Feasibility and Structural Feature on Monotone Second-Order Cone Linear Complementarity Problems in Hilbert Space

Given a real finite-dimensional or infinite-dimensional Hilbert space H with a Jordan product, the second-order cone linear complementarity problem(SOCLCP)is considered. Some conditions are investigated, for which the SOCLCP is feasible and solvable for any element q?H. The solution set of a monotone SOCLCP is also characterized. It is shown that the second-order cone and Jordan product are interconnected.

A Class of Second-Order Cone Eigenvalue Complementarity Problems for Higher-Order Tensors

In this paper,we consider the second-order cone tensor eigenvalue complementarity problem(SOCTEiCP)and present three different reformulations to the model under consideration.Specifically,for the general SOCTEiCP,we first show its equivalence to a particular variational inequality under reasonable conditions.A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem.Then,for the symmetric and sub-symmetric SOCTEiCPs,we reformulate them as appropriate nonlinear programming problems,which are extremely beneficial for designing reliable solvers to find solutions of the considered problem.Finally,we report some preliminary numerical results to verify our theoretical results.

A New Complementarity Function and Applications in Stochastic Second-Order Cone Complementarity Problems

This paper considers the so-called expected residual minimization(ERM)formulation for stochastic second-order cone complementarity problems,which is based on a new complementarity function called termwise residual complementarity function associated with second-order cone.We show that the ERM model has bounded level sets under the stochastic weak R0-property.We further derive some error bound results under either the strong monotonicity or some kind of constraint qualifications.Then,we apply the Monte Carlo approximation techniques to solve the ERM model and establish a comprehensive convergence analysis.Furthermore,we report some numerical results on a stochastic second-order cone model for optimal power flow in radial networks.

GUS-property for Lorentz cone linear complementarity problems on Hilbert spaces

Given a real(finite-dimensional or infinite-dimensional) Hilbert space H with a Jordan product,we consider the Lorentz cone linear complementarity problem,denoted by LCP(T,Ω,q),where T is a continuous linear operator on H,ΩH is a Lorentz cone,and q ∈ H.We investigate some conditions for which the problem concerned has a unique solution for all q ∈ H(i.e.,T has the GUS-property).Several sufficient conditions and several necessary conditions are given.In particular,we provide two suficient and necessary conditions of T having the GUS-property.Our approach is based on properties of the Jordan product and the technique from functional analysis,which is different from the pioneer works given by Gowda and Sznajder(2007) in the case of finite-dimensional spaces.

Quadratic Optimization over a Second-Order Cone with Linear Equality Constraints

This paper studies the nonhomogeneous quadratic programming problem over a second-order cone with linear equality constraints.When the feasible region is bounded,we show that an optimal solution of the problem can be found in polynomial time.When the feasible region is unbounded,a semidefinite programming(SDP)reformulation is constructed to find the optimal objective value of the original problem in polynomial time.In addition,we provide two sufficient conditions,under which,if the optimal objective value is finite,we show the optimal solution of SDP reformulation can be decomposed into the original space to generate an optimal solution of the original problem in polynomial time.Otherwise,a recession direction can be identified in polynomial time.Numerical examples are included to illustrate the effectiveness of the proposed approach.

Primal-dual Interior-point Algorithms for Second-order Cone Optimization Based on a New Parametric Kernel Function

A class of polynomial primal-dual interior-point algorithms for second-order cone optimization based on a new parametric kernel function, with parameters p and q, is presented. Its growth term is between linear and quadratic. Some new tools for the analysis of the algorithms are proposed. The complexity bounds of O（√Nlog N log N/ε） for large-update methods and O（√Nlog N/ε） for smallupdate methods match the best known complexity bounds obtained for these methods. Numerical tests demonstrate the behavior of the algorithms for different results of the parameters p and q.

Exact Computable Representation of Some Second-Order Cone Constrained Quadratic Programming Problems

Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation,which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints.In this paper,we provide exact computable representations for some more subclasses of the QCQP problem,in particular,the subclass with one secondorder cone constraint and two special linear constraints.

