On Several Composite Quadratic Lyapunov Functions for Switched Systems

Hu, Tingshu; Ma, Lixiang; Lin, Zongli

doi:10.1109/cdc.2006.377788

Cited by 16 publications

(13 citation statements)

References 21 publications

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“…[32], [13], [15], [36], [38], [35], [37], [30], [29], [20]. Perhaps one of the earliest references is the work on "piecewise quadratic Lyapunov functions" in [32].…”

Section: Contributions and Organizationmentioning

confidence: 99%

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

Ahmadi¹,

Jungers²,

Parrilo³

et al. 2014

SIAM J. Control Optim.

105

133

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Abstract. We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratic, and maximum/minimumof-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We derive approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs. This provides worst-case perfomance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

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“…[32], [13], [15], [36], [38], [35], [37], [30], [29], [20]. Perhaps one of the earliest references is the work on "piecewise quadratic Lyapunov functions" in [32].…”

Section: Contributions and Organizationmentioning

confidence: 99%

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

Ahmadi¹,

Jungers²,

Parrilo³

et al. 2014

SIAM J. Control Optim.

105

133

View full text Add to dashboard Cite

show abstract

“…V2(x) satisfy the inequalities of any of the graphs in (e), (f), and (h), then min{V1(x), V2(x)} is a common Lyapunov function. These arguments serve as alternative proofs of stability and in the case where V1 and V2 are quadratic functions, they correspond to the works in [13], [12], [10], [14]. The next two corollaries prove these statements in a more general setting.…”

Section: |N |} Is a Piecewise Lyapunov Function Associated With Gmentioning

confidence: 99%

“….e, the pointwise maximum, the pointwise minimum, and the convex envelope of a set of quadratic functions [13], [12], [10], [14]. These works are mostly done in continuous time for analysis of linear differential inclusions, but they have obvious discrete time counterparts.…”

Section: Contributions and Organizationmentioning

confidence: 99%

“…The converse is not true. The works in [13], [12], [10], [14] have additional multipliers in (15) and (16) [17] on "path-dependent quadratic Lyapunov functions", and the work in [8] When we have so many different ways of imposing conditions for stability, it is natural to ask which ones are better. This seems to be a hard question in general, but we have studied in detail all the path-complete graphs with two nodes that imply stability for the case of a switched system with two matrices.…”

Section: V1(x) Andmentioning

confidence: 99%

“…[15], [7], [13], [12], [10]. Perhaps one of the earliest references is the work on "piecewise quadratic Lyapunov functions" in [15].…”

Section: Contributions and Organizationmentioning

confidence: 99%

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