# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 1 Introduction  
Over the past few decades, electro-hydraulic systems have been extensively used in various civilian and industrial applications due to many outstanding achievements [1; 2; 3; 4]. There exist different research studies, and many results have been introduced to obtain the force or position control requirements, including model-based feedback control [5], sliding-mode control [6], backstepping control [7], adaptive robust control [8], and so on. However, in the abovementioned works, many simultaneous problems of EHAs, such as uncertainties, disturbances, faults, and optimal performance, have not been well investigated. It is noted that the uncertainties include parametric uncertainties (i.e., parameter deviation, modeling errors, etc.) and uncertain nonlinearities (i.e., nonlinear friction force, and external load) [5; 6; 7; 8]. Meanwhile, the most popular fault in the EHA is the internal leakage fault that comes from the piston seal and/or internal flow loss [9; 10]. Such problems may degrade tracking performance or even cause instability of the whole EHA system if they are not tackled in a timely manner. Thus, the fault-tolerant capability is vital for guaranteeing the reliable control system and obtaining optimal performance of the EHA system. To overcome the adverse effects of disturbance and uncertainty, the actuator fault issue, many effective FTC methods have been suggested to ensure system stability at an acceptable level [11]. Most notably, observers or an intelligent approximator-based FTC approach are widely adopted to react to faults, i.e., disturbance observer (DO) [12], state-dependent delay (SDD) [13], fixed-time observer [14], extended state observer [15; 16], neural networks (NN) [17; 18], fuzzy logic system (FLS) [19; 20], and so on. It is noted that the effect of fault is significantly larger than lumped disturbance; many works merged the disturbance and bias fault into lumped disturbance. To deal with this issue, a designed compensator is then presented in Refs. [12; 13; 14; 15; 16; 17; 18; 19; 20]. In Ref. [21], by using the time delay estimation-based FTC control approach, the effects of disturbances and an internal leakage fault are compensatedas lumped uncertainties. The authors in Ref. [22] proposed an adaptive NN actuator failure compensation control scheme for the vehicle active suspension systems. In Ref. [23], the fault estimation and FTC scheme were suggested for the hydraulic manipulator in the presence of the internal leakage fault and lumped disturbance, by using two DOs and a control reconfiguration law. In Ref. [24], the combination of an adaptive law and the DO was adopted to cope with the internal leakage fault. However, in the existing literature, the FTC approach based on optimal performance for the nonlinear system has not been well addressed, especially for EHA systems. How to obtain reliable control of the EHA whilst ensuring optimal performance is still challenging in the control field. In most industrial fields, certain optimal performance of the EHA is required. Thus, the optimal control emerges as an effective method and there has been more attention paid to not only handle the control issue but also guarantee the optimal performance for the nonlinear system. To obtain the optimal control performance, solving the solution of the Hamilton-Jacobi-Bellman (HJB) equation is necessary [25; 26; 27; 28]. Nonetheless, the HJB equation is solved complexity due to the inherent nonlinearity and intractability, which results in reducing its feasibility and applicability. Then, NN-based reinforcement learning (RL) is an effective tool to overcome the above challenge. A typical structure, namely actor-critic architecture, was used to implement the RL algorithm [29; 30]. In Ref. [31], the optimal tracking control problems for an unknown unmanned surface vehicle have been investigated via using an actor-critic reinforcement learning framework. In Ref. [32], actor-critic RL has been successfully applied for the EHA.