A pointwise criterion for controller robustness

Schumacher, Johannes

doi:10.1016/0167-6911(92)90102-x

Cited by 39 publications

(21 citation statements)

References 31 publications

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“…This point of view can be found, e.g., in [31], [39]. Actually, we will show in the next section that the generic 1-input, 3-output system of order 6 is 1-nondegenerate.…”

Section: Consider a Left Coprime Factorization D;l(s)nl(s) G(s)mentioning

confidence: 95%

On Dynamic Feedback Compensation and Compactification of Systems

Rosenthal¹

1994

SIAM J. Control Optim.

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This paper introduces a compactification of the space of proper $p \times m$ transfer functions with a fixed McMillan degree $n$. Algebraically, this compactification has the structure of a projective variety and each point of this variety can be given an interpretation as a certain autoregressive system in the sense of Willems. It is shown that the pole placement map with dynamic compensators turns out to be a central projection from this compactification to the space of closed-loop polynomials. Using this geometric point of view, necessary and sufficient conditions are given when a strictly proper or proper system can be generically pole assigned by a complex dynamic compensator of McMillan degree $q$. ©1994 Society for Industrial and Applied Mathematics SIAM J. CONTROL AND OPTIMIZATION Vol. 32, No. 1, pp. 279-296, January 1994 Abstract. This paper introduces a compactification of the space of proper p x m transfer functions with a fixed McMillan degree n. Algebraically, this compactification has the structure of a projective variety and each point of this variety can be given an interpretation as a certain autoregressive system in the sense of Willems. It is shown that the pole placement map with dynamic compensators turns out to be a central projection from this compactification to the space of closedloop polynomials. Using this geometric point of view, necessary and sufficient conditions are given when a strictly proper or proper system can be generically pole assigned by a complex dynamic compensator of McMillan degree q.

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“…This point of view can be found, e.g., in [31], [39]. Actually, we will show in the next section that the generic 1-input, 3-output system of order 6 is 1-nondegenerate.…”

Section: Consider a Left Coprime Factorization D;l(s)nl(s) G(s)mentioning

confidence: 95%

On Dynamic Feedback Compensation and Compactification of Systems

Rosenthal¹

1994

SIAM J. Control Optim.

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“…If this condition holds, another formula for the minimal angle is given by (see again e.g. Gohberg and Krein, 1969, p. 339) Schumacher, 1992;Cevik and Schumacher, 1994) that sin <f>(97'(oo), cg(oo)) is positive as well. It follows from Martin and Hermann (1978) (see also de Does and Schumacher, 1994b;Cevik and Schumacher, 1994) that the functions SH9Jl(s) and SH C6'(s) are COntinUOUS mappings from c+ to the Grassmannian manifolds G"'(6Y x 6U) and GP(OJJ x 6U) respectively.…”

Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

“…Another way to express the above result is that ~s) and ~(s) should be complementary at each point SE c+. lt has been shown by Schumacher (1992) that the minimal angle is the appropriate measure of the robustness of complementarity of two subspaces OJI and 1£, in the sense that it gives exactly the distance (in the sense of the gap) of OJI to the set of subspaces OJI' that are not complementary to 1£. As a measure of robustness of stability, we shall therefore take…”

Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

“…For this, we use the formulation in terms of subspace-valued functions, which can be traced back to Martin and Hermann (1978) and Brockett and Byrnes (1981). It has already been demonstrated (Qiu and Davison 1992;Schumacher 1992) that subspace-valued functions are excellently suited to describe robust stability properties. In this paper we employ the same framework for the regulator problem.…”

Section: Introductionmentioning

confidence: 99%

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The Regulator Problem with Robust Stability

Cevik

Schumacher

1995

IFAC Proceedings Volumes

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The maximally achievable degree of robustness of stability, in the sense of coprime factor perturbations, is determined for linear servo systems.Abstract-The design of a controller such that the closed-loop system will track reference signals or reject disturbance signals from a specified class is known as the 'servomechanism problem' or the 'regulator problem'. For the regulator problem to be solvable with robust closed-loop stability, the plant obviously needs to be such that the regulation problem and the robust stabilization problem are solvable separately. In this paper we determine the extra conditions that are necessary and sufficient for the two problems to be solved simultaneously. It turns out that these conditions can be given a simple geometric interpretation in terms of a multivariable version of the Nyquist curve of the plant.

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“…They arise, for instance, in the context of controller robustness [Schumacher (1992) and Zhu (1994)]. Roughly speaking, the spaces associated with the controller and the plant (a system described by a set of differential equations) are complementary subspaces.…”

Section: Introductionmentioning

confidence: 99%

The Angle Between Complementary Subspaces

Ipsen¹,

Meyer²

1995

The American Mathematical Monthly

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Although the concept of angles between general subspaces may be too advanced for an undergraduate linear algebra course, the angle between complementary subspaces can be readily understood from basic properties of projectors and matrix norms. The purpose of this article is to derive some simple formulas for the angle between a pair of complementary subspaces by employing only elementary techniques.

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A pointwise criterion for controller robustness

Cited by 39 publications

References 31 publications

On Dynamic Feedback Compensation and Compactification of Systems

On Dynamic Feedback Compensation and Compactification of Systems

The Regulator Problem with Robust Stability

The Angle Between Complementary Subspaces

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