- Email: [email protected]
On Equivalence at Infinity and Row-By-Row Decoupling for Linear Systems With Delays
Copyright © IFAC Linear Time Delay Systems, Grenoble, France, 1998
ON EQUIVALENCE AT INFINITY AND ROW-BY-ROW DECOUPLING FOR LINEAR SYSTEMS WITH DELAYS Rabah RABAH
1
lnstitut de Recherche en Cybernetique de Nantes, UMR 6597 1, rue de la Noe, BP 92 101 44321 Nantes Cedex 03, France E-mail: [email protected]
Abstract: For linear systems with delays the problem of non interacting control (rowby-row decoupling problem) is considered. Using the notion of equivalence at infinity, a characterization of the solution by a compensator design is given. The solution is non-anticipative. A easily checkable necessary condition is established. Connections with other conditions are discussed. Copyright © 1998 IFAC Keywords: Linear delay systems, structure at infinity, row-by-row decoupling.
1. INTRODUCTION
terns the structure at infinity, as a canonical form, cannot be defined in the general case. When the structure at infinity is well defined for infinite dimensional systems, then one has a good and simple characterization for the solvability of some control problems. In this paper we develop this approach using the notion of equivalent systems at infinity without requiring a precise canonical form at infinity. This allows, via a simple calculus, to describe the solvability of some control problems. However the problem of the definition of a canonical form is still open. A weak concept of canonical form was introduced in [6] and used recently in [11] to characterize the problem of disturbance rejection via generalized state feedback. This question is not developed here for row-by-row decoupling problem. We limit ourself to compensator design. The realization of this compensator need further investigations.
The geometric approach developed by several authors (see [2], [5], [8], [7], [13] and the references there) is an attempt to extend the concept of (A, B)-invariant subspaces to infinite dimensional systems. Since, for a large class of systems, this extension is not possible, several problems cannot be characterized using this approach. Recently, an extension of the geometric approach based on Hautus' description of (A, B)-invariant subspaces has been developed in [10]. The characterizations of row-by-row decoupling, disturbance rejection, model matching via compensator design was given. The results look like finite dimensional results but the problem of the realizability of compensators cannot always be solved in the general cases. For some systems, conditions are given to describe solutions in the state space approach. Another important tool used for finite dimensional systems is the structure at infinity which allows to describe the behavior of the system at t = 0 via the behavior of the transfer function matrix at infinity. For infinite dimensional sys-
2. STRUCTURE AT INFINITY The geometric conditions given in [10] may be formulated in the structural approach for some infinite dimensional systems, for example for some
I Also with Ecole des Mines de Nantes. BP 20722.4, rue Alfred Kastler. 44307. Nantes Cedex 3.
19
classes of delay systems. In this case the conditions are easily checkable and the feedback in the original state space formulation is computable via the well known techniques of finite dimensional systems (see [9]). This approach was used in [6] for the model matching and the disturbance rejection problems. However, the canonical form defined there is not sufficient to insure non-anticipativity of the control laws (see also [12]) for the rowby-row decoupling problem). In [9], this problem was discussed and a strong structure at infinity is defined, but the canonical form introduced there is not defined for all systems. In order to introduce the notion of structure at infinity, we first need the notion of properness for infinite dimensional systems.
and ni,j
= 1, ... , k,
The obtained matrix is called the strong canonical form of the system at infinity. Conditions for the existence of this strong canonical form are given in [9]. They are expressed in term of the structures at infinity of matrices 'Tj (s): the orders of zeros at infinity are increasing with J.
Let us consider two illustrative examples. The first one is the following:
T(s, e- S) = [S-l
s-2 e-s].
A simple calculation gives:
Definition 1. An analytiefunction I(s) for Res 2: So is called strong proper if for Re s -+ 00 there exists a fini te limit of 1(s). The function is strictly strong proper if sl(s) has a finite limit. A matrix B(s) is strong biproper if it is strong proper, invertible and B- 1 (s) is also strong proper.
And the matrix
[~
1e -S]
S-1
is clearly strong biproper. Let now T(s, e- S ) be given by:
Consider the delay systems
i(t) = Aoz(t) { y(t) = Coz(t)
~ ni,j+1, i
T(s, e- S) = [s-2
+ A 1 z(t -
1) + Bou(t) (1)
S-l e -S].
