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A second multivariable normal form for model reduction of discrete-time systems
Systems & Control North-Holland
Letters
4 (1984)
1099117
April
1984
A second multivariable normal form for model reduction of discrete-time systems E. BADREDDIN Swiss Federal Switzerland Received Revised
and M. MANSOUR
Institute
of TechnologV
(ETH-
Ziirich),
Institute
of Automatic
Control
and
Industrial
Electronics,
CH -8092
Ziirich,
7 May 1983 12 December 1983
Starting by the block-controllability form, a normal form for multivariable discrete-time systems is developed. The construction of this normal form is made straight-forward by the introduction of the matrix Schur-Cohn table. It shows a one-to-one (scalar-to-matrix) correspondence with the single-input Mansour form [2]. Examples for the construction of the proposed form as well as its application to model reduction are also given. Keywords:
Linear
multivariable
systems,
Canonical
forms,
Discrete-time
systems,
Model
reduction.
1. Introduction In a previous paper [l], a multivariable normal form for discrete-time systems was developed starting by the Luenberger first form. That form was specially developed for model reduction. Aspects like simple computation, structural matching and structure preservation when applying the reduction method described in [3] were the main concern in the development of that first form. Here, we shall introduce a second form which, also, can be considered a direct extension of the Mansour form [2]. Whilst the first form in [l] shows a block-to-block correspondence with the Luenberger first form, the form proposed here shows a one-to-one, i.e. scalar-to-matrix correspondence with the Mansour form [2] for single-input systems. This is achieved by first transforming the Luenberger first form to the block-controllability form, by means of elementary similarity transformations, and then applying a generalized matrix Schur-Cohn table. The proposed form is then constructed in the same manner as the single-input case. The resulting form possesses similar structural properties to those of the first form and is also suitable for model reduction. For the full comprehension and appreciation of the developments presented in this paper, the reader is advised to review the previous paper [l] especially for the construction of the scalar Schur-Cohn table and for the definition of the Luenberger first form. 2. The matrix Schur-Cohn If the controllable form (A,, 4)
A,=
0167-6911/84/$3.00
0 z,
0 0
0
1,
b
...
table and the second multivariable normal form
n-th order linear system with m inputs can be described by the block controllability
__. 0 .. . 0
-A, -A,-, 7
0 . z,
0 1984, Elsevier
-A,
Science
Publishers
B,=
zm 0 :
(1)
b B.V. (North-Holland)
109
Volume
4, Number
2
SYSTEMS
& CONTROL
LETTERS
April
where 1, is the m x m unity matrix, 0 is the m x m zero matrix, A, through A, are q = n/m (integer ) 0), then it can be shown (see [4] for the proof ‘) that F, = T,A,T,-‘,
m x m
1984
matrices and
G, = T,B,
where G, = B,, -4
(I-e:)
-0, F,=
(z-e;)
- 44
0
:
9
-e,-, _ -0,
-e,_,e,
.. .
...
-eqe,
.. .
...
(2)
(-;-I) -e,e,-,
T,= [G,,F,G, ,...,
(3)
The elements 8,, &,. _. ,8, are m x m matrices obtained from the matrix Schur-Cohn table which can be constructed e.g. for q = 3 as shown in Table 1. If the diagonal blocks in the Luenberger first form are of the same size q, it can be transformed into the block controllability form by elementary similarity transformations [4]. Now, if the size of the diagonal blocks of the Luenberger first form is not the same for all blocks, which is definitely the case if the system order is not a multiple of the number of inputs, some modifications will be necessary to obtain the multivariable normal form by means of the matrix Schur-Cohn table. We shall extend every block with a size smaller than q by adding the proper number of zero rows and columns such that the structure of the block is maintained. This operation corresponds to adding a series of unity-delay blocks at the input port of the subsystem represented by that particular block which needs extension. Since the Luenberger first’form represents a number of subsystems connected in one direction only, we may consider any two subsystems, say S, and S,, and let the feed forward paths from S, to S, be denoted by a vector /? of the proper dimension. Now, suppose that S, is to be extended. Because of the newly inserted unity-delay blocks at its input port, the output of S, reaching S, through p is no longer in ‘synchronization’ with S2’s own input. Actually, the sequences coming from S, are reaching S, as many ‘units’ too late as have been inserted at the input of S,. This delay is compensated for by shifting the entry points of the P-ports in S, ‘forwards’ as many ‘units’ as required.
