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An Algorithm for the Computation of the Transfer Function Matrix of Two-dimensional Systems
An Algorithm for the Computation of the Transfer Function Matrix of Twodimensional Systems by
B. G. MERTZIOS
Democritus University of Thrace, Engineering, Xanthi, Greece
Department
of Electrical
Engineering,
School of
An algorithm is developed for the computation of the transfer function matrix of a two-dimensional system, which is given in its state-space form, without inverting a polynomial matrix. A new transformation has been considered so that the well known Fadeeva’s algorithm for regular systems can be used as the basisfor the derivation of the present algorithm. The above transformation can be generally used in the reduction of many two-dimensional problems to the corresponding one-dimensional ones. The algorithm presented is well-suitedfor computer use. ABSTRACT :
I. Zntvoduction Two-dimensional (2-D) systems have drawn considerable attention in recent years, since they provide the mathematical framework for the study of 2-D digital filters which find numerous crucial applications in image processing, medical imaging and processing of geophysical and seismic data (l-3). This paper establishes a new algorithm for the direct computation of the transfer function matrix of a linear, shift-invariant, discrete, multi-variable 2-D system, from its state-space description, without inverting a polynomial matrix in two variables. This problem has been considered by Koo and Chen (4) who extended Fadeeva’s algorithm (5) for the inversion of the resolvent matrix (d-A) in the 2-D case. Moreover, a formula has been presented which allows the determination of the transfer function matrix of a multivariable 2-D system in terms of the state transition matrix and the characteristic equation (6). The proposed algorithm in the present paper uses the simple, well known 1-D Fadeeva’s algorithm (also called Leverrier’s algorithm) and does not involve the 2-D nature of the system and the relevant complicated notation. The algorithm reduces computational cost and is well suited for computer use. It is useful in the analysis, synthesis and control of 2-D and multidimensional systems, since it can be extended to more dimensions.
ZZ.Algorithm Consider
0
the linear, shift-invariant,
The Franklin Institute 0016a32/86
$3.00 + 0.00
discrete, multivariable
2-D system, described 73
B. G. Mertzios in state-space
as follows (7) : (la)
Y(i,j) = CC1/ w
[
xh(i, 3 x”(i, j)1
(lb)
or, more compactly, x’ = Ax+Bu
(W
y = cx
(2b)
where xh E R,, is the horizontal state-vector, X”E R,, is the vertical state-vector, x E RN, (N = nl + n,) is the local state-vector, u E R, is the input vector, y E R, is the output vector and A, B, C are constant matrices of appropriate dimensions. If we take the 2-D z-transform of the system (l), the following transfer function matrix results H(z,,z,) = CIS-A]-lB
(3)
where (4) and i denotes the direct sum of matrices. The transfer function matrix H(z,, zJ can be always written H(z,,z,)
in the form
= C[I,(~-[I,(~-SSA]]-~B = CII,(p-;i]-lB
(5)
A =&p-S++
(6)
where
and cp is a pseudovariable, which does not affect H(z,,z,) since it is eliminated by construction. In (5) considering the pseudovariable cpas variable and the matrix ;i in the place of the constant matrix A, Fadeeva’s algorithm can be directly applied to compute the inverse of I,cp - A. To this end H(z,, z2) may be written as
Wz,, zz) =
CW,, z,)B cd%4
(7)
where N-l
R(z,,z,) = Adj [I,(p-A]
= c
cpiR,-l-i(z,,z,)
(8)
i=O N-l
&I,ZZ)
= det [lN(P-Al
= ‘pN+ 1
qi4N-i(z1,Z2)*
(9)
i=O
74
Journal of the Franklin Institute Pergamon Press Ltd.
Transfer Function Matrix of 2-D Systems Then Fadeeva’s R&z,)
algorithm
gives
= I,,
4r(z1, G) =
R&I>4 = ~R,hz,) +&I, 4&w R,(z,,z,) . . . . . . . . . . . . .=. . .-&h . . . . . . . . . .4 . . . .+ . . .eh . . . . . . .zz)I,, . . . . .. .
-&I
&1, 4 - 4 trC&h 41 . .dzl, . . . . . . . .4. . . .= . . . -. . . 3 . . trC-%zI, . . . . . . . . . . . . . .4. . . .
