https://www.math.bgu.ac.il/~paf/homomorphisms_and_equivalence.pdf

On behavior homomorphisms and system equivalence

External equivalence of behavioral representations is investigated within an impulsive-smooth framework. As an attendant problem, initial-value consistency is also considered. Explicit necessary and sufficient conditions are derived. Applying the external equivalence result to conventional system representations reveals some interesting interpretations.

A test for behavioral equivalence

Solvability and general solutions to dynamic systems in polynomial matrix descriptions are investigated. The polynomial matrix descriptions need not necessarily be regular. Consistency of initial values and inputs are discussed. The characterization of the general solution for the systems in polynomial matrix descriptions has an explicit separation of proportional, integral and differential terms with respect to initial values and inputs. The derivation is based on a decomposition form of a singular matrix pencil.

General solution to systems in polynomial matrix form

Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore-Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

/pdf/on-generalized-inverses-of-singular-matrix-pencils-1svoaawmvt.pdf

On generalized inverses of singular matrix pencils

Presents a generalization of the chain-scattering representation to the case of general plants. Through the notion of input-output consistency, the conditions under which the generalized chain-scattering representation (GCSR) and the dual-generalized chain-scattering representation (DGCSR) exist are proposed. The generalized chain-scattering matrices are formulated into a general parameterized form by using the generalized inverse of matrices. Some algebraic system properties, such as the cascade structure property, the symmetry (duality) of the GCSR's and DGCSR's, are studied.

A generalized chain-scattering representation and its algebraic system properties

The concept of a transfer function is extended to general dynamic systems described by non-regular linear algebraic and integro-differential equations. The input consistency and uniqueness of the system output for the given consistent input are discussed. These two properties are closely related to the existence of a transfer function which itself does not need to be unique. Input-output equivalence in the sense that two systems have the same output for the given consistent input is proved to be necessary and sufficient for the existence of an input-output isomorphic map between the solution spaces of the systems under zero initial conditions.

Generalized transfer functions and input-output equivalence

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Generalized transfer functions and input-output equivalence

General solution to systems in polynomial matrix form