A tensor phase theory with applications in multilinear control

The purpose of this paper is to initiate a phase theory for tensors under the Einstein product, and explore its applications in multilinear control systems. Firstly, the sectorial tensor decomposition for sectorial tensors is derived, which allows us to define phases for sectorial tensors. A numerical procedure for computing phases of a sectorial tensor is also proposed. Secondly, the maximin and minimax expressions for tensor phases are given, which are used to quantify how close the phases of a sectorial tensor are to those of its compressions. Thirdly, the compound spectrum, compound numerical ranges and compound angular numerical ranges of two sectorial tensors $A,B$ are defined and characterized in terms of the compound numerical ranges and compound angular numerical ranges of the sectorial tensors $A,B$. Fourthly, it is shown that the angles of eigenvalues of the product of two sectorial tensors are upper bounded by the sum of their individual phases. Finally, based on the tensor phase theory developed above, a tensor version of the small phase theorem is presented, which can be regarded as a natural generalization of the matrix case, recently proposed in Ref. [10]. The results offer powerful new tools for the stability and robustness analysis of multilinear feedback control systems.

💡 Research Summary

This paper pioneers a comprehensive phase theory for tensors defined under the Einstein product and demonstrates its significant applications in the analysis of multilinear control systems. The work bridges a critical gap in multilinear algebra by extending the well-established concept of matrix phase, central to linear system analysis, to the higher-order tensor domain.

The foundation is laid by defining the numerical range for even-order square tensors. A tensor is termed sectorial if its numerical range does not contain the origin, analogous to sectorial matrices. The first major theoretical contribution is the Sectorial Tensor Decomposition Theorem (Theorem 3.1), which proves that any sectorial tensor A can be decomposed as A = Q^H * D * Q, where Q is nonsingular and D is a unitary diagonal tensor. This decomposition is fundamental, as it allows for the definition of tensor phases: the phases of a sectorial tensor A are defined as the arguments (angles) of the complex numbers on the diagonal of D (Definition 3.4). A numerical algorithm for computing these phases is also provided.

Subsequent sections delve into characterizing the properties of these tensor phases. Key results include establishing minimax and maximax expressions for the phases (Lemma 3.6), which are then used to analyze how the phases of a tensor relate to the phases of its compressions onto subspaces (Theorems 3.3, 3.4). The paper further explores concepts like the compound spectrum, compound numerical range, and compound angular numerical range for pairs of tensors, linking them to the individual tensors’ properties (Theorems 3.5, 3.6). An important bound is established in Theorem 3.7, showing that the angles of the eigenvalues of the product of two sectorial tensors are bounded above by the sum of their individual phases, a generalization of the matrix multiplicative phase property.

The culmination of the theoretical development is the Tensor Small Phase Theorem (Theorem 4.1), presented in the application section. This theorem provides necessary and sufficient conditions for the internal stability of a multilinear feedback system composed of two stable multilinear components. The condition is phrased in terms of the phases of the open-loop transfer tensors, stating that the closed-loop system is stable if and only if the sum of their phases remains within a specific interval (