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The disturbance decoupling and the simulteneous disturbance and input – output decoupling problems for singular systems are considered in the context of the matrix fraction description (MFD) of the system. Solvability conditions are obtained in terms of the composite matrix of a column reduced MFD of the system, a characterisation of the fixed poles of both problems is given and it is shown that the remaining poles can be arbitrarily assigned.

Early stages modelling of processes involves issues of classification of variables into inputs, outputs and internal variables, referred to as Model Orientation Problem (MOP) which may be addressed on state space implicit, or matrix pencil descriptions. Defining orientation is equivalent to producing state space models of the regular or singular type. Studying the conditions, under which such models may be derived, as well as the structural properties of the resulting oriented problems, is one of the issues considered here. Oriented models of the S(A,B,C,D) type have transfer functions which may not be well behaving as far as properties of input, output regularity and nondegeneracy. Defining subsystems by reduction of the number of inputs, outputs such that the resulting transfer function is well behaved as far as preservation of degeneracy, is a problem considered in the context of systems having physical input, output variables, which have to be preserved.

Transfer function models used for early stages of design are large dimension, contain all possible physical inputs, outputs, may be badly conditioned and possibly degenerate. The problem considered here is the selection of maximal cardinality subsets of the physical input, output sets, such as the resulting model is nondegenerate and satisfies additional properties such as controllability and observability and avoids the existence of high order infinite zeros. Conditions for characterisation of such properties are given. A selection procedure for natural subsets of inputs and outputs, which guarantee well conditioning in the above sense, may be based on these criteria.

The input-output decoupling problem for singular systems via state feedback and regular input transformation is considered. The treatment of the problem is in the frequency domain and it is an extension of the state-space systems case. The polynomial matrix approach is shown to be considerably simpler than the existing methods and the solvability conditions are readily obtained from the "numerator" matrix of a coprime and column reduced MFD of the given system. Pole placement issues are also considered and a characterisation of the fixed poles of the system is given.

Resume Nous considerons une forme canonique du type Popov pour des systemes singuliers avec sorties, sous l'action de l'equivalence restreinte des systemes. Nous decrivons les transformations qui conduisent a cette forme canonique et nous montrons que celle-ci est directement liee a la matrice composee, issue d'une factorisation de la matrice du transfert du systeme. Elie met en evidence, de maniere tout a fait transparente, les indices d'atteignabilite du systeme et permet de classifier, par simple inspection, les indices de commandabilite propres et nonpropres.

For linear systems, a family of cover problems of the geometric theory are introduced as extensions of the standard cover problem and a matrix pencil formulation of such problems is given. It is shown that the solvability of such problems is reduced to a problem of Kronecker Invariant Transformation by Matrix Pencil Augmentation and a Matrix Pencil Realisation Problem. Necessary, as well as sufficient conditions for solvability of both problems are given, which lead to a number of conditions for solvability of the partial, as well as standard cover problem. The special cases of left regular, regular solutions of the cover problem are investigated and a parametrisation of such families of solutions is given.