Systems & Control Letters 15 (1990) 397-403 North-Holland
397
A minimal lattice realization for multivariable systems Kazumi Horiguchi, Takuya Nishimura and Akira Nagata Department of Electronic Engineering, Faculty of Science and Technology, Kinki University, 3-4-1 Kowakae, Higashiosaka, Osaka 577, Japan Received 1 July 1990 Revised 17 September 1990
Abstract: An algorithm for constructing a minimal lattice realization of a multivariable stable linear discrete-time system is proposed. The algorithm consists of four steps: (i) realize the objective system in an input normal Hessenberg form, (ii) construct an LBR system associated with the objective system, (iii) construct a lattice realization of the LBR system from its input normal Hessenberg realization, (iv) transform the lattice realization of the LBR system into one of the objective system.
Keywords: Minimal realization; lattice realization; Hessenberg realization; LBR system; stable linear discrete-time system.
1. Introduction The purpose of this paper is to construct a minimal lattice realization, a lattice realization with a minimal degree, of a multivariable stable linear discrete-time system. The lattice realization is one of the most remarkable realizations for linear discrete-time systems from the theoretical viewpoint as well as from the practical viewpoint, and it is applied in many fields [4]. For scalar systems, a minimal lattice realization has already been given and analyzed in detail based on a state-space description [8]. For multivariable systems, however, we have no algorithms that always give a minimal lattice realization of an arbitrary stable system. On the lattice realization for multivariable systems, a well-known result is the L W R algorithm [7]. The L W R algorithm is a fast recursive algorithm which constructs a multivariable A R model whose finite sequence of autocorrelation
coefficients matches the prescribed one. This algorithm gives a lattice realization of the multivariable A R model, but has no concern with its degree. It is also well-known that a synthesis procedure for J-lossless systems leads to a lattice realization where each lattice section has first degree, and that this procedure is essentially equivalent to the LWR algorithm [2,3,5]. These results, however, have not been applied to constructing a lattice realization of an arbitrary stable system. Vaidyanathan and Mitra developed a minimal lattice realization for LBR systems [10]. They presented a procedure for constructing a lattice realization of an arbitrary stable system as well as an LBR system, but this procedure does not always give a minimal lattice realization of an arbitrary stable system. In their report, it was difficult to construct a minimal lattice realization, because multivariable systems were treated based on a matrix fraction description. In this paper, we shall propose an algorithm which constructs a minimal lattice realization of a multivariable stable linear discrete-time system based on a state-space description.
2. Input normal Hessenberg realization Let us describe a reachable and observable representation of an r-input m-output stable linear discrete-time system as
x ( t + 1) = A x ( t ) + Bu(t),
(la)
y ( t ) = Cx(t) + Du(t).
(lb)
Hereafter, we assume that (A B C D) of the objective system (1) is an input normal Hessenberg realization. It is known that a reachable and observable realization of a stable linear system can always b e transformed into this form [1,9]. The input normal Hessenberg realization is defined as follows.
0167-6911/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
K. Horiguchi et al. / Minimal lattice realization
398
Definition. A system (A B C D) with r inputs and m outputs is said to be a Hessenberg realization if A
=
01
Aa2
[ All
Ap_ lp Apl
R(z) = DR + C(zI-A)-~B
R.
(7)
Then it is easy to show that (5) is the state-space representation of the relation
'
½{R(z)+RT(z
1)}-H(z)HT(z-1)=O.
(8)
app
(2a) C=[C a 0
-.-
0],
(2b)
with h i j : m i × m j, Bi: m i × r, Ca: m × ml, and rank C1 = ma,
m i
and let
= rank
rank Ai_m~= mi,
•
- rank
L cA'i- 1 J
= m I +
,
3. LBR system
CA i- 2 J
i = 2 . . . . . p, dim A ~ n
•
(3a)
This means that when the power spectrum function of the system (1), H ( z ) H T ( z - 1 ) , is described as the sum of the causal function ½R(z) (a function analytic in I z [ >/1) and the anticausal function ½RT(z -1) (a function analytic in I zl < 1), then (A B R C DR) is a realization of the causal function R (z).
