear functions. Reference [4] proposes a robust control algorithm for a three-axis stabilized flexible spacecraft in the presence of control input nonlinearity/dead-zone. In [5, 6], an adaptive neurofuzzy control is proposed by describing nonlinear affine systems as a Takagi-Sugeno fuzzy model. An adaptive fuzzy backstepping control approach is considered in [7] for a class of nonlinear strict-feedback systems with unknown functions, unknown dead zones, and immeasurable states. In [8], step tracking control problem for discrete-time nonlinear systems, which are represented by a Takagi-Sugeno fuzzy system, is investigated in a networked environment with a limited capacity. In [9], the problem of adaptive fuzzy tracking control via output feedback for a class of uncertain strict-feedback nonlinear systems with unknown time-delay functions is investigated. In [10], the problem of tracking control for a class of large-scale nonlinear systems with unmodeled dynamics is addressed by designing the decentralized adaptive fuzzy output feedback controller to guarantee that all the signals in the closed-loop system are bounded. Reference [11] deals with the adaptive sliding-mode control problem for nonlinear active suspension systems with varying sprung and unsprung masses, unknown actuator nonlinearity, and suspension performances.In HST, besides nonlinearity, train traction module may fail to work due to various kinds of reasons [12–14], such as overvoltage in traction transformer, overcurrent in traction converter, and overheat in asynchronous motor. AF in HST will cause problems, that is, inaccurate position tracking, inaccurate velocity tracking, or even train accident. It is significant to design the controller addressing AF to realize fault-tolerant control (FTC) in HST [15–21]. In control society, there are many remarkable results on FTC for AF. In [22], a class of nonlinear systems with faults, parametric uncertainties, and without full state measurements are considered. A novel observer is designed whose estimation error is not affected by faults and an observer-based fault-tolerant tracking controller is proposed to make the outputs asymptotically track the reference signals while the states are bounded. In [23], a FTC scheme based on the adaptive control technique for near-space-vehicle attitude dynamics is considered. In [24], a FTC scheme using backstepping and neural network methodology is proposed for a class of nonlinear systems with known structure and unknown faults. In [25, 26], a direct adaptive feedback controller is developed for linear time-invariant plants with AF and the closed-loop stability and asymptotic tracking are ensured. Reference [27] proposes an adaptive compensation for parametric strict-feedback systems. A fault-tolerant robust control for a class of nonlinear systems is investigated in [28]. A robust FTC will switch itself between robust control strategies designed under normal operation and faulty condition. Reference [29] investigates the reliable H∞ control problem for discrete-time piecewise linear systems with time delays and AF. In [30], a fault-tolerant controller is presented for Lipschitz nonlinear continuous-time systems in the presence of disturbances and noises. An integrated design of the adaptive robust control and the fault identification for a linear system with AF is proposed in [31]. Reference [32] investigates the FTC problem for near-space vehicle attitude dynamics with AF, which is described by a Takagi-Sugeno fuzzy model. In [33], a new FTC scheme is proposed by incorporating integral sliding modes, unknown input observers, and a fixed control allocation scheme, where only measured system outputs are assumed to be available. The problem of FTC for a class of nonlinear systems with AF is discussed, and an observer-based FTC scheme is proposed in [34]. Adaptive fuzzy observers are proposed to provide a bank of residuals for fault detection and isolation and an accommodation scheme is proposed to compensate for the effect of the fault. Reference [35] presents a FTC for nonlinear systems which are connected in a networked control system.CIS, which implies that the output of serving motors is constrained, is another problem in HST, and a few studies have been done on this field [36, 37]. Study on designing controllers by considering CIS beforehand is promoted for system in which higher performance is expected. Karason and Annaswamy [38] deal with the problem of adaptive control for a linear time-invariant plant in the presence of constraints on the input amplitude. In [39], a nonlinear small gain theorem is presented that provides formalism for analyzing the behavior of certain control systems that contain or utilize saturation. Polycarpou et al. [40] address the issue of CIS in on-line approximation based control for nonlinear systems, and a modified control design framework is presented for preventing CIS from destroying the learning capabilities and memory of an on-line approximation. In [41], the robust control of an induction motor is investigated, and a parameter-dependent model is addressed. Wu et al. [42] present a method, for designing output feedback laws that stabilize a linear system subject to actuator saturation with a large domain of attraction, which applies to general linear systems including strictly unstable ones and is presented in both continuous-time and discrete-time setting. Zhou et al. [43, 44] develop some significative methods in this direction using Raccati equations as a basic tool. Fridman and Dambrine [45] consider quantized and delayed state feedback control of linear systems with given constant bounds on the quantization error and on the time-varying delay. Reference [46] deals with the problem of tracking and stabilization control of internally damped mobile robots with unknown parameters and subject to input torque saturation and external disturbances.Recently, a novel I&I adaptive method for nonlinear systems is presented to realize performance-oriented control [47–49]. In this paper, based on a nonlinear dynamic model for a class of HST, an adaptive FTC with I&I AMFDD is presented. An AMFDD module using I&I adaptive state observer is designed to detect AF, and direct adaptive controller based on on-line AMFDD information is achieved to handle nonlinearity, unknown AF, and CIS in HST. A new framework to design a monotone mapping is proposed in I&I method, that is, P(x)-monotone. The stability of HST systems is proved theoretically.The rest of this paper is organized as follows. Section2 introduces problem formulation in HST. In Section 3, an AMFDD module is introduced and corresponding I&I adaptive FTC is presented for AF and CIS. The simulation is shown in Section 4. Section 5 draws the conclusion.


## 2. Problem Formulation

In this section, we propose a nonlinear dynamic model for a class of HST with and without AF.


### 2.1. Dynamic Model of a Single Carriage

The HST consists of a number of carriages, and couplers are used to couple adjacent carriages. A carriage during travelling is subjected to various kinds of forces, such as traction and braking forces, forces between carriages, and resistance forces. The force analysis of a single carriage in HST is illustrated in Figure1.Figure 1
The force analysis of a single carriage.Traction and Braking Forces.
u
i denotes traction force of the ith carriage, that is, ui>0, and braking force of the ith carriage, that is, ui<0. λi denotes whether the ith carriage is powered or nonpowered.Assumption 1.
The class of HST that we considered is distributed driving type [50], that is, λi=1 for all carriages.Interacting Forces between Carriages. Connection modules between two carriages in HST are nonlinear subsystems. Hence, it is difficult to obtain accurate mechanical model of interacting forces between two carriages. A widely used mechanical model for couplers consists of elastic forces and damping forces. In [51], nonlinear elastic force acting between carriages is described as follows:
(1)fk(t)=k(t)ϵi(t),
where relative displacement between two carriages is as
(2)ϵi(t)=xi(t)-xi-1(t),
where xi(t) is position of the ith carriage. k(t) denotes nonlinear relationship between displacements and forces and
(3)k(t)=k0(1+ϵi(t)σ).
If σ=0, coupler between carriages is linear, that is, k(t)=k0, which does not exist in practice. If σ<0 or σ>0, the coupler is soft coupling or hard coupling, respectively. Khalil and Grizzle [52] and Franklin and Powell [53] have proved that a system with a soft spring is more likely to have unsatisfying control performance. We take the minimum value k of k(t) in controller design, that is, k(t)=k. The damping force is described as
(4)fb(t)=b(t)(x˙i(t)-x˙i-1(t)),
where b(t) is damping coefficient and x˙i(t) is velocity of the ith carriage. We take the minimal value b of the coefficient b(t), that is, b(t)=b. The interacting force between carriages is as
(5)fic(t)=fk(t)+fb(t).Resistance Forces of HST.During travelling, carriages of HST are subjected to various kinds of resistance forces mainly including mechanical resistance and air resistance. The two forces are as follows [54]:
(6)fip(t)=mi(c0+cvx˙i(t)+cax˙i2(t)),
where mi represents mass of the ith carriage and mi(c0+cvx˙i(t)) and micax˙i2(t) denote mechanical resistance and air resistance, respectively. c0,cv, and ca are coefficients which may not be obtained accurately during train locomotion.Slope and curve resistance forces of HST depend on railway condition. Slope and curve resistance forces are normally considered by Garg and Dukkipati [55] as follows:
(7)fie(t)=migsinθi+0.002midaRt,
where da is axle length of a carriage and Rt is turning radius. Based on the above analysis, dynamic model of a single carriage in HST is given as follows:
(8)mix¨i(t)=λiui(t)+fi+1c(t)-fic(t)-fip(t)-fie(t)=λiui(t)-mi(c0+cvx˙i(t))-micax˙i2(t)-k(xi(t)-xi-1(t))-b(x˙i(t)-x˙i-1(t))-k(xi(t)-xi+1(t))-b(x˙i(t)-x˙i+1(t))-migsinθi-0.002midaRt.


