Achieving an $\mathcal{L}_{2}$ string stable one vehicle look-ahead platoon with heterogeneity in time-delays

A methodology is proposed to design stabilising and robust fixed-order decentralised controllers for heterogeneous vehicular platoons with Cooperative Adaptive Cruise Control (CACC). We consider Linear Time Invariant (LTI) models with constant time-delays at state, input and output. The closed-loop systems of (identical) local controllers and heterogeneous parameter vehicles are modelled by a system of delay differential algebraic equations. The proposed frequency domain approach uses the non-conservative direct optimisation approach towards stabilisation and robustness optimisation of delay systems. In this paper, the design problem of stabilising (identical) controllers achieving $\mathcal{L}_{2}$ string stability for one vehicle look-ahead platoon is reduced to a simultaneous controller design problem for a parameterised (sub)system, where the allowable values of the parameters correspond to heterogeneity (including time-delays) of the vehicles. By treating the heterogeneity in parameters as perturbations contained in specific intervals or regions, we determine the values for pseudo-spectral abscissa and robust induced $\mathcal{L}_{2}$ norm. Hence, we ensure that the achieved exponential stability and string stability properties along with the overall computational complexity (of designing the controller) are independent of the number of vehicles. The application of CACC is simulated in MATLAB software.