Control Of Systems With Time Varying Unilateral Constraints Using Robust And Optimal Controllers: A Homogeneity Framework

A tracking problem is studied in this article where a time-varying unilateral constraint and disturbances are present. An extension to the method of nonsmooth state transformation is developed. The proposed framework enables synthesis and analysis of control of a rigid body colliding with a time-varying unilateral constraint. Both the translational and rotational dynamics in three dimensions are covered. The control problem encompasses continuous and discontinuous controllers in the presence of impacts and discontinuous disturbances. This leads to new results in the area of control of systems with impacts and dry friction, a well-known challenging direction in nonsmooth mechanics. Major contribution of the article is in achieving finite-time tracking for a class of unilaterally constrained systems where the system dynamics with impacts are converted into one without impacts. Stability results are obtained via the homogeneity of differential inclusions. Numerical simulations are shown to demonstrate specific examples of the theoretical development.