Regularized ${\rm RLS}$-$\lambda$ and DHOBE: An Adaptive Feedforward for a Solenoid Valve

To allow a stable and fast acting hydraulic pressure control on a continuously variable transmission (CVT) for road vehicles, an adaptive feedforward strategy is used. The Dasgupta-Huang outer bounding ellipsoid (DHOBE) and recursive least squares (RLS) with exponential forgetting factor(RLS-λ) adaptation algorithms are compared to the non-adaptive feedforward. The experiments show a clear advantage for the adaptive over the non-adaptive version by compensating for the slow drift of the valve pressure gain during the warm-up period of the transmission. Because of highly correlated input data, the adaptation algorithms offer deceiving performances with oscillating identified parameters. A regularization procedure is added to both adaptation algorithms, giving the rRLS-λ and rDHOBE. The regularized algorithms offer significantly better performances and stability than their non-regularized counterparts. Because of its implicit parametric uncertainty calculation while keeping an equivalent convergence rate, and a lower number of updates, the rDHOBE algorithm is regarded as the best solution for the application. By adapting a simple linear model, the rDHOBE adaptive feedforward succeeds in responding to an abrupt change of the external pressure setpoint with no added actuation delay while keeping the pressure error under 0.5 bar.