On the realization of periodic functions

Int. J. Robust Nonlinear Control

2006

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SUMMARYWe design a controller for infinite-dimensional linear systems (with bounded control, observation and feedthrough operators) which, under certain assumptions, achieves asymptotic tracking of arbitrary bounded uniformly continuous reference signals in the presence of disturbances. The proposed controller is of feedforward-feedback type: The dynamic feedback part is used to stabilize the closed-loop system consisting of the plant and the controller, whereas the feedforward part is tuned using the regulator equations to achieve the regulation of desired signals. We also completely solve the regulator equations for SISO systems, and we discuss robustness properties of the proposed controller. A useful feature in our design is that the feedforward part of the controller can be designed independently of the feedback part. This automatically leads to a degree of robustness in the stabilizing part of the controller, which is not present in the existing state feedback controllers solving the same output regulation problem.

Section: The Exogenous Systemmentioning

confidence: 99%

A feedforward–feedback controller for infinite-dimensional systems and regulation of bounded uniformly continuous signals

Int. J. Robust Nonlinear Control

2006

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“…the search for a space W, operators S and Q, and a suitable initial state w 0 -for a specific function u(t) is called realization. This procedure is discussed in the articles [5,6,13,16,18]. The following example shows that the function generator approach includes inputs in various subspaces of BUC(R + , U) [2,7,23].…”

Section: Function Generators In Banach Spacesmentioning

confidence: 99%

Function generators, the operator equation ΠS=AΠ+BQ and inherited dynamics in inhomogenous Cauchy problems

Journal of Differential Equations

2005

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This article concerns the interrelation between the existence of a bounded solution to the operator equation S = A + BQ in D(S) and the asymptotic behaviour of the mild solutions z(t) of the abstract Cauchy problemż(t) = Az(t) + Bu(t), t 0, in a Banach space Z. Here B and Q are bounded, whereas A and S generate C 0 -semigroups T A (t) and T S (t) on Z and W (W is a Banach space), respectively. Banach space valued inputs u(t) ∈ U are generated by linear dynamical systems. We define asymptotically inherited dynamics of z(t) and show that for strongly stable semigroups T A (t), z(t) asymptotically inherits the dynamics of the inputs if there exists ∈ L(W, Z) such that S = A + BQ in D(S). If T A (t) and T S (t) are bounded, then z(t) is bounded and uniformly continuous provided that S = A + BQ in D(S). For the converse we show that if z(t) asymptotically inherits the dynamics of the inputs and if T S (t)is a suitable C 0 -group, then T A (t) is strongly stable and there exists ∈ L(W, Z) such that S = A + BQ in a subspace of D(S). We also discuss why inputs u(t) frequently completely determine the asymptotic properties of z(t) if T A (t) is exponentially stable. As an application, we consider almost periodic inputs u(t) in Sobolev spaces H (U, f n , n ).

“…Such realizations play a central role in some regulator problems of modern control theory [8]. The realizationsoccasionally also called state space realizations-that we consider are of the forṁ Here x(t) is the state of the system (which lies in some Hilbert space H ) while y(t) is the output of the system.…”

Section: Introduction and Some Definitionsmentioning

confidence: 99%

“…System (1.1) is denoted Σ(A, −, C, −) [3]. One possible approach to constructing an infinite-dimensional realization for a given 2π -periodic function y ∈ C 1 [0, 2π] is as follows [8]. Choose the state space H to be L 2 [0, 2π], select the C 0 -semigroup T (·) to be the 2π -periodic left shift on H (such that T (τ )f = f (· + τ − 2π ·+τ 2π ) for each f ∈ H ), select x 0 = y and finally set the observation operator C to be the (ordinary) point evaluation operator at the origin.…”

Section: Introduction and Some Definitionsmentioning

confidence: 99%

“…Then CT (t)x 0 = y(ξ + t − 2π ξ +t 2π )| ξ =0 = y(t) for t 0 as is required, and based on this information, a system (1.1) is easy to construct. It turns out that by only altering the observation operator-and, in the process, the concept of a point value-we can enlarge the class of functions which are realizable by the system in question [8].…”

Section: Introduction and Some Definitionsmentioning

confidence: 99%

See 1 more Smart Citation

On the concept of point value in the infinite-dimensional realization theory

Journal of Mathematical Analysis and Applications

2004

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In this article, we study the effect of the chosen representation of a point value (and point evaluation) on the class of periodic signals realizable using a certain type of infinite-dimensional linear system. By suitably representing the point evaluation at the origin in a Hilbert space, we are able to give a complete characterization of its extensions. These extensions involve a new concept called δ-sequence, the use of which as an observation operator of an infinite-dimensional linear system is studied in this article. In particular, we consider their use in the realization of periodic signals. We also investigate how the use of δ-sequences affects the convergence properties of such realizations; we consider the rate and character of convergence and the removal of the Gibbs phenomenon. As still a further demonstration of the significance of the chosen concept of a point value, we discuss the use of distributional point values in the realization of periodic distributions. The possible applications of this work lie in regulator problems of infinite-dimensional control theory, as is indicated by the well-known internal model principle.  2004 Elsevier Inc. All rights reserved.