(3b) • ..
+
mp,
m>~ma>~ " ' " > ~ m p > O .
(3c)
Moreover, (A B C D) in (2) is said to be an input normal Hessenberg realization if the reachability Gramian is the identity matrix, that is, if
We refer to a system whose transfer function is an LBR function (Lossless Bounded Real function) as an LBR system. In this section as the first procedure for constructing a minimal lattice realization of the system (1), an LBR system is introduced. Let us construct an (r + m)-input m-output system, denoted by (F G=[G,
AA T - I + BB T = 0
G2] H J = [ J ,
J2]),
(4) from (A B C D) and ( B R D R ) according to the following transformation:
holds. It seems to be clear that an input normal Hessenberg realization of a stable system is reachable and observable. In the following sections, we shall develop a procedure for constructing a minimal lattice realization of the system (1), where (A B C D) is an input normal Hessenberg realization. As a preliminary, we introduce two matrices, B R : n × m and D R : m × m, such that
F = A -- B R ( I + D R ) - l C , [G 1
(9a)
G2I=[B-BR(I+DR)-'D - - B R ( I + DR)-1],
H = 2( I + D R ) - I C ,
(9b) (9c)
[Jl J21 = [ 2 ( I + D R ) - 1 D ( I + D R ) - ' ( I - D I ¢ ) ] , (9d)
A A T - I + B B T = O,
(5a)
½BR - BD T - A C T = 0,
(5b)
where the existence of ( I + DR)-1 is guaranteed because (5c) assures that D R >/0. Since it is clear that
D R - DD T - CC T = 0.
(5c)
( I + Jz) = 2 ( I + DR) -1
Let H ( z ) be the transfer function of the system (1), that is, H(z) = D + C(zI-
A)-IB,
(6)
is nonsingular, the inverse transformation of (9) can be given by A = F-
Gz(I+Jz)-IH,
(10a)
K. Horiguchi et al. / Minimal lattice realization
[ B BR] = [ G , -
and let us derive an input normal Hessenberg realization
G2(l + J2)-'Ja
(10b)
-2G2(I-I-J2)-I],
[D Dnl=[(l+J2)-lJ1
(I+J2)-'(l-J2)
of an LBR system with a lower degree from (F(1) G(O) H(O) j(o)).
].
(10d) Substituting (10) into (5), we have F F T - I + GG T = 0,
(lla)
GJ T + F H T = 0,
(llb)
I - j j T _ H H T = 0,
(llc)
which means that the system ( F G H J ) becomes an LBR system [6]. As stated in Section 2, (A B C D) is a Hessenberg realization described as (2) and (3)• This implies that ( F G H J ) given by (9) is also in a Hessenberg realization, that is, ( F G H J ) can be written in the following form:
0] [
F,2
[i1
,
0
...
rank H 1 = rn 1,
(12b)
0],
rank F,. u = m i.
(13)
In this section we construct a lattice realization of the LBR system from its input normal Hessenberg realization (12). We shall develop the procedure step by step below. First, set H (°)&H,
j(o)&j,
zx
u('>= u '>j
Ha(°) &//1,
is an mo-dimensional orthogonal matrix. Substituting (15) into (11c), we have 1 - j(o)j(O)T _ H(O)H(0)T =
v t l ) ( I -- ZlaAT ) V( ')T - HI(°)H1(°)T
=0.
(16)
Since H1(°) is an m o x m 1 matrix with full rank, that is, rank H1(°) = ml, (16) implies that
This is the reason why j(0) has m I singular values less than 1 and m 0 - m 1 singular values equal to 1 as stated above. Moreover, (16) implies that there exists an ml-dimensional orthogonal matrix Ta such that H1(°) = VIO)( I - AIA~)I/ZTa.
