Modal Strong Structural Controllability: Graph-Based Analysis in LTI Systems

Abstract

This paper introduces a new concept of modal strong structural controllability in linear time-invariant (LTI) systems. An eigenvalue of a system matrix is considered controllable if it can be directly influenced by the control inputs. We focus on an arbitrary set Δ⊆C and define a specific family of systems as modal strongly structurally controllable with respect to Δ if, for all systems in this family, every λ∈Δ is a controllable eigenvalue. In this family of LTI systems, the zero/nonzero/arbitrary pattern of system matrices is known, along with additional information about the spectrum of the subsystems, derived from their physical properties. To establish controllability conditions, we define a Δ-graph and use a coloring process to relate the set of control subsystems to zero forcing sets. We define efficient Δ, and for networks of one-dimensional subsystems, we prove that the graph-theoretic condition is both necessary and sufficient when Δ is efficient. Moreover, for cases where Δ is not efficient, we establish conditions under which it can be partitioned into efficient subsets. Furthermore, we demonstrate how our approach can derive existing results on strong structural controllability when Δ={0} or Δ=C∖{0}.

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