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A New Approach to Multilinear Dynamical Systems and Control*

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A New Approach to Multilinear Dynamical Systems and Control* A New Approach to Multilinear Dynamical Systems and Control* Randy C. Hoover1, Kyle Caudle2 and Karen Braman2 Abstract—The current paper presents a new approach to mul- algorithms have played a central role in extending many of the tilinear dynamical systems analysis and control. The approach existing machine learning algorithms [1]–[12]. However, as is based upon recent developments in tensor decompositions their popularity has gained more traction over the last decade, and a newly defined algebra of circulants. In particular, it is shown that under the right tensor multiplication operator, a they have made their way into the dynamical systems and third order tensor can be written as a product of third order controls community as well [13]–[25]. While most applica- tensors that is analogous to a traditional matrix eigenvalue de- tions of Tucker/CP in the dynamical systems and controls composition where the “eigenvectors” become eigenmatrices and community revolve around the reduction of certain classes the “eigenvalues” become eigen-tuples. This new development of nonlinear systems to multilinear counterparts [15]–[21], allows for a proper tensor eigenvalue decomposition to be defined and has natural extension to linear systems theory through a [25], others have focused on time-series modeling [24], [26], tensor-exponential. Through this framework we extend many fuzzy inference [23], or identification/modeling of inverse of traditional techniques used in linear system theory to their dynamics [22]. multilinear counterpart. In the current paper, we describe a new approach to multilinear dynamical systems analysis and control through I. INTRODUCTION Fourier theory and an algebra of circulants as outlined in [27]– Traditional approaches to the analysis and control of lin- [31]. It is shown that under the right tensor multiplication ear time invariant (LTI) systems is well known and well operator, a third order tensor can be written as a product of understood. However as systems become increasingly com- third order tensors in which the left tensor is a collection plex, and multi-dimensional measurement devices become of eigenmatrices, the middle tensor is a front-face diagonal more commonplace, extensions from the linear system to a (denoted as f-diagonal) tensor of eigen-tuples, and the right multilinear system framework needs to be developed. While tensor is the tensor inverse of the eigenmatrices resulting there have been several approaches developed to investigate in a tensor-tensor eignevalue decomposition that is similar multilinear dynamical systems, most rely on decompositions to its matrix counterpart. Moreover, using the aformentioned revolving around either the Tucker or Canonical Decompo- decomposition, [32], [33] illustrates that a multilinear system sition/Parallel Factors (commonly referred to collectively as of ordinary differential equations (MODEs) can be effectively the CP decomposition). Tucker/CP provides a framework for solved via a tensor-version of the matrix exponential (referred decomposing a high order tensor into a collection of factor to as the t-exponential). matrices multiplying a “core tensor”. The structure of the Building on the work of [27]–[33], the contributions of core-tensor depends on which factorization strategy is being the current paper are four fold: (1) we extend the results used (Tucker produces a “dense” core whereas CP produces of [32], [33] to include the zero-state response to the mul- a “diagonal” core). Regardless of the decomposition being tilinear dynamical system in an effort to introduce multi- applied, both are regarded as form of higher order singular linear feedback control, (2) we develop a stability criterion value decomposition [1]–[4]. for the multilinear dynamical system to include exponential As a form of high-order singular value decomposition, convergence of system trajectories, (3) we introduce a new arXiv:2108.13583v1 [cs.LG] 31 Aug 2021 Tucker/CP algorithms have a natural fit within the machine approach to validate controllability of multilinear systems learning community where data naturally arises as two- using a block-Krylov subspace condition, and finally (4) we dimensional structures, e.g. digital image data. As such, these present a method to design multilinear state-feedback control using the developments of (1) - (3). *The current research was supported in part by the Department of the The remainder of this paper is organized as follows: In Navy, Naval Engineering Education Consortium under Grant No. (N00174- Section II we discuss the relevant tensor algebra and the 19-1-0014), the NASA Space Grant Consortium and the National Science Foundation under Grant No. (2007367). Any opinions, findings, and conclu- newly defined tensor multiplication operator. In Section III sions or recommendations expressed in this material are those of the authors we present the tensor-tensor eigenvalue decomposition and and do not necessarily reflect the views of the Naval Engineering Education show how it can be used to define functions on tensors Consortium, NASA or the National Science Foundation. 