Fractional circuits have attracted extensive attention of scholars and researchers for their superior performance and potential applications. Fractional circuits constitute a new challenge for the analysis and synthesis methods of traditional circuits theory. Passivity is the fundamental property of traditional circuits (integer order electric circuits). As is known to all, passivity is equivalent to positive realness in traditional linear circuits. However, this equivalence is broken down by introducing fractional elements into electrical networks in s-domain. To address this issue, on the basis of s-W transformation, we study the passive criteria of fractional circuits with rational order elements in this paper. Definitions of positive-real (matrix) function in W-domain are given, and the equivalence conditions of positive realness are derived. In addition, a conclusion is proposed in which the immittance (matrix) function of passive fractional circuits with rational order elements is positive real in Wdomain. The applications of passive criteria in circuit synthesis are shown. and can be found in the literature, 16,17 such as the Hamiltonian matrix method and the frequency sweeping method. If a model of a passive system is passive, it is then known that the model itself cannot result in unstable simulations. 18 When described by the fractional expression, the immittance of passive fractional circuits is with passive property. Analysis of passivity is vital for the in-depth research of fractional circuits theory, fractional modeling and inherent characteristics of electrical materials, etc. In traditional linear network theory, the passivity of an electrical circuit or a network is equivalent to the positive-real property of the immittance functions or matrices. By using the positive-real criterion, we can judge whether a network is passive or not. However, this equivalence is broken down by introducing fractional elements into an electrical circuit or a network. Such as the "positiveness"of s α is satisfied while the "realness" is not with the order α = 0.5; hence, the positive-real criterion is not suitable anymore. Therefore, it is necessary to explore the passive equivalence conditions of fractional circuits, which lay the foundation for the passive circuit synthesis and passive model building.At present, the relevant literatures about passivity and passive criteria of fractional circuits are few. A method based on the positive semidefinite Hermitian part of impedance matrix was proposed, 19 which was limited for the uncertain existence of impedance matrix. No other methods for fractional circuit passivity analysis have been reported so far. Hence, it is necessary to obtain more general criteria of passive fractional circuits. We research the passive criteria of fractional circuits with rational order elements in this paper.The rest of this paper is organized as follows. Section 2 briefly reviews some preliminaries. In Section 3, definitions of positive-real (matrix) function in W-domain are given, and equi...

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Considering the under-actuated characteristic of the carangiform robotic fish whose forward movement and turning are carried out by its single caudal fin, this paper investigates its shortest path planning problem subjected to the terminal constrains. First, a simplified kinematics model is proposed in view of its locomotion characteristic. With this model, the shortest path planning is transformed into the shortest time optimal control problem and the maximum principle is employed to obtain the necessary condition. Furthermore, based on the analysis on the relationship between the length of all feasible paths and the terminal condition, the sufficient and necessary conditions are derived for the shortest path under any terminal constrains. Finally, a program developed with Matlab verifies the effectiveness of the proposed results.

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