(14a)
G]°> G,
(r + ml) matrix such that
d i m ( I - AIAT1 ) = r a n k ( I -- aaAT) = m 1.
4. Lattice realization of the LBR system
G (°)&G,
X
(15)
ml
In addition, (11a) implies that ( F G H J ) is an input normal Hessenberg realization.
Fu,
where A1 is an m I I - z~lA~ > 0,
F.
with F~j: m i X m j , Gi: m i X (r + m ) , 1-11: m X ml, and
°)
J(°)=[vii)V(1)][t 0][ Utl)]IJ[Uj2{1)
G=
(12a)
F (°)&F,
( l l c ) claims that j(o) has no singular values greater than 1. j(o) has m 1 singular values less than I and m o - m I singular values equal to 1 (the reason is stated below). Hence, the singular-value decomposition of j(0) can be expressed as
is an (r + mo)-dimensional orthogonal matrix, and
?],
Fp_l p
1
H=[H~
(F(1) G(1) H(1) j(1) )
(10c)
C = (I+J2)-'H,
F=
399
(17)
From (17), T1 is given by (14b) (14c)
I"1 = ( I -
alAT)-I/2VI(')TH(a°).
(18)
400
K• H o r i g u c h i
e t al. / M i n i m a l
lattice realization
a~°)]
Substituting (15) and (17) into (llb), we have G(°)J( °)T + F ( ° ) H (°)T
Q1 =
"
U(1)T-
(25)
G;o,j Now, let us define ( F (1) G (1) H (]) j ( 1 ) ) as
+
[<' "
r~(°, r#)
F(I> =
Tf(l-A11ff11)l12Vl O)T
IF;1°' equation
by
Vl.>(s
=-
H ( 1 ) = [T1F1 (O> 0 j(,) = TaPl(i
i lUl<,>~a](s-a,a])-'12r
1.
F; O)
(20)
Post-multiplying (19) by V2°) yields G~°) (21)
LG; °,
_
...
0],
ATAa ) - 1 / 2 .
(26c) (26d)
F(l)F O)T - I + G(1)G (1)T = 0,
(27a)
GO)J (1)T + F(1)H O)T = 0,
(27b)
I - j(1)j(1)T _ H(a)H(1)T = 0.
(27c)
Using (20), (22), (23) and (26), F (°) and G (°) can be written as
This implies that there exist appropriate matrices /°1 and Q1 such that G~°) = P, U1(1),
(26b)
We shall verify that this is an input normal Hessenberg realization of an LBR system. Since it is clear that ( F (1) G (1) H (1) j ( 1 ) ) is a Hessenberg realization, we have only to verify that it satisfies the relation:
Gj0,j
U2(1)T = 0.
(26a)
-
G}°) ]
•
Fp(O>? ' . . . . . .
Post-multiplying this A1A~)-I/ET 1 yields
[
0 t
Lr; °' a<"= Q,(/-a~al)1/2,
(19)
=0.
F;1°'
[
F(O) =
(22)
- TITj(a)ATT1 __ G (1)ATT1
T T H O) ] F (') ],
(28a)
G2°) =
•
(23)
Q 1 U 1 (1> .
°,
G (o) =
QIU1(1) ]
a<,,(i_ a~a,),/2v(, ]
T1TJO) ( I - a~a 1)l/2/_71(1)
From (22) and (23), P1 and Ol are respectively given by P1
P1UI(I) ]
= G~°>U1(1)T,
(24)
Substituting (28) into (lla), the relation of the _1
( I - Atr At)uU* (I}
0
J:i )t/ (F~°)G<°)H~md
~°~) ;'
" 2 ' O"
o
~l! :~ I
(28b)
(F"~G(t>H<')J~'~)~ - -
[2 ImF~ xt
-< 1"
~ Vltt~ (1 -- /I, Air) ½ Fig. 1. Relationship between ( F (°) G (°) H (°) j(o)) and ( F O) G O) H (1) j(1)).
