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Finite spectrum assignment for input delay systems1
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FINITE SPECTRUM ASSIGNMENT FOR INPUT DELAY SYSTEMS 1 S. Mondie·,2 and J.J. Loiseau·· • Departamento de Control Automcitico C1NVESTA \'-IPN, Av. 1PN 2508, A.P. 14-740, 07300 Mexico, D.F., Mexico, [email protected]. mx •• 1nstitut de Recherche en Nantes, UMR CNRS 6597, de Nantes, Ecole des Mines Cedex 03, France.
Communications et Cybernetique de Ecole Centrale de Nantes, Unit1ersite de Nantes. BP 92101. 44321 Nantes [email protected]'
Abstract: The problem of polynomial invariant. factors assignment. of input delay systems with classical spectrum assigment control laws with distributed delays is adressed. The multiplicities of the invariant fadors are shown to be restricted by specified Rosenbrock type inequalities. The results are proved with t.he help of an equivalent linear assignment problem with no delay, and within the Bezout domain E. A bidimensional illustrative example is given. Copyright © 2001 1FAC Keywords: delay systems, finite spectrum assignment, invariant factors. 1. I TRODUCTION
loop, are crucial for the closed loop dynamics. The question that. arises is to characterize the freedom in assigning the invariant. factors.
Consider a linear system with delay described by K
x(t)
=L
K
Aix(t - ih)
i=O
+L
The answer to this query was given in the recent result in Loiseau (2001): the limitation is that the sum of the degrees of the invariant factors must. be equal to n.
BiU(t - ih) . (1)
i=O
where the control input u(t) E lE. m and the inst.antaneous state x(t) E m. n for t 2: 0, and the family of control laws described by integral Volterra equations of the second kind,
J +J Kh
u(t)
=
f(r)u(t - r)dr +
K
L giX(t -
o
In this work, we restrict our attention to input delay systems. This subfamily of the systems introduced above includes the models of a wide class of applications where the delay is due to transport phenomenons. time consuming information processing, sensors design among others. These systems are described by
ih)
1=0
Kh
g(r)x(t - r)dr.
x(t)
= Ax(t) + ~~oBiU(t -
ih).
(3)
(2)
As shown in Manitius and Olbrot (1979), Olbrot (1978), these systems are n-assignable by control laws that are simpler that those described by (2), those of the form
o Such a control law was introduced in Olbrot (1978). where the spectral controllability of system (1) is shown to be a necessary and sufficient. condition for the control law (2) to freely assign a closed loop finite spectrum. However, not only the location of the roots. but also their multiplicit.ies, or equivalently the invariant. factors of the closed
J t
u(t)
= I\'[x(t) + