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

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta \subseteq \mathbb {C}$</tex-math></inline-formula> and define a specific family of systems as modal strongly structurally controllable with respect to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta$</tex-math></inline-formula> if, for all systems in this family, every <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lambda \in \Delta$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta$</tex-math></inline-formula>-graph and use a coloring process to relate the set of control subsystems to zero forcing sets. We define efficient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \Delta$</tex-math></inline-formula>, and for networks of one-dimensional subsystems, we prove that the graph-theoretic condition is both necessary and sufficient when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \Delta$</tex-math></inline-formula> is efficient. Moreover, for cases where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \Delta$</tex-math></inline-formula> is not efficient, we establish conditions under which it can be partitioned into efficient subsets. Furthermore, we demonstrate how our approach can derive existing results on strong structural controllability when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta =\lbrace 0\rbrace$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta =\mathbb {C}\setminus \lbrace 0\rbrace$</tex-math></inline-formula>.