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\mathrm{S}\mathrm{I}\mathrm{A}\mathrm{M} \mathrm{J}. \mathrm{C}
\mathrm{O}\mathrm{N}\mathrm{T}\mathrm{R}\mathrm{O}\mathrm{L}
\mathrm{O}
\mathrm{P}\mathrm{T}\mathrm{I}\mathrm{M}
.
©
2023 \mathrm{S}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{t}\mathrm{y} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{I}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{A}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s}
\mathrm{V}\mathrm{o}\mathrm{l}. 61, \mathrm{N}\mathrm{o}. 3, \mathrm{p}\mathrm{p}. 1113--1135
NEAR-OPTIMAL DISTRIBUTED LINEAR-QUADRATIC
REGULATOR FOR NETWORKED SYSTEMS
*
\mathrm{S}\mathrm{U}\mathrm{N}\mathrm{G}\mathrm{H}\mathrm{O} \mathrm{S}\mathrm{H}\mathrm{I}\mathrm{N}
\dagger
, \mathrm{Y}\mathrm{I}\mathrm{H}\mathrm{E}\mathrm{N}\mathrm{G} \mathrm{L}\mathrm{I}\mathrm{N}
\ddagger
, \mathrm{G}\mathrm{U}\mathrm{A}\mathrm{N}\mathrm{N}\mathrm{A}\mathrm{N} \mathrm{Q}\mathrm{U}
\S
, \mathrm{A}\mathrm{D}\mathrm{A}\mathrm{M} \mathrm{W}\mathrm{I}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{N}
\ddagger
,
AND
\mathrm{M}\mathrm{I}\mathrm{H}\mathrm{A}\mathrm{I} \mathrm{A}\mathrm{N}\mathrm{I}\mathrm{T}\mathrm{E}\mathrm{S}\mathrm{C}\mathrm{U}
\P
\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft .
\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s} \mathrm{p}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r} \mathrm{s}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{e}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{e}-\mathrm{o}ff \mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{t}\mathrm{h}\mathrm{e}
\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{a} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r} \mathrm{i}\mathrm{n} \mathrm{a} \mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l} \mathrm{s}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}. \mathrm{W}\mathrm{e} \mathrm{s}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{y} \mathrm{a} \mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m} \mathrm{o}\mathrm{f}
\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{a}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s} \mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r} \mathrm{a} \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{a} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r}, \mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{d}
\kappa
-\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}, \mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}
\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{a}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s} \mathrm{m}\mathrm{a}\mathrm{k}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s} \mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{d} \mathrm{o}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e} \mathrm{i}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}
\kappa
\mathrm{o}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e}
\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}. \mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{a}\mathrm{n} \mathrm{t}\mathrm{u}\mathrm{n}\mathrm{e} \mathrm{i}\mathrm{t}\mathrm{s} \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}
\kappa
\mathrm{a}\mathrm{n}\mathrm{d}
\mathrm{t}\mathrm{h}\mathrm{u}\mathrm{s} \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{s} \mathrm{a} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p} \mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{n} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}. \mathrm{W}\mathrm{e} \mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}
\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r} \mathrm{m}\mathrm{i}\mathrm{l}\mathrm{d} \mathrm{a}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}, \mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}, \mathrm{d}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{a} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y} \mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}, \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{d}\mathrm{i}ff\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{n}
\kappa
-\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d} \mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l}
\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l} \mathrm{b}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{s} \mathrm{e}\mathrm{x}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y} \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l} \mathrm{i}\mathrm{n}
\kappa
. \mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{t} \mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l} \mathrm{c}\mathrm{a}\mathrm{n} \mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{e}\mathrm{v}\mathrm{e}
\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}-\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{a} \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{t}\mathrm{h}\mathrm{u}\mathrm{s} \mathrm{i}\mathrm{t} \mathrm{i}\mathrm{s} \mathrm{a}\mathrm{n} \mathrm{e}ff\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}
\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r} \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}-\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e} \mathrm{n}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{d} \mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{s}.
\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs .
\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}, \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}, \mathrm{n}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{d} \mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{s}
\bfM \bfS \bfC \bfc \bfo \bfd \bfe \bfs .
49\mathrm{N}10, 93\mathrm{A}14, 93\mathrm{B}70
\bfD \bfO \bfI .
10.1137/22\mathrm{M}1489836
1. Introduction.
Because of the increasing complexity of networked systems,
distributed control has gained substantial attention in the literature [17, 3, 30, 2, 13,
46]. Distributed control aims to design a set of local controllers that cooperatively
optimize the systemwide performance while having access only to local information
*
\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{y} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s} \mathrm{A}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{l} 12, 2022; \mathrm{a}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} (\mathrm{i}\mathrm{n} \mathrm{r}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{d} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}) \mathrm{N}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} 2,
2022; \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{d} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y} \mathrm{M}\mathrm{a}\mathrm{y} 11, 2023. \mathrm{G}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{L}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}: \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{m}\mathrm{a}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{t} \mathrm{h}\mathrm{a}\mathrm{s}
\mathrm{b}\mathrm{e}\mathrm{e}\mathrm{n} \mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{y} \mathrm{U}\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{o} \mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}, \mathrm{L}\mathrm{L}\mathrm{C}, \mathrm{O}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e} \mathrm{N}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{L}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y} (``\mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}"").
\mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}, \mathrm{a} \mathrm{U}.\mathrm{S}. \mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{O}ffi\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{S}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{l}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}, \mathrm{i}\mathrm{s} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r} \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t} \mathrm{N}\mathrm{o}.
\mathrm{D}\mathrm{E}\mathrm{A}\mathrm{C}02-06\mathrm{C}\mathrm{H}11357. \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{U}.\mathrm{S}. \mathrm{G}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{r}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{s} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s} \mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{o}\mathrm{n} \mathrm{i}\mathrm{t}\mathrm{s} \mathrm{b}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f}, \mathrm{a}
\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{p} \mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}, \mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{e} \mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e} \mathrm{i}\mathrm{n} \mathrm{s}\mathrm{a}\mathrm{i}\mathrm{d} \mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e} \mathrm{t}\mathrm{o} \mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}, \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}
\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{s}, \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{e}\mathrm{s} \mathrm{t}\mathrm{o} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c}, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{y} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{y}, \mathrm{b}\mathrm{y} \mathrm{o}\mathrm{r} \mathrm{o}\mathrm{n} \mathrm{b}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f} \mathrm{o}\mathrm{f}
\mathrm{t}\mathrm{h}\mathrm{e} \mathrm{G}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}. \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{w}\mathrm{i}\mathrm{l}\mathrm{l} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{e} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c} \mathrm{a}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{t}\mathrm{o} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{s} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}
\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h} \mathrm{i}\mathrm{n} \mathrm{a}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{D}\mathrm{O}\mathrm{E} \mathrm{P}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c} \mathrm{A}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}. \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{m}\mathrm{a}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{t} \mathrm{h}\mathrm{a}\mathrm{s}
\mathrm{b}\mathrm{e}\mathrm{e}\mathrm{n} \mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{y} \mathrm{U}\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{o} \mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}, \mathrm{L}\mathrm{L}\mathrm{C}, \mathrm{O}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e} \mathrm{N}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{L}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y} (""\mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}"").
\mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}, \mathrm{a} \mathrm{U}.\mathrm{S}. \mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{O}ffi\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{S}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{l}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}, \mathrm{i}\mathrm{s} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r} \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t} \mathrm{N}\mathrm{o}.
\mathrm{D}\mathrm{E}-\mathrm{A}\mathrm{C}02-06\mathrm{C}\mathrm{H}11357. \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{U}.\mathrm{S}. \mathrm{G}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{r}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{s} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s} \mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{o}\mathrm{n} \mathrm{i}\mathrm{t}\mathrm{s} \mathrm{b}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f}, \mathrm{a}
\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{d}-\mathrm{u}\mathrm{p} \mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}, \mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{e} \mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e} \mathrm{i}\mathrm{n} \mathrm{s}\mathrm{a}\mathrm{i}\mathrm{d} \mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e} \mathrm{t}\mathrm{o} \mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}, \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}
\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{s}, \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{e}\mathrm{s} \mathrm{t}\mathrm{o} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c}, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{y} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y} \mathrm{p}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{y}, \mathrm{b}\mathrm{y} \mathrm{o}\mathrm{r} \mathrm{o}\mathrm{n} \mathrm{b}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f} \mathrm{o}\mathrm{f}
\mathrm{t}\mathrm{h}\mathrm{e} \mathrm{G}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}.
\mathrm{h}\mathrm{t}\mathrm{t}\mathrm{p}\mathrm{s}://\mathrm{d}\mathrm{o}\mathrm{i}.\mathrm{o}\mathrm{r}\mathrm{g}/10.1137/22\mathrm{M}1489836
\bfF \bfu \bfn \bfd \bfi \bfn \bfg :
\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l} \mathrm{i}\mathrm{s} \mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{d} \mathrm{u}\mathrm{p}\mathrm{o}\mathrm{n} \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k} \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{y} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{U}.\mathrm{S}. \mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}, \mathrm{O}ffi\mathrm{c}\mathrm{e}
\mathrm{o}\mathrm{f} \mathrm{S}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}, \mathrm{O}ffi\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{A}\mathrm{d}\mathrm{v}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{d} \mathrm{S}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}fi\mathrm{c} \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{R}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h} (\mathrm{A}\mathrm{S}\mathrm{C}\mathrm{R}) \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r} \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t} \mathrm{D}\mathrm{E}\mathrm{A}\mathrm{C}02-
06\mathrm{C}\mathrm{H}11347.
\dagger
\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{S}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{D}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}, \mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e} \mathrm{N}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{L}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}, \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{t}, \mathrm{I}\mathrm{L} 60439
\mathrm{U}\mathrm{S}\mathrm{A} (\mathrm{s}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{n}@\mathrm{a}\mathrm{n}\mathrm{l}.\mathrm{g}\mathrm{o}\mathrm{v}).
\ddagger
\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{i}\mathrm{a} \mathrm{I}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{T}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{y}, \mathrm{P}\mathrm{a}\mathrm{s}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{a}, \mathrm{C}\mathrm{A} 91125 \mathrm{U}\mathrm{S}\mathrm{A} (\mathrm{y}\mathrm{i}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{l}@\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{h}.\mathrm{e}\mathrm{d}\mathrm{u},
\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{w}@\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{h}.\mathrm{e}\mathrm{d}\mathrm{u}).
\S
\mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{E}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{E}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}, \mathrm{C}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{e} \mathrm{M}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{n} \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}, \mathrm{P}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{s}\mathrm{b}\mathrm{u}\mathrm{r}\mathrm{g}\mathrm{h},
\mathrm{P}\mathrm{A} 15213 \mathrm{U}\mathrm{S}\mathrm{A} (\mathrm{g}\mathrm{q}\mathrm{u}@\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{r}\mathrm{e}\mathrm{w}.\mathrm{c}\mathrm{m}\mathrm{u}.\mathrm{e}\mathrm{d}\mathrm{u}).
\P
\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{S}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{D}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}, \mathrm{A}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e} \mathrm{N}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{L}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}, \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{t}, \mathrm{I}\mathrm{L}
60439 \mathrm{U}\mathrm{S}\mathrm{A}, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s}, \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{o}\mathrm{f} \mathrm{C}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{o}, \mathrm{C}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{o}, \mathrm{I}\mathrm{L} 60637 \mathrm{U}\mathrm{S}\mathrm{A}
(\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{u}@\mathrm{m}\mathrm{c}\mathrm{s}.\mathrm{a}\mathrm{n}\mathrm{l}.\mathrm{g}\mathrm{o}\mathrm{v}).
Copyright
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1113
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1114
\mathrm{S}\mathrm{H}\mathrm{I}\mathrm{N}, \mathrm{L}\mathrm{I}\mathrm{N}, \mathrm{Q}\mathrm{U}, \mathrm{W}\mathrm{I}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{N}, \mathrm{A}\mathrm{N}\mathrm{D} \mathrm{A}\mathrm{N}\mathrm{I}\mathrm{T}\mathrm{E}\mathrm{S}\mathrm{C}\mathrm{U}
within a prescribed range. This architecture has advantages over its centralized coun-
terpart in robustness (failure in a component does not cause system failure), com-
putation (online computation load is small), privacy (global information sharing is
not needed), and implementation (less and shorter communication). Furthermore,
it has advantages over its decentralized counterpart in performance (communication
between agents mitigates the impact of decentralization).
However, the synthesis of an optimal distributed controller is a challenging prob-
lem. In particular, the optimal distributed policy can be nonlinear even for a simple
stochastic linear-quadratic control setting [47], and synthesizing an optimal distrib-
uted controller is intractable in general [5]. Several works have studied making the
problem tractable by imposing additional structural assumptions, for example, nested
information structure [17], finite-dimensional linear policy [3, 26], quadratic invariance
[38], and sparsity invariance [13]. Other works have studied linear matrix inequality
conditions for the well-posedness, stability, and performance of distributed controllers
[9, 12, 20]. Additionally, recent works on system-level synthesis have proposed a tech-
nique to synthesize localized control policy in a scalable manner [1, 46]. Even for such
optimal distributed policies, however, the nominal performance can be significantly
worse than that of the centralized optimal policy because of the structural constraints
imposed on the local controllers [19].
An important gap in the literature between centralized optimal control and dis-
tributed controller synthesis is the limited understanding of how much performance
loss is incurred by decentralization. The general relationship between decentralization
and control performance has been studied in various contexts, such as decentralized
control [6, 19, 44], cooperative coverage problems [27], and multiagent optimization
[45], but the trade-off relationship between
the degree of decentralization
(determined
by the structural constraint imposed on the controller) and
the distributed controller's
performance
has not been reported in the literature. Since myriad possible communi-
cation structures for distributed control exist, without a guiding principle for trading
off the degree of decentralization against performance, it is difficult to appropriately
balance the level of decentralization and control performance. This situation moti-
vates an important open question:
Can one quantify the trade-off between decentral-
ization and performance?
