Non-existence conditions of local bifurcations for rational systems with structured uncertainties

Kishida, Masako; Braatz, Richard D.

doi:10.1109/acc.2014.6858689

Cited by 5 publications

(2 citation statements)

References 34 publications

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“…See [3] for the construction of an LFT for this example. Example 3 Possible parameter values for p 1 , p 2 , and p 3 (a higher‐dimensional case) A predator‐prey model is given by [34, 35]

\begin{matrix} right left1em4pt \\ {\dot{x}}_{1} & = x_{1} (1 - x_{1}) - \frac{p_{1} x_{1} x_{2}}{p_{3} + x_{1}}, \\ {\dot{x}}_{2} & = - p_{2} x_{2} + \frac{p_{1} x_{1} x_{2}}{p_{3} + x_{1}}, \end{matrix}

where x 1 and x 2 are scaled population numbers, and p 1 , p 2 , p 3 are parameters that characterise the behaviour of the system. For this system to avoid bifurcations, the parameters must satisfy two conditions: one to avoid a steady‐state bifurcation and one to avoid a Hopf bifurcation; these conditions can be simplified to two scalar conditions [36]. …”

Section: Numerical Examplesmentioning

confidence: 99%

Quality‐by‐design by skewed spherical structured singular value

Kishida

Braatz

2015

IET Control Theory & Applications

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This study develops numerical algorithms to compute an ellipsoidal set of input parameters, called a design space, that ensures that the system outputs lie within a set of design specifications in quality-by-design. The algorithm is based on a proposed skewed spherical structured singular value ν s , for which this study derives upper bounds and proves the (scaled) main loop theorems and small-gain theorem for the Frobenius norm. Three examples are included to illustrate applications of the numerical algorithms. 2 Mathematical preliminaries The set of real numbers, real vectors of length n, and real matrices of size n × m are denoted by R, R n , and R n×m , respectively. The set of complex numbers, real vectors of length n, and real matrices of size n × m are denoted by C, C n , and C n×m , respectively. For a vector v ∈ R n , v T denotes the transpose of the vector v, v i denotes the ith element of v, and v 2 = √ v T v. For a matrix M ∈ R n×m , M T denotes the transpose of the matrix M , for a matrix

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“…See [3] for the construction of an LFT for this example. Example 3 Possible parameter values for p 1 , p 2 , and p 3 (a higher‐dimensional case) A predator‐prey model is given by [34, 35]

\begin{matrix} right left1em4pt \\ {\dot{x}}_{1} & = x_{1} (1 - x_{1}) - \frac{p_{1} x_{1} x_{2}}{p_{3} + x_{1}}, \\ {\dot{x}}_{2} & = - p_{2} x_{2} + \frac{p_{1} x_{1} x_{2}}{p_{3} + x_{1}}, \end{matrix}

Section: Numerical Examplesmentioning

confidence: 99%

Quality‐by‐design by skewed spherical structured singular value

Kishida

Braatz

2015

IET Control Theory & Applications

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“…The class of positive matrices like level symmetric matrices and Hermitian positive definite matrices are widely used in mathematics and in various applications of engineering. For instance, computer vision (Nemirovskii, 1993), the mechine learning (Kishida and Braatz, 2014) and in the area of convex optimization (Braatz et al, 1994). The Linear Matrix Inequality (LMI) technique based on the positive definiteness nature of matrix is widely used to study the stability analysis of feedback systems in linear control.…”

Section: Introductionmentioning

confidence: 99%

An analytical approach to compute lower bounds of 𝝁-values

al.¹

2020

Int. j. adv. appl. sci.

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The lower bounds of 𝜇 −values for a family of square real and complex valued matrices are computed analytically. The proposed methodology consists of factorizing an admissible perturbation from a set of block diagonal matrices into a block diagonal matrix. The computation of the lower bounds of 𝜇 −values is then carried out by computing the spectral radius and numerical radius for matrix under consideration. The lower bounds of 𝜇 −value provide the conditions which guarantee the instability analysis of the linear feedback system.

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Networks — An Overview

Malby¹,

Anderson-Wallace²

2016

Networks in Healthcare

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Non-existence conditions of local bifurcations for rational systems with structured uncertainties

Cited by 5 publications

References 34 publications

Quality‐by‐design by skewed spherical structured singular value

Quality‐by‐design by skewed spherical structured singular value

An analytical approach to compute lower bounds of 𝝁-values

Networks — An Overview

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