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1

A multi-model approach to Saint-Venant equations: A stability study by LMIs

Dos Santos Martins V., Rodrigues M., Diagne M.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 3

539-550

EN

This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities (LMIs) and an Internal Model Boundary Control (IMBC) structure. The method is applied both in simulations and real experiments through a microchannel, illustrating thus the theoretical results developed in the paper.

2

High-gain feedback and sliding modes in infinite dimensional systems

Levaggi L.

Control and Cybernetics

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2004

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Vol. 33, no 1

33-50

EN

This paper focuses on the connection between sliding motions and low frequency modes of high-gain feedback systems in an infinite dimensional framework. We study a particular class of abstract control systems in a Hilbert space setting and analyse their high-gain behaviour through singular perturbations. We show that the "slow" motion derived from the reduced model approximates the evolution of the closed loop after a fast transient. Moreover we prove a relation between this slow component of the high-gain feedback system and sliding motions, in the spirit, of the analogous result in the finite dimensional setting by Young, Kokotovic and Utkin (Young et al., 1977).

3

Sturm-Liouville Systems Are Riesz-Spectral Systems

Delattre C., Dochain D., Winkin J.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 4

481-484

EN

The class of Sturm-Liouville systems is defined. It appears to be a subclass of Riesz-spectral systems, since it is shown that the negative of a Sturm-Liouville operator is a Riesz-spectral operator on L2(a,b) and the infinitesimal generator of a C0-semigroup of bounded linear operators.

4

Constrained Controllability of Dynamic Systems

Klamka J.

International Journal of Applied Mathematics and Computer Science

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1999

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Vol. 9, no 2

231-244

EN

The present paper is devoted to a study of constrained controllability and controllability for linear dynamic systems if the controls are taken to be non-negative. By analogy to the usual definition of controllability it is possible to introduce the concept of positive controllability. Weshall concentrate on approximate positive controllability for linear infinite-dimensional dynamic systems when the values of controls are taken from a positive closed convex cone and the operator of the system is normal and has pure discrete point spectrum. Special attention is paid to positive infinite-dimensional linear dynamic systems. General approximate constrained controllability results are then applied to distributed-parameter dynamic systems described by linear partial-differential equations of parabolic type with various kinds of boundary conditions. Several remarks and comments on the relationships between different concepts of controllability are given. Finally, a simple illustrative example is also presented.

5

Unit sliding mode control in infinite dimensional systems

Orlov Y., Utkin V. I.

Applied Mathematics and Computer Science

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1998

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Vol. 8, no 1

7-20

EN

In contrast to the conventional component-wise design of sliding mode controI, a new approach is developed for infinite-dimensional systems. The conventional approach is not applicable since, generally speaking, the infinite-dimensional controI may not be represented in the component form as well as a sliding manifold. The concept of "unit controI", previously introduced for finite-dimensional systems, does not depend on the dimension of controI and is generalized for the dynamic processes governed by differential equations in Banach and Hilbert spaces. The design methods for heat and mechanical distributed processes are given.

6

Controllability of second-order semilinear infinite-dimensional dynamical systems

Klamka J.

Applied Mathematics and Computer Science

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1998

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Vol. 8, no 3

459-470

EN

In the paper, the approximate controllability of semilinear abstract second-order infinite-dimensional dynamical systems is considered. It is proved by using the frequency-domain and functional-analysis methods that the approximate controllability of second-order semilinear dynamical system can be verified by the approximate controllability conditions for a simplified suitably-defined firstorder linear dynamical system. General results are then applied to a semilinear mechanical flexible-structure vibratory dynamical system. Some special cases are also considered. Moreover, remarks and comments on the relationships between different concepts of controllability are given. The paper extends the results presented in (Klamka, 1992; Triggiani, 1978) to a more generaI class of second-order abstract dynamical systems.