A Direct-Construction Approach to ... - Semantic Scholar

Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 24-28, 2006

ThA11.4

A Direct-Construction Approach to Multidimensional Realization and LFR Uncertainty Modeling Li Xu∗, Huijin Fan†, Zhiping Lin‡ and N. K. Bose§ ∗ Akita Prefectural University, Akita, Japan † Huazhong University of Science and Technology, Wuhan, P. R. China ‡ Nanyang Technological University, Singapore § The Pennsylvania State University, PA 16802, USA Abstract This paper proposes a direct-construction realization procedure which simultaneously treats all the involved variables and/or uncertain parameters and directly generates an overall Roesser model realization or LFR model for a given nD causal transfer matrix with rational function or polynomial entries. It is shown that the nD realization problem for an nD transfer matrix G(z1 , . . . , zn ), which is assumed without loss of generality to be strictly causal and given in the form of G(z1 , . . . , zn ) = N (z1 , . . . , zn )D(z1 , . . . , zn )−1 with D(0, ..., 0) = I and N (0, ..., 0) = 0, can be essentially reduced to the construction of an admissible nD polynomial matrix Ψ for which there exist matrices A, B, C such that N (z1 , . . . , zn ) = CZΨ and ΨD(z1 , . . . , zn )−1 = (I − AZ)−1 B with Z being the corresponding variable and/or uncertainty block structure, i.e., Z = diag{z1 Ir1 , . . . , zn Irn }. This important fact reveals a substantial difference between the 1D and nD (n ≥ 2) realization problems as in the 1D case Ψ can only be a monomial matrix and never a polynomial one, and implies the possibility to achieve a realization with further lower order than the constructive realization method given recently by the authors. Necessary and sufficient conditions that ensure Ψ to be admissible are given and, based on these conditions, algorithms are proposed for construction of an admissible Ψ with lower order and the corresponding realization. Illustrative examples are presented to illustrate the basic ideas and the effectiveness of the proposed method.

1

Introduction

The prevailing framework for robustness analysis and synthesis of uncertain systems requires modeling the underlying polynomial or rational parametric uncertainty in the so-called linear fractional representation (LFR). It has been observed that the LFR uncertainty modeling problem is algebraically equivalent to the realization problem for MIMO (multi-input, multi-output) multidimensional (nD) systems by Roesser state space model [1–8]. Therefore, an effective and efficient nD realization procedure will not only contribute significantly to nD system theory but also to robust control theory. In addition, as it is difficult in general to obtain a minimal realization for the nD (n ≥ 2) cases, it is particularly important to develop procedures which can generate nD realizations or LFRs with lowest possible order. Several general realization approaches have been documented in the literature (see, e.g., [1, 5, 7, 8, 10] and the references therein). The approaches established in [5, 7, 8, 11] are mainly based on the following ideas. Decompose a given transfer matrix to sums and products of simple factors consisting of 1D monomial or even

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a single variable itself, then construct the desired realization step by step, i.e., first create the elementary LFRs for each factor or variable, and next combine these elementary LFRs to obtain the overall LFR by utilizing the coupling formulas of LFRs. It has been shown that the tree decomposition approach proposed in [7] is the most effective one among this kind of approaches [8]. However, as the tree decomposition is not unique, heuristics are usually necessary to get the best decomposition especially for complicated cases [7, 8]. A common feature for step-by-step approaches is that the combination of the elementary LFRs requires lots of matrix calculations and complicated permutations for grouping together and sorting lexicographically the variables, which obviously brings considerable computational burden [7, 8]. On the other hand, the procedures given in [6, 17] directly generate an overall realization. The basic idea is to construct two vectors of variables, say w and z, from the nD monomials appearing in the given nD polynomial matrix, such that w = ∆z with ∆ being a diagonal uncertainty block structure for the LFR realization to be found. It can be shown, however, as this procedure is only based on the structural property of ∆ rather than the general properties of Roesser model itself, redundant variables are usually introduced in w and z which makes the order of the resultant realization unnecessarily high. It has been shown recently in [1] that the realization problem for an nD transfer matrix G(z1 , . . . , zn ), which is assumed without loss of generality to be strictly causal and given by a right matrix fractional description (MFD), i.e., G(z1 , . . . , zn ) = N (z1 , . . . , zn )D(z1 , . . . , zn )−1 with D(0, ..., 0) = I and N (0, ..., 0) = 0, can be essentially reduced to the construction of an nD monomial matrix Ψ such that N (z1 , . . . , zn ) = CZΨ and ΨD(z1 , . . . , zn )−1 = (I − AZ)−1 B where A, B, C are certain real matrices, which give a realization for G(z1 , . . . , zn ), and Z is the corresponding variable and/or uncertainty block structure, i.e., Z = diag{z1 Ir1 , . . . , zn Irn }. By investigating the structural properties of Roesser model, the following conditions that a specific Ψ must satisfy in order to meet the above restrictions have been clarified. (a) The entries of the jth column of ZΨ contain all the monomials occurring in the polynomial entries of the ˜ 1 , · · · , zn )T ]T with D(z ˜ 1 , . . . , zn ) = I−D(z1 , . . . , zn ). jth column of F (z1 , · · · , zn ) , [N (z1 , · · · , zn )T D(z (b) For each column of Ψ, there is at least a unit entry (i.e., an entry equal to 1). (c) Every non-unit entry of Ψ can be obtained from another entry in the same column by multiplying zi (i ∈ {1, . . . , n}), or equivalently, is equal to another entry of ZΨ in the same column. Based on these conditions, then, constructive algorithms have been given for the construction of such a Ψ with lowest possible order and the corresponding realization. An obvious advantage of this approach is that it is simple both conceptually and methodologically, which makes it computationally more efficient than the existing procedures. Moreover, it applies to both rational and polynomial transfer matrices with numerical or symbolic coefficients, and thus requires no further LFR operations, which are usually necessary for most of the existing approaches to transform a result for polynomial case to its rational counterpart. Illustrative examples have demonstrated that the procedure of [1] may generate realizations with much lower order than the other existing ones. In this paper, we will show that further significant improvements over the results of [1] can be achieved. A fact is first shown that it is not necessary to restrict the entries of Ψ to be nD monomials. Indeed, Ψ may be a matrix of nD polynomials, which obviously includes the monomial one of [1] as a special case. In this case, the conditions of (a) and (b) shown above remain its original forms except that Ψ is now a polynomial matrix, while the condition of (c) should be changed to (c’) Every non-unit entry of Ψ can be expressed as a linear combination of the entries of ZΨ in the same column.

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This is a very important property which has been unknown up to now. It reveals a substantial difference between the 1D and nD realization problems as in the 1D case we never encounter such situation, and it means that some monomials in the matrix Ψ obtained by the method of [1] may be combined into a single polynomial entry which in turn implies that a realization of lower order may be achieved. Then, several conditions will be shown for the matrix Ψ with monomial entries obtained in [1] to be combined into a polynomial one subject to the restrictions of (a), (b) and (c’). These conditions clarify that the combinability of the entries of Ψ depends not only on the structure of the given transfer matrix but also on the coefficient values of the related monomials. Constructive algorithms will be given to carry out such combination and to construct the corresponding realization (A, B, C) with reduced order. Non-trivial symbolic and numerical examples will be presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.

