In addition, the cylinder inlet/outlet flows Q1, Q2 can be represented byQ1=Ax�B+Va��edp1dt+L(p1?p2),Q2=Ax�B?Vb��edp2dt?L(p2?p1),(2)where A is pressurized area so of hydraulic piston, L is internal leakage coefficient of the cylinder, and p1, p2 are the cylinder inlet/outlet chamber pressures, respectively. In this study, the external leakage of the cylinder is neglected.In this study, Va and Vb are assumed to be equal due to the double rod, and double acting hydraulic cylinder is used in EHA system. Also, Va and Vb can be expressed with the nominal volume of each EHA chamber V0 which equals the mean volume of pipe and cylinder chamber as follows:Va=V0+Ax,Vb=V0?Ax.(3)Since a solid tubing is used in the prototype, the pressure impact of pipe expansion and flexibility on the effective bulk modulus is assumed to be negligible.

Therefore, the relationship pump port flows (Qa, Qb) and cylinder chamber flows (Q1, Q2) can be expressed asQ1=Qa,Q2=Qb.(4)Then, the load QL can be defined as follows [21]:QL=Q1+Q22=Qa+Qb2.(5)The relationship between the pump port pressures (pa, pb) and cylinder chamber pressures (p1, p2) can then be expressed aspa=p1+ppipe,pb=p2?ppipe,(6)where ppipe is pressure drop.Due to the short length of tubing, pump port pressures are assumed to be equal; that is, the pressure drop can be neglected, to the actuator inlet and outlet pressures such that [7, 8, 11, 17, 22]pa=p1,pb=p2.(7)Since the symmetrical double-rod hydraulic cylinder is used in this EHA system, the following equation for the cylinder chamber pressures can be established:dp1dt=?dp2dt.

(8)By Carfilzomib combining (1), (2), (5), (7), and (8), a simplified pump/cylinder model equation can be expressed asDp��p=Ax�B+V0��ep�BL+CTpL+f1,(9)where pL = p1 ? p2, CT = �� + L + (Cp/2) is the equivalent leakage coefficient and f1 is unmodeled dynamics of hydraulic part of the EHA system.The actuator force and the displacement of the load can be represented asF=pLA=Mx��+Bx�B+f2,(10)where A is pressurized area of hydraulic cylinder, M is mass of load, B is viscous friction coefficient and f2 is lumped uncertain nonlinearities due to external disturbance, the unmodeled friction forces, and other hard-to-model terms of mechanical part of the EHA system.By combining (9) and (10) and neglecting f1 and f2, the linear transfer function between the angular velocity of the pump and the displacement of the main cylinder can be expressed +��e(BCTDp+A2)MV0s)?1,(11)where??????????asGh(s)=x(s)��p(s)=(ADp��eMV0)(s3+(BM+CTDp��eV0)s2 s is the Laplace operator.