R,-dz,,zJ = WR,-,(z,,z,)+q,-,(z,,z,)I,, CL&~,4 = -
! (10)
I
ktrCAR,-1(~1,~2)1 1
0 = AR,-,+a&,z,P,
where the last equation serves as a check. The matrices Ri(zl, z,), i = 0, 1, , . . , IV- 1 and the scalars qi(zl, zJ, i = 1,2,. . . , N, are coefficients of the corresponding powers of cp according to (8) and (9). However, our objective is to determine the coefficients of the Adjoint matrix R(zl, z2) and the characteristic polynomial q(z,, z2) w.r.t. the powers of zl, z2. We note that Ri(zl, z2), z1,z2. qi(z1,z2) are not constants any more, but depend on the variables Furthermore, since Eq. (5) is independent of cp, in the following analysis we choose cp = 0 for simplicity. Then the relations (6x9) become ;z = -S-i-A
WZIJ,) =
CR- ,hzJB
= det
(12)
4N(Zlr z2)
R(z,, z2) = Adj [S-A]
&,,z,)
(11)
= R,_ l(zl, ZJ
(13)
[S-AI = a&1,4.
(14)
It is pointed out that each specific value of the pseudovariable cp results in a different form of b; however, the same transfer function matrix H(z,, ZJ is obtained. From (10) it is seen that for the determination of R,_ l(zl, z2) and qN(zl, z,), which are involved in (12), the matrices Ri, i = 0, 1,. . . , N - 2 and the scalars qi, i = 1,. . . , N - 1 are needed. The degree of the polynomial matrix Ri(zl,z2) and of the polynomial qi(z,,zJ w.r.t. both variables zl, z2 are at most equal to i. Therefore, they can be written in the form R,(z,,z,)
=
1
R;,lz;z;,
i = 0,1,. . . , N-
1
(15)
k+l=O
&(Z,,Z,)
=
c
&z:z:,
i =
1,2,. . . ,N
(16)
k+l=O
where
and i denotes an index and not an exponent. Note that the matrices Rj+-k, i = 0, 1,. . . , NVol. 321, No. 2, pp. 73-80, February 1986 Printed in Great Britain
1 and k = 0, 1,. . . , i, which are the 75
B. G. Mertzios coefficients of the greatest powers of the monomial z!z\ w.r.t. both zl,zZ are diagonal. This can be seen from (lo), since A = -S + A where S is diagonal. The general form of the matrices Rt,i_k has been found by induction to be as follows : R;,i_k
=
(17)
where (18)
(19) From (17)-( 19) the following
interesting
properties
are directly
obtained
(r~l,i-nl)l
=
0
(20)
(rf-nz,n2)2
=
0
(21)
and Ri,i-k=O
Now, combining
if
n, Q kg
i
or
n, < i-k<
(13) (14) and (15), (16) respectively,
i.
(22)
we obtain
N-l
R(z,,z,)
=
1
R;;lztz;
=
f
5
R%;lzfz;
(23)
k=O l=O (k,0 # (n~,nz)
k+l=O
(24) k+I=O
Thus the matrices determination of H(z,, matrix R(z,, z2) and substituting (15), (16) powers of z:zi on both
k=O
l=O
R&l and the scalars c&, which are needed for the zJ by (12), are coefficients of the powers z:zi of the Adjoint
the characteristic polynomial q(z,, z2) respectively. Now in (10) and equating the coefficients of the corresponding sides of each equation, the following recursive relations result
R;,, = AR;, ’ - [(R;:1,,,)1*o + (R:,’
Jo,‘] +&I,
(25)
& = - f tr[AR& ’ - [(R:I1,,,)l,o + (R&A l)“,l]]
(26)
where
___y___;___e___ I (RbL1 (R;r,1)3
= (R;,l)l~o+(R:,,)‘,l 76
(27) Journal of the Franklin Institute Pergamon Press Ltd.
Transfer Function Matrix of 2-D Systems and (Ri&, (R$,, (R&, (Ri& are the submatrices of dimensions n1 x nl, n1 x n,, n2 x n1 and n2 x n2 respectively. Relations (25), (26) permit the easy calculation of Rc; I, & which appear in (23), (24), in terms of R:,,, i = 0, 1, . . . , IV- 2 and L&, i = 0, 1, . . . , N-l and k+l< i. Note that relations (25), (26) are reduced to Fadeeva’s algorithm for regular systems if we consider S = ~1. In the sequel, a lemma will be established which gives the set of initial and “boundary” conditions of the recursive equations (25), (26). Lemma. The initial conditions for the propagation of the recursive equations (25), (26) are (28a)
R8,o = 1, Rf,,, and the “boundary”
= RL,_, = 0
conditions (R:,i&l
k, E 2 1
for
i = 0, 1,. . . , N - 1 (28b)
and
are
= 0
for
;I;‘;2 3
(RL,,,)l*’ = 0
for
+ ‘I;; N - ’
(294
2
,..-,
i = n,,n,+l,...,N-1 l=
O,l,...,i-n,
’
ProoJ: Relations (28a, b) are obvious. For the proof of (29a, b), we consider following recursive equations of the form (25) i+l R kg,*+
I =
AR:,,,+
I -
CW:-
l,nz+
I)‘,’
(29b) the
+(RhJ”*ll + d;:+ ,Iiv for
i = n,,n,+l,...,N-1 i k=Ol 3 I...> i-n,
(30)
Note that = R:,,, 1 Ri- l,nz+ 1 0, L$$:+ = in z2 order greater than n2 cannot appear. Therefore (R:,,)OP1 = 0. Relation (29b) can be similarly proved. The main result can be concisely included in the following theorem. transfer function matrix of the 2-D system given in its state-space form (1) is given by 1 WZ,,
Z2)
=
C[R,, _ l,nz~;l- ‘~2;
~ 4h
4
+R,,,,,_lz”,‘z~-l+~~~+Ri,j~~zj,+~~~+Ro,o]B
(31)
where q(Z,,Z2)
=
Z~‘Z~++4nl-l,n2Z;1-1Z~+qnl,nz_1Z;1Z~-1+’.’+qi,jZ~Zjz+“‘+q0,0.
(32) The Ri,j, qi,j in (31), (32) are the matrices RzJr ’ and the scalars q~j, which are given by (25), (26) in conjunction with the conditions of the lemma. Note that the indices N - 1 and N are omitted from (3 l), (32) respectively due to simplicity. Vol. 321, No. 2, pp. 73-80, Fekmmy Primed in Great Britain
1986
77
B. G. Mertzios III. Illustrative example Consider
a 2-D system of the form (l), with n, = n2 = 2 and (6)
and c = [C,
I C,] =
1 O-l/l
1
I 0
-1
1
1’
For this system, we obtain R:,o = I,,
78
Joumalof
the Franklin Institute Pergamon Press Ltd.
Transfer Function Matrix of 2-D Systems
R;,,
=
-;_I;-. E 1 2
Similarly,
we find
C&,0= C&l = 4A.1 = 2, q& = -7,
q:,,, = 6,
=
CT;,1
q&J = 5, qf,2
=
=
&o = q&z = 1, q;,,, =4,
q:,l =6,
d,z = -2,
2,
&
-2,
41.1 = 4,
q;,l = -12,
q:,o = -1,
q;,o = -14, 4:,2
&,I = 0,
= -12, &,I
q:,1 = 2,
4,
=
44.2
=
4!,1 = -1%
q;,o = 4,
4,2
1 =
03
1.
Substituting R& = R,,, and q& = qk,l in (31), (32), we finally obtain function matrix H(z,,z,)
=
~ q(z,,
hl(zl?zz) z2)
hz(zl~z2)
C h,,(z,,z,)
hzz(z1,
z2)
the transfer
1
where h,,(z,,
z2) = - z:z2 + 6z,z, - 222 + 22; + 132, + 72,
h&z,,
Z2) =
zlZf
h,,(z,,
z2)
32~2, -3z,_z,
h,,(z,,z,)
=
+
5Z,Z2
= z:z2-zlzz
+
Z;
+
62, +
32,
+ 62: - 32; - 132, - 142, - 8
+2z:-z;-32,
-5z,-4.