2

Preliminary Formulation and Motivation Examples

The nD Roesser model is described by [14]     x1 (i1 + 1, · · · , in ) x1 (i1 , · · · , in )     ··· ···   = A  + Bu(i1 , . . . , in ) xn (i1 , · · · , in + 1) xn (i1 , · · · , in )   x1 (i1 , · · · , in )   y(i1 , . . . , in ) = C  ···  + Du(i1 , . . . , in ) xn (i1 , · · · , in )

(1)

(2)

where xk ∈ Rrk , k = 1, · · · , n is the kth state vector; u ∈ Rl is the input vector, y ∈ Rm is the output vector, and A, B, C, D are all real matrices of dimensions r × r, r × l, m × r and m × l, respectively, with Pn k=1 rk .

r,

The m × l transfer matrix of the above system is G(z1 , · · · , zn ) = CZ (I − AZ)−1 B + D

(3)

with Z = diag{z1 Ir1 , · · · , zn Irn }. Consider the realization problem for a given m × l nD causal rational transfer matrix G(z1 , · · · , zn ) having a right matrix fraction description (MFD) N (z1 , · · · , zn )D(z1 , · · · , zn )−1 . If G(z1 , · · · , zn ) is a polynomial transfer matrix, one can choose N (z1 , · · · , zn ) = G(z1 , · · · , zn ) and D(z1 , · · · , zn ) = Il . As D = G(0, . . . , 0) by (3), we can assume, without loss of generality, that G(z1 , . . . , zn ) is strictly causal, and in particular N (0, · · · , 0) = 0, D(0, · · · , 0) = I. The realization problem then is to find real matrices A, B, C such that the relation G(z1 , . . . , zn ) = CZ(I − AZ)−1 B ˜ 1 , . . . , zn ) = I − D(z1 , . . . , zn ) and is satisfied. Further, let D(z " # N (z1 , . . . , zn ) F (z1 , . . . , zn ) = ˜ . D(z1 , . . . , zn )

(4)

(5)

Denote the nD z-transformation of the input and output of the given system by Y (z1 , . . . , zn ), U (z1 , . . . , zn ), respectively. Then, express the input-output relation as Y (z1 , . . . , zn )

=

G(z1 , . . . , zn )U (z1 , . . . , zn )

=

N (z1 , . . . , zn )D−1 (z1 , . . . , zn )U (z1 , . . . , zn )

=

N (z1 , . . . , zn )Ξ(z1 , . . . , zn )

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(6)

where Ξ(z1 , . . . , zn ) , D−1 (z1 , . . . , zn )U (z1 , . . . , zn )

(7)

is the introduced pseudo state. Suppose, at this point, that there exists a r ×l matrix Ψ consisting of n-variate monomial entries in the form z1i1 z2i2 · · · znin (ij ∈ {0, 1, 2, . . .}, j = 1, . . . , n) such that ˜ 1 , . . . , zn ) = DHT ZΨ, D(z

(8)

N (z1 , . . . , zn ) = NHT ZΨ,

(9)

Ψ = A0 ZΨ + B,

(10)

Ψ = (I − A0 Z)−1 B

(11)

and

or equivalently

for certain matrices DHT ∈ Rl×r , NHT ∈ Rm×r , A0 ∈ Rr×r , B ∈ Rr×l , and Z = diag{z1 Ir1 , . . . , zn Irn } where r = r1 + · · · + rn . Note that, for a transfer matrix G(z1 , . . . , zn ) which does not contain a zero column, i.e., a column whose entries are all zero, the conditions of (8) – (10) require implicitly that B does not contain a zero column, either. It then follows that ˜ 1 , . . . , zn )Ξ(z1 , . . . , zn ) + U (z1 , . . . , zn ) Ξ(z1 , . . . , zn ) = D(z = DHT ZΨΞ(z1 , . . . , zn ) + U (z1 , . . . , zn )

(12)

and ΨΞ(z1 , . . . , zn ) = (I − A0 Z)−1 BΞ(z1 , . . . , zn ) = (I − A0 Z)−1 B(DHT ZΨΞ(z1 , . . . , zn ) + U (z1 , . . . , zn ))

(13)

which implies that  −1 ΨΞ(z1 , . . . , zn ) = I − (I − A0 Z)−1 BDHT Z (I − A0 Z)−1 BU (z1 , . . . , zn ) = [I − (A0 + BDHT )Z]−1 BU (z1 , . . . , zn ) = (I −AZ)−1 BU (z1 , . . . , zn )

(14)

with A = A0 + BDHT . Now, it is straightforward to see that Y (z1 , . . . , zn ) = N (z1 , . . . , zn )Ξ(z1 , . . . , zn ) = NHT ZΨΞ(z1 , . . . , zn ) = NHT Z(I − AZ)−1 BU (z1 , . . . , zn ) = CZ(I −AZ)−1 BU (z1 , . . . , zn )

(15)

with C = NHT . The above result can be obtained more directly from (8), (9), and (10) or (11) without involving the relations among Y (z1 , . . . , zn ), U (z1 , . . . , zn ) and Ξ(z1 , . . . , zn ) as follows. ˜ 1 , . . . , zn ))−1 = Ψ(I − DHT ZΨ)−1 = (I − ΨDHT Z)−1 Ψ ΨD−1 (z1 , . . . , zn ) = Ψ(I − D(z  −1 = I − (I − A0 Z)−1 BDHT Z (I − A0 Z)−1 B = [I − (A0 + BDHT )Z]−1 B , (I −AZ)−1 B

(16)

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Letting C = NHT , we have that G(z1 , . . . , zn ) = N (z1 , . . . , zn )D−1 (z1 , . . . , zn ) = CZΨD−1 (z1 , . . . , zn ) = CZ(I − AZ)−1 B.

(17)

It has been shown in the above that a realization can be obtained for a strictly causal nD transfer matrix G(z1 , . . . , zn ) = N (z1 , . . . , zn )D(z1 , . . . , zn )−1 , if an nD monomial matrix Ψ can be constructed such that the conditions of (8) – (10) hold. Moreover, the order of the obtained realization is just the same as the dimension of Ψ. Therefore, the problem we have now is how to construct such a matrix Ψ that has dimension as low as possible. When n = 1, i.e., for the 1D case, it is very easy to have such a Ψ [15]. Consider a strictly causal 1D transfer function given by G(z) =

n(z) n1 z + n2 z 2 + · · · + nr z r = d(z) 1 + d1 z + d2 z 2 + · · · + dr z r

(18)

where n(z) and d(z) are assumed to be coprime, and deg G(z) = r. For simplicity, let r = 4 here. It is well known that we can simply set Ψ = [z 3 z 2 z 1]T ,

(19)

and let Z = zI4 . Then, it immediately follows that ˜ = 1 − d(z) = −d1 z − d2 z 2 − d3 z 3 − d4 z 4 = [−d4 − d3 − d2 − d1 ]ZΨ , DHT ZΨ d(z) 2

3

4

n(z) = n1 z + n2 z + n3 z + n4 z = [n4 n3 n2 n1 ]ZΨ , NHT ZΨ

(20) (21)

and  3  z 0  z 2  0     =  z  0 1 0

1 0 0 0

0 1 0 0

 0 z  0  0  1  0 0 0

0 z 0 0

0 0 z 0

  3   0 z 0  2   0  z  0   +   0  z  0 1 z 1

, A0 ZΨ + B.

(22)

The controllable (canonical) realization of G(z) is now given by     0 1 0 0 0  0  0 0 1 0     A = A0 + BDHT =  , B =  ,  0 0 0 0 1  −d4 −d3 −d2 −d1 1

h C = NHT = n4

n3

n2

i n1 .

(23)

The order of the obtained realization (A, B, C) is equal to the Macmillan degree of G(z), which is the minimal one we can achieve for G(z). It is worth noting that for a 1D transfer function of degree r having some powers of z absent, say ˜ G(z) =

n4 z 4 1 + d4 z 4

(i.e., ni = di = 0, i = 1, 2, 3 in (18)), we still have the same Ψ, A0 and B. The only difference is that the ˜ coefficients ni , di , i = 1, 2, 3 appearing in A and C are replaced by their values in G(z) (0 here). This fact tells that, for the 1D case, we only need to consider the general form of G(z) given by (18), and the realization (A, B, C) of (23) can be obtained by just inspection as shown above (without employing any complicated ˜ algorithm), and further the resultant realization can be applied to any particular transfer function G(z) of ˜ the same degree by just substituting the coefficient values of G(z) into the obtained A and C. Or in the

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other words, to find a realization for a transfer function of certain degree r, all we need is to know the resultant realization of (23), and we never need to take the trouble to carry out the realization process for each particularly specified transfer function. However, the situation for the general nD (n ≥ 2) cases will be far complicated and totally different. Let us first have a look at several simple examples to see some substantial differences to the conventional 1D case. Example 1 Consider the strictly causal 2D transfer function given by G(z1 , z2 ) =

n(z1 , z2 ) n10 z1 + n01 z2 + n20 z12 + n11 z1 z2 + n21 z12 z2 , = d(z1 , z2 ) 1 + d10 z1 + d01 z2 + d20 z12 + d11 z1 z2 + d21 z12 z2

