Identification and estimation of parameters defining a class of hybrid systems

In this paper, we consider the problem of parameter estimation in an air brake system. In an air brake system, the pressure of air in the brake chamber and the displacement of the pushrod and their derivatives form a set of states that characterize the system. The position of a valve or mass flow rate of air is an input and the pressure is the measured variable or the output. The pressure acting on the pushrod of the brake chamber causes motion, and the mode in which the system operates depends on the displacement of the pushrod. The mode-dependent nature of the system is a result of different sets of spring compliances associated with the piston in different ranges of its displacement. The mode to mode transition in the air brake system is governed by a parameter which is the clearance between the brake pads and the drum. The clearance between the brake pads and the drum can vary due to a variety of factors — for example, brake pad wear or brake fade. In these applications, characterizing the transition from one mode to another requires a lot of constitutive assumptions, and it can be difficult to calibrate the parameters associated with the constitutive assumptions. We therefore treat the air brake system as a system in which the parameter governing the transition from one mode to another (clearance between the brake pads and the drum) is not known exactly. Clearly, this parameter dictates the time delay and lag between the command and delivery of the brake torque at the wheels and affects the stopping distance of the vehicles considerably. The problem of identification considered in this paper is as follows. Suppose that the pressure of the fluid were to be measured and that the motion of the piston is not measured. Is it possible to estimate the final displacement of the piston without knowing the parameters that govern the system to transition from one mode to another?