Oblique Projected Dynamical Systems and Incremental Stability Under State Constraints

Projected dynamical systems (PDS) are discontinuous dynamical systems obtained by projecting a vector field on the tangent cone of a given constraint set. As such, PDS provide a convenient formalism to model constrained dynamical systems. When dealing with vector fields, which satisfy certain monotonicity properties, but not necessarily with respect to usual Euclidean norm, the resulting PDS does not necessarily inherit this monotonicity, as we will show. However, we demonstrate that if the projection is carried out with respect to a well-chosen norm, then the resulting "oblique PDS" preserves the monotonicity of the unconstrained dynamics. This feature is especially desirable as monotonicity allows to guarantee important (incremental) stability properties and stability of periodic solutions (under periodic excitation). These properties can now be guaranteed based on the unconstrained dynamics using "smart" projection instead of having to carry out a difficult a posteriori analysis on a constrained discontinuous dynamical system. To illustrate this, an application in the context of observer re-design is presented, which guarantees that the state estimate lies in the same state set as the observed state trajectory.