(24)

and the following special cases of G(z1 , z2 ): G1 (z1 , z2 ) =

n10 z1 + n01 z2 + n11 z1 z2 + n21 z12 z2 , 1 + d10 z1 + d01 z2 + d11 z1 z2 + d21 z12 z2

G2 (z1 , z2 ) =

n10 z1 + n01 z2 + n11 z1 z2 + cn10 z12 z2 , 1 + d10 z1 + d01 z2 + d11 z1 z2 + cd10 z12 z2

G3 (z1 , z2 ) =

n10 z1 + n11 z1 z2 + n21 z12 z2 , 1 + d10 z1 + d11 z1 z2 + d21 z12 z2

G4 (z1 , z2 ) =

n10 z1 + n01 z2 + n21 z12 z2 . 1 + d10 z1 + d01 z2 + d21 z12 z2

Denote by M1 and M2 the partial degree of G(z1 , z2 ) in z1 and z2 , respectively. Then, we have that M1 = degz1 G(z1 , z2 ) = 2, M2 = degz2 G(z1 , z2 ) = 1. Define M = deg G(z1 , z2 ) = M1 + M2 = 3 and call this ”total degree” or, for brevity, ”degree” of G(z1 , z2 ). The order of the minimal realization of G(z1 , z2 ) cannot be less ˜ 1 , z2 )]T where d(z ˜ 1 , z2 ) = 1 − d(z1 , z2 ). To get a realization of than M . Further, let F (z1 , z2 ) = [n(z1 , z2 ) d(z G(z1 , z2 ), we can construct Ψ, which is a column vector for SISO case, by the following method: collect all monomials z1h z2k , (h, k) ≤ (M1 , M2 ), in F (z1 , z2 ) which leads here to {z1 , z2 , z12 , z1 z2 , z12 z2 }; use the monomials ˜ 1 and Ψ ˜ 2 , which results here in in, respectively, z1 and z2 , to construct the column vectors Ψ ˜ 1 = [z12 z1 ]T , Ψ

˜ 2 = [z2 ]; Ψ

˜ 1 or Ψ ˜ 2 and define Ψ = [ΨT1 ΨT2 ]T with Ψi = z −1 Ψ ˜i then put the monomials that contain both z1 and z2 into Ψ i 2 ˜ (i = 1, 2). Therefore, choosing to put z1 z2 and z1 z2 into Ψ1 , we have that ˜ 1 = [z12 z2 z1 z2 z12 z1 ]T , Ψ ˜ 2 = [z2 ], Ψ h iT . Ψ = [ΨT1 ΨT2 ] = z1 z2 z2 z1 1 1 It is now easy to see that n(z1 , z2 ) = n10 z1 + n01 z2 + n20 z12 + n11 z1 z2 + n21 z12 z2  z1 0  0 z 1 i h  0 = n21 n11 n20 n10 n01  0   0 0 0 0

0 0 z1 0 0

0 0 0 z1 0

0 0 0 0 z2

      

z1 z2 z2 z1 1 1

    ,  

, NHT ZΨ,

(25)

˜ 1 , z2 ) = −d10 z1 − d01 z2 − d(z

d20 z12

− d11 z1 z2 −

d21 z12 z2 

h =

−d21

−d11

−d20

−d10

−d01

, DHT ZΨ,

 i    

z1 0 0 0 0

0 z1 0 0 0

0 0 z1 0 0

0 0 0 z1 0

0 0 0 0 z2

      

z1 z2 z2 z1 1 1

      

(26)

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and     Ψ=  

z1 z2 z2 z1 1 1





      =    

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 1 0 0

0 1 0 0 0

      

z1 0 0 0 0

0 z1 0 0 0

0 0 z1 0 0

0 0 0 z1 0

0 0 0 0 z2

      

z1 z2 z2 z1 1 1





      +    

0 0 0 1 1

      

, A0 ZΨ + B

(27)

which is equivalent to Ψ = (I − AZ)−1 B. Due to (14) and (15), or equivalently, (16) and (17), we have then the  0 0 1 0 0  0 0 0 0 1   0 0 1 0 A = A0 + BDHT =  0   −d21 −d11 −d20 −d10 −d01 −d21 −d11 −d20 −d10 −d01 h i C = NHT = n21 n11 n20 n10 n01 .

following realization for G(z1 , z2 ).    0   0        , B =  0       1  1 (28)

The method employed above can be viewed as a straightforward generalization of the 1D method, which has been in fact considered by Kung et al. in [2]. It should be noted that there is no monomial z1h z2k with (h, k) ≤ (M1 , M2 ) = (2, 1) absent in G(z1 , z2 ) (or F (z1 , z2 )), which is the key point that makes the construction of Ψ simple. Obviously, the order of the resultant realization is M1 + M2 + M1 M2 which may be much higher than the degree of G(z1 , z2 ), i.e., M = M1 + M2 . For our example, the order of the realization is 5 while the degree of G(z1 , z2 ) is 3. If we are not concerned with the order of the realization, we can immediately obtain a realization of order 5 for each of the special cases Gi (z1 , z2 ) i = 1, . . . , 4 by just substituting the values of the corresponding coefficients into (A, B, C) given by (28). However, the things would be rather complicated if we want to have a realization with lowest possible order. G1 (z1 , z2 ) differs from G(z1 , z2 ) by just n20 = d20 = 0. It is ready to see, by the same method used for G(z1 , z2 ), that a column vector Ψ of dimension 4 (rather than 5) can be constructed as h Ψ=

iT z1 z2

z2

1

1

which leads to the following realization for G1 (z1 , z2 ):    0 0 0 1 0  0  0 1  0 0    A1 =   , B1 =   1  −d21 −d11 −d10 −d01  −d21 −d11 −d10 −d01 1 It is interesting to observe that if n21 /n10 = d21 /d10 , c    −d10 c − d11 −d01    0 1  , B2 =  A2 =  0 −d10 −d11 −d01

   , 

i

h C1 =

n21

n11

n10

n01

.

as shown in G2 (z1 , z2 ), then we can verify that  1 h i  0  , C2 = n10 n11 n01 1

gives a (an absolutely) minimal realization of G2 (z1 , z2 ). Note that this fact cannot be seen and the construction of this kind of realization can never be done by the above method, which will be considered later. For G3 (z1 , z2 ), if we directly apply the above method, then we get a vector Ψ = [ΨT1 [z1 z2

z2

1]T and Ψ2 = ∅, or Ψ1 = [1] and Ψ2 = [z12

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ΨT2 ]T with Ψ1 =

z1 ]T , by either of which it is impossible to find

some matrices A0 and B such that (10) holds. However, if we have luckily chosen that Ψ1 = [z1 z2 1]T and Ψ2 = [z1 ], i.e., h Ψ=

iT z1 z2

then it would be straightforward to see that    z1 z2 0 0    Ψ= 1 = 0 0 z1 0 1

1

z1

 1 z1  0  0 0 0

0 z1 0

,

    0 z1 z2 0     0  1  +  1  z2 z1 0

and to have the following minimal realization for G3 (z1 , z2 ):     0 0 1 0     A3 =  −d21 −d10 −d11  , B3 =  1  , 0 1 0 0

i

h C3 =

n21

n11

n10

.

It seemed, from the above cases, always possible to obtain a realization if we properly arrange the monomials having non-zero coefficients in the given transfer function to get a vector Ψ. However, this is indeed not true in general. It can be seen that, for G4 (z1 , z2 ), no matter how we arrange the monomials {z1 , z2 , z12 z2 }, we cannot get a suitable Ψ of dimension 3 such that (10) holds. However, it is observed that if we put an additional monomial 0 · z1 z2 or 0 · z12 into G4 (z1 , z2 ), then we can get realizations of order 4 for each of the cases as follows. For the former case, we only need to substitute n11 = d11 = 0 into the realization obtained for G1 (z1 , z2 ) to get the result:  0 1 0  0 0 0  A4 =   −d21 0 −d10 −d21 0 −d10

0 1 −d01 −d01

   , 

   B4 =  

0 0 1 1

   , 

h C4 =

i n21

0

n10

n01

.

