0957-4158/$ - see front matter © 2012 Elsevier Ltd。 All rights reserved。 http://dx。doi。org/10。1016/j。mechatronics。2012。03。008

W。 Kim et al。 / Mechatronics 22 (2012) 766–777 767

of the nonlinear controllers, which are designed for the strict feed- back form, are not seperated from those of the observers, closed- loop stability should be studied。 Several observer-based nonlinear controllers were proposed [30,31]。 In [30] only acceleration was estimated by the observer。 A proportional integral (PI) observer was designed to estimate full state [31]。 Although the estimation performance was validated, the closed-loop stability was not proven。

The main contribution of this paper is the design and imple- mentation of an output feedback nonlinear control for position tracking in EHS。 Our previous research [32] studied for the rotary type EHS and was based on simulations with focus on the design of the controller and the observer。 This paper solves the problem for the linear type EHS, and focuses on the gain selection of the passivity-based controller for the output feedback control and the proof of the closed-loop stability。 Moreover, its performance is validated via experiments。 The high gain observer is designed to estimate the full state, and the high gain technique is used to re- duce the effects of the nonlinear terms。 The passivity-based con- troller [22] is implemented for position tracking。 Although the passivity-based control is simpler and more straightforward than the backstepping algorithm [22], the passivity-based controller guarantees only asymptotic stability of the tracking error dynam- ics。 Therefore, it is difficult for the passivity-based controller to be designed with a high gain observer since the exponential stabil- ities of the quasi-steady-state model and the boundary layer sys-

includes a servo-valve to control the movement of the actuator, ar- ranged as an open-loop system without  spool  position  feedback and operated by the electrical input  current  of  a  torque motor。 The flow of hydraulic fluid into the cylinder chambers, which are connected to a servo valve through cylinder ports, is controlled by the servo-valve and is influenced by load pressure, which is the difference between the pressures of  two  actuators:  ports  A and B。 The force of the actuator is produced by the load pressure。 In this section, we introduce a nonlinear mathematical model of the EHS shown in Fig。 1。 For the modeling of EHS [1,2], we assume that

●The valve is matched  symmetrically。

●The spool of the valve is ideal, with zero    lapping。

●The spool valve radial-clearance leakage is   negligible。

●The valve restriction areas are linearly proportional to the spool valve opening。

●The loading force, FL is linearly proportional to the piston dis- placement, and the load pressure, PL where PL is the differential pressure between PA  and PB  [N/m2] is defined as follows:

FL ¼ AP ðPA — PB Þ; 1

PL ¼ PA — PB ¼ FL =AP ; ð Þ

where AP  is the pressure area of the piston   [m2]。

●The volume of the chamber are VA = V0 + AxP and VB = V0 — AxP, where xp  is the piston position [m], and V0  is initial volume of

tem  are  required  for  the  stability  of  the  singular  perturbed

system [34]。 The problem is solved in this paper by selecting the

the chamber, including hoses, when xp ¼ xp

max

=2,  and xp

max

controller gain such that the origin of the tracking error dynamics is exponentially stable。 We prove that the closed-loop system is asymptotically stable using the singular perturbation method if the condition is satisfied。 Therefore, the observer gain is chosen with consideration of the controller gain。 The performance of the proposed method is validated through simulation and experiments。