Controllability and Observability in Linear Time-Variable Networks With Arbitrary Symmetry Groups

Abstract

This paper presents a unified treatment of linear time‐variable networks displaying arbitrary geometrical symmetries by incorporating group theory into an analysis scheme. Symmetric networks have their elements arranged so that certain permutations of the network edges result in a configuration which is identical with the original. These permutations lead to a group of monomial matrices which are shown to commute with the network A‐matrix and the state transition matrix of the normal form equation. The representation theory of groups facilitates the study of those network properties which are determined solely by symmetry. By using group theory, a simple arithmetic condition is derived which, when satisfied, implies that the network is noncontrollable or nonobservable because of symmetry alone. The results allow the determination by inspection of linear combinations of the original state variables which result in noncontrollable variables. It is shown that networks displaying axial point group symmetry are generally only weakly controllable.

Effects of symmetry on the structural controllability of neural networks: A perspective

“…For linear systems containing group symmetries, Rubin and Meadows [4] used a similarity transform T to change the coordinates of the n -dimensional system (1) to an orthogonal basis defined by the group action of the symmetry on ℝ n (called the symmetry basis). Furthermore, Ref.…”

“…Furthermore, Ref. [4] demonstrated how group representation theory [5] is used to construct the symmetry basis for a symmetric group from the irreducible representations of the group which transforms the system matrix A into block diagonal form. In some cases the type of symmetry would cause the network to be non-controllable due to symmetries (termed NCS), evident by inspection of the structure of the transformed system.…”

“…Structural controllability [3] did not explicitly cover symmetry, so for any structurally controllable pair ( A, B ) that contains no dilations or isolated nodes, the presence of symmetry could still cause the network to be NCS (as shown in [4]), as the act of transforming the network to the symmetry basis would redefine the structure to one that is un-controllable. These two theorems together paint a more complete picture of controllability than either alone as shown in [8], where both are used in concert to explain and understand why certain neural networks were not controllable from particular inputs.…”

“…In [4] the similarity transformation that takes the system into the coordinates of the symmetry basis is defined from the irreducible representations Γ ( p ) ( R ) of the group symmetry present in the network. The number of irreducible representations p = 1 … k can be determined from the character of the representation χ ( R ).…”

“…Finally, the similarity transform T that takes the system (1) into the coordinates of the symmetry basis is defined in [4] as a projection of the irreducible representations onto the space of the system (1) on ℝ n : $$G_i^{false(pfalse)}=true\sum _R{\mathrm{\Gamma }}^{false(pfalse)}{false(Rfalse)}_{\mathit{ii}}^{\ast }Dfalse(Rfalse)$$ where $G_i^{false(pfalse)}$ generates basis vectors on ℝ n from the p th irreducible representation Γ ( p ) , * indicates the complex conjugate, and i indicates the ( i, i )-th entry of the irreducible representation for i = 1 … l p . Once $G_i^{false(pfalse)}$ is computed for each p and i , the similarity transform is constructed from the normalized linearly independent vectors that span ℝ n .…”