Then it is easy to show that if B(s, e- S ) is a strong biproper matrix such that
T(s,e-S)B(s,e- S) = [t(s,e- S)
where z(t) E ~n, y(t) E ~P and u(t) E ~m. The transfer function matrix of the system is
with strictly strong proper t (s, e- S) first row of B- 1 (s,e- s ) is
¥
0] 0 and the
[a(s,e- S) b(s,e- S)], In being the identity in decomposed as follows
jRn
T(s, e- S ) may be
then a simple calculation gives:
b(s, e- S) = sa(s, e-")e- s . It is clear that a(s, e- S) and b(s, e- S) are proper or strictly strong proper and they cannot be both strictly strong proper because of strong biproperness of B(s,e- S). Suppose that a(s, e- S ) is just strong proper, then
00
T(s,e- S)
= LTj(s)e- jS ,
(2)
j=O
where
b(s, e-') --'a(s.e----'-----'S)- = se -s
Definition 2. We say that the structure at infinity of the system (1) is well defined if there exist strong biproper matrices B 1 (s, e- S ) and B 2 (s, e- S ) such that
must be strong proper. But it is not possible, as the limit of se- S when Re s -+ 00 does not exist. Suppose now that a(s, e-') is strictly strong proper. We have
BI(s, e-S)T(s, e- S)B2(s, e- S) = .6.(s, e- S) = .6. o(s)
0
o
.6.ds)e- S
o o
o o
0 0
a(s,e-')
= diag [s-n,."
S
and eSb(s, e-') is strong proper.We can write
0 0
e'b(s,e- S) = ,B+w(s,e- S), with constant,B and strictly strong proper w(s, e- S ). This gives
b(s.e- S) = e-·'(13+;..;(s,e- S)),
where
.6. i (s)
= s-l e b(s,e-')
which means that b(s, e-') is strictly strong proper.
.... S-n,),]
20
T(s, e-S)K(s, e- S) = diag{ Tds, e- S), ... , Tm(S, e- S)},
In both cases B(s, e- S ) is not strong biproper. Hence, in this example, it is not possible to reduce the matrix T(s, e- S ) by strong biproper operations. The canonical form defined in [9] is the strong version of the weak canonical form given in [6] where the properness is defined when s -+ 00, with sE lR (see Section 4). The strong canonical form, when it exists, allows to check the conditions of solvability of the control problems considered in [9). As shown in the examples, it is not clear how to define a canonical form which could exist for all delay systems in order to solve control problems using static state feedback without anticipation or realizable compensators. Because of this difficulty, we shall use structural equivalence at infinity without needing explicitly the canonical form. It seems to be the main tool under consideration for the the structure at infinity for classical systems. The first application considered here is the row-by-row decoupling problem with disturbance rejection.
that is: each row is affected by one and only one input. It is clear that it means that T(s,e- S ) is equivalent at infinity to a diagonal strictly strong proper matrix. For the finite dimensional case, the functions Tj(s,e- S) may be chosen as Tj(s,e- S) = s-n,.
In [10) a geometric condition which characterize this control problem was given. However this condition is not easy to verify. The main result of this section is the following one. The verification of the condition given here is easer as we shall see in the following section.
Theorem 5. Let t,(s, e- S) denote the i-th row, i = 1, ... ,m of T(s,e- S). The system (1) with m = p is row-by-row decouplable if and only if: i) T( s, e- S )
Definition 3. Let T(s, e- S) and 6(s, e- S ) be transfer function matrices. We say that T(s, e- S ) and 0(s,e- S) are equivalent at infinity iff there exist strong biproper matrices Bds, e- S ) and B 2 (s, e- S ) such that:
~
ii) tj (s, e- S)
°
~
diag{ TI( s, e- s ), ... , T m (s, e- S ) }
Tj (s, e- S)
[0
°],
with Tj ::f In other words the global behavior at infinity is equivalent to the union of the behavior of each row.
Proof. If the system is decouplable then The strong biproper matrices B 1 (s, e- ) and B 2 (s, e- S) consist in fact in elementary operations on rows and columns: permutations, scaling and addition to a row (column) of another row (column) multiplied by a proper functions. These operations do not modify the behavior of the transfer function matrix at infinity (when Re s -+ (0). S
T(s, e-S)K(s, e- S) = diag{ Tds, e- S), ... , Tm(S, e-')}, which means that condition i) is verified. On the other hand, 0],
The relation introduced in Definition 3 is an equivalence relation. i. e. if we note this relation by"', we have: T(s, e- S) ~ T(s, e- S );
i. e. the condition ii) also holds. Suppose now that the conditions of the theorem are verified. Using column reducing procedures by right strong biproper operations, we get:
T(s, e- S) ~ 0(s, e- S) ::} 6(s, e- S ) ~ T(s, e- S) T(s,e- S) ~ 0(s,e- s ),and0(s,e- S) ~3(s,e-S)::} T(s, e- S) ~ 3(s, e- S).
