Browsing by Subject "control of infinite-dimensional systems"

Now showing 1 - 4 of 4

Access status: Open Access ,
Dynamical model of propagation of pollutants in a river
(Wydawnictwa AGH, 2008) Żołopa, Elżbieta; Grabowski, Piotr
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 $H^{\infty}$ ($\mathbb{C}^{+}$, $L(\mathbb{C}^{M}, \mathbb{C}^{K})$). In the final part we also formulate the LQ problem with infinite-time horizon.
Access status: Open Access ,
Small-gain theorem for a class of abstract parabolic systems
(Wydawnictwa AGH, 2018) Grabowski, Piotr
We consider a class of abstract control system of parabolic type with observation which the state, input and output spaces are Hilbert spaces. The state space operator is assumed to generate a linear exponentially stable analytic semigroup. An observation and control action are allowed to be described by unbounded operators. It is assumed that the observation operator is admissible but the control operator may be not. Such a system is controlled in a feedback loop by a controller with static characteristic being a globally Lipschitz map from the space of outputs into the space of controls. Our main interest is to obtain a perturbation theorem of the small-gain-type which guarantees that null equilibrium of the closed-loop system will be globally asymptotically stable in Lyapunov's sense.
Access status: Open Access ,
The lq-controller synthesis problem for infinite-dimensional systems in factor form
(2013) Grabowski, Piotr
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.
Access status: Open Access ,
The LQ/KYP problem for infinite-dimensional systems
(2017) Grabowski, Piotr
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.