We seek to address this question by analyzing the performance of a
\kappa
-distributed
linear-quadratic regulator
(LQR). This controller allows the user to tune the degree
of decentralization using the user-defined parameter
\kappa
. The system under study is
a linear system composed of agents interconnected over a graph, which appears in a
wide range of applications [2, 32] and has been considered in many works on distrib-
uted control [30, 46]. The graph allows us to construct a limited-range communication
structure based on the distance on the graph. In particular,
\kappa
-distributed LQR imple-
ments truncated state feedback, which considers only the agents within a prescribed
distance
\kappa
, while ignoring the agents beyond that distance (Figure 1). This controller
becomes more decentralized for small
\kappa
and less decentralized for large
\kappa
. Thus,
\kappa
-distributed LQR provides a ground for investigating the trade-off between decen-
tralization and performance. A similar decentralized overlapping control architecture
was studied in [43, Chapter 8]. In this direction, the suboptimality bounds that we
discuss here have been studied [43]. However, an explicit relationship between the
degree of decentralization and the suboptimality bound, as we do in this work, has
not been established.
In this work we show that the nominal performance of the
\kappa
-distributed policy,
measured by a quadratic performance criterion, becomes exponentially close to that
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by SIAM. Unauthorized reproduction of this article is prohibited.
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\mathrm{N}\mathrm{E}\mathrm{A}\mathrm{R}-\mathrm{O}\mathrm{P}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{A}\mathrm{L} \mathrm{D}\mathrm{I}\mathrm{S}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{B}\mathrm{U}\mathrm{T}\mathrm{E}\mathrm{D} \mathrm{L}\mathrm{Q}\mathrm{R} \mathrm{F}\mathrm{O}\mathrm{R} \mathrm{N}\mathrm{E}\mathrm{T}\mathrm{W}\mathrm{O}\mathrm{R}\mathrm{K}\mathrm{E}\mathrm{D} \mathrm{S}\mathrm{Y}\mathrm{S}\mathrm{T}\mathrm{E}\mathrm{M}\mathrm{S}
1115
i
\scrN
1
\scrG
[
i
]
\scrN
2
\scrG
[
i
]
j
\scrN
1
\scrG
[
j
]
\scrN
2
\scrG
[
j
]
Fig. 1.
Illustration of the communication structure of
\kappa
-distributed control for
\kappa
= 1
and
2
.
(Color available online.)
of the centralized optimal policy as
\kappa
is increased. Thus, one can indeed improve the
performance of the distributed controller by
enforcing a less restrictive information
exchange structure
. Moreover, the exponential relationship reveals that distributed
control can achieve
near-optimal performance
with a moderate degree of decentral-
ization. This result manifests the effectiveness of the distributed control architecture
for the control of large-scale networked systems. In addition, our result is obtained
without making restrictive structural assumptions; rather, it relies only on the stan-
dard assumptions in linear system theory (stabilizability and detectability) and a
mild assumption on the graph topology (subexponential growth condition). These
assumptions are substantially weaker than the structural assumptions employed by
prior works on near-optimal distributed control, such as decentrally optimizability
[50] and linear periodic controller [28]. Thus, our result can be applied to a wide
range of networked system control problems.
1.1. Contributions.
The main contribution of this work is the quantitative
characterization of the performance of
\kappa
-distributed LQR. We show that the optimal-
ity gap of
\kappa
-distributed LQR, compared with centralized optimal control, is exponen-
tially small in
\kappa
. Specifically, under several mild assumptions including stabilizability,
detectability, and subexponentially growing graph assumptions, we show that the
\kappa
-
distributed LQR is stabilizing for sufficiently large
\kappa
and that the optimality gap
decays exponentially in
\kappa
(Theorems 4.1 and 4.2). That is, one can achieve near-
optimal performance in the sense that the optimality gap can be made arbitrarily
small by choosing a sufficiently large
\kappa
. This result rigorously quantifies the trade-off
between performance and decentralization and, in turn, provides a guiding principle
for balancing the degree of decentralization and control performance.
The key technical result underlying our analysis is the exponential decay property
of the optimal gain (Theorem 3.3). By leveraging the state-of-the-art perturbation
bounds in graph-structured optimization [30, 35, 39], we show that the optimal gain
from one agent's state to another agent's control decays exponentially with respect
to the distance between the two agents. This result provides an intuitive explanation
for the near-optimality: As the gains from the agents more than
\kappa
hops away are
exponentially small in
\kappa
, truncating them does not cause significant performance loss.
This approach is novel in that, to the best of our knowledge, the exponential decay
property has not been used for the performance analysis of distributed control.
We complement the stability and performance results by establishing sufficient
conditions for the uniformity of the main assumptions (stabilizability and detectabil-
ity) in the system size (Theorem 5.3). These conditions in turn guarantee the uni-
formity of the stability and performance results in the system size. The established
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by SIAM. Unauthorized reproduction of this article is prohibited.
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1116
\mathrm{S}\mathrm{H}\mathrm{I}\mathrm{N}, \mathrm{L}\mathrm{I}\mathrm{N}, \mathrm{Q}\mathrm{U}, \mathrm{W}\mathrm{I}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{N}, \mathrm{A}\mathrm{N}\mathrm{D} \mathrm{A}\mathrm{N}\mathrm{I}\mathrm{T}\mathrm{E}\mathrm{S}\mathrm{C}\mathrm{U}
t
= 0
t
= 1
t
= 2
t
=
T
-
2
t
=
T
-
1
t
=
T
\cdot \cdot \cdot
t
= 0
t
= 1
t
=
T
-
1
t
=
T
(Dense Coupling)
\cdot \cdot \cdot
Fig. 2.
Illustration of the temporal graph (top) and space-time graph (bottom).
sufficient conditions allow us to apply our main results to an even broader class of sys-
tems whose satisfaction of uniform stabilizability and detectability conditions might
not be immediately clear. These situations may arise when only a subset of nodes
are controlled or only a subset of states are observed by the performance index. The
proposed sufficient conditions can serve as a design principle for large-scale systems
that facilitates the application of distributed control.
1.2. Related works.
Our main results are built on the exponential decay prop-
erty in the perturbation analysis of graph-structured optimization problems. This
property was first studied in early works on block-banded matrices [10], wherein the
authors showed the exponential decay of the entries of the inverse of band matrices.
Recent works [39, 42] have extended these results to analyze the sensitivity of the so-
lutions of graph-structured optimization problems. In the context of optimal control,
the exponential decay property on the time domain has been studied in [16, 24, 31,
48]. The connection between exponential decay in graph-structured optimization and
the exponential decay in optimal control problems has been made in [40]. However,
these results do not establish the exponential decay of optimal gain of LQR with
respect to the spatial distance (i.e., distance on the graph).
