Stability Analysis in Continuous and Discrete Time, using the Cayley Transform

Besseling, N.C.; Zwart, Hans

doi:10.1007/s00020-010-1805-8

Cited by 7 publications

(5 citation statements)

References 6 publications

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“…In the previous two theorems, we have related the growth of a semigroup with a perturbed one. For the discrete counterparts of these results, we refer to [3]. Combining Theorem 4.6 and 4.11, we obtain the following corollary.…”

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confidence: 76%

Growth of semigroups in discrete and continuous time

2011

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We show that the growth rates of solutions of the abstract differential equationsẋ(t) = Ax(t),ẋ(t) = A −1 x(t), and the difference equation x d (n + 1) = (A + I)(A − I) −1 x d (n) are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup (e A −1 t) t≥0 is O(4 √ t), and for ((A + I)(A − I) −1) n it is O(4 √ n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of ((A + I)(A − I) −1) n is O(1), i.e., the operator is power bounded.

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confidence: 76%

Growth of semigroups in discrete and continuous time

2011

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“…By the Cayley transform we map the unbounded operators Λ and B into the bounded operators in the discrete-time counterpart [19]. Therefore, by partitioning the extended space of the square summable sequences into the slow and fast modal discrete states, the overall state evolution is expressed by the following discrete abstract state space form.…”

Section: B Boundary Transformationmentioning

confidence: 99%

Linear matrix inequalities (LMIs) based observer and controller design for second order parabolic PDE

Yang

Dubljević

2014

2014 American Control Conference

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In this work, a linear matrix inequality (LMI) methodology is used to design a controller to stabilize the second order parabolic PDE in the presence of input constraints. The novel feature of the proposed synthesis is to construct the LMI formulation within the modal space by using the spectrum analysis, which transfers the original infinite dimensional PDE state into the modal state, described by the abstract state space representation. The state feedback controller and Luenberger observer are developed by taking the entire infinite number of modal states into account, thereby stabilizing the system and reconstructing the state rigorously. The manipulated variable constraints, naturally existing in most real applications, are also considered in the design framework. Finally, since the initial value of modal states are hardly known in advance, the LMI formulation is designed to maximize the region of attraction (ROA) for augmented state space, including the modal state, input, estimation error, together with the fast modal output bound, such that the controller and the observer are robust enough to the error associated with initial conditions. Using a numerical example of an unstable second order parabolic PDE, we demonstrate that if feasible, the state feedback controller and observer, generated by LMI, have enough capability to drive the process modal state into the equilibrium.

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“…Nice identities between the powers of the cogenerator and integral expressions which involve generalized Laguerre polynomials appear in [17, Theorem 1], [6,Lemma 4.4] and [5,Lemma 6.7].…”

Section: -Semigroups and Resolvent Operators Given By Laguerre Expans...mentioning

confidence: 99%

$C_0$-semigroups and resolvent operators approximated by Laguerre expansions

Abadías¹,

Miana²

2013

Preprint

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In this paper we introduce Laguerre expansions to approximate vector-valued functions expanding on the well-known scalar theorem. We apply this result to approximate C0semigroups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions, its Laplace transforms and the convergence of Laguerre series in Lebesgue spaces. The concluding section of this paper is devote to consider some examples of C0-semigroups: shift, convolution and holomorphic semigroups where some of these results are improved.

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Stability Analysis in Continuous and Discrete Time, using the Cayley Transform

Cited by 7 publications

References 6 publications

Growth of semigroups in discrete and continuous time

Growth of semigroups in discrete and continuous time

Linear matrix inequalities (LMIs) based observer and controller design for second order parabolic PDE

$C_0$-semigroups and resolvent operators approximated by Laguerre expansions

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