454 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 3, MARCH 1999 A Recursive Method for Solving Unconstrained Tangential Interpolation Problems Tibor Boros, Ali H. Sayed, Member, IEEE, and Thomas Kailath, Life Fellow, IEEE Abstract— An efficient recursive solution is presented for the tangential interpolation conditions of the form one-sided unconstrained tangential interpolation problem. The method relies on the triangular factorization of a certain struc- tured matrix that is implicitly defined by the interpolation data. The recursive procedure admits a physical interpretation in where are given complex numbers and where and terms of discretized transmission lines. In this framework the are and complex vectors, respectively. generating system is constructed as a cascade of first-order This paper deals with a generalization of these problems that sections. Singular steps occur only when the input data is contra- dictory, i.e., only when the interpolation problem does not have imposes tangential interpolation conditions both on and a solution. Various pivoting schemes can be used to improve on its derivatives. We describe an efficient recursive algorithm numerical accuracy or to impose additional constraints on the for computing the rational interpolants and show how to handle interpolants. The algorithm also provides coprime factorizations minimality constraints recursively. This will be achieved by for all rational interpolants and can be used to solve polynomial using a generalized Schur-type algorithm originally developed interpolation problems such as the general Hermite matrix inter- polation problem. A recursive method is proposed to compute a for the fast triangular factorization of structured matrices and column-reduced generating system that can be used to solve the by exploiting some degrees of freedom in its description (see minimal tangential interpolation problem. [13] for a review on matrix factorization). Relations to earlier Index Terms—Interpolation, matrix decomposition, numerical work will be presented after a formal problem statement. stability, polynomial matrices, rational functions, rational matri- ces. A. Problem Statement Let denote the field of scalar rational functions of I. INTRODUCTION a single variable and the linear space of EVERAL problems in control, circuit theory, and digital rational matrix functions defined over Furthermore, Sfilter design can be reduced to the solution of matrix ra- let be an operator that maps into an tional interpolation problems which have been widely studied upper triangular Toeplitz block-matrix, , (see, especially, [1]–[8]). This paper treats left-sided tan- as shown in (1) at the bottom of the next page, where gential interpolation problems with and without minimality stands for the th derivative of constraints. Applications occur, for example, in minimal partial Problem 1.1 (Unconstrained Tangential Interpolation): realization [1]–[3] and in the -parameterization of stabilizing Consider a set of points and two sets of row vectors controllers for unstable plants [9]–[12]. and such that In its simplest form, an interpolation problem would ask for (2a) rational functions that meet the interpolation conditions for given complex numbers and (2b) The interpolants can further be required to (2c) have minimal complexity measured in terms of their McMillan degree. An extension of this problem to the vector case would ask for rational matrix functions that satisfy 1) Given the nodes and the associated vectors and find all rational interpolants that are analytic at and satisfy the interpolation Manuscript received November 4, 1994; revised December 6, 1995 and August 4, 1996. Recommended by Associate Editor, P. Van Dooren. This conditions work was supported by the Army Research Office under contract DAAH- 93-G-0029 and by a grant from NSF under Award MIP-9409319. This work for all (3) was initiated while A. H. Sayed was a Research Associate in the Information Systems Laboratory. 2) Given the extraction points evaluate T. Boros is with ArrayComm, Inc., San Jose, CA 95134 USA. for any particular solution A. H. Sayed was with Stanford University, on leave from Escola Polit´ecnica, da Universidade de Sao˜ Paulo, Brazil. He is now with the Department of In the problem statement, we use the adjective uncon- Electrical Engineering, University of California, Los Angeles, CA 90095 USA. strained because no other restrictions, such as minimality T. Kailath is with the Information Systems Laboratory, Stanford University, Stanford, CA 94305 USA. or boundedness, are imposed on the rational interpolants. Publisher Item Identifier S 0018-9286(99)02125-X. In particular, apart from the analyticity conditions at 0018–9286/99$10.00 1999 IEEE BOROS et al.: SOLVING UNCONSTRAINED TANGENTIAL INTERPOLATION PROBLEMS 455 there are no other constraints on the location of poles of the a rather different problem, viz., the triangular factorization interpolants. At the same time, in this paper we also investigate problem for non-Hermitian matrices possessing