Discrete-time negative imaginary control systems using zero-order hold sampling

Abstract

We develop a discrete-time negative imaginary (NI) framework that reflects how continuous-time NI systems behave under zero-order hold (ZOH) sampling, which is commonly used in digital control implementations. Continuous-time NI theory is built around an analog interconnection involving the instantaneous relationship between inputs and the derivatives of the outputs. After ZOH sampling, however, the controller interacts only with piecewise-constant inputs and sampled outputs. This leads to a fundamentally different input–output structure, which requires a discrete-time notion of NI specifically tied to ZOH sampling. We introduce such a notion, called the ZOH-NI property, and establish a nonlinear feedback stability result showing that a ZOH-NI plant can be asymptotically stabilized using a step-advanced weakly state strictly negative imaginary controller under suitable assumptions. For linear time-invariant systems, we provide necessary and sufficient conditions for the ZOH-NI property in both state-space and frequency-domain forms. In the linear case, the stability conditions reduce to a DC-gain condition. The proposed control framework is illustrated using a nonlinear example.