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For efficient analysis it is important to choose the proper model that fits the problem that needs to be solved. This paper discusses three pragmatic simulation models for longitudinal behaviour of a passenger car tyre (Steady State, Transient and Rigid Ring) that may be used in e.g. an ABS simulation. The characteristics of the simulation models are evaluated using some simple simulations. Simulations with a quarter vehicle model that includes load transfer effects are carried out to determine the deviation in results between the mentioned tyre models for an ABS application. The results show that the Steady State model may only be used below 10 Hz and that the Transient model is valid up to about 30 Hz. The results from the ABS simulation with the Rigid Ring model are most reliable and are clearly different from the Steady State and Transient model, which indicates that ABS simulations should be carried out with the Rigid Ring model. Additionally it is demonstrated that for tyre behaviour on uneven roads the influence of the tyre belt cannot be neglected.

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In this paper the systematic development of an integrated braking controller for a vehicle driven by an electric motor on the front axle is presented. The objective is to engage the electric motor only during braking, up to the point at which the vehicle reaches its manoeuvrability and stability limit. The control challenges are to distribute the braking effort correctly between the hydraulic brakes at the four tyres and the electric motor, to handle the tyre saturation and motor constraints effectively and to adapt the control allocation based on the vehicle’s states. The controller is designed using the state-dependent Riccati equation control technique, the vehicle state estimation and the ‘magic formula’ tyre model. The state-dependent Riccati equation control technique is a suboptimal control design technique for non-linear systems. A novel method for constructing the state-dependent coefficient formulation of the system dynamics is proposed. Soft constraints in the state dynamics are described, while an augmented penalty approach is suggested for handling the system’s hard constraints. The performance of the controller was evaluated for different braking scenarios using simulations in a MATLAB/Simulink environment. An eight-degree-of-freedom non-linear vehicle model was utilized. The numerical results show that the controller suboptimizes the regenerative braking effort while considering the tyre force saturation, the motor torque limits, the vehicle yaw rate and the slip angle error. A comparison with a constrained linear quadratic regulator shows the advantages of the proposed controller.