Imbalance in torque generation leads to engine speed fluctuation. To improve the idle engine speed performance, the torque balancing control problem is addressed in this paper for multi-cylinder SI engines. To evaluate cylinder-to-cylinder imbalance, the average torque in ignition-event scale is introduced as controlled output, which enables a feedback control to be performed without measurement of instantaneous torque, and the individual spark advances are chosen as control inputs. A linear discrete time model with single input and single output is proposed to represent the dynamics of the imbalance, where a sequentially switching function is introduced to describe the spark advance signals delivered to each cylinder and the differences in torque generation caused by the individual cylinder characteristics are equivalently modelled as unknown offset in the inputs. An estimation algorithm with the proof of convergence is presented to provide on-line estimation of the unknown offset under the passivity assumption of the system. Furthermore, a feedback control law which combines the unknown offset estimation and the model predictive control is proposed. Finally, the unknown offset estimation and the feedback control approach are validated based on the experimental results carried out on a six-cylinder gasoline engine test bench.

The axial offset joint has two rotating axes that do not intersect but have a specific offset in space. It is used widely in parallel manipulators (PMs). The offset-joint workspace can directly affect the PM workspace. This study performed a theoretical derivation and workspace analysis of a class of axial offset joints. First, a theoretical parametric model describing the rotation range of the offset joint is established that considers the interference of the offset joint because of the contact between the upper- and lower-joint brackets during movement. Second, the analytical expressions of the offset-joint workspace are formulated based on the coordinate system transformation. The offset-joint workspace is theoretically calculated in this study using formulations. Then, through a comparative analysis, the superiority of the offset joint compared with the universal joint is verified. The theoretical formulations in this paper can be used to calculate the workspace of a class of axial offset joints. Finally, based on a workspace analysis of three types of PMs using offset, universal, and spherical joints, the offset-joint PM workspace is much larger than those of the other two types.