On BIBO stability of infinite-dimensional systems
Alexander Wierzba is a PhD student in the department Mathematical Systems Theory. (Co)Promotors are prof.dr. H.J. Zwart and dr. F.L. Schwenninger from the faculty of Electrical Engineering, Mathematics and Computer Science from the University of Twente.
This thesis is concerned with the notion of bounded-input bounded-output stability or short BIBO stability of infinite-dimensional systems. An input-output system is BIBO stable if uniformly bounded inputs in time give rise to outputs bounded uniformly in time and furthermore there exists a uniform relative bound between the respective supremum norms.
For linear systems, this notion can be related to boundedness of a linear operator with respect to (essential) supremum norms. Problems of this type, in particular when the operator maps time signals to time signals, are omnipresent in functional analysis, but also have significance in many other fields such as harmonic analysis or control theory. Much of the complexity of this question arises from the non-Hilbert and non-reflexivity properties of the L∞ spaces or spaces of continuous functions. If these were instead replaced by L2, the question would become much simpler as such operators have been extensively studied for instance in the context of well-posed systems.
BIBO stability of linear systems described by finite-dimensional state-space representations is a classical and well-studied notion with applications for instance in signal processing and controller design. Turning to linear systems described by infinite-dimensional state-space representations, such as they arise from boundary-controlled partial differential equations, the situation becomes more complex. In this case even the question of a proper definition of BIBO stability itself becomes non-trivial, due to the existence of multiple different solution concepts. In this work we derive a characterisation of BIBO stability of such linear infinite-dimensional systems and therefore clarify on the subtleties of defining BIBO in this case. Furthermore, we obtain sufficient conditions for BIBO stability and investigate to which extent this property is preserved under additive and multiplicative perturbations of the system.
Second, we consider the behaviour of BIBO stability under duality transformations of the system. This investigation gives rise to the notion of LILO stability that serves as the dual concept to BIBO. This property can be characterised by the boundedness of the input-output mapping with respect to the normed function space L1, instead of L∞ as in the case of BIBO. It turns out that in some cases the more suitable structure of L1 compared to L∞ makes LILO stability easier to check than BIBO stability thus aiding the study of the latter.
Third, we discuss BIBO stability of distributed port-Hamiltonian systems. This class of systems plays an important role in the modelling of physical systems involving vibrations, flows and transport phenomena. We derive sufficient conditions for BIBO stability in terms of matrix conditions and in relation to exponential stability on an altered state space.
Fourth, we investigate BIBO stability of semilinear state-space systems. Employing the results for linear systems we derive sufficient conditions based upon different Lipschitz continuity assumptions for the nonlinearity with respect to interpolation spaces.
Finally we turn towards control-theoretic applications, discussing the role BIBO stability plays in the application of funnel control to relative-degree systems and how the derived results can be used in this context. Further we describe the potential use of this control technique (and thus of BIBO stability) within the setting of the Predictive Avatar Control and Feedback project (PACOF) which is aimed at mitigating the detrimental effects of time-delays in telerobotics setups.
