Finite Iteration Steps Research Articles

An iterative method for updating undamped structural systems with connectivity constraints

Model updating is a method of correcting analytical models, such as the finite element models, by improving the correlation between the measured data and the analytical modal data. In this paper, we develop an iterative method to update undamped structural systems by using measured modal parameters. The method allows mass and stiffness matrices to be updated simultaneously. And we can obtain optimal updated matrices within finite iteration steps by choosing a special kind of initial matrix pair. More importantly, the method can preserve the physical connectivity of the original model and the updated model is in good agreement with the experimental modal data. Two real numerical examples show that the proposed method is feasible and efficient.

An iterative method for updating finite element models with connectivity constraints

It is well known that the analytical matrices arising from the discretization of distributed parameter systems using the finite element technique are usually symmetric and banded. How to preserve the coefficient matrices of the updated model being of the same band structure is an important yet difficult challenge for model updating in structural dynamics. In this paper, an iterative method for updating mass, damping and stiffness matrices simultaneously based on partial modal measured data is provided. By the method, the optimal updated matrices can be obtained within finite iteration steps by choosing a special kind of initial matrix triplet. The proposed approach not only preserves the physical connectivity of the original model, but also the updated model reproduces the measured modal data, which can be utilized for various finite element model updating problems. Numerical examples confirm the effectiveness of the introduced method.

Generalized conjugate direction algorithm for solving general coupled Sylvester matrix equations

In this paper, a generalized conjugate direction algorithm (GCD) is proposed for solving general coupled Sylvester matrix equations. The GCD algorithm is an improved gradient algorithm, which can realize gradient descent by introducing matrices Pj(k) and Tj(k) to construct parameters α(k) and β(k). The matrix Pj(k) and Tj(k) are iterated in a cross way to accelerate the convergence rate. In addition, it is further proved that the algorithm converges to the exact solution in finite iteration steps in the absence of round-off errors if the system is consistent. Also, the sufficient conditions for least squares solutions and the minimum F-norm solutions are obtained. Finally, numerical examples are given to demonstrate the effectiveness of the GCD algorithm.

An index search method based inner-outer iterative algorithm for solving nonnegative least squares problems

We propose an index search method (ISM) for solving nonnegative least squares problems (NNLS). This method uses inner and outer schemes to find the index set corresponding to the nonzero component of the optimal solution. The outer iteration updates the approximate index set such that the objective value of the sequence generated by ISM is monotonically decreasing. Hence, the index set generated by ISM does not repeat, and the optimal solution can be achieved with finite iteration steps. Some normal equations need to be solved in the inner iteration, which is the dominant computational complexity in ISM. Numerical experiments are provided to support the theoretical results.

Control Design for Iterative Methods in Solving Linear Algebraic Equations

In the interaction between control and mathematics, mathematical tools are fundamental for all the control methods, but it is unclear how control impacts mathematics. This is the first part of our paper that attempts to give an answer with focus on solving linear algebraic equations (LAEs) from the perspective of systems and control, where it mainly introduces the controllability-based design results. By proposing an iterative method that integrates a learning control mechanism, a class of tracking problems for iterative learning control (ILC) is explored for the problem solving of LAEs. A trackability property of ILC is newly developed, by which analysis and synthesis results are established to disclose the equivalence between the solvability of LAEs and the controllability of discrete control systems. Hence, LAEs can be solved by equivalently achieving the perfect tracking tasks of resulting ILC systems via the classic state feedback-based design and analysis methods. It is shown that the solutions for any solvable LAE can all be calculated with different selections of the initial input. Moreover, the presented ILC method is applicable to determining all the least squares solutions of any unsolvable LAE. In particular, a deadbeat design is incorporated to ILC such that the solving of LAEs can be completed within finite iteration steps. The trackability property is also generalized to conventional two-dimensional ILC systems, which creates feedback-based methods, instead of the common used contraction mapping-based methods, for the design and convergence analysis of ILC.

Conjugate gradient-based iterative algorithm for solving generalized periodic coupled Sylvester matrix equations

This paper focuses on constructing a conjugate gradient-based (CGB) method to solve the generalized periodic coupled Sylvester matrix equations in complex space. The presented method is developed from a point of conjugate gradient methods. It is proved that the presented method can find the solution of the considered matrix equations within finite iteration steps in the absence of round-off errors by theoretical derivation. Some numerical examples are provided to verify the convergence performance of the presented method, which is superior to some existing numerical algorithms both in iteration steps and computation time.

