IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 2, MARCH 2020 517 Optimal State Estimation and Fault Diagnosis for a Class of Nonlinear Systems Hamed Kazemi and Alireza Yazdizadeh, Senior Member, IEEE Abstract—This study proposes a scheme for state estimation and, consequently, fault diagnosis in nonlinear systems. Initially, an optimal nonlinear observer is designed for nonlinear systems subject to an actuator or plant fault. By utilizing Lyapunov’s direct method, the observer is proved to be optimal with respect to a performance function, including the magnitude of the observer gain and the convergence time. The observer gain is obtained by using approximation of Hamilton-Jacobi-Bellman (HJB) equation. The approximation is determined via an online trained neural network (NN). Next a class of affine nonlinear systems is considered which is subject to unknown disturbances in addition to fault signals. In this case, for each fault the original system is transformed to a new form in which the proposed optimal observer can be applied for state estimation and fault detection and isolation (FDI). Simulation results of a single- link flexible joint robot (SLFJR) electric drive system show the effectiveness of the proposed methodology. Index Terms—Differential geometry, fault detection and iso- lation (FDI), fault diagnosis, neural network (NN), nonlinear observer and filter design, optimal state estimation. I. I NTRODUCTION O BSERVER design for nonlinear systems is a popular problem in control theory that has been investigated in many aspects [1], [2]. State estimation of nonlinear systems in the presence of unknown input (UI) is another interesting and relevant topic in the modern control theory [3]−[6]. One of the main observer design techniques for mentioned systems is UI decoupling through state transformation [7], [8]. In this method the state estimation is performed on a reduced system corresponding to the UI free subsystem. For that, the state equation is splitted into two parts, one being sensitive to the UI, the other being decoupled from this input. It is then possible, under specific conditions, to eliminate the UI influence on the state and the measurement equations [9]. An UI can represent the impact of disturbance or failure of actuators or plant components and thus worth to be used in the field of fault diagnosis consists of fault detection and isolation (FDI). On the other hand, model-based FDI is a well-established technique in [10], [11]. Among the model-based FDI, Manuscript received January 24, 2017; revised May 25, 2017; accepted May 25, 2017; updated January 16, 2020. Recommended by Associate Editor Changhua Hu. (Corresponding author: Hamed Kazemi). Citation: H. Kazemi and A. Yazdizadeh, “Optimal state estimation and fault diagnosis for a class of nonlinear systems,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 2, pp. 517-526, Mar. 2020. The authors are with the Department of Electrical Engineering, Shahid Beheshti University, Tehran 1983963113, Iran (e-mail: ha kazemi@sbu.ac.ir; a yazdizadeh@sbu.ac.ir). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2020.1003051 observer-based techniques have been more developed and implemented [12], [13]. So far, various observer-based FDI design approaches, including FDI via Kalman filter [14], and high gain adaptive observers [15], have been reported in the literature. Most of these techniques are developed for linear systems. However, during the past two decades, a number of observer-based FDI approaches for nonlinear systems have been presented [16]−[18]. De Persis and Isidori [19] have designed a nonlinear filter for a class of affine nonlinear systems for FDI using a differential geometry approach. The proposed method is applicable to nonlinear systems that have conditioned invariant distributions which is an extension of the concept of unobservability subspace in linear systems [20] and leads to a coordinate transformation of the original system. The design of a new nonlinear observer for FDI under the mentioned transformation is considered in this study. The basic idea behind the use of the observer for fault detection is to estimate states of the system by using some type of observers, and then construct a residual by a properly weighted output error [10], [21]. For nonlinear systems, the theory of observer design is not nearly complete or successful, as it is for the linear case [22], and nonlinear observers related to FDI of nonlinear systems are restricted. Most of them use sliding mode approach for detecting or estimating the fault [23]−[25]. Sliding mode observer is limited to a class of nonlinear system in a standard form in which the nonlinear term, that is a separate term, must satisfies Lipschitz function assumptions. Also appearance chattering phenomenon in sliding mode observer is another restriction that should be avoided. Designing and utilizing other observers in this field could be worthwhile. Nonlinear observers are limited to a class of nonlinear systems and some of them use linearized model of the system [4]. It is also noted that most of the observer gains are very high or they depend on estimation error which is initially very high. However, high value of the observer gain increases the sensitivity to noise. Also finite time convergence is an important feature that should be considered in designing procedure. In [22] according to a method presented in [26] for optimal control of nonlinear systems, an optimal nonlinear observer using Hamilton-Jacobi- Bellman (HJB) equation based formulation is proposed for state estimation of a class of affine nonlinear system that is not subject to fault or unknown disturbance. Since it is difficult to find solution of the HJB equation, neural network (NN) has been used to approximate it. Here we want to use this method and consider it for a system subject to fault signal and also utilizing the geometric coordinate transformation, apply the proposed method to a class of nonlinear system with unknown disturbances.