Applications of Matrix Equations in Linear Time-Invariant Systems

ZHOU Yan-ping, CHEN Yan-ping, ZHANG Juan

doi:10.13371/j.cnki.chin.q.j.m.2025.03.001

Chinese Quarterly Journal of Mathematics >

2025 , Vol. 40 >Issue 3: 221 - 237

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2025.03.001

Applications of Matrix Equations in Linear Time-Invariant Systems

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1. School of Mathematics and Physics, Three Gorges Mathematical Research Center, Three Gorges University, Yichang 443002, China; 2. School of Science, Posts and Telecommunications University, Nanjing 210023, China 3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

ZHOU Yan-ping (1980-), female, native of Yichang, Hubei, associate professor of China Three Gorges University, mainly engages in convex geometric analysis and partial differential equations; CHEN Yan-ping (1963-), female, native of Nanjing, Jiangsu, professor of Nanjing University of Posts and Telecommunications, mainly engages in finite elements and adaptive algorithms; ZHANG Juan (1983-), female, native of Xiangtan, Hunan, professor of Xiangtan University, mainly engages in research on numerical algebra, control theory, and optimization algorithms.

CHEN Yan-ping (1963-), female, native of Nanjing, Jiangsu, professor of Nanjing University of Posts and Telecommunications, mainly engages in finite elements and adaptive algorithms.

Received date: 2025-05-17

Online published: 2025-09-16

Supported by

Supported by National Natural Science Foundation of China (Grant No. 12571388); the Visiting Scholar Program of National Natural Science Foundation of China (Grant No. 12426616); Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications (Grant No. NY223127).

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Abstract

With the development of science and technology, the design and optimization of control systems are widely applied. This paper focuses on the application of matrix equations in linear time-invariant systems. Taking the inverted pendulum model as an example, the algebraic Riccati equation is used to solve the optimal control problem, and the system performance and stability are achieved by selecting the closed-loop pole and designing the gain matrix. Then, the numerical methods for solving the stochastic algebraic Riccati equations are applied to practical problems, with Newton’s iteration method as the outer iteration and the solution of the mixed-type Lyapunov equations as the inner iteration. Two methods for solving the Lyapunov equations are introduced, providing references for related research.

Key words： Algebraic Riccati equation; Linear time-invariant system; LQR method; Solution of mixed Lyapunov equation

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ZHOU Yan-ping, CHEN Yan-ping, ZHANG Juan . Applications of Matrix Equations in Linear Time-Invariant Systems[J]. Chinese Quarterly Journal of Mathematics, 2025 , 40(3) : 221 -237 . DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.001

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