Dynamic Feedback Decoupling Problem for Delay-differential Systems via Systems over Rings

Dynamic Feedback Decoupling Problem for Delay-differentialSystems via Systems over Rings

G. Contea,*, A.M. Perdonb, A. Lombardoa

aDip. Elettronica e Automatica, UniversitaÂ di Ancona, Via Brecce Bianche, 60131 Ancona, ItalybDip. Matematica `̀ V. Volterra'',UniversitaÂ di Ancona, Via Brecce Bianche, 60131 Ancona, Italy

Abstract

Systems with coefficients in a ring are employed for studying the Decoupling Problem for delay-differential systems. Using

geometric methods, the existence of dynamic state feedback solutions is completely characterized. # 1998 IMACS/Elsevier

Science B.V.

Keywords: Delay differential systems; Systems over rings; Decoupling Problem; Geometric methods

1. Introduction

It is known (see [2,3]) that linear systems with coefficients in a ring can be used for studyingnoninteracting control problems which involve continuous-time, delay-differential systems. Basically,this fact relies on the possibility of exploiting a simple correspondence between the framework ofdelay-differential systems with a finite number of incommensurable delays and that of systems withcoefficients in a ring. Problems and related solutions can be transferred from one framework to theother, with the advantage of using finite dimensional algebra in the study of delay-differential system.

In this paper, we follow the approach mentioned above for studying Decoupling Problems for delay-differential systems. From a design point of view, the specific problem we consider is that of finding astate feedback which decomposes the input/output relation of a given system � into an assignednumber of input/output blocks, each one decoupled from the others, maintaining the outputcontrollability properties of the original system.

In Section 2 we state formally the problem, called Block Decoupling Problem, for systems withcoefficients in a ring. A characterization of the existence of solutions to this problem was obtained forthe case of systems with coefficients in a field in [10], using tools and methods of the geometricapproach. Results concerning the case of systems with coefficients in a Unique Factorization Domain

Mathematics and Computers in Simulation 45 (1998) 235±244

ÐÐÐÐ

* Corresponding author. Fax: (+39) 071 22804334; E-mail: [email protected]

0378-4754/98/$19.00 # 1998 IMACS/Elsevier Science B.V. All rights reserved

PII S 0 3 7 8 - 4 7 5 4 ( 9 7 ) 0 0 1 0 3 - 1

were obtained, using an input/output approach, in [6]. In the same line of [10], after the geometricapproach was extended to the framework of systems with coefficients in a ring ([7,8,2]), the solvabilityof the Block Decoupling Problem with static state feedback was characterized in geometric terms in [9]in the case of systems with coefficients in a Principal Ideal Domain. The results of [9] were improved in[2] and in [5], by removing the limitation on the ring of coefficients and by providing simplerconditions for the existence of solutions.

Here, after recalling briefly for completeness the results of [5], we give in Section 3 acomplete characterization of the solvability of the Block Decoupling Problem, for systems withcoefficients in a very general ring, in the case in which dynamic state feedback is allowed. Thecharacterization is given in purely geometric terms and it coincides with that given for systems withcoefficients in a field in [10]. The main result of Section 3 improves a partial result, previouslyannounced in [4], for systems with coefficients in a Principal Ideal Domain. Finally, in Section 4 wedescribe, by means of an example, the application of the result of Section 3 to delay-differentialsystems.

2. The Decoupling Problem

Let R denote a commutative ring with identity. By a system with coefficients in R, or a system over R,we mean a discrete-time, linear dynamical system whose evolution is described by a set of equations ofthe form

x�t � 1� � Ax�t� � Bu�t�y�t� � Dx�t�

�(1)

where x belongs to the free state module X�Rn; u belongs to the free input module U�Rm; y belongs tothe free output module Y�Rp; and where A, B and D are matrices of suitable dimensions with entries inR. Assuming that the output of (1) is split into k blocks, k�2, and writing yi2Yi�Rpi , i�1, 2, . . . , k,with

Pki�1 pi � p and Y�Y1 �� Yk, the output equation of (1) reads as

yi�t� � Dix�t� i � 1; 2; . . . ; k; (2)

where Di: X! Yi, i�1, 2, . . . , k, are matrices of suitable dimension with entries in R. Then, following[10], Section 9.4, we consider the noninteracting control problem described below.

2.1. Block Decoupling Problem

Given a system � of the form Eqs. (1) and (2), find, if possible, integers na and mi and a feedback lawof the form

xa�t � 1� � A1x�t� � A2xa�t� �Pk

i�1 Gaivi�t�;u�t� � Fx�t� � Hxa�t� �

Pki�1 Givi�t�;

�(3)

where xa2Xa�Rna ; vi2Rmi , i�1, 2, . . . , k; A1, A2, F, H, Gi and Gai are matrices of suitabledimensions with entries in the ring R, such that in the compensated system �F,G, given by the

236 G. Conte et al. / Mathematics and Computers in Simulation 45 (1998) 235±244

equations

x�t � 1� � �A� BF�x�t� � BHaxa�t� �Pk

i�1 BGivi�t�xa�t � 1� � A1x�t� � A2xa�t� �

Pki�1 Gaivi�t�

yi�t� � Dix�t�; i � 1; . . . ; k

8<: (4)

each new block input vi completely controls the corresponding block output yi, but has no influence onyj for j6�i, i�1, . . . , k.

We will speak of Restricted Decoupling Problem when the feedback (3) is constrained to be static,namely na�0, and of Extended Decoupling Problem otherwise.

A basic tool for dealing with the Decoupling Problem in the case of systems over a field, as well as inthe case of systems over a ring, is the notion of controllability submodule (see [10,9]).

Definition 1 A submodule R of the state module X is called a controllability submodule if there existR-linear maps F: X ! U and G: U ! U such that R�h(A�BF)|ImBGi.

It can be proved, following the line of [10], Proposition 5.1, that R is a controllability submodule ifand only if there exists an R-linear feedback map F: X ! U such that

R � h�A� BF�jImB \Ri: (5)

Recalling (see [7]) that a submodule V of the state module X is called (A, B)-invariant if and only ifAV�V�ImB and that it is called (A,B)-invariant of feedback type if and only if there exists an R-linearmap F: X ! U such that (A� BF)V�V, it is easy to see that controllability submodules are (A,B)-invariant submodules of feedback type. In particular (5) holds for every friend F of R, i.e. for every R-linear map F: X! U such that (A�BF)R�R. Since (A,B)-invariance of feedback type is a very strongrequirement for a submodule, it turns out to be useful to introduce a notion weaker than that ofcontrollability submodule, which, nevertheless, captures its geometric meaning.

Definition 2 (See [1], Definition 4) A submodule R of the state module X is called a pre-controllabilitysubmodule if

(i) R is (A,B)-invariant;(ii) R is the minimum element of the family SR defined by

SR � fS � X such that S � R \ �AS� ImB�g:Controllability submodules are pre-controllability submodules. The converse is true for (A,B)-

invariant submodules of feedback type. A proof of these facts, as well as an algorithm for verifying if,for a given R, the condition (ii) above holds, has been given in [1], Propositions 2 and 3.

The family of all pre-controllability submodules of X which are contained in a given submodule Khas a maximum element, denoted by R*(K), which plays a key role in the solution of the RestrictedDecoupling Problem. In fact, denoting by Ki the submodules of the state space X defined, for i�1, . . . ,k, by Ki �

Tkj�1;j6�i Ker Dj, and reasoning as in [10], Section 9, one can see that the decoupling of the

i-th input vi from the outputs yj, j6�i, and, at the same time, the complete controllability of yi by meansof vi, may be achieved, in the compensated system, if and only if there exist controllability submodulesRi of � which satisfy, for i�1, . . . , k, the following conditions

G. Conte et al. / Mathematics and Computers in Simulation 45 (1998) 235±244 237

(i) Ri�Ki;(ii)Ri�Ker Di�X;(iii) (A�BF)Ri�Ri for some feedback F: X ! U or, in other terms, the Ri's have a commonfriend F.