不确定奇异时滞系统的时滞相关型鲁棒H_∞弹性控制

研究了含范数有界参数不确定性的奇异时滞系统的时滞相关型鲁棒H_∞弹性控制问题.文中所考虑的控制器增益摄动包括加性摄动和乘性摄动两种形式.在标称奇异时滞系统的时滞相关型稳定性判据的基础上,引入了一个新的时滞相关型有界实引理(BRL),进而给出了时滞相关型鲁棒H_∞弹性控制器存在的充分性条件,并利用LMIs和锥补线性化算法给出了控制器的表达式.从数值算例可以看出,所给出的设计算法是有效的.

计及时滞影响的广域附加阻尼控制

针对广域测量信号的时滞对系统动态性能的不利影响,提出了一种电力系统广域附加区间阻尼控制器设计的直接迭代方法。由已知的线性矩阵不等式定理,推得一种作为时滞系统渐近稳定性判据的基于状态反馈的矩阵不等式定理。利用变量替换法,将定理中的非线性矩阵不等式转变为线性矩阵不等式,进而将阻尼控制器设计转化为隶属于线性矩阵不等式的锥补问题并进行迭代求解。该方法可直接获得状态反馈控制矩阵,并具有良好的收敛性。测试系统的仿真结果表明,广域附加区间阻尼控制器,具有一定的时滞不敏感性,能够较好地抑制区间振荡。

大延时网络化控制系统保性能控制

针对事件驱动方式下的大延时网络化控制系统控制性能下降的情况,建立了时钟驱动方式的网络化控制系统模型,提出了一种基于状态预估的保性能控制设计方法。基于李亚普诺夫稳定性理论,设计了大延时时钟驱动方式的保性能控制器,使得系统在大延时的情况下仍能保证稳定并具有一定的控制性能,仿真算例验证了这种控制器的可行性。

基于状态预估的大延时网络化控制系统H_∞控制

针对一类具有随机大延时(延时大于系统的采样周期)的网络化控制系统,提出一种基于状态预估和时钟驱动的H∞控制器的设计方法,通过状态预估获取当前控制周期的预估控制状态,并通过相应的控制律计算出控制量输出。利用锥补线性化算法和LMI(Linear Ma-trix Inequality)给出了系统稳定的H∞次优控制器。仿真算例验证了本文结论的有效性和可用性。

不确定性线性系统的H_∞输出反馈鲁棒重复控制

为了抑制乘性不确定性线性系统中的外部扰动,提出了一种重复控制器与输出反馈控制器参数同时优化的方法.引入一个假想摄动将系统的扰动抑制问题归结为具有结构型摄动的系统鲁棒稳定性问题.根据小增益定理得到该结构型摄动系统鲁棒稳定性的充分条件.引入一个稳定的定标阵,得到该结构型摄动系统低保守性的鲁棒稳定性充分条件.基于该充分条件和线性矩阵不等式方法(LMI)、锥补线性化方法(CCL),提出的迭代算法可以同时计算出重复控制器中低通滤波器剪切频率的最大值及其对应的输出反馈控制器的参数.最后,低频线振动台系统验证了该方法的有效性。

解三维摩擦接触问题的一个二阶锥线性互补法

针对三维摩擦接触问题的求解,给出了一种基于参变量变分原理的二阶锥线性互补法。首先,基于三维Coulomb摩擦锥在数学表述上属于二阶锥的事实,利用二阶锥规划对偶理论,建立了三维Coulomb摩擦接触条件的参变量二阶锥线性互补模型,它是二维Coulomb摩擦接触条件参变量线性互补模型在三维情形下的自然推广;随后,利用参变量变分原理与有限元方法,建立了求解三维摩擦接触问题的二阶锥线性互补法。较之于将三维Coulomb摩擦锥进行显式线性化的线性互补法,该方法无需对三维Coulomb摩擦锥进行线性化,因而在保证精度的前提下所解问题的规模要小很多。最后通过算例展示了该方法的特点。