displacement the following constrained interpolation problem. structure [9], [14]–[13]. Problem 1.2 (Minimal Tangential Interpolation): Given The factorization of a non-Hermitian matrix can be naturally the nodes and the direction vectors and as in associated with two -input -output feedforward (2), find all rational matrix functions so that cascade systems denoted by and Each step of the • is analytic at and satisfies the interpolation algorithm determines first-order (lattice) sections in each of the conditions (3); two cascades. The elementary sections obtained this way have • the complexity of (measured in terms of its McMil- transmission zeros: certain inputs at certain frequencies yield lan degree) is as small as possible. zero outputs (this is a general property of any linear system). When the sections are designed appropriately, these “local” B. Connections to Earlier Work transmission zeros combine to yield a “global” transmission zero (see Proposition 4.2 below) which can be used to solve Prior work on the unconstrained tangential interpolation unconstrained rational interpolation problems. This approach problem has been largely carried out by Ball et al. [4], [5]. has been successfully used in various other interpolation The main result in [4] states that the family of all rational problems as well (see, e.g., [6] for Schur-type and [15] for functions that satisfy (3) can be parameterized in terms of unconstrained interpolation problems). a certain linear fractional map. Specifically, it is possible to The matrix that we factor here is implicitly determined translate the interpolation data into a so-called left null pair via a non-Hermitian displacement equation of the form1 that describes the zero structure of a rational matrix function denoted by (5) where and are constructed directly from the inter- polation data [as shown in (10)], while and are free parameters that can be chosen to guarantee that no breakdowns Reference [4, Th. 5.1.2] then states that satisfies the occur in the recursive algorithm. In contrast to the methods in interpolation conditions (3) if and only if, one can write [4] and [15], the pair does not have to be known in advance; the relevant entries can be chosen “on the fly” (4) when they are needed in the algorithm (see Algorithm 6.1). The additional degrees of freedom in can be used for some rational matrices and To compute a to impose various constrains on the rational interpolants (see suitable generating system (which incidentally is called Section VIII). We further note that in this approach does the resolvent matrix by the Odessa school of operator theory), not have to be invertible or strongly regular, or even explicitly one has to first construct a so-called right pole pair so known. that the solution of the Sylvester equation The main results on minimal interpolation problems to this date appear in [1] and [3] where it is shown that, in the special case when the transfer function is a column-reduced polynomial matrix, it is possible to extract the admissible is invertible. Then, can be obtained from a global state- degrees of complexity as well as the minimal degree of space formula that involves and (see [5, pp. complexity from the linear fractional parameterization formula 23–24, 74, and 103] for the exact definition of left null pairs, (4). In the scalar case, Antoulas et al. suggested first finding right pole pairs, and null-pole triples). the Lagrange interpolating polynomial and then applying long In this paper, we present a different recursive method division (Euclidean algorithm) to obtain in column- which can be used to compute the generating system as a product of elementary first-order rational matrix functions. 1 In principle, the signature matrix t on the right-hand side of (5) could be The recursive technique allows us to update whenever merged into q or ffX However, the present form allows us to remain consistent with Schur-type interpolation problems where the underlying displacement a new interpolation point is added to the input data set. The equation can be written as pppp a qqttqtqY i.e., e a p and f a q aforementioned algorithm was first studied in connection with [13], [6]. (1) . .. 456 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 3, MARCH 1999 reduced form [1], [2]. Since it is difficult to extend this method The next problem is borrowed from [12] (see also [5, Ch. to the tangential case, one is obliged first to find a general 23], [10], [11], and [17]). transfer matrix by using an unconstrained algorithm. In Problem 1.4 -Parameterization for Unstable Plants): Let the next step the generating system must be transformed be a strictly proper rational plant, into column-reduced form via a sequence of elementary (uni- having unstable poles at with corresponding multiplicities modular) transformations [16]. A detailed algorithm for the Assume, furthermore, that the Laurent construction of a column-reduced rational matrix function expansion of at is given by from a given null-pole triple has recently appeared in [7].