Large-scale algebraic Riccati equations with high-rank constant terms

Yu, Bo; Fan, Hung Yuan; Chu, Eric King-wah

doi:10.1016/j.cam.2019.04.014

Cited by 11 publications

(6 citation statements)

References 26 publications

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“…Although the deflation of the low-rank factors and kernels in the last section can reduce dimensional growth, the exponential increment of the undeflated part is still rapid, making large-scale computation and storage infeasible. Conventionally, one efficient way to shrink the column number of low-rank factors is by truncation and compression (TC) [17,18], which, unfortunately, is hard to be applied to our case due to the following two main obstacles.…”

Section: Partial Truncation and Compressionmentioning

confidence: 99%

“…However, when the constant term H in the DARE equation has a high-rank structure, the stabilizing solution is no longer numerically low-ranked, making its storage and outputting difficult. To solve this issue, an adapted version of the doubling algorithm, named SDA_h, was proposed in [18]. The main idea behind SDA_h is to take advantage of the numerical low-rank of the stabilizing solution in the dual equation to estimate the residual of the original DARE.…”

Section: Introductionmentioning

confidence: 99%

“…• A deflation process of the low-rank factors is proposed to reduce the column number of the low-rank part. The conventional truncation and compression in [17,18] for the whole low-rank factor does not to work as it destroys the implicit structure and makes the subsequent deflation infeasible. Instead, a partial truncation and compression (PTC) technique is then devised to impose merely on the exponentially increasing part (after deflation), effectively slimming the dimensions of the low-rank factors.…”

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confidence: 99%

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Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System

Dong

2023

Fractal Fract

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In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures of the state matrix in these systems (e.g., being banded-plus-low-rank) could be beneficial for large-scale computation. In this paper, a factorized structure-preserving doubling algorithm (FSDA) is developed under the assumptions that the non-linear and constant terms are positive semidefinite and banded-plus-low-rank. The detailed iteration scheme and a deflation process for FSDA are analyzed. Additionally, a technique of partial truncation and compression is introduced to reduce the dimensions of the low-rank factors. The computation of residual and the termination condition of the structured version are also redesigned. Illustrative numerical examples show that the proposed FSDA outperforms SDA with hierarchical matrices toolbox (SDA_HODLR) on CPU time for large-scale problems.

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Section: Partial Truncation and Compressionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System

Dong

2023

Fractal Fract

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“…Riccati equations are a recurrent and important feature in many theoretical control design results and have been the subject of study for a long time, including [1] and more recently [2]. When the unique positive semidefinite solution of a Riccati equation is time invariant we then fall back into the particular, albeit ubiquitous, subset known as algebraic Riccati equations (AREs) [3]- [5]. AREs have played a central role in many control design synthesis procedures, [6]- [8], including H 2 optimal control [9] and H ∞ optimal control [10]- [12].…”

Section: Introductionmentioning

confidence: 99%

Modified Algebraic Riccati Equation Closed-Form Stabilizing Solution

Rojas¹

2021

IEEE Access

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“…where ω(x) is an unknown complex function to be found, μ(x) : [0, T] → C and R(x, t, ω(t)) : [0, T] 2 × C → C are continuous and Lipschitzian periodic functions such that as Maxwell's equations, biological, radiative energy, engineering problems, potential theory, and transfers problems of oscillations that can be formulated by this equation and fractional integro-differential equations; see [5][6][7]. Some numerical algorithms that discuss the approximation of the solution of IDE can be listed such as the nonsmooth initial data arising method [8], Haar and RH methods [9][10][11], cubic B-spline finite element method [12], Runge-Kutta-Nystrom methods [13,14], and high-rank constant terms [15]. Furthermore, in [16,17], by using a system of Cauchy type and numerical method with graded meshes, singular integral equations were solved.…”

Section: Introductionmentioning

confidence: 99%

Using of PQWs for solving NFID in the complex plane

2020

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We approximate the solution of the nonlinear Fredholm integro-differential equation (NFID) in the complex plane by periodic quasi-wavelets (PQWs). This kind of wavelets possesses orthonormality properties, the numbers of terms in the decomposition and reconstruction formulas are strictly limited, and the localization is not emphasized. To the best of our knowledge, there are no numerical methods to obtain the solution of the NFID by PQWs. Here, we attempt to obtain the numerical solution of the NFID based on B-spline functions. Finally, the simulation results are shown for three examples. Keywords: Nonlinear integro Fredholm integral equation; PQWswhere M is a Lipschitz constant.We approximate the solution of NFIDE by using the B-spline basis functions and applying the iterative method [4] in each iteration on the complex plane. Every integro-differential equation (IDE) is an ordinary differential equation in which one of the variables is integral. There are many equations in mathematical modeling, such © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.

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Large-scale algebraic Riccati equations with high-rank constant terms

Cited by 11 publications

References 26 publications

Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System

Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System

Modified Algebraic Riccati Equation Closed-Form Stabilizing Solution

Using of PQWs for solving NFID in the complex plane

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