本研究旨在设计一种针对高维分数阶非线性系统的滑模追踪控制器,使得系统输出在预定时间内收敛到给定的期望轨迹上。首先,为了便于滑模面的设计,本文利用传统的高阶滑模控制的方法,将复杂系统转化为更为简单的链式系统。然后,将传统的...本研究旨在设计一种针对高维分数阶非线性系统的滑模追踪控制器,使得系统输出在预定时间内收敛到给定的期望轨迹上。首先,为了便于滑模面的设计,本文利用传统的高阶滑模控制的方法,将复杂系统转化为更为简单的链式系统。然后,将传统的整数阶固定时间滑模控制策略进行改进,设计了两种分数阶滑模面,使其改进的滑模控制方法能够适用于分数阶系统。通过对滑模面的求导和利用Lyapunov稳定性定理,最终所设计的两类分数阶滑模控制器能够使系统的输出在预定时间内追踪上期望轨迹,与传统的固定时间滑模策略相比,该方法可以随意控制系统的最大收敛时间,因而控制效果更优。最后,两个仿真结果证明了这两类控制策略的可行性和有效性。This research is dedicated to designing a sliding mode tracking controller for high-dimensional fractional-order nonlinear systems, with the objective of making the system output converge to a given desired trajectory within a prescribed-time. In order to facilitate the design of the sliding mode surface, this paper uses the traditional high-order sliding mode control method to transform the complex system into a simpler chained-form system. Subsequently, this paper modifies the traditional integer-order fixed-time sliding-mode control strategy and designs two types of fractional-order sliding mode surfaces, so that the improved sliding-mode control approach can be applied to fractional-order systems. By differentiating the sliding mode surface and leveraging the Lyapunov stability theorem, the two classes of fractional-order sliding mode controllers designed can ensure that the system output tracks the desired trajectory within the prescribed-time. Compared with the traditional fixed-time sliding mode strategy, the proposed method has a significant advantage in that it can freely control the maximum convergence time of the system. Finally, two simulation results demonstrate the feasibility and effectiveness of these two types of control strategies.展开更多
研究了一类不确定分数阶非线性系统有限时间稳定性及自适应滑模同步控制,通过构造有效的分数阶滑模面及自适应规则,设计了主动控制器,并证明了在满足系统所有变量有界的情况下误差系统能够在有限时间内趋于滑模面。数值仿真中实现了分数...研究了一类不确定分数阶非线性系统有限时间稳定性及自适应滑模同步控制,通过构造有效的分数阶滑模面及自适应规则,设计了主动控制器,并证明了在满足系统所有变量有界的情况下误差系统能够在有限时间内趋于滑模面。数值仿真中实现了分数阶Duffing-Holmes系统和分数阶Van der Pol系统的异结构有限时间同步,进一步验证了该方法的有效性和鲁棒性。展开更多
In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)...In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),where s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s is the fractional critical exponent in dimension 3,V_(λ)(x)=λV(x)+1 withλ>0.Under the case of steep potential well,we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma.Furthermore,we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution.Our results improve the recent results in the literature.展开更多
We study the boundary value problem of a coupled differential system of fractional order, and prove the existence and uniqueness of solutions to the considered problem. The underlying differential system is featured b...We study the boundary value problem of a coupled differential system of fractional order, and prove the existence and uniqueness of solutions to the considered problem. The underlying differential system is featured by a fractional differential operator, which is defined in the Riemann-Liouville sense, and a nonlinear term in which different solution components are coupled. The analysis is based on the reduction of the given system to an equivalent system of integral equations. By means of the nonlinear alternative of Leray-Schauder,the existence of solutions of the factional differential system is obtained. The uniqueness is established by using the Banach contraction principle.展开更多
文摘本研究旨在设计一种针对高维分数阶非线性系统的滑模追踪控制器,使得系统输出在预定时间内收敛到给定的期望轨迹上。首先,为了便于滑模面的设计,本文利用传统的高阶滑模控制的方法,将复杂系统转化为更为简单的链式系统。然后,将传统的整数阶固定时间滑模控制策略进行改进,设计了两种分数阶滑模面,使其改进的滑模控制方法能够适用于分数阶系统。通过对滑模面的求导和利用Lyapunov稳定性定理,最终所设计的两类分数阶滑模控制器能够使系统的输出在预定时间内追踪上期望轨迹,与传统的固定时间滑模策略相比,该方法可以随意控制系统的最大收敛时间,因而控制效果更优。最后,两个仿真结果证明了这两类控制策略的可行性和有效性。This research is dedicated to designing a sliding mode tracking controller for high-dimensional fractional-order nonlinear systems, with the objective of making the system output converge to a given desired trajectory within a prescribed-time. In order to facilitate the design of the sliding mode surface, this paper uses the traditional high-order sliding mode control method to transform the complex system into a simpler chained-form system. Subsequently, this paper modifies the traditional integer-order fixed-time sliding-mode control strategy and designs two types of fractional-order sliding mode surfaces, so that the improved sliding-mode control approach can be applied to fractional-order systems. By differentiating the sliding mode surface and leveraging the Lyapunov stability theorem, the two classes of fractional-order sliding mode controllers designed can ensure that the system output tracks the desired trajectory within the prescribed-time. Compared with the traditional fixed-time sliding mode strategy, the proposed method has a significant advantage in that it can freely control the maximum convergence time of the system. Finally, two simulation results demonstrate the feasibility and effectiveness of these two types of control strategies.
文摘研究了一类不确定分数阶非线性系统有限时间稳定性及自适应滑模同步控制,通过构造有效的分数阶滑模面及自适应规则,设计了主动控制器,并证明了在满足系统所有变量有界的情况下误差系统能够在有限时间内趋于滑模面。数值仿真中实现了分数阶Duffing-Holmes系统和分数阶Van der Pol系统的异结构有限时间同步,进一步验证了该方法的有效性和鲁棒性。
基金supported by the Natural Science Foundation of Sichuan(No.2023NSFSC0073)。
文摘In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),where s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s is the fractional critical exponent in dimension 3,V_(λ)(x)=λV(x)+1 withλ>0.Under the case of steep potential well,we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma.Furthermore,we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution.Our results improve the recent results in the literature.
基金supported by National Natural Science Foundation of China(Grant Nos.11471274,11421110001 and 91130002)Natural Science Foundation of Guizhou Province(Grant No.LKS[2013]04)
文摘We study the boundary value problem of a coupled differential system of fractional order, and prove the existence and uniqueness of solutions to the considered problem. The underlying differential system is featured by a fractional differential operator, which is defined in the Riemann-Liouville sense, and a nonlinear term in which different solution components are coupled. The analysis is based on the reduction of the given system to an equivalent system of integral equations. By means of the nonlinear alternative of Leray-Schauder,the existence of solutions of the factional differential system is obtained. The uniqueness is established by using the Banach contraction principle.
