Comparison of Linear Systems Across Time Domains: Continuous-time vs. Discrete-time

We develop a formal framework for the behavioral comparison of linear systems across different time domains. We accomplish this by introducing the notion of system interpolation, which determines whether the input-state trajectories of a continuous-time system can be realized as piecewise polynomial interpolations of the input-state trajectories of a discrete-time system. In this context, a piecewise polynomial interpolation of a discrete-time signal is characterized as a continuous-time function that coincides with the discrete-time signal at given sampling instants and can be realized as a polynomial of a prescribed degree over intervals between these instants. By representing piecewise polynomial functions as linear combinations of shifted Legendre polynomials, we characterize system interpolation as a subspace inclusion that is completely in terms of system parameters. This therefore allows for a computationally efficient comparison of the input-state behavior of a continuous-time system with that of a discrete-time one. We then exploit this characterization to discretize a given continuous-time system into a discrete-time one. Lastly, given a control specification, we exploit system interpolation to synthesize controllers that ensure satisfaction at each given sampling instant, while they measure the extent of (possible) violation over intervals between these instants.