Table
1 A3 I (A2
normalized: ==a
A2 Al
- AlA,)
(A, - A,A,)(I A 2.2
- A:)-’
I
(A,,,
normalized: ==a
110
A3
0, = A,-,
(A, - A,A,)
(1-A:) I
42
I
4.2
A
(I- A<.,)-’
’ The proof is based on the relation controllability assumption.
I
A2
(A,-A,A,)(I-A:)-’
- 4,z4,,)
(A,,, - A,.2A2,2)(1 A 1.1
A,
between
02
= A,
= A 2.2 =(A,-A,A,)(I-A;)-’
2.2
4.2)
I I
A, and 0, from
8, = A,., = A,,,(1
the matrix
Schur-Cohn
+ ,42,2)-l table.
The regularity
of T is guaranteed
by the
Volume
4, Number
2
SYSTEMS
SK CONTROL
April
LE’ITERS
1984
If S, is to be extended, the P-ports will be shifted ‘backwards’ as many ‘units’ as have been added at its input port. The algorithm will be illustrated by an example that should also demonstrate the way in which it could be generalized. Example 1. Consider the linear system in the Luenberger first form,
By means of elementary similarity transformations
0 1 0 0 1 -P3
-a32
1
0
0
-a22
0
1
0
--a11
0
1
-&
0 0 0 - a2, A=
F 0
-&
0 B=
0 -a12
it can be transformed 1
0
0 0
1 0
0 0
0 0
to (A, B),
(4)
The block diagram representation of the linear discrete time system x( k + 1) = Ax(k) + Bu( k) is shown in Figure 1. Comparing (4) with (1) we notice that A, in (4) is only a 1 X 2 matrix. We shall extend the matrix A, by zero elements, the matrix A by a row and a column and the matrix B by a row with permutation as follows: 0000 0 0
0
0
0 -yj
-a32
0
0 1
1 0
1 0
0 1
0 0
0 0
-a21 -yz
-a22
0 0
0 0
0 0
0 0
1 0
0 1
-a,, -y,
0 -aI2
0 0
0 0
0
(5)
Notice that the elements yi, y2 y3 replaced &, p2, p3. This must be done since the extension of A, by an upper zero row means that a delay block (represented by z-’ in the block diagram) has been connected to the input of the first subsystem. This means that sequences of x2(t) and x4(k) of the first subsystem will
Fig.
1. 111
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SYSTEMS
& CONTROL
LETTERS
April
1984
appear ‘one step’ later and are denoted by iZ(k) and Z4(k). To compensate for this delay on the 2nd subsystem, the P-ports should be advanced by one delay block. These block diagram modifications are shown in Figure 2. The resulting dynamic equations for x5, x3, x, are Xl(k + 1) = -%*(-a)
-P&t(k))
= -h%(k) x3(k + 1) =x,(k) =x,(k)
-a,,x,(k) -%(x,(k)& -Y*%(k)
+ u1 = %P,-f’,(~>
-U,,%(k)
+ u,(k)
+4(k), - w4)
-P,%(k)
= x,(k)
-(P,
- ~,,P,F4(~)
-%*x,(k)
-%,X,(k),
x,(~+1)=x,(~)-PZ~-l(~)-~,2(Xs(~)-P,~.4(~))=X3(~)-(P2-~lZP1)~:4(~)-~,2xS(~) =X3(k)-Y1~4(k)-ulZX5(k) and
In general, if p is the dimension of the 1st subsystem and v is the dimension of the 2nd subsystem, and v > IL, and if the initial coupling elements are represented by the v-dimensional vector @, then the 1st subsystem will have to be expanded (Y - p) times to match the size of the 2nd subsystem. The final coupling vector y can be computed as follows: y = A:“,-“‘fi.
16)
Now, in Example 1 the matrix Schur-Cohn
table will be applied with
to give
and the resulting multivariable
normal form will be
(7)
Fig. 2. 112
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Remark. Noting that [b,, b,, Ab,, Ab,, A’b,] = I, the transformation F2g,] where b,, g, are the i-th column of B and G respectively.