Now, denoting by R�i the maximum pre-controllability submodule contained in Ki, conditions (i)and (ii) above can be substituted by the following one

R�i � Ker Di � X; for i � 1; . . . ; k: (6)

For systems over a field, for which pre-controllability submodules are controllability submodules,the additional condition

\ki�1

Ker Di � f0g (7)

assures the existence of a feedback F which is a common friend of all the R�i 's, and therefore (6) and(7) guarantee the solvability of the Restricted Decoupling Problem.

For systems over a ring, since pre-controllability submodules are not always controllabilitysubmodules, the situation is technically more involved. However, as shown in ([4], Proposition 5),assuming only the very general condition that the ring R is Noetherian (in other terms: every ideal in Rhas a finite set of generators), formally the same result holds.

Proposition 1 ([4], Proposition 5) Given a system � of the form (1)±(2) over a Noetherian ring R,assume that condition (7) holds for �. Then, the Restricted Decoupling Problem for � is solvable if andonly if condition (6) is satisfied, i.e.

R�i � Ker Di � X; for i � 1; . . . ; k; (8)

where R�i denotes the maximum pre-controllability submodule contained in Ki.

The conditions of Proposition 1 are purely geometric and they coincide with those known in the caseof systems with coefficients in a field (Compare with [10], Theorem 9.1). From a theoretic point ofview, the characterization of solvability of the Restricted Decoupling Problem given in Proposition istherefore the best result one can hope to obtain. However, for practical applications, the computation ofpre-controllability submodules required for testing conditions (8) may be too complex (see [5]). Acondition much simpler to test can be derived from the following Lemma.

Lemma 1 ([5], Lemma 1) Assume that for the system � over the Noetherian ring R condition (7) and

(8) hold. Then R�i�Ki for every i�1, . . . , k.

By the Lemma one has, in fact, the following result.

Proposition 2 ([5], Proposition 6) Assume that for the system � over the Noetherian ring Rconditions (7) holds. Then, the Restricted Decoupling Problem is solvable if and only if the followingconditions are satisfied


(i) Ki is a pre-controllability submodule for all i�1, . . . , k(ii)Ki�Ker Di�X for all i�1, . . . , k.

3. Solution of the Extended Decoupling Problem

Condition (7) is quite restrictive. In case it does not hold, the existence of solutions to the RestrictedDecoupling Problem is difficult to be characterized and one usually prefers to look for dynamicfeedback solutions. This leads us to consider the Decoupling Problem in its Extended formulation. TheExtended Decoupling Problem for a system � of the form (1) and (2) is clearly equivalent to aRestricted Decoupling Problem for an extended system �e defined by

�e � xe�t � 1� � Aexe�t� � Beue�t�;y�t� � Dexe�t�;

�(9)

whose state and input modules are given, respectively, by Xe�X�Xa, Ue�U�Ua, with Xa�Ua�Rna

for a suitable integer na; and for which the matrices Ae, Be and De are given by

Ae � A 0

0 0

� �; Be � B 0

0 Ina

� �; De � D 0� � (10)

In fact, a static feedback which solves the Restricted Decoupling Problem for �e has the form

ua�t� � A1x�t� � A2xa�t� �Pk

i�1 Gaivi�t�u�t� � Fx�t� � Hxa�t� �

Pki�1 Givi�t�

�(11)

for some matrices A1, A2, F, H,Gi and Gai. Hence, remarking that xa(t�1)�ua(t) by (9) and (10), thefeedback (11) defines a dynamic feedback of the form (3) which solves the Decoupling Problem for �.

According to what has been said in the previous Section, the solvability of the Extended DecouplingProblem is therefore equivalent to the existence of an extended system �e for which there arecontrollability submodules Rie, for i�1, . . . , k, verifying the conditions:

(i) Rie�Kie:�Tk

j�1;j6�i(Ker Dj�Xa);(ii) Rie�Ker Die�Xe;(iii) (Ae�BeFe) Rie�Rie for some feedback Fe: Xe ! Ue or, in other terms, the Rie's have acommon friend Fe.

