Technical note Robust feedback linearization using an adaptive PD regulator for a sensorless control of a throttle valve Paolo Mercorelli * University of Applied Sciences Wolfsburg, Faculty of Automotive Engineering, Robert-Koch-Platz 8-a., 38440 Wolfsburg, Germany article info Article history: Received 18 May 2008 Accepted 25 August 2009 Keywords: Feedback linearization Robust tracking Minimum variance control Throttle valve PD regulators Hardware-in-the-loop abstract With classic gasoline injection systems, engine efficiency and emissions are affected by the control of the throttle plate, in particular its angular position. Depending on the current engine load, the angular posi- tion must track a trajectory as determined by the accelerator. This paper considers two problems. The first one is the design of a state observer. A velocity estimator is proposed based on measurements of cur- rent. If the effect of the noise is minimized, the angular position can be achieved through a cascade struc- ture between a particular velocity estimator and an inversion of the electrical system. This approach allows us to avoid a more complex structure for the observer, and yields an acceptable performance and the elimination of bulky position sensor systems. The elimination of the position sensor system sim- plifies the production system of the valve. The second problem, the robustness of the tracking, is addressed using a minimum variance control approach. This paper presents feasible real-time self-tuning of an approximated proportional derivative (PD) regulator, which compensates for the tracking error caused by inexact feedback linearization. It is interesting to note that the structure of the approximated PD regulator is similar to the velocity estimator. Robustness in the proposed loop control is achieved. Measured results on a real experimental setup with hardware-in-the-loop are shown. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction and motivation A robust throttle valve control has been an attractive problem since throttle by wire systems were established in the mid-90s. Moreover, a sensorless control is always been a challenging prob- lem in order to avoid bulky position sensor systems. To realize that it is necessary an observer structure. Decisive for that is the choice of the model which is assumed for the observation. For instance in previous models, the joint between the motor and the throttle flap is oft considered rigid. However, the assumption of rigidity of the joint is not always satisfied. In fact, a backlash is always present, which is caused by the space between the gear teeth. These consid- erations lead us to the proposed approach, which eliminates the gearbox model in the estimator and generally allows a reduction in the dimensions of the observer. For example, Consoli et al. [1] develop an algorithm for a sensorless position control actuator, using an approach based on the complete model. This approach would not be suitable in our case due the presence of the gears, be- cause of problems related to robustness as explained above. With regard to realizability, their idea appears plausible. In fact, the authors tested the algorithm in an 8-bit microcontroller, which currently is the typical solution for a controller-area network (CAN-bus) used in automotive applications. Other research is based on robust observability, such as with sliding mode control observers. Some new and elegant algorithms have been presented [2,3], but they present two problems in application. First, because of the relatively high switch frequency of the input voltage, the derivative of the current will be considerably large. This generates an ‘‘amplification” of the error in the model due to a multiplication with the uncertainty of the inductance. Second, due to the non-dif- ferentiability of the function ‘‘sign”, an implementation of the algo- rithm requires an approximation of the function ‘‘sign” with a differentiable function, such as ‘‘tan” or a polynomial function. This approximation produces errors. This is particularly true at the point where the velocity is equal to zero [2]. In the example pre- sented here, the stationary case (with velocity equal to zero) is the most important. Significant problems are caused by the robust- ness of the algorithm given the noise and the parameter uncertain- ties, together with the cost of implementation and the cost of the devices. In [4], a different approach is presented in which the ob- server is based on a mechanical model. Unfortunately, the mechan- ical model suffers from the backlash effect. The motivation behind the approach in this paper arises from the robustness and precision of the algorithm, the cost of its implementation, and the identifica- tion of compact and inexpensive devices with which to implement it. Proportional derivative (PD) regulators are largely applied in industrial applications, for example, PD regulators in a cascade 0957-4158/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2009.08.008 * Tel.: +49 (0)5361 831615; fax: +49 (0)5361 831602. E-mail address: p.mercorelli@fh-wolfsburg.de Mechatronics 19 (2009) 1334–1345 Contents lists available at ScienceDirect Mechatronics journal homepage: www.elsevier.com/locate/mechatronics