April
1984
matrix will be T = [g,, g,, Fg,, Fgz,
At this point, we still ought to show that the pairs {A, B} and {F, G} are similar with the transformation matrix T. The proof is, however, quite lengthy and is therefore omitted here. Interested readers may refer to [4]. ’ To explain the algorithm further, we consider the 3-inputs case with a different block structure. Example 2. For the pair (A,,, B,} in the Luenberger first form
A,,=
B,, =
By means of elementary similarity transformations,
0 0 0 1 0
0 1 0 0 0
1 0 0 0 0
we obtain
(8)
Then we extend {A, B} to the block-companion
j-C
Fig.
-00 0
00 0
00 0
-id -p2
1 0 0
0 1 0
0 0 1
-c -p3 -&
-d,0 -a2 -d, --a1
descrir
00 1
I 00 B=
-1 {A, B} as follows: 01
01 1 (9)
0 -a,
3.
’ The proof is based, in principle, on constructing an extended transformation matrix, T, from the extended pair { F, G}. Applying r on the extended state vector, X, and omitting the dispensable components, will result in omitting the associated rows and columns in -T, F and G and restores the original dimension. To complete the proof is then a matter of algebraic manipulation and induction. 113
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1984
The-- block diagram of (8) is shown in Figure 3 by solid lines and the dotted lines indicate the extension to { A, B} and the shifting of the feed-forward paths. After shifting, the resulting equivalent coupling elements will be
Since the a-path has not been shifted, {(Ye, (Y,} remain unchanged. Then, we may write
A2-[siI g. A,=[iI1:g. Applying
the matrix Schur-Cohn 0
table with A,, A,, we get
1
0 7 A 23 A 8, =A,(Z+
A,)-’
=
(ii -An&) ,(S,-A,,~,)-~,(~Y,-A,,CU,)/(~+A~~)
where
A,,=c, A,,=&,
A,,=&, 2
0
0
A
0
(;-A,d,(l+A,,)
A,,
1
A,,=&, A,,=a,, 2
and the resulting normal form {F, G} will be
where E is the elimination matrix [0 Z,lT and Z2 is the (2 X 2) unity matrix. Since [b2, b,, b,, Ab,, Ab,] = I, then the transformation matrix will be T = [g,, g,, g,, Fg2, Fg,]. Up till now we have considered the case where the size of the first block in the lower-block triangular version of the Luenberger first form is smaller than the second and the size of the second block is smaller than the third, etc. But what happens when the size of one block is larger than the next one? The answer to this question will be given by the next example. Example 3.
A,=
rO 1 0
0
-a3
0 1
-a2 -a,
0
cl 0
-P2
-0 114
-P,
-I7
B, =
0 0 0 1 0-1
1 0 0. 0 0
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By means of elementary similarity transformations,
B=
A=
LETTERS
April
the pair {A,, I?, } could be transformed
0
1
0 1
0 0.
0 0
0 0
1984
into {A, B}.
The ‘extended’ matrices A, B will be 0 010 0000
O/-a,
0
0
0
1000-a,
0
1
0
0
0
-y2
-d,
0010-a, 0 0 0 t
1
-y,
-d,
0
Figure 4 shows the block diagram of the system {A, B} in solid lines and the extended together with the shifting of the /S-ports by one step ‘backward’ in dotted lines. The ‘backward’ shifting of the P-ports is explained by keeping the ‘synchronization’ inputs to the subsystem. But since the control input U, has been delayed by one unit due step to achieve block-size matching, the feed forward paths which are the b-ports, should by one unit, i.e. shifted one step ‘backwards’. The equivalent coupling elements after the shifting are
The elements of the matrix Schur-Cohn
A3=[;3 ;],
A2=[;;
-system {A, B} between all the to the extension be also delayed
table in this case are
%].A,=[;:;,I
e,=A,,
8,=(A,-A,A3)(I-A;)-‘=
1
l/(1
1 2
0
-d;)
0
‘I9
1
Fig. 4. 115
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LETTERS
April
1984
0
A,, e2=
& CONTROL
-A$)
(~2-A31~1)/(1
I’
A22
A,,
0
Y,(~+A,,A,,)-Y,(A,~+A~,)
A
B,=(A,-A~A,)[(I-&)+(A,-A,A,)]-‘=
, 12
(1 -&)(l
+A,,)
Then the matrix F is built as follows:
F=
(I-0:)
-4 -e,
0.e
I -eT-e3
1 0I ’
(I-ef)+e
- e,e, - eT ae3e1
G=B.