K. Horiguchi et al. / Minimal lattice reaflzation (2,2)-block element yields (27a), the relation of the (2,1)-block element yields (27b), and the relation of the (1,1)-block element yields (27c), respectively. Thus we have derived an input normal Hessenberg realization ( F (1) G O) H (1) j o ) ) of an LBR system with degree m : + . . . +rap from (F(0) G(O)H(0)j(0)), which is an input normal Hessenberg realization of an LBR system with degree m 1 + . . . +rap. Next, we shall verify that the relationship of these two LBR systems is as shown in Figure 1. In Figure 1, let us describe the system seen from terminals 1-1' as x(1)(/+ 1) = FO)x(1)(t) + G(')u(1)(t),
(29a)
y(a)(t) = UO)x(1)(t) + J(1)u(a)(t).
(29b)
Consider the system seen from terminals 0 - 0 ' , that is, the system formed by attaching a lattice section L (a) to the system (29). From Figure 1, we have U(1)(t) = ( I - - AT1A1)I/2u(1)u(O)(t) -- a T x I ( t ) ,
(30a)
Xa(l + 1) = y ( 1 ) ( t ) ,
(30b)
y(0)(/) __ V2(1)U(1)u(O)( t ) q- vI(X)(I-- A 1 A T ) - I / 2
• {AlU(1)(t)+Xl(t)}.
(30C)
Eliminate u(l)(t) and y(1)(t) from (29) and (30), define
x(O)(t)a=[ T T X I ( t ) ]
[ x(,(t)
'
and observe (15), (17) and (28); then it follows that the system seen from terminals 0 - 0 ' can be described as x(°)( t + 1) = F(°)x(°)( t) + G(°)u(°)(t),
(31a)
y(°)(t) = U(°)x(°)(t) + J(°)u(°)(t).
(31b)
It has been verified that in Figure 1 if the system seen from terminals 1-1' is ( F °) G O) H °) j o ) ) then the system seen from terminals 0 - 0 ' is (F(O) G(0) H(0) j(o)). The reverse can also be verified.
(F G H d) ~
401
L ~p~
d~
±
( 1 - AATAk)2U~ ~> /
Fig. 2. Lattice realization of (F G H J).
Accordingly, when we extract a lattice section L (1) from the LBR system ( F (°) G (°) H (°) j(0)) with degree ml + . . . + m_, we have an LBR system ( F (~) G (~) H (~) J(f)) with degree m2 +..-+rap as a remainder. Similar, for k = 2 , . . . , p - 1, when we extract a lattice section L (k) from the LBR system ( F (*-~) G (k-~) H (k-l) j ( k - ~ ) ) with degree i n k + . . - +m_, we have an LBR system ( F (k) G (k) H (k) J ( k f i with degree mk + ~ + • • • +rap degree as a remainder. When we extract a lattice section L (p) from the LBR system (F(p-1) G(p-1) H(p-1) j ( p - 1 ) ) with degree rap, we have a terminal parameter J(P) with I - j ( p ) j ( p ) T = 0 as a remainder. Finally, a lattice realization of the LBR system ( F G H J ) is represented in Figure 2. The algorithm for constructing the lattice realization of ( F G H J ) is stated as follows. (i) Set ( F (°) G (°) H (°) j(0)) as in (14). (i.i) For k = 1 . . . . . p. (a) Find Ak, Uk and Vk from the singularvalue decomposition of j(k-1). This constructs the lattice section L (k). (b) Form ( F (k) G (k) H (k) j ( k ) ) according to a similar relation to (26). When k = p , calculate J(P) only. The procedure (ii) constructs a series of lattice sections L(1),..., L (p) step by step to get J(P) in the end. (iii) Terminate the lattice section L (p) by J(P). Remark. The present procedure for constructing a lattice realization of an LBR system is equivalent to that developed in [10], where a lattice realization is derived based on a matrix fraction description. We can construct another type of lattice realization by applying the result developed in [5].
