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The multidimensional ($n$-D) systems described by Roesser model are presented in this paper. These $n$-D systems consist of discrete systems and continuous fractional order systems with fractional order $ u$, $0< u<1$. The stability and Robust stability of such $n$-D systems are investigated.
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The celebrated GKYP is widely used in integer-order control system. However, when it comes to the fractional order system, there exists no such tool to solve problems. This paper prove the FGKYP which can be used in the analysis of problems in fractional order system. The $H_infty$ and $L_infty$ of fractional order system are analysed based on the FGKYP.
As an essential characteristics of fractional calculus, the memory effect is served as one of key factors to deal with diverse practical issues, thus has been received extensive attention since it was born. By combining the fractional derivative with memory effects and grey modeling theory, this paper aims to construct an unified framework for the commonly-used fractional grey models already in place. In particular, by taking different kernel and normalization functions, this framework can deduce some other new fractional grey models. To further improve the prediction performance, the four popular intelligent algorithms are employed to determine the emerging coefficients for the UFGM(1,1) model. Two published cases are then utilized to verify the validity of the UFGM(1,1) model and explore the effects of fractional accumulation order and initial value on the prediction accuracy, respectively. Finally, this model is also applied to dealing with two real examples so as to further explain its efficacy and equally show how to use the unified framework in practical applications.
This paper focuses on some properties, which include regularity, impulse, stability, admissibility and robust admissibility, of singular fractional order system (SFOS) with fractional order $1<alpha<2$. The finitions of regularity, impulse-free, stability and admissibility are given in the paper. Regularity is analysed in time domain and the analysis of impulse-free is based on state response. A sufficient and necessary condition of stability is established. Three different sufficient and necessary conditions of admissibility are proved. Then, this paper shows how to get the numerical solution of SFOS in time domain. Finally, a numerical example is provided to illustrate the proposed conditions.
In this paper, we introduce two new matrix stochastic processes: fractional Wishart processes and $varepsilon$-fractional Wishart processes with integer indices which are based on the fractional Brownian motions and then extend $varepsilon$-fractional Wishart processes to the case with non-integer indices. Both of two kinds of processes include classic Wishart processes when the Hurst index $H$ equals $frac{1}{2}$ and present serial correlation of stochastic processes. Applying $varepsilon$-fractional Wishart processes to financial volatility theory, the financial models account for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets returns but also between their volatilities and for stochastic serial correlation of the relevant assets.
This paper proposes a fractional order gradient method for the backward propagation of convolutional neural networks. To overcome the problem that fractional order gradient method cannot converge to real extreme point, a simplified fractional order gradient method is designed based on Caputos definition. The parameters within layers are updated by the designed gradient method, but the propagations between layers still use integer order gradients, and thus the complicated derivatives of composite functions are avoided and the chain rule will be kept. By connecting every layers in series and adding loss functions, the proposed convolutional neural networks can be trained smoothly according to various tasks. Some practical experiments are carried out in order to demonstrate fast convergence, high accuracy and ability to escape local optimal point at last.
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