IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 5, SEPTEMBER 2012 1311 Brief Papers Regularized - and DHOBE: An Adaptive Feedforward for a Solenoid Valve Jean-Philippe Gauthier and Philippe Micheau Abstract—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 Das- gupta-Huang outer bounding ellipsoid (DHOBE) and recursive least squares (RLS) with exponential forgetting factor - adaptation algorithms are compared to the non-adaptive feedfor- ward. 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 adap- tation algorithms offer deceiving performances with oscillating identified parameters. A regularization procedure is added to both adaptation algorithms, giving the - and . 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. Index Terms—Adaptive, bounding ellipsoids, control, continu- ously variable transmission (CVT), hydraulics, least squares, open- loop, transmission. I. INTRODUCTION I N VEHICLE control systems, the actuation forces are often created by hydraulic pressure. The hydraulic technology, for its compactness, low cost, high power density, self lubri- cation, and quick response time is particularly well suited for vehicle control applications [1]. Despite the trend to include a growing number of electric actuators, hydraulic systems are still needed in numerous applications where higher force and torque are needed [2]. A way to get the flexibility of electrical control while keeping the high power density of hydraulics is to use electro-hydraulic valves. In the particular case of a steel belt continuously variable transmission (CVT) transmission, the required extreme thrust forces make it a good candidate for an electro-hydraulic actua- tion system. The electro-hydraulic valves used are pressure reg- Manuscript received December 02, 2010; accepted May 18, 2011. Manuscript received in final form June 20, 2011. Date of publication July 25, 2011; date of current version June 28, 2012. Recommended by Associate Editor M. Guay. The authors are with the Department of Mechanical Engineering, Uni- versité de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada (e-mail: jean- philippeg@gmail.com). Digital Object Identifier 10.1109/TCST.2011.2160641 ulators that convert an electrical input signal into oil pressure. According to various factors like the oil and solenoid tempera- tures, the equation describing the valve pressure versus its input signal changes significantly. To provide good shifting perfor- mances, the valve has to provide quick pressure changes with minimal pressure error. Numerous authors have proposed different techniques to con- trol electro-hydraulic valves. Aside from the classic closed-loop proportional-integral derivative (PID) control, most of the more advanced control techniques have been developed for motion control hydraulic systems. The adaptive closed-loop control, which determines the closed-loop controller gain in function of the valve characteristics, has been proposed [3]. For flow con- trol valves with a dead zone, adaptive fuzzy compensation has been investigated and proved effective [4]. Again for flow con- trol valves and motion controlled piston type actuators, robust adaptive control has been tested with satisfactory results [5]. For pressure control systems, open-loop and linear closed- loop control are mostly used. While the open-loop control cre- ates unacceptable pressure errors, the closed-loop control brings additional dynamics to the valve itself. To minimize the added dynamics, high controller gains have to be used, lessoning the robustness of the system [6]. For example, the experimental re- sults from Amirante [7] show that a closed-loop controlled valve brings much more pressure oscillations than the same valve controlled in open loop. An adaptive open-loop controller is proposed here as an alternative for the control. The proposed controller works the same way as a basic open-loop controller would, but an adaptive inverse model of the steady state valve characteristics is used. Since the adaptation loop does not gen- erate the command itself, filtering of the feedback signal has no effect on the system dynamics. The problem is to design and implement an efficient online identification scheme to track the slow variation of the parameters. The inverse model adaptation is made using one of two adap- tation algorithms: The recursive least squares with exponential forgetting - and the Dasgupta Huang Outer Bounding Ellipsoids (DHOBE). - has been used in the past for model reference adaptive control (MRAC) of hydraulic systems and is considered the state of the art for hydraulic adaptive control [8]. It is included in this study to provide a comparison basis to evaluate the DHOBE adaptation scheme. DHOBE is a refine- ment of the - algorithm [9] which brings the advantages of implicitly calculating the uncertainties of the adapted param- eters and accepting a bounded signal noise regardless of its fre- quency spectrum. DHOBE has been used successfully in var- 1063-6536/$26.00 © 2011 IEEE