Discrete-time Negative Imaginary Systems from ZOH Sampling

TL;DR

Addresses digital control of negative imaginary systems by defining a discrete-time NI (DT-NI) property aligned with zero-order hold (ZOH) sampling. Proposes a nonlinear time-domain DT-NI framework, proves that ZOH sampling of a continuous-time NI system yields DT-NI, and develops Lyapunov-based interconnection stability results for NI plants with SAOSNI/SANI and OSNI controllers; in the linear case, stability reduces to the DC loop-gain condition $\lambda_{\max}(G(1)H(1))<1$ and applicable LMI tests. Specializations yield necessary-and-sufficient LMIs for linear DT-NI via a quadratic storage $V(x)=\frac12 x^T P x$ and related DC-domain criteria, along with frequency-domain characterizations. The work contrasts with bilinear-transform discretizations by providing a time-domain dissipativity framework and direct discretization links, offering a practical toolkit for the digital control of NI systems in engineering applications such as flexible structures and nano-positioning. Overall, it delivers a robust, theory-grounded approach to ensure stability of DT-NI interconnections under ZOH sampling and linear-case verifications via simple DC tests.

Abstract

A new definition of discrete-time negative imaginary (NI) systems is provided. This definition characterizes the dissipative property of a zero-order hold sampled continuous-time NI system. Under some assumptions, asymptotic stability can be guaranteed for the closed-loop interconnection of an NI system and an output strictly negative imaginary system, with one of them having a one step advance. In the case of linear systems, we also provide necessary and sufficient frequency-domain and LMI conditions under which the definition is satisfied. Also provided is a simple DC gain condition for the stability results in the linear case.

Figures (4)

• Figure 1: Closed-loop interconnection of the NI plant $H_1$ and the SAOSNI controller $\widetilde{H}_2$.
• Figure 2: Open-loop interconnection of the NI plant $H_1$ and the SAOSNI controller $\widetilde{H}_2$.
• Figure 3: Closed-loop interconnection of the SANI controller $\overline H_1$ and the OSNI plant $H_2$.
• Figure 4: Open-loop interconnection of the SANI plant $\overline H_1$ and the OSNI controller $H_2$.

Theorems & Definitions (16)

• Definition 1: discrete-time NI systems
• Definition 2: continuous-time NI systems
• Lemma 1
• Definition 3: continuous-time NI systems
• Remark 1
• Definition 4: discrete-time OSNI systems
• Definition 5
• Theorem 1
• Definition 6
• Theorem 2