The derivation of exponential decay in LQR gain (Theorem 3.3) from the results
in [39] is nontrivial; several novel ideas are required. First, we consider a finite-horizon
LQR problem for a networked system as a graph-structured optimization problem in-
duced by a space-time graph (Figure 2, bottom). Second, we replace the terminal
cost with an infinite-horizon cost-to-go function to make the solution equal to the
infinite-horizon counterpart. This allows applying the result in [39] to analyze the
infinite-horizon LQR solution. Third, we establish the relationship between the uni-
form regularity conditions [39, Assumption 5.2--5.3] and system-theoretical properties
to express the decay constants in terms of constants related to system-theoretical
properties.
The exponential decay in the optimal LQR gain has been previously studied
in the work on
spatially decaying systems
[30]. The authors aimed to show that
for the continuous-time linear-quadratic optimal control setting and under several
technical assumptions, there is exponential decay in the optimal gain. Although
a counterexample was later reported [7], it was subsequently reported that some
weaker forms of the main theorem still hold true [8, 29]. These results differ from
ours in several ways. (i) Their results are asymptotic (the graph is infinite), whereas
ours are nonasymptotic (the graph is finite); the nonasymptotic and explicit nature
of our results allow the subsequent stability and performance analysis. (ii) Their
setting is continuous time, whereas ours is discrete. (iii) Their proof is based on
infinite-dimensional operator theory, whereas ours is based purely on linear algebra
in finite-dimensional spaces.
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by SIAM. Unauthorized reproduction of this article is prohibited.
Downloaded 07/05/23 to 132.174.253.218 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy
\mathrm{N}\mathrm{E}\mathrm{A}\mathrm{R}-\mathrm{O}\mathrm{P}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{A}\mathrm{L} \mathrm{D}\mathrm{I}\mathrm{S}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{B}\mathrm{U}\mathrm{T}\mathrm{E}\mathrm{D} \mathrm{L}\mathrm{Q}\mathrm{R} \mathrm{F}\mathrm{O}\mathrm{R} \mathrm{N}\mathrm{E}\mathrm{T}\mathrm{W}\mathrm{O}\mathrm{R}\mathrm{K}\mathrm{E}\mathrm{D} \mathrm{S}\mathrm{Y}\mathrm{S}\mathrm{T}\mathrm{E}\mathrm{M}\mathrm{S}
1117
Beyond the LQR setting, the exponential decay property has been studied in
various other contexts, including combinatorial optimization [14], multiagent rein-
forcement learning [25, 34, 35], and statistical physics [4, 23], and is known to lead
to effective distributed algorithm designs based on the truncation of long-range in-
teractions [14, 35]. While sharing some similar flavors, the analysis technique in our
work deals with continuous variables, which is different from the techniques in [4, 14,
23, 25, 34, 35] that deal with discrete variables. Moreover, the exponential decay in
discrete variable settings (in particular, the Markov decision process) derives from
discount factors [35] or bounded total variation [34], whereas the exponential decay
in this paper derives from the regularity of the problem.
1.3. Notation.
The set of real numbers and the set of integers are denoted by
\BbbR
and
\BbbI
, respectively. We define
\BbbI
A
:=
\BbbI
\cap
A
,
\BbbI
>
0
:=
\BbbI
(0
,
\infty
)
, and
\BbbI
\geq
0
:=
\BbbI
[0
,
\infty
)
. We use the
syntax [
M
1
;
\cdot \cdot \cdot
;
M
n
] := [
M
\top
1
\cdot \cdot \cdot
M
\top
n
]
\top
;
\{
M
i
\}
i
\in \scrI
:= [
M
i
1
;
\cdot \cdot \cdot
;
M
i
m
];
\{
M
ij
\}
i
\in \scrI
,j
\in \scrJ
:=
\{ \{
M
\top
ij
\}
\top
j
\in \scrJ
\}
i
\in \scrI
for strictly ordered sets
\scrI
=
\{
i
1
<
\cdot \cdot \cdot
< i
m
\}
and
\scrJ
=
\{
j
1
<
\cdot \cdot \cdot
< j
n
\}
.
Furthermore,
v
[
i
] is the
i
th component of vector
v
;
M
[
i,j
] is the (
i,j
)th component
of matrix
M
;
v
[
\scrI
] :=
\{
v
[
i
]
\}
i
\in \scrI
; and
M
[
\scrI
,
\scrJ
] :=
\{
M
[
i,j
]
\}
i
\in \scrI
,j
\in \scrJ
. Vector 2-norms and
induced 2-norms of matrices are denoted by
\| \cdot \|
. For matrices
A
and
B
,
A
\succ
(
\succeq
)
B
indicates that
A
-
B
is positive (semi)definite. The largest and smallest eigenvalues
of symmetric matrices are denoted by
\lambda
(
\cdot
)
,\lambda
(
\cdot
). The identity matrix is denoted by
\bfitI
,
and the zero matrix or vector is denoted by
0
. A graph
\scrG
= (
\scrV
,
\scrE
) is a pair of a node
set
\scrV
and an edge set
\scrE \subseteq \{ \{
i,j
\} \subseteq \scrV
:
i
\not
=
j
\}
. The closed neighborhood of
i
\in \scrV
on
graph
\scrG
is denoted by
\scrN
\scrG
[
i
] :=
\{
i
\} \cup \{
j
\in \scrV
:
\{
i,j
\} \in \scrE \}
. The distance between
i
and
j
on graph
\scrG
= (
\scrV
,
\scrE
), denoted by
d
\scrG
(
i,j
), is the number of edges in the shortest path
connecting
i
and
j
.
2. Problem setting.
We study a networked system that can be expressed by a
discrete-time, linear, time-invariant system over a graph:
x
i
(
t
+ 1) =
\sum
j
\in \scrN
\scrG
[
i
]
(
A
ij
x
j
(
t
) +
B
ij
u
j
(
t
))
\forall
i
\in \scrV
, t
\in
\BbbI
\geq
0
.
(2.1a)
Here
\scrG
:= (
\scrV
:=
\BbbI
[1
,N
]
,
\scrE
) is a graph;
x
i
(
t
)
\in
\BbbR
n
x
i
and
u
i
(
t
)
\in
\BbbR
n
u
i
are the state and
control at node
i
\in \scrV
and time index
t
\in
\BbbI
\geq
0
;
A
ij
\in
\BbbR
n
x
i
\times
n
x
j
; and
B
ij
\in
\BbbR
n
x
i
\times
n
u
j
.
The stage performance index at node
i
is given by a quadratic function
\ell
i
(
x
i
,u
i
) := (1
/
2)
x
\top
i
\left(
\sum
j
\in \scrN
\scrG
[
i
]
Q
ij
x
j
\right)
+ (1
/
2)
u
\top
i
\left(
\sum
j
\in \scrN
\scrG
[
i
]
R
ij
u
j
\right)
\forall
i
\in \scrV
,
(2.1b)
where
Q
ij
\in
\BbbR
n
x
i
\times
n
x
j
;
R
ij
\in
\BbbR
n
u
i
\times
n
u
j
; and
Q
ij
=
Q
\top
ji
and
R
ij
=
R
\top
ji
for all
i,j
\in \scrV
.