### 2.2.  Dynamic Model of HST

Dynamic model of HST withn carriages can be derived as follows:
(9)m1x¨1(t)=λ1u1(t)-k(x1(t)-x2(t))-b(x˙1(t)-x˙2(t))-m1(c0+cvx˙1(t))-m1cax˙12(t)-m1gsinθ1-0.002m1daRt,m2x¨2(t)=λ2u2(t)-k(x2(t)-x1(t))-b(x˙2(t)-x˙1(t))-k(x2(t)-x3(t))-b(x˙2(t)-x˙3(t))-m2(c0+cvx˙2(t))-m2cax˙22(t)-m2gsinθ2-0.002m2daRt,⋮mix¨i(t)=λiui(t)-k(xi(t)-xi-1(t))-b(x˙i(t)-x˙i-1(t))-k(xi(t)-xi+1(t))-b(x˙i(t)-x˙i+1(t))-mi(c0+cvx˙i(t))-micax˙i2(t)-migsinθi-0.002midaRt,⋮mnx¨n(t)=λnun(t)-k(xn(t)-xn-1(t))-b(x˙n(t)-x˙n-1(t))-mn(c0+cvx˙n(t))-mncax˙n2(t)-mngsinθn-0.002mndaRt,
where xi subsystem represents dynamics of the ith carriage in HST. To simplify the controller design, (9) is rewritten as
(10)x˙(t)=Ax(t)+Bu(t)+C0+Cvx(t)+Cax2(t)=[0n×nIn×nA21A22]x(t)+[0n×nB2]u(t)+[0n×1E02+C02]+[0n×n0n×n0n×nCv2]x(t)+[0n×n0n×n0n×nCa2]x2(t),y=Cx(t),
where x=[x1⋯xnx˙1⋯x˙n]T is the state vector. A∈R2n×2n and B∈R2n×n are known, and C0∈R2n, Cv∈R2n×2n, and Ca∈R2n×2n are unknown matrixes, and we have
(11)A12=[-km1km10⋯⋯⋯0122km2-2km2km20⋯⋯00km3122-2km3km30⋯00⋯⋯⋯⋯⋯00⋯⋯⋯0kmnkmn],A22=[-bm1bm10⋯⋯⋯0122bm2-2bm2bm20⋯⋯00bm3122-2bm3bm30⋯00⋯⋯⋯⋯⋯00⋯⋯⋯0bmnbmn],B2=diag⁡(λ1m1λ2m2⋯λnmn),E02=diag⁡(gsinθ1+0.002daRt⋯⋯gsinθn+0.002daRt),C02=[-c0-c0⋯⋯-c0]T,Cv2=diag⁡(-cv-cv⋯⋯-cv),Ca2=diag⁡(-ca-ca⋯⋯-ca).


### 2.3. Fault Analysis of Traction Systems in HST

In HST, velocity cruise control is accomplished through the traction system which consists of high-voltage circuit (included by pantograph, current transformer, main circuit breaker, etc.), traction transformer, traction converter, asynchronous motor, wheels, and so on [12–14]. The principle of velocity cruise control is shown in Figure 2.Figure 2
The principle of velocity cruise control in HST.Traction converter is used to control traction motor to achieve velocity cruise control of HST. The probability of incorrect voltage and current, such as malfunction of electronic components in traction converter, is great [56, 57]. The output of ith actuator in failure status can be described as uf(t) and its elements is shown as follows:
(12)uif(t)=fi(t)ui(t)(i=1,…,n),
where ui(t) is control input without AF. fi(t)∈(fi_,f¯i) is unknown piecewise hopping function, with known boundary fi_,f¯i, which represents malfunction of electronic components in traction system and
(13)fi(t)={1t<t1fi1t1≤t<t2fi2t2≤t<t3⋮⋮
where fij is unknown constant for j=1,2,…,l and tj for j=1,2,…,l satisfies
(14)tj-tj-1≥tj,min⁡,
where t0=0 and tj,min⁡, which is the minimum switch time for fij, will be discussed later. It is assumed that E02 is unknown constant matrix for t∈(tj-1,tj). After AF, dynamic model (10) will be
(15)x˙(t)=Ax(t)+Bf(t)u(t)+C0+Cvx(t)+Cax2(t),
or suppose there are p failed actuators; we have
(16)x˙(t)=Ax(t)+∑i=1,…,m‍bifi(t)ui(t)+C0+Cvx(t)+Cax2(t)=Ax(t)+∑i≠j1,…,jp‍biui(t)+∑j=j1,…,jp‍fj(t)bjuj(t)+C0+Cvx(t)+Cax2(t),
where bi is the column of B, f(t)∈Rn×n, and
(17)f(t)=diag⁡(f1(t)f2(t)f3(t)⋯fm(t)).
Therefore, after AF, HST system is multiplied by an unknown AF which impacts on stability of HST system. The control objective is to design a feedback control such that all signals in the closed-loop system are bounded, and x(t) asymptotically tracks a given reference xm(t) which is generated from the reference system:
(18)x˙m=Amxm(t)+Bmr(t),
where xm∈R2n, Am∈R2n×2n, and Bm∈R2n×1 are known constant matrices such that all the eigenvalues of Am are in the open left-half complex plane, and r(t)∈R is bounded and piecewise continuous.


