Structure of finite dimensional exact estimation algebra on state dimension 3 and linear rank 2*. Issue 2 (1st February 2023)

• Record Type: Journal Article
• Authors: Jiao, Xiaopei; Yau, Stephen S.-T.
• Is Part Of: International journal of control. Volume 96:Issue 2(2023)
• Page Start: 362
• Page End: 373
• Publication Date: 2023-02-01
• Subjects: Finite dimensional filters -- exact estimation algebra -- nonmaximal rank -- estimate; Automatic control -- Periodicals; Electronic journals; 629.8
• DOI: 10.1080/00207179.2021.1992673
• Languages: English
• ISSNs: 0020-7179

Abstract

Abstract : The estimation algebra plays an important role in classification of finite dimensional filters. When finite dimensional estimation algebra has maximal rank, Yau et al. [Yau (2003). Complete classification of finite-dimensional estimation algebras of maximal rank. International Journal of Control, 76 (7), 657–677; Yau & Hu (2005). Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state-space dimension and Mitter conjecture. International Journal of Control, 78 (10), 689–705.] have proved that η must be a degree 2 polynomial. In this paper, we study the structure of finite dimensional exact estimation algebra with state dimension 3 and rank 2. We establish a sufficient and necessary condition for estimation algebra with nonmaximal rank to be finite dimensional. Importantly, in the new filtering system, η needs not to be a degree 2 polynomial and can be of any degree 4 n 1 + 2, n 1 ∈ Z + . It is the first time to systematically analyse nonmaximal rank exact estimation algebra in which η is a polynomial of any degree 4 n 1 + 2, n 1 ∈ Z + . For Riccati-type equation, estimates have been done from the viewpoints of both classical solution and weak solution respectively. Finally, finite dimensional filters of Benés type are constructed successfully.