The mathematical model of a hydrostatic transsmision for controller design Janusz Kwaśniewski, Agata Piotrowska, Waldemar Rączka, Marek Sibielak Department of Process Control, University of Mining and Metallurgy. In the paper, model of hydrostatic transmission is shown. First part of the paper using of hydrostatic transmission and problems connected with control of hydraulic motor shaft rotation is presented. Next, mathematical model of the plant is shown. Verification of the model was described in the next part of the paper. Following, The test results of the model and real plant are presented. Formulated model was simplified because of its future usage in controller synthesis. Key words: hydrostatic transmission, mathematical model, simulation tests. Introduction A controlled hydrostatic transmission represents hydraulic systems in power transmission systems of working machines. It is applied mainly in kinematic chains of such machines' power transmission, but also in power transmission of their additional equipment. The advantages of this transmission, such as small volume and mass, easy method of assembling, large power transmitted, or a possibility of stepless change to rotational speed, make many engineers use it in their constructions. Though smooth change of speed in a controlled hydrostatic transmission is not a problem, stabilising this speed at a given level is not a trivial task. It has turned out, however, that in power transmission systems in which such a transmission is used, several parametric and non-parametric disturbances occur. In particular, in transportation machines, there are large changes to moment of inertia. Change to reduced moment of inertia is related with large changes to the mass of transported load as well as change gearbox ratio, which is often placed in kinematic chains of such machines' power transmission to extend their potential. Another type of disturbances hindering the regulation process is strongly changeable load of such machines, which, due to non-linearities encountered in a transmission, creates significant problems in selecting the controller's parameters. Additionally, non-stationary features of a transmission are caused by changes to the working fluid parameters, created by variability of its temperature, or impurities. Another problem is a change to parameters of constructional elements being a transmission's part, due to mechanical wear. At present in majority of cases PID controllers are used, but increasing requirements concerning quality of machine speed controllers call for reaching for better solutions. Consequently, a number of research centres carry relevant research in this subject, e.g. [Lennevi et al., 1994] suggests adaptive controller with gain- scheduling. Precise speed control is particularly important in such machines, as bridge cranes, airport tractors, etc. Such machines require speed stabilisation at variable object parameters and load signal. The main purpose of this paper is formulation of the mathematical model of the hydrostatic transsmision. This nonlinear model give us good description of a real plant however the identification of parameters of the model is easy. Mathematical model of the transmission A hydrostatic transmission is a very complex object. Having in mind a huge knowledge accumulated on this subject and the constructed model's target application, certain simplicative assumptions were adopted. These assumptions aim at writing a model in a possibly simplest form, at the same time preserving the significant features of a real object. Accordingly, it has been assumed that: • a hydrostatic transmission is a system with lumped constants, • static and dynamic features of a transmission do not depend upon the direction of hydraulic engine's rotation – mathematical model was developed for one rotation direction only; • a transmission is in thermal balance state, • modulus of volume elasticity is constant, • angular velocity of the main pump shaft is constant, • pressure drop in hydraulic cables is negligible, • leaks in pump and engine can be summed, • a pump's efficiency does not depend upon its shaft's rotation angle, • absorptivity of hydraulic engine does not depend upon its shaft's rotation angle, • safety valve is closed all the time.