IV. Conclusions In this paper a new algorithm is derived for the computation of the transfer function matrix of a linear, 2-D system from its state-space description. This algorithm is suitable for computer use and reduces the computation cost, since the inversion of a 2-D polynomial matrix is avoided. Furthermore, it can be more easily programmed on the computer than the already available 2-D Fadeeva’s algorithm (4) since the relevant, complicated notation is avoided. The main advantage of the proposed algorithm is its generality. Specifically, a simple transformation has been proposed, which allows the use of Fadeeva’s algorithm. This transformation can be easily applied to other classes of complicated systems (i.e. multidimensional, singular, delay-differential) where various analysis and control problems can be reduced to the simpler regular case by using this transformation (8). Vol. 321, No. 2, pp. 73-80, February Printed in Great Britain
1986
79
B. G. Mertzios
References (1) N. K. Bose, “Multidimensional Systems : Theory and Applications”, IEEE Press, New York, 1979. (2) N. K. Bose, “Applied Multidimensional Systems Theory”, Van Nostrand Reinhold, New York, 1982. (3) S. K. Mitra and M. P. Ekstrom, “Two-dimensional Digital Signal Processing”, Dowden, Hutchinson and Ross, Stroudsburg, 1978. (4) C. S. Koo and C. T. Chen, “Fadeeva’s algorithm for spatial dynamical equations”, Proc. IEEE, Vol. 65, pp. 975-976, June 1977. (5) L. A. Zadeh and C. A. Desoer, “Linear System Theory”, McGraw-Hill, New York, 1963. (6) B. G. Mertzios and P. N. Paraskevopoulos, “Transfer function matrix of 2-D systems”, ZEEE Trans. Aut. Control, Vol. AC-26, pp. 722-724, June 1981. (7) R. P. Roesser, “A discrete state-space model for linear image processing”, IEEE Trans. Aut. Control, Vol. AC-20, pp. l-10, Feb. 1975. (8) B. G. Mertzios, “Decoupling of 2-D systems by dynamic state feedback controllers”, Proc. 9th Int. Fed. Aut. Control World Congr., Vol. 9, pp. 144-149, Budapest, Hungary, July 1984.
80
Journal
of the Franklin Institute Pergamon Press Ltd.
B. G. MERTZIOS
Democritus University of Thrace, Engineering, Xanthi, Greece
Department
of Electrical
Engineering,
School of
An algorithm is developed for the computation of the transfer function matrix of a two-dimensional system, which is given in its state-space form, without inverting a polynomial matrix. A new transformation has been considered so that the well known Fadeeva’s algorithm for regular systems can be used as the basisfor the derivation of the present algorithm. The above transformation can be generally used in the reduction of many two-dimensional problems to the corresponding one-dimensional ones. The algorithm presented is well-suitedfor computer use. ABSTRACT :
I. Zntvoduction Two-dimensional (2-D) systems have drawn considerable attention in recent years, since they provide the mathematical framework for the study of 2-D digital filters which find numerous crucial applications in image processing, medical imaging and processing of geophysical and seismic data (l-3). This paper establishes a new algorithm for the direct computation of the transfer function matrix of a linear, shift-invariant, discrete, multi-variable 2-D system, from its state-space description, without inverting a polynomial matrix in two variables. This problem has been considered by Koo and Chen (4) who extended Fadeeva’s algorithm (5) for the inversion of the resolvent matrix (d-A) in the 2-D case. Moreover, a formula has been presented which allows the determination of the transfer function matrix of a multivariable 2-D system in terms of the state transition matrix and the characteristic equation (6). The proposed algorithm in the present paper uses the simple, well known 1-D Fadeeva’s algorithm (also called Leverrier’s algorithm) and does not involve the 2-D nature of the system and the relevant complicated notation. The algorithm reduces computational cost and is well suited for computer use. It is useful in the analysis, synthesis and control of 2-D and multidimensional systems, since it can be extended to more dimensions.
