Constrained traffic equilibrium in routing

Use qij to denote the (i; j)-entry of the matrix Q (i; j = 1; 2). Then (Q) reads as (Q) = 0(1 + p 05)r 0 2 0(2 0 p 05)r + (1 + p 05) 2r + (1 0 p 05) 0(1 + p 05)r 0 2 (2) where r = (01+ p 05)q 11 0(2+ p 05)q 12 +2q 21 + (01+ p 05)q 22 : (3) Because any stabilizing controller can be obtained from the (2, 1)-and (2, 2)-entries of (2), we are interested in the set of r's of (3), that is, the ideal generated by (01 + p 05); 0(2 + p 05), and 2. By letting q 11 = q 22 = 0; q 12 = p 05;q 21 = 02 + p 05, we see that r is equal to 1. This implies that the set of r's of (3) is identical to A. Hence, the parameterization of stabilizing controllers of p is given as 2r + (1 0 p 05) 0(1 + p 05)r 0 2 (4) where r is a parameter of A. The denominator of (4) cannot be zero because in this case, r = 0(1=3)(1 0 p 05), which is not in A. The form of (4) is similar to the Youla–Kučera parameterization. However, we note that as polynomials of r, the numerator and the denominator of the fraction have constant terms that are not coprime, and the same is the case for the coefficients of r. IV. CONCLUSION All stabilizing controllers of the example given in [1] can be param-eterized by only one parameter even though it does not have a coprime factorization. From this result, even in the case where there is no doubly coprime factorization, we observe that the controller parameterization may be in the form of the Youla–Kučera parameterization and that the number of parameters may be smaller than (m + n) 2. Based on the result of this note, we need to further investigate under what condition the parameterization adopts the form of the Youla-Kučera parameterization as in (4). Moreover, the minimum number of parameters should be explained. So far we know that the number of parameters is less than or equal to (m + n) 2 [4], but we do not yet have a method of determining the minimum number. ACKNOWLEDGMENT The author would like to thank the Associate Editor for his …