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.
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.
The well-known GKYP is widely used in system analysis, but for singular systems, especially singular fractional order systems, there is no corresponding theory, for which many control problems for this type of system can not be optimized in the limited frequency ranges. In this paper, a universal framework of finite frequency band GKYP lemma for singular fractional order systems is established. Then the bounded real lemma in the sense of L is derived for different frequency ranges. Furthermore, the corresponding controller is designed to improve the L performance index of singular fractional order systems. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results.
We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.