Stability of control systems and graphs of linear systems

New conditions for internal stability of a closed-loop control system are given in terms of the graphs of the multiplication operators induced by the transfer functions of the plant and the controller. These conditions can be given a geometrical interpretation. This relates closed-loop stability to the minimal angle between the graph space associated with the system and the graph space associated with the controller. The maximally stabilizing controller is defined as the controller that maximizes the minimum angle between the graph space associated with the system and the graph space associated with the controller. It is shown that this controller can be calculated as a Nehari extension of the coprime factors of the system.     

Continuous time-varying linear systems

We discuss implicit systems of ordinary linear differential equations with (time-) variable coefficients, their solutions in the signal space of hyperfunctions according to Sato and their solution spaces, called time-varying linear systems or behaviours, from the system theoretic point of view. The basic result, inspired by an analogous one for multidimensional constant linear systems, is a duality theorem which establishes a categorical one–one correspondence between time-varying linear systems or behaviours and finitely generated modules over a suitable skew-polynomial ring of differential operators. This theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic (hyper-)functions or of distributions on . It is used to obtain various results on key notions of linear system theory. Several new algorithms for modules over rings of differential operators and, in particular, new Gröbner basis algorithms due to Insa and Pauer make the system theoretic results effective.     

On the Moser- and super-reduction algorithms of systems of linear differential equations and their complexity

The notion of irreducible forms of systems of linear differential equations with formal power series coefficients as defined by Moser [Moser, J., 1960. The order of a singularity in Fuchs’ theory. Math. Z. 379–398] and its generalisation, the super-irreducible forms introduced in Hilali and Wazner [Hilali, A., Wazner, A., 1987. Formes super-irréductibles des systèmes différentiels linéaires. Numer. Math. 50, 429–449], are important concepts in the context of the symbolic resolution of systems of linear differential equations [Barkatou, M., 1997. An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system. Journal of App. Alg. in Eng. Comm. and Comp. 8 (1), 1–23; Pflügel, E., 1998. Résolution symbolique des systèmes différentiels linéaires. Ph.D. Thesis, LMC-IMAG; Pflügel, E., 2000. Effective formal reduction of linear differential systems. Appl. Alg. Eng. Comm. Comp., 10 (2) 153–187]. In this paper, we reduce the task of computing a super-irreducible form to that of computing one or several Moser-irreducible forms, using a block-reduction algorithm. This algorithm works on the system directly without converting it to more general types of systems as needed in our previous paper [Barkatou, M., Pflügel, E., 2007. Computing super-irreducible forms of systems of linear differential equations via Moser-reduction: A new approach. In: Proceedings of ISSAC’07. ACM Press, Waterloo, Canada, pp. 1–8]. We perform a cost analysis of our algorithm in order to give the complexity of the super-reduction in terms of the dimension and the Poincaré-rank of the input system. We compare our method with previous algorithms and show that, for systems of big size, the direct block-reduction method is more efficient.