# A Control-Oriented Linear Parameter-Varying Model of a Commercial Vehicle Air Brake System
## 1 Introduction
A brake system model was developed and used to tune the associated PID controller to optimize the ABS performance [13]. [PERSON] et al. [14] researched the proportional relay valve and control algorithm for the purpose of pneumatic electric braking system (EBS) control strategy development. [PERSON] et al. [15] developed an autonomous collision avoidance system (CAS) by electric braking torque of the electric vehicle. Commercial vehicles electronically controlled technology is developed slowly. As vehicle technology EBS is gaining its popularity, especially for electric vehicles [16], other vehicle control features, also require the brake system to work with them to achieve their functionality. Therefore, designing a brake control system with satisfactory performance becomes extremely important. During the development of the brake control system, it is dangerous to directly carry out on the real vehicle braking test. Establishing a mathematical model is vital for the design of controller parameters and simulation tests. To develop a model-based control strategy for reducing brake system response time, a control-oriented model is necessary. There are many brake system modeling tools, for example, AMEsim and Matlab/Simulink. Most of the existing researches on modeling focus on the original brake system components. For instance, [PERSON] et al. [17] developed a nonlinear brake system model to predict the brake chamber transient pressure, and model-based longitudinal control schemes are also designed to improve the transient braking performance and maintain the desired brake chamber pressure. Note that modeling the spool of a proportional valve is not a simple task, and the existing brake system model is either nonlinear or linearized based on the nonlinear one. There are many literature on the brake system model linearization and associated linear control design for ABS brake systems but not many for pneumatic brake systems. [PERSON] [18] presented an adaptive neural network-based hybrid controller for the ABS using feedback linearization, but it was a challenge to handle the brake system nonlinearity and uncertain time-varying parameters. [PERSON] [19] also showed a nonlinear output feedback control law for ABS. The proposed control algorithm yields robust performance and guarantees the control action for unknown road conditions such as road grade. [PERSON] [20] proposed a new brake system architecture using proportional servo-valves and designed a fuzzy logic control strategy to achieve the optimal pressure in the brake circuit. To study the nonlinear characteristics of the brake system, [PERSON] [21] used the nonlinear vehicle brake model to predict the brake noise and showed that the linearized model could be used to predict closed-loop system instability. Since it is challenging to design a controller with guaranteed performance for nonlinear systems due to a lack of available nonlinear control design methodologies, linearizing the nonlinear system model is one option to design a linear controller. An alternative method to linearization is using system identification, such as the PRBS (Pseudo-Random Binary Signal) q-Markov Cover [22], to obtain a linear model. The PRBS q-Markov Cover method was used for an electric variable valve timing system with satisfactory results in [23]. By comparing with the PID-type approaches, the model-based control action is 'intelligent' and has advantages in uniformity, disturbance rejection, and setpoint tracking [24]. To further improve the closed-loop brake system performance, model-based gain-scheduling control is preferred, which requires a control-oriented LPV (Linear Parameter-Varying) system model. In this paper, a nonlinear pneumatic brake system model with electro-pneumatic proportional valves was developed.