Discrete-Time Control Design Based on Symplectic Integration: Linear Systems

Abstract

We propose a novel approach to design discrete-time state feedback controllers for sampled control systems with guaranteed stability under arbitrary sampling times that fulfill the Nyquist-Shannon condition. The key idea is borrowed from symplectic integration and backward error analysis of Hamiltonian systems: The closed-loop target system is the discretized version of a continuous-time system with appropriately shaped Hamiltonian, where a symplectic scheme is used for discretization. We adopt this argumentation for a systematic discrete-time design procedure for controllable linear systems based on the implicit midpoint rule. We motivate the approach on the example of two basic linear systems under zero order hold sampling, we show the construction of target systems with desired eigenvalues based on the discrete-time controller canonical form, and we illustrate the quality of the main result on a random sixth order system.