In the case of systems over a field, the above conditions are equivalent, in terms of the originaldata, to condition (8) (see [10], Theorem 9.3), namely to R�i�Ker Di�X for i�1, . . . , k. Inour situation, the following analogous result, which represents the main result of the paper, can bestated.

Proposition 3 Given a system � of the form (1)±(2) over a Noetherian ring R and denoting by R�i themaximum pre-controllability submodule contained in Ki�

Tkj�1;j6�i Ker Dj, the Extended Decoupling

Problem for � is solvable if and only if the following conditions hold:

R�i � Ker Di � X for i � 1; . . . ; k: (12)


The proof of Proposition 3 requires two preliminary Lemmas.

Lemma 2 Assume that Re�Xe is a controllability submodule for �e and let R denote its image in X bythe canonical projection P: Xe�X�Xa ! X, that is R�PRe. Then, R is a pre-controllability

submodule for �.

Proof For every element x in Xe we have that APx�PAex. Then, the (Ae,Be)-invariance of Re impliesthe (A,B)-invariance of its projection R, since AR�APRe�PAeRe�P(Re�Im Be)�PRe�PImBe�R�ImB. The fact that Re is, in particular, a pre-controllability submodule implies (using [1],Proposition 3) that the sequence {Sn}, defined by

S0 � f0g; Sk�1 � Re

\�AeSk � ImBe�; for k � 1; 2; . . . (13)

converges in a finite number of steps to Re. In particular, Re�Re

T(AeRe�ImBe). This implies that

R�PRe�P(Re

T(AeRe�Ime))�PRe

T(APRe�PImBe)�R

T(AR�ImB). Now, the sequence fS0ng

defined by

S00 � f0g; S0k�1 � R\�AS0k � ImB�; for k � 1; 2; . . . (14)

converges in a finite number of steps to a submodule S0� � R and, assuming that PSk�S0k, we havePSk�1�P(Re

T(AeSk�ImBe)�R

T(APSk�PImBe)�R

T(AS0k�ImB)�S0k�1. By induction, we have there-

fore that the limits Re of {Sn} and S0� of {S0n} verify PRe�R�S0�. Then, R�S0�, and by [1], Proposition2, it is a pre-controllability submodule for �.

Lemma 3 Assume that R�X is a pre-controllability submodule for �, and assume that Re is any

submodule of Xe such that R�PRe, P: Xe ! X being the canonical projection. Then, Re is a pre-controllability submodule for �e.

Proof The (A,B)-invariance of R implies the (Ae,Be)-invariance of Re, since AeRe�APRe�{0}�(R�ImB)�{0}�Re�ImBe. Consider now the sequence {S0n} defined as in (14) whichconverges to R, and the sequence {Sn} defined as in (13) in which, by construction, Sk�Re for every k.Given x2Re such that Px2S01, we have x2S1. In fact, x can be written as x�x0�a�Bu�a, with x02S01 anda2Xa, then x2Re

TIm Be�S1. Assuming that x2Re and Px2S0k imply x2Sk, given x2Re such that

Px2S0k�1, we have x2Sk�1. In fact, by Px�Ax0�Bu, with x02Sk�R, there exists b2Xa and a2Ua suchthat (x0�b)2Sk�Re, and x�x00�a�Ae(x

0�b)�Be(u�a)2(AeSk�ImBe)�Sk�1. Since the sequence {S0n}converges to R, we have that R�S0

kfor some k, then x2Re implies Px2S0

k. Hence, we have Re�Sk and

Re is the limit of the sequence {Sn}. Therefore, by [1], Proposition 2, Re turns out to be a pre-controllability submodule for �e.