402
K. Horiguehi et al. / Minimal lattice realization I
tion of an arbitrary stable system will be constructed and our purpose will be accomplished. Let us describe the state equation of the system
(1 - Air At) 2 F"
U,t ~"
( F G = [ G I G2] H J = [ J : Xl
U~
a
as
U~2 u~]
I"
J2])
x(t + 1) = Fx(t) = O:u,(t) + G2uz(t), Ul(t ) ~ R r, U2(/) ~ n m .
m
(32)
t
(I r/
Utt u)
Here, we set uz(t ) as follows:
Atr A,) 2
u2(t) = - ( I + J 2 ) - l { H x ( t ) +J,u](t)}. ~---.)>-----~y = - u 2 ~ " ,at
~'~
I /:l+d,)-'
Considering (10), (32) can be reduced to
\
Xt]
x(t+l)=Ax(t)+gu(t),
"
Atr A,) 2
u20)
/'
Ht
(34)
= - { Cx(t)
+
Du(t)},
(35)
and thus an output of the system (1) can be given by
/ll r
C
u(t)~=u,(t).
This is the very state-space equation of system (1). While considering (10), (33) can be reduced to
!
(1
(33)
y(t) -- Cx(t) + Du(t) = -u2(t ).
Xt
(36)
. i I +d~) :
It has been shown that the system ( F G H J ) can be transformed into the system (A B C D) according to (33) and (36). Recalling that ( F G H J ) is an input normal Hessenberg realization of (12), (33) can be rewritten as
Fig. 3. Transformation from ( F G H J ) into (A B C D): (a) Lattice realization of ( F G H J ) (without the output being composed). (b) Setting y = - u 2 = ( I + J 2 ) - l ( H l x l + Jlu]). (c) Lattice realization of (A B C D) (simplification of figure (b)).
5. Lattice realization of the stable system
u2(t)=-(I+J2)
i
[/'tl (H
/:( 1 +d2) -~
ZJl'At) 2
,/~
Nt
(37)
where x~(t) is the state variable consisting in the lattice section L (a). Therefore, we have only to change the lattice section L (1) to transform a lattice realization of ( F G H J ) into one of (A B C D). The process of this transformation is shown in Figure 3. Finally, a lattice realization of the system (1) is represented in Figure 4. In Figure 4, it should be noted that this lattice realization has degree n A
In Section 3 we have constructed an LBR system ( F G H J ) from the objective system (A B C D) and (B R DR) according to the transformation (9). Both systems were an input normal Hessenberg realization. In Section 4 we have constructed a lattice realization of the LBR system ( F G H J). In this section we shall transform a lattice realization of ( F G H J ) into one of the objective system (A B C D). As a result, a minimal lattice realiza-
(l
'{H:x](t)+J:ua(t)},
_
_
. . . . . . . .
Xt
Fig. 4. Lattice realization of (A B C D).
K. Horiguchi et al. / Minimal lattice realization + m p . Since we have assumed that (A B C D) is realized in an input normal Hessenberg form of (2) with a minimal degree, we find that this lattice realization is minimal. As developed in the previous sections and this section, the algorithm for constructing a minimal lattice realization of the system (1) has been completed. m 1 + ...
6. Conclusion
An algorithm which constructs a minimal lattice realization of a multivariable stable linear discrete-time system has been proposed. The algorithm consisted of four steps. (i) Transform a reachable and observable realization of the objective system into an input normal Hessenberg realization (A B C D). (ii) Construct an input normal Hessenberg realization ( F G H J ) of an LBR system from (A BCD).
(iii) Construct a lattice realization of the LBR system ( F G H J). (iv) Transform the lattice realization of ( F G H J ) into a lattice realization of the objective system (A B C D). The step (iii) was the main procedure of the algorithm. Each step only required a few simple matrix operations.
403
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