Youla–Kučera Parametrization with no Coprime Factorization—Single-Input Single-Output Case

We present a generalization of the Youla—Kučera parametrization to obtain all stabilizing controllers for single-input and single-output plants. This uses three parameters and can be applied to plants that may not admit coprime factorizations. In this generalization, at most two rational expressions of plants are required, while the Youla–Kučera parametrization requires precisely one rational expression.


Introduction
So far, the coprime factorization has played a central role to obtain stabilizing controllers in the factorization approach [1]. The factorization approach to control systems has the advantage that it includes, within a single framework, numerous linear systems such as continuous-time as well as discrete-time systems, lumped as well as distributed systems, one-dimensional as well as multidimensional systems, etc. [1,2]. A transfer function of this approach is considered as the ratio of two stable causal transfer functions. One of the attractive points of the factorization approach is the fact that all stabilizing controllers can be obtained by the Youla-Kučera parametrization with coprime factorization [3][4][5]. This Youla-Kučera parametrization has been used in a wide variety of applications for a long time (e.g., [6][7][8][9][10]).
Unfortunately, the Youla-Kučera parametrization cannot be applied to the plants that do not admit coprime factorizations. Mori, so far, gave the method to obtain part of stabilizing controllers by some different factorizations [11,12]. The objective of this paper is to generalize the Youla-Kučera parametrization to be applicable even for single-input single-output plants that may not admit the coprime factorization. This generalization employs three parameters and requires at most two rational expressions of plant, while the Youla-Kučera parametrization requires only one parameter and one rational expression of plant. The generalization will be expressed with an extension of Bézout identity. We will show that this generalization is equivalent to the parameterization method of [13], which does not require coprime factorization.
This paper is started with preliminaries from Section 2 to recall the notion of the factorization approach. We next state the main results of this paper, generalization of the Bézout identity and the Youla-Kučera parametrization, in Section 3. Then we review, in Section 4, the parametrization of stabilizing controllers of plants which may not admit coprime factorizations [13]. The proofs of the main results are given in Section 5. In Section 6, we give examples for the main results of Section 3. First example will be the plants that admit coprime factorizations. The next one will be Anantharam's example [14]. Third one will be the discrete-time systems without the unit-delay element. We consider that the set of stable causal transfer functions is an integral domain with identity, denoted by A. The total field of fractions of A is denoted by F ; that is, F = {n/d | n, d ∈ A, d = 0}. This F is considered as the set of all possible transfer functions. Let Z be a prime ideal of A with Z = A. Define the subsets P and P s of F as follows: Then, a transfer function in P (P s ) is called causal (strictly causal).
Throughout the paper, the plant we consider has single-input and single-output, and its transfer function, which is also called a plant itself simply, is denoted by p and belongs to P (that is, p is causal).
For p ∈ P and c ∈ F , a matrix H(p, c) ∈ F 2×2 is defined as provided that 1 + pc is nonzero. This H(p, c) is the transfer matrix from [ u 1 u 2 ] t to [ e 1 e 2 ] t of the feedback system of Figure 1. If 1 + pc is nonzero and H(p, c) ∈ A 2×2 , then we say that the plant p is stabilizable, p is stabilized by c, and c is a stabilizing controller of p. In the definition above, we do not mention the causality of the stabilizing controller. Even so, it is known that if a causal plant is stabilizable, there always exists a causal stabilizing controller of the plant, and further if a strictly causal plant is stabilizable, any stabilizing controller of the plant is causal [16] [Propositions 6.1 and 6.2]. We denote by S(p) the set of all stabilizing controllers of the plant p, and by H(p) the set of H(p, c)'s with all stabilizing controllers c of p. The relationship between S(p) and H(p) is as follows [17]: Thus, obtaining S(p) and obtaining H(p) are equivalent to each other.

Main Results
We present three main results. The first one is a generalization of the notions of Bézout identity and coprime factorization. The others are generalizations of the Youla-Kučera parametrization that can be applied to stabilizable plants even with no coprime factorization. Theorem 1. Let p be a causal plant (p ∈ P). Then p is stabilizable if and only if there exist n 1 and n 2 of A, and d 1 and d 2 of A − {0} such that with y 1 , x 1 , y 2 , x 2 of A.
Theorem 2. Let p be a stabilizable causal plant of P with symbols in Theorem 1 satisfying (4) and (5). Then the set S(p) of all stabilizing controllers of p is given as Theorem 3. Let p be a stabilizable causal plant of P and c its stabilizing controller. Denote Then the set S(p) of stabilizing controllers of p is given as Remark 1. The fraction in (6) can be rewritten as and, by noting that n 1 d 2 = n 2 d 1 , Observing the fractions above, we might add new parameter s of A as follows: where s and s are s of (8) and (9), respectively. Even so, rearranging the numerator and denominator, we have Since s and s appear as in the form s + s only, we can remove the parameter s and consider s only.

Remark 3.
An attractive point of Theorem 2 is that the set of stabilizing controllers can be obtained in the generalization form of the Youla-Kučera parametrization without the computation of coprime factorization, once the Equations (4) and (5) are obtained. On the other hand, Theorem 3 has the same attractive point once exactly one stabilizing controller is obtained.