For the latter case, if we choose a Ψ as iT

h Ψ=

z1

z12

1

1

which again cannot be obtained by the method used for G1 (z1 , z2 ) and G2 (z1 , z2 ), we can have the realization:     0 0 0 0 1  1   0 −d h i −d21 −d01  10    ¯4 =  ¯4 = 0 n10 n21 n01 . A¯4 =  , B  , C  0   1 0 0  0 0 −d10 −d21 −d01 1 To show some further difficulties of nD realization problem, let us see another simple example. Example 2 Consider the following 2D transfer functions of degree 2: G5 (z1 , z2 ) =

z1 + z2 + z1 z2 , 1 + z1 + z2 + z1 z2

G6 (z1 , z2 ) =

z1 + z2 + z1 z2 , 1 + z1 + z2 − z1 z2

which differs from each other by only a sign. It is easy to verify that G5 (z1 , z2 ) has a minimal realization given by " A5 =

# −1 −1

0 −1

" ,

# 1 1

B5 =

h ,

C5 =

i 1

1

.

On the other hand, however, we can show by using the same method of [2] that there exists no (real-gain) realization of order 2 for G6 (z1 , z2 ), and a realization of order 3 of G6 (z1 , z2 ) can be obtained by the method used in Example 1 as  0  A6 =  1 1

0 −1 −1

 1  −1  , −1



 0   B6 =  1  , 1

1878

h C6 =

i 1

1

1

.

It should be clear from the above examples that the realization order for nD (n ≥ 2) cases depends not only on the degree of the given transfer function G(z1 , . . . , zn ), but also on the coefficient values of monomials in G(z1 , . . . , zn ). More precisely, two kinds of facts have been confirmed: (i) Transfer functions that have the same degree but different structures, i.e., some monomials below the degree are absent, may admit realizations of different orders. In addition, a transfer function that has more monomials absent may admit a realization of lower order. (ii) Transfer functions that have the same degree and structure, but different coefficient values, may also have realizations of different orders. These properties are substantially different from the 1D case, and strongly suggest that, in order to get a realization of lowest possible order, we have to consider the realization for each particularly specified nD (n ≥ 2) system, rather than directly using the fixed realization obtained for a general form of the system as we have done in the 1D case.

3

Construction of Ψ based on System Structure

It has been seen that the key point is how to find a suitable Ψ, as the order of the realization is determined by the dimension of Ψ and (A, B, C) directly follows from Ψ. Though it is easy to construct a Ψ satisfying (8) – (10) for a transfer function that has no monomial absent as shown in Example 1, it is a highly non-trivial problem to systematically construct a Ψ with lowest possible dimension when there are some monomials absent or possessing different coefficient values, which has never been encountered in the 1D case. To solve this problem is in fact the main motivation of our research shown in this paper. There arise two questions at this moment: what are the conditions that ensure a specific Ψ to satisfy (8) – (10); and how can we systematically construct a Ψ of lowest possible dimension such that (8) – (10) holds. By investigating the structural properties of Roesser model, we found that, for an nD monomial matrix Ψ to meet the restrictions of (8) – (10), it is necessary to satisfy the following conditions [1]: (a) The entries of the jth column of ZΨ contain all the monomials occurring in the polynomial entries of the jth column of F (z1 , · · · , zn ). (b) For each column of Ψ, there is at least a unit entry (i.e., an entry equal to 1). (c) For every non-unit entry Ψ(i, j) (i ∈ {1, . . . , r}) in the jth column of Ψ (j = 1, . . . , l), there exists another entry in the same column, say Ψ(hk , j) (hk ∈ {1, . . . , r}, k ∈ {1, . . . , n}), such that Ψ(i, j) = zk Ψ(hk , j). Furthermore, among all the matrices satisfying the above conditions, the desired Ψ should be a minimal one, that is, no entry of Ψ can be removed without violating these conditions. In the sequel, we order the nD monomials z1h1 z2h2 · · · znhn by the total degree lexicographic order. It is assumed, without loss of generality, that the order of zj is higher than zi with j > i. Two algorithms can be given to construct Ψ and (A, B, C), respectively [1]. In Algorithm 1, the construction of Ψ starts from the monomials occurring in the polynomial entries of each column of F (z1 , · · · , zn ), then appropriate monomials will be inserted into Ψ until conditions (a) - (c) are all satisfied. Algorithm 1. S1.1 d = 0. S1.2 d = d + 1. If d > l, go to Step S1.8. Otherwise, proceed to Step S1.3.

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S1.3 Collect all the monomials z1h1 · · · znhn with non-zero coefficients occurring in the dth column of F (z1 , · · · , zn ), ˜ dk ˜ d1 , · · · , Ψ ˜ dn and Ψ ˜ d0 by putting the collected monomials z hk into Ψ and construct column vectors Ψ k

according to the descending total degree lexicographic order and z1h1 · · · znhn , which has at least two ˜ d0 , according to the ascending total degree lexicographic non-zero indexes among {h1 , · · · , hn }, into Ψ ˜ dk , k = 1, · · · , n and rdk , rd0 be the dimensions of Ψdk and order, respectively. Let Ψdk = zk−1 Ψ ˜ d0 , respectively, i.e., Ψdk ∈ Rrdk , Ψ ˜ d0 ∈ Rrd0 , k = 1, · · · , n. Denote the jth entry of Ψdk by Ψ h

Ψdk (j), j = 1, · · · , rdk . Note that Ψdk (1) = zk dk have the highest order among the entries of Ψdk , respectively. In the case that there is no collected power product to be put into, e.g., Ψdk , we denote it as an empty vector by Ψdk =[∅] and set the dimension of Ψdk to zero, i.e., rdk = 0. S1.4 For k = 1, · · · , n, fill all absent power products zkh , 0 ≤ h < hdk into Ψdk following the descending total degree lexicographic order. Thus the dimension of each Ψdk is rdk = hdk + 1. In the case when Ψdk is empty, however, do not carry out this filling operation. ˜ d0 6= [∅], proceed to Step S1.6. Otherwise, go to Step S1.2. S1.5 j = 0. If Ψ S1.6 j = j + 1. If j > rd0 , go to Step S1.2. ˜ d0 , say Ψ ˜ d0 (j) = z h1 · · · znhn , verify whether there exist k1 , k and jk Otherwise, for the jth entry of Ψ 1 with 1 ≤ k1 , k ≤ n and 1 ≤ jk ≤ rdk such that ˜ d0 (j) = zk Ψdk (jk ). zk−1 Ψ 1 hk −1

˜ d0 (j) = z h1 · · · z 1 • If yes, then insert zk−1 Ψ 1 k1 1

(29)

· · · znhn into Ψdk1 according to the descending total

degree lexicographic order, and set rdk1 = rdk1 + 1. For the case that condition (29) is satisfied for more than one k1 , see for example, k1 , k2 , · · · , ks with 1 ≤ s ≤ n, denote kt as the minimal one among the index set {κ} with {hκ } = max{hk1 , · · · , hks }, ˜ d0 (j) into Ψdkt at an appropriate position and set rdkt = rdkt + 1. Repeat Step S1.6. insert z −1 Ψ kt

• If no, go to Step S1.7. ˜ d0 (j) into Ψdkt according to the descending total degree lexicographic order and set rdkt = S1.7 Insert zk−1 Ψ t rdkt +1, where kt is the minimal one among the index set {κ} with {hκ } = max{h1 , · · · , hn }. Meanwhile, ˜ d0 (j) into Ψ ˜ d0 as the (j +1)th entry Ψ ˜ d0 (j +1) and set rd0 = rd0 +1, without considering also insert z −1 Ψ kt

its total degree lexicographic order here. Return to Step S1.6. S1.8 Denote



 0 ..   .   Ψl1   ..  . .    0  ..   .  Ψln

(30)

Note that there is only one non-zero entry in the each row of Ψ.



Ψ11  .  .  .   0   . Ψ =  ..  Ψ  1n  .  .  . 0

··· .. . ··· .. . ··· .. . ···

Once Ψ has been constructed by Algorithm 1, D(z1 , · · · , zn ) and N (z1 , · · · , zn ) can be easily expressed in

1880

the forms of D(z1 , · · · , zn ) =DHT ZΨ

(31)

N (z1 , · · · , zn ) =NHT ZΨ

(32)

Z = diag{z1 Ir1 , · · · , zn Irn },

(33)

where

DHT ∈ Rl×r and NHT ∈ Rm×r are matrices whose entries are the corresponding coefficients of the entries of Pl P D(z1 , · · · , zn ) and N (z1 , · · · , zn ), respectively, where r = n d=1 rdk . k=1 rk with rk , The system matrices A, B, C can be constructed as follow. Algorithm 2. S2.1 Construct a matrix A0 ∈ Rr×r by the following method. Set initially all the entries of A0 to zero. For i = 1, · · · , r, let A0 (i, j) = 1, if the only non-zero entry in the ith row of Ψ, say Ψ(i, d), d ∈ {1, . . . , l}, equals to the (j, d)th entry of ZΨ, where Z is defined in (33). S2.2 Construct the matrix B ∈ Rr×l by the following method. Set initially all the entries of B to zero. For each i = 1, · · · , r, j = 1, · · · , l, reset B(i, j) = 1 if Ψ(i, j) = 1. S2.3 Let A = A0 + BDHT . It is then easy to see that (10) or (11) holds, and to get the result of (16). S2.4 Let C = NHT .



The constructed (A, B, C) directly gives a realization of G(z1 , . . . , zn ) as shown in (17).

4

Construction of Ψ based on System Coefficients

It will be revealed here for the first time that Ψ, defined as an nD monomial matrix in the previous section, can be generalized to an nD polynomial one, which in turn leads to significant reduction of the realization order. First of all, we claim the following result. In the sequel, let Ψ be an nD polynomial matrix having the form shown in (30). We call Ψ admissible if there exist matrices DHT , NHT , A0 and B of suitable size such that (8) – (10) holds. Theorem 4.1 An nD polynomial matrix Ψ given in the form of (30) is admissible iff it satisfies the following conditions: (a) The entries of the jth column of ZΨ contain all the monomials occurring in the polynomial entries of the jth column of F (z1 , · · · , zn ). (b) For each column of Ψ, there is at least a unit entry (i.e., an entry equal to 1). (c’) Every non-unit entry Ψ(i, j) can be expressed as a linear combination of the entries of the jth column of ZΨ (j = 1, . . . , l), i.e., there exists some real row vector Aij such that Ψ(i, j) = Aij ZΨj , h where Aij ∈ R1×r , Ψj , Ψ(1, j) Proof.

···

(34)

iT Ψ(r, j) , i = 1, . . . , r. In particular, Aij = 01×r if Ψ(i, j) = 0.

It is trivial to see that (a) is necessary and sufficient for (8) and (9) to hold true. If (b) is not

satisfied, then B will contain a zero column which means that the input-output relation of (15) cannot be

1881

established. The condition (c’) obviously includes (c) as a special case. Therefore, if (c), so (c’), is not satisfied, one can never find, due to the structural properties of (10), a matrix A0 such that (10) holds. Now, what we need to do is show that if (b) and (c’) are satisfied, we can always have some A0 and B to ¯ , Ψ − B contains only non-unit entries of Ψ. Define further satisfy (10). Let B = Ψz1 =···=zn =0 . Then, Ψ Aij = 01×r , if Ψ(i, j) = 1.

(35)

In view of that Ψ has the form of (30), Aij can always be restricted to have the forms: i

h Ai1 =

0

0

···

0

···

Ai1n

0

Ai22

0

···

0

···

0

0

···

···

0

0

···

0

Ai12

0

Ai12

0

···

0

0

0

···

Ail1

Ai11

0

0

···

0

0

Ai1n

0

···

0

0

0

···

i

h Ai2 = .. . h Ail =

i 0

0

Ail2

0

Ailn

where Aijk (i = 1, . . . , r; j = 1, . . . , l; k = 1, . . . , n) are real row vector of suitable dimension. Then, it follows that h

i h i ¯ 1) Ψ(i, ¯ 2) · · · Ψ(i, ¯ l) = Ai1 ZΨ1 Ai2 ZΨ2 · · · Ail ZΨl Ψ(i, h i = Ai11 z1 Ψ11 + Ai12 z2 Ψ12 + · · · + Ai1n zn Ψ1n · · · Ail1 z1 Ψl1 + Ail2 z2 Ψl2 + · · · + Ailn zn Ψln  Ψ11 · · ·  . ..  .. .     0 · · · z I i  1 r1 h  .. .. ..  = Ai11 Ai21 · · · Ail1 · · · Ai1n Ai2n · · · Ailn  . .    .  zn Irn  Ψ1n · · ·   .. ..  . . 0 ···

0 .. . Ψl1 .. . 0 .. . Ψln

              

l X =( Aij )ZΨ , A0i ZΨ. j=1

It is now ready to see that 

 A01   ¯ = Ψ − B =  ..  ZΨ , A0 ZΨ Ψ  .  A0r

(36)

which gives the result of (10). Example 3 Consider a 3D transfer function given by G(z1 , z2 , z3 ) =

N (z1 , z2 , z3 ) = n1 z 1 + n2 z 2 + n3 z 1 z 3 + n4 z 2 z 3 D(z1 , z2 , z3 )

with N (z1 , z2 , z3 ) = G(z1 , z2 , z3 ) and D(z1 , z2 , z3 ) = 1. The set of monomials occurring in N (z1 , z2 , z3 ) and D(z1 , z2 , z3 ) is {z1 , z2 , z1 z3 , z2 z3 }. By Algorithm 1, it is easy to have that Ψ = [ΨT11 ΨT12 ΨT13 ]T with " # z2 Ψ11 = [1], Ψ12 = [1], Ψ13 = . z1 Thus, a realization of order 4 can be obtained by Algorithm 2.

1882

Now, note that N (z1 , z2 , z3 ) can be expressed as N (z1 , z2 , z3 ) = n1 z1 + n2 z2 + n3 z3 (z1 +  z1 0 0  = [n1 n2 n3 ]  0 z2 0 0 0 z3

n4 z2 ) n3 

 1   1   n4 z1 + n z 2 3

˜ , NHT Z Ψ ˜ T13 ]T , and ˜ T12 Ψ ˜ = [Ψ ˜ T11 Ψ with Ψ ˜ 11 = [1], Ψ

˜ 12 = [1], Ψ

˜ 13 = [z1 + n4 z2 ]. Ψ n3

˜ 1 , z2 , z3 ) = 1 − D(z1 , z2 , z3 ) = 0, we have that On the other hand, as D(z ˜ 1 , z2 , z3 ) = DHT Z Ψ ˜ D(z with DHT = [0 0 0]. Further, it is ready to see that ˜ ˜ 13 (1) = z1 + n4 z2 Ψ(3, 1) = Ψ n3  z1 0 n4  = [1 0]  0 z2 n3 0 0

  1 0   1 0   n4 z1 + n3 z2 z3

˜ , A31 Z Ψ. ˜ ˜ Let A11 = A21 = [0 0 0] as Ψ(1, 1) = Ψ(2, 1) = 1,    0 0 A11    A0 = A21  = 0 0 n4 1 n A31 3

and  0  0 , 0

˜ z1 =z2 =z3 =0 B = Ψ|

  1   = 1 . 0

It is then easy to see that a realization of order 3 for G(z1 , z2 , z3 ) is given by     0 0 0 1     A = A0 + BDHT = 0 0 0 , B = 1 , C = NHT = [n1 n2 n3 ]. n4 1 n 0 0 3 It has been clarified above that if we can find a suitable Ψ satisfying the conditions of (a), (b) and (c”), then we can construct a realization which may have lower order than that obtained based on the conditions of (a), (b) and (c). The problem now is how to construct such a suitable Ψ. For simple case as shown in Example 3, we may obtain such a Ψ by directly choosing certain suitable factorizations of the given transfer function. However, for a complicated case, it is in general a difficult task to see which factorization will satisfy the required conditions and how to choose a better (or the best) one which leads to a lower (or the lowest) realization order from the possible factorizations. Further, it is usually difficult to implement such a factorization procedure by a program. On the other hand, it is observed that from the monomial matrix Ψ constructed by Algorithm 1, it is easy to identify the monomials of G(z1 , . . . , zn ) that have certain common factors and thus can be considered as the potential candidates to be combined together. For example, consider G(z1 , z2 , z3 ) = z1 + z2 + z1 z3 + z2 z3 + z1 z2 z3 + z22 z3 .

1883

The corresponding Ψ1k , k = 1, 2, 3, obtained by Algorithm 1, are " # " # " # z2 z3 z2 z3 z2 Ψ11 = , Ψ12 = , Ψ13 = . 1 1 z1 It is easy to see that z2 , z1 in Ψ13 mean that the monomials of z2 z3 and z1 z3 in G(z1 , z2 , z3 ) have the common factor z3 , while z2 z3 exists in both Ψ11 and Ψ12 , and is a common factor of z1 z2 z3 and z22 z3 in G(z1 , z2 , z3 ). Therefore, these monomials are probably able to be combined. Note that some cases are not so explicit but still can be identified from Ψ. Consider the example: G(z1 , z2 , z3 ) = z1 + z2 + z3 + z1 z2 + z1 z3 . Say we have had particularly the following results for G(z1 , z2 , z3 ): " # " # h i z3 z1 Ψ11 = , Ψ12 = , Ψ13 = 1 . 1 1 Though here we cannot see any common factor directly from Ψ1i , i = 1, 2, 3, we note that z1 in Ψ12 corresponds to the monomial z1 z2 in G(z1 , z2 , z3 ) and thus can be moved into Ψ11 such that   z3 h i h i   Ψ11 = z2  , Ψ12 = 1 , Ψ13 = 1 . 1 Now, it is easy to see that z1 is a common factor of the monomials of z1 z2 and z1 z3 . That is, an entry contains zi in Ψjk (k 6= i) means that it has a common factor zi with the entries in Ψji . In the sequel, we first consider the possibility of combining the entries in the same Ψjk for certain j ∈ {1, . . . , l} and k ∈ {1, . . . , n}, by clarifying several facts. If two nD monomials φ1 and φ2 satisfying the relation φ2 = αφ1 for a certain nD monomial α, then we say that φ1 generates φ2 , or equivalently, φ2 is generated from φ1 . For an nD monomial entry φjki in Ψjk for certain integer i, let cf(φjki ) denote the coefficient vector of the corresponding monomial zk φjki in the jth column of F (z1 , . . . , zn ) which is defined as in the previous section. For example, if we have that " # a1 z1 + a2 z2 + a3 z1 z3 + a4 z2 z3 + a5 z1 z2 z3 + a6 z22 z3 F (z1 , z2 , z3 ) = b1 z1 + b2 z2 + b3 z1 z3 + b4 z2 z3 + b5 z1 z2 z3 + b6 z22 z3 and φ120 = z2 z3 , φ121 = z1 z3 are entries of Ψ12 , then cf(φ120 ) and cf(φ121 ) denote the coefficient vectors of the monomials z2 φ120 = z22 z3 and z2 φ121 = z1 z2 z3 in (the first column of) F (z1 , z2 , z3 ), respectively, i.e., " # " # a6 a5 cf(φ120 ) = , cf(φ121 ) = . b6 b5 Fact 1. For a certain Ψjk , k ∈ {1, . . . , n} obtained by Algorithm 1, the entries in Ψjk that are not used to generate (or not involved in generation of) any other entry in the jth column of Ψ, i.e., Ψj , can be combined together as a linear combination of these entries themselves, provided the coefficients of the monomials in G(z1 , . . . , zn ) corresponding to these entries satisfy certain proportional conditions. More precisely, let φjk0 , φjk1 , . . . , φjkq be the (non-unit) monomial entries in Ψjk which do not generate any other entries in Ψj . If there exist constants ci 6= 0 (i = 1, . . . , q) such that cf(φjki ) = ci cf(φjk0 ),

(37)

then φjk0 , φjk1 , . . . , φjkq in Ψjk can be replaced by the following linear combination of these entries:   φjk0  h i φjk1   (38) 1 c1 · · · cq   ..  = φjk0 + c1 φjk1 + · · · + cq φjkq  .  φjkq

1884

Proof. As Ψ obtained by Algorithm 1 satisfies the condition of (c), for each non-unit monomial φjki (i = 0, 1, . . . , q) we have that φjki = zsi Ψ(hsi , j), si ∈ {1, . . . , n}, hsi ∈ {1, . . . , r} which can be expressed as  h φjki = 0

···

1

···



 Ψ1,j   ..   .      Ψ(hsi , j)     ...  

z1

  i  0    

..

. zs i ..

. zn

Ψ(r, j)

, A˜i ZΨj . Therefore, for any constants c˜i (i = 0, 1, . . . , q), we have that c˜0 φjk0 + c˜1 φjk1 + · · · + c˜q φjkq = (˜ c0 A˜0 + c˜1 A˜1 + · · · + c˜q A˜q )ZΨj , A¯j ZΨj with A¯j ∈ R1×r . That is, (c’) is always satisfied for any linear combination of the non-unit entries in Ψjk . The problem is whether or not the condition of (a) can be satisfied when we replace φjk0 , φjk1 , . . . , φjkq by a linear combination of them. Let F˜j (z1 , . . . , zn ) be an nD polynomial vector constructed from the jth column of F (z1 , . . . , zn ) by removing the other monomials except zk φjki (i = 1, . . . , q). Obviously, F˜j (z1 , . . . , zn ) can be expressed as F˜j (z1 , . . . , zn ) = cf(φjk0 )zk φjk0 + cf(φjk1 )zk φjk1 + · · · + cf(φjkq )zk φjkq . If the condition of (37) is satisfied, then we have that F˜j (z1 , . . . , zn ) = cf(φjk0 )zk (φjk0 + c1 φjk1 + · · · + cq φjkq ) which means that the linear combination φjk0 + c1 φjk1 + · · · + cq φjkq can be treated as a single polynomial ˜ 1 , . . . , zn ) and N (z1 , . . . , zn ) into the forms of (8) and (9), respectively. See the entry when expressing D(z examples given below for more details. It is not difficult to see the following simple but very useful fact. Fact 2. It is impossible to combine an entry that does not generate any other entry with an entry that generate some other entry (or entries). Now, several examples are given to show some details on the above facts. Example 4 Consider again the 3D polynomial transfer function given in Example 3: G(z1 , z2 , z3 ) = n1 z1 + n2 z2 + n3 z1 z3 + n4 z2 z3 . Then, by Algorithm 1 we have that " Ψ11 = [1]×,

Ψ12 = [1]×,

Ψ13 =

# z2 z1

◦ ◦

where we marked each of the entries that generate some other entries by ×, and each of the entries that do not generate any other entry by ◦.

1885

By Fact 1, it is easy to see that φ130 , z1 and φ131 , z2 in Ψ13 are candidates for possible combination. Since G(z1 , z2 , z3 ) is a polynomial transfer function, it is always possible to choose c1 =

cf(φ131 ) n4 = , cf(φ130 ) n3

and this leads to the following updated results for Ψ: h Ψ11 = [1],

Ψ12 = [1],

Ψ13 =

i z 1 + c1 z 2

.

Using this updated Ψ = [ΨT1 ΨT2 ΨT3 ]T , it is ready to get the realization obtained in Example 3. The difference of the proposed method to the method of Example 3 is that we do not need to do any factorization here, which can therefore be used to complicated cases and implemented by a computer program much more easily. Also note that, by Example 4, we see that for an nD polynomial transfer function, all the entries in the same Ψ1k (k ∈ {1, . . . , n}) that do not generate any other entry can always be combined as the proportional condition of (37) is always satisfied. To be clearer, let us see another slightly complicated example. Example 5 Consider the realizations for G1 (z1 , z2 , z3 ) = n1 z1 + n2 z2 + n3 z3 + n4 z12 + n5 z1 z2 + n6 z1 z3 + n7 z2 z3 + n8 z13 + n9 z12 z2 + n10 z12 z3 + n11 z1 z2 z3 ; G2 (z1 , z2 , z3 ) =

n1 z1 + n2 z2 + n3 z3 + n4 z12 + n5 z1 z2 + n6 z1 z3 + n7 z2 z3 + n8 z13 + n9 z12 z2 + n10 z12 z3 + n11 z1 z2 z3 . 1 + d1 z1 + d2 z2 + d3 z3 + d4 z12 + d5 z1 z2 + d6 z1 z3 + d7 z2 z3 + d8 z13 + d9 z12 z2 + d10 z12 z3 + d11 z1 z2 z3

First, we have that F1 (z1 , z2 , z3 ) = G1 (z1 , z2 , z3 ) and # " n1 z1 + n2 z2 + n3 z3 + n4 z12 + n5 z1 z2 + n6 z1 z3 + n7 z2 z3 + n8 z13 + n9 z12 z2 + n10 z12 z3 + n11 z1 z2 z3 . F2 (z1 , z2 , z3 ) = −d1 z1 − d2 z2 − d3 z3 − d4 z12 − d5 z1 z2 − d6 z1 z3 − d7 z2 z3 − d8 z13 − d9 z12 z2 − d10 z12 z3 − d11 z1 z2 z3 As they have the same set of monomials: {z1 , z2 , z3 , z12 , z1 z2 , z1 z3 , z2 z3 , z13 , z12 z2 , z12 z3 , z1 z2 z3 }, they share the same Ψ = [ΨT11 ΨT12 ΨT13 ]T generated by Algorithm 1:   z2 z3 ◦ z z  ◦  1 3    z1 z2  ◦ " #  2   z1  ◦ z3 ×   Ψ11 =  × , Ψ12 = 1 × , z 3    z ×  2     z1 × 1 ×

h i Ψ13 = 1 × .

