Algebraic Riccati Tensor Equations with Applications in Multilinear Control Systems
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911064833130496 |
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| author | Wang, Yuchao Wei, Yimin Zhang, Guofeng Chang, Shih Yu |
| author_facet | Wang, Yuchao Wei, Yimin Zhang, Guofeng Chang, Shih Yu |
| contents | In a recent paper by Chen et al. [8], the authors initiated the control-theoretic study of a class of discrete-time multilinear time-invariant (MLTI) control systems, where system states, inputs, and outputs are all tensors endowed with the Einstein product. They established criteria for fundamental system-theoretic notions such as stability, reachability, and observability through tensor decomposition. Building on this new research direction, the purpose of our paper is to extend the study to continuous-time MLTI control systems. Specifically, we define Hamiltonian tensors and symplectic tensors, and we establish the Schur-Hamiltonian tensor decomposition and the symplectic tensor singular value decomposition (SVD). Based on these concepts, we propose the algebraic Riccati tensor equation (ARTE) and demonstrate that it has a unique positive semidefinite solution if the system is stabilizable and detectable. To find numerical solutions to the ARTE, we introduce a tensor-based Newton method. Additionally, we establish the tensor versions of the bounded real lemma and the small gain theorem. A first-order robustness analysis of the ARTE is also conducted. Finally, we provide a numerical example to illustrate the proposed theory and algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13491 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic Riccati Tensor Equations with Applications in Multilinear Control Systems Wang, Yuchao Wei, Yimin Zhang, Guofeng Chang, Shih Yu Optimization and Control 15A69, 93B35, 93C05, 93D15 In a recent paper by Chen et al. [8], the authors initiated the control-theoretic study of a class of discrete-time multilinear time-invariant (MLTI) control systems, where system states, inputs, and outputs are all tensors endowed with the Einstein product. They established criteria for fundamental system-theoretic notions such as stability, reachability, and observability through tensor decomposition. Building on this new research direction, the purpose of our paper is to extend the study to continuous-time MLTI control systems. Specifically, we define Hamiltonian tensors and symplectic tensors, and we establish the Schur-Hamiltonian tensor decomposition and the symplectic tensor singular value decomposition (SVD). Based on these concepts, we propose the algebraic Riccati tensor equation (ARTE) and demonstrate that it has a unique positive semidefinite solution if the system is stabilizable and detectable. To find numerical solutions to the ARTE, we introduce a tensor-based Newton method. Additionally, we establish the tensor versions of the bounded real lemma and the small gain theorem. A first-order robustness analysis of the ARTE is also conducted. Finally, we provide a numerical example to illustrate the proposed theory and algorithms. |
| title | Algebraic Riccati Tensor Equations with Applications in Multilinear Control Systems |
| topic | Optimization and Control 15A69, 93B35, 93C05, 93D15 |
| url | https://arxiv.org/abs/2402.13491 |