All Solutions of a Bitangential Interpolation Problem that Includes Boundary Points

Stefanovski, Jovan

doi:10.1137/120893380

Cited by 6 publications

(15 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the formulated general aBIP (which includes the cases of existent imaginary axis eigenvalues of A, possibly singular matrix R and the nonstrict inequality U ∞ ≤ 1), a necessary and sufficient condition for the existence of a solution is the condition (5.12) in [3], while all problem solutions are given in Theorem 1.1 in [23].…”

Section: Technical Details On the Abipmentioning

confidence: 99%

“…In the example of this section, we have shown how to choose a nonsingular matrix R > 0. The interpolation problem with these matrices A, C =: C 1 C 2 and R > 0 is solved in Theorem 1.1 of [23] and all its solutions are given by:…”

Section: H ∞ Optimal Control With Jω Invariant Zerosmentioning

confidence: 99%

“…One of the most general interpolation problems, which includes all of the above mentioned cases, is the augmented basic interpolation problem (aBIP) (see [8] for the problem statement, see [3,4,[6][7][8]23] for some results, and see [7] for some examples, which illustrate the application of the aBIP to bitangential interpolation). Formulation of aBIP.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A reformulation of augmented basic interpolation problem and an application to H∞ control

Stefanovski

2015

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

MSC:In this paper we present a new formulation of the augmented basic interpolation problem (aBIP) with rational matrices, in terms of the stability of four rational matrices, so that the aBIP transforms into a purely linear-algebraic problem. Actually, the existing interpolation condition, given by an integral, is replaced by stability of a rational matrix. The new condition is applied to the H ∞ optimal control of one-block plants having imaginary axis invariant zeros. A new parameter in the parametrization of H ∞ optimal controllers is revealed, which is given in terms of the derivative of the closed-loop system at the imaginary axis invariant zeros of the plant. The H ∞ optimal control algorithm is illustrated by an example, and compared to the existing algorithms.

show abstract

Section: Technical Details On the Abipmentioning

confidence: 99%

Section: H ∞ Optimal Control With Jω Invariant Zerosmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A reformulation of augmented basic interpolation problem and an application to H∞ control

Stefanovski

2015

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…A motivation for the introduction of this condition is the fact, proven in Stefanovski (2013b), that if U is an interpolant of the interpolation problem without zeros at infinity, then condition (CIV) holds.…”

Section: An Equivalent Formulation Of the Interpolationmentioning

confidence: 99%

Optimal boundary interpolation and one-block optimal control problem with invariant zeros on the imaginary axis and infinity

Stefanovski¹

2014

International Journal of Control

Self Cite

View full text Add to dashboard Cite

We formulate a matrix interpolation problem with existing interpolation points on the imaginary axis and infinity and existing equal left and right interpolation points, using the concept of parametrisation of stabilising controllers. Then, we solve the problem of obtaining all its solutions. If interpolation points at infinity are absent, we show that the introduced problem is equivalent to the existing one. We apply this result to solve the problem of optimal interpolation with existent interpolation points on the imaginary axis and infinity. We show by an example that the solution of optimal interpolation is directly applicable to the one-block H ∞ optimal control with existent invariant zeros on the imaginary axis and infinity. It is seen from the example that not only the transfer matrix of the closed-loop system is constrained on the extended imaginary axis, but also its derivatives.

show abstract

“…The latter identity defines a necessary interpolation condition in the point jω 0 on the derivative of Φ, although the derivative of Φ is not present in the original conditions (1.1) and (1.2). It is proved in [23] (and illustrated by Example 2 in [23]) that this kind of necessary condition for all such ω 0 can be unified into a single necessary condition,…”

Section: Introductionmentioning

confidence: 97%

${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

Stefanovski¹

2015

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

For given rational matrices V a, U a, V b , U b , we find necessary and sufficient conditions for existence of a stable rational matrix Φ satisfying Φ ∞ ≤ 1, V aΦ = U a, and ΦV b = U b . A condition is the positive semidefiniteness of a matrix, denoted by R. We present a parametrization of all problem solutions. A property of the proposed algorithm is, as a first step, to reduce the problem to a minimal realization, by an orthogonal transformation. Another property is the ability to transform the problem into one with constant matrices, by another orthogonal transformation. A problem motivation is the optimal H∞ control problem of descriptor systems. We show by an example that the existing numerical H∞ control optimization algorithms, which solve the problem of obtaining stabilizing controllers such that Φ ∞ ≤ γ, where Φ is the closed loop transfer matrix, and γ is slightly greater than the optimal one, compute controllers such that the closed loop system suffers from lack of stability robustness. Our proposed algorithm doesn't require γ to be greater than the optimal one, and the closed loop system has the property of stability robustness. Consequences of the singularity of matrix R, which property generically appears for the optimal γ, are (1) that Φ(jω) = γ for all real ω and all optimal controllers, and (2) if the measured output is single, or the control input is single, then the H∞ controller is unique. A numerical example is presented to illustrate the H∞ control optimization algorithm. Introduction.Consider rational matrices (rm's) V a (s) ∈ R(s) μ×m , U a (s) ∈ R(s) μ×r , V b (s) ∈ R(s) r×ρ , and U b (s) ∈ R(s) m×ρ , where by R(s) μ×m we denote the set of rm's with real coefficients, of dimension μ × m, which are possibly improper. We state the following problem.Problem 1. Find a proper stable rm Φ(s) ∈ R(s) m×r such that Φ ∞ ≤ 1 and the following conditions are satisfied:

show abstract

All Solutions of a Bitangential Interpolation Problem that Includes Boundary Points

Cited by 6 publications

References 10 publications

A reformulation of augmented basic interpolation problem and an application to H∞ control

A reformulation of augmented basic interpolation problem and an application to H∞ control

Optimal boundary interpolation and one-block optimal control problem with invariant zeros on the imaginary axis and infinity

${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

Contact Info

Product

Resources

About