Infinite-dimensional Sylvester equations: Basic theory and application to observer design

Emirsajlow, Zbigniew

doi:10.2478/v10006-012-0018-5

Cited by 9 publications

(4 citation statements)

References 7 publications

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“…Remark 3: Unlike the classical finite-dimensional pole placement problem where, in order to impose the invertibility of the solution of the Sylvester equation, an observability property is required [20], [21], such a necessary condition is not a sufficient in the infinite dimension case. We illustrate this using a counter example at the end of this paper.…”

Section: Resultsmentioning

confidence: 99%

Harmonic Pole Placement

Riedinger¹,

Daafouz²

2022

Preprint

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In this paper, we propose a method to design state feedback harmonic control laws that assign the closed loop poles of a linear harmonic model to some desired locations. The procedure is based on the solution of an infinite-dimensional harmonic Sylvester equation under an invertibility constraint. We provide a sufficient condition to ensure this invertibility and show how this infinite-dimensional Sylvester equation can be solved up to an arbitrary small error. The results are illustrated on an unstable linear periodic system. We also provide a counter-example to illustrate the fact that, unlike the classical finite dimensional case, the solution of the Sylvester equation may not be invertible in the infinite dimensional case even if an observability condition is satisfied.

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Section: Resultsmentioning

confidence: 99%

Harmonic Pole Placement

Riedinger¹,

Daafouz²

2022

Preprint

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“…For more results on the properties of the implemented semigroup and its generator see e.g. [8], [5], [6] and references cited therein.…”

Section: Homogeneous Dsementioning

confidence: 99%

“…In this paper we use the implemented semigroup concept (see, e.g., [8], [5]) to study the infinite-dimensional differential Sylvester equation (DSE) of the forṁ…”

Section: Introductionmentioning

confidence: 99%

Remarks on the existence and uniqueness of solutions to the infinite dimensional sylvester equations

Emirsajlow

2014

2014 19th International Conference on Methods and Models in Automation and Robotics (MMAR)

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In various control problems for infinite-dimensional systems (see, e.g., [1], [2], [3], [4], [7]) one can found usefulness of Sylvester type operator equations. The aim of the paper is to popularize the implemented semigroup framework (e.g. [8], [5], [6]) by showing how it helps to deal with the infinite-dimensional algebraic Sylvester equation both in differential (DSE) and in the algebraic (ASE) form. The main emphasis is put on some basic results concerning the existence and uniqueness of solutions. The collected results are presented from the point of view of potential applications in the systems and control theory.

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“…In the case of boundary control systems or, more precisely, systems described by partial differential equations with boundary control and observation, the popular backstepping method seems to be most successful (e.g., Smyshlayev and Krstic, 2005;2008;Hasana et al, 2016). More direct methods, exploring the relation between the boundary control models and the more familiar state space models or simply dealing with the latter can be found in the works of Demetriou and Rosen (2005), Vries et al (2010), Demetriou (2013), Ferrante et al (2020), Emirsajłow (2012; and the references cited therein. For a short survey on the observer design methods for infinite-dimensional control systems, see the work of Hidayat et al (2011).…”

Section: Introductionmentioning

confidence: 99%

Discrete-time output observers for boundary control systems

Emirsajlow

2021

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The paper studies the output observer design problem for a linear infinite-dimensional control plant modelled as an abstract boundary input/output control system. It is known that such models lead to an equivalent state space description with unbounded control (input) and observation (output) operators. For this class of infinite-dimensional systems we use the Cayley transform to approximate the sophisticated infinite-dimensional continuous-time model by a discrete-time infinitedimensional one with all involved operators bounded. This significantly simplifies mathematical aspects of the observer design procedure. As is well known, the essential feature of the Cayley transform is that it preserves various system theoretic properties of the control system model, which may be useful in analysis. As an illustration, we consider an example of designing an output observer for the one-dimensional heat equation with measured controls (inputs) in the Neumann boundary conditions, measured outputs in the Dirichlet boundary conditions and an unmeasured output at a fixed point within the domain. Numerical simulations of this example show that the interpolated continuous-time signal, obtained from the discrete-time observer, can be successfully used for tracking the continuous-time plant output.

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Infinite-dimensional Sylvester equations: Basic theory and application to observer design

Cited by 9 publications

References 7 publications

Harmonic Pole Placement

Harmonic Pole Placement

Remarks on the existence and uniqueness of solutions to the infinite dimensional sylvester equations

Discrete-time output observers for boundary control systems

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