The underlying idea of controllers for systems with an input dead zone is very similar。 Because the cascading of a dead zone and a linear dynamical system (Hammerstein model) can be described as linear in the unknown parameters, a standard recursive least-squares algorithm (RLS) can be used to estimate these parameters。 Thus, a controller for the linear part of the system can be designed, and to compensate for the input non-linearity, an additional inverse operator must be defined。 The main drawback of these controllers is that they work only for one particular type of static non-linearity, which must be exactly known。 In general,

Z。 Knohl, 5。 Unbehauen ] Wechatvonicz 10 (2000) 127−143 l29

however, these characteristics are not fully known and often vary slowly with time。

In this paper an ANN is used for the on-line approximation of the non-linear behaviour of the valve。 Uith the introduction of an ANN, this controller can deal with unknown input non-linearities that change their structure and are not linearly parameterized [l3]。 Because of the combination of an ANN and non-linear adaptive (NA) control, this new control scheme is called ANNNA control。 The organization of the paper is as follows„ Section 2 deals with the properties of the plant and the actuating unit based on the linearized diRerential equations derived by physical modelling。 The ANNNA control scheme and its design procedure are described in Section 3。 The experimental set-up with regard to its hardware and software realization and the results of the real-time control are discussed in Section 4。 In Section 5 conclusions are drawn。

2。Dessription of the plant

The hydraulic system shown in Fig。 l is comprised of a cylinder, a 4]3 way proportional valve and a variable load force。 The load consists of a damping cylinder, a spring and a weight。 The damping and the initial tension of the spring as well as the weight are adjustable。 Therefore, a wide variety of practically relevant problems can be tested under experimental conditions„ positioning of a working piece carrier with diRerent external loads, system behaviour with diRerent initial or variable conditions。 The supply pressure P is assumed to be constant,

Fig。 l。 Schematic diagram of the hydraulic system。

l3O Z。 Knohl, 5。 Unbehauen ] Wechatvonicz 10 (2000) 127−143

and the control objective is the positioning of the pay load。 The proportional valve used in this plant is a low-cost product, which can be characterized by a relatively large and asymmetric dead zone。 Generally, in real plants the dead zone assumes a non-linear form。 An example is depicted in Fig。 2, where 1 is the input,

1- the  output,  [Ul,  U2]  the  upper]lower  break  points  and  [ fl(1 ),  f2(1 )]  the  non-

linear function in the upper]lower area, respectively。 For this kind of non-linear dead zone, the standard control schemes [l4,l2,l5] are not applicable because they are based on an ordinary dead zone model。

A complete mathematical model of such an electrohydraulic system, for example, has been given by [l6]。 However, these equations are highly complex and diAcult to utilize in control design。 A more practical model may be obtained through the linearization of the non-linear functions。

A mathematical model of the plant can be derived from the flow equation of the valve, the continuity equation and balance of forces at the piston。 The valve- flow-rate equation is highly non-linear and dependent on the valve  displacement 论文网

from neutral, which is proportional to the input voltage 1-  and the pressure drop

across the load PL。 A Taylor series linearization leads to

QL  = Kq1¯-KSPL,