Generalized conjugate direction algorithm for solving generalized coupled Sylvester transpose matrix equations over reflexive or anti-reflexive matrices

The paper studies the iterative solutions of the generalized coupled Sylvester transpose matrix equations over the reflexive (anti-reflexive) matrix group by the generalized conjugate direction algorithm. The convergence analysis shows that the solution group can be obtained within finite iterative steps in the absence of round-off errors for any initial given reflexive (anti-reflexive) matrix group. Furthermore, we can get the minimum-norm solution group by choosing special kinds of initial matrix group. Finally, some numerical examples are given to demonstrate the algorithm considered is quite effective in actual computation.

An iterative algorithm for generalized Hamiltonian solution of a class of generalized coupled Sylvester-conjugate matrix equations

In this work, we present an iterative algorithm to solve a class of generalized coupled Sylvester-conjugate matrix equations over the generalized Hamiltonian matrices. We show that if the equations are consistent, a generalized Hamiltonian solution can be obtained within finite iteration steps in the absence of round-off errors for any initial generalized Hamiltonian matrix by the proposed iterative algorithm. Furthermore, we can obtain the minimum-norm generalized Hamiltonian solution by choosing the special initial matrices. Finally, numerical examples show that the iterative algorithm is effective.

New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms

Abstract Positive definite polynomials are important in the field of optimization. ℋ-tensors play an important role in identifing the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose an iterative scheme for identifying ℋ-tensor and prove that the algorithm can terminate within finite iterative steps. Some numerical examples are provided to illustrate the efficiency and validity of methods.

Developing iterative algorithms to solve Sylvester tensor equations

This paper is concerned with solving high order Sylvester tensor equation arising in control theory. We propose the tensor forms of the bi-conjugate gradient and bi-conjugate residual methods for solving the tensor equation. To improve their performance, two preconditioned iterative algorithms based on the nearest Kronecker product are developed for finding its solution. We also prove that the proposed algorithms are convergent to an exact solution within finite iteration steps for any initial tensor in the absence of round-off errors. At last, some numerical examples are provided to illustrate the feasibility and validity of the algorithms proposed.

An iterative algorithm for generalized periodic multiple coupled Sylvester matrix equations

The paper is indicated to constructing a modified conjugate gradient iterative (MCG) algorithm to solve the generalized periodic multiple coupled Sylvester matrix equations. It can be proved that the proposed approach can find the solution within finite iteration steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm solution of the system. Some numerical examples are illustrated to show the performance of the proposed approach and its superiority over the existing method CG.

An iterative scheme for identifying the positive semi-definiteness of even-order real symmetric H-tensor

H-tensor plays an important role in identifying the positive (semi-) definiteness of even-order real symmetric tensor. In this paper, we propose an algorithm for identifying general H-tensor and prove the algorithm will terminate within finite iterative steps. As application, we give an iterative method for identifying positive semi-definiteness of the even order symmetric tensor. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.

FINITE ITERATIVE (R, S)-CONJUGATE SOLUTIONS OF THE GENERALIZED COMPLEX COUPLED SYLVESTER-TRANSPOSE EQUATIONS

The iterative method of generalized complex coupled Sylvester-transpose equations $ AXB+CY^TD = E,\; MX^TN+GYH = F $ over (R, S)-conjugate matrix solution $ (X,Y) $ is proposed. Usually, the type of matrix arises from some physical problems with some form of generalized reflexive symmetry. On the condition that the coupled matrix equations are consistent, we show the solution pair $ (X^*, Y^*) $ can be obtained by generalization of CG iterative method within finite iterative steps in the absence of roundoff-error for any initial guess chosen by the (R, S)-conjugate matrix. Moreover, the optimal approximation (R, S)-conjugate matrix solutions can be derived by searching the least Frobenius norm solution of the novel generalized complex coupled Sylvester-transpose matrix equations. Finally, some numerical examples are given to illustrate the presented iterative algorithm is efficient.

