Contents I. Introduction
Observers are techniques that are used to estimate and detect the faults of the systems. In the design of observers there are two key elements that should be taken into account, theses are as follows 1) the type and size of the fault which is either multiplicative fault (parameter faults) or additive faults (actuator or sensor faults); 2) the disturbance characteristics. Over the past three decades, much attention has been paid to the problem of fault detection and diagnosis in dynamic systems because of an increase demand for high reliability industrial processes. Luenberger observer was firstly proposed and developed in [1] and further developed in[2]. Since these early studies which concentrated on purely deterministic continuous time-invariant linear systems, the theory of the observer design has been extended by many researchers to include time-variant, discrete and stochastic issues. The observers have been formed in design of an integral part of numerous control systems. In general, Luenberger Observer possesses a relative simple design that makes it an attractive general design technique[3], [4]. Later, the Luenberger observer was extended to form a Kalman filter [5]. Although the Kalman filter is in use for more than 30 years and has been described in many papers and books, its design is still an area of concern for many researches and studies. It could be argued that the Kalman filter is one of the good observers against a wide range of disturbances [4], [6]. The problem of estimating a state of a dynamical system driven by unknown inputs has been the subject of a large number of studies in the past three decades. An observer that is capable of estimating the state of a linear system with unknown inputs can also be of tremendous use when dealing with the problem of instrument fault detection, since in such systems most actuator faults can be generally modeled as unknown inputs to the system [7], [8]. A new methodology for fault detection and identification subject to plant parameter uncertainties is presented in [7]. A full-order observer procedure was developed for linear systems with unknown inputs using straightforward matrix calculations in [9]. Estimating using a reduced order disturbance de-coupled observer was presented in [10]. A full-order unknown input and output structure is used in order to generate residuals, which can be used to detect fault and isolate on a vertically taking-off and landing aircraft dynamic model in [11]. Designing the unknown input and output observer was reported by considering the unknown constant disturbance of parameters in chaotic systems in [12]. However, when the numbers of sensors and unknown inputs are equal, the observer may not exist. Hence, the unknown input observer method is not always feasible for fault detection. To overcome this drawback, several studies have been developed and implemented for augmented observers by given bounds of plant uncertainty. The fault detection scheme facilitates determining free matrices in the partial state observer, by which the residual function can be identified and distinguished for the sensor and actuator faults. An augmented system model that a residual function could be generated according to partial state observers was presented in [13], [14]. The generated residual signals, which disclose the fault, are sensitive to faults while insensitive to uncertainties. To derive residual functions, existence conditions and its design procedure are presented in [15].
