# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 2 System Descriptions and Preliminaries  
The schematic diagram of the EHA with actuator faults is shown in Figure 1. The internal leakage fault of the hydraulic driven actuator is studied. The dynamics of the actuator's moving part can be derived by [PERSON]'s second law  
\[m\ddot{z}=P_{L}A-k_{b}\dot{z}+\Delta\big{(}t,z,\dot{z}\big{)} \tag{1}\]  
where \(z\) is the piston's displacement; \(\Delta(\bullet)\) represents the disturbance terms, i.e., load's force, friction; \(P_{L}=P_{1}\)-\(P_{2}\) is the load pressure, with \(P_{1},P_{2}\) being the pressure inside the extend and retract chambers; and \(m\) and \(k_{b}\) describe the mass of the moving part and the equivalent viscous damping coefficient. We suppose that the valve's voltage input is related to the spool valve displacement by the directly proportional. Then, the load flow rate \(Q_{L}\) is related to the control signal \(u\) by  
\[Q_{L}=k_{s}u\sqrt{P_{s}-sign(u)P_{L}} \tag{2}\]  
where \(P_{s}\) is the supply pressure, \(k_{s}\) is a servo-valve's proportional gain, and \(sign(\bullet)\) is the standard signum function. The fluid flow distribution into the two chambers of the cylinder is given by Refs. [8; 36]:  
\[Q_{L}=A\dot{z}+\frac{V_{c}}{4\beta_{e}}P_{L}+Q_{l}-q_{Le} \tag{3}\]  
where \(\beta_{e},~{}A\) are the Bulk modulus, a piston effective area, and \(Q_{l}\) is the internal leakage. \(V_{c}\) is a fixed control volume of the cylinder. \(q_{Le}\) denotes the time-varying disturbances (i.e., external flow leakage, lumped modeling error). Considering the system suffering from an internal leakage fault, the \(Q_{l}\) can be represented as follows:  
\[Q_{l}=C_{0}P_{L}+v \tag{4}\]  
where \(C_{0}\) is the nominal coefficient, and the time-varying bias fault signal \(v\) is leakage at the piston seal or internal flow loss. It can be formulated as follows in Refs. [37; 38]  
\[v=-C_{f}\sqrt{\frac{1}{A}}\sqrt{|AP_{L}|}sign(AP_{L}) \tag{5}\]  
where the time-varying internal leakage fault \(C_{f}\) is considered with time profiles modeled by  
\[C_{f}=\left\{ \begin{array}{ll}0&\mbox{if $t<T_{0}$}\\ (1-e^{-a(t-T_{0})})C_{f0}&\mbox{otherwise}\end{array} \right. \tag{6}\]  
Figure 1: Schematic of hydraulic system with faults. where \(T_{0},\alpha,C_{f0}\) denote, in turn, the unknown time that a fault occurs, the unknown fault evolution rate, and the leakage fault coefficient. Choosing \(\zeta=[\zeta_{1},\zeta_{2},\zeta_{3}]^{T}\triangleq[z,\dot{z},AP_{L}]^{T}\) as the state variables, from (3)-(6), the system with actuator faults modeled can then be written as:  
\[\left\{ \begin{array}{l}\dot{\zeta}_{1}=\zeta_{2}\\ m\dot{\zeta}_{2}=f_{2}(\zeta_{2})+\zeta_{3}+\sigma_{1}(\zeta_{1},\zeta_{2},t) \\ \dot{\zeta}_{3}=f_{3}(\zeta_{2},\zeta_{3})+g_{3}(\zeta_{3},u)u+\sigma_{2}(\zeta _{3},t)\end{array} \right. \tag{7}\]  
where  
\[f_{2}=-k_{b}\zeta_{2},\sigma_{1}=\Delta,\sigma_{2}=\frac{4A\beta_{e}}{V_{c}} \bigg{(}q_{Li}+\frac{C_{f}}{\sqrt{A}}\sqrt{\xi_{3}}sign(\zeta_{3})\bigg{)},\]  
\[f_{3}=-\frac{4A\beta_{e}C_{0}}{V_{c}}\zeta_{3}-\frac{4\beta_{e}A^{2}}{V_{c}} \zeta_{2},g_{3}=\frac{4A\beta_{e}k_{t}}{V_{c}}\sqrt{P_{s}-\frac{1}{A}\zeta_{3 }}sign(u).\]  
**Assumption 1**. :
1. _The tracking reference signal_ \(\zeta_{1d}\)_, its derivative_ \(\dot{\zeta}_{1d}\)_, and all states are bounded._
2. _Under normal working conditions, the pressures_ \(P_{1},P_{2}\)_, and_ \(|P_{L}|\) _are sufficiently smaller than_ \(P_{s}\)_._
3. _There exist upper bounds for_ \(\sigma_{1},\sigma_{2}\) _and their derivatives_ \(\dot{\sigma}_{1},\dot{\sigma}_{2}\)_, such that they satisfy_ \(|\sigma_{i}|\leq d_{im},\left|\dot{\sigma_{i}}\right|\leq D_{i};i=1,2\)_, where_ \(d_{im},\ D_{i}\)_, are positive constants._  
**Remark 1**. : _For assumptions 1a and c: These are popular assumptions that have been used in previous works, for instance Refs. [1; 7; 38], to facilitate validating the stability problem of the closed-loop system. For assumption 1b: The pressures \(P_{1},P_{2}\) originate from \(P_{s}\), whilst \(P_{L}=P_{1}\)-\(P_{2}\) also does not exceed \(P_{s}\).