Piecewise affine dynamical systems: well-posedness, controllability and stabilizability

PhD ceremony: Mr. Q.T. Le, 16.15 uur, Academiegebouw, Broerstraat 5, Groningen

Dissertation: Piecewise affine dynamical systems: well-posedness, controllability and stabilizability

Promotor(s): prof. A.J. van der Schaft

Faculty: Mathematics and Natural Sciences

In this thesis, we consider a subclass of hybrid systems, namely piecewise affine dynamical systems.

A piecewise affine dynamical system is a special type of finite-dimensional, nonlinear input/state/output systems with the distinguishing feature that the functions describing the system’s differential equations and output equations are piecewise affine functions. In the context of such systems, we study various fundamental system-theoretic problems: Non-Zenoness, well-posedness, controllability and stabilizability. Particularly, we show that continuous piecewise affine systems without inputs do not exhibit Zeno behavior.

This result opens new possibilities in studying controllability and stabilizability for continuous piecewise affine systems. Under a certain right invertibility assumption, we derive algebraic necessary and sufficient conditions, which are of Popov-Belevitch-Hautus type, for both controllability and stabilizability of continuous piecewise affine systems.

We also study well-posedness (i.e. existence and uniqueness of solutions) issue for bimodal discontinuous piecewise affine systems and provide a set of necessary and a set of sufficient conditions for existence and uniqueness of Filippov solutions. These conditions yield a number of stronger results for particular cases such as bimodal piecewise linear systems as well as in the context of Zeno behavior. In the context of bimodal piecewise linear systems with possibly discontinuous vector fields, we again study controllability and stabilizability issues. In the absence of continuity, well-posedness imposes certain geometric properties which, in turn, lead to algebraically verifiable necessary and sufficient conditions for both controllability and stabilizability.