## 2.1. Dynamic Model of a Single Carriage

The HST consists of a number of carriages, and couplers are used to couple adjacent carriages. A carriage during travelling is subjected to various kinds of forces, such as traction and braking forces, forces between carriages, and resistance forces. The force analysis of a single carriage in HST is illustrated in Figure1.Figure 1
The force analysis of a single carriage.Traction and Braking Forces.
u
i denotes traction force of the ith carriage, that is, ui>0, and braking force of the ith carriage, that is, ui<0. λi denotes whether the ith carriage is powered or nonpowered.Assumption 1.
The class of HST that we considered is distributed driving type [50], that is, λi=1 for all carriages.Interacting Forces between Carriages. Connection modules between two carriages in HST are nonlinear subsystems. Hence, it is difficult to obtain accurate mechanical model of interacting forces between two carriages. A widely used mechanical model for couplers consists of elastic forces and damping forces. In [51], nonlinear elastic force acting between carriages is described as follows:
(1)fk(t)=k(t)ϵi(t),
where relative displacement between two carriages is as
(2)ϵi(t)=xi(t)-xi-1(t),
where xi(t) is position of the ith carriage. k(t) denotes nonlinear relationship between displacements and forces and
(3)k(t)=k0(1+ϵi(t)σ).
If σ=0, coupler between carriages is linear, that is, k(t)=k0, which does not exist in practice. If σ<0 or σ>0, the coupler is soft coupling or hard coupling, respectively. Khalil and Grizzle [52] and Franklin and Powell [53] have proved that a system with a soft spring is more likely to have unsatisfying control performance. We take the minimum value k of k(t) in controller design, that is, k(t)=k. The damping force is described as
(4)fb(t)=b(t)(x˙i(t)-x˙i-1(t)),
where b(t) is damping coefficient and x˙i(t) is velocity of the ith carriage. We take the minimal value b of the coefficient b(t), that is, b(t)=b. The interacting force between carriages is as
(5)fic(t)=fk(t)+fb(t).Resistance Forces of HST.During travelling, carriages of HST are subjected to various kinds of resistance forces mainly including mechanical resistance and air resistance. The two forces are as follows [54]:
(6)fip(t)=mi(c0+cvx˙i(t)+cax˙i2(t)),
where mi represents mass of the ith carriage and mi(c0+cvx˙i(t)) and micax˙i2(t) denote mechanical resistance and air resistance, respectively. c0,cv, and ca are coefficients which may not be obtained accurately during train locomotion.Slope and curve resistance forces of HST depend on railway condition. Slope and curve resistance forces are normally considered by Garg and Dukkipati [55] as follows:
(7)fie(t)=migsinθi+0.002midaRt,
where da is axle length of a carriage and Rt is turning radius. Based on the above analysis, dynamic model of a single carriage in HST is given as follows:
(8)mix¨i(t)=λiui(t)+fi+1c(t)-fic(t)-fip(t)-fie(t)=λiui(t)-mi(c0+cvx˙i(t))-micax˙i2(t)-k(xi(t)-xi-1(t))-b(x˙i(t)-x˙i-1(t))-k(xi(t)-xi+1(t))-b(x˙i(t)-x˙i+1(t))-migsinθi-0.002midaRt.


## 2.2.  Dynamic Model of HST

Dynamic model of HST withn carriages can be derived as follows:
(9)m1x¨1(t)=λ1u1(t)-k(x1(t)-x2(t))-b(x˙1(t)-x˙2(t))-m1(c0+cvx˙1(t))-m1cax˙12(t)-m1gsinθ1-0.002m1daRt,m2x¨2(t)=λ2u2(t)-k(x2(t)-x1(t))-b(x˙2(t)-x˙1(t))-k(x2(t)-x3(t))-b(x˙2(t)-x˙3(t))-m2(c0+cvx˙2(t))-m2cax˙22(t)-m2gsinθ2-0.002m2daRt,⋮mix¨i(t)=λiui(t)-k(xi(t)-xi-1(t))-b(x˙i(t)-x˙i-1(t))-k(xi(t)-xi+1(t))-b(x˙i(t)-x˙i+1(t))-mi(c0+cvx˙i(t))-micax˙i2(t)-migsinθi-0.002midaRt,⋮mnx¨n(t)=λnun(t)-k(xn(t)-xn-1(t))-b(x˙n(t)-x˙n-1(t))-mn(c0+cvx˙n(t))-mncax˙n2(t)-mngsinθn-0.002mndaRt,
where xi subsystem represents dynamics of the ith carriage in HST. To simplify the controller design, (9) is rewritten as
(10)x˙(t)=Ax(t)+Bu(t)+C0+Cvx(t)+Cax2(t)=[0n×nIn×nA21A22]x(t)+[0n×nB2]u(t)+[0n×1E02+C02]+[0n×n0n×n0n×nCv2]x(t)+[0n×n0n×n0n×nCa2]x2(t),y=Cx(t),
where x=[x1⋯xnx˙1⋯x˙n]T is the state vector. A∈R2n×2n and B∈R2n×n are known, and C0∈R2n, Cv∈R2n×2n, and Ca∈R2n×2n are unknown matrixes, and we have
(11)A12=[-km1km10⋯⋯⋯0122km2-2km2km20⋯⋯00km3122-2km3km30⋯00⋯⋯⋯⋯⋯00⋯⋯⋯0kmnkmn],A22=[-bm1bm10⋯⋯⋯0122bm2-2bm2bm20⋯⋯00bm3122-2bm3bm30⋯00⋯⋯⋯⋯⋯00⋯⋯⋯0bmnbmn],B2=diag⁡(λ1m1λ2m2⋯λnmn),E02=diag⁡(gsinθ1+0.002daRt⋯⋯gsinθn+0.002daRt),C02=[-c0-c0⋯⋯-c0]T,Cv2=diag⁡(-cv-cv⋯⋯-cv),Ca2=diag⁡(-ca-ca⋯⋯-ca).


## 2.3. Fault Analysis of Traction Systems in HST

In HST, velocity cruise control is accomplished through the traction system which consists of high-voltage circuit (included by pantograph, current transformer, main circuit breaker, etc.), traction transformer, traction converter, asynchronous motor, wheels, and so on [12–14]. The principle of velocity cruise control is shown in Figure 2.Figure 2
The principle of velocity cruise control in HST.Traction converter is used to control traction motor to achieve velocity cruise control of HST. The probability of incorrect voltage and current, such as malfunction of electronic components in traction converter, is great [56, 57]. The output of ith actuator in failure status can be described as uf(t) and its elements is shown as follows:
(12)uif(t)=fi(t)ui(t)(i=1,…,n),
where ui(t) is control input without AF. fi(t)∈(fi_,f¯i) is unknown piecewise hopping function, with known boundary fi_,f¯i, which represents malfunction of electronic components in traction system and
(13)fi(t)={1t<t1fi1t1≤t<t2fi2t2≤t<t3⋮⋮
where fij is unknown constant for j=1,2,…,l and tj for j=1,2,…,l satisfies
(14)tj-tj-1≥tj,min⁡,
where t0=0 and tj,min⁡, which is the minimum switch time for fij, will be discussed later. It is assumed that E02 is unknown constant matrix for t∈(tj-1,tj). After AF, dynamic model (10) will be
(15)x˙(t)=Ax(t)+Bf(t)u(t)+C0+Cvx(t)+Cax2(t),
or suppose there are p failed actuators; we have
(16)x˙(t)=Ax(t)+∑i=1,…,m‍bifi(t)ui(t)+C0+Cvx(t)+Cax2(t)=Ax(t)+∑i≠j1,…,jp‍biui(t)+∑j=j1,…,jp‍fj(t)bjuj(t)+C0+Cvx(t)+Cax2(t),
where bi is the column of B, f(t)∈Rn×n, and
(17)f(t)=diag⁡(f1(t)f2(t)f3(t)⋯fm(t)).
Therefore, after AF, HST system is multiplied by an unknown AF which impacts on stability of HST system. The control objective is to design a feedback control such that all signals in the closed-loop system are bounded, and x(t) asymptotically tracks a given reference xm(t) which is generated from the reference system:
(18)x˙m=Amxm(t)+Bmr(t),
where xm∈R2n, Am∈R2n×2n, and Bm∈R2n×1 are known constant matrices such that all the eigenvalues of Am are in the open left-half complex plane, and r(t)∈R is bounded and piecewise continuous.


