Article
Mathematics, Applied
Minghua Lin, Teng Zhang
Summary: In this paper, we investigate the eigenvalues of the solution to the continuous algebraic Riccati equation without assuming the positivity of A+A(T). We also propose a possible improvement and provide a numerical example to illustrate the new result.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Automation & Control Systems
Mohammad Akbari, Bahman Gharesifard, Tamas Linder
Summary: The article provides a set of counterexamples to illustrate the monotonicity of the Newton-Hewer method in solving the discrete-time algebraic Riccati equation in dynamic settings, contrasting it with the Riccati difference equation.
Article
Mathematics, Interdisciplinary Applications
Bo Yu, Chengxu Jiang, Ning Dong
Summary: This paper presents a computational method for a class of discrete-time algebraic Riccati equations (DAREs) with a low-ranked coefficient matrix G and a high-ranked constant matrix H, using a structured doubling algorithm to solve large-scale problems. Compared to the existing doubling algorithm with O(2(k)n) flops at the k-th iteration, the newly developed version requires only O(n) flops for preprocessing and O((k+1)(3)m(3)) flops for iterations, making it more suitable for large-scale computations when mMUCH LESS-THANn. The convergence and complexity of the algorithm are subsequently analyzed. Illustrative numerical experiments show that the presented algorithm, which consists of a dominant time-consuming preprocessing step and a trivially iterative step, is capable of efficiently computing the solution for large-scale DAREs.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Xiongyu Shao, Yimin Wei, Eric King-wah Chu
Summary: Inspired by the study of complex nonsymmetric algebraic Riccati equations, this paper investigates nonsymmetric algebraic Riccati equations for quaternionic matrices. The comparison matrix of a quaternionic matrix is introduced, extending that of a complex matrix, and is used to provide a condition for the existence and uniqueness of the extremal solution. The solution can be obtained by a fixed-point iteration.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Applied
Yongdo Lim
Summary: In this study, we investigate the Lie semigroup of symplectic Hamiltonians acting on positive definite matrices via linear fractional transformations. Our findings include the strict contraction of the invariant Finsler metric and the Thompson metric for each member of the interior, as well as the existence of a unique positive definite fixed point.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics, Applied
Changli Liu, Wei-Guo Wang, Jungong Xue, Ren-Cang Li
Summary: This paper investigates a M-matrix algebraic Riccati equation, decomposes it into multiple coupled algebraic Riccati equations, and solves them using a doubling algorithm. It is shown that these equations have minimal nonnegative solutions during the doubling iterations, and a highly accurate implementation of the doubling algorithm is designed to compute high relative accuracies for the solutions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Hung-Yuan Fan, Chun-Yueh Chiang
Summary: In this paper, we study a class of conjugate discrete-time Riccati equations that arise from the linear quadratic regulation problem for discrete-time antilinear systems. We provide a constructive proof for the existence of the maximal solution to the conjugate discrete-time Riccati equation under mild assumptions and the fixed-point iteration framework. The control weighting matrix in the maximal solution is nonsingular and has a Hermitian constant term. We also show that the fixed-point iteration generates a nonincreasing sequence that converges at least linearly to the maximal solution of the Riccati equation. An example is provided to demonstrate the correctness of the main theorem and offer insights into the study of other meaningful solutions.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Jinrui Guan, Zhixin Wang
Summary: This paper studies the numerical solution of nonsymmetric algebraic Riccati equations and proposes a generalized ALI iteration method to solve the equation. Theoretical analysis and numerical experiments demonstrate the feasibility and effectiveness of the proposed method compared to the ALI iteration method and the MALI iteration method.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Akbar Shirilord, Mehdi Dehghan
Summary: In this work, a closed-form expression for the solution of the non-symmetric algebraic Riccati equation (NARE) is obtained, under certain conditions on the matrices involved. It is shown that the solution of this NARE is given by X = (root A(2) + DC - A)C-1, where root M denotes the square root of matrix M. The formula is then applied to some NAREs.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Peter Chang-Yi Weng
Summary: The article discusses the solution to the large-scale nonsymmetric algebraic Riccati equation using a structure-preserving doubling algorithm. By applying the appropriate mathematical formulas and sparse plus-low-rank representations, the algorithm achieves a computational complexity of O(n) and essentially quadratic convergence.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Automation & Control Systems
Jiachen Qian, Peihu Duan, Zhisheng Duan, Guanrong Chen, Ling Shi
Summary: This paper investigates the performance of measurement-based distributed filtering with finite consensus fusion operations. By introducing a modified discrete-time algebraic Riccati equation and novel techniques, the convergence of estimation error covariance matrix is guaranteed, and the relation between performance degradation and reduced fusion frequency is established. Moreover, it is shown that the estimation error covariance matrix exponentially converges to the optimal steady-state covariance matrix with infinite fusion steps during each sampling interval.
Article
Automation & Control Systems
Li Wang, Yuli Zhu
Summary: In this paper, a new inverse-free iterative algorithm is proposed to obtain the positive definite solution of the discrete algebraic Riccati equation (DARE). Monotone convergence is proved and convergence rate analysis is presented for the derived algorithm. Numerical examples demonstrate the feasibility and effectiveness of the proposed algorithm.
IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION
(2023)
Article
Automation & Control Systems
Jianzhou Liu, Zheng Wang, Zhiming Xie, Li Wang
Summary: In practical engineering, many control problems can be transformed into solutions of the discrete algebraic Riccati equation (DARE), which involves matrix inverse operations. This paper proposes a method to transform a DARE with multiple inversions into an equivalent form with a single inversion, and presents an iterative algorithm for solving it. The paper also introduces a new iterative algorithm for a special case of DARE, which avoids matrix inversions. The monotone convergence and error analysis of the algorithms are proven, and numerical examples validate the superiority and effectiveness of the proposed methods.
IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION
(2022)
Article
Automation & Control Systems
P. I. E. R. R. E. DEL MORAL, E. M. M. A. HORTON
Summary: In this article, a self-contained study of discrete time Riccati matrix difference equations is provided, including a novel Riccati semigroup duality formula and a new Floquet-type representation. The impact of these formulae is illustrated through an explicit description of the solution of time-varying Riccati difference equations and its fundamental-type solution, as well as uniform upper and lower bounds on the Riccati maps.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2022)
Article
Mathematics
Ivan G. Ivanov, Hongli Yang
Summary: This paper investigates iterative methods for solving various types of nonlinear matrix equations. Specifically, it focuses on iterative methods for finding the minimal nonnegative solution of a set of Riccati equations, the nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation X+A*X-1A=Q. The paper studies recent iterative methods for solving these specific types of equations, proposes more effective modifications to these methods, and demonstrates their effectiveness through illustrative examples.
