Wyniki wyszukiwania - BazTech

1

The LQ/KYP problem for infinite-dimensional systems

Grabowski P.

Opuscula Mathematica

|

2017

|

Vol. 37, no. 1

21--64

EN

Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur’e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system.

2

The lq-controller synthesis problem for infinite-dimensional systems in factor form

Grabowski P.

Opuscula Mathematica

|

2013

|

Vol. 33, no. 1

29-79

EN

The general lq-problem with infinite time horizon for well-posed infinite-dimensional systems has been investigated by George Weiss and Martin Weiss and by Olof Staffans with a complement by Kalle Mikkola and Olof Staffans. Our aim in this paper is to present a solution of a general lq-optimal controller synthesis problem for infinite-dimensional systems in factor form. The systems in factor form are an alternative to additive models, used in the theory of well-posed systems, which rely on leading the analysis exclusively within the basic state space. As a result of applying the simplified analysis in terms of the factor systems and an another derivation technique, we obtain an equivalent, however, astonishingly not the same formulae expressing the optimal controller in the time-domain and the method of spectral factorization. The results are illustrated by two examples of the construction of both the optimal control and optimal controller for some standard lq-problems met in literature: a control problem for a class of boundary controlled hyperbolic equations initiated by Chapelon and Xu, to which we give full solution and an example of the synthesis of the optimal control/controller for the standard lq-problem with infinite-time horizon met in the problem of improving a river water quality by artificial aeration, proposed by Zołopa and the author.

3

Dynamical model of propagation of pollutants in a river

Żołopa E., Grabowski P.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

|

2008

|

T. 12, z. 2

169-182

EN

In this paper a dynamical model of propagation of pollutants in a river with M point controls in the form of aerators and K point measurements is being transformed to an abstract model on a suitably chosen Hilbert space. Our model belongs to the class of abstract models of the factor-type. It is shown that the semigroup generated by the state operator A has a property of decaying in a finite-time, the observation operator is admissible, and the system transfer function is in the space H∞ (C+, L(CM, CK)). In the final part we also formulate the LQ problem with infinite-time horizon.

PL

W artykule przekształcany jest model dynamiczny rozprzestrzeniania się zanieczyszczeń w rzece, z punktowymi sterowaniami w postaci M aeratorów i K punktami pomiarowymi, do modelu abstrakcyjnego na odpowiednio dobranej przestrzeni Hilberta. Model abstrakcyjny jest typu sfaktoryzowanego. Pokazano, że półgrupa generowana przez operator stanu A ma własność zanikania w skończonym czasie, operator obserwacji jest dopuszczalny i transmitancja systemu należy do przestrzeni H∞ (C+, L(CM, CK)). W końcowej części pracy formułuje się problem liniowo-kwadratowy z nieskończonym horyzontem czasowym.