011
T(s,e
-s
)B(s,e
_.
021
)=
:
[
Om1
The characterization of the canonical form corresponding to this equivalence relation is still open.
where the argument (s. e- S ) is omitted in Ojj. This is possible, otherwise by left elementary operations one could not obtain a diagonal form as specified by the condition i) of the Theorem. Moreover Ojj(s, e- S ) ::f 0. We assume that the column procedure is complete, i.e. the column j cannot be reduced by a column jf with j' > j. This means that if Ojj(s, e- S) ::f 0, then Ojj may not be written III the form
3. THE ROW-BY-ROW DECOUPLING PROBLEM Let us first define the considered problem.
Definition 4. The system (1) with m = pis rowby-row decouplable iff there exists a biproper compensator K(s,e- S ) such that
21
with strictly proper functions q;j(s, e-') "# 0 and Pij(S, e-·). Indeed, ifit is not the case, the column reducing procedure would give Pij(S, e-') in place of l1ij (s, e-') by multiplying the j'-th column by -qij(S, e-') and adding this column to the j-th column. As we assume that the column reducing procedure is complete, l1ij(s, e-') is not reducible. Hence, l1ij (s, e-') is either 0 or cannot be reduced by another column operation. Then for each row t;(s, e-') of the transfer matrix function we have
This form was used in [11] to solve the disturbance rejection problem in a general form. The designed precompensator is weak proper but it may be realized by generalized static state feedback. That feedback use the derivative of the delayed disturbance. Here this form gives a necessary condition of row-by-row decoupling problem. We have the following result. Theorem 7. The row-by-row decoupling problem is solvable only if
t;(S, e-')B(s, e-') = [11;ds,e-')
l1;i(s,e-')
.6.(s, e- S) = [JI{s,:e-') ]
0] .
0
Jm(s, e-')
This gives
t;(s,e- S) ~ [l1iJts,e-')
where o;(s, e- ) are the weak canonical forms of the rows of the matrix T(s, e- S ) (with reordered columns) and ~(s,e-') and denotes the weak canonical form at infinity for T(s,e- S). S
0] .
and l1ij, j < i cannot be reduced by column operations. According to the condition iij, this means that l1ij = 0, j < i. Then,
Proof. As the system is decouplable, then
T(s,e-S)K(s,e-') = diag{ Tl (s, e-')' ... , Tm(S, e- S)},
T(s, e-S)K(s, e- S) = diag{ Tl (s, e- S), ... , Tm(S, e- S )},
with T;(S, e- S) = l1i;(s, e- S) for i = 1, ... , m. This means that the system is decouplable by a strong biproper compensator. •
and I«s, e- S ) is biproper which implies that this matrix is weak biproper. The structure of the transfer matrix function T(s, e- S ) (see the decomposition (2)) implies that the functions Ti(s,e- S) are of the same type as the entries of T(s, e-·). Moreover, for each row of the transfer matrix function, one has
4. PRACTICAL COMPUTATION First let us note that the use of the weak structure at infinity (cL [9]) gives a necessary condition. This structure is defined for all delay systems.
t;(s, e-')K(s, e-') =
Definition 6. A complex valued function Its) is called weak proper if lim /( s) is finite when s E lR tends to 00. It is called strictly weak proper if this limit is O. A matrix B(s) is weak biproper if it is a weak proper and its inverse is also weak proper.
ti(S, e-S)K(s, e-')B;{s, e- S) = [0 0 s-n'e-k,s 0
where
~(s,
0],
with
B i (s, e- S)
[6]): ~(s,
0],
and then
For the delay system (1) there exist weak biproper matrices BJts, e- S) and B 2(s, e- S) such that (see
B I (s, e-S)T(s, e-')B2(S, e- S) =
0 T;(s,e- S) 0
[0
e- S),
= diag{ 1, ... , 1, 1 +l1i (s, e- S), 1, ... ,I}.
with strictly weak proper functions l1i(s, e- S). Then
e-' is the same matrix as in Definition
2.
T(s, e-S)K(s, e-S)B(s, e-') = diag{s-",e-k,., ... , s-nme-k~s}.
