On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems A.I.G. Vardulakis, N.P. Karampetakis and E.N. Antoniou Department of Mathematics Faculty of Sciences Aristotle University of Thessaloniki Thessaloniki , Greece

Introduction  Review of the Realization Theory of Polynomial Transfer Function Matrices via “Pure" Generalized State Space Models  Study of associated concepts and features  Comparison to results from the classical State Space realization theory  Key topics: Generalized order of GSS realizations Cancellations of decoupling zeros at ∞ Irreducibility at infinity & Minimality Dynamic & Non-dynamic variables Isomorphism of spectral structures at ∞ A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki

Introduction Given a polynomial transfer function matrix One may obtain its generalized state space realization of the form: and nilpotent, where Such that A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki

Introduction Remark. A GSS realization of a polynomial matrix A(s) can be obtained from a state space realization of the strictly proper rational matrix A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Example. Given where then

Generalized Order of a GSS Realization A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Definition. Let be GSS realization of A(s). The generalized order f g of Σ g is defined as follows: where δ Μ (.) denotes the McMillan degree. Example. Notice that the generalized order of the example in the previous slide is f g =3.

Irreducibility at infinity A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Let and let be a GSS realization of A(s) Definition. The input decoupling zeros (i.d.z.) at s= ∞ of Σ g are defined as the zeros at s= ∞ of the pencil: Respectively, the output decoupling zeros (o.d.z.) at s= ∞ of Σ g are defined as the zeros at s= ∞ of the pencil: Finally, the input-output decoupling zeros (i.o.d.z.) at s= ∞ of Σ g are the common zeros at s= ∞ of the pencils:

Irreducibility at infinity A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Let where Then the Smith- McMillan form of the matrix pencil at s=∞ is

Irreducibility at infinity A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Remark. Candidates for (i.d.z.) and (o.d.z) at s= ∞ of a GSS realization of of a polynomial matrix A(s) are the zeros at s=∞ of A(s) Example. Let i.e. A(s) has one pole at s=∞ of order q 1 =2 and no zeros at s= ∞ Continued…

Irreducibility at infinity A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki …Continued A GSS realization is with The generalized order of Σ g is Continued… J ∞3 C ∞1 C ∞2 C ∞3 J ∞1 J ∞2 B ∞1 B ∞2

Irreducibility at infinity A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki …Continued Continued… It can be seen that both have a zero of order 1 at s=∞, and thus Σ g has an i.d.z., an o.d.z. and i.o.d.z.at s=∞. Now since we may easily obtain a “smaller” GSS realization of A(s), by simply eliminating the “middle” blocks from Σ g,, with

Irreducibility at infinity A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki …Continued It can be seen that both have no zeros at s=∞, which leads to the following definition

Irreducibility at infinity A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Let and be a GSS realization of A(s) with J ∞ in Jordan normal form. Then: Corollary. (i) has no zeros at s=∞, iff (ii) has no zeros at s=∞, iff Definition. A GSS realization of a polynomial matrix A(s) is called irreducible at s=∞, iff has no input and no output decoupling zeros at s= ∞.

Minimal GSS realizations of a polynomial matrix A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Definition. A GSS realization of a polynomial matrix A(s) is called minimal, iff it has the least number of generalized states. Theorem. Let be the Smith – McMillan form of A(s) at s=∞. A GSS realization of A(s), with J ∞ in Jordan canonical form is minimal iff:

Minimal GSS realizations of a polynomial matrix A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Remark. (i)A necessary condition for the minimality of a GSS realization is must be irreducible at s=∞ must not have non-dynamic variables (ii)The least dimension of a GSS realization is Where q i are the non-zero orders of the poles at s=∞, of A(s). Corollary. A GSS realization of a polynomial matrix A(s) is a minimal GSS realization iff it is an irreducible at s= ∞ GSS realization and has no non-dynamic variables.

Minimal GSS realizations of a polynomial matrix A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Example. Continuing the previous example, an irreducible at infinity GSS realization of A(s), was Obviously this realization is not minimal, since the expected least dimension of a GSS realization of A(s) should be (Recall that ) Continued…

Minimal GSS realizations of a polynomial matrix A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki This realization Σ g can be further reduced by moving its non-dynamic variables, from the state vector to the D ∞ matrix. The resulting minimal GSS realization is then given by …Continued

Minimal GSS realizations of a polynomial matrix A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Proposition. Let A(s) be a column proper polynomial matrix with column degrees where are the columns of A(s). Then an irreducible at s=∞ GSS realization of A(s) can be obtained by inspection and is given by:

Minimal GSS realizations of a polynomial matrix A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki Theorem. Let be the Smith-McMillan form at s=∞ of A(s). If the GSS realization is irreducible at s=∞ then the zero structure at is isomorphic to the zero structure of the GSS Rosenbrock system matrix and its Smith-McMillan form at s=∞ is given by

Conclusions  We have investigated the mechanism of cancellations of zeros at s=∞.  We discussed the concepts of irreducibility and minimality of pure GSS realization.  The role of dynamic and non-dynamic variables has been examined.  The isomorphism between zeros at s=∞ of the "infinite pole pencil" and the Rosenbrock system matrix has been presented. A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki