LQG control of large networks: A clustering-based approach

Abstract

In this paper we present a set of projection-based designs for constructing simplified linear quadratic regulator (LQR) controllers for large-scale network systems. When such systems have tens of thousands of states, the design of conventional LQR controllers becomes numerically challenging, and their implementation requires a large number of communication links. Our proposed algorithms bypass these difficulties by clustering the system states using structural properties of its closed-loop transfer matrix. The assignment of clusters is defined through a structured projection matrix P , which leads to a significantly lower-dimensional LQR design. The reduced-order controller is finally projected back to the original coordinates via an inverse projection. The problem is, therefore, posed as a model matching problem of finding the optimal set of clusters or P that minimizes the H2-norm of the error between the transfer matrix of the full-order network with the full-order LQR and that with the projected LQR. We derive a tractable relaxation for this model matching problem, and design a P that solves the relaxation. The design is shown to be implementable by a convenient, hierarchical two-layer control architecture, requiring far less number of communication links than full-order LQR.Index Terms-Clustering, Large-scale networks, Projection, LQR, H2 performance.2) Reduced-order LQR design: We similarly project the LQR parameters byQ = P QP T ∈ R r×r , and letR = R such thatG :=BR −1B = P GP T ∈ R r×r . An LQR problem for the reduced-order model (6) is then posed as to minimizẽ J := ∞ 0

Synchronization and Aggregation of Nonlinear Power Systems with Consideration of Bus Network Structures

“…for any δ i ,δ i , and δ k , k = i, j. Choosingδ i = 0 and δ k = δ i , condition (10) follows. Takingδ i = 1 and δ k = δ i , we find condition (9). Lastly, with δ i − δ k = π 2 for some k = i, j and δ i − δ ℓ = 0 for ℓ = i, j, k, condition (11) follows for the chosen k.…”

Control Inversion: A Clustering-Based Method for Distributed Wide-Area Control of Power Systems

“…Therefore, by minimizing ξ the matching error g ǫ (s) − g ǫ (s) H2,ω bounded below can be made small as well, which then helps in attaining the stability ofĝ ǫ (s). Given that the derivation for (RL-1) follows from [7] except for a few discrepancies in proofs due to the different norm metric. For a complete understanding of how (RL-1) follows from (27), and the associated stability condition forĝ ǫ (s), we refer the reader to Appendix B.…”

“…Preliminary results on this design were presented in our recent paper [7] for a generic LTI system. The design in this paper, however, is different than that in [7] to suit the specialties of WAC.…”

“…For example, unlike [7] the definition of H 2 norm for solving the model matching here is only limited to a selected frequency range that targets the suppression of low-frequency oscillations (also known as inter-area oscillations) arising from the slow electro-mechanical dynamics of the synchronous generators. This distinction results in different relaxation and solution strategies than those reported in [7]. The second constraint arises from power balance between the generators as dictated by Kirchhoff's law.…”

Influence Models on Layered Uncertain Networks: A Guaranteed-Cost Design Perspective

“…In this work, we employ the LQ theory as a potential methodology to design state-feedback controllers for systems with layered structures. LQ methods have been applied in the literature for the control of large-scale systems and social networks [17], [18]. In general, the applicability of the LQ framework is reasonable when the resources used for social influence are restricted.…”