- eTe3e2. e
Combinations of ‘forward’ and ‘backward’ shifting might become necessary in some cases, and can be treated in the same manner as illustrated by the previous examples,
3. Reduction of the second normal form Example 4. Consider the pair { F, G} of Example 1, -4,
0
-6 F=
(1 ‘A:,) -W,,
-A,2
-A,,
0
+A,,)
-4421
0
0
0
1
(l-42)
0
1
0
0
0
0 0
0. 0
0
0
-6,
-A22
-(A&
+
Ad21
-442,
-83
-A32
- (AA4
+ A,,&)
-44x
G=
(1 -A;,) -Ad-‘,,
Let the output matrix be
Al222
h4323
h4424
I
h 25 . h,
Substituting the quasi-steady-state approximation 4th-order reduced model described by
of x5 in the rest of the dynamic equations, we obtain the
0 (1
-A:,)
0 -Ad22
(hn-hl)
116
(h,d%~,+A,,h,))
(h,,-414)
(4,
(h,, -4,hz)
-(bh;
+ AA))
I
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SYSTEMS
& CONTROL
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April
1984
where
h = A;2+A3* 22 1 +A22432
h, = ’
h, =
~2=P2-~22Pb
4 1 +A22432
A 1 + AZA,,
h2 = ’
h
15’
h,=
4 1 + A,,A,, A 1 +A;2A,,
h
25’
h
“’
As for the single-input case [3,4], the reduced model will preserve stability properties and the steady-state step response.of the original system. The reduced model approximates, in general, the ‘slow’ response of the system [4].
4. Conclusions and final remarks A multivariable normal form for discrete-time systems, as an extension of the Mansour-form [2] for the single-input case, has been developed. This form is called the second multivariable normal form to be distinguished from the first form presented iA an earlier paper [l]. Starting by the block-controllability form, a generalized matrix Schur-Cohn table is constructed, resulting in coefficients 13,,i = 1, 2,. . . , 4, where q = n/m = integer > 0, m is the number of inputs and n is the system order. 8, are (m X m) matrices rather than scalars as in the single-input case. The proposed second form is then constructed in the same manner as the single-input form, but using 8, instead of the scalar Schur-Cohn coefficients 6, [l]. When the system order is not a multiple of the number of inputs, i.e. q # integer, the system matrix is extended by zero rows/columns which correspond to the addition of unity-delay elements. The additional delay is compensated for by ‘forward’ and/or ‘backward’ shifting of the coupling paths. After constructing the normal form for the extended system, proper rows and columns are omitted resulting in the final form for the unextended system matrix. This normal form will always exist for all controllable {F, G} and is unique once the Luenberger first form is given. The construction of the Luenberger first form is, however, dependent on the order in which the transformation basis is chosen [l]. The second normal form discussed here avoids separately computing the coupling elements as in the first form [l]. Beside the one-to-one correspondence to the single-input case, the second form possesses similar structural properties to those of the first form, such as structure preservation when applying the model reduction method described in [3]. Beside model reduction, for which this form was specially developed, the proposed second form may also be useful in other applications. For example, if the block-controllability form is obtained from the Luenberger first form, the resulting 0, will be triangular matrices and the system will be stable iff the diagonal elements of 6, are less than one in magnitude.
References [l]
E. Badreddin
and M. Mansour,
A multivariable
normal-form
for model
reduction
of discrete-time
systems,
Sysfems
& Control
Left.
2(1983)271-285.
M. Mansour, [3] E: Badreddin
[2]
Die Stabilitat linearer Abtastsysteme and M. Mansour, Model reduction
und die zweite Methode of discrete time systems
van Lyapunov, Regelungsrechnik (1965) 592-596. using the Schwarz canonical form, Electron. Lett.
16
(1980)782-783. [4]
E. Badreddin,
A time-scale
method
for model
reduction
of discrete-time
systems,
Ph.D.
thesis
No. 7207,
ETH-Zurich
(1982).