## 3. I&I Adaptive Fault-Tolerant Control Based on AMFDD

In this section, an adaptive controller combined with fault-tolerant module based on I&I AMFDD technology is developed. I&I AMFDD module is designed to detect AF and direct adaptive controller handles AF of HST. The system (10) can be rewritten as follows:
(19)x˙(t)=Ax(t)+Bu(t)+θΦ(x),
where θ=[C0CvCa], and Φ(x)=[1x(t)x2(t)]T is Lipschitz with G as
(20)|Φ(x1)-Φ(x2)|=|[1x1(t)x12(t)]T-[1x2(t)x22(t)]T|=|[0x1(t)-x2(t)x12(t)-x22(t)]T|=|x1(t)-x2(t)||[01x1(t)+x2(t)]T|≤|[012x¯]T||x1(t)-x2(t)|=G|x1(t)-x2(t)|,
where x¯ is the maximum value of position and velocity of HST. For the control problem considered in this paper, the following is assumed.Assumption 2.
The state of system (19) is available at every instant, and the maximum value of position and velocity of HST, that is, x¯, is known before AF (12).Assumption 3.
If system parameters and AF (up ton-1 failures) are known, the remaining actuators can still achieve a desired control objective.Assumption 4.
All pairs{A,Bf(t)} are uniformly completely controllable for fi(t)∈(fi_,f¯i), i=1,…,n.


### 3.1. Adaptive Model-Based Fault Detection and Diagnosis (AMFDD)

AMFDD module is combined with train control system in HST. Taking China Railways High-speed (CRH) trains [58–60] as an example, AMFDD module is designed to detect AF in Train Control and Management System. For t∈(tj-1,tj), consider the following AMFDD:
(21)x^˙(t)=Ax^(t)+Θ^Φ(x^(t))+Bu(t)-L(y(t)-y^(t)),
where x^, Θ^, and y^ are the estimate of x, θ, and y. I&I adaptive control is a noncertainty equivalence technique with
(22)Θ^=θ^+β(x),
where the estimation objective is to render the manifold
(23){(x,θ^),∣θ^+β(x)-θ=0}
invariant and (asymptotically) attractive, and the continuous function β(x), which is added to the estimated parameter vector θ^, is also a design parameter. Estimation of θ is achieved by driving the estimate error
(24)z=θ^-θ+β(x)
to zero. Define parameterized function:
(25)Qp(θ)=dβ(x)dxθΦ(x),
which satisfies P(x)-monotone, that is,
(26)(a-b)P(x)(Qp(a)-Qp(b))≥0,
where P(x) is a design function, which simplifies computational complexity of β(x), satisfying
(27)P(x)≥0,dP(x)dt≤0.Theorem 5.
AMFDD (21) and the adaptive law
(28)θ^˙=-dβ(x)dx(Ax(t)+(θ^+β(x(t)))Φ(x(t))+Bu(t))
can realise that (1)
the estimate error (24) converges to zero, that is, lim⁡t→∞z=0;(2)
the observational errorex satisfies
(29)ex(t)≤emax⁡(t),where
(30)ex(t)=x(t)-x^(t),emax⁡(t)=eλmax⁡(A¯)te0jjjjjjjjjjj+e-λmin⁡(A¯)tθ¯Φ(x¯)A¯-1(eλmax⁡(A¯)t-1),
where e0 is initial state vector of ex, θ¯ is the upper bound of z, and λmax⁡() and λmin⁡() denote the maximum and minimum value of matrix. (3)
Decision on the occurrence of AF in carriages is made if at least one term of the estimation errorex(t) exceeds its corresponding error bound emax⁡(t).Proof.
The derivative ofz is as follows:
(31)z˙=θ^˙+dβ(x)dx(Ax(t)+θΦ(x(t))+Bu(t)),
which clearly yields
(32)z˙=θ^˙+dβ(x)dx(Ax(t)+θΦ(x(t))+Bu(t))=-dβ(x)dx(θ^+β(x(t)))Φ(x(t))+dβ(x)dxθΦ(x(t))=Qp(θ)-Qp(θ^+β(x(t))).
Consider a positive-definite function V=(1/2)zTP(x)z, and using (26), (27), and (32), we have
(33)V˙=zTP(x)z˙+12zTP˙(x)z=zTP(x)(θ^˙+dβ(x)dx(Ax(t)+θΦ(x(t))ffffffggggggh+Bu(t))dβ(x)dx)+12zTdP(x)dtz≤(θ^-θ+β(x))TP(x)×(Qp(θ)-Qp(θ^+β(x)))≤0
from which convergence of z(t) follows; that is,
(34)lim⁡t→∞(θ^+β(x))=θ.
The derivative of ex is as follows:
(35)e˙x(t)=x˙(t)-x^˙(t)=Ax(t)+θΦ(x)+Bu(t)-Ax^(t)-(θ^+β(x^))Φ(x^)-Bu(t)-L(y(t)-y^(t))=(A-LC)ex(t)+θΦ(x)-(θ^+β(x^))Φ(x^)=(A-LC)ex(t)+θΦ(x)-(θ^+β(x^))Φ(x)+(θ^+β(x^))Φ(x)-(θ^+β(x^))Φ(x^)≤(A-LC)ex(t)-zΦ(x)+(θ^+β(x^))Gex(t)=(A-LC+(θ^+β(x^))G)ex(t)-zΦ(x)=A¯ex(t)-zΦ(x).
Consequently, when no AF occurs, we have
(36)ex(t)=exA¯te0+∫0t‍ex-A¯(t-τ)(-z(τ)Φ(x(τ)))dτ≤exλmax⁡(A¯)te0+ex-λmin⁡(A¯)t∫0t‍exA¯τ(-z(τ)Φ(x))dτ≤exλmax⁡(A¯)te0+ex-λmin⁡(A¯)tθ¯Φ(x¯)A¯-1∫0t‍exA¯τdA¯τ=exλmax⁡(A¯)te0+ex-λmin⁡(A¯)tθ¯Φ(x¯)A¯-1×(exλmax⁡(A¯)t-1)=emax⁡.
Hence, decision on the occurrence of AF in carriages is made if at least one term of the estimation error ex(t) exceeds its corresponding error bound emax⁡(t).


