Grabowski, Piotr
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automatyka, elektronika, elektrotechnika i technologie kosmiczne
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Item type:Article, Access status: Open Access , Small-gain theorem for a class of abstract parabolic systems(Wydawnictwa AGH, 2018) Grabowski, PiotrWe 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.Item type:Article, Access status: Open Access , The lq-controller synthesis problem for infinite-dimensional systems in factor form(2013) Grabowski, PiotrThe 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.Item type:Article, Access status: Open Access , The motion planning problem and exponential stabilization of a heavy chain. Part 2(2008) Grabowski, PiotrThis is the second part of paper [P. Grabowski, <i>The motion planning problem and exponential stabilization of a heavy chain. Part I</i>, to appear in International Journal of Control], where a model of a heavy chain system with a punctual load (tip mass) in the form of a system of partial differential equations was interpreted as an abstract semigroup system and then analysed on a Hilbert state space. In particular, in [P. Grabowski, <i>The motion planning problem and exponential stabilization of a heavy chain. Part I</i>, to appear in International Journal of Control] we have formulated the problem of exponential stabilizability of a heavy chain in a given position. It was also shown that the exponential stability can be achieved by applying a stabilizer of the colocated-type. The proof used the method of Lyapunov functionals. In the present paper, we give other two proofs of the exponential stability, which provides an additional intrinsic insight into the exponential stabilizability mechanism. The first proof makes use of some spectral properties of the system. In the second proof, we employ some relationships between exponential stability and exact observability.Item type:Article, Access status: Open Access , Dynamical model of propagation of pollutants in a river(Wydawnictwa AGH, 2008) Żołopa, Elżbieta; Grabowski, PiotrIn 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.Item type:Article, Access status: Open Access , Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory(2006) Grabowski, PiotrThe modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [12, 13] two groups of similarity criteria for a given hybrid closed-loop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions.