On the application of different numerical methods to obtain null-spaces of polynomial matrices. Part 1: block Toeplitz algorithms. J.C. Zu´niga˜ and D. Henrion Abstract— Four different algorithms are designed to obtain a polynomial matrix using these Toeplitz methods, in the the null-space of a polynomial matrix. In this first part we present work we develop four Toeplitz algorithms to obtain present two algorithms. These algorithms are based on classi- null-spaces of polynomial matrices. cal methods of numerical linear algebra, namely the reduction into the column echelon form and the LQ factorization. Both First, we develop two algorithms based on classical algorithms take advantage of the block Toeplitz structure of numerical linear algebra methods, and in the second part the Sylvester matrix associated with the polynomial matrix. of this work [17] we present two other algorithms based on We present a full comparative analysis of the performance of the displacement structure theory [9], [10]. The motivation both algorithms and also a brief discussion on their numerical for the two algorithms presented in [17] is to apply them to stability. large dimension problems for which the classical methods I. INTRODUCTION could be inefficient. In the next section 2 we define the null-space structure By analogy with scalar rational transfer functions, matrix of a polynomial matrix. In section 4 we present the two transfer functions of multivariable linear systems can be algorithms to obtain the null-space of A(s), based on the N(s)D 1(s) written in polynomial matrix fraction form as − numerical methods described in section 3. In section 5 we D(s) where is a non-singular polynomial matrix [1], [8], analyze the complexity and the stability of the algorithms. [11]. Therefore, the eigenstructure of polynomial matrices N(s) or D(s) contains the structural information on the represented linear system and it can be used to solve several II. NULL-SPACE STRUCTURE control problems. Consider a k l polynomial matrix A(s) of degree d. In this paper we are interested in obtaining the null-spaces We define the null-space× of A(s) as the set of non-zero of polynomial matrices. The null-space of a polynomial polynomial vectors v(s) such that matrix A(s) is the part of its eigenstructure which contains all the non-zero vectors v(s) which are non-trivial solutions A(s)v(s) = 0: (1) of the polynomial equation A(s)v(s) = 0. By analogy with the constant case, obtaining the null- ρ = rank A(s) A(s) space of a polynomial matrix A(s) yields important infor- Let . A basis of the null-space of l ρ mation such as the rank of A(s) and the relation between is formed by any set of linearly independent vectors δ − its linearly dependent columns or rows. satisfying (1). Let i be the degree of each vector in the basis. If the sum of all the degrees δ is minimal then we Applications of computational algorithms to obtain the i have a minimal null-space basis in the sense of Forney [2]. null-space of a polynomial matrix can be found in the poly- nomial approach to control system design [11] but also in The set of degrees δi and the corresponding polynomial others related disciplines. For example, in fault diagnostics vectors are called the right null-space structure of A(s). the residual generator problem can be transformed into the In order to find each vector of a minimal basis of the null- problem of finding a basis of the null-space of a polynomial space of A(s) we have to solve equation (1) for vectors v(s) matrix [12]. with minimal degree. Using the Sylvester matrix approach Pursuing along the lines of our previous work [15] where we can write the following equivalent system of equations we showed how the Toeplitz approach can be an alternative A0 0 to the classical pencil approach [14] to obtain infinite 2 3 . structural indices of polynomial matrices, and [16] where . A0 v0 6 7 2 3 we showed how we can obtain all the eigenstructure of 6 . .. 7 v1 6 Ad . 7 = 0: (2) 6 7 6 . 7 J.C. Zu´niga˜ and D. Henrion are with LAAS-CNRS, 7 Avenue du Colonel 6 Ad A0 7 6 . 7 Roche, 31077 Toulouse, France. 6 7 6 7 6 . 7 4 vδ 5 D. Henrion is also with the Institute of Information Theory and Au- 6 .. 