On common diagonal Lyapunov solutions

Gümüş, Mehmet; Xu, Jianhong

doi:10.1016/j.laa.2016.05.032

Cited by 8 publications

(5 citation statements)

References 17 publications

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“…where

P \in ℜ^{n \times n}

is a p.d. diagonal matrix referred to as the diagonal Lyapunov solution 42 . The existence of such a diagonal CLF for a given nonlinear system (2), which ensures g.a.s of the equilibrium point at the origin, cannot be guaranteed.…”

Section: Problem Formulation and Relation To Diagonal Stabilitymentioning

confidence: 99%

“…diagonal matrix referred to as the diagonal Lyapunov solution. 42 The existence of such a diagonal CLF for a given nonlinear system (2), which ensures g.a.s of the equilibrium point at the origin, cannot be guaranteed. However, such a diagonal CLF would be capable of assuring asymptotic stability of the feedback system over a certain domain in state space.…”

Section: Formulation Of Ioc As a D-stability Problemmentioning

confidence: 99%

“…Thus D‐stability of

\overset{true^}{A} false(x false)

S_{δ}

ensures: (i) existence of a diagonal CLF of the form (7), and (ii) determination of a domain

ψ = false{x \in ℜ^{n} : \dot{V} false(x false) < 0 false}

where

ψ > > S_{δ}

. While D‐stability of real symmetric matrices has been widely studied, 32,33,36–42 D‐stability of SDC matrices has not been studied to the best of our knowledge. In the next section, the criteria for the D‐stability of the SDC matrix

\overset{true^}{A} false(x false)

are established followed by the systematic construction of IOC feedback for the fulfillment of these criteria.…”

Section: Problem Formulation and Relation To Diagonal Stabilitymentioning

confidence: 99%

“…In Reference 41, Kalman–Yacubovich–Popov lemma is used to derive the results in Reference 36 which also leads to other generalizations with respect to D‐stability and Hurwitz stability of typical class of matrices. Another shorter proof for these results along with extensions are presented in Reference 42. Basically, the results are rephrased using the notion of Schur complements which imparts both theoretical and computational advantages (by avoiding explicit matrix inversion).…”

Section: Notations and Mathematical Preliminariesmentioning

confidence: 99%

“…In the present work, the IOC problem is transformed to a D‐stability problem of SDC factored system matrix. Therefore, the matrix theoretic characterization of D‐stability in Reference 42 is extended and redefined to propose a novel concept of D‐stability characterization of SDC matrices. Now, the reformulation of the IOC problem as a D‐stability problem is presented in the following section.…”

Section: Notations and Mathematical Preliminariesmentioning

confidence: 99%

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A novel domain of attraction based synthesis of inverse optimal control

Prasanna

Jacob

Nandakumar

2021

Intl J Robust & Nonlinear

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This article proposes a novel and systematic methodology for the inverse optimal control (IOC) of a class of input affine nonlinear systems. First, the system is represented in the pseudo-linear state-dependent coefficient (SDC) form so as to facilitate the use of matrix algebraic concepts in the design, while preserving the nonlinear system dynamics. Second, the IOC problem which seeks the solution to the Hamilton-Jacobi-Bellman equation is translated to a diagonal stability (D-stability) problem. Therefore, the general notion of D-stability, which deals with the characterization of a matrix A ∈ ℜ n×n for the existence of a diagonal matrix Γ ∈ ℜ n×n such that A T Γ + ΓA < 0, is redefined with respect to SDC factored matrices in this work. Consequently, the criteria for the existence (necessary condition) and determination of a diagonal solution (sufficient condition) are formulated in terms of the SDC matrices. It is shown in this work that these criteria can be readily met by the closed-loop system matrix Â(x) = A(x) − D, for any arbitrary domain S 𝛿 in state-space, by manipulating only the diagonal elements using the diagonal matrix D constituted by the IOC feedback. In short, the IOC feedback is designed such that the criteria for D-stability by the closed-loop system matrix Â(x) is fulfilled. Hence this novel design methodology is meaningfully named as "Inverse optimal control via diagonal stabilization" (IOC-D).The main advantages of IOC-D include: (i) guaranteed estimate of the domain of attraction (DOA), (ii) Immense flexibility of design, and (iii) guaranteed closed form solution which makes the design procedure analytic and computationally efficient. Numerical simulations validate the theoretical developments and the overall efficiency of the proposed approach.

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“…where