117
Letters
4 (1984)
1099117
April
1984
A second multivariable normal form for model reduction of discrete-time systems E. BADREDDIN Swiss Federal Switzerland Received Revised
and M. MANSOUR
Institute
of TechnologV
(ETH-
Ziirich),
Institute
of Automatic
Control
and
Industrial
Electronics,
CH -8092
Ziirich,
7 May 1983 12 December 1983
Starting by the block-controllability form, a normal form for multivariable discrete-time systems is developed. The construction of this normal form is made straight-forward by the introduction of the matrix Schur-Cohn table. It shows a one-to-one (scalar-to-matrix) correspondence with the single-input Mansour form [2]. Examples for the construction of the proposed form as well as its application to model reduction are also given. Keywords:
Linear
multivariable
systems,
Canonical
forms,
Discrete-time
systems,
Model
reduction.
1. Introduction In a previous paper [l], a multivariable normal form for discrete-time systems was developed starting by the Luenberger first form. That form was specially developed for model reduction. Aspects like simple computation, structural matching and structure preservation when applying the reduction method described in [3] were the main concern in the development of that first form. Here, we shall introduce a second form which, also, can be considered a direct extension of the Mansour form [2]. Whilst the first form in [l] shows a block-to-block correspondence with the Luenberger first form, the form proposed here shows a one-to-one, i.e. scalar-to-matrix correspondence with the Mansour form [2] for single-input systems. This is achieved by first transforming the Luenberger first form to the block-controllability form, by means of elementary similarity transformations, and then applying a generalized matrix Schur-Cohn table. The proposed form is then constructed in the same manner as the single-input case. The resulting form possesses similar structural properties to those of the first form and is also suitable for model reduction. For the full comprehension and appreciation of the developments presented in this paper, the reader is advised to review the previous paper [l] especially for the construction of the scalar Schur-Cohn table and for the definition of the Luenberger first form. 2. The matrix Schur-Cohn If the controllable form (A,, 4)
A,=
0167-6911/84/$3.00
0 z,
0 0
0
1,
b
...
table and the second multivariable normal form
n-th order linear system with m inputs can be described by the block controllability
__. 0 .. . 0
-A, -A,-, 7
0 . z,
0 1984, Elsevier
-A,
Science
Publishers
B,=
zm 0 :
(1)
b B.V. (North-Holland)
109
Volume
4, Number
2
SYSTEMS
& CONTROL
LETTERS
April
where 1, is the m x m unity matrix, 0 is the m x m zero matrix, A, through A, are q = n/m (integer ) 0), then it can be shown (see [4] for the proof ‘) that F, = T,A,T,-‘,
m x m
1984
matrices and
G, = T,B,
where G, = B,, -4
(I-e:)
-0, F,=
(z-e;)
- 44
0
:
9
-e,-, _ -0,
-e,_,e,
.. .
...
-eqe,
.. .
...
(2)
(-;-I) -e,e,-,
T,= [G,,F,G, ,...,
(3)
The elements 8,, &,. _. ,8, are m x m matrices obtained from the matrix Schur-Cohn table which can be constructed e.g. for q = 3 as shown in Table 1. If the diagonal blocks in the Luenberger first form are of the same size q, it can be transformed into the block controllability form by elementary similarity transformations [4]. Now, if the size of the diagonal blocks of the Luenberger first form is not the same for all blocks, which is definitely the case if the system order is not a multiple of the number of inputs, some modifications will be necessary to obtain the multivariable normal form by means of the matrix Schur-Cohn table. We shall extend every block with a size smaller than q by adding the proper number of zero rows and columns such that the structure of the block is maintained. This operation corresponds to adding a series of unity-delay blocks at the input port of the subsystem represented by that particular block which needs extension. Since the Luenberger first’form represents a number of subsystems connected in one direction only, we may consider any two subsystems, say S, and S,, and let the feed forward paths from S, to S, be denoted by a vector /? of the proper dimension. Now, suppose that S, is to be extended. Because of the newly inserted unity-delay blocks at its input port, the output of S, reaching S, through p is no longer in ‘synchronization’ with S2’s own input. Actually, the sequences coming from S, are reaching S, as many ‘units’ too late as have been inserted at the input of S,. This delay is compensated for by shifting the entry points of the P-ports in S, ‘forwards’ as many ‘units’ as required.
Table
1 A3 I (A2
normalized: ==a
A2 Al
- AlA,)
(A, - A,A,)(I A 2.2
- A:)-’
I
(A,,,
normalized: ==a
110
A3
0, = A,-,
(A, - A,A,)
(1-A:) I
42
I
4.2
A
(I- A<.,)-’
’ The proof is based on the relation controllability assumption.
I
A2
(A,-A,A,)(I-A:)-’
- 4,z4,,)
(A,,, - A,.2A2,2)(1 A 1.1
A,
between
02
= A,
= A 2.2 =(A,-A,A,)(I-A;)-’
2.2
4.2)
I I
A, and 0, from
8, = A,., = A,,,(1
the matrix
Schur-Cohn
+ ,42,2)-l table.
The regularity
of T is guaranteed
by the
Volume
4, Number
2
SYSTEMS
SK CONTROL
April
LE’ITERS
1984
If S, is to be extended, the P-ports will be shifted ‘backwards’ as many ‘units’ as have been added at its input port. The algorithm will be illustrated by an example that should also demonstrate the way in which it could be generalized. Example 1. Consider the linear system in the Luenberger first form,
By means of elementary similarity transformations
0 1 0 0 1 -P3
-a32
1
0
0
-a22
0
1
0
--a11
0
1
-&
0 0 0 - a2, A=
F 0
-&
0 B=
0 -a12
it can be transformed 1
0
0 0
1 0
0 0
0 0
to (A, B),
(4)
The block diagram representation of the linear discrete time system x( k + 1) = Ax(k) + Bu( k) is shown in Figure 1. Comparing (4) with (1) we notice that A, in (4) is only a 1 X 2 matrix. We shall extend the matrix A, by zero elements, the matrix A by a row and a column and the matrix B by a row with permutation as follows: 0000 0 0
0
0
0 -yj
-a32
0
0 1
1 0
1 0
0 1
0 0
0 0
-a21 -yz
-a22
0 0
0 0
0 0
0 0
1 0
0 1
-a,, -y,
0 -aI2
0 0
0 0
0
(5)
Notice that the elements yi, y2 y3 replaced &, p2, p3. This must be done since the extension of A, by an upper zero row means that a delay block (represented by z-’ in the block diagram) has been connected to the input of the first subsystem. This means that sequences of x2(t) and x4(k) of the first subsystem will
Fig.
1. 111
Volume
4, Number
2
SYSTEMS
& CONTROL
LETTERS
April
1984
appear ‘one step’ later and are denoted by iZ(k) and Z4(k). To compensate for this delay on the 2nd subsystem, the P-ports should be advanced by one delay block. These block diagram modifications are shown in Figure 2. The resulting dynamic equations for x5, x3, x, are Xl(k + 1) = -%*(-a)
-P&t(k))
= -h%(k) x3(k + 1) =x,(k) =x,(k)
-a,,x,(k) -%(x,(k)& -Y*%(k)
+ u1 = %P,-f’,(~>
-U,,%(k)
+ u,(k)
+4(k), - w4)
-P,%(k)
= x,(k)
-(P,
- ~,,P,F4(~)
-%*x,(k)
-%,X,(k),
x,(~+1)=x,(~)-PZ~-l(~)-~,2(Xs(~)-P,~.4(~))=X3(~)-(P2-~lZP1)~:4(~)-~,2xS(~) =X3(k)-Y1~4(k)-ulZX5(k) and
In general, if p is the dimension of the 1st subsystem and v is the dimension of the 2nd subsystem, and v > IL, and if the initial coupling elements are represented by the v-dimensional vector @, then the 1st subsystem will have to be expanded (Y - p) times to match the size of the 2nd subsystem. The final coupling vector y can be computed as follows: y = A:“,-“‘fi.
16)
Now, in Example 1 the matrix Schur-Cohn
table will be applied with
to give
and the resulting multivariable
normal form will be
(7)
Fig. 2. 112
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Remark. Noting that [b,, b,, Ab,, Ab,, A’b,] = I, the transformation F2g,] where b,, g, are the i-th column of B and G respectively.