ZZ.Algorithm Consider
0
the linear, shift-invariant,
The Franklin Institute 0016a32/86
$3.00 + 0.00
discrete, multivariable
2-D system, described 73
B. G. Mertzios in state-space
as follows (7) : (la)
Y(i,j) = CC1/ w
[
xh(i, 3 x”(i, j)1
(lb)
or, more compactly, x’ = Ax+Bu
(W
y = cx
(2b)
where xh E R,, is the horizontal state-vector, X”E R,, is the vertical state-vector, x E RN, (N = nl + n,) is the local state-vector, u E R, is the input vector, y E R, is the output vector and A, B, C are constant matrices of appropriate dimensions. If we take the 2-D z-transform of the system (l), the following transfer function matrix results H(z,,z,) = CIS-A]-lB
(3)
where (4) and i denotes the direct sum of matrices. The transfer function matrix H(z,, zJ can be always written H(z,,z,)
in the form
= C[I,(~-[I,(~-SSA]]-~B = CII,(p-;i]-lB
(5)
A =&p-S++
(6)
where
and cp is a pseudovariable, which does not affect H(z,,z,) since it is eliminated by construction. In (5) considering the pseudovariable cpas variable and the matrix ;i in the place of the constant matrix A, Fadeeva’s algorithm can be directly applied to compute the inverse of I,cp - A. To this end H(z,, z2) may be written as
Wz,, zz) =
CW,, z,)B cd%4
(7)
where N-l
R(z,,z,) = Adj [I,(p-A]
= c
cpiR,-l-i(z,,z,)
(8)
i=O N-l
&I,ZZ)
= det [lN(P-Al
= ‘pN+ 1
qi4N-i(z1,Z2)*
(9)
i=O
74
Journal of the Franklin Institute Pergamon Press Ltd.
Transfer Function Matrix of 2-D Systems Then Fadeeva’s R&z,)
algorithm
gives
= I,,
4r(z1, G) =
R&I>4 = ~R,hz,) +&I, 4&w R,(z,,z,) . . . . . . . . . . . . .=. . .-&h . . . . . . . . . .4 . . . .+ . . .eh . . . . . . .zz)I,, . . . . .. .
-&I
&1, 4 - 4 trC&h 41 . .dzl, . . . . . . . .4. . . .= . . . -. . . 3 . . trC-%zI, . . . . . . . . . . . . . .4. . . .
R,-dz,,zJ = WR,-,(z,,z,)+q,-,(z,,z,)I,, CL&~,4 = -
! (10)
I
ktrCAR,-1(~1,~2)1 1
0 = AR,-,+a&,z,P,
where the last equation serves as a check. The matrices Ri(zl, z,), i = 0, 1, , . . , IV- 1 and the scalars qi(zl, zJ, i = 1,2,. . . , N, are coefficients of the corresponding powers of cp according to (8) and (9). However, our objective is to determine the coefficients of the Adjoint matrix R(zl, z2) and the characteristic polynomial q(z,, z2) w.r.t. the powers of zl, z2. We note that Ri(zl, z2), z1,z2. qi(z1,z2) are not constants any more, but depend on the variables Furthermore, since Eq. (5) is independent of cp, in the following analysis we choose cp = 0 for simplicity. Then the relations (6x9) become ;z = -S-i-A
WZIJ,) =
CR- ,hzJB
= det
(12)
4N(Zlr z2)
R(z,, z2) = Adj [S-A]
&,,z,)
(11)
= R,_ l(zl, ZJ
(13)
[S-AI = a&1,4.
(14)
It is pointed out that each specific value of the pseudovariable cp results in a different form of b; however, the same transfer function matrix H(z,, ZJ is obtained. From (10) it is seen that for the determination of R,_ l(zl, z2) and qN(zl, z,), which are involved in (12), the matrices Ri, i = 0, 1,. . . , N - 2 and the scalars qi, i = 1,. . . , N - 1 are needed. The degree of the polynomial matrix Ri(zl,z2) and of the polynomial qi(z,,zJ w.r.t. both variables zl, z2 are at most equal to i. Therefore, they can be written in the form R,(z,,z,)
=
1
R;,lz;z;,
i = 0,1,. . . , N-
1
(15)
k+l=O
&(Z,,Z,)
=
c
&z:z:,
i =
1,2,. . . ,N
(16)
k+l=O
where
and i denotes an index and not an exponent. Note that the matrices Rj+-k, i = 0, 1,. . . , NVol. 321, No. 2, pp. 73-80, February 1986 Printed in Great Britain
1 and k = 0, 1,. . . , i, which are the 75
B. G. Mertzios coefficients of the greatest powers of the monomial z!z\ w.r.t. both zl,zZ are diagonal. This can be seen from (lo), since A = -S + A where S is diagonal. The general form of the matrices Rt,i_k has been found by induction to be as follows : R;,i_k
=
(17)
where (18)
(19) From (17)-( 19) the following
interesting
properties
are directly
obtained
(r~l,i-nl)l
=
0
(20)
(rf-nz,n2)2
=
0
(21)
and Ri,i-k=O
Now, combining
if
n, Q kg
i
or
n, < i-k<
(13) (14) and (15), (16) respectively,
i.
(22)
we obtain
N-l
R(z,,z,)
=
1
R;;lztz;
=
f
5
R%;lzfz;
(23)
k=O l=O (k,0 # (n~,nz)
k+l=O
(24) k+I=O
Thus the matrices determination of H(z,, matrix R(z,, z2) and substituting (15), (16) powers of z:zi on both
k=O
l=O
R&l and the scalars c&, which are needed for the zJ by (12), are coefficients of the powers z:zi of the Adjoint
the characteristic polynomial q(z,, z2) respectively. Now in (10) and equating the coefficients of the corresponding sides of each equation, the following recursive relations result
R;,, = AR;, ’ - [(R;:1,,,)1*o + (R:,’
Jo,‘] +&I,
(25)
& = - f tr[AR& ’ - [(R:I1,,,)l,o + (R&A l)“,l]]
(26)
where
___y___;___e___ I (RbL1 (R;r,1)3
= (R;,l)l~o+(R:,,)‘,l 76
(27) Journal of the Franklin Institute Pergamon Press Ltd.
Transfer Function Matrix of 2-D Systems and (Ri&, (R$,, (R&, (Ri& are the submatrices of dimensions n1 x nl, n1 x n,, n2 x n1 and n2 x n2 respectively. Relations (25), (26) permit the easy calculation of Rc; I, & which appear in (23), (24), in terms of R:,,, i = 0, 1, . . . , IV- 2 and L&, i = 0, 1, . . . , N-l and k+l< i. Note that relations (25), (26) are reduced to Fadeeva’s algorithm for regular systems if we consider S = ~1. In the sequel, a lemma will be established which gives the set of initial and “boundary” conditions of the recursive equations (25), (26). Lemma. The initial conditions for the propagation of the recursive equations (25), (26) are (28a)
R8,o = 1, Rf,,, and the “boundary”
= RL,_, = 0
conditions (R:,i&l
k, E 2 1
for
i = 0, 1,. . . , N - 1 (28b)
and
are
= 0
for
;I;‘;2 3
(RL,,,)l*’ = 0
for
+ ‘I;; N - ’
(294
2
,..-,
i = n,,n,+l,...,N-1 l=
O,l,...,i-n,
’
ProoJ: Relations (28a, b) are obvious. For the proof of (29a, b), we consider following recursive equations of the form (25) i+l R kg,*+
I =
AR:,,,+
I -
CW:-
l,nz+
I)‘,’
(29b) the
+(RhJ”*ll + d;:+ ,Iiv for
i = n,,n,+l,...,N-1 i k=Ol 3 I...> i-n,
(30)
Note that = R:,,, 1 Ri- l,nz+ 1 0, L$$:+ = in z2 order greater than n2 cannot appear. Therefore (R:,,)OP1 = 0. Relation (29b) can be similarly proved. The main result can be concisely included in the following theorem. transfer function matrix of the 2-D system given in its state-space form (1) is given by 1 WZ,,
Z2)
=
C[R,, _ l,nz~;l- ‘~2;
~ 4h
4
+R,,,,,_lz”,‘z~-l+~~~+Ri,j~~zj,+~~~+Ro,o]B
(31)
where q(Z,,Z2)
=
Z~‘Z~++4nl-l,n2Z;1-1Z~+qnl,nz_1Z;1Z~-1+’.’+qi,jZ~Zjz+“‘+q0,0.
