Adaptive Generalized Eigenvector Estimating Algorithm for Hermitian Matrix Pencil

Generalized eigenvector plays an essential role in the signal processing field. In this paper, we present a novel neural network learning algorithm for estimating the generalized eigenvector of a Hermitian matrix pencil. Differently from some traditional algorithms, which need to select the proper values of learning rates before using, the proposed algorithm does not need a learning rate and is very suitable for real applications. Through analyzing all of the equilibrium points, it is proven that if and only if the weight vector of the neural network is equal to the generalized eigenvector corresponding to the largest generalized eigenvalue of a Hermitian matrix pencil, the proposed algorithm reaches to convergence status. By using the deterministic discrete-time (DDT) method, some convergence conditions, which can be satisfied with probability 1, are also obtained to guarantee its convergence. Simulation results show that the proposed algorithm has a fast convergence speed and good numerical stability. The real application demonstrates its effectiveness in tracking the optimal vector of beamforming.      

Global convergence analysis of a discrete time nonnegative ICA algorithm

When the independent sources are known to be nonnegative and well-grounded, which means that they have a nonzero pdf in the region of zero, Oja and Plumbley have proposed a "Nonnegative principal component analysis (PCA)" algorithm to separate these positive sources. Generally, it is very difficult to prove the convergence of a discrete-time independent component analysis (ICA) learning algorithm. However, by using the skew-symmetry property of this discrete-time "Nonnegative PCA" algorithm, if the learning rate satisfies suitable condition, the global convergence of this discrete-time algorithm can be proven. Simulation results are employed to further illustrate the advantages of this theory.     

具有干扰的多领航者系统的群集运动的快速收敛

针对具有多领航者网络化系统的离散时间群集运动问题,提出了一阶/二阶网络化系统的包容控制算法。运用现代控制理论、代数图论和线性矩阵不等式等分析工具对所提出的控制算法进行理论分析,得到了具有干扰的多领航者网络化系统在离散时间情况下有限时间内实现群集运动的收敛条件。最后,利用LMI工具箱数值仿真求得正定矩阵范围,进而确定线性系统的稳定性。系统仿真验证了所得结论的正确性。     

Convergence characterisation of an iterative algorithm for periodic Lyapunov matrix equations

This paper is concerned with convergence characterisation of an iterative algorithm for a class of reverse discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. Firstly, a simple necessary condition is given for this algorithm to be convergent. Then, a necessary and sufficient condition is presented for the convergence of the algorithm in terms of the roots of polynomial equations. In addition, with the aid of the necessary condition explicit expressions of the optimal parameter such that the algorithm has the fastest convergence rate are provided for two special cases. The advantage of the proposed approaches is illustrated by numerical examples.     

离散高阶分布式一致性算法

为了提高分布式一致性算法的收敛速度, 提出了一种离散高阶分布式一致性算法。该算法通过单跳通信, 利用二跳邻接节点的前多步信息来加速分布式一致性算法的收敛速度。对无向通信拓扑下该算法的收敛性能和收敛速度, 以及带通信延时的该算法的收敛性能进行了分析和仿真比较, 结果显示, 该算法在满足条件下能收敛到初始状态的平均值, 与同样利用二跳邻接节点信息的算法相比, 具有通信量小, 收敛速度更快的特点, 但是能容忍的通信延时变小。     

Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations

The solution of coupled discrete-time Markovian jump Lyapunov matrix equations (CDMJLMEs) is important in stability analysis and controller design for Markovian jump linear systems. This paper presents a simple and effective iterative method to produce numerical solutions to this class of matrix equations. The gradient-based algorithm is developed from an optimization point of view. A necessary and sufficient condition guaranteeing the convergence of the algorithm is established. This condition shows that the algorithm always converges provided the CDMJLMEs have unique solutions which is evidently different from the existing results that converge conditionally. A simple sufficient condition which is easy to test is also provided. The optimal step size in the algorithm such that the convergence rate of the algorithm is maximized is given explicitly. It turns out that an upper bound of the convergence rate is bounded by a function of the condition number of the augmented coefficient matrix of the CDMJLMEs. Some parameters are introduced to the algorithm that will potentially reduce the condition number and thus increase the convergence rate of the algorithm. A numerical example is used to illustrate the efficiency of the proposed approach.     