Let φ110 = z12 , φ111 = z1 z2 , φ112 = z1 z3 , φ113 = z2 z3 . For F1 (z1 , z2 , z3 ), define c˜10 = cf(φ110 ) = n8 , c˜11 = cf(φ111 ) = n9 , c˜12 = cf(φ112 ) = n10 , c˜13 = cf(φ113 ) = n11 , while for F2 (z1 , z2 , z3 ), define " # " # " # " # n8 n9 n10 n11 c˜20 = cf(φ110 ) = , c˜21 = cf(φ111 ) = , c˜22 = cf(φ112 ) = , c˜23 = cf(φ113 ) = . d8 d9 d10 d11 Then, for G1 (z1 , z2 , z3 ), we can always set c11 =

c˜11 n9 = , c˜10 n8

c12 =

c˜12 n10 = , c˜10 n8

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c13 =

c˜13 n11 = , c˜10 n8

˜ = [Ψ ˜ T1 Ψ ˜ T2 Ψ ˜ T3 ]T to distinguish, for arbitrary n8 , n9 , n10 and n11 such that the updated Ψ, denoted here by Ψ is   φ˜ z   3  ˜1 =  Ψ  z2  ,    z1  1

" # z3 ˜ Ψ2 = , 1

h i ˜3 = 1 . Ψ

where φ˜ = c13 z2 z3 + c12 z1 z3 + c11 z1 z2 + z12 . It follows then that F1 (z1 , z2 , z3 ) = G1 (z1 , z2 , z3 ) = n8 z1 (z12 + c11 z1 z2 + c12 z1 z3 + c13 z2 z3 ) + n6 z1 z3 + n5 z1 z2 + n4 z12 + n1 z1 + n7 z2 z3 + n2 z2 + n3 z3    ˜1 Ψ h i z1 I 5 ˜   = n8 n6 n5 n4 n1 n7 n2 n3  z2 I 2  Ψ 2  ˜3 Ψ z3 ˜HT Z Ψ ˜ ,N and        ˜ = Ψ      

φ˜ z3 z2 z1 1 z3 1 1





            =            

0 0 0 0 0 0 0 0

c12 0 0 0 0 0 0 0

c11 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0

c13 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 1 0 0 0 1 0 0





     z1 I 5        

z2 I 2

        z3     

φ˜ z3 z2 z1 1 z3 1 1





            +            

0 0 0 0 1 0 1 1

             

˜ + B. ˜ , A˜0 Z Ψ ˜ HT = 01×8 here, we have the realization for G1 (z1 , z2 , z3 ) determined by A˜ = A˜0 , As it is easy to see that D ˜ and C ˜=N ˜HT . B On the other hand, for G2 (z1 , z2 , z3 ), we cannot conduct the combination for arbitrary values of ni , di (i = 8, 9, 10, 11). However, if there exist some constants c21 , c22 c23 such that c˜21 = c21 c˜20 , c˜22 = c22 c˜20 , c˜23 = c23 c˜20 , or equivalently, c21 =

n9 d9 n10 d10 n11 d11 = , c22 = = , c23 = = , n8 d8 n8 d8 n8 d8

then we can have an updated Ψ, denoted here by   φ¯ z   3  ¯1 =  Ψ  z2  ,    z1  1

¯ = [Ψ ¯ T1 Ψ ¯ T2 Ψ ¯ T3 ]T , as: Ψ " # z3 ¯ Ψ2 = , 1

where φ¯ = c23 z2 z3 + c22 z1 z3 + c21 z1 z2 + z12 .

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h i ¯3 = 1 . Ψ

It is now ready to see that # # " # " # " " n4 n5 n6 n8 z12 z1 z2 + z1 z3 + z1 (z12 + c21 z1 z2 + c22 z1 z3 + c23 z2 z3 ) + F2 (z1 , z2 , z3 ) = −d4 −d5 −d6 −d8 # " # " # " # " n7 n2 n3 n1 z1 + z2 z3 + z2 + z3 + −d7 −d2 −d3 −d1    " # z1 I 5 ¯1 Ψ n8 n6 n5 n4 n1 n7 n2 n3 ¯   = z2 I 2  Ψ 2   −d8 −d6 −d5 −d4 −d1 −d7 −d2 −d3 ¯3 Ψ z3 " # ¯HT N ¯ , ¯ Z Ψ. DHT ¯ has the same structure as Ψ, ˜ we can obtain A¯0 by just replacing c11 , c12 , c13 in A˜0 by c21 , c22 , c23 , As Ψ ¯ = B. ˜ Now, we have the realization for G2 (z1 , z2 , z3 ) as follows: and B     0 c22 c21 1 0 0 c23 0 0  0  0  0 0 1  0 0 0 0          0  0  0 0 0 0 0 1 0         0 0 0 1 0 0 0  ¯D ¯ HT =  0 ¯ =  0 , , B A¯ = A¯0 + B  −d8 −d6 −d5 −d4 −d1 −d7 −d2 −d3   1       0  0  0 0 0 0 0 0 1           −d8 −d6 −d5 −d4 −d1 −d7 −d2 −d3   1  −d8 −d6 −d5 −d4 −d1 −d7 −d2 −d3 1 h i ¯=N ¯HT = n8 n6 n5 n4 n1 n7 n2 n3 . C The following example shows that we may do such combination simultaneously in different Ψk . Example 6 Let G(z1 , z2 , z3 ) =z1 + z2 + z3 + z12 + z22 + z32 + z1 z2 + z1 z3 + z2 z3 + z13 + z23 + z33 + z12 z2 + z12 z3 + z1 z2 z3 + z1 z22 + z22 z3 + z1 z32 + z2 z32 . Then, we have that Ψ = [ΨT1 ΨT2 ΨT3 ]T with   z2 z3 ◦   z z ◦ z2 z3  1 3     z1 z2  z1 z2  ◦     z2  z12  ◦  2   , Ψ = Ψ1 =   2   z3  z3  ×   z ×  z2   2    z1  × 1 1 ×



◦ ◦  ◦   , ×  × ×

    Ψ3 =   

z2 z3 z1 z3 z32 z3 1



◦ ◦   ◦  × ×

It is not difficult to confirm that we can combine all the entries marked by ◦ in Ψ1 , Ψ2 and Ψ3 , respectively. That is, 

 z12 + z1 z2 + z1 z3 + z2 z3   z3     Ψ1 =  z2 ,     z1 1

   Ψ2 =  

 z22 + z1 z2 + z2 z3  z3  ,  z2 1

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

 z32 + z1 z3 + z2 z3   Ψ3 =  z3  1

where Ψ(1) = Ψ1 (1) = z12 + z1 z2 + z1 z3 + z2 z3 

h =

0

1

1

1

0

0

1

0

0

0

0

         i  0           

z1 (z12 + z1 z2 + z1 z3 + z2 z3 ) z1 z3 z1 z2 z12 z1 z2 (z22 + z1 z2 + z2 z3 ) z2 z3 z22 z2 z3 (z32 + z1 z3 + z2 z3 ) z32 z3

                      

, A1 ZΨ, Ψ(6) = Ψ2 (1) = z22 + z1 z2 + z2 z3 h = 0 0 1 0 0 0 1

1

0

0

0

0

Ψ(10) = Ψ3 (1) = z32 + z1 z3 + z2 z3 h = 0 1 0 0 0 0 1

0

0

0

1

0

i ZΨ, i ZΨ.