Proof of Proposition 3Necessity: Assume that the Extended Decoupling Problem for � is solvable. This means that for an

extended system �e there exist controllability submodules Rie satisfying the conditions (i)±(iii) recalledbefore stating Proposition 3. Hence, denoting by P the canonical projection P: Xe�X�Xa!X and byDie the matrices defined, for i�1, . . . , k, by Die� Di 0� �, we have, for i�1, . . . , k, PRie�PKie�Ki

and X�PXe�P(Rie�Ker Die)�PRie�P Ker Die�PRie�Ker Di. Therefore, since PRie is, by Lemma2, a pre-controllability submodule for, (12) follows from the above equalities.


Sufficiency: Assume that, for i�1, . . . , k, R�i is generated by the columns of the n�ni matrix Vi, thenthe (A,B)-invariance of Ri implies the existence of matrices Li and Mi such that AVi�ViLi�BMi. Lettingna��i�1, . . . , k, ni, assume that Rie is the submodule of X�Rna spanned by the columns of thefollowing matrix Wi. Then, by Lemma 3, each Rie is a pre-controllability submodule for the extendedsystem �e, in particular we have

Wi �

Vi

0n1�ni

..

.

Ini�ni

..

.

0nk�ni

2666666664

3777777775; AeWi � WiLi � Be

Mi

0n1�ni

..

.

Li

..

.

0nk�ni

2666666664

3777777775Since each Rie is by construction a free module and a direct summand of Xe, it is an (A,B)-invariant

of feedback type by [5], Proposition 3 and a controllability submodule. Again by construction and by(12), we have Rie�

Tkj�1;j6�iKer Dje and Rie�KerDie�Xe. Moreover, a basis of Xe is given by the

columns of the matrix

Wi � W1 . . . WkIn�n

0na�n

� �� and it is easy to see that the feedback

Fe �ÿM1 ÿM2 . . . ÿMk 0

L1 0 . . . 0 0

. . . . . . . . . . . . . . .0 0 . . . Lk 0

26643775Wÿ1

is a common friend of the Rie's, since (Ae�BeFe)Wi�WiLi for i�1, . . . , k. Then, the conclusion follows.

4. Application to delay-differential systems

In order to describe how the results of the previous Sections apply to continuous time, delay-differential systems, let us make use of a comprehensive example. In particular, let us consider, for sakeof illustration, the delay-differential system � given by

� �

_x1�t� � x1�t� � x2�t ÿ �1� � u1�t�;_x2�t� � x1�t ÿ �2� � x3�t� � u2�t�;_x3�t� � x2�t ÿ �1 ÿ �2� � u1�t ÿ �1�;y1�t� � x1�t�;y2�t� � x2�t ÿ �1�;

8>>>><>>>>:where �1 and �2 represent two fixed delays. By introducing the delay operators �i, i�1,2,defined for any function f(t) by �if(t)�f(tÿ�i), we can formally associate to � the system � given


by

� �

x1�t � 1� � x1�t� ��1x2�t� � u1�t�x2�t � 1� � �2x1�t� � x3�t� � u2�t�x3�t � 1� � �1�2x2�t� ��1u1�t�y1�t� � x1�t�y2�t� � �1x2�t�

8>>>><>>>>:which can be viewed as a system with coefficients in the ring R[�1,�2] of polynomials in twoindeterminates.

The two systems � and � are different objects, both from a mathematical and from a dynamicalpoint of view, but they have the same signal flow graph. This fact implies that the problemof decomposing the input/output relation of � into two input/output blocks, each one decoupledfrom the other, by means of a state feedback (possibly a dynamic state feedback which involves delayson inputs and states) may be solved if the corresponding Block Decoupling Problem for � has asolution.

In considering the Block Decoupling Problem for �, we are interested in the maximum pre-controllability submodules R�i , contained, respectively in Ki�Ker D3ÿi, for i�1, 2, withD1� 1 0 0� � and D2� 0 �1 0� �. We have that R�1, R�2 are generated, respectively, by the columnsof the matrices

V1 � 1 0 0

0 0 �1

� �T

and V2 � 0 1 0

0 0 �1��2 ÿ�1�� T

Condition (7), namely Ker D1

TKer D2�{0} is not satisfied, while condition (12), namely R�i�Ker Di

�X for i�1, 2, is satisfied. Hence, by Proposition 3, the Block Decoupling Problem for � is solvableby a dynamic feedback.

