Controllability and Observability of Linear Quaternion-valued Systems
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Controllability and Observability of Linear Quaternion-valued Systems Bang Xin JIANG
Yang LIU
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, P. R. China E-mail : [email protected] [email protected]
Kit Ian KOU1) Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, P. R. China E-mail : [email protected]
Zhen WANG College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China E-mail : [email protected] Abstract The aim of this paper is to define an extension of the controllability and observability for linear quaternion-valued systems (QVS). Some criteria for controllability and observability are derived, and the minimum norm control and duality theorem are also investigated. Compared with real-valued or complex-valued linear systems, it is shown that the classical Caylay–Hamilton Theorem as well as Popov–Belevitch–Hautus (PBH) type controllability and observability test do not hold for linear QVS. Hence, a modified PBH type necessary condition is studied for the controllability and observability, respectively. Finally, some examples are given to illustrate the effectiveness of the obtained results. Keywords
Linear system, controllability, observability, quaternion
MR(2010) Subject Classification
1
34A30, 93B05, 93B07, 93C05
Introduction
Controllability and observability are significant and useful in controlled systems, and most of the systems in practice are supposed to be completely controllable and observable. The concepts of controllability and observability were given by R. E. Kalman in 1961 ([17]). The relationship between controllability and observability is formalized by means of the principle of duality and contributions. These concepts of the optimal control theory are also provided by R. E. Kalman Received April 18, 2018, revised March 2, 2020, accepted June 5, 2020 Supported by the Natural Science Foundation of Zhejiang Province of China (Grant Nos. LR20F030001 and LD19A010001), by the National Natural Science Foundation of China (Grant No. 11671361), by the University of Macau (Grant No. MYRG2019-00039-FST), and by the Science and Technology Development Fund, Macau SAR (Grant No. FDCT/085/2018/A2) 1) Corresponding author
Jiang B. X. et al.
1300
(see [17] and [16]). By this motivation and inspiration, the controllability and observability have been studied for various kinds of systems, such as distributed parameter control systems ([38]), neuronal networks ([35]), complex networks ([30]), impulsive systems ([26]) and Boolean networks ([20, 28, 29]) etc. Moreover, the improved controllability and the relevant properties of networked systems have been well investigated in [13, 23, 34, 36, 39]. Quaternions were introduced by Hamilton as an extension of complex numbers and as an instrument for manipulating 3-dimensional vectors ([9, 10]). However
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