大系统关联时滞分散鲁棒H∞控制

研究了一类关联时滞系统的分散鲁棒H∞控制问题,其中时滞是时变的。首先,设计了分散状态反馈H∞控制器,引入一种积分不等式方法,结合Lyapunov-Krasovskii泛函方法、积分矩阵不等式技巧和锥补法(CCL)导出了此类系统的关联时滞分散H∞控制的非线性矩阵不等式(NMI)和线性矩阵不等式(LMI)充分条件。接着,将结果扩展到分散输出反馈中。最后,数值例子说明了方法的有效性。

网络化控制系统的建模与控制

针对具有不确定时变延时以及数据包丢失的网络化控制系统,提出了一种建模方式来研究该系统的稳定性以及镇定控制器设计的问题.用自由权矩阵结合Lyapunov-Krasovskii方法给出了使得网络化控制系统渐进稳定的充分条件,自由权矩阵的引入降低了采用固定权矩阵方法所得结果的保守性.给出了镇定控制器以及保证系统渐进稳定的最大延时的求解算法,数值算例验证了所提方法的可行性.

带有时延和丢包的变增益网络化预测控制

本文研究了带有时变时延和丢包的网络化控制系统变增益预测控制器设计问题。通过切换系统理论得到了闭环系统渐近稳定的充分条件并得到了一组非线性矩阵不等式。考虑到非线性矩阵不等式难以求解,采用锥补线性化的方法将非线性矩阵不等式转化为线性矩阵不等式。通过求解一组带约束的优化问题,可以得到在约束条件下对不同时延情况都满足控制系统稳定的参数,从而得到了变增益预测控制方案下的控制器参数来补偿网络中的时延和丢包。最后通过仿真实例验证了所设计方案的有效性。

具有输出延迟的网络化控制系统稳定性分析

将网络化控制系统看成一个具有输出时延的采样控制系统,运用具有时延的输出信号设计了一种新的观测器和相应的控制器,从而得到含有时滞的闭环网络化控制系统和使该闭环系统稳定的充分条件。最后给出了求解观测器增益和控制器增益的锥补线性化方法,仿真实例说明了所提方法的有效性。

柔性梁降阶H_∞控制实验研究

研究了压电传感器、作动器非同位配置情况下柔性悬臂梁的降阶H∞振动控制问题。采用频域辨识方法获取低阶名义模型,合理选取权函数,将鲁棒H∞控制问题转化为标准H∞控制问题。采用CCL(Cone Complementarity Lin-earization)算法设计降阶H∞控制器。比较了全阶H∞控制器和降阶H∞控制器的控制效果,实验结果表明,设计的降阶H∞控制器能够有效抑制柔性梁的前三阶模态振动,而且不会产生溢出问题。

离散网络化控制系统的鲁棒H∞控制

提出了一种新的建模方式研究同时含有时变延时以及数据包丢失的离散网络化控制系统的鲁棒H∞控制问题。引入自由权矩阵并结合Lyapunov-Krasovskii方法推导出使得网络化控制系统鲁棒渐近稳定并满足具有H∞扰动衰减度γ的充分条件,自由权矩阵的引入可以很大程度上降低用传统固定权矩阵来分析网络化控制系统所带来的保守性。由于条件中非线性项的存在,利用锥补线性化的方法将其转化为基于线性矩阵不等式的非线性最小化的问题,并分别给出了求解最大延时以及次优γ的算法。仿真算例说明了本文建模方式的有效性和优越性。

模糊双线性系统的鲁棒控制器设计新方法

针对一类具有范数有界参数不确定性的连续模糊双线性系统,提出了一种求解其鲁棒控制器的新方法。首先,基于T-S模糊模型对模糊双线性系统建模,采用并行分布补偿算法设计模糊控制器。结合Lyapunov稳定性理论,得到了闭环模糊双线性系统全局稳定的充分条件。然后,基于一种改进的锥补线性化算法的思想,将模糊控制器的设计转化成为一个受线性矩阵不等式约束的最小化问题。最后的仿真结果验证了设计方法的有效性。