Remark 5.
Suppose that a plant p ∈ P has two rational representations n 1 /d 1 and n 2 /d 2 with n 1 , n 2 , d 1 , d 2 ∈ A. Suppose further that we have found y 1 and x 2 of A such that In this case, we can apply Theorems 1 and 2 to the plant with y 2 = x 1 = 0 as special cases of Theorems 1 and 2, so that a stabilizing controller can be obtained and the set of stabilizing controllers can also be obtained. See Sections 6.2 and 6.3 for examples. Note that, in (10), we consider only numerator and denominator of (possibly) different rational expressions. Also, (10) does not mean the coprimeness of the plant. Evan so, once we have (10), we can obtain all stabilizing controllers.

Parametrization without Coprime Factorizability
Here, we briefly review the parameterization method of [13], which does not require coprime factorization. This is used to give the proof of Theorem 2.
This theorem gives the parameterization with a parameter matrix Q without coprime factorizability of the plant. The parameterization by Ω(Q) is independent of the choice of stabilizing controller c.
Consider the case where x 1 d 1 + x 2 d 2 is zero. This is equal to 1 − (y 1 n 1 + y 2 n 2 ), so that y 1 n 1 + y 2 n 2 = 1. It follows that at least one of y 1 n 1 and y 2 n 2 is nonzero. Assume, without loss of generality, that y 1 n 1 is nonzero, which means that both y 1 and n 1 are nonzero. Because d 1 is nonzero, n 1 d 1 is a nonzero. Thus, From the previous paragraph, we observe that the expression is nonzero by appropriate choice of (r 0 , t 0 ) from (0, 0), (1, 0), (0, 1). In the following, we suppose that (20) is nonzero with (r 0 , t 0 ) being one of (0, 0), (1, 0), (0, 1).
From now, we show that the following c is a stabilizing controller of p: This is done by showing that H(p, c) with c of (21) is over A, which consists of h 11 , h 12 , h 21 of (19).
Observe that which are all in A, so that H(p, c) is over A. Further, 1 + pc is 1/( , which is nonzero. Hence, c is a stabilizing controller of p. Therefore, p is stabilizable.
Proof of Theorem 2. Suppose that p is stabilizable. We denote by S(p) the right-hand side of (6). We also introduce H(p), by virtue of the relation (2), as follows: This H(p) is expressed as follows:

Thus, the proof of Theorem 2 is achieved by showing H(p) = H(p), which is done by showing H(p) ⊃ H(p) and H(p) ⊂ H(p).
In the following, based on the proof of Theorem 1, we assume, without loss of generality, that which is a stabilizing controller of p. Then H(p, c) is as follows: Based on this c, we consider an element of H(p), that is, a matrix below in the set of the right-hand side of (17) with the equations of (14), (15), and (16): Now we let q 12 = (n 2 1 + d 1 n 2 1 (x 1 + r 0 n 1 ) + d 1 n 1 n 2 (x 2 + t 0 n 2 ))(r + r 0 ) + n 1 n 2 s + (n 2 2 + d 2 n 1 n 2 (x 1 + r 0 n 1 ) + d 2 n 2 2 (x 2 + t 0 n 2 ))(t + t 0 ), Then, a straightforward but tedious computation shows that the matrix of (24) becomes equal to the matrix in the right-hand side of (22). Hence we have H(p) ⊃ H(p).
(H(p) ⊂ H(p)). Suppose an element of H(p) of (22). Then we let By a straightforward but tedious computation again, the matrix in the right-hand side of (22) becomes equal to Ω(Q) of (11) with Q = q 11 q 12 q 21 0 ,

Therefore, we have H(p) ⊂ H(p).
Proof of Theorem 3. We consider two cases: c = 0 and c = 0.
(c = 0). In this case, p is in A. Then, h 11 = 1, h 12 = −p, and h 21 = 0. The fraction in (7) is expressed as r 1−rp provided 1 − rp is nonzero. This is just the Youla-Kučera parametrization of the plant p in A by noting that the coprime factorization of p ∈ A is p = n/d with n = p, d = 1, and the Bézout identity 0 · n + 1 · d = 1.

Coprime Factorization
Suppose that a plant admits a coprime factorization, say p = n/d and yn + xd = 1 with n, d, y, x ∈ A. Letting n 1 = n 2 = n, d 1 = d 2 = d, y 1 = y, x 1 = x, y 2 = x 2 = 0, one can apply Theorems 1 and 2 to the plant in order to obtain stabilizing controllers. The expression in the right-hand side of (6) is expressed as By replacing (r + s + t) by new parameter u of A, we have which is equivalent to the Youla-Kučera parametrization.

Discrete-Time Systems without Unit-Delay Element
Mori [16] considered the case A = R[z 2 , z 3 ], where R denotes the set of real numbers. We also let Z be { f ∈ A | f has zero constant term.}. This ring is an integral domain but not a unique factorization domain. In fact, z 6 ∈ A has two factorizations, z 2 · z 2 · z 2 and z 3 · z 3 .

Conclusions and Future Work
This paper has presented a generalization of the Youla-Kučera parametrization to obtain all stabilizing controllers without coprime factorization. This is based on two rational expressions of a given plant. Alternative parametrization is also given by one stabilizing controller.
As future work, we will aim to investigate further generalization of the Youla-Kučera parametrization for multi-input multi-output plants with no coprime factorizations as well. Also, the possibility to extend Theorems 1-3 will be investigated.