An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

Information criterion based fast PCA adaptive algorithm

The novel information criterion （NIC） algorithm can find the principal subspace quickly, but it is not an actual principal component analysis （PCA） algorithm and hence it cannot find the orthonormal eigen-space which corresponds to the principal component of input vector. This defect limits its application in practice. By weighting the neural network＇s output of NIC, a modified novel information criterion （MNIC） algorithm is presented. MNIC extractes the principal components and corresponding eigenvectors in a parallel online learning program, and overcomes the NIC＇s defect. It is proved to have a single global optimum and nonquadratic convergence rate, which is superior to the conventional PCA online algorithms such as Oja and LMSER. The relationship among Oja, LMSER and MNIC is exhibited. Simulations show that MNIC could converge to the optimum fast. The validity of MNIC is proved.

HOSVD-based LPV modeling and mixed robust H_2/H_∞ control design for air-breathing hypersonic vehicle

This paper focuses on synthesizing a mixed robust H_2/H_∞ linear parameter varying（LPV） controller for the longitudinal motion of an air-breathing hypersonic vehicle via a high order singular value decomposition（HOSVD） approach.The design of hypersonic flight control systems is highly challenging due to the enormous complexity of the vehicle dynamics and the presence of significant uncertainties.Motivated by recent results on both LPV control and tensor-product（TP） model transformation approach,the velocity and altitude tracking control problems for the air-breathing hypersonic vehicle is reduced to that of a state feedback stabilizing controller design for a polytopic LPV system with guaranteed performances.The controller implementation is converted into a convex optimization problem with parameterdependent linear matrix inequalities（LMIs） constraints,which is intuitively tractable using LMI control toolbox.Finally,numerical simulation results demonstrate the effectiveness of the proposed approach.