This canonical form always exists. For the example given in Section 2:
T(s, e-')
= [s-2
s-le- s ],
where
we have
T(S,e-')[~ s~~S]=[s_2 and
[~ s~~']
B(s,e-')
0].
diag{l
=
+ l1I{s, e-
S
), ••• ,
1 + I1 m (s, e-·)}.
which is a weak biproper matrix. This means that the condition of the Theorem is satisfied. •
is clearly weak biproper.
22
Corollary 8. If the condition of the theorem 7 is satisfied and the corresponding structures at infinity are obtained by (strong) biproper operations, then the system is decouplable.
6. REFERENCES [1] D. Brethe (1997), Contribution a l'etude de la stabilisation des systemes lineaires avec retards, PhD Thesis, Nantes. [2] R. F. Curtain (1986), Invariance concepts in infinite dimension, SIAM J. on Control and Optimization, 24, pp.1009-103l. [3] M.L.J. Hautus (1975), The formal Laplace transform for smooth linear systems, In: Proceeding of the International Symposium on Mathematical Systems Theory. Udine, Italy, June 1975, Lecture Notes in Economics and Mathematical Systems. 131, pp.29-47, Springer-Verlag, 1975. [4] J.-J. Loiseau, D. Brethe (1998), Algebraic tools for the control and stabilization of time-delay systems, This Workshop. [5] M. Malabre, R. Rabah (1990), On infinite zeros for infinite dimensional systems, In: Progress in Systems and Control Theory 3, Realization and Modelling in System Theory, Volume 1, Birkhauser, Boston, pp. 199206. [6] M. Malabre, R. Rabah (1993), Structure at infinity, model matching and disturbance rejection for linear systems with delays. f{ ybernetika, 29, No 5, pp. 485-498. [7] N. Otsuka, H. Inaba, T. Oide (1991), Decoupling by state feedback in infinite dimensional systems, IMA 1. of Mathematical Control and Information, 7, pp. 125-14l. [8] L. Pandolfi (1986), Disturbance decoupling and invariant subspaces for delay systems, Applied Mathematzcs and Optimization, 14, pp. 55-72. [9] R. Rabah, R. Malabre (1996), Structure at infinity for delay systems revisited, IMACS and IEEE-SMC Multiconference CESA'96, Symposium on Modelling, Analysis and Simulation, Lille-France, July 9-12, pp. 87-90. [10] R. Rabah, R. Malabre (1997), A note on decoupling for linear infinite dimensional systems. In: Proc. 4-th IFAC Conference on System Structure and Control, Bucharest (Romania), pp. 78-83. [11] R. Rabah, R. Malabre (1998), On the structure at infinity of linear delay systems with application to decoupling problems, In: Proc. of the 6th IEEE Mediterranean Conf. on Control and Systems, 9-11 June 1998, Alghero, Italia.(to appear). [12] O. Sename, R. Rabah, J.F. Lafay (1995), Decoupling without prediction of linear systems with delays: a structural approach. Syst. Contr. Letters, 25, pp 387-395. [13] H. Zwart (1989), Geometric theory for infinite dimensional systems. Lecture Notes in Control and Information Sciences, 115.
Proof. The condition of the theorem and the strong biproperness of the operations imply that the conditions of the Theorem 5 are satisfied and then the system is decouplable. •
Let us consider two examples. Let the system be given by: T(
s, e
-s
= [s-20e -S
S-Ie-S] s-2'
Then a simple calculation (see Section 2) shows that there is no strong biproper matrix B(s, e- S ) such that tJ(s,e-S)B(s,e- S) [rds,e-S) 0].
=
This system is not decouplable. If the system is given by: T(s, e- S )
= [s-20e -S
Then
and tJ(s, e- S) ~ [s-2 e -S t2(s,e-S)~[0
0],
s-2].
This system is decouplablea and the precompensator is ." -s S - I e -S]-I I\(s,e )= 0 1 '
[1
which is clearly strong biproper. In the general case, the good equivalent form at infinity is not easy to check. This problem is under investigation. Some recent results by Brethe and Loiseau (see the PhD thesis by Brethe and [4]) give a good framework in order to develop some algebraic tools which certainly may be used in order to compute equivalent forms and extended canonical forms at infinity.
5. CONCLUSION For delay systems, the equivalence at infinity may be used in place of the structure at infinity which need a canonical form. This notion is used to characterize the regular row-by-row decoupling problem. For other control problems this notion may be used also (disturbance rejection, model matching). It is expected that this notion may as well be extended to more general infinite dimensional systems. The problem of the good definition of a canonical form corresponding to this equivalence relation is still open.
23
ON EQUIVALENCE AT INFINITY AND ROW-BY-ROW DECOUPLING FOR LINEAR SYSTEMS WITH DELAYS Rabah RABAH
1
lnstitut de Recherche en Cybernetique de Nantes, UMR 6597 1, rue de la Noe, BP 92 101 44321 Nantes Cedex 03, France E-mail: [email protected]
Abstract: For linear systems with delays the problem of non interacting control (rowby-row decoupling problem) is considered. Using the notion of equivalence at infinity, a characterization of the solution by a compensator design is given. The solution is non-anticipative. A easily checkable necessary condition is established. Connections with other conditions are discussed. Copyright © 1998 IFAC Keywords: Linear delay systems, structure at infinity, row-by-row decoupling.
1. INTRODUCTION
terns the structure at infinity, as a canonical form, cannot be defined in the general case. When the structure at infinity is well defined for infinite dimensional systems, then one has a good and simple characterization for the solvability of some control problems. In this paper we develop this approach using the notion of equivalent systems at infinity without requiring a precise canonical form at infinity. This allows, via a simple calculus, to describe the solvability of some control problems. However the problem of the definition of a canonical form is still open. A weak concept of canonical form was introduced in [6] and used recently in [11] to characterize the problem of disturbance rejection via generalized state feedback. This question is not developed here for row-by-row decoupling problem. We limit ourself to compensator design. The realization of this compensator need further investigations.
The geometric approach developed by several authors (see [2], [5], [8], [7], [13] and the references there) is an attempt to extend the concept of (A, B)-invariant subspaces to infinite dimensional systems. Since, for a large class of systems, this extension is not possible, several problems cannot be characterized using this approach. Recently, an extension of the geometric approach based on Hautus' description of (A, B)-invariant subspaces has been developed in [10]. The characterizations of row-by-row decoupling, disturbance rejection, model matching via compensator design was given. The results look like finite dimensional results but the problem of the realizability of compensators cannot always be solved in the general cases. For some systems, conditions are given to describe solutions in the state space approach. Another important tool used for finite dimensional systems is the structure at infinity which allows to describe the behavior of the system at t = 0 via the behavior of the transfer function matrix at infinity. For infinite dimensional sys-
2. STRUCTURE AT INFINITY The geometric conditions given in [10] may be formulated in the structural approach for some infinite dimensional systems, for example for some
I Also with Ecole des Mines de Nantes. BP 20722.4, rue Alfred Kastler. 44307. Nantes Cedex 3.
19
classes of delay systems. In this case the conditions are easily checkable and the feedback in the original state space formulation is computable via the well known techniques of finite dimensional systems (see [9]). This approach was used in [6] for the model matching and the disturbance rejection problems. However, the canonical form defined there is not sufficient to insure non-anticipativity of the control laws (see also [12]) for the rowby-row decoupling problem). In [9], this problem was discussed and a strong structure at infinity is defined, but the canonical form introduced there is not defined for all systems. In order to introduce the notion of structure at infinity, we first need the notion of properness for infinite dimensional systems.
and ni,j
= 1, ... , k,
The obtained matrix is called the strong canonical form of the system at infinity. Conditions for the existence of this strong canonical form are given in [9]. They are expressed in term of the structures at infinity of matrices 'Tj (s): the orders of zeros at infinity are increasing with J.
Let us consider two illustrative examples. The first one is the following:
T(s, e- S) = [S-l
s-2 e-s].
A simple calculation gives:
Definition 1. An analytiefunction I(s) for Res 2: So is called strong proper if for Re s -+ 00 there exists a fini te limit of 1(s). The function is strictly strong proper if sl(s) has a finite limit. A matrix B(s) is strong biproper if it is strong proper, invertible and B- 1 (s) is also strong proper.
And the matrix
[~
1e -S]
S-1
is clearly strong biproper. Let now T(s, e- S ) be given by:
Consider the delay systems
i(t) = Aoz(t) { y(t) = Coz(t)
~ ni,j+1, i
T(s, e- S) = [s-2
+ A 1 z(t -
1) + Bou(t) (1)
S-l e -S].
