Proper stabilisation of multidimensional input/output behaviours

• Authors: Martin Scheicher;Ulrich Oberst
• Venue: Multidimensional Systems and Signal Processing
• Year: 2009

Most papers in multidimensional stabilisation theory use discrete signals and a suitable integral domain $${\fancyscript{S}}$$ of "SISO stable plants" which consists of multivariate rational functions over a field. In this theory a system is described by its transfer matrix H of rational functions and is called stable if the entries of H belong to $${\fancyscript{S}}$$ . In most cases the transfer matrix is assumed proper. The significance of properness in the one-dimensional theory was already recognized by Kalman and is well-known to engineers. In the multidimensional theory properness and stability of the transfer matrix often enable its interpretation as input/output (IO) map or (convolution) operator and the proof of BIBO stability. Important contributions to this multidimensional stabilisation theory, in particular for our own work, are due to Bose, Guiver, Lin, Quadrat, Shankar, Sule, Xu, Ying, Zerz et al., some of whom were, in turn, essentially influenced by the seminal papers of Desoer, Francis, Ku驴era, Vidyasagar, Youla, Zames and their co-authors in the one-dimensional case and on the fractional representation approach. In a recent paper we developed a stabilisation theory for multidimensional IO behaviours both in the continuous and the discrete cases, but did neither require nor prove the properness of the given or derived transfer matrices. The transfer matrix describes only the controllable part of a system whereas its behavioural description also enables the finer discussion of the internal stability of its autonomous part which is not determined by its transfer matrix. The latter fact was one of Kalman's reasons for the development of the one-dimensional state space theory. In the present paper we treat proper stabilisation of multidimensional IO behaviors, i.e., we characterise those proper or non-proper plants which admit a proper or non-proper stabilising compensator such that the resulting closed loop system is not only stable, but also has a proper transfer matrix. Under a weak pole condition on the plant which implies its properness we also construct strictly proper stabilising compensators for a stabilisable plant. For discrete systems described by transfer matrices such a result had been proven by Lin. For the construction of stabilising compensators we describe various algorithms and demonstrate these by examples. The parametrisation of all stabilising compensators was given in our previous paper. Our results seem interesting for one-dimensional IO behaviours too.