Coprime factorization and internal stability

In the previous section the internal stability of the LFT configuration was discussed through the system state space description. In this and the successive section however, stability is analysed in a different framework, involving the coprime factorization of the constituent systems of the feedback interconnection over the set of proper and stable transfer matrices. Studying the stability problem in this framework is a special case of a more general approach in which the analysis and synthesis problems are formulated based on the fractional representation of the systems, principally developed in [11, 12, 13, 20]. This approach, which has its roots in abstract algebra, considers in the SISO case the transfer functions with the prescribed properties as a ring , and then models a given system as the ratio of two transfer functions in . This casts the synthesis problem as designing a feedback system which lies in a desired ring of operators when both the plant and compensator are modelled as a quotient of operators from that ring [11]. What makes the procedure highly interesting is that the synthesis problem yields a complete characterization of all compensators which place the feedback system in the ring . This approach could be readily extended to the MIMO systems when the transfer matrix has all its entries in . The operations of matrix addition and multiplication induced on the set of matrices over by the associated addition and multiplication operations with , correspond to parallel and cascade interconnection of such systems [15, 17].

For the purpose of this section, we are only concerned with those aspects of the fractional representation theory pertaining to feedback stabilization. The central idea is that of expressing each constituent elements of the feedback interconnection as the irreducible quotient of two proper and stable elements. [15, 13] This is accomplished by considering

the notion of coprime factorization and its characteristics relevant to internal stability theory, which forms the foundation for developing a parameterization of stabilizing controllers in the next section.

Let be a ring and × be the set of × matrices whose elements all belong to . Every element � of × , the set of × transfer matrices, can be factored as an element of the field of fractions associated with the ring and expressed as the ratio of two matrices and , as � = − where , ∈ × and det ≠ . The pair , is referred to as a right fractional representation of �. In a similar way, the left fractional representation of every � ∈ × is defined as � = ̃− ̃ where again ̃, ̃ ∈ × and

det ̃ ≠ [11, 13, 20, 165, 166].

Definition 2.5.1 [13] Two matrices , ∈ × are called right coprime if there exists matrices , ∈ × such that:

− = (2.5.1)

which can be stated equivalently as:

Definition 2.5.2 [19] Two matrices , ∈ × are right coprime if they have equal number of columns and there exists matrices , ∈ × such that:

[ _{] [− ] = (2.5.2)}

This is equivalent to the matrix [ − ] being left-invertible in × .

The equality (2.5.1) is known as the right Bezout identity or right Diophantine identity.

It extends the notion of relatively prime (or coprime) integers, i.e. the Euclid’s algorithm,

to matrices. If and are two integers, i.e. , ∈ ℤ, there exists , ∈ ℤ such that +

= GCD , , with GCD denoting the greatest common divisor of and . and are

Definition 2.5.3 [13] In definition 2.5.2, if is non-singular, � = − is referred to right coprime factorization (r.c.f) of �.

The notions of left coprime and left coprime factorization can be defined analogously.

Definition 2.5.4 [19] For � ∈ × , ̃, ̃ ∈ × with equal number of rows, and ̃ non-singular, � = ̃− ̃ is called the left coprime factorization (l.c.f) of � if there exists matrices ̃, ̃ ∈ × such that:

[̃ ̃] [ ̃

− ̃] = (2.5.3)

This is equivalent to the matrix [̃ ̃] being right-invertible in × . The Bezout identity corresponding to (2.5.3):

̃ ̃ − ̃ ̃ = (2.5.4)

is referred to left Bezout identity or left Diophantine identity.

The ring concerning the internal stability problem being considered here, is the set of proper and stable rational transfer matrices, namely the ring _∞ [19, 165, 166]. The setting in which = _∞, not only catches the usual notion of instability as the result of existing unstable closed-loop poles, but also excludes the possibility of unstable pole-zero cancellations between the plant and controller. These will become clear as we proceed. From coprime fractional representation over _∞, some significant properties imposed by coprimeness can be inferred, which reveals the benefits of studying stabilization problem in a ring theoretic setting.

In view of Lemma 2.4.4 which establishes the equivalence between the stabilization of the plant and that of (figure 2.3.1), all the subsequent discussion is made about . Suppose that has a r.c.f over _∞ as = − , where , ∈ _∞. Rewriting the right Bezout identity of (2.5.1) (in accordance to the r.c.f of ) as − = , in which , ∈ _∞, and using the identity − = − , it can be inferred that

Remark 2.5.5 [15] Instabilities of are completely characterized by the denominator of a r.c.f of , i.e. the unstable poles of are precisely the unstable zeros of .

Theorem 2.5.6 [20] The pairs , ∈ _∞ and , ∈ _∞ define right coprime factorizations of as = − = − , if and only if:

[ ] = [ ]

where , − ∈ _∞. In other words, is _∞-unimodular.

Definition 2.5.7 [46] Let ℛ be a ring and ℛ× denote the × matrices with elements from ℛ. A matrix ∈ ℛ × is unimodular, if and only if det is a unit in ℛ, i.e. it is a

matrix in ℛ × whose inverse belongs to ℛ × too. Such matrices are termed

ℛ-unimodular and designated as [ℛ].

Theorem 2.5.6 may be stated in the equivalent form of the following remark.

Remark 2.5.8 [22] A right coprime factorization is unique up to a unimodular common right divisor.

This remark, in turn, leads to the following important observation:

Remark 2.5.9 [17] Cancellation of instabilities, between the numerator and denominator of a r.c.f is not allowed. In this sense, a r.c.f is irreducible.

This makes clear the notion of irreducible quotient of stable elements mentioned earlier. A similar theorem and remarks hold analogously for a l.c.f of = − .

It is worth mentioning that for rational transfer matrices existence of right and left coprime factorizations is assured. However, this is not the case in the general ring theoretic formulation [24, 23, 6].

The following theorem establishes the connection between the right and left coprime factorizations.

Theorem 2.5.10 [23] Assume that admits both right and left coprime factorization as

= − ₌ − _{, with , , , ∈}

∞, for which there exists , ∈ ∞

such that − = . Then, there exist , ∈ _∞ such that:

[₋ − ] [ ] = [ ] (2.5.5)

This is referred to as the doubly coprime factorization or generalized Bezout identity, and is the cornerstone in the parameterization of all stabilizing controllers, as will be shown in the next section.

The notion of coprimeness readily extends to continuous-time as well as discrete-time systems, lumped and distributed systems, and one- and multi-dimensional systems. Therefore, all these situations can be captured within the single framework of stable factorization approach [15, 13].