Piotr Grabowski

Admissible observation operators. Semigroup criteria of admissibility

We investigate admissible observation operators. Semigroup criteria of admissibility are derived. We also discuss the recent Russell-Weiss necessity condition of exact observability.

Admissibility of observation functionals

The concept of infinite-time admissibility of unbounded observation functionals is introduced. Under the assumption of exponential stability of the semigroup, it is equivalent to finite-time admissibility recently investigated in Weiss (1988 b, 1989). Necessary and sufficient criteria for admissibility are given. In particular, it is shown that the Ho-Russell-Weiss (1988 b) test for admissibility of observation functionals/control vectors can be derived in an elementary way without invoking the geometric interpretation of the Carleson measure, while the criterion in Weiss (1991) can easily be deduced from the Carleson Embedding Theorem. Some practically applicable sufficient conditions guaranteeing admissibility are discussed in § 3. The results are illustrated by a feedback system containing RLCG transmission line.

Stability of a heat exchanger feedback control system using the circle criterion

The standard linearized model of a controlled heat exchanger in the form of a system of two first-order PDEs is interpreted as an abstract semigroup system on an appropriately selected Hilbert state space. We analyse some its basis properties: exponential stability, admissibility of an observation functional and regularity of the system impulse response. Next we give a proof of a version of the circle criterion which bases on the use of the Leray–Schauder fixed point theorem jointly with a uniqueness result. The circle criterion is then implemented to a feedback control system of the heat exchanger with a proportional controller and a non-linear valve in the feedback loop. All considerations are illustrated by a numerical example of the heat exchanger installed in central heating system at the AGH University of Science and Technology, Cracow, Poland, for which we estimate feasible settlements of the proportional controller.

The motion planning problem and exponential stabilisation of a heavy chain. Part I

A model of a heavy chain system with a punctual load (tip mass) in the form of a system of partial differential equations is interpreted as an abstract semigroup system on a Hilbert state space. Our aim is to solve the output motion planning problem of the same nature as in the case of an unloaded heavy chain (Grabowski, P. (2003), ‘Abstract Semigroup Model of Heavy Chain System with Application to a Motion Planning Problem’, in Proceedings of 9th IEEE International Conference: Methods and Models in Automation and Robotics, 25–28 August, Międzyzdroje, Poland, pp. 77–86 (IS1-2-3.PDF)). In order to solve this problem we first analyse its well-posedness and some basic properties. Next, we solve the output motion planning problem using a substitute of the inverse of the input–output operator represented in terms of the Laplace transforms. A problem of exponential stabilisation is also formulated and solved using a stabiliser of the colocated type. The exponential stabilisation is proved using the method of Lyapunov functionals combined with some frequency-domain tools. The method of Lyapunov functionals can be replaced by the spectral or exact controllability approach as shown in the second part (Grabowski, P. (2008), ‘The Motion Planning Problem and Exponential Stabilisation of a Heavy Chain. Part II’, Opuscula Mathematica, 28 (2008) (Special issue dedicated to the memory of Professor Andrzej Lasota), 481–505) of the present article. A laboratory setup which allows verification of the results in practice is described in detail. Its dynamical model is used as an example to illustrate the theoretical results. †Dedicated to Frank M. Callier on the occasion of his 65th birthday.

Stabilization of Wave Equation Using Standard/Fractional Derivative in Boundary Damping

We discuss the problem of stabilization of wave equation by means of the standard or fractional derivative in boundary damping. The problem is being reduced to a selection between the proportional or fractional integrator of order 1 − α feedback controllers. The fractional integration leads to the strong asymptotic stability only, while the proportional feedback control can ensure the exponential stability. This means that exponential stability is not robust around the value α = 1. We shall discuss mathematical and control theory aspects of this fact.

Evaluation of quadratic cost functionals for neutral systems: the frequency-domain approach

Abstract As shown in Grabowski (1983), the problem of evaluation of the quadratic performance index for delay-differential systems of neutral type can be effectively solved by the use of Lyapunov functionals. In this paper another solution to that problem, based on the frequency-domain approach, is proposed. With the use of Plancherels theorem and other tools of complex analysis, this task is reduced to a Riemann-Hilbert problem, an explicit solution to which is found by elementary methods. Our results can be regarded as a generalization of those due to Walton and Marshall (1984, 1987). Two simple examples of application are also provided.