Iterative method to the coupled operator matrix equations with sub-matrix constraint and its application in control

A finite iterative algorithm is presented for solving the numerical solutions to the coupled operator matrix equations in Zhang (2017b). In this paper, a new finite iterative algorithm is presented for solving the constraint solutions to the coupled operator matrix equations [Formula: see text], where the constraint solutions include symmetric solutions, bisymmetric solutions and reflexive solutions as special cases. If this system is consistent, for any initial constraint matrices, the exact constraint solutions can be obtained by the introduced algorithm within finite iterative steps in the absence of the roundoff errors. Also, if this system is not consistent, the least-norm constraint solutions can be obtained within the finite iteration steps in the absence of the roundoff errors. Furthermore, if a group of suitable matrices are given, the optimal approximation solutions can be derived. Finally, several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Policy Iteration Approach to the Infinite Horizon Average Optimal Control of Probabilistic Boolean Networks

This article studies the optimal control of probabilistic Boolean control networks (PBCNs) with the infinite horizon average cost criterion. By resorting to the semitensor product (STP) of matrices, a nested optimality equation for the optimal control problem of PBCNs is proposed. The Laurent series expression technique and the Jordan decomposition method derive a novel policy iteration-type algorithm, where finite iteration steps can provide the optimal state feedback law, which is presented. Finally, the intervention problem of the probabilistic Ara operon in E. coil, as a biological application, is solved to demonstrate the effectiveness and feasibility of the proposed theoretical approach and algorithms.

Modified CGLS iterative algorithm for solving the generalized Sylvester-conjugate matrix equation

By introducing the real inner product, this paper offers an modified conjugate gradient least squares iterative algorithm (MCGLS)for solving the generalized Sylvester-conjugate matrix equation. The properties of this algorithm are discussed and the finite convergence of this algorithm is proven. This new iterative method can obtain the symmetric least squares Frobenius norm solution within finite iteration steps in the absence of roundoff errors. Finally, two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.

AN ITERATIVE ALGORITHM FOR SOLVING A CLASS OF GENERALIZED COUPLED SYLVESTER-TRANSPOSE MATRIX EQUATIONS OVER BISYMMETRIC OR SKEW-ANTI-SYMMETRIC MATRICES

This paper presents an iterative algorithm to solve a class of generalized coupled Sylvester-transpose matrix equations over bisymmetric or skew-anti-symmetric matrices. When the matrix equations are consistent, the bisymmetric or skew-anti-symmetric solutions can be obtained within finite iteration steps in the absence of round-off errors for any initial bisymmetric or skew-anti-symmetric matrices by the proposed iterative algorithm. In addition, we can obtain the least norm solution by choosing the special initial matrices. Finally, numerical examples are given to demonstrate the iterative algorithm is quite efficient. The merit of our method is that it is easy to implement.

Iterative algorithm for the reflexive solutions of the generalized Sylvester matrix equation

In this paper, the generalized Sylvester matrix equation AV + BW = EVF + C over reflexive matrices is considered. An iterative algorithm for obtaining reflexive solutions of this matrix equation is introduced. When this matrix equation is consistent over reflexive solutions then for any initial reflexive matrix, the solution can be obtained within finite iteration steps. Furthermore, the complexity and the convergence analysis for the proposed algorithm are given. The least Frobenius norm reflexive solutions can also be obtained when special initial reflexive matrices are chosen. Finally, numerical examples are given to illustrate the effectiveness of the proposed algorithm.

Minimum norm partial quadratic eigenvalue assignment for vibrating structures using receptance method

Abstract In this paper, we consider the minimum norm partial quadratic eigenvalue assignment problem (MNPQEAP) for multi-input vibration control systems. Our work is based on the recent paper proposed by Bai et al. (2018) who solve this problem by using receptances and system matrices. By using receptance method, we transform this problem into finding the minimum norm solutions of the matrix equations and propose a modified conjugate gradient (MCG) method for solving this problem. Our method can solve the MNPQEAP with finite iteration steps in the absence of roundoff errors. The numerical examples show the efficiency of our method.

Optimal control of Boolean control networks with average cost: A policy iteration approach

This paper deals with the infinite horizon optimal control problem for deterministic Boolean control networks (BCNs) with average cost. Based on the semi-tensor product of matrices and Jordan decomposition technique, a nested optimality equation for the average infinite horizon problem of BCNs is presented. By resorting to Laurent series expression, a novel policy iteration algorithm, which can find the optimal state feedback controller in finite iteration steps, is proposed. Finally, as a practical application, the optimal intervention problem of Ara operon in E. coil is addressed.