That is, the system is composed of multiple agents, interconnected over graph
\scrG
, and
the internodal couplings are made through the nonzero
A
ij
,
B
ij
,
Q
ij
, and
R
ij
.
2.1. Centralized LQR.
The centralized LQR problem formulation for the net-
worked system in (2.1a) is as follows:
min
\{
\bfitx
(
t
)
\}
\infty
t
=0
,
\{
\bfitu
(
t
)
\}
\infty
t
=0
\infty
\sum
t
=0
(1
/
2)
\bfitx
(
t
)
\top
\bfitQ \bfitx
(
t
) + (1
/
2)
\bfitu
(
t
)
\top
\bfitR \bfitu
(
t
)(2.2a)
s.t.
\bfitx
(0) =
\bfitx
,
(2.2b)
\bfitx
(
t
+ 1) =
\bfitA \bfitx
(
t
) +
\bfitB \bfitu
(
t
)
\forall
t
= 0
,
1
,....
(2.2c)
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1118
\mathrm{S}\mathrm{H}\mathrm{I}\mathrm{N}, \mathrm{L}\mathrm{I}\mathrm{N}, \mathrm{Q}\mathrm{U}, \mathrm{W}\mathrm{I}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{N}, \mathrm{A}\mathrm{N}\mathrm{D} \mathrm{A}\mathrm{N}\mathrm{I}\mathrm{T}\mathrm{E}\mathrm{S}\mathrm{C}\mathrm{U}
Here we let
\bfitx
(
t
) :=
\{
x
i
(
t
)
\}
i
\in \scrV
,
\bfitu
(
t
) :=
\{
u
i
(
t
)
\}
i
\in \scrV
,
\bfitA
:=
\{
A
ij
\}
i,j
\in \scrV
,
\bfitB
:=
\{
B
ij
\}
i,j
\in \scrV
,
\bfitQ
:=
\{
Q
ij
\}
i,j
\in \scrV
,
\bfitR
:=
\{
R
ij
\}
i,j
\in \scrV
(for convenience,
A
ij
:=
0
,
B
ij
:=
0
,
Q
ij
:=
0
, and
R
ij
:=
0
for
\{
i,j
\}
/
\in \scrE
). Note that (2.2a) is the summation of the nodal performance
index (2.1b) over the infinite horizon and (2.2b) is the concatenation of the nodal
system dynamics (2.1a) into that of the full system. It is well known that under
\bfitQ
\succeq
0,
\bfitR
\succ
0, (
\bfitA
,
\bfitB
)-stabilizability, and (
\bfitA
,
\bfitQ
1
/
2
)-detectability, the optimal policy
\bfitpi
\star
(
\cdot
) for (2.2) can be expressed by a linear state feedback law
\bfitpi
\star
(
\bfitx
) :=
-
\bfitK
\star
\bfitx
, where
\bfitK
\star
=
\{
K
\star
ij
\}
i,j
\in \scrV
is the optimal gain matrix [22, Theorem 2.4-2]. We define the state
transition mapping under the optimal gain by
\bfitphi
\star
(
\bfitx
) :=
\Phi
\star
\bfitx
, where
\Phi
\star
:=
\bfitA
-
\bfitB \bfitK
\star
.
2.2.
\bfitkappa
-distributed control.
The proposed
\kappa
-distributed control policy
\bfitpi
\kappa
(
\cdot
) is
defined as
\bfitpi
\kappa
(
\bfitx
) :=
-
\bfitK
\kappa
\bfitx
, where the optimal gain
\bfitK
\star
is replaced by the
\kappa
-truncated
gain
\bfitK
\kappa
:=
\{
K
\kappa
ij
\}
i,j
\in \scrV
with
K
\kappa
ij
:=
K
\star
ij
if
d
\scrG
(
i,j
)
\leq
\kappa
and
0
otherwise. That is, the
\kappa
-
distributed control employs a limited-range state feedback law by taking into account
only the state information within the
\kappa
-hop neighborhood
\scrN
\kappa
\scrG
[
i
] :=
\{
j
\in \scrV
:
d
\scrG
(
i,j
)
\leq
\kappa
\}
(Figure 1). The
\kappa
-distributed control reduces to a purely decentralized control if
\kappa
= 0, in the sense that the local controllers only have access to the state information
of themselves, and reduces to a purely centralized control if
\kappa
\geq
max
i,j
\in \scrV
d
\scrG
(
i,j
), in
the sense that the local controllers have access to the full state information. The state
transition mapping is defined by
\bfitphi
\kappa
(
\bfitx
) :=
\Phi
\kappa
\bfitx
, where
\Phi
\kappa
:=
\bfitA
-
\bfitB \bfitK
\kappa
. In summary,
the nodal feedback law of centralized and
\kappa
-distributed LQR can be expressed as
follows:
u
i
(
t
) =
\sum
j
\in \scrV
-
K
\star
ij
x
j
(
t
) (centralized);
u
i
(
t
) =
\sum
j
\in \scrN
\kappa
\scrG
[
i
]
-
K
\star
ij
x
j
(
t
) (
\kappa
-distributed)
.
That is,
\kappa
-distributed LQR is a truncated version of centralized LQR.
2.3. Examples.
We now discuss two motivating examples. We use these exam-
ples to demonstrate the theoretical developments in the subsequent sections. Besides
the examples below, the networked system control problems also appear in various
consensus problems for robotic networks (e.g., attitude alignment, flocking, coordi-
nated decision making) [2, 36] as well as the operation of storage networks [21].
Example
2.1 (building temperature control [32]). Heating, ventilation, and air
conditioning (HVAC) systems in large buildings with multiple zones can be modeled
as networked systems. The control goal is to maintain the temperature of each zone at
the desired setpoint while dealing with disturbances. We assume that all the constant
disturbances (e.g., heat generation, effect of ambient conditions) are eliminated by
expressing the state/control variables as deviation variables around the desired steady
state; thus, the control goal becomes driving the system to the origin. The model and
nodal performance index are as follows:
\.
U
i
=
T
i
,
\.
T
i
=
-
\sum
j
\in \scrN
\scrG
[
i
]
k
ij
(
T
i
-
T
j
) +
\eta
1
u
i
, \ell
i
(
U
i
,T
i
,u
i
) =
\eta
2
2
U
2
i
+
\eta
2
3
T
2
i
+
u
2
i
,
where
T
i
is the temperature of zone
i
;
U
i
is its integrator of
T
i
;
\ell
i
(
\cdot
) is the stage perfor-
mance index of zone
i
;
k
ij
characterizes the degree of coupling (heat exchange) between
zones
i
and
j
;
u
i
is the manipulated heat generation/absorption of zone
i
; and
\eta
1
,\eta
2
,\eta
3
are the constants we will use later. The integrator is not necessary for the nominal
setting, but it is used in practice to deal with the offsets caused by disturbances.
