Several types of factorizations solve the main problems of system theory (e.g., identification, estimation, system inversion, system approximation, and optimal control). The factorization type depends on what kind of operator is factorized, and what form the factors should have. This and the following chapter are, therefore, devoted to the two main types of factorization: this chapter treats what is traditionally called coprime factorization, while the next is devoted to inner–outer factorization. Coprime factorization, here called “external factorization” for more generality, characterizes the system’s dynamics and plays a central role in system characterization and control issues. A remarkable result of our approach is the derivation of Bezout equations for time-variant and quasi-separable systems, obtained without the use of Euclidean divisibility theory. From a numerical point of view, all these factorizations reduce to recursively applied QR or LQ factorizations, applied on appropriately chosen operators.

Two chapters conclude the basic course.

This chapter presents an alternative theory of external and coprime factorization, using polynomial denominators in the noncommutative time-variant shift Z rather than inner denominators as done in the chapter on inner–outer theory. “Polynomials in the shift Z” are equivalent to block-lower matrices with a support defined by a (block) staircase, and are essentially different from the classical matrix polynomials of module theory, although the net effect on system analysis is remarkably similar. The polynomial method differs substantially and in a complementary way from the inner method. It is computationally simpler but does not use orthogonal transformations. It offers the possibility of treating highly unstable systems using unilateral series. Also, this approach leads to famous Bezout equations that, as mentioned in the abstract of Chapter 7, can be derived without the benefit of Euclidean divisibility methods.