### 3.2.  Adaptive Control for Unknown AF

After detection of AF, the key task of AF compensation control is to design a direct adaptive control law such that all trajectories of the closed-loop system (16) are bounded and lim⁡t→∞x(t)=xm. Using the AMFDD (21), the AF system (16) is as follows:
(37)x˙(t)=Ax(t)+∑i=1nbifi(t)ui(t)+zΦ(x)=Ax(t)+∑i≠j1,…,jpbiui(t)+∑j=j1,…,jpfj(t)bjuj(t)+zΦ(x)=Ax(t)+B′(t)u(t)+zΦ(x),
where B′(t)=[b1′b2′b3′⋯bn′]∈R2n×n is unknown constant matrix with f(t), t∈(tj-1,tj), and z, which is exponential convergence, is defined in (24).Before addressing the control problem for unknown AF, we must first derive the existence of controllers for the system (37) with known AF to obtain some basic plant-model matching conditions which are useful for controller parameterization in adaptive designs for unknown AF. Without AF, matching equations of HST system (37) are satisfied:
(38)A+∑i=1n‍biK1i*T=Am,∑i=1n‍bik2i*=Bm,
and K1*=[K11*K12*⋯K1n*]∈R2n×n and k2*=[k21*k22*⋯k2n*]T∈Rn. We see Assumptions 3 implies that there exist constant vectors Ks1j* and nonzero constant ks2j* such that when only one AF occurs, the following matching equations are satisfied:
(39)A+∑i≠jnbiK1i*T+bj′Ks1j*T=Am,∑i≠jnbik2i*+bj′ks2j*=Bm.
Using (38) and (39), we obtain
(40)bj′=-k2j*ks2j*bj=-K1j*TKs1j*Tbj,K1j*Tk2j*=Ks1j*Tks2j*.Now we develop an adaptive control scheme for the system (37) with unknown AF. We propose the controller structure:
(41)u(t)=K1T(t)x(t)+k2(t)r(t)+k3(t),
where
(42)K1(t)=[K11(t)K12(t)⋯K1n(t)]∈R2n×n,k2(t)=[k21(t)k22(t)⋯k2n(t)]T∈Rn
are adaptive estimates of the unknown parameters K1* and k2*, and
(43)k3(t)=[k31(t)k32(t)⋯k3n(t)]T∈Rn
is a design vector. Define the parameter errors
(44)e(t)=x(t)-xm(t),K~1i(t)=K1i(t)-K1i*,k~2i(t)=k2i(t)-k2i*,
for i=1,…,n. Substituting the control law (41) into the system (37), we obtain
(45)x˙(t)=Ax(t)+B′(t)u(t)+zΦ(x)=Ax(t)+B′K1T(t)x(t)+B′k2(t)r(t)+zΦ(x)+B′k3(t).
Using (40) and (44), the system (45) is as follows:
(46)x˙(t)=(A+B′K1T)x(t)+B′k2r(t)+zΦ(x)+B′k3=(A+B′K1*T)x(t)+B′k2*r(t)+B′K~1T(t)x(t)+B′k~2(t)r(t)+zΦ(x)+B′k3(t)=Amx(t)+Bmr(t)+B′K~1T(t)x(t)+B′k~2(t)r(t)+zΦ(x)+B′k3(t)=Amx(t)+Bmr(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+zΦ(x)+∑i=1n‍-k2i*ks2i*bik3i(t).
Then, we have the tracking error equation:
(47)e˙(t)=x˙(t)-x˙m(t)=Ame(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+zΦ(x)+∑i=1n‍-k2i*ks2i*bik3i(t).
Consider a positive definite function
(48)V=eT(t)Pe(t)+∑i=1n‍1|ks2i*|K~1iT(t)Γ1i-1K~1i(t)+∑i=1n‍1|ks2i*|k~22i(t)γ2i-1.
According to [25], we assume that the sign of parameter ks2j* is known, and ks2j0 is a known upper bound on |ks2j*|. Choose the adaptive laws as
(49)K˙1i(t)=-sgn⁡[ks2i*]Γ1ixeT(t)PBm,k˙2i(t)=-sgn⁡[ks2i*]γ2ir(t)eT(t)PBm,
where P∈R2n×2n, P=PT>0 such that
(50)PAm+AmTP=-Q
for any constant Q∈R2n×2n such that Q=QT>0, Γ1i is constant matrix such that Γ1i=Γ1iT>0, and γ2i>0 is constant. Then, we have
(51)V˙≤-eT(t)Qe(t)+2eT(t)Pθ¯Φ(x)iiiii+2eT(t)P∑i=1n-k2i*ks2i*bik3i(t),
where θ¯ is defined in (29). We choose
(52)k3i(t)=sgn⁡[ks2i*]ki0|Φ(x)|2eT(t)PBm,
and we obtain
(53)V˙≤-eT(t)Qe(t)+2eT(t)Pθ¯Φ(x)-2eT(t)PB∑i=1n‍k2i*ks2i*k3i(t)=-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2-|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2+2|eT(t)Pθ¯Φ(x)|-2|eT(t)PBm|2∑i=1n‍1|ks2i*|ki0|Φ(x)|2=-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2-(|θ¯|∥P∥2|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|ddhgd-2|eT(t)PBm|2∑i=1n‍1|ks2i*|ki0|Φ(x)||θ¯|∥P∥2|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0)2≤-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2≤-eT(t)Qe(t)+ϕ0,
where
(54)ϕ0=|θ¯|22|Bm|2∑i=1n(1/|ks2i0|)ki0.
Hence, V˙≤0 whenever |e|>ϕ0/λmin⁡(Q), which means that the tracking error e(t) will converge to a set of a small size if ki0 is chosen to be large.Theorem 6.
For HST system (37) with Assumptions 1–4 and AF (12) satisfying (56), the controller is given by the control law (41) and the adaptation laws (49) and (52); then we have the following. (1)
HST system (37) is internally stable and all closed-loop signals are bounded.(2)
Ifki0 is chosen to be large, the tracking error e(t) will converge to a set of a small size.Remark 7.
The convergence timete is defined by
(55)te={te∣|e(t)|≤δ},
where δ is a design parameter, and we assume the minimum switch time tmin⁡ must satisfy
(56)tj,min⁡>te.


### 3.3. Adaptive Control for Unknown AF and CIS

In this subsection, we consider unknown AF (12) and CIS simultaneously for HST system (37). In many practical applications, CIS occurs after AF (12), for example, when signal u(t) generated by the adaptive fault-tolerant control law (41) cannot be implemented due to some physical constraints in HST traction system.Due to CIS, adaptive fault-tolerant control law, which is different from (41), is as follows:
(57)us(t)=sat⁡(u(t),umin⁡,umax⁡),
where umin⁡ and umax⁡ are the minimum and maximum values of u(t), and u(t) is defined in (41). Saturation function sat⁡(u(t),umin⁡,umax⁡) is linear with unity slope between its lower and upper bound; that is,
(58)sat⁡(u(t),umin⁡,umax⁡)={umin⁡,u(t)<umin⁡,u(t),umin⁡≤u(t)≤umax⁡,umax⁡,u(t)>umax⁡,
and we consider CIS error η=[η1η2⋯ηn]T∈Rn described by
(59)ηi(t)=uis(t)-ui(t),
and |ηi(t)|<η¯. Now we develop an adaptive control scheme for the system (37) with unknown AF (12) and CIS (57). We suppose there are q CIS in HST, that is, j=j1,…,jq, and propose the controller structure:
(60)ujs(t)={K1jT(t)x(t)+k2j(t)r(t)+k4j(t)+k3j(t)hhhhhhhhhhhhhhhhhhhhhj≠j1,…,jqK1jT(t)x(t)+k2j(t)r(t)+ηj(t)+k3j(t)hhhhhhhhhhhhhhhhhhhhhj=j1,…,jq,
where K1j(t), k2j(t), and k3j(t) are defined in (49) and (52). k4=[k41k42⋯k4n]T∈Rn. We obtain
(61)x˙(t)=Ax(t)+B′(t)us(t)+zΦ(x)=Amx(t)+Bmr(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+∑i=1n‍-k2i*ks2i*bik3i(t)+zΦ(x)+∑j≠j1,…,jq‍-k2j*ks2j*bjk4j(t)+∑j=j1,…,jq‍-k2j*ks2j*bjηj(t).
We have the tracking error equation:
(62)e˙(t)=x˙(t)-x˙m(t)=Ame(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+∑i=1n‍-k2i*ks2i*bik3i(t)+zΦ(x)+∑j≠j1,…,jq‍-k2j*ks2j*bjk4j(t)+∑j=j1,…,jq‍-k2j*ks2j*bjηj(t).
Consider a positive definite function:
(63)V=eT(t)Pe(t)+∑i=1n‍1|ks2i*|K~1iT(t)Γ1i-1K~1i(t)+∑i=1n‍1|ks2i*|k~22i(t)γ2i-1,
and using (49) and (52), we have
(64)V˙≤-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i0|)ki0|e(t)|2+2eT(t)PB(∑j≠j1,…,jq‍-k2j*ks2j*k4j(t)+∑j=j1,…,jq‍-k2j*ks2j*ηj(t))≤-eT(t)Qe(t)+ϕ0+2eT(t)PB(∑j≠j1,…,jq‍-k2j*ks2j*k4j(t)gggggggggggggggggggggggggggg+∑j=j1,…,jq‍-k2j*ks2j*ηj(t)),
and we assume
(65)ηp=max⁡j1,…,jq∈{1,…,n}|∑j=j1,…,jq‍-k2j*ks2j*ηj(t)|(1≤q≤n-1),η0=max⁡{η1,…,ηn-1},ks2j0>|ks2j*|.
We choose the design signal k4j(t) as
(66)k4j(t)=ks2j0η0sgn⁡[eT(t)PBm]sgn⁡[ks2j*].
Since the control signal k4j(t) in (66) is not continuous, according to [61, 62], to avoid system chatterings caused by such discontinuous control laws, the following common approximations will be used for sgn⁡(·) function:
(67)sgn⁡[x]≈x|x|+ɛ,ɛ>0.
We obtain
(68)V˙≤-eT(t)Qe(t)+ϕ0-2|eT(t)PBm|2|eT(t)PBm|+ɛii×∑j≠j1,…,jq‍ks2j0|ks2j*|η0+2|eT(t)PBm|ηp≤-eT(t)Qe(t)+ϕ0-2|eT(t)PBm|ii×(|eT(t)PBm||eT(t)PBm|+ɛ∑j≠j1,…,jq‍ks2j0|ks2j*|η0-ηp)≤-eT(t)Qe(t)+ϕ0+2η0|eT(t)PBm|ii×ɛ|eT(t)PBm|+ɛ∑j≠j1,…,jq‍ks2j0|ks2j*|≤-eT(t)Qe(t)+ϕ0+2β0ɛ,
where β0=∑j≠j1,…,jq(ks2j0/|ks2j*|)η0 is a constant. Hence V˙<0, whenever
(69)|e|>ϕ0+2β0ɛλmin⁡(Q),
which means |e| decreases to a lower bound proportional to ɛ and ki0.Theorem 8.
For HST system (37) with Assumptions 1–4 and AF (12) and CIS (57), the controller is given by the control law (60) and the adaptation laws (49), (52), and (66); then we have the following.(1)
HST system (37) is internally stable and all closed-loop signals are bounded.(2)
Ifɛ is chosen to be small and ki0 is chosen to be large, the tracking error e(t) will converge to a set of a small size.


## 3.1. Adaptive Model-Based Fault Detection and Diagnosis (AMFDD)

AMFDD module is combined with train control system in HST. Taking China Railways High-speed (CRH) trains [58–60] as an example, AMFDD module is designed to detect AF in Train Control and Management System. For t∈(tj-1,tj), consider the following AMFDD:
(21)x^˙(t)=Ax^(t)+Θ^Φ(x^(t))+Bu(t)-L(y(t)-y^(t)),
where x^, Θ^, and y^ are the estimate of x, θ, and y. I&I adaptive control is a noncertainty equivalence technique with
(22)Θ^=θ^+β(x),
where the estimation objective is to render the manifold
(23){(x,θ^),∣θ^+β(x)-θ=0}
invariant and (asymptotically) attractive, and the continuous function β(x), which is added to the estimated parameter vector θ^, is also a design parameter. Estimation of θ is achieved by driving the estimate error
(24)z=θ^-θ+β(x)
to zero. Define parameterized function:
(25)Qp(θ)=dβ(x)dxθΦ(x),
which satisfies P(x)-monotone, that is,
(26)(a-b)P(x)(Qp(a)-Qp(b))≥0,
where P(x) is a design function, which simplifies computational complexity of β(x), satisfying
(27)P(x)≥0,dP(x)dt≤0.Theorem 5.
AMFDD (21) and the adaptive law
(28)θ^˙=-dβ(x)dx(Ax(t)+(θ^+β(x(t)))Φ(x(t))+Bu(t))
can realise that (1)
the estimate error (24) converges to zero, that is, lim⁡t→∞z=0;(2)
the observational errorex satisfies
(29)ex(t)≤emax⁡(t),where
(30)ex(t)=x(t)-x^(t),emax⁡(t)=eλmax⁡(A¯)te0jjjjjjjjjjj+e-λmin⁡(A¯)tθ¯Φ(x¯)A¯-1(eλmax⁡(A¯)t-1),
where e0 is initial state vector of ex, θ¯ is the upper bound of z, and λmax⁡() and λmin⁡() denote the maximum and minimum value of matrix. (3)
Decision on the occurrence of AF in carriages is made if at least one term of the estimation errorex(t) exceeds its corresponding error bound emax⁡(t).Proof.
The derivative ofz is as follows:
(31)z˙=θ^˙+dβ(x)dx(Ax(t)+θΦ(x(t))+Bu(t)),
which clearly yields
(32)z˙=θ^˙+dβ(x)dx(Ax(t)+θΦ(x(t))+Bu(t))=-dβ(x)dx(θ^+β(x(t)))Φ(x(t))+dβ(x)dxθΦ(x(t))=Qp(θ)-Qp(θ^+β(x(t))).
Consider a positive-definite function V=(1/2)zTP(x)z, and using (26), (27), and (32), we have
(33)V˙=zTP(x)z˙+12zTP˙(x)z=zTP(x)(θ^˙+dβ(x)dx(Ax(t)+θΦ(x(t))ffffffggggggh+Bu(t))dβ(x)dx)+12zTdP(x)dtz≤(θ^-θ+β(x))TP(x)×(Qp(θ)-Qp(θ^+β(x)))≤0
from which convergence of z(t) follows; that is,
(34)lim⁡t→∞(θ^+β(x))=θ.
The derivative of ex is as follows:
(35)e˙x(t)=x˙(t)-x^˙(t)=Ax(t)+θΦ(x)+Bu(t)-Ax^(t)-(θ^+β(x^))Φ(x^)-Bu(t)-L(y(t)-y^(t))=(A-LC)ex(t)+θΦ(x)-(θ^+β(x^))Φ(x^)=(A-LC)ex(t)+θΦ(x)-(θ^+β(x^))Φ(x)+(θ^+β(x^))Φ(x)-(θ^+β(x^))Φ(x^)≤(A-LC)ex(t)-zΦ(x)+(θ^+β(x^))Gex(t)=(A-LC+(θ^+β(x^))G)ex(t)-zΦ(x)=A¯ex(t)-zΦ(x).
Consequently, when no AF occurs, we have
(36)ex(t)=exA¯te0+∫0t‍ex-A¯(t-τ)(-z(τ)Φ(x(τ)))dτ≤exλmax⁡(A¯)te0+ex-λmin⁡(A¯)t∫0t‍exA¯τ(-z(τ)Φ(x))dτ≤exλmax⁡(A¯)te0+ex-λmin⁡(A¯)tθ¯Φ(x¯)A¯-1∫0t‍exA¯τdA¯τ=exλmax⁡(A¯)te0+ex-λmin⁡(A¯)tθ¯Φ(x¯)A¯-1×(exλmax⁡(A¯)t-1)=emax⁡.
Hence, decision on the occurrence of AF in carriages is made if at least one term of the estimation error ex(t) exceeds its corresponding error bound emax⁡(t).


