Distributed Fault Accommodation for a Class of Interconnected Nonlinear Systems with Event-Triggered Inter-Communications

In this paper, the distributed fault accommodation problem is studied for a class of interconnected nonlinear systems with event-triggered control and inter-subsystem communications. Each subsystem is subject to potential faults resulting from the local dynamics and/or transmitted from neighboring subsystems. The time periods before and after the fault detection in each subsystem are considered and corresponding event-triggered controllers are proposed, where the neural network based adaptive approximation technique is used to estimate the fault effect online. The closed-loop stability of the interconnected system with the proposed event-triggered fault accommodation controllers is rigorously analyzed.


I. INTRODUCTION
To enhance the reliability and safety of interconnected systems, various fault-tolerant control and fault accommodation methods have been reported under different fault and system scenarios. To mention a few, the cyclic-small-gain methodology is used in [1] to form a decentralized fault accommodation scheme, the backstepping method with adaptive approximation is employed to deal with the delayed process fault [2], the dead zone phenomenon and bias actuator fault are considered in [3], a neural network-based decentralized fault accommodation is proposed in [4,5], fuzzy approximationbased actuator fault-tolerant is proposed in [6], and partial and coordinated communication based fault accommodation schemes are proposed in [7,8], respectively. Note that the above mentioned results are all based on continuous intersubsystem measurement and control implementation.
For the distributed control scheme of the interconnected system, inter-subsystem communication is essential. As each subsystem may not be collocated, the inter-subsystem communication burden may increase significantly with increasing the scale of interconnected systems. Moreover, the actuator has to respond to the control signal updating all the time. To reduce communication burden and extend the lifetime of the actuators, there is a need to balance the control performance, communication resources, and signal quality. To address this problem, the event-triggering approach provides a promising alternative since signals are aperiodically sampled and transmitted when needed, instead of all the time. The basic stability and triggering scheme dynamics (nonzero inter-event time or exclusion of Zeno behavior) of event-triggered control have been studied, e.g., see [9][10][11][12].
In this study, we focus on balancing the communication burden, control performance, and, most critically, fault tolerance; thus, an event-triggered fault accommodation scheme for a class of nonlinear uncertain systems is first proposed. In contrast to [11,13], both the inter-subsystem communication and the controller to actuator communication are eventtriggered. Each subsystem is subject to modeling uncertainty, interconnections with neighboring subsystems, and potential process faults arising from the local dynamics or transmitted from a neighboring subsystem. A full lifecycle of fault accommodation is designed, where the time periods before the occurrence of a fault, after the detection of a fault, and within the occurrence time and detection time of a fault (fault detection delay) are considered. An event-triggered nominal controller is proposed for the time period before the detection of a fault, and the event-triggered fault accommodation control scheme is proposed for the time period after the detection of a fault, where the neural network based adaptive approximation technique is employed to approximate the unknown fault effect with the event-triggered inter-subsystem measurement. The fault accommodation system stability during the overall lifecycle is rigorously analyzed. The main contribution of this work is the fault accommodation strategy design and analysis for the interconnected system with event-triggered inter-subsystem and control signal communications.
The rest of the paper is organized as follows. The system description, triggering scheme, and problem to be solved are presented in Section II. The event-triggered nominal controller design and analysis are given in Section III, and the details of fault accommodation scheme are given in Section IV. Concluding remarks are given in the last section.

