Stability analysis and stabilization of systems presenting nested saturations

This note addresses the problems of stability analysis and stabilization of systems presenting nested saturations. Depending on the open-loop stability assumption, the global stability analysis and stabilization problems are considered. In the (local) analysis problem, the objective is the determination of estimates of the basin of attraction of the system. Considering the stabilization problem, the goal is to design a set of gains in order to enlarge the basin of attraction of the closed-loop system. Based on the modelling of the system presenting nested saturations as a linear system with dead-zone nested nonlinearities and the use of a generalized sector condition, linear matrix inequality (LMI) stability conditions are formulated. From these conditions, convex optimization strategies are proposed to solve both problems.     

Distributed Event-triggered Containment Control for Discrete-time Multi-agent Systems

This paper investigates the event-triggered containment control for discrete-time multi-agent systems. Event-triggered control strategies are employed in order to reduce the number of controller actuation updates for multi-agent systems with limited resources. It is assumed that each follower in the system updates its state only at some instants which are determined by the proposed event-triggered condition. Both centralized and decentralized event-triggered strategies are proposed to solve the containment control problem. Convergence analysis is given with the help of matrix theory and Lyapunov method, and it is showed that the proposed strategy does not exhibit Zeno behaviors. Numerical simulations are given to illustrate the effectiveness of the theoretical results.     

Distributed block independent set algorithms and parallel multilevel ILU preconditioners

We present a class of parallel preconditioning strategies utilizing multilevel block incomplete LU (ILU) factorization techniques to solve large sparse linear systems. The preconditioners are constructed by exploiting the concept of block independent sets (BISs). Two algorithms for constructing BISs of a sparse matrix in a distributed environment are proposed. We compare a few implementations of the parallel multilevel ILU preconditioners with different BIS construction strategies and different Schur complement preconditioning strategies. We also use some diagonal thresholding and perturbation strategies for the BIS construction and for the last level Schur complement ILU factorization. Numerical experiments indicate that our domain-based parallel multilevel block ILU preconditioners are robust and efficient.     

The parallel complexity of embedding algorithms for the solution ofsystems of nonlinear equations

Embedding algorithms used to solve nonlinear systems of equations do so by constructing a continuous family of systems and solving the given system by tracking the continuous curve of solutions to the family. Solving nonlinear equations by a globally convergent embedding algorithm requires the evaluation and factoring of a Jacobian matrix at many points along the embedding curve. Ways to optimize the Jacobian matrix on a hypercube are described. Several static and dynamical strategies for assigning components of the Jacobian to processors on the hypercube are investigated. It is found that a static rectangular grid mapping is the preferred choice for inclusion in a robust parallel mathematical software package. The static linear mapping is a viable alternative when there are many common subexpressions in the component evaluation, and the dynamic assignment strategy should only be considered when there is large variation in the evaluation times for the components, leading to a load imbalance on the processors     

粗糙集理论中的求核与约简

约简与核是粗糙集理论的两个重要概念，而直接由定义来计算约倚与核是一个典型的NP难题。发现了分辨矩阵的若干有用性质，利用这些性质使粗糙集理论中的求核与约简问题得以解决。进而分别讨论了无决策信息系统的约简和有决策信患系统的约简问题。最后举例说明了所得结果的有效性。     

基于扩展状态观测器的随机隐Markov正跳变系统有限时间异步控制

针对一类含不匹配扰动的随机隐Markov跳变系统, 本文研究了基于扩展状态观测器(ESO)的有限时间异步 控制问题. 首先, 引入一组扩展变量将隐Markov跳变系统转换成一组新的随机扩展系统, 补偿不匹配扰动对系统控 制输出的影响. 基于Lyapunov–Krasovskii泛函方法, 给出使得基于ESO的闭环随机隐Markov增广跳变系统是正系 统, 且有限时间有界的充分条件. 进而得到直接求解观测器增益和控制器增益的线性矩阵不等式. 最后, 通过仿真结 果验证了本文所设计的异步状态反馈控制器和观测器的有效性和可行性.     

