Interconnection of (Q,S,R)-Dissipative Systems in Discrete Time

TL;DR

This work develops a comprehensive dissipativity framework for discrete-time systems, showing that quadratic supply rates $s(y,u)=y^ op Q y + 2 y^ op S u + u^ op R u$ inherently capture virtual-output effects and enable direct, decentralized stability analysis of interconnected nonlinear subsystems with arbitrary topology. It provides two LMIs-based approaches for linear DT control—the primal and the dual—yielding controllers that make the closed-loop $(Q,S,R)$-dissipative, and extends these results to nonlinear subsystems with zero-state detectability through a global interconnection condition involving the interconnection matrix $H$. For networks of DT subsystems, the paper offers decentralized sufficient conditions (including Laplacian-based criteria) that guarantee network stability, and demonstrates the methodology on islanded microgrid models with 100 DGUs, quantifying the gains from DT dissipativity over naive discretization. The practical impact lies in enabling robust, energy-based design of large-scale, interconnected DT systems (e.g., microgrids, distributed energy resources) without sacrificing sparsity or requiring onerous bookkeeping of continuous-time passivity under discretization, thus supporting scalable, distributed control in digital implementations.

Abstract

Discrete-time systems cannot be passive unless there is a direct feedthrough from the input to the output. For passivity-based control to be exploited nevertheless, some authors introduce virtual outputs, while others rely on continuous-time passivity and then apply discretization techniques that preserve passivity in discrete time. Here we argue that quadratic supply rates incorporate and extend the effect of virtual outputs, allowing one to exploit dissipativity properties directly in discrete time. We derive decentralized (Q,S,R)-dissipativity conditions for a set of nonlinear systems interconnected with arbitrary topology, so that the overall network is guaranteed to be stable. For linear systems, we develop dissipative control conditions that are linear in the supply rate matrices. To demonstrate the validity of our methods, we provide numerical examples in the context of islanded microgrids.

Figures (7)

• Figure 1: Power balance for a DT system. When the internal accumulation is less than the energy supplied, the system is said to be dissipative.
• Figure 2: Block representation of a DT system (a) with feedthrough $y=h(x,u)$, (b) without feedthrough $y=h(x)$, and (c) with virtual output $z=y+\hat{R}u$. In the case (b), where $y$ does not depend directly on $u$, the system cannot be passive with respect to $y^\top u$. The system in configuration (c), instead, can be passive with respect to the virtual supply $z^\top u$.
• Figure 3: Model of a distributed generation unit (DGU), comprising an input voltage, a Buck converter and a local load, connected to the rest of the microgrid.
• Figure 4: Graph with $N=100$ nodes representing the microgrid model, randomly generated by using a preferential attachment mechanism.
• Figure 5: Eigenvalues distribution of (a) the CT system \ref{['microgridglobal']}, (b) the corresponding DT system discretized with forward Euler and stepsize $h=0.005$, and (c) the DT system equipped with our dissipative controllers and for different discretization stepsizes $h$.

Theorems & Definitions (31)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 1
• Lemma 4
• Lemma 5