Automotive applications of explicit non-linear model predictive control

This thesis presents automotive applications of explicit non-linear model predictive control. Model predictive control is a state-of-the-art control methodology that allows for systematic incorporation of physical and operational constraints in the control system design. Multiple-input multiple-output systems can be easily employed in the optimal control problem. The receding horizon method combines optimisation and feedback adjustment. However, for systems with non-linear and sufficiently fast dynamics, the online optimisation can be a challenging task to accomplish in real-time on embedded hardware. The non-linear optimisation problem is formulated in parametric form and a sub-optimal solution thereof is computed offline. Local accurate multi-parametric quadratic approximations are used together with iterative and recursive orthogonal partitioning of the parameter exploration space. A piecewise affine state feedback law defined on polyhedral regions is obtained. The reduction in online software complexity leads to computing times lower than the required sampling times. The explicit solution, with guaranteed levels of sub-optimality, allows for a priori verification and functional safety validation being viable for safety-critical applications. Three representative automotive control problems are selected. Each case study includes a detailed analysis of the control system based on explicit non-linear model predictive control (NMPC). The design and performance assessment of an explicit NMPC-based vehicle stability controller with an electro-hydraulic braking system is presented in the first case study. A systematic simulation-based investigation on the prediction model complexity shows the necessity of a non-linear lateral tyre force model for acceptable controller behaviour. A load transfer model considering the side-slip angle rate is important for an accurate prediction of lateral tyre forces and their yaw moment contributions. The modelling of longitudinal and lateral tyre force coupling significantly influences the front-to-rear braking force distribution. The second case study proposes traction controllers (TC) based on explicit NMPC for electric vehicles with in-wheel motors and compares these with more conventional TC strategies based on proportional-integral control. The NMPC allows for longitudinal slip tracking improvement during variable tyre-road friction scenarios simulated with a high fidelity vehicle model. Experimental validation on a fully electric vehicle prototype demonstrates real-time operation. An explicit NMPC-based TC for combined driving and cornering conditions is presented for a front-wheel-drive electric vehicle with in-wheel motors. The controller based on a combined slip tyre force model shows enhanced slip tracking performance compared to a pure longitudinal slip tyre force model in a TC scenario. A proof-of-concept design of an explicit NMPC for anti-lock braking systems (ABS) is demonstrated. The study includes an implementation on an industrial electro-hydraulic braking unit. The experimental comparison on a hardware-in-the-loop test-rig shows that the NMPC ABS consistently outperforms a proportional-integral-derivative ABS. In the third case study, the design, optimisation-based tuning, and systematic comparison of six different explicit NMPC anti-jerk controllers for a front-wheel-drive electric vehicle with on-board motors are presented. The inclusion of a non-linear backlash model improves in all cases the controller behaviour. The modelling of the wheel dynamics and the tyre dynamics with constant relaxation length does not bring significant performance enhancements. The detailed analysis of the explicit solutions shows the best suitability of the approach for systems with a low number of states, minimal number of additional parameters, limited number of control inputs, strong non-linearities, and fast dynamics for applications that are safety-critical, subjected to strict performance requirements, and characterised by low computational power but sufficient memory capacity. The influence on the controller complexity in terms of polyhedral critical regions and memory requirements of various aspects, e.g. number of parameters (states and varying parameters), number of control inputs, definition of the parameter exploration space, formulation and number of constraints, etc. is investigated. The saturation of parameters, degrading closed-loop performance, and the exploitation of symmetry properties and state transformations, reducing complexity, are analysed. The role of soft constraints, promoting feasibility, is discussed. The influence of approximation tolerances, requiring a compromise between acceptable performance degradations and controller complexity, is shown. Methods to reduce complexity are embedded in the proposed post-processing algorithm that achieves good performance. These include memory-optimised binary search tree generation, reducing the memory requirements and online complexity, disjoint optimal and suboptimal merging procedures and clipping-based complexity reduction, both leading to lower numbers of polyhedral regions in the explicit approximate receding horizon feedback law. An analysis on the memory requirements shows great variations depended on the specific application. The empirically assessed execution times of the explicit controllers on a rapid control prototyping unit are in the sub-millisecond range and prove real-time feasibility. The applications of the explicit NMPC demonstrate the incorporation of non-linear system dynamics in the controller design, reduction of online computing times, the systematic constraint satisfaction, and the possibility of a priori verification for safety-critical applications. The analysis also shows increased memory requirements with the suitability of the explicit NMPC approach depending on the specific system including design choices and the number of states and parameters, respectively.