## 3.2.  Adaptive Control for Unknown AF

After detection of AF, the key task of AF compensation control is to design a direct adaptive control law such that all trajectories of the closed-loop system (16) are bounded and lim⁡t→∞x(t)=xm. Using the AMFDD (21), the AF system (16) is as follows:
(37)x˙(t)=Ax(t)+∑i=1nbifi(t)ui(t)+zΦ(x)=Ax(t)+∑i≠j1,…,jpbiui(t)+∑j=j1,…,jpfj(t)bjuj(t)+zΦ(x)=Ax(t)+B′(t)u(t)+zΦ(x),
where B′(t)=[b1′b2′b3′⋯bn′]∈R2n×n is unknown constant matrix with f(t), t∈(tj-1,tj), and z, which is exponential convergence, is defined in (24).Before addressing the control problem for unknown AF, we must first derive the existence of controllers for the system (37) with known AF to obtain some basic plant-model matching conditions which are useful for controller parameterization in adaptive designs for unknown AF. Without AF, matching equations of HST system (37) are satisfied:
(38)A+∑i=1n‍biK1i*T=Am,∑i=1n‍bik2i*=Bm,
and K1*=[K11*K12*⋯K1n*]∈R2n×n and k2*=[k21*k22*⋯k2n*]T∈Rn. We see Assumptions 3 implies that there exist constant vectors Ks1j* and nonzero constant ks2j* such that when only one AF occurs, the following matching equations are satisfied:
(39)A+∑i≠jnbiK1i*T+bj′Ks1j*T=Am,∑i≠jnbik2i*+bj′ks2j*=Bm.
Using (38) and (39), we obtain
(40)bj′=-k2j*ks2j*bj=-K1j*TKs1j*Tbj,K1j*Tk2j*=Ks1j*Tks2j*.Now we develop an adaptive control scheme for the system (37) with unknown AF. We propose the controller structure:
(41)u(t)=K1T(t)x(t)+k2(t)r(t)+k3(t),
where
(42)K1(t)=[K11(t)K12(t)⋯K1n(t)]∈R2n×n,k2(t)=[k21(t)k22(t)⋯k2n(t)]T∈Rn
are adaptive estimates of the unknown parameters K1* and k2*, and
(43)k3(t)=[k31(t)k32(t)⋯k3n(t)]T∈Rn
is a design vector. Define the parameter errors
(44)e(t)=x(t)-xm(t),K~1i(t)=K1i(t)-K1i*,k~2i(t)=k2i(t)-k2i*,
for i=1,…,n. Substituting the control law (41) into the system (37), we obtain
(45)x˙(t)=Ax(t)+B′(t)u(t)+zΦ(x)=Ax(t)+B′K1T(t)x(t)+B′k2(t)r(t)+zΦ(x)+B′k3(t).
Using (40) and (44), the system (45) is as follows:
(46)x˙(t)=(A+B′K1T)x(t)+B′k2r(t)+zΦ(x)+B′k3=(A+B′K1*T)x(t)+B′k2*r(t)+B′K~1T(t)x(t)+B′k~2(t)r(t)+zΦ(x)+B′k3(t)=Amx(t)+Bmr(t)+B′K~1T(t)x(t)+B′k~2(t)r(t)+zΦ(x)+B′k3(t)=Amx(t)+Bmr(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+zΦ(x)+∑i=1n‍-k2i*ks2i*bik3i(t).
Then, we have the tracking error equation:
(47)e˙(t)=x˙(t)-x˙m(t)=Ame(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+zΦ(x)+∑i=1n‍-k2i*ks2i*bik3i(t).
Consider a positive definite function
(48)V=eT(t)Pe(t)+∑i=1n‍1|ks2i*|K~1iT(t)Γ1i-1K~1i(t)+∑i=1n‍1|ks2i*|k~22i(t)γ2i-1.
According to [25], we assume that the sign of parameter ks2j* is known, and ks2j0 is a known upper bound on |ks2j*|. Choose the adaptive laws as
(49)K˙1i(t)=-sgn⁡[ks2i*]Γ1ixeT(t)PBm,k˙2i(t)=-sgn⁡[ks2i*]γ2ir(t)eT(t)PBm,
where P∈R2n×2n, P=PT>0 such that
(50)PAm+AmTP=-Q
for any constant Q∈R2n×2n such that Q=QT>0, Γ1i is constant matrix such that Γ1i=Γ1iT>0, and γ2i>0 is constant. Then, we have
(51)V˙≤-eT(t)Qe(t)+2eT(t)Pθ¯Φ(x)iiiii+2eT(t)P∑i=1n-k2i*ks2i*bik3i(t),
where θ¯ is defined in (29). We choose
(52)k3i(t)=sgn⁡[ks2i*]ki0|Φ(x)|2eT(t)PBm,
and we obtain
(53)V˙≤-eT(t)Qe(t)+2eT(t)Pθ¯Φ(x)-2eT(t)PB∑i=1n‍k2i*ks2i*k3i(t)=-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2-|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2+2|eT(t)Pθ¯Φ(x)|-2|eT(t)PBm|2∑i=1n‍1|ks2i*|ki0|Φ(x)|2=-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2-(|θ¯|∥P∥2|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|ddhgd-2|eT(t)PBm|2∑i=1n‍1|ks2i*|ki0|Φ(x)||θ¯|∥P∥2|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0)2≤-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i*|)ki0|e(t)|2≤-eT(t)Qe(t)+ϕ0,
where
(54)ϕ0=|θ¯|22|Bm|2∑i=1n(1/|ks2i0|)ki0.
Hence, V˙≤0 whenever |e|>ϕ0/λmin⁡(Q), which means that the tracking error e(t) will converge to a set of a small size if ki0 is chosen to be large.Theorem 6.
For HST system (37) with Assumptions 1–4 and AF (12) satisfying (56), the controller is given by the control law (41) and the adaptation laws (49) and (52); then we have the following. (1)
HST system (37) is internally stable and all closed-loop signals are bounded.(2)
Ifki0 is chosen to be large, the tracking error e(t) will converge to a set of a small size.Remark 7.
The convergence timete is defined by
(55)te={te∣|e(t)|≤δ},
where δ is a design parameter, and we assume the minimum switch time tmin⁡ must satisfy
(56)tj,min⁡>te.