Then it is easy to show that if B(s, e- S ) is a strong biproper matrix such that
T(s,e-S)B(s,e- S) = [t(s,e- S)
where z(t) E ~n, y(t) E ~P and u(t) E ~m. The transfer function matrix of the system is
with strictly strong proper t (s, e- S) first row of B- 1 (s,e- s ) is
¥
0] 0 and the
[a(s,e- S) b(s,e- S)], In being the identity in decomposed as follows
jRn
T(s, e- S ) may be
then a simple calculation gives:
b(s, e- S) = sa(s, e-")e- s . It is clear that a(s, e- S) and b(s, e- S) are proper or strictly strong proper and they cannot be both strictly strong proper because of strong biproperness of B(s,e- S). Suppose that a(s, e- S ) is just strong proper, then
00
T(s,e- S)
= LTj(s)e- jS ,
(2)
j=O
where
b(s, e-') --'a(s.e----'-----'S)- = se -s
Definition 2. We say that the structure at infinity of the system (1) is well defined if there exist strong biproper matrices B 1 (s, e- S ) and B 2 (s, e- S ) such that
must be strong proper. But it is not possible, as the limit of se- S when Re s -+ 00 does not exist. Suppose now that a(s, e-') is strictly strong proper. We have
BI(s, e-S)T(s, e- S)B2(s, e- S) = .6.(s, e- S) = .6. o(s)
0
o
.6.ds)e- S
o o
o o
0 0
a(s,e-')
= diag [s-n,."
S
and eSb(s, e-') is strong proper.We can write
0 0
e'b(s,e- S) = ,B+w(s,e- S), with constant,B and strictly strong proper w(s, e- S ). This gives
b(s.e- S) = e-·'(13+;..;(s,e- S)),
where
.6. i (s)
= s-l e b(s,e-')
which means that b(s, e-') is strictly strong proper.
.... S-n,),]
20
T(s, e-S)K(s, e- S) = diag{ Tds, e- S), ... , Tm(S, e- S)},
In both cases B(s, e- S ) is not strong biproper. Hence, in this example, it is not possible to reduce the matrix T(s, e- S ) by strong biproper operations. The canonical form defined in [9] is the strong version of the weak canonical form given in [6] where the properness is defined when s -+ 00, with sE lR (see Section 4). The strong canonical form, when it exists, allows to check the conditions of solvability of the control problems considered in [9). As shown in the examples, it is not clear how to define a canonical form which could exist for all delay systems in order to solve control problems using static state feedback without anticipation or realizable compensators. Because of this difficulty, we shall use structural equivalence at infinity without needing explicitly the canonical form. It seems to be the main tool under consideration for the the structure at infinity for classical systems. The first application considered here is the row-by-row decoupling problem with disturbance rejection.
that is: each row is affected by one and only one input. It is clear that it means that T(s,e- S ) is equivalent at infinity to a diagonal strictly strong proper matrix. For the finite dimensional case, the functions Tj(s,e- S) may be chosen as Tj(s,e- S) = s-n,.
In [10) a geometric condition which characterize this control problem was given. However this condition is not easy to verify. The main result of this section is the following one. The verification of the condition given here is easer as we shall see in the following section.
Theorem 5. Let t,(s, e- S) denote the i-th row, i = 1, ... ,m of T(s,e- S). The system (1) with m = p is row-by-row decouplable if and only if: i) T( s, e- S )
Definition 3. Let T(s, e- S) and 6(s, e- S ) be transfer function matrices. We say that T(s, e- S ) and 0(s,e- S) are equivalent at infinity iff there exist strong biproper matrices Bds, e- S ) and B 2 (s, e- S ) such that:
~
ii) tj (s, e- S)
°
~
diag{ TI( s, e- s ), ... , T m (s, e- S ) }
Tj (s, e- S)
[0
°],
with Tj ::f In other words the global behavior at infinity is equivalent to the union of the behavior of each row.
Proof. If the system is decouplable then The strong biproper matrices B 1 (s, e- ) and B 2 (s, e- S) consist in fact in elementary operations on rows and columns: permutations, scaling and addition to a row (column) of another row (column) multiplied by a proper functions. These operations do not modify the behavior of the transfer function matrix at infinity (when Re s -+ (0). S
T(s, e-S)K(s, e- S) = diag{ Tds, e- S), ... , Tm(S, e-')}, which means that condition i) is verified. On the other hand, 0],
The relation introduced in Definition 3 is an equivalence relation. i. e. if we note this relation by"', we have: T(s, e- S) ~ T(s, e- S );
i. e. the condition ii) also holds. Suppose now that the conditions of the theorem are verified. Using column reducing procedures by right strong biproper operations, we get:
T(s, e- S) ~ 0(s, e- S) ::} 6(s, e- S ) ~ T(s, e- S) T(s,e- S) ~ 0(s,e- s ),and0(s,e- S) ~3(s,e-S)::} T(s, e- S) ~ 3(s, e- S).
