Structured Doubling Algorithm for Discrete-Time Algebraic Riccati Equations With Singular Control Weighting Matrices

Chiang, Chun Yueh; Fan, Hung Yuan; Lin, Wen Wei

doi:10.11650/twjm/1500405875

Cited by 12 publications

(10 citation statements)

References 19 publications

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“…The Euclidean norm of the ECT V (P, α) bounds the asymptotic convergence rates of many matrix splitting iteration methods such as (i) the alternating direction implicit (ADI) method [58,1,36,61], the Hermitian and skew-Hermitian splitting (HSS) method [13,10], the normal and skew-Hermitian splitting (NSS) method [14], the positive-definite and skew-Hermitian splitting (PSS) method [12], the shift-splitting preconditioning method [18] and the triangular skew-Hermitian splitting method [50,60,17,51] for solving large sparse and non-Hermitian positive-definite systems of linear equations, (ii) the preconditioned HSS (PHSS) method [15,11], the accelerated HSS (AHSS) method [9,4], the dimensional split preconditioning method [20] and the block alternating splitting implicit (BASI) method [7] for solving large sparse saddle-point linear systems, (iii) the modulus method [22,47,57,52], the modified modulus method [28], the extrapolated modulus method [38,40,37] and the modulus-based splitting methods [6] for solving large sparse linear complementarity problems, and (iv) the alternately linearized implicit (ALI) method [16], the structure-preserving doubling algorithm [35,54,43,23] and the inexact Newton methods based on doubling iteration scheme [32] for computing the minimal nonnegative solutions of large sparse nonsymmetric algebraic Riccati equations; see also [31,15,21,…”

Section: Introductionmentioning

confidence: 99%

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

Bai

Hadjidimos

2014

Linear Algebra and its Applications

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For the extrapolated Cayley transform, we give necessary and sufficient conditions for guaranteeing its convergence and contraction (in the Euclidean norm). We derive upper bounds for the convergence and the contraction factors, and compute the optimal parameters minimizing these upper bounds and the corresponding optimal values of these upper bounds. Numerical computations show that these upper bounds are reasonably sharp compared with the exact convergence and the exact contraction factors of the extrapolated Cayley transform, respectively.

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Section: Introductionmentioning

confidence: 99%

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

Bai

Hadjidimos

2014

Linear Algebra and its Applications

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“…It is interesting to point out that (22), obtained by applying CR to the solution of the UQME (18), coincides with SDA of [16,32,41]. In the critical case where H is singular and µ = 0, the convergence of the doubling algorithms is linear as shown in [15,25].…”

Section: Doubling Algorithmsmentioning

confidence: 92%

“…The research activity concerning the analysis of NAREs associated with Mmatrices and the design of numerical algorithms for their solution has had a strong acceleration in the last decade. Important progress has been made concerning theoretical properties of this class of matrix equations and new effective algorithms relying on the properties of M-matrices have been designed and analyzed [6,10,11,13,15,18,20,21,23,24,25,26,32,35,41,42,43].…”

Section: Introductionmentioning

confidence: 99%

“…, x 2p be the roots of a(z) = det(A 0 + zA 1 + z 2 A 2 ), including roots at the infinity if deg a(z) < 2p, ordered by increasing modulus. Suppose that |x p | 1 |x p+1 | and |x p | < |x p+1 |, and that a solution G exists to(15) such that ρ(G) = |x p |. Then, G is the unique solution to (15) with minimal spectral radius, moreover, if CR (16) can be carried out with no breakdown, the sequence…”

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confidence: 99%

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Nonsymmetric Algebraic Riccati Equations Associated with an M-Matrix: Recent Advances and Algorithms

Бини

Iannazzo

Meini

et al. 2010

Matrix Methods: Theory, Algorithms and Applications

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We survey theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton's iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence.

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“…For large-scale DRREs, when R(X, 0) = R > 0, the methods in [43] can be used. When R is singular, the shifting technique in [14] can first be applied to remove the singularity, before the application of the methods in [43]. More efficiently, we may replace the zero eigenvalues in R, justified by [23], and then apply [43] to obtain a stabilizing X 0 .…”

Section: Computing Issues and Operation Countsmentioning

confidence: 99%

Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control

Zhang

Fan²,

Chu

et al. 2015

SIAM J. Sci. Comput.

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We consider the numerical solution of the rational algebraic Riccati equations in R n , arising from stochastic optimal control in continuous and discrete time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of O(n 3 ) computational complexity and O(n 2 ) memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided. Introduction.In this paper, we consider the numerical solution of rational Riccati equations (RREs) arising in stochastic optimal control. Linear time-invariant (LTI) systems [2,46,53] with stochastic components will be considered. These can be considered to be local approximations to more general nonlinear and time-variant systems. The theory of LTI systems has been vital in control system theory and has contributed successfully to many real-life applications. In other words, the solution of the difficult RREs, which leads to the optimal control of the corresponding LTI systems with stochastic components, is of some importance to control engineers in many applications.

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Structured Doubling Algorithm for Discrete-Time Algebraic Riccati Equations With Singular Control Weighting Matrices

Cited by 12 publications

References 19 publications

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

Nonsymmetric Algebraic Riccati Equations Associated with an M-Matrix: Recent Advances and Algorithms

Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control

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