(32) The Ri,j, qi,j in (31), (32) are the matrices RzJr ’ and the scalars q~j, which are given by (25), (26) in conjunction with the conditions of the lemma. Note that the indices N - 1 and N are omitted from (3 l), (32) respectively due to simplicity. Vol. 321, No. 2, pp. 73-80, Fekmmy Primed in Great Britain
1986
77
B. G. Mertzios III. Illustrative example Consider
a 2-D system of the form (l), with n, = n2 = 2 and (6)
and c = [C,
I C,] =
1 O-l/l
1
I 0
-1
1
1’
For this system, we obtain R:,o = I,,
78
Joumalof
the Franklin Institute Pergamon Press Ltd.
Transfer Function Matrix of 2-D Systems
R;,,
=
-;_I;-. E 1 2
Similarly,
we find
C&,0= C&l = 4A.1 = 2, q& = -7,
q:,,, = 6,
=
CT;,1
q&J = 5, qf,2
=
=
&o = q&z = 1, q;,,, =4,
q:,l =6,
d,z = -2,
2,
&
-2,
41.1 = 4,
q;,l = -12,
q:,o = -1,
q;,o = -14, 4:,2
&,I = 0,
= -12, &,I
q:,1 = 2,
4,
=
44.2
=
4!,1 = -1%
q;,o = 4,
4,2
1 =
03
1.
Substituting R& = R,,, and q& = qk,l in (31), (32), we finally obtain function matrix H(z,,z,)
=
~ q(z,,
hl(zl?zz) z2)
hz(zl~z2)
C h,,(z,,z,)
hzz(z1,
z2)
the transfer
1
where h,,(z,,
z2) = - z:z2 + 6z,z, - 222 + 22; + 132, + 72,
h&z,,
Z2) =
zlZf
h,,(z,,
z2)
32~2, -3z,_z,
h,,(z,,z,)
=
+
5Z,Z2
= z:z2-zlzz
+
Z;
+
62, +
32,
+ 62: - 32; - 132, - 142, - 8
+2z:-z;-32,
-5z,-4.
IV. Conclusions In this paper a new algorithm is derived for the computation of the transfer function matrix of a linear, 2-D system from its state-space description. This algorithm is suitable for computer use and reduces the computation cost, since the inversion of a 2-D polynomial matrix is avoided. Furthermore, it can be more easily programmed on the computer than the already available 2-D Fadeeva’s algorithm (4) since the relevant, complicated notation is avoided. The main advantage of the proposed algorithm is its generality. Specifically, a simple transformation has been proposed, which allows the use of Fadeeva’s algorithm. This transformation can be easily applied to other classes of complicated systems (i.e. multidimensional, singular, delay-differential) where various analysis and control problems can be reduced to the simpler regular case by using this transformation (8). Vol. 321, No. 2, pp. 73-80, February Printed in Great Britain
1986
79
B. G. Mertzios
References (1) N. K. Bose, “Multidimensional Systems : Theory and Applications”, IEEE Press, New York, 1979. (2) N. K. Bose, “Applied Multidimensional Systems Theory”, Van Nostrand Reinhold, New York, 1982. (3) S. K. Mitra and M. P. Ekstrom, “Two-dimensional Digital Signal Processing”, Dowden, Hutchinson and Ross, Stroudsburg, 1978. (4) C. S. Koo and C. T. Chen, “Fadeeva’s algorithm for spatial dynamical equations”, Proc. IEEE, Vol. 65, pp. 975-976, June 1977. (5) L. A. Zadeh and C. A. Desoer, “Linear System Theory”, McGraw-Hill, New York, 1963. (6) B. G. Mertzios and P. N. Paraskevopoulos, “Transfer function matrix of 2-D systems”, ZEEE Trans. Aut. Control, Vol. AC-26, pp. 722-724, June 1981. (7) R. P. Roesser, “A discrete state-space model for linear image processing”, IEEE Trans. Aut. Control, Vol. AC-20, pp. l-10, Feb. 1975. (8) B. G. Mertzios, “Decoupling of 2-D systems by dynamic state feedback controllers”, Proc. 9th Int. Fed. Aut. Control World Congr., Vol. 9, pp. 144-149, Budapest, Hungary, July 1984.
80
Journal
of the Franklin Institute Pergamon Press Ltd.