# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

March 2021 , Volume 11 , Issue 1

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2021, 11(1): 1-12 doi: 10.3934/naco.2020011 +[Abstract](1199) +[HTML](525) +[PDF](424.52KB)
Abstract:

A method of designing observer-based feedback controller against actuator failures for uncertain singular fractional order systems (SFOS) is presented in this paper. By establishing actuator fault model and state observer, an observer-based fault-tolerant state feedback controller is developed such that the closed-loop SFOS is admissible. The controller designed by the proposed method guarantees that the closed-loop system is regular, impulse-free and stable in the event of actuator failures. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.

2021, 11(1): 13-26 doi: 10.3934/naco.2020012 +[Abstract](1025) +[HTML](562) +[PDF](351.1KB)
Abstract:

The decoupling of polynomial matrix system is to diagonalize its system matrix. In this paper, decoupling problems for cubic polynomial matrix system are considered. The decoupling conditions for a class of cubic polynomial matrix systems are derived under strict equivalence transformation. By using linearization, isospectral decoupling method for cubic polynomial matrix system is proposed. To be specific, necessary and sufficient conditions of isospectral diagonalization for nonsingular cubic polynomial matrix are given. These results are extended to singular cubic polynomial matrix. Solving processes are given to obtain isospectral diagonal cubic polynomial matrix for nonsingular and singular cases. Finally, illustrating examples are provided to verify the main results.

2021, 11(1): 27-43 doi: 10.3934/naco.2020013 +[Abstract](1147) +[HTML](508) +[PDF](1515.36KB)
Abstract:

In this study, we present a generalization of the successive overrelaxation (GSOR) iteration method to find the solution of the image restoration problem. Moreover, an improved version of the GSOR (IGSOR) method is also given to solve the proposed problem. Convergence of the GSOR and IGSOR methods are investigated. Three numerical examples are given to illustrate the effectiveness and accuracy of the methods.

2021, 11(1): 45-61 doi: 10.3934/naco.2020014 +[Abstract](1450) +[HTML](526) +[PDF](569.23KB)
Abstract:

A bang-bang iteration method equipped with a component-wise line search strategy is introduced to solve unconstrained optimization problems. The main idea of this method is to formulate an unconstrained optimization problem as an optimal control problem to obtain an optimal trajectory. However, the optimal trajectory can only be generated by impulsive control variables and it is a straight line joining a guessed initial point to a minimum point. Thus, a priori bounds are imposed on the control variables in order to obtain a feasible solution. As a result, the optimal trajectory is made up of bang-bang control sub-arcs, which form an iterative model based on the Lyapunov function's theorem. This is to ensure monotonic decrease of the objective function value and convergence to a desirable minimum point. However, a chattering behavior may occur near the solution. To avoid this behavior, the Newton iterations are then applied to the proposed method via a two-phase approach to achieve fast convergence. Numerical experiments show that this new approach is efficient and cost-effective to solve the unconstrained optimization problems.

2021, 11(1): 75-86 doi: 10.3934/naco.2020016 +[Abstract](998) +[HTML](499) +[PDF](345.18KB)
Abstract:

In this paper we derive the extremal ranks and inertias of the matrix \begin{document}$X+X^{\ast}-P$\end{document}, with respect to \begin{document}$X$\end{document}, where \begin{document}$P\in\mathbb{C} _{H}^{n\times n}$\end{document} is given, \begin{document}$X$\end{document} is a least rank solution to the matrix equation \begin{document}$AXB = C$\end{document}, and then give necessary and sufficient conditions for \begin{document}$X+X^{\ast}\succ P$\end{document} \begin{document}$\left( \geq P\text{, }\prec P\text{, }\leq P\right)$\end{document} in the Löwner partial ordering. As consequence, we establish necessary and sufficient conditions for the matrix equation \begin{document}$AXB = C$\end{document} to have a Hermitian Re-positive or Re-negative definite solution.

2021, 11(1): 87-98 doi: 10.3934/naco.2020017 +[Abstract](1618) +[HTML](587) +[PDF](2599.31KB)
Abstract:

Feature selection is a valuable tool in supervised machine learning research fields, such as pattern recognition or classification problems. Feature selection used to eliminate irrelevant and noise features that adversely affect results. Swarm algorithms are usually used in feature selection problem; these algorithms need transfer functions that change search space from continuous to the discrete. However, transfer functions are the backbone of all binary swarm algorithms. Transfer functions in the current formula cannot provide binary swarm algorithms with a fit balance between exploration and exploitation stages. In this work, a feature selection approach based on the binary whale optimization algorithm with different kinds of updating techniques for the time-varying transfer functions is proposed. To evaluate the performance of the proposed method, three of each chemical and biological binary datasets are used. The results proved that BWOA-TV2 has consistency in feature selection and it gives rise to the high accuracy of the classification with more congruent in the convergence. It worth mentioning that the proposed method is proved advance in performance over competitor optimization algorithms, such as particle swarm optimization (PSO) and firefly optimization (FO) that commonly used in this field.

2021, 11(1): 99-115 doi: 10.3934/naco.2020018 +[Abstract](1054) +[HTML](393) +[PDF](555.47KB)
Abstract:

In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.

2021, 11(1): 117-126 doi: 10.3934/naco.2020019 +[Abstract](1036) +[HTML](451) +[PDF](342.11KB)
Abstract:

A PID control method which combined optimal control strategy is proposed in this paper. The posterior unmodeled dynamics measurement data information are made full use to compensate the unknown nonlinearity of the system, and the unknown increment of the unmodeled dynamics is estimated. Then, a nonlinear PID controller with compensation of the posterior unmodeled dynamics measurement data and the estimation of the increment of the unmodeled dynamics is designed. Finally, through the numerical simulation, the effectiveness of the proposed method is vertified.

2021, 11(1): 127-142 doi: 10.3934/naco.2020020 +[Abstract](972) +[HTML](426) +[PDF](424.91KB)
Abstract:

This paper investigates the problem of dissipative control for a class of uncertain singular Markovian jump systems. Different from the traditional control strategy, a derivative gain and impulsive control part are added in the proposed controller. A linearization approach via congruence transformations is proposed to solve the feedback design problem. In addition, the derived results contain \begin{document}$H_{\infty}$\end{document} and passive control as special cases. Finally, examples are provided to illustrate the effectiveness and applicability of the proposed methods.