011
T(s,e
-s
)B(s,e
_.
021
)=
:
[
Om1
The characterization of the canonical form corresponding to this equivalence relation is still open.
where the argument (s. e- S ) is omitted in Ojj. This is possible, otherwise by left elementary operations one could not obtain a diagonal form as specified by the condition i) of the Theorem. Moreover Ojj(s, e- S ) ::f 0. We assume that the column procedure is complete, i.e. the column j cannot be reduced by a column jf with j' > j. This means that if Ojj(s, e- S) ::f 0, then Ojj may not be written III the form
3. THE ROW-BY-ROW DECOUPLING PROBLEM Let us first define the considered problem.
Definition 4. The system (1) with m = pis rowby-row decouplable iff there exists a biproper compensator K(s,e- S ) such that
21
with strictly proper functions q;j(s, e-') "# 0 and Pij(S, e-·). Indeed, ifit is not the case, the column reducing procedure would give Pij(S, e-') in place of l1ij (s, e-') by multiplying the j'-th column by -qij(S, e-') and adding this column to the j-th column. As we assume that the column reducing procedure is complete, l1ij(s, e-') is not reducible. Hence, l1ij (s, e-') is either 0 or cannot be reduced by another column operation. Then for each row t;(s, e-') of the transfer matrix function we have
This form was used in [11] to solve the disturbance rejection problem in a general form. The designed precompensator is weak proper but it may be realized by generalized static state feedback. That feedback use the derivative of the delayed disturbance. Here this form gives a necessary condition of row-by-row decoupling problem. We have the following result. Theorem 7. The row-by-row decoupling problem is solvable only if
t;(S, e-')B(s, e-') = [11;ds,e-')
l1;i(s,e-')
.6.(s, e- S) = [JI{s,:e-') ]
0] .
0
Jm(s, e-')
This gives
t;(s,e- S) ~ [l1iJts,e-')
where o;(s, e- ) are the weak canonical forms of the rows of the matrix T(s, e- S ) (with reordered columns) and ~(s,e-') and denotes the weak canonical form at infinity for T(s,e- S). S
0] .
and l1ij, j < i cannot be reduced by column operations. According to the condition iij, this means that l1ij = 0, j < i. Then,
Proof. As the system is decouplable, then
T(s,e-S)K(s,e-') = diag{ Tl (s, e-')' ... , Tm(S, e- S)},
T(s, e-S)K(s, e- S) = diag{ Tl (s, e- S), ... , Tm(S, e- S )},
with T;(S, e- S) = l1i;(s, e- S) for i = 1, ... , m. This means that the system is decouplable by a strong biproper compensator. •
and I«s, e- S ) is biproper which implies that this matrix is weak biproper. The structure of the transfer matrix function T(s, e- S ) (see the decomposition (2)) implies that the functions Ti(s,e- S) are of the same type as the entries of T(s, e-·). Moreover, for each row of the transfer matrix function, one has
4. PRACTICAL COMPUTATION First let us note that the use of the weak structure at infinity (cL [9]) gives a necessary condition. This structure is defined for all delay systems.
t;(s, e-')K(s, e-') =
Definition 6. A complex valued function Its) is called weak proper if lim /( s) is finite when s E lR tends to 00. It is called strictly weak proper if this limit is O. A matrix B(s) is weak biproper if it is a weak proper and its inverse is also weak proper.
ti(S, e-S)K(s, e-')B;{s, e- S) = [0 0 s-n'e-k,s 0
where
~(s,
0],
with
B i (s, e- S)
[6]): ~(s,
0],
and then
For the delay system (1) there exist weak biproper matrices BJts, e- S) and B 2(s, e- S) such that (see
B I (s, e-S)T(s, e-')B2(S, e- S) =
0 T;(s,e- S) 0
[0
e- S),
= diag{ 1, ... , 1, 1 +l1i (s, e- S), 1, ... ,I}.
with strictly weak proper functions l1i(s, e- S). Then
e-' is the same matrix as in Definition
2.
T(s, e-S)K(s, e-S)B(s, e-') = diag{s-",e-k,., ... , s-nme-k~s}.