基于LMI方法的保性能迭代学习算法设计

研究基于性能的迭代学习算法设计与优化问题.首先定义了迭代域二次型性能函数,然后针对线性离散系统给出了迭代域最优迭代学习算法;基于线性矩阵不等式(LMI)方法,针对不确定线性离散系统给出了保性能迭代学习算法及其优化方法.对于这两类迭代学习算法,只要调整性能函数中的权系数矩阵,便可很好地调整迭代学习收敛速度.另外,保性能迭代学习算法设计及优化过程,可利用MATLAB工具箱很方便地求解.     

Convergence analysis of the extended Kalman filter used as anobserver for nonlinear deterministic discrete-time systems

In this paper, convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented. Based on a new formulation of the first-order linearization technique, sufficient conditions to ensure local asymptotic convergence are established. Furthermore, it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly. The efficiency of this approach, compared to the classical version of the EKF, is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete-time systems     

Stability and convergence analysis of multi-agent consensus with information reuse

In this article, we study multi-agent consensus algorithms with information reuse by intentionally introducing the outdated state information into the traditional consensus algorithms. In the continuous-time case, we first show that the outdated state information combined with the current state information does not necessarily jeopardise the stability of a single system, but may improve the convergence speed without increasing the maximal control effort. Then this idea is extended from the single-agent case to the multi-agent case. When the directed communication graph is fixed, the corresponding Laplacian matrix and the outdated state information satisfy certain conditions, we show that the consensus algorithm with both the current and outdated states can achieve a faster convergence speed than the standard one. We also consider the case of a switching directed communication graph and derive corresponding conditions. In the discrete-time case, we propose a discrete-time consensus algorithm with both the current and outdated states under an undirected fixed communication graph. We then derive conditions on the communication graph, the sampling period and the outdated state information such that the proposed algorithm can achieve a faster convergence speed than that using the standard one. In both the continuous-time and discrete-time settings, we show that the maximum control efforts for the proposed consensus algorithms are identical to those for the standard ones. Several simulation examples are presented as a proof of concept.     

Direct model reference adaptive control using Kp = LDU factorization for multivariable discrete-time plants

For a large class of discrete-time multivariable plants with arbitrary relative degrees, the design and analysis of the direct model reference adaptive control scheme are investigated under less restrictive assumptions. The algorithm is based on a new parametrization derived from the high frequency gain matrix factorization Kp=LDU under the condition that the signs of the leading principal minors of/fp are known. By reproving the discrete-time Lp and L2σ norm relationship between inputs and outputs, establishing the properties of discrete-time adaptive law, defining the normalizing signal, and relating the signal with all signals in the closed-loop system, the stability and convergence of the discrete-time multivariable model reference adaptive control scheme are analyzed rigorously in a systematic fashion as in the continuous-time case.     

离散时间Itˆo 型跳变系统Lyapunov 方程的有限次迭代求解算法

针对离散时间Itˆo 型马尔科夫跳变系统Lyapunov 方程的求解给出一种迭代算法. 经证明, 在误差允许的范围内, 该算法可以在确定的有限次数内收敛到系统的精确解, 收敛速度较快, 具有良好的数值稳定性, 并且该算法为显式迭代, 可避免迭代过程中求解其他矩阵方程对结果精度产生的影响. 最后通过一个数值算例对该算法的有效性进行了验证.

     

Brazilian Society of Simulation

Estimation of the parameters of a reducible (inflated common denominator) model for the transfer function matrix of MIMO systems is well known. However, the reduction of the model to the minimal form by pole-zero cancellation is possible only in the noise-free case. This paper presents an algorithm for the estimation of the minimal continuous-time transfer function matrix model. Monte Carlo simulation results are presented for discrete-time and continuous-time models. Least-squares and generalized least-squares methods have been used in both cases. An asymptotic analysis of convergence has also been provided for these models in the noise-free case. The computation times and space complexities of different variants of the algorithm are compared. The results show that in noisy situations, obtaining a discrete-time model by discretizing an estimated continuous-time model may be a viable proposition     

Continuous-time stochastic model reference adaptive control

The authors establish global convergence and asymptotic properties of a direct adaptive controller for continuous-time stochastic linear systems by presenting a direct adaptive control algorithm and an associated proof of convergence. This result is comprehensive and covers many other existing results as special cases. It has practical implications for the discrete-time case since it reveals how the existing discrete-time results must be modified so that they have meaningful limits as the sampling period decreases     

Structure-Preserving Algorithms for Periodic Discrete-Time Algebraic Riccati Equations

In this paper we investigate structure-preserving algorithms for computing the symmetric positive semi-definite solutions to the periodic discrete-time algebraic Riccati equations (P-DAREs). Using a structure-preserving swap and collapse procedure, a single symplectic matrix pair in standard symplectic form is obtained. The P-DAREs can then be solved via a single DARE, using a structure-preserving doubling algorithm. We develop the structure-preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability. With several numerical results, the algorithm is shown to be efficient, out-performing other algorithms on a large set of benchmark problems.     