Note that, we have not yet shown any thing about the possibility on the combination of the entries all marked by ×, which will be in fact much difficult than the case considered above. We shall show the details of the results on this part and the full direct-construction realization algorithm in a journal version of this paper.

5

Comparative Examples

To compare the effectiveness of the proposed method with the existing ones, two more examples are presented here, which are worked out by using the full-version algorithm. Example 7 Consider a 2D system G(z1 , z2 ) = n(z1 , z2 )/d(z1 , z2 ) given by n(z1 , z2 ) = n00 + n10 z1 + n20 z12 + n30 z13 , d(z1 , z2 ) = 1 + d10 z1 + d01 z2 + d11 z1 z2 + d20 z12 + ad01 z22 + ad11 z1 z22 + d21 z12 z2 . By the method proposed in this paper we can directly obtain a minimal realization (A, B, C, D) for G(z1 , z2 ) without order reduction as follows.  0 1 0  0 0 1   −d20 −d10 A= 0   0 d21 /(ad11 ) 0 0 −d01 d20 /d11 1 − d01 d10 /d11 h C = n30 n20 − n00 d20 n10 − n00 d10

0 0 −ad11 0 −ad01

0 0 −d11 1 −d01

−an00 d11

    ,  

    B=  

−n00 d11

i ,

0 0 1 0 d01 /d11

    ,  

D = n00

In contrast, the realization orders due to the method of [1] are 8 before reduction and 7 after reduction. By the method based on Fact 1, only a realization of order 6 can be obtained.

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Example 8 Consider the well-known example of a29 (m, Xcg , Zcg , VA ) used in [9, 12]. The method proposed in this paper generates a realization of order 15 without order reduction, and a realization of order 13 after order reduction, where the corresponding block structures are (3, 6, 1, 5) and (3, 4, 1, 5), respectively. A comparison with the results obtained by the existing methods is given in Table 1. It can be seen that the order of 13 obtained here is very close to the lowest order of 11 for this example obtained in [12] by some symbolic preprocessing techniques. It should be noted that our results are automatically generated by a symbolic software implementation of the proposed algorithms without applying any preprocessing techniques requiring heuristics, although our approach never precludes such techniques. In fact, adopting, if possible and efficient, such preprocessing techniques [7, 12] on a high level before applying our realization algorithms will be certainly helpful to achieve realizations with lower order [1].

Table 1: Comparison of the realization orders:

Method

Order Reduction

(r1 , r2 , r3 , r4 )

r∆

Object-oriented LFT realization without any preprocessing [9] Common subexpression optimization [9] Horner factorization [9] Method of Sugie [1, 6] Tree decomposition [1, 7] Common subexpression optimization [9] Horner factorization[9] Our previous method [1] Our previous method [1] The proposed method The proposed method

No

(49, 72, 36, 136)

293

No

(19, 6, 1, 83)

109

No No No Yes

(3, 6, 9, 41) (3, 6, 9, 39) (4, 9, 6, 9) (4, 4, 1, 18)

59 53 28 27

Yes No Yes No Yes

(3, 2, 3, 18) (4, 12, 3, 7) (4, 2, 3, 7) (3, 6, 1, 5) (3, 4, 1, 5)

26 26 16 15 13

–

(2, 2, 3, 4)

11

Variable splitting factorization [12]

6

Conclusions

A direct-construction realization procedure has been proposed, which simultaneously treats all the involved variables and/or uncertain parameters and directly generates an overall Roesser model realization or LFR model for a given nD polynomial or causal rational function matrix with symbolic or numerical coefficients.

In particular, it has been shown that the nD realization problem for an nD transfer matrix

G(z1 , . . . , zn ) = N (z1 , . . . , zn )D(z1 , . . . , zn )−1 with D(0, ..., 0) = I and N (0, ..., 0) = 0, can be essentially reduced to the construction of an admissible nD polynomial matrix Ψ such that N (z1 , . . . , zn ) = CZΨ and ΨD(z1 , . . . , zn )−1 = (I − AZ)−1 B for certain real matrices A, B, C and the corresponding variable and/or uncertainty block structure Z = diag{z1 Ir1 , . . . , zn Irn }. This important fact is clarified for the first time in this paper, which reveals a substantial difference between the 1D and nD (n ≥ 2) realization problems

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as in the 1D case Ψ can only be a matrix with monomial entries. Necessary and sufficient conditions that ensure admissibility of Ψ have been given and, based on these conditions, algorithms have been proposed for construction of an admissible Ψ with lower order and the corresponding realization. Illustrative examples have been presented to illustrate the basic ideas as well as the effectiveness of the proposed method. Though we only considered the case for a transfer matrix described by a right MFD, similar results can be easily given to the case using left MFD description. In general, the proposed direct-construction realization approach can be considered as a generalization of the 1D controller-form or observer-form realization [15]. However, as the realization order in nD (n ≥ 2) case depends not only on the degree of the given transfer function, but also on its structure and even coefficient values, the construction of an admissible Ψ brings up a totally new research topic, which makes the generalization far from trivial. The results in Section 3 give a solution with respect to the system structure, while those in Section 4 provide a promise approach for treating the coefficient-dependent order realization problem of the given multidimensional system.

References [1] L. Xu, H. Fan, Z. Lin, Y. Xiao and Y. Anazawa, “A constructive procedure for multidimensional realization and LFR uncertainty modelling”, Proc. ISCAS2005, pp.2044–2047, Kobe, Japan, 2005. [2] S. Y. Kung, R. Levy, M. Morf, T. Kailath, “New results in 2-D systems theory, part II: 2-D state space models – realization and the notions of controllability, observability and minimality,” Proc. IEEE, Vol. 65, pp. 945–961, 1977. [3] R. Eising, “Realization and stabilization of 2-D systems,” IEEE Trans. Automat. Contr., Vol. 23, no. 5, pp. 793–799, 1978. [4] P. Lambrechts, J. Terlouw, and M. Steinbuch, “Parametric uncertainty modelling using LFTs,” Proc. ACC, pp. 267-272, San Francisco, 1993. [5] Y. Cheng, and B. DeMoor, “A Multidimensional realization algorithm for parametric uncertainty modelling and multiparameter margin problems,” Int. J. Control, Vol. 60, pp. 789–807, 1994. [6] T. Sugie and M. Kawanishi, “µ analysis/synthesis based on exact expression of physical parameter variations and its application,” Proc. ECC , pp. 159–164, 1995. [7] J. C. Cockburn and B. G. Morton, “Linear fractional representations of uncertain systems,” Automatica, Vol. 33, pp. 1263–1271, 1997. [8] J. F. Magni, “Linear fractional representation toolbox: modelling, order reduction, gain scheduling,” Technical Report, France, July, 2004. (http://www.cert.fr/dcsd/idco/perso/Magni/download /lfrt manual v130.pdf) [9] A. Varga and G. Looye, “Symbolic and numerical software tools for LFT-based low order uncertainty modelling,” Proc. IEEE CACSD, pp. 176-181, Hawaii, USA, Aug. 1999. [10] J. C. Cockburn, “Multidimensional realizations of systems with parameter uncertainty ,” Proc. MTNS, France, June, 2000. [11] B. Morton, “New applications of µ to real-parameter variation problems,” Proc. CDC, pp. 233–238, Florida, Dec. 1985. [12] S. Hecker and A. Varga, “Symbolic techniques for low order LFT-modelling”, Proc. of 16th IFAC Congress, Prague, Czech Republic, 2005.

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[13] S. Hecker, A. Varga and J. F. Magni, “Enhanced LFR-Toolbox for MATLAB,” Aerospace Science and Technology, Vol. 9, pp. 173–180, 2005. [14] D. Givone and R. Roesser, “Minimization of multidimensional linear iterative circuits,” IEEE Trans. Computer, Vol. 22, pp. 673–678, 1973. [15] T. Kailath, Linear Systems, USA: Prentice-Hall, Inc., 1980. [16] C. Beck, and R. D’Andrea, “Minimality, controllability and observability for uncertain systems,” Proc. ACC, pp. 3130-3135, New Mexico, 1997. [17] E. Zerz, “LFT Representations of parameterized polynomial systems,” IEEE Trans. Circuits and Systems I, Vol. 46, pp. 410-416, 1999.

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