A neural network solution for fixed-final time optimal control of nonlinear systems

In this paper, fixed-final time optimal control laws using neural networks and HJB equations for general affine in the input nonlinear systems are proposed. The method utilizes Kronecker matrix methods along with neural network approximation over a compact set to solve a time-varying HJB equation. The result is a neural network feedback controller that has time-varying coefficients found by a priori offline tuning. Convergence results are shown. The results of this paper are demonstrated on an example.     

机械臂逆运动学模型控制新方法

根据机械臂关节轴线方向建立了连杆坐标系,采用Denavit-Hartenberg（D-H）法得到连杆坐标系变换矩阵,通过坐标系变换矩阵得到机械臂正运动学模型,由正运动模型得到机械臂逆运动学模型。该模型是一个非线性约束优化问题。为了求解模型,基于多种群和多变异策略,提出了多变异策略的多种群差分演化算法。多种群策略可以提升个体共享群体信息的能力,多变异模式策略可以提升个体间的差异性,数值试验表明新算法可以有效求解机械臂逆运动学模型。     

Discrete-time output feedback sliding-mode control design for uncertain systems using linear matrix inequalities

An output feedback-based sliding-mode control design methodology for discrete-time systems is considered in this article. In previous work, it has been shown that by identifying a minimal set of current and past outputs, an augmented system can be obtained which permits the design of a sliding surface based upon output information only, if the invariant zeros of this augmented system are stable. In this work, a procedure for realising discrete-time controllers via a particular set of extended outputs is presented for non-square systems with uncertainties. This method is applicable when unstable invariant zeros are present in the original system. The conditions for existence of a sliding manifold guaranteeing a stable sliding motion are given. A procedure to obtain a Lyapunov matrix, which simultaneously satisfies both a Riccati inequality and a structural constraint, is used to formulate the corresponding control to solve the reachability problem. A numerical method using linear matrix inequalities is suggested to obtain the Lyapunov matrix. Finally, the design approach given in this article is applied to an aircraft problem and the use of the method as a reconfigurable control strategy in the presence of sensor failure is demonstrated.     

Guaranteed estimates of the domain of attraction for a class of hybrid systems

This paper addresses the estimation of the domain of attraction for a class of hybrid nonlinear systems where the state space is partitioned into several regions. Each region is described by polynomial inequalities, and the union of all the regions is a complete cover of the state space. The system dynamics are defined on each region independently from the others by polynomial functions. First, the problem of computing the largest sublevel set of a Lyapunov function included in the domain of attraction is considered. An approach is proposed for addressing this problem based on linear matrix inequalities, which provides a lower bound of the sought estimate by establishing negativity of the Lyapunov function derivative on each region. Second, a sufficient and necessary condition is provided for establishing optimality of the found lower bound, which requires to solve linear algebra operations in typical cases. Third, the problem of looking for Lyapunov functions that maximize the estimate of the domain of attraction is considered, describing several strategies where the proposed approach can be readily adopted. Copyright © 2014 John Wiley & Sons, Ltd.     

Characterization of feedback Nash equilibria for multi-channel systems via a set of non-fragile stabilizing state-feedback solutions and dissipativity inequalities

We consider the problem of state-feedback stabilization for a multi-channel system in a game-theoretic framework, where the class of admissible strategies for the players is induced from a solution set of the individual objective functions that are associated with certain dissipativity properties of the system. In such a framework, we characterize the feedback Nash equilibria via a set of non-fragile stabilizing state-feedback solutions corresponding to a family of perturbed multi-channel systems. Moreover, we show that the existence of a weak-optimal solution to a set of constrained dissipativity problems is a sufficient condition for the existence of a feedback Nash equilibrium, whereas the set of non-fragile stabilizing state-feedbacks solutions is described in terms of a set of dilated linear matrix inequalities.     

On the verification of polynomial system solvers

We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of well-known polynomial systems. Our experimental results illustrate the high efficiency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve.     

Parallel Strategies for Computing the Orthogonal Factorizations Used in the Estimation of Econometric Models

Parallel strategies based on compound disjoint Givens rotations are proposed for computing the main two factorizations that are used in the solution of seemingly unrelated regression and simultaneous equations models. The first factorization requires the triangularization of a set of upper-trapezoidals after deleting columns. The second factorization is equivalent to updating a lower-triangular matrix with a matrix having a block lower-triangular structure. Theoretical measures of complexity and examples are used for comparing and investigating the various parallel strategies. Received June 14, 1997; revised January 19, 1998.     

Applications of the methods of weighted residuals in system science

Piecewise linear functions arc applied to solve the state equations of linear system science with the introduction of the methods of weighted residuals (MWR). There are four different weighted procedures of MWR to be derived: (a) collocation, (b) subdomain, (c) Galerkin's and (d) least-squares methods. Applying the MWR, a linear system can be represented by a set of simultaneous linear algebraic equations, and a recursive algorithm and/or an operational matrix are available. Analysis and parameter identification of stale space equations of linear systems can thus be converted into a set of linear algebraic equations and solved using a recursive algorithm.     

Development of scheduling strategies in the TITAN operating system

Scheduling strategies are evolved to solve problems of selection in operating systems. Such strategies frequently undergo evolution in time as details of the specification are changed or, more frequently, as inadequacies in them are shown up by use in a heavily loaded system. This paper describes a number of scheduling strategies used in the TITAN system, and their development to meet changing circumstances or to overcome inadequacies.     

Simultaneous LQ control of a set of LTI systems using constrained generalized sampled-data hold functions

In this paper, sampled-data control of a set of continuous-time LTI systems is considered. It is assumed that a predefined guaranteed continuous-time quadratic cost function, which is, in fact, the sum of the performance indices for all systems, is given. The main objective here is to design a decentralized periodic output feedback controller with a prespecified form, e.g., polynomial, piecewise constant, exponential, etc., which minimizes the above mentioned guaranteed cost function. This problem is first formulated as a set of matrix inequalities, and then by using a well-known technique, it is reformulated as a LMI problem. The set of linear matrix inequalities obtained provides necessary and sufficient conditions for the existence of a decentralized optimal simultaneous stabilizing controller with the prespecified form (rather than a general form). Moreover, an algorithm is presented to solve the resultant LMI problem. Finally, the efficiency of the proposed method is demonstrated in two numerical examples.     

H∞ model reduction of Markovian jump linear systems

In this paper, the H_∞ model reduction problem for linear systems that possess randomly jumping parameters is studied. The development includes both the continuous and discrete cases. It is shown that the reduced order models exist if a set of matrix inequalities is feasible. An effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions. Using the numerical solutions of the matrix inequalities, the reduced order models can be obtained. An example is given to illustrate the proposed model reduction method.     

A heuristic approach to solving a class of bilinear matrix inequality problems

Many canonical and modern control problems can be recast into the problem of a group of matrix inequalities. Some of them are in the form of linear matrix inequalities (LMIs), which can be solved very efficiently by the powerful LMI toolbox in Matlab, but some others are in the form of bilinear matrix inequalities. The characteristic of this latter class of problems is that when the so called “communicating variables” are fixed, the overall problem will be reduced to the problem in LMIs. Thus, how to find the communicating variables is the key to solve the whole problem. In this paper, an optimal estimate for the communicating variables is presented. We will illustrate our method by completely solving the problems of overshoot bound control and reachable set analysis for uncertain systems. Numerical examples are provided to show the effectiveness of the proposed method.     

基于策略迭代算法的连续时间线性Markov跳变系统非零和微分反馈Nash控制

针对一类连续时间线性Markov跳变系统,本文提出了一种新的策略迭代算法用于求解系统的非零和微分反馈Nash控制问题.通过求解耦合的数值迭代解,以获得具有线性动力学特性和无限时域二次成本的双层非零和微分策略的Nash均衡解.在每一个策略层,采用策略迭代算法来计算与每一组给定的反馈控制策略相关联的最小无限时域值函数.然后,通过子系统分解将Markov跳变系统分解为N个并行的子系统,并将该算法应用于跳变系统.本文提出的策略迭代算法可以很容易求解非零和微分策略所对应的耦合代数Riccati方程,且对高维系统有效.最后通过仿真示例证明了本文设计方法的有效性和可行性.     

A tau approach for solution of the space fractional diffusion equation

Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.