Valoración subjetiva de la conformación

Finally, a simulation example is used to demonstrate the proposed prognostics scheme.

I. Introduction

Quantitative methodology-based fault detection (FD) schemes have become popular due to the low implementation cost when compared to other techniques [1]. In the quantitative method, a model representative of the system is used in conjunction with the actual system output for residual generation and fault detection. The system model could be derived from either first principles or borrowed from control scientists/engineers. Normally, a predefined threshold on the residual is utilized to declare the presence of a fault and initiate diagnosis. Although, the selection of the fault detection threshold is important to improve detection while minimizing false alarms, a rigorous analytical procedure is now available to identify the fault detection threshold [1-3].

Many available model-based FD and diagnosis methods [4-13] use some sort of residual signal. Such methods for linear systems use structured and fixed directional residuals [1], parity relations [2], geometric approach [3], and eigenstructure assignment [2] etc. However, the prognostics component is not addressed so far.

Recently, the FD and diagnosis schemes are extended to nonlinear continuous- time systems [4-13]. In particular, in [7, 11, 12], a nonlinear sliding mode observer-based FD design is proposed whereas in [10] a nonlinear diagonal observer method is introduced. On the other hand, in [13], geometric relationship is employed. Moreover, a good survey of fault detection and isolation (FDI) schemes for hydraulic systems, flight controls etc., are given in [14]. On the other hand, a recent survey [15] on model-based

FD techniques presents an excellent overview of the state-of-the art developments. A common issue that has been gaining interest in the literature is stability analysis using Lyapunov theory for the design of FD schemes [7-9]. However, the FD schemes [7-9] render only uniform ultimate boundness (UUB) of the closed-loop signals due to the presence of system uncertainties. By contrast, in the recent work [16], asymptotic convergence of the identification error in continuous-time is demonstrated for robot manipulators with actuator faults. However, the time to failure (TTF) determination is not discussed for prognostics although a TTF scheme is essential for next generation complex dynamic systems.

By contrast, certain TTF schemes using data-driven framework [17-19], assumed a specific degradation model which has been found to be limited to the system or material type under consideration. Another scheme [20] employs a deterministic polynomial and a probabilistic method for prognosis by assuming that certain parameters are affected by the fault while others [21] use a black box approach using neural network (NN) on the failure data. All these schemes [17-21] while being data-driven address only TTF prediction, require offline training, and do not offer performance guarantees. Also, no analytical results are included. Therefore, it is envisioned that a combined FI and TTF determination scheme or else referred to as prognostics would not only provide the remaining useful life but also identify the fault occurred. Besides, analytical performance guarantees of the FI and TTF schemes are normally required.

It is reported in [22-23] that a direct conversion of continuous-time FD schemes [4-13] to discrete-time requires high sampling rate whereas when implemented using low sampled embedded hardware results in stability problems. Therefore, FD of discrete-time

systems is explicitly addressed in [22-24] while ensuring that the detection residual is guaranteed to be bounded. However, prognostics component is not studied. Additionally, to the best knowledge of the authors, there are no previously reported discrete-time schemes that can detect, isolate, and estimate TTF for systems with simultaneous and multiple faults. Hence, in this paper, prognostics framework, in discrete-time with guarantees of asymptotic convergence of the FI residual is introduced for a class of nonlinear discrete-systems with simultaneous and multiple faults.

Simultaneous and multiple faults imply that for an n-dimensional system, the fault could occur in more than one state at the same time (multiple faults) and also more than one fault can occur on the same state (multiple fault types). Therefore, in this paper, first, the FD estimator from [24] is revisited for the purpose of fault detection. Subsequently, the online approximator in discrete-time (OLAD) and the robust adaptive term in the FD estimator are initiated to learn the unknown fault dynamics. Upon detection, the fault is identified by using a novel FI estimator. Each state of the FI estimator corresponds to a particular type of fault combination. As a consequence, simultaneous and multiple faults occurring on the states are identified if the corresponding FI residual converges to zero asymptotically. Unlike other schemes [8, 9, 22, 23], asymptotic convergence is guaranteed even in the presence of system uncertainties due to the robust adaptive term in the FI estimator.

Subsequently, after isolating the fault, its magnitude is estimated online using a parameter update law, which is used for determining TTF. A mathematical equation is derived to estimate TTF at each time instant by projecting the current value of the FI

parameters to their corresponding limits. The limits provided by the designer indicate that the system is unsafe to operate beyond these limits. Moreover, for most practical systems, the parameters could be tied to physical quantities that have a safe range of values. Alternatively, the state trajectories could be used for TTF determination due to asymptotic convergence of the residual. Finally, a simulation example is used to demonstrate the performance of the prognostics scheme.

Therefore, the important contributions of this paper include an online prognostics scheme, which includes fault isolation and TTF determination, for a class of nonlinear discrete-time systems with abrupt or incipient faults which can occur simultaneously and more than one fault can occur on the same state. Unlike other schemes [8, 9, 22, 23], the proposed scheme delivers asymptotic stability in discrete-time, which means guaranteed isolation and reliable TTF determination in the presence of unstructured system uncertainties [1, 2, 10].

The paper is organized as follows: Section II introduces the system under investigation whereas Section III revisits the fault detection scheme. In Section IV, the prognostic scheme is introduced. Finally, in Section V, a simulation example is used to illustrate the performance of the proposed prognostics scheme. Section VI presents some concluding remarks and discusses future work.