The explicit Euler discretization with
x
i
:= [
U
i
;
T
i
] and time interval \Delta
t
yields
A
ii
:=
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\mathrm{N}\mathrm{E}\mathrm{A}\mathrm{R}-\mathrm{O}\mathrm{P}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{A}\mathrm{L} \mathrm{D}\mathrm{I}\mathrm{S}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{B}\mathrm{U}\mathrm{T}\mathrm{E}\mathrm{D} \mathrm{L}\mathrm{Q}\mathrm{R} \mathrm{F}\mathrm{O}\mathrm{R} \mathrm{N}\mathrm{E}\mathrm{T}\mathrm{W}\mathrm{O}\mathrm{R}\mathrm{K}\mathrm{E}\mathrm{D} \mathrm{S}\mathrm{Y}\mathrm{S}\mathrm{T}\mathrm{E}\mathrm{M}\mathrm{S}
1119
\biggl[
1
\Delta
t
0 1
-
(\Delta
t
)
\sum
j
\in \scrN
\scrG
[
i
]
k
ij
\biggr]
,
A
ij
:=
\biggl[
0
0
0 (\Delta
t
)
k
ij
\biggr]
,
B
ii
:=
\biggl[
0
\eta
1
(\Delta
t
)
\biggr]
,
Q
ii
:=
\biggl[
\eta
2
2
\eta
2
3
\biggr]
,
and
R
ii
:= 1 (off-diagonal blocks of
\bfitB
,
\bfitQ
,
\bfitR
are zero). Here we aim to design a dis-
tributed proportional-integral controller of the form
u
i
=
\sum
j
\in \scrN
\kappa
\scrG
[
i
]
-
(
K
U
ij
U
j
+
K
T
ij
T
j
),
where
K
U
ij
and
K
T
ij
are the gains.
Example
2.2 (frequency control of power systems [2]). The frequency control
problem in AC power networks can be expressed as a networked system control prob-
lem. Here we consider a DC-approximated model, which assumes that the voltage
magnitudes are constant, the phase angle differences between adjacent buses are small,
and the network is dominantly inductive. The control goal is to minimize the deviation
of frequencies from the common reference frequency while dealing with constant dis-
turbances. We assume that all the constant disturbances (e.g., loads) are eliminated
by expressing the state/control variables as deviation variables around the desired
steady state. The model and nodal performance indexes are as follows:
\.
\theta
i
=
\omega
i
,
\.
\omega
i
=
-
\sum
j
\in \scrN
\scrG
[
i
]
k
ij
(
\theta
i
-
\theta
j
) +
\eta
1
u
i
, \ell
i
(
\theta
i
,\omega
i
,u
i
) =
\eta
2
2
\theta
2
i
+
\eta
2
3
\omega
2
i
+
u
2
i
,
where
\theta
i
is the phase angle of bus
i
(relative to a moving reference frame),
\omega
i
is
the frequency of bus
i
,
\ell
i
(
\cdot
) is the stage performance index of bus
i
,
k
ij
is the line
susceptance, multiplied by the voltage magnitudes on both ends and divided by in-
ertia of the generator,
u
i
is the power injection, and
\eta
1
,\eta
2
,\eta
3
are the constants we
will use later. The explicit Euler discretization with
x
i
:= [
\theta
i
;
\omega
i
] and time interval
\Delta
t
yields
A
ii
:=
\biggl[
1
\Delta
t
-
(\Delta
t
)
\sum
j
\in \scrN
\scrG
[
i
]
k
ij
1
\biggr]
,
A
ij
:=
\biggl[
0
0
(\Delta
t
)
k
ij
0
\biggr]
,
B
ii
:=
\biggl[
0
\eta
1
(\Delta
t
)
\biggr]
,
Q
ii
:=
\biggl[
\eta
2
2
\eta
2
3
\biggr]
, and
R
ii
:= 1 (off-diagonal blocks of
\bfitB
,
\bfitQ
,
\bfitR
are zero). Here we
aim to design a distributed controller of the form
u
i
=
\sum
j
\in \scrN
\kappa
\scrG
[
i
]
-
(
K
\theta
ij
\theta
j
+
K
\omega
ij
\omega
j
),
where
K
\theta
ij
and
K
\omega
ij
are the gains. Typically, the phase angle cannot be obtained by
simply integrating the frequency but must be measured via a phasor measurement
unit [33].
3. Exponential decay in optimal gain.
A natural requirement for the
\kappa
-
truncated gain
\bfitK
\kappa
to be a good approximation of the optimal gain
\bfitK
\star
is that the
block entries
K
\star
ij
for
i,j
\in \scrV
such that
d
\scrG
(
i,j
)
>\kappa
are sufficiently small. In this section
we show that under
\bfitQ
\succeq
0,
\bfitR
\succ
0, (
\bfitA
,
\bfitB
)-stabilizability, and (
\bfitA
,
\bfitQ
1
/
2
)-detectability,
the optimal gain
K
\star
ij
from
x
j
to
u
i
decays exponentially in the distance
d
\scrG
(
i,j
) be-
tween
i
and
j
. We first define three basic concepts: stability, stabilizability, and
detectability. Although these are commonly used concepts in the control literature,
we restate them here to explicitly introduce the associated parameters.
Definition 3.1.
We define the following for
L>
0
and
\alpha
\in
[0
,
1)
:
(a)
\Phi
is
(
L,\alpha
)
-stable if
\|
\Phi
t
\| \leq
L\alpha
t
for
t
\in
\BbbI
\geq
0
.
(b) (
\bfitA
,
\bfitB
)
is
(
L,\alpha
)
-stabilizable if
\exists
\bfitK
such that
\|
\bfitK
\| \leq
L
and
\bfitA
-
\bfitB
\bfitK
is
(
L,\alpha
)
-
stable.
(c) (
\bfitA
,
\bfitC
)
is
(
L,\alpha
)
-detectable if
(
\bfitA
\top
,
\bfitC
\top
)
is
(
L,\alpha
)
-stabilizable.
Note that Definition 3.1 is not altering the standard definitions. For example,
the standard definition of stability (in discrete-time setting) is
sr
(
\Phi
)
<
1, where
sr
(
\cdot
) denotes the spectral radius. By [18, Theorem 5.6.12 and Corollary 5.6.13],
there always exist
L >
0 and
\alpha
\in
(0
,
1) such that
\|
\Phi
t
\| \leq
L\alpha
t
if and only if
\Phi
is
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1120
\mathrm{S}\mathrm{H}\mathrm{I}\mathrm{N}, \mathrm{L}\mathrm{I}\mathrm{N}, \mathrm{Q}\mathrm{U}, \mathrm{W}\mathrm{I}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{N}, \mathrm{A}\mathrm{N}\mathrm{D} \mathrm{A}\mathrm{N}\mathrm{I}\mathrm{T}\mathrm{E}\mathrm{S}\mathrm{C}\mathrm{U}
stable. Similarly, the standard definition of (
\bfitA
,
\bfitB
)-stabilizability is the existence of
\bfitK
such that
sr
(
\bfitA
-
\bfitB
\bfitK
)
<
1, and there always exist
L >
0 and
\alpha
\in
(0
,
1) such that
\|
\bfitK
\| \leq
L
and
\|
(
\bfitA
-
\bfitB
\bfitK
)
t
\| \leq
L\alpha
t
. We call
\bfitK
in Definition 3.1(b) (
L,\alpha
)-stabilizing
feedback gain for (
\bfitA
,
\bfitB
). Note that stabilizability and detectability are relaxations
of controllability and observability. We now formally introduce the main assumption.