7 6 7 tomation, Academy of Sciences of the Czech Republic, Pod vodarenskou´ 0 A veˇzˇ´ı 4, 18208 Prague, Czech Republic, and with the Department of Control 4 d 5 Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka´ 2, 16627 Prague, Czech Republic. Notice that we can also consider the transposed linear system more used method to reveal the rank of a matrix because of its robustness [3], [7]. See section 5 for a brief discussion vT vT vT 0 1 δ about the stability of the algorithms presented in this paper. T T ··· × A0 Ad 0 2 A···T AT 3 0 ··· d B. Column Echelon Form (CEF) 6 . 7 = 0: (3) 6 .. .. 7 6 7 For the definition of the reduced echelon form of a matrix 4 0 AT AT 5 0 ··· d see [3]. Let L be the column echelon form of M. The If we are interested in the left null-space structure of particular form that could have L depends on the position A(s), i.e. the set of vectors w(s) such that w(s)A(s) = 0, of the r linearly independent rows of M. For example, let us then we can consider the transposed relation AT (s)wT (s) = take a 5 4 matrix M. Consider that after the Householder × 0. Because of this duality, in the remainder of this work we transformations Hi, applied onto the rows of M in order to only explain in detail how to obtain the right null-space obtain its LQ factorization, we obtain the column echelon structure of A(s), that we call simply null-space of A(s). form III. LQ FACTORIZATION AND COLUMN ECHELON FORM 11 0 0 0 2 × 3 21 0 0 0 From equations (2) or (3) we can see that the null-space T × L = MQ = MH1H2H3 = 6 31 32 0 0 7 : of A(s) can be recovered from the null-space of some 6 × × 7 6 41 42 43 0 7 suitable block Toeplitz matrices as soon as the degree δ 6 × × × 7 of the null-space of A(s) is known. 4 51 52 53 0 5 × × × (4) Let us first assume that degree δ is given. The methods This means that the second and the fifth rows of M are that we propose to obtain the constant null-spaces of the linearly dependent. corresponding Toeplitz matrices consist in the application of The column echelon form L allows us to obtain a basis of successive Householder transformations in order to reveal the left null-space of M. From equation (4) we can see that their ranks. We consider two methods: the LQ factorization the coefficients of linear combination for the fifth row of and the Column Echelon Form. M can be recovered by solving the linear triangular system A. LQ factorization 11 0 0 The LQ factorization is the dual of the QR factorization, 2 × 3 k1 k3 k4 31 32 0 = see [3]. Consider a k l constant matrix M. We define × × × 4 41 42 43 5 the LQ factorization of M as M = LQ where Q is an × × × 51 52 53 ; (5) orthogonal l l matrix and L = [Lr 0] is a k l matrix − × × × that we can always× put in a lower trapezoidal form× with a row permutation P : and then k1 0 k3 k4 1 M = 0: w11 2 . × 3 . .. Also notice that when obtaining the basis of the left null- 6 7 T PL = 6 wr1 wrr 0 7 = [PLr 0] space of M from matrix L it is not necessary to store Q = 6 ··· 7 6 . 7 H1H2:::. This fact can be used to reduce the algorithmic 6 . 7 6 7 complexity. 4 wk1 wkr 5 ··· We apply the CEF method to solve the system of equa- where r min(k; l) is the rank of M. tions (3). Since the number of columns of the Toeplitz Notice≤ that L has l r zero columns. Therefore, an matrix can be large, we do not store orthogonal matrix Q. orthogonal basis of the− null-space of M is given by the We only obtain the CEF and we solve the corresponding last l r rows of Q, namely, MQ(r + 1 : l; :)T = 0 and triangular systems as in (5) to compute the vectors of − moreover, we can check that M = LrQ(1 : r; :). the null-space. For more details about the CEF method We apply the LQ factorization to solve system (2). Here described above see [6]. the number of columns of the Toeplitz matrix is not very large, so we obtain the orthogonal matrix Q and, as a IV. BLOCK TOEPLITZ ALGORITHMS by-product, the vectors of the null-space. This method is a cheap version of the one presented in [15] where we For the algorithms presented here we consider that the used the singular value decomposition (SVD) to obtain the rank ρ of A(s) is given. There are several documented rank and the null-spaces of the Toeplitz matrices.