April
1984
matrix will be T = [g,, g,, Fg,, Fgz,
At this point, we still ought to show that the pairs {A, B} and {F, G} are similar with the transformation matrix T. The proof is, however, quite lengthy and is therefore omitted here. Interested readers may refer to [4]. ’ To explain the algorithm further, we consider the 3-inputs case with a different block structure. Example 2. For the pair (A,,, B,} in the Luenberger first form
A,,=
B,, =
By means of elementary similarity transformations,
0 0 0 1 0
0 1 0 0 0
1 0 0 0 0
we obtain
(8)
Then we extend {A, B} to the block-companion
j-C
Fig.
-00 0
00 0
00 0
-id -p2
1 0 0
0 1 0
0 0 1
-c -p3 -&
-d,0 -a2 -d, --a1
descrir
00 1
I 00 B=
-1 {A, B} as follows: 01
01 1 (9)
0 -a,
3.
’ The proof is based, in principle, on constructing an extended transformation matrix, T, from the extended pair { F, G}. Applying r on the extended state vector, X, and omitting the dispensable components, will result in omitting the associated rows and columns in -T, F and G and restores the original dimension. To complete the proof is then a matter of algebraic manipulation and induction. 113
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The-- block diagram of (8) is shown in Figure 3 by solid lines and the dotted lines indicate the extension to { A, B} and the shifting of the feed-forward paths. After shifting, the resulting equivalent coupling elements will be
Since the a-path has not been shifted, {(Ye, (Y,} remain unchanged. Then, we may write
A2-[siI g. A,=[iI1:g. Applying
the matrix Schur-Cohn 0
table with A,, A,, we get
1
0 7 A 23 A 8, =A,(Z+
A,)-’
=
(ii -An&) ,(S,-A,,~,)-~,(~Y,-A,,CU,)/(~+A~~)
where
A,,=c, A,,=&,
A,,=&, 2
0
0
A
0
(;-A,d,(l+A,,)
A,,
1
A,,=&, A,,=a,, 2
and the resulting normal form {F, G} will be
where E is the elimination matrix [0 Z,lT and Z2 is the (2 X 2) unity matrix. Since [b2, b,, b,, Ab,, Ab,] = I, then the transformation matrix will be T = [g,, g,, g,, Fg2, Fg,]. Up till now we have considered the case where the size of the first block in the lower-block triangular version of the Luenberger first form is smaller than the second and the size of the second block is smaller than the third, etc. But what happens when the size of one block is larger than the next one? The answer to this question will be given by the next example. Example 3.
A,=
rO 1 0
0
-a3
0 1
-a2 -a,
0
cl 0
-P2
-0 114
-P,
-I7
B, =
0 0 0 1 0-1
1 0 0. 0 0
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By means of elementary similarity transformations,
B=
A=
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the pair {A,, I?, } could be transformed
0
1
0 1
0 0.
0 0
0 0
1984
into {A, B}.
The ‘extended’ matrices A, B will be 0 010 0000
O/-a,
0
0
0
1000-a,
0
1
0
0
0
-y2
-d,
0010-a, 0 0 0 t
1
-y,
-d,
0
Figure 4 shows the block diagram of the system {A, B} in solid lines and the extended together with the shifting of the /S-ports by one step ‘backward’ in dotted lines. The ‘backward’ shifting of the P-ports is explained by keeping the ‘synchronization’ inputs to the subsystem. But since the control input U, has been delayed by one unit due step to achieve block-size matching, the feed forward paths which are the b-ports, should by one unit, i.e. shifted one step ‘backwards’. The equivalent coupling elements after the shifting are
The elements of the matrix Schur-Cohn
A3=[;3 ;],
A2=[;;
-system {A, B} between all the to the extension be also delayed
table in this case are
%].A,=[;:;,I
e,=A,,
8,=(A,-A,A3)(I-A;)-‘=
1
l/(1
1 2
0
-d;)
0
‘I9
1
Fig. 4. 115
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0
A,, e2=
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-A$)
(~2-A31~1)/(1
I’
A22
A,,
0
Y,(~+A,,A,,)-Y,(A,~+A~,)
A
B,=(A,-A~A,)[(I-&)+(A,-A,A,)]-‘=
, 12
(1 -&)(l
+A,,)
Then the matrix F is built as follows:
F=
(I-0:)
-4 -e,
0.e
I -eT-e3
1 0I ’
(I-ef)+e
- e,e, - eT ae3e1
G=B.