Convergence properties of the softassign quadratic assignment algorithm.

The softassign quadratic assignment algorithm is a discrete-time, continuous-state, synchronous updating optimizing neural network. While its effectiveness has been shown in the traveling salesman problem, graph matching, and graph partitioning in thousands of simulations, its convergence properties have not been studied. Here, we construct discrete-time Lyapunov functions for the cases of exact and approximate doubly stochastic constraint satisfaction, which show convergence to a fixed point. The combination of good convergence properties and experimental success makes the softassign algorithm an excellent choice for neural quadratic assignment optimization.     

基于迭代神经动态规划的数据驱动非线性近似最优调节

利用数据驱动控制思想,建立一种设计离散时间非线性系统近似最优调节器的迭代神经动态规划方法.提出针对离散时间一般非线性系统的迭代自适应动态规划算法并且证明其收敛性与最优性.通过构建三种神经网络,给出全局二次启发式动态规划技术及其详细的实现过程,其中执行网络是在神经动态规划的框架下进行训练.这种新颖的结构可以近似代价函数及其导函数,同时在不依赖系统动态的情况下自适应地学习近似最优控制律.值得注意的是,这在降低对于控制矩阵或者其神经网络表示的要求方面,明显地改进了迭代自适应动态规划算法的现有结果,能够促进复杂非线性系统基于数据的优化与控制设计的发展.通过两个仿真实验,验证本文提出的数据驱动最优调节方法的有效性.     

基于随机牛顿算法的离散系统自适应参数估计

针对一类离散系统,提出一种基于随机牛顿算法的自适应参数估计新框架,相较于已有的参数估计算法,所提出方法仅要求系统满足有限激励条件,而非传统的持续激励条件.所提出算法的核心思想在于通过对原始代价函数的修正,在使用当前时刻误差信息的基础上融入历史误差信息,进而通过对历史信息和历史激励的复用使得持续激励条件转化为有限激励条件;然后,为了解决传统算法收敛速度慢的问题并避免潜在的病态问题,采用随机牛顿算法推导出参数自适应律,并引入含有历史信息的海森矩阵作为时变学习增益,保证参数估计误差指数收敛;最后,基于李雅普诺夫稳定性理论给出不同激励条件下所提出算法的收敛性结论和证明,并通过对比仿真验证所提出算法的有效性和优越性.     

Determining continuous-time state equations from discrete-time state equations via the principalqth root method

Fast computational methods are developed for finding the equivalent continuous-time state equations from discrete-time state equations. The computational methods utilize the direct truncation method, the matrix continued fraction method, and the geometric-series method in conjunction with the principalqth root of the discrete-time system matrix for quick determination of the approximants of a matrix logarithm function. It is shown that the use of the principalqth root of a matrix enables us to enlarge the convergence region of the expansion of a matrix logarithm function and to improve the accuracy of the approximants of the matrix logarithm function.     

Parameter identification in a class of nonlinear systems

Two approaches are proposed for on-line identification of parameters in a class of nonlinear discrete-time systems. The system is modeled by state equations in which state and input variables enter nonlinearly in general polynomial form, while unknown parameters and random disturbances enter linearly. All states and inputs must be observed with measurement errors represented by white Gaussian noise having known covariance. System disturbances are also white and Gaussian with finite, but unknown, covariance. One method of parameter estimation is based upon a least squares approach, the second is a related stochastic approximation algorithm (SAA). Under fairly mild conditions the estimate derived from the least squares algorithm (LSA) is shown to converge in probability to the correct parameter; the SAA yields an estimate which converges in mean square and with probability 1. Examples illustrate convergence of the LSA which even in recursive form requires inversion of a matrix at each step. The SAA requires no matrix inversions, but experience with the algorithm indicates that convergence is slow relative to that of the LSA.     

Adaptive control via consistent estimation for deterministic systems

For multidimensional discrete-time deterministic systems the optimal adaptive control has been derived by use of a probabilistic method so that when the reference signal is an arbitrary bounded random sequence, the tracking error and the estimation error based on a projection algorithm go to zero with a near-exponential convergence rate. For this, the basic step is to prove the consistency of estimates when the condition number of the matrix consisting of regressors diverges to infinity; in other words, when the persistent excitation condition is not satisfied.