1Randy C. Hoover is with the department of Computer Sci- (namely the tensor exponential). In Section IV we provide ence and Engineering, South Dakota Mines, Rapid City, SD, USA several extensions of traditional linear systems theory to their [email protected] multilinear counterpart. Section V we provide an illustrative 2Kyle Caudle and Karen Braman the Department Math- ematics, South Dakota Mines, Rapid City, SD, USA example of the newly developed theory and finally, Section VI fkyle.caudle,karen.bramang@sdsmt,edu presents some discussion and provides some insight into future research directions. If A 2 R`×m×n with ` × m frontal slices then 2 A(1) A(n) A(n−1) :::A(2) 3 II. MATHEMATICAL FOUNDATIONS OF TENSORS (2) (1) (n) (3) 6 A A A :::A 7 6 7 bcirc(A) = 6 . 7 ; In the current section we discuss the mathematical founda- 6 . .. .. .. 7 tions of the tensor decompositions used in the current work. 4 5 (n) (n−1) .. (2) (1) While most of the theory in this section is outlined in [10], A A . A A [27]–[29], [32], [33], we summarize this theory here to keep is a block circulant matrix of size `n × mn. the current work self contained. We anchor the MatVec command to the frontal slices of The term tensor, as used in the context of this paper, refers the tensor. MatVec(A) takes an ` × m × n tensor and returns to a multi-dimensional array of numbers, sometimes called a block `n × m matrix `×m×n an n-way or n-mode array. If, for example, A 2 R 2 A(1) 3 then we say A is a third-order tensor where order is the (2) 6 A 7 number of ways or modes of the tensor. Thus, matrices and MatVec(A) = 6 7 : 6 . 7 vectors are second-order and first-order tensors, respectively. 4 . 5 Fundamental to the results presented in this paper is a recently A(n) defined multiplication operation on third-order tensors which The operation that takes MatVec(A) back to tensor form is itself produces a third-order tensor [27], [28]. the fold command: Further, it has been shown in [29] that under this multiplica- `×m×n fold(MatVec(A)) = A: tion operation, R is a free module over a commutative ring with unity where the “scalars” are R1×1×n tuples. In With these two operations in hand, we introduce the t- addition, it has been shown in [29] and [30] that all linear product between two, third-order tensors [27], [28]: transformations on the space R`×m×n can be represented by multiplication by a third-order tensor. Thus, even though Definition 2. Let A 2 R`×p×n and B 2 Rp×m×n be two third `×m×n R`×m×n is not strictly a vector space, many of the familiar order tensors. Then the t-product A ∗ B 2 R is defined tools of matrix linear algebra can be applied in this new as context, including the basic building blocks for dynamical A ∗ B = fold (bcirc(A) · MatVec(B)) : systems and control of multilinear systems. For a more in depth discussion on this topic, the reader is referred to [29]. Note that the tensor t-product enables the multiplication First, we review the basic definitions from [28] and [27] of two third order tensors via mod-n circular convolution. and introduce some basic notation. It will be convenient to Moreover, in general, the t-product of two tensors will not `×m×n break a tensor A in R up into various slices and tubal commute, with the exception in which ` = p = m = 1, i.e., th elements, and to have an indexing on those. The i lateral when the tensors are tubal-scalars. As a matter of illustration, th slice will be denoted Ai whereas the j frontal slice will be Example 1 details the application of the t-product on two (j) denoted A . In terms of MATLAB indexing notation, this third order tensors. (j) means Ai ≡ A(:; i; :) while A ≡ A(:; :; j). th We use the notation aik to denote the i; k tube inA; that Example 1: Suppose A 2 R`×p×3 and B 2 Rp×m×3. th (j) is aik = A(i; k; :). The j entry in that tube is aik . Indeed, Then these tubes have special meaning for us in the present work, 02 A(1) A(3) A(2) 3 2 B(1) 31 as they will play a role similar to scalars in . Thus, we make R A ∗ B = fold @4 A(2) A(1) A(3) 5 4 B(2) 5A : the following definition: A(3) A(2) A(1) B(3) 1×1×n Definition 1. An element c 2 is called a tubal-scalar m×m×n R Definition 3.
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