- eTe3e2. e
Combinations of ‘forward’ and ‘backward’ shifting might become necessary in some cases, and can be treated in the same manner as illustrated by the previous examples,
3. Reduction of the second normal form Example 4. Consider the pair { F, G} of Example 1, -4,
0
-6 F=
(1 ‘A:,) -W,,
-A,2
-A,,
0
+A,,)
-4421
0
0
0
1
(l-42)
0
1
0
0
0
0 0
0. 0
0
0
-6,
-A22
-(A&
+
Ad21
-442,
-83
-A32
- (AA4
+ A,,&)
-44x
G=
(1 -A;,) -Ad-‘,,
Let the output matrix be
Al222
h4323
h4424
I
h 25 . h,
Substituting the quasi-steady-state approximation 4th-order reduced model described by
of x5 in the rest of the dynamic equations, we obtain the
0 (1
-A:,)
0 -Ad22
(hn-hl)
116
(h,d%~,+A,,h,))
(h,,-414)
(4,
(h,, -4,hz)
-(bh;
+ AA))
I
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where
h = A;2+A3* 22 1 +A22432
h, = ’
h, =
~2=P2-~22Pb
4 1 +A22432
A 1 + AZA,,
h2 = ’
h
15’
h,=
4 1 + A,,A,, A 1 +A;2A,,
h
25’
h
“’
As for the single-input case [3,4], the reduced model will preserve stability properties and the steady-state step response.of the original system. The reduced model approximates, in general, the ‘slow’ response of the system [4].
4. Conclusions and final remarks A multivariable normal form for discrete-time systems, as an extension of the Mansour-form [2] for the single-input case, has been developed. This form is called the second multivariable normal form to be distinguished from the first form presented iA an earlier paper [l]. Starting by the block-controllability form, a generalized matrix Schur-Cohn table is constructed, resulting in coefficients 13,,i = 1, 2,. . . , 4, where q = n/m = integer > 0, m is the number of inputs and n is the system order. 8, are (m X m) matrices rather than scalars as in the single-input case. The proposed second form is then constructed in the same manner as the single-input form, but using 8, instead of the scalar Schur-Cohn coefficients 6, [l]. When the system order is not a multiple of the number of inputs, i.e. q # integer, the system matrix is extended by zero rows/columns which correspond to the addition of unity-delay elements. The additional delay is compensated for by ‘forward’ and/or ‘backward’ shifting of the coupling paths. After constructing the normal form for the extended system, proper rows and columns are omitted resulting in the final form for the unextended system matrix. This normal form will always exist for all controllable {F, G} and is unique once the Luenberger first form is given. The construction of the Luenberger first form is, however, dependent on the order in which the transformation basis is chosen [l]. The second normal form discussed here avoids separately computing the coupling elements as in the first form [l]. Beside the one-to-one correspondence to the single-input case, the second form possesses similar structural properties to those of the first form, such as structure preservation when applying the model reduction method described in [3]. Beside model reduction, for which this form was specially developed, the proposed second form may also be useful in other applications. For example, if the block-controllability form is obtained from the Luenberger first form, the resulting 0, will be triangular matrices and the system will be stable iff the diagonal elements of 6, are less than one in magnitude.
References [l]
E. Badreddin
and M. Mansour,
A multivariable
normal-form
for model
reduction
of discrete-time
systems,
Sysfems
& Control
Left.
2(1983)271-285.
M. Mansour, [3] E: Badreddin
[2]
Die Stabilitat linearer Abtastsysteme and M. Mansour, Model reduction
und die zweite Methode of discrete time systems
van Lyapunov, Regelungsrechnik (1965) 592-596. using the Schwarz canonical form, Electron. Lett.
16
(1980)782-783. [4]
E. Badreddin,
A time-scale
method
for model
reduction
of discrete-time
systems,
Ph.D.
thesis
No. 7207,
ETH-Zurich
(1982).
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