R T

m2 (t)  m*2k −  m*2k − m2 (tk ) − α t − t2 2 ,

m*2k   Pz   A2  (L − x1k ) (7)

R T

After the time t3 a new balance has to be achieved, while pis-ton is approaching desired stroke x. A relation for time t3 is complex but after some simplification can be approximated by a relation

α 2t32 1 − χ  2αt3   m*2k − m2  −   m*1k − m1 − m1 − χm2   0

χ  x1k

L − x

1k

(8) where a suitable solution (t3>tk) for above equation has to be used. Application of above simplified rules has given some transients, but at the laboratory test still some overshoot has existed. It was observed that better results could be obtained, when at the second time interval valves activities for volumes 1 and 2, the filling of the under piston volume 2 will be fi-nished before the filling of the volume 1, Fig.4. The second filling was started at the same time for both volumes (only for the drawing was a little shift)

Filling

Volume 1

tk t2 t3 t

Emptying

Volume 2

Fig. 4. Filling and emptying phases for the displacement control within the movement

The further investigations in the lab have confirmed, that the displacement x can be controlled by the time tk and the most suitable time instant for the start of braking is a time of the pressure equilibrium in both volumes. In the next presentation time tk is replaced by t1. In order to reach the set position with desired level of the displacement error and acceptable over-shoot, it was crucial to adjust the timing (t1, t2, t3). For discov-ering of proper rules many experiments have to be conducted

– the timing depends on the: initial position, cylinder length, desired piston stroke, supply air pressure, moved mass, valve characteristics and what is very important but hard to deter-mine – friction conditions in the cylinder. In order to reduce the research time, the first part of this investigation has been performed with a fast prototyping approach. A simulation package PExSim (Process Explorer and Simulation) [7] has been used for the simulation of different control strategies [2].

In pneumatic positioning systems, the basic is the precise

(6)

handling of the friction force in a cylinder, what is very hard to be measured. To perform a simulation more realistic, the transients were calculated with the Stribeck model of a fric-

tion force (fitted to the measured displacement transients). The optimization was performed with the application of the PExSim package.

Some results representing a relation between the values of time t1, t3 and the piston stroke x are presented below.

0,3

t1 [s] a.

0,25 5

0,2 4

0,15 3