Assumption
3.2. There exist
L>
1,
\alpha
\in
(0
,
1), and
\gamma
\in
(0
,
1) such that
(a)
\|
\bfitA
\|
,
\|
\bfitB
\|
,
\|
\bfitQ
\|
,
\|
\bfitR
\| \leq
L
;
(b)
\bfitR
\succeq
\gamma
\bfitI
;
(c) (
\bfitA
,
\bfitB
) is (
L,\alpha
)-stabilizable;
(d)
\bfitQ
\succeq
0
, and (
\bfitA
,
\bfitQ
1
/
2
) is (
L,\alpha
)-detectable.
Here we use common parameters (e.g.,
\|
\bfitA
\| \leq
L
,
\|
\bfitB
\| \leq
L
,
\|
\bfitK
\| \leq
L
,
...
) rather
than introducing parameters for each bound (e.g.,
\|
\bfitA
\| \leq
L
\bfitA
,
\|
\bfitB
\| \leq
L
\bfitB
,
\|
\bfitK
\| \leq
L
\bfitK
,
...
) to reduce the notational burden. Note that we consistently use
L
for upper
bounds,
\gamma
for strictly positive lower bounds, and
\alpha
for the upper bounds strictly less
than 1. We assume
L >
1,
\gamma
\in
(0
,
1), and
\alpha
\in
(0
,
1), rather than
L,\gamma ,\alpha >
0 to derive
the results in a simple form. Note that Assumption 3.2 is a standard assumption
used in the classical LQR literature. We are now ready to state the exponential decay
result.
Theorem 3.3.
Under Assumption
3.2
,
\|
K
\star
ij
\| \leq
\Upsilon
\rho
d
\scrG
(
i,j
)
for
i,j
\in \scrV
, where
\gamma
\bfitF
:=
(1
-
\alpha
)
2
L
2
(1 +
L
)
2
, \gamma
\bfitG
:=
(1
-
\alpha
)
2
\gamma
2
L
4
(1 +
L
)
2
, L
\bfitP
:=
L
3
(1 +
L
2
)
1
-
\alpha
2
,
(3.1a)
\mu
:= (2
L
2
\bfitH
/\gamma
\bfitG
+
\gamma
\bfitG
+
L
\bfitH
)
/\gamma
\bfitF
,
(3.1b)
\gamma
\bfitH
:=
\biggl(
2
\gamma
\bfitG
+
\biggl(
1 +
4
L
\bfitH
\gamma
\bfitG
+
4
L
2
\bfitH
\gamma
2
\bfitG
\biggr)
L
\bfitH
(1 +
\mu L
\bfitH
)
\gamma
\bfitF
+
\mu
\biggr)
-
1
,
(3.1c)
L
\bfitH
:= max(2
L
+ 1
,L
\bfitP
+ 1)
, \rho
:=
\biggl(
L
2
\bfitH
-
\gamma
2
\bfitH
L
2
\bfitH
+
\gamma
2
\bfitH
\biggr)
1
/
2
,
\Upsilon :=
L
\bfitH
/\gamma
2
\bfitH
\rho .
(3.1d)
Furthermore,
\rho
\in
(0
,
1)
, and
\Upsilon
\geq
1
.
The proof is in Appendix B.1. The proof of Theorem 3.3 relies on the exponential
decay property of the inverse of graph-induced banded matrices (Appendix A). To
apply this property, we derive an equivalent finite-horizon formulation for (2.2) by
using the infinite-horizon cost-to-go function, which can be obtained by the solution
of the discrete-time algebraic Riccati equation (DARE). An important property of
the equivalent formulation is that it is
graph-structured
, in the sense that the sparsity
structure of the Karush--Kuhn--Tucker (KKT) matrix is induced by the space-time
graph (Figure 2, bottom). Then, we observe that the optimal gain
\bfitK
\star
can be obtained
as a submatrix of the inverse of the graph-induced banded KKT matrix, which we
show satisfies the exponential decay property. In other words, the block component
of the inverse of the KKT matrix decays exponentially with respect to the distance on
the space-time graph, and the decay rate depends on the singular values of the KKT
matrix, which we show are uniformly upper and lower bounded by using Assumption
3.2. This exponential decay property in the inverse of the KKT matrix directly leads
to the exponential decay in the optimal gain and finishes the proof of Theorem 3.3.
Based on Theorem 3.3, we now can show that the
\kappa
-truncated gain
\bfitK
\kappa
is an
exponentially accurate approximation of the optimal gain
\bfitK
\star
. We additionally require
the subexponentially growing graph assumption.
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\mathrm{N}\mathrm{E}\mathrm{A}\mathrm{R}-\mathrm{O}\mathrm{P}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{A}\mathrm{L} \mathrm{D}\mathrm{I}\mathrm{S}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{B}\mathrm{U}\mathrm{T}\mathrm{E}\mathrm{D} \mathrm{L}\mathrm{Q}\mathrm{R} \mathrm{F}\mathrm{O}\mathrm{R} \mathrm{N}\mathrm{E}\mathrm{T}\mathrm{W}\mathrm{O}\mathrm{R}\mathrm{K}\mathrm{E}\mathrm{D} \mathrm{S}\mathrm{Y}\mathrm{S}\mathrm{T}\mathrm{E}\mathrm{M}\mathrm{S}
1121
Assumption
3.4. There exists a subexponential function
p
(
\cdot
) such that
| \{
j
\in \scrV
:
d
\scrG
(
i,j
) =
d
\} | \leq
p
(
d
)
\forall
i
\in \scrV
.
Under Assumption 3.4, one can derive the following corollary of Theorem 3.3.
Corollary 3.5.
Under Assumptions
3.2
and
3.4
,
\|
\bfitK
\star
-
\bfitK
\kappa
\| \leq
\Psi
\delta
\kappa
, where
\delta
:= (
\rho
+ 1)
/
2
,
\Psi :=
\biggl(
sup
d
\in
\BbbI
\geq
0
p
(
d
)(
\rho /\delta
)
d
\biggr)
\Upsilon
\delta /
(1
-
\delta
)
,
(3.2)
and
\rho ,
\Upsilon
are defined in
(3.1)
. Moreover,
\delta
\in
(0
,
1)
.
