A note on the extremal properties of the discrete-time Lyapunov matrix equation

Fahmy, Mahmoud; Hanafy, A.

doi:10.1016/s0019-9958(79)90775-7

Cited by 3 publications

(4 citation statements)

References 3 publications

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“…Observe that we have assumed |σ * i σ j | < 1, ∀i, j while using the Taylor series expansion, which has been known previously [5]. Substituting (9) in (8), we have…”

Section: Closed-form Solution Of the Dlementioning

confidence: 99%

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An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

Patel¹,

Khan²

2021

Preprint

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A simple closed-form solution to the discrete Lyapunov equation (DLE) is established for certain families of matrices. This solution is expressed in terms of the eigen decomposition (ED) for which closed-form solutions are known for all 2 × 2, 3 × 3 and certain families of matrices. For general matrices, the proposed ED based closed-form solution can be used as an efficient numerical solution when the ED can be computed. The result is then extended to give closed-form solutions for a generalization of the DLE, called the Stein equation. The proposed explicit solution's complexity is of the same order as iterative solutions and significantly smaller than known closed-form solutions. These solutions may prove convenient for analysis and synthesis problems related to these equations due to their compact form.

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“…Observe that we have assumed |σ * i σ j | < 1, ∀i, j while using the Taylor series expansion, which has been known previously [5]. Substituting (9) in (8), we have…”

Section: Closed-form Solution Of the Dlementioning

confidence: 99%

“…For an arbitrary A, the integral forms [6] and the matrix power form [7] were proposed. Sensitivity of the Lyapunov equations was studied in [8]- [10].…”

Section: Introductionmentioning

confidence: 99%

An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

Patel¹,

Khan²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…For an arbitrary A, the integral forms [6] and the matrix power form [7] were proposed. Sensitivity of the Lyapunov equations was studied in [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

Patel

Khan

2022

Preprint

View full text Add to dashboard Cite

A simple closed-form solution to the discrete Lyapunov equation(DLE) is established for certain families of matrices. This solution isexpressed in terms of the eigen decomposition (ED) for which closed-form solutions are known for all 2 × 2, 3 × 3 and certain familiesof matrices. For general matrices, the proposed ED based closed-formsolution can be used as an efficient numerical solution when the EDcan be computed. The result is then extended to give closed-formsolutions for a generalization of the DLE, called the Stein equation. The proposed explicit solution’s complexity is of the same order asiterative solutions and significantly smaller than known closed-formsolutions. These solutions may prove convenient for analysis and syn-thesis problems related to these equations due to their compact form.

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“…When the matrix A is in arbitrary form, perhaps the well known ones are the integral forms in [7] of the unique solution to the continuous‐time Lyapunov matrix equation, and the matrix power form in [8] of the unique solution to the discrete‐time Lyapunov matrix equation. In [9], a lower bound of the minimal eigenvalue of the solution to the discrete‐time Lyapunov equation was given in terms of the coefficient matrices. The sensitivity of the solution of the stable continuous‐time Lyapunov equations was analysed in [10] by using the concept of doubly stochastic linear maps.…”

Section: Introductionmentioning

confidence: 99%

Explicit solutions to the matrix equation E X F − AX = C

2013

IET Control Theory & Applications

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The matrix equation EX F − AX = C is investigated in this study, and three approaches are provided to solve this equation. The first approach is to transform it into a real matrix equation with the help of real representation of complex matrices. In the second approach, the solution is given in terms of characteristic polynomial of a constructed matrix pair. In the third approach, the solution can be neatly expressed in terms of controllability matrices and observability matrices. By specialising the obtained solutions of EX F − AX = C, some new expressions of the generalised Sylvester matrix equations are also provided.

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A note on the extremal properties of the discrete-time Lyapunov matrix equation

Cited by 3 publications

References 3 publications

An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

Explicit solutions to the matrix equation E X F − AX = C

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