In order to construct a solution, since n1�number of generators of R�1�2 and n2�number ofgenerators of R�2�2, take na�4 and consider the extended system �e of the form (9), withXa�(R[�1,�2]).

The extended pre-controllability submodules R1e and R2e to be considered here, as in the proof ofProposition 3, are generated, respectively, by the columns of the matrices

W1 � 1 0 0 1 0 0 0

0 0 �1 0 1 0 0

� �T

and W2 � 0 1 0 0 0 1 0

0 0 �1��2 ÿ�1� 0 0 0 1

� �T

Choosing the matrices L1, M1, L2, M2 as follows,

L1 � 0 0

ÿ1 0

� �; M1 � 1 0

�2 �1

� �; L2 � 0 �1��2 ÿ�1�

1 0

� �; M2 � �1 0

0 0

� �we have

AeW1 � W1L1 � Be

M1

ÿL1

02x2

24 35 and AeW2 � W2L2 � Be

M2

02x2

ÿL2

24 35


Denoting by W the matrix whose columns form a basis of Xe, a common friend of R1e and R2e is givenby the feedback

Fe �ÿM1 ÿM2 0

L1 0 0

0 L2 0

24 35Wÿ1; where W � W1 W2I3�3

04�3

� �� (15)

Denoting by r1 and r2, respectively, the generators of Rie

TImBe for i�1, 2, we have

r1� 1 0 �1 1 1 0 0� �T and r2� 0 1 0 0 0 1 0� �T, that is r1�Be u1�Be

1 0 1 1 0 0� �T and r2� Be u2�Be 0 1 0 0 0 1 0� �T. Let Ge be the matrix G�[Ge1

Ge2]�[u1 u2], then we finally get for �e the feedback

u1�t�u2�t�

ua

0@ 1A � Fe

x1�t�x2�t�x3�t�xa�t�

0BB@1CCA� Ge1v1 � Ge2v2 (16)

which defines for � the dynamic feedback

u1�t�u2�t�

xa�t � 1�

0@ 1A � Fe

x1�t�x2�t�x3�t�xa�t�

0BB@1CCA� Ge1v1 � Ge2v2 (17)

Now, recalling the definition of �i, we get from (17) the dynamic feedback

u1�t� � ÿxa1�t� ÿ xa3�t ÿ �1� � v1�t�;u2�t� � ÿxa1�t ÿ �2� ÿ xa2�t ÿ �1� � v2�t�;xa1�t � 1� � v1�t�;xa2�t � 1� � ÿxa1�t� � v1�t�;xa3�t � 1� � ÿxa4�t ÿ 2�1� � xa4�t ÿ �1 ÿ �2� � v2�t�;xa4�t � 1� � xa3�t�;

8>>>>>><>>>>>>:(18)

for the delay-differential system �. Compensating � by (18) one gets, after a tedious but simplecomputation, a system whose transfer function reads as

T�s; eÿs� � 1=s 0

0 seÿd2s=�s2 ÿ eÿ�d1�d2�s � eÿ2d1s��

References

[1] G. Conte, A.M. Perdon, The Decoupling Problem for Systems over a Ring, Proceedings IEEE-CDC'95 (New-Orleans,

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[2] G. Conte, A.M. Perdon, The disturbance decoupling problem for systems over a ring, SIAM Journal on Control and

Optimization, 33 (1995).


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Proceedings Colloque on Analyse et commande des systemes avec retards (Ecole de Mines de Nantes, Nantes, 1996).

[4] G. Conte, A.M. Perdon, A. Lombardo, The Static and Dynamic Block Decoupling Problem for Systems over a Ring,

Proceedings IFAC World Conference (San Francisco, CA, 1996).

[5] G. Conte, A. M.Perdon, A. Lombardo, The Block Decoupling Problem for Systems Over A Ring, IEEE Trans.

Automatic Control (1996) submitted.

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Dynamic Feedback Decoupling Problem for Delay-differential Systems via Systems over Rings