The proof is in Appendix B.2. One can show that even if there are multiple
nodes more than
\kappa
hops away, their accumulated effect is still exponentially small
in
\kappa
, since the exponential decay in magnitude (Theorem 3.3) is stronger than the
subexponential increase in their number (Assumption 3.4). The supremum in (3.2)
is bounded, since the product of a subexponential function and an exponentially
decaying function converges.
The results in this section establish the fundamental basis for the stability and
performance analysis in the next section. Intuitively, Theorem 3.3 suggests that the
far-ranging interactions are negligible, so neglecting them in the control policy does
not significantly degrade the control performance. In addition, since we have the
explicit bounds for the difference from the optimal policy (Corollary 3.5), the result
can be directly applied to analyze the stability and performance in the next section.
Remark
3.6. Our exponential decay result is different from that in [29, 30], where
the authors have aimed to establish various
spatial decay
conditions in an
asymptotic
sense. In the particular case of exponential decay, the authors aimed to establish that
there exists
\rho
\in
(0
,
1) such that
\sum
j
\in \scrV
\rho
-
d
\scrG
(
i,j
)
\|
K
\star
ij
\|
<
+
\infty
for all
i
\in \scrV
(originally
presented in [30, Theorem 6] and corrected in [29, Modified Version of Theorem 6]),
where
\scrV
is an infinite node set. A key limitation is that one cannot obtain the
exponentially decaying upper bound in the full operator space; that is,
\|
\bfitK
\star
-
\bfitK
\kappa
\|
cannot be bounded as in Corollary 3.5, which serves as a crucial intermediate result
for the stability and performance analysis.
Remark
3.7. Since Theorem 3.3 establishes exponential decay in the
distance
on
the graph, the results can be complemented by showing the uniformity of the pa-
rameters
L,\gamma ,\alpha
in the system size
N
(the number of nodes). Typically, additional
conditions are necessary to guarantee such uniformity because otherwise these param-
eters may tend to infinity or zero as the size of the system grows. Thus, sufficient
conditions for the uniformity of Assumption 3.2 are of interest. Addressing this issue
is important, but we postpone the discussion on this matter to section 5 in order not
to bury the main points under the technical details.
Remark
3.8. Note that for any given finite graph, one can find
p
(
\cdot
) that satisfies
Assumption 3.4. However, if we consider a
family of systems
(and associated graphs)
and want to obtain the bounds \Psi and
\delta
that apply uniformly to each system, we need
p
(
\cdot
) to be uniform as well, and obtaining such
p
(
\cdot
) may be impossible for certain cases.
For example, consider a family of systems obtained by subgraphs of an infinite regular
tree. For these systems, the number of nodes in distance
d
can grow exponentially
with
d
; thus, we cannot find a uniform subexponential function
p
(
\cdot
). One of the
sufficient conditions for the existence of such a uniform
p
(
\cdot
) is that the graphs are
obtained as subgraphs of a mesh in a finite-dimensional space. For instance, if the
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1122
\mathrm{S}\mathrm{H}\mathrm{I}\mathrm{N}, \mathrm{L}\mathrm{I}\mathrm{N}, \mathrm{Q}\mathrm{U}, \mathrm{W}\mathrm{I}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{N}, \mathrm{A}\mathrm{N}\mathrm{D} \mathrm{A}\mathrm{N}\mathrm{I}\mathrm{T}\mathrm{E}\mathrm{S}\mathrm{C}\mathrm{U}
parent graph is a 1-dimensional mesh, the number of nodes in distance
d
is always not
greater than 2. In a more general case of
D
-dimensional mesh, the number of nodes
in distance
d
is
O
(
d
D
-
1
); thus, one can always find
p
(
\cdot
) satisfying Assumption 3.4.
Remark
3.9. We now discuss the intuitive interpretation of Theorem 3.3. One
possible misinterpretation is that the sparsity in (2.1) causes the delay in the prop-
agation of perturbations, and it causes exponential decay. This ignores that even if
the sparsity in the system causes the delay in propagation, the initial condition of any
node has effects on every agent's decision at time 0 (assuming the graph is connected).
This follows from the fact that the optimal decision from (2.2) takes into account the
anticipated propagation of the perturbations from distant agents and takes proactive
actions; mathematically this can be seen from the fact that the inverse of the KKT
matrix, which maps changes in data to changes in the optimal control, is dense. While
this proactive action can be arbitrarily large in principle, Theorem 3.3 establishes that
the magnitude of such optimal proactive action decays exponentially as long as As-
sumption 3.2 holds. This indicates that the exponential decay does not derive from
the delayed propagation of perturbations but rather comes from the regularity of
the problem (imposed by Assumption 3.2), which naturally damps the magnitude of
proactive control actions against the effects of distant agents.
Remark
3.10. An interesting open question is whether comparable results hold
for continuous-domain (continuous-time or space) optimal control problems. The
temporal exponential decay for continuous-domain problems was recently established
[15, 16], but the spatial counterpart has not been reported. Studying continuous-
domain problems as limiting cases of discrete-domain problems is nontrivial because
the uniform regularity does not hold when refining the discretization mesh size. We
leave the analysis of continuous-domain problems as a topic of future work.
Remark
3.11. Since many practical control problems have nonlinearity or con-
straints, establishing comparable results for nonlinear, constrained settings is of in-
terest. Generalizing our results to a nonlinear setting can be done by applying the
classical perturbation analysis results for nonlinear programs and generalized equa-
tions [11, 31, 37, 39], and analyzing constrained problems can be done via active set
analysis [41, 48, 49]. However, the perturbation analysis for nonlinear, constrained
problems is local in nature, and strong assumptions are necessary to obtain the per-
turbation bound over a neighborhood of interest. We leave the analysis of nonlinear,
constrained settings as a topic of future work.
4. Stability and performance analysis.
We now establish the exponential
stability and the near optimality of
\kappa
-distributed control. Stability is a prerequisite
for the performance analysis because if the system is unstable, the performance index
trivially becomes +
\infty
. The following theorem establishes that under Assumptions 3.2
and 3.4 and sufficiently large
\kappa
, the state transition matrix
\Phi
\kappa
induced by
\bfitpi
\kappa
(
\cdot
) is
exponentially stable.
Theorem 4.1.
Under Assumptions
3.2
and
3.4
and
\kappa
\geq
\kappa
,
\Phi
\kappa
is
(\Omega
,\beta
)
-stable,
where
\beta
:=
\bigl(
1
-
(1
-
\rho
2
)
/
2\Upsilon
2
\bigr)
1
/
2
,
\Omega :=
\bigl(
\Upsilon
2
/
(1
-
\rho
2
)
\bigr)
1
/
2
,
(4.1a)
\kappa
:= log
\biggl(
1
-
\rho
2
2\Upsilon
2
\Psi
L
(
L
\Psi + 2
L
(1 +
L
\bfitP
L
2
/\gamma
))
\biggr)
/
log
\delta ,
(4.1b)
and
\Upsilon
,
\rho
,
L
\bfitP
,
\delta
, and
\Psi
are defined in
(3.1)
and
(3.2)
. Furthermore,
\beta
\in
(0
,
1)
.
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