This canonical form always exists. For the example given in Section 2:
T(s, e-')
= [s-2
s-le- s ],
where
we have
T(S,e-')[~ s~~S]=[s_2 and
[~ s~~']
B(s,e-')
0].
diag{l
=
+ l1I{s, e-
S
), ••• ,
1 + I1 m (s, e-·)}.
which is a weak biproper matrix. This means that the condition of the Theorem is satisfied. •
is clearly weak biproper.
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Corollary 8. If the condition of the theorem 7 is satisfied and the corresponding structures at infinity are obtained by (strong) biproper operations, then the system is decouplable.
6. REFERENCES [1] D. Brethe (1997), Contribution a l'etude de la stabilisation des systemes lineaires avec retards, PhD Thesis, Nantes. [2] R. F. Curtain (1986), Invariance concepts in infinite dimension, SIAM J. on Control and Optimization, 24, pp.1009-103l. [3] M.L.J. Hautus (1975), The formal Laplace transform for smooth linear systems, In: Proceeding of the International Symposium on Mathematical Systems Theory. Udine, Italy, June 1975, Lecture Notes in Economics and Mathematical Systems. 131, pp.29-47, Springer-Verlag, 1975. [4] J.-J. Loiseau, D. Brethe (1998), Algebraic tools for the control and stabilization of time-delay systems, This Workshop. [5] M. Malabre, R. Rabah (1990), On infinite zeros for infinite dimensional systems, In: Progress in Systems and Control Theory 3, Realization and Modelling in System Theory, Volume 1, Birkhauser, Boston, pp. 199206. [6] M. Malabre, R. Rabah (1993), Structure at infinity, model matching and disturbance rejection for linear systems with delays. f{ ybernetika, 29, No 5, pp. 485-498. [7] N. Otsuka, H. Inaba, T. Oide (1991), Decoupling by state feedback in infinite dimensional systems, IMA 1. of Mathematical Control and Information, 7, pp. 125-14l. [8] L. Pandolfi (1986), Disturbance decoupling and invariant subspaces for delay systems, Applied Mathematzcs and Optimization, 14, pp. 55-72. [9] R. Rabah, R. Malabre (1996), Structure at infinity for delay systems revisited, IMACS and IEEE-SMC Multiconference CESA'96, Symposium on Modelling, Analysis and Simulation, Lille-France, July 9-12, pp. 87-90. [10] R. Rabah, R. Malabre (1997), A note on decoupling for linear infinite dimensional systems. In: Proc. 4-th IFAC Conference on System Structure and Control, Bucharest (Romania), pp. 78-83. [11] R. Rabah, R. Malabre (1998), On the structure at infinity of linear delay systems with application to decoupling problems, In: Proc. of the 6th IEEE Mediterranean Conf. on Control and Systems, 9-11 June 1998, Alghero, Italia.(to appear). [12] O. Sename, R. Rabah, J.F. Lafay (1995), Decoupling without prediction of linear systems with delays: a structural approach. Syst. Contr. Letters, 25, pp 387-395. [13] H. Zwart (1989), Geometric theory for infinite dimensional systems. Lecture Notes in Control and Information Sciences, 115.
Proof. The condition of the theorem and the strong biproperness of the operations imply that the conditions of the Theorem 5 are satisfied and then the system is decouplable. •
Let us consider two examples. Let the system be given by: T(
s, e
-s
= [s-20e -S
S-Ie-S] s-2'
Then a simple calculation (see Section 2) shows that there is no strong biproper matrix B(s, e- S ) such that tJ(s,e-S)B(s,e- S) [rds,e-S) 0].
=
This system is not decouplable. If the system is given by: T(s, e- S )
= [s-20e -S
Then
and tJ(s, e- S) ~ [s-2 e -S t2(s,e-S)~[0
0],
s-2].
This system is decouplablea and the precompensator is ." -s S - I e -S]-I I\(s,e )= 0 1 '
[1
which is clearly strong biproper. In the general case, the good equivalent form at infinity is not easy to check. This problem is under investigation. Some recent results by Brethe and Loiseau (see the PhD thesis by Brethe and [4]) give a good framework in order to develop some algebraic tools which certainly may be used in order to compute equivalent forms and extended canonical forms at infinity.
5. CONCLUSION For delay systems, the equivalence at infinity may be used in place of the structure at infinity which need a canonical form. This notion is used to characterize the regular row-by-row decoupling problem. For other control problems this notion may be used also (disturbance rejection, model matching). It is expected that this notion may as well be extended to more general infinite dimensional systems. The problem of the good definition of a canonical form corresponding to this equivalence relation is still open.
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