## 3.3. Adaptive Control for Unknown AF and CIS

In this subsection, we consider unknown AF (12) and CIS simultaneously for HST system (37). In many practical applications, CIS occurs after AF (12), for example, when signal u(t) generated by the adaptive fault-tolerant control law (41) cannot be implemented due to some physical constraints in HST traction system.Due to CIS, adaptive fault-tolerant control law, which is different from (41), is as follows:
(57)us(t)=sat⁡(u(t),umin⁡,umax⁡),
where umin⁡ and umax⁡ are the minimum and maximum values of u(t), and u(t) is defined in (41). Saturation function sat⁡(u(t),umin⁡,umax⁡) is linear with unity slope between its lower and upper bound; that is,
(58)sat⁡(u(t),umin⁡,umax⁡)={umin⁡,u(t)<umin⁡,u(t),umin⁡≤u(t)≤umax⁡,umax⁡,u(t)>umax⁡,
and we consider CIS error η=[η1η2⋯ηn]T∈Rn described by
(59)ηi(t)=uis(t)-ui(t),
and |ηi(t)|<η¯. Now we develop an adaptive control scheme for the system (37) with unknown AF (12) and CIS (57). We suppose there are q CIS in HST, that is, j=j1,…,jq, and propose the controller structure:
(60)ujs(t)={K1jT(t)x(t)+k2j(t)r(t)+k4j(t)+k3j(t)hhhhhhhhhhhhhhhhhhhhhj≠j1,…,jqK1jT(t)x(t)+k2j(t)r(t)+ηj(t)+k3j(t)hhhhhhhhhhhhhhhhhhhhhj=j1,…,jq,
where K1j(t), k2j(t), and k3j(t) are defined in (49) and (52). k4=[k41k42⋯k4n]T∈Rn. We obtain
(61)x˙(t)=Ax(t)+B′(t)us(t)+zΦ(x)=Amx(t)+Bmr(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+∑i=1n‍-k2i*ks2i*bik3i(t)+zΦ(x)+∑j≠j1,…,jq‍-k2j*ks2j*bjk4j(t)+∑j=j1,…,jq‍-k2j*ks2j*bjηj(t).
We have the tracking error equation:
(62)e˙(t)=x˙(t)-x˙m(t)=Ame(t)+∑i=1n‍-k2i*ks2i*biK~1iT(t)x(t)+∑i=1n‍-k2i*ks2i*bik~2i(t)r(t)+∑i=1n‍-k2i*ks2i*bik3i(t)+zΦ(x)+∑j≠j1,…,jq‍-k2j*ks2j*bjk4j(t)+∑j=j1,…,jq‍-k2j*ks2j*bjηj(t).
Consider a positive definite function:
(63)V=eT(t)Pe(t)+∑i=1n‍1|ks2i*|K~1iT(t)Γ1i-1K~1i(t)+∑i=1n‍1|ks2i*|k~22i(t)γ2i-1,
and using (49) and (52), we have
(64)V˙≤-eT(t)Qe(t)+|θ¯|2∥P∥22|eT(t)PBm|2∑i=1n(1/|ks2i0|)ki0|e(t)|2+2eT(t)PB(∑j≠j1,…,jq‍-k2j*ks2j*k4j(t)+∑j=j1,…,jq‍-k2j*ks2j*ηj(t))≤-eT(t)Qe(t)+ϕ0+2eT(t)PB(∑j≠j1,…,jq‍-k2j*ks2j*k4j(t)gggggggggggggggggggggggggggg+∑j=j1,…,jq‍-k2j*ks2j*ηj(t)),
and we assume
(65)ηp=max⁡j1,…,jq∈{1,…,n}|∑j=j1,…,jq‍-k2j*ks2j*ηj(t)|(1≤q≤n-1),η0=max⁡{η1,…,ηn-1},ks2j0>|ks2j*|.
We choose the design signal k4j(t) as
(66)k4j(t)=ks2j0η0sgn⁡[eT(t)PBm]sgn⁡[ks2j*].
Since the control signal k4j(t) in (66) is not continuous, according to [61, 62], to avoid system chatterings caused by such discontinuous control laws, the following common approximations will be used for sgn⁡(·) function:
(67)sgn⁡[x]≈x|x|+ɛ,ɛ>0.
We obtain
(68)V˙≤-eT(t)Qe(t)+ϕ0-2|eT(t)PBm|2|eT(t)PBm|+ɛii×∑j≠j1,…,jq‍ks2j0|ks2j*|η0+2|eT(t)PBm|ηp≤-eT(t)Qe(t)+ϕ0-2|eT(t)PBm|ii×(|eT(t)PBm||eT(t)PBm|+ɛ∑j≠j1,…,jq‍ks2j0|ks2j*|η0-ηp)≤-eT(t)Qe(t)+ϕ0+2η0|eT(t)PBm|ii×ɛ|eT(t)PBm|+ɛ∑j≠j1,…,jq‍ks2j0|ks2j*|≤-eT(t)Qe(t)+ϕ0+2β0ɛ,
where β0=∑j≠j1,…,jq(ks2j0/|ks2j*|)η0 is a constant. Hence V˙<0, whenever
(69)|e|>ϕ0+2β0ɛλmin⁡(Q),
which means |e| decreases to a lower bound proportional to ɛ and ki0.Theorem 8.
For HST system (37) with Assumptions 1–4 and AF (12) and CIS (57), the controller is given by the control law (60) and the adaptation laws (49), (52), and (66); then we have the following.(1)
HST system (37) is internally stable and all closed-loop signals are bounded.(2)
Ifɛ is chosen to be small and ki0 is chosen to be large, the tracking error e(t) will converge to a set of a small size.


## 4. Simulations

To prove the effectiveness of proposed controller, a train model of CRH2 [59], which consists of two powered carriages, is used. The parameters in this simulation are given in Table 1.Table 1
Parameters of high-speed trains.
Symbol
Implication
Value
n
Number of carriages
2
m
i
(
i
=
1,2
)
Mass of powered carriages
8500 kg
k
Elasticity coefficient
2
×
1
0
7 N/m
b
Damping coefficient
5
×
1
0
6 N·s/mThe unknown parameters, which should not be ignored, are shown as follows:(70)c0=5.2N/kg,cv=0.038N·s/(m·kg),ca=0.00112N2·s/(m2·kg).
We consider the AF in the first carriage is as follows:
(71)f1(t)u1(t)={1,0≤t<30,0.6,30≤t≤50
and CIS in the second carriage is as follows:
(72)u2(t)∈(-40kN,40kN).In order to exhibit the advantage of the proposed adaptive fault-tolerant control, a fault-tolerant control using neural networks [63] (FTCNN) is introduced. Using FTCNN, performance of the system (37) under AF (71) and CIS (72) is shown in Figures 3 and 4. The maximum position and velocity tracking errors after AF attain to 2 m and 2.5 m/s, respectively.The position tracking error of the carriages.
(a)
(b)The velocity tracking error of the carriages.
(a)
(b)The AMFDD (21) and adaptive fault-tolerant control law (60) with adaptation laws (49), (52), and (66) are applied to handle unknown AF (71) and CIS (72) in HST system. Figures 5 and 6 show the system responses which indicate that even when there are unknown AF, the tracking errors still converge to zero. It is shown that the developed adaptive fault-tolerant control law (60) with adaptation laws (49), (52), and (66) ensures that, in addition to closed-loop signal boundedness, the position and velocity tracking errors converge to zero as the time goes by, despite of the plant uncertainties, AF, and CIS.The position tracking error of the carriages.
(a)
(b)The velocity tracking error of the carriages.
(a)
(b)


## 5. Conclusion

In this paper, an adaptive state feedback and tracking control with AMFDD module is proposed to deal with a class of HST in the presence of unknown CIS and AF. That is, some of the plant inputs are influenced by hopping function. An AMFDD module using I&I observer is introduced based on the model of HST to detect AF, and a corresponding adaptive control law based on the AMFDD information is switched on to realize the fault-tolerant control. The proposed adaptive FTC guarantees the boundedness of all signals in the HST system.



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*Source: 481315-2014-07-08.xml*