# A Control-Oriented Linear Parameter-Varying Model of a Commercial Vehicle Air Brake System
## 1 Introduction  
The vehicle brake system is a crucial component of commercial vehicles and it is closely related to driving safety, especially under downhill braking and other emergency conditions to avoid an accident. Over 90% of vehicle rear-end accidents and 60% of frontal collisions could be avoided effectively if the brake response time could be reduced by one second based on the National Transportation Safety Board (NTSB) special investigation report [1]. The engineering solution for decreasing the rear-end accidents is to develop an advanced brake system [2]. Most commercial vehicles, such as buses, heavy- and mediate-duty trucks, are equipped with an air brake system. Compressed air is used as the actuation power in the air brake system of commercial vehicles. The air brake system used in commercial vehicles is quite different from the hydraulic brake system used in passenger cars, where hydraulic power is used, and air brake system response time is much slower than that of hydraulic brake system since air is compressible. Anothermajor difference is the required braking torque, where commercial vehicle braking torque is much higher than that of passenger cars. Since air pressure is relatively difficult to regulate, controlling the air brake system to achieve the desired performance is a challenge. There are many literature in the air brake system and associated control methods [3; 4]. Nowadays, electronically controlled pneumatic braking systems are widely used for commercial vehicles to improve their performance to make it possible to install a longitudinal driving assistance system, where the braking air pressure is electronically controlled using electro-pneumatic proportional valves. As a result, studying the brake system characteristics is of great importance. The vital dynamic characteristics of the brake system centers on the pedal force, pedal travel, response time and vehicle stopping distance [5]. The key to improving air brake system performance is to reduce response time with smooth brake torque. The driver braking response time, the brake line layout, the actuation solenoid valve response time, and the master valve response time affect brake system response time, leading to undesired braking performance. The developed model was used to predict the brake chamber transient performance and estimate brake chamber actuator displacement, and it is also used to detect brake system leaks. This model is a useful tool for a control system to reduce response time and stopping distance [6]. [PERSON] et al. [7] developed a simple mathematical model describing the air flow dynamic characteristics for a pneumatic brake system. [PERSON] [8] analyzed the effect of driver pedal force to brake distance through multi-body dynamic simulations of a Single Unit Truck (SUT) based on a validated vehicle model and showed that it is feasible to use the air brake system model to predict the transient brake chamber pressure performance. [PERSON] [9] proposed a mathematical model of electro-hydro-mechanical brake system for passenger cars to describe the braking force generated by the hydraulic master boost-cylinder. [PERSON] [10] developed a mathematical model using AMEsim software to evaluate the brake system dynamic characteristics. [PERSON] [11] designed the full brake system model to make a study of the performance of the brake system and individual brake system component. [PERSON], et al. [12] adopted a servo-device to simulate the brake pedal operations and study the hysteresis characteristics of pneumatic brake systems. Active safety technology such as anti-lock brake system (ABS), adaptive cruise system (ACC), active collision avoidance system (CAS), Electronic Stability Control (ESC), Electronic braking system (EBS) and brake force distribution(EBD) are widely adopted. ABS is essential to prevent tire slipping and hence to maximize brake torque, and brake torque redistribution are also very important for reducing brake distance.