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

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 amonotonemapping is proposed in I&Imethod, 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.


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/deadzone.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][13][14], such as overvoltage in traction transformer, overcurrent in traction 2 Mathematical Problems in Engineering 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][16][17][18][19][20][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 faulttolerant 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 strictfeedback 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  ∞ 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][48][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, ()-monotone.The stability of HST systems is proved theoretically.
The rest of this paper is organized as follows.Section 2 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.

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

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 Figure 1.
Traction and Braking Forces.  denotes traction force of the th carriage, that is,   > 0, and braking force of the th carriage, that is,   < 0.   denotes whether the th carriage is powered or nonpowered.
Assumption 1.The class of HST that we considered is distributed driving type [50], that is,   = 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 The force analysis of a single carriage.mechanical model for couplers consists of elastic forces and damping forces.In [51], nonlinear elastic force acting between carriages is described as follows: where relative displacement between two carriages is as where   () is position of the th carriage.() denotes nonlinear relationship between displacements and forces and If  = 0, coupler between carriages is linear, that is, () =  0 , which does not exist in practice.If  < 0 or  > 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  of () in controller design, that is, () = .The damping force is described as where () is damping coefficient and ẋ  () is velocity of the th carriage.We take the minimal value  of the coefficient (), that is, () = .The interacting force between carriages is as 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]: where   represents mass of the th carriage and   ( 0 +  V ẋ  ()) and     ẋ 2  () denote mechanical resistance and air resistance, respectively. 0 ,  V , and   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: where   is axle length of a carriage and   is turning radius.Based on the above analysis, dynamic model of a single carriage in HST is given as follows:

Dynamic Model of HST. Dynamic model of HST with 𝑛
carriages can be derived as follows: where   subsystem represents dynamics of the th carriage in HST.To simplify the controller design, ( 9) is rewritten as where T is the state vector. ∈  2×2 and  ∈  2× are known, and  0 ∈  2 ,  V ∈  2×2 , and   ∈  2×2 are unknown matrixes, and we have

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][13][14].The principle of velocity cruise control is shown in Figure 2.
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 th actuator in failure status can be described as   () and its elements is shown as follows: where   () is control input without AF.  () ∈ (  ,   ) is unknown piecewise hopping function, with known boundary   ,   , which represents malfunction of electronic components in traction system and where   is unknown constant for  = 1, 2, . . .,  and   for  = 1, 2, . . .,  satisfies where  0 = 0 and  ,min , which is the minimum switch time for   , will be discussed later.It is assumed that  02 is unknown constant matrix for  ∈ ( −1 ,   ).After AF, dynamic model (10) will be or suppose there are  failed actuators; we have where   is the column of , () ∈  × , and 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 () asymptotically tracks a given reference   () which is generated from the reference system: where   ∈  2 ,   ∈  2×2 , and   ∈  2×1 are known constant matrices such that all the eigenvalues of   are in the open left-half complex plane, and () ∈  is bounded and piecewise continuous.

I&I Adaptive Fault-Tolerant Control Based on AMFDD
In this section, an adaptive controller combined with faulttolerant 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: where Lipschitz with  as where  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, , is known before AF (12).
Assumption 3. If system parameters and AF (up to  − 1 failures) are known, the remaining actuators can still achieve a desired control objective.

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][59][60] as an example, AMFDD module is designed to detect AF in Train Control and Management System.For  ∈ ( −1 ,   ), consider the following AMFDD: where x, Θ, and ŷ are the estimate of , , and .I&I adaptive control is a noncertainty equivalence technique with where the estimation objective is to render the manifold invariant and (asymptotically) attractive, and the continuous function (), which is added to the estimated parameter vector θ, is also a design parameter.Estimation of  is achieved by driving the estimate error to zero.Define parameterized function: which satisfies () − , that is, where () is a design function, which simplifies computational complexity of (), satisfying can realise that (1) the estimate error (24) converges to zero, that is, lim  → ∞  = 0; (2) the observational error   satisfies where where  0 is initial state vector of   ,  is the upper bound of , and  max () and  min () 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 error   () exceeds its corresponding error bound  max ().
Proof.The derivative of  is as follows: Consider a positive-definite function  = (1/2)  (), and using ( 26), (27), and (32), we have from which convergence of () follows; that is, lim The derivative of   is as follows: Hence, decision on the occurrence of AF in carriages is made if at least one term of the estimation error   () exceeds its corresponding error bound  max ().

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 closedloop system ( 16) are bounded and lim  → ∞ () =   .Using the AMFDD ( 21), the AF system ( 16) is as follows: where is unknown constant matrix with (),  ∈ ( −1 ,   ), and , 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: Using ( 38) and ( 39), we obtain Now we develop an adaptive control scheme for the system (37) with unknown AF.We propose the controller structure: where 42) are adaptive estimates of the unknown parameters  * 1 and  * 2 , and is a design vector.Define the parameter errors for  = 1, . . ., .Substituting the control law (41) into the system (37), we obtain Using ( 40) and (44), the system ( 45) is as follows: Then, we have the tracking error equation: Consider a positive definite function According to [25], we assume that the sign of parameter  * 2 is known, and  0 2 is a known upper bound on | * 2 |.Choose the adaptive laws as where  ∈  2×2 ,  =   > 0 such that for any constant  ∈  2×2 such that  =   > 0, Γ 1 is constant matrix such that Γ 1 = Γ  1 > 0, and  2 > 0 is constant.Then, we have where  is defined in (29).We choose and we obtain where Hence, V ≤ 0 whenever || > √ 0 / min (), which means that the tracking error () will converge to a set of a small size if  0  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) If  0  is chosen to be large, the tracking error () will converge to a set of a small size.Remark 7. The convergence time   is defined by where  is a design parameter, and we assume the minimum switch time  min must satisfy

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 () 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: where  min and  max are the minimum and maximum values of (), and () is defined in (41) and we consider CIS error  = [ 1  2 ⋅ ⋅ ⋅   ]  ∈   described by and |  ()| < .Now we develop an adaptive control scheme for the system (37) with unknown AF (12) and CIS (57).We suppose there are  CIS in HST, that is,  =  1 , . . .,   , and propose the controller structure: We have the tracking error equation: Consider a positive definite function: and using ( 49) and ( 52), we have We choose the design signal  4 () as Since the control signal  4 () 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: We obtain where which means || decreases to a lower bound proportional to √ and  0  .
(1) HST system (37) is internally stable and all closed-loop signals are bounded.
(2) If √ is chosen to be small and  0  is chosen to be large, the tracking error () will converge to a set of a small size.

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.
The unknown parameters, which should not be ignored, are shown as follows: 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 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.

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.

Figure 2 :
Figure 2: The principle of velocity cruise control in HST.

Figure 3 :
Figure 3: The position tracking error of the carriages.

Figure 4 :
Figure 4: The velocity tracking error of the carriages.

Figure 5 :
Figure 5: The position tracking error of the carriages.

Figure 6 :
Figure 6: The velocity tracking error of the carriages.

Table 1 :
Parameters of high-speed trains.