Large-scale discrete-time algebraic Riccati equations— Doubling algorithm and error analysis

Chu, Eric King-wah; Weng, Peter Chang-Yi

doi:10.1016/j.cam.2014.09.005

Cited by 12 publications

(7 citation statements)

References 20 publications

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“…To develop the doubling Smith method, we first use the generalized Cayley transform [17,20,25] (Section 2) to convert the Sylvester Equation ( 12) to the Stein equation. By introducing two positive parameters µ and ν, Equation ( 12) can be rewritten as…”

Section: Doubling Smith Methods With the Optimal Parametersmentioning

confidence: 99%

Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

Dong

2023

Fractal Fract

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The implicit difference approach is used to discretize a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and of nonsingular M-matrix structure, respectively. A low-ranked doubling Smith method with determined optimally iterative parameters is presented for solving the corresponding matrix equation. In comparison to the existing Krylov solver with Fast Fourier Transform (FFT) for the sequence Toeplitz linear system, numerical examples demonstrate that the proposed method is more effective on CPU time for solving large-scale problems.

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Section: Doubling Smith Methods With the Optimal Parametersmentioning

confidence: 99%

Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

Dong

2023

Fractal Fract

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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%

“…In many large-scale control problems, the matrix G = BR −1 B in the non-linear term and H = C T −1 C in the constant term are of low-rank with B ∈ R N×m g , R ∈ R m g ×m g , C ∈ R m h ×N , T ∈ R m h ×m h , and m g , m h N. Then the unique positive definite stabilizing solution in the DARE (3) or its dual equation can be approximated numerically by a lowrank matrix [16,17]. 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.…”

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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“…In large scale systems it is then advised to apply numerical algorithms adjusted for calculations involving sparse matrices, e.g. [14].…”

Section: Remarkmentioning

confidence: 99%

Augmented state space approach for solving infinite-horizon linear-quadratic problem in discrete-time systems with mulitple time-delays

Ignaciuk

2015

2015 19th International Conference on System Theory, Control and Computing (ICSTCC)

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In this paper, the infinite-horizon optimal control problem with quadratic performance index in discrete-time multi-delay systems is addressed. The classical approach of using an augmented state space for single-delay systems is extended to the multi-delay case. The solvability conditions for the considered state-space representation are established and control system robustness to uncertainties is formally evaluated. The optimal solution is given in a state-feedback form with the design equations consistent with the standard ones for undelayed systems. Thus, the problem can be treated with the existing numerical algorithms and solver packages. The analytical findings are illustrated in a simulation scenario concerning optimal control of a logistic network.

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Large-scale discrete-time algebraic Riccati equations— Doubling algorithm and error analysis

Cited by 12 publications

References 20 publications

Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

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

Augmented state space approach for solving infinite-horizon linear-quadratic problem in discrete-time systems with mulitple time-delays

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