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Multilinear Control Systems Theory
- Source :
- SIAM Journal on Control and Optimization. 59:749-776
- Publication Year :
- 2021
- Publisher :
- Society for Industrial & Applied Mathematics (SIAM), 2021.
-
Abstract
- In this paper, we provide a system theoretic treatment of a new class of multilinear time-invariant (MLTI) systems in which the states, inputs and outputs are tensors, and the system evolution is governed by multilinear operators. The MLTI system representation is based on the Einstein product and even-order paired tensors. There is a particular tensor unfolding which gives rise to an isomorphism from this tensor space to the general linear group, i.e. the group of invertible matrices. By leveraging this unfolding operation, one can extend classical linear time-invariant (LTI) system notions including stability, reachability and observability to MLTI systems. While the unfolding based formulation is a powerful theoretical construct, the computational advantages of MLTI systems can only be fully realized while working with the tensor form, where hidden patterns/structures can be exploited for efficient representations and computations. Along these lines, we establish new results which enable one to express tensor unfolding based stability, reachability and observability criteria in terms of more standard notions of tensor ranks/decompositions. In addition, we develop a generalized CANDECOMP/PARAFAC decomposition and tensor train decomposition based model reduction framework, which can significantly reduce the number of MLTI system parameters. We demonstrate our framework with numerical examples.<br />27 pages, 2 figures, 3 tables, SIAM Journal on Control and Optimization, accepted to appear. arXiv admin note: text overlap with arXiv:1905.07427
- Subjects :
- Multilinear map
Control and Optimization
Applied Mathematics
Stability (learning theory)
Numerical Analysis (math.NA)
Systems and Control (eess.SY)
Electrical Engineering and Systems Science - Systems and Control
Algebra
Optimization and Control (math.OC)
Reachability
Control system
ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION
FOS: Mathematics
FOS: Electrical engineering, electronic engineering, information engineering
Mathematics - Numerical Analysis
Observability
Mathematics - Optimization and Control
Mathematics
Subjects
Details
- ISSN :
- 10957138 and 03630129
- Volume :
- 59
- Database :
- OpenAIRE
- Journal :
- SIAM Journal on Control and Optimization
- Accession number :
- edsair.doi.dedup.....8f631d9c7ae2b80d89d9c1ada8cc2f3d