II. PROBLEM FORMULATION
We consider a class of nonlinear systems, comprised of m interconnected subsystems. The i-th subsystem, i ∈ M = {1, ..., m}, is described by: where x i = col{x i1 , ...x ini } ∈ R ni and u i ∈ R are the state and input of the i-th subsystem, respectively. Let M i be the index set containing all the neighboring subsystems affecting the i-th subsystem, thenM i = M i ∪ {i} is the index containing all the subsystems affecting the i-th subsystem. Based on the index sets,x i = col{..., x j , ...} ∈ Rn i is the state of the neighbors of the i-th subsystem, where j ∈ M i ; x a i = col{..., x l , ...} ∈ Rn a i is the augmented neighboring state of the i-th subsystem, where l ∈M i . The functions f i : R ni → R and g i : R ni → R are continuously differentiable functions representing the nominal dynamics of the i-th subsystem (g i = 0), while η i : R ni × R + → R is a continuous function representing the unknown and unstructured modeling uncertainty of the i-th subsystem; γ i : Rn i → R is a continuous function characterizing the interconnections between the i-th subsystem and its neighbors; h i (x a i ) : Rn a i → R represents the fault effect in the i-th subsystem resulting from its local dynamics and/or neighboring dynamics due to the interconnections; and scalar function β i (t − T i 0 ) characterizes the time profile function for the potential system fault in the i-th subsystem and T i 0 is the fault occurrence time in the ith subsystem. In this study, we consider abrupt faults, where In this paper we consider the fault accommodation design problem, which is initiated after a fault is detected. Let the fault detection time be T i d for the i-th subsystem. The following assumptions will be used for control design.
Assumption 1 means that the magnitude of the possible fault in each subsystem remains bounded before the fault is detected. This is a reasonable assumption for control design as in the literature [14]. Assumption 2 presents a general requirement about the structure of the interconnections. To focus on the fault accommodation controller design and save computation resources, the global Lipschitz condition is assumed; however, the piecewise Lipschitz condition also works for the proposed controllers as can be seen in the following. Assumption 3 characterizes the upper bound of the local modeling uncertainty for the i-th subsystem.
Considering the distributed location of each subsystem, the event-triggered scheme for the inter-subsystem information transmission is adopted to save the communication resources. Thus, for the local controller, its neighbors' state information is not continuously accessible, but determined by an eventtriggered scheme. To ensure the control performance, the local measurement is available to the local controller. At the same time, to reduce the updating of the actuator and extend the lifetime of actuators, an event-triggered control signal transmission/updating scheme is considered. To characterize the event-triggered signal transmission, two monotonically increasing sequences for the i-th subsystem are introduced: where τ i s,k is the time sequence for the state information transmission and τ i c,k is the time sequence for the control signal transmission. S i s and S i c are Boolean variables denoting the corresponding triggering conditions: is the received state by the controller of the connected subsystems with the i-th subsystems,û i is the received control signal by the i-th subsystem (actuator output or system input of the i-th subsystem), and u i since here is the controller output as the event-triggered control implementation is used. Nonnegative scalars w i s , z i s , w i c , and z i c are the eventtriggering parameters and w i c ∈ [0, 1). Based on (2) and (3), for the i-th subsystem with j ∈ M i , we obtain: The definitions in (3) and (5) imply two triggering scenarios of control signal transmission in the local controller. The first one is triggered by the neighboring system state information updating in the local controller, and the second one is triggered by the local controller output evolution. For the considered distributed control scheme, the neighboring system state and the local control signal are the input and output of the local controller, respectively. Hence, both the input and output changing of the local controller can trigger events for local control signal transmission. This triggering scheme design is motivated by the distributed control scheme and the dynamics of the controller.
In this study, an event-triggered fault accommodation strategy will be proposed to cover three time periods for each subsystem: fault-free period (t < T i 0 ), fault detection delay period (T i 0 ≤ t < T i d ), and post-fault detection period (t ≥ T i d ). For the event-triggered fault accommodation strategy, a nominal controller will be designed for each subsystem for the period before the detection of any fault. Based on the adaptive approximation approach, an event-triggered fault accommodation law will be initialized to reconfigure the nominal controller after a fault is detected in the local subsystem. The stability under event-triggering properties of the overall closed-loop system will be presented for each of the three time periods of control.

III. EVENT-TRIGGERED NOMINAL CONTROLLER
In this section, an event-triggered control design is proposed for each subsystem in the time period before any fault is detected (t < T i 0 and T i 0 ≤ t < T i d ). The closed-loop stability analysis will be established as well.

A. Event-Triggered Nominal Controller Design
For the i-th subsystems, where i ∈ M , the original system description is rewritten as: where Note thatû i is utilized in (6), due to the event-triggered control implementation.
For the event-triggered controller design, the event-induced discrepancies in the controller input and system input for each subsystem will be analyzed. Based on (2), we have Based on (7), the overall event-induced discrepancy in the input of the i-th local controller is characterized as Regarding the control input, based on the definition of S i c in (3), we obtain that where the "or" logic operation in S i c ensures the existence of the upper bound for |u i −û i |. Based on (9), we havê where δ i 1 , δ i 2 ∈ [−1, 1]. Here, the unknown parameters δ i 1 and δ i 2 are introduced to denote the discrepancy in system input, compared to the corresponding controller output. Resulting from the relative and constant triggering portion design in S i c , both multiplicative and additive discrepancy portions can be found for system inputû i . Thus, taking (8) and (10) into consideration, a nominal controller for the i-th subsystem is designed as follows: where e i = B T P i x i , ω i,l ∈ R + (l = 1, 2, 3, 4) are the controller parameters, and K i is the stabilization gain such that A−BK i is Hurwitz. For a given Q i > 0, positive definite

B. Event-Triggered Nominal Controller Analysis
In this section, the stability of the considered interconnected nonlinear system will be analyzed.
Consider the Lyapunov function V i = 1 2 x T i P i x i for the i-th subsystem. First, we will focus on the time period t < T i 0 for the i-th subsystem. Based on the actuator output and the system dynamics in (6), the time derivative of V i is deduced asV The following lemma will be used in the sequel. Lemma 1 [15]. For any a ∈ R + and b ∈ R, it holds that where c satisfies that c = e −(c+1) (c ≈ 0.2785).
Based on the designed event-triggered nominal controller in (11) and Assumption 2, the following inequalities can be obtained: Next, we replaceû i in (14) by (10). Since the parameters δ i 1 and δ i 2 are unknown in (10), we will quantify their effect first. Now, let's check the term e i g i (x i )(1 + δ i 1 w i c )u i . Recalling the property that "tanh" is an increasing function with tanh(0) = 0, we have that e i tanh(e i ) ≥ 0. Since w i c ∈ [0, 1), based on (11), we obtain Thus, we havė Based on Lemma 1, the controller given in (11), the eventinduced discrepancy in (7), and Assumption 3, the following result can be obtained from (16): where where δ p,i = σmin(Qi) σmax(Pi) > 0. Based on (18), we know that the overall interconnected system is uniformly ultimately bounded for the time period when all the subsystems are fault-free and controlled by the nominal controllers given in (11), since cω i is bounded. Now, considering the fault detection delay case for the i-th Based on (17), the time derivative of V i satisfies thaṫ Referring to Assumption 1, we obtain where where δ χ,i = σmin(Qχ,i) σmax(Pi) and cω χ,i = cω i + 1 2 χ −1 i ε 2 f,i B T P 2 B. Since the fault occurrence time and fault detection time among subsystems are different, the case considered here is to verify the stability of the overall interconnected system, where some subsystems are subject to fault detection delay and controlled by the corresponding event-triggered nominal controllers as in (11). As some subsystems are faulty, based on (18), we havė Comparing (22) and (21) with (18) and (17), respectively, one knows that the bounded stability conclusion for the interconnected system during the fault-free time period holds before any fault is detected, but may correspond to a larger state upper bound. We summarize the above discussion of closed-loop stability for the interconnected nonlinear system by the following Theorem.
Theorem 1. Let Assumptions 1-3 hold. The event-triggered nominal controllers in (11) with the triggering scheme (2)- (5) guarantees that the state of the interconnected nonlinear system (1) is uniformly ultimately bounded, prior to the detection of a fault in any subsystems.

IV. EVENT-TRIGGERED FAULT ACCOMMODATION CONTROLLER
In this section, the event-triggered fault accommodation controller design and analysis will be presented. Adaptive approximation approach with the event-triggered measurement will be used to deal with the fault effect for any the i-th

A. Event-Triggered Fault Accommodation Controller Design
For the i-th subsystem, to get sufficient information of the occurred fault after t ≥ T i d , we employ the following neural network-based approximation model: whereĥ i is the approximation of h i , θ i ∈ R n θ,i is the neural network weight vector, and π i is the preselected Radial Basis Function vector.
To measure the performance of the adaptive approximation based on (23), an optimal weight vector θ * i is introduced: where X a i ⊂ Rn a i . To facilitate the fault accommodation controller design, based on the optimal approximation weight vector, the approximation error or parameter uncertainty is further characterized by the following assumption.
Regarding the unknown bounding parameter ρ i and the optimal weight vector θ * i , their estimationρ i andθ i are introduced for a feasible fault accommodation controller design, respectively. Due to the online estimation of these parameters, the fault accommodation controller will be dynamic, which is different from the nominal controller. and fault accommodation controller may be used simultaneously for the interconnected system. Comparing (33) with (21) and (17), it can be known that the controller switch will not destabilize the corresponding subsystem and also the overall interconnected system. The difference is that the residual cω χ,i may be unbounded without the fault accommodation after the detection of any fault, and its effect can be decreased by using the fault accommodation scheme.
The following theorem provides a summary for the eventtriggered fault accommodation controller.
Theorem 2. Let Assumptions 2-4 hold. The event-triggered nominal controllers in (25) with (11) under the triggering scheme (2)-(5) guarantees that the state of the nonlinear interconnected system (1) is uniformly ultimately bounded, after the detection of a fault in any subsystems.

V. CONCLUSIONS
The event-triggered fault accommodation for interconnected nonlinear systems has been studied. The inter-subsystem and controller to actuator communications are determined by the predefined events. The target control schemes for fault-free, fault detection delay, and post-fault detection time periods have been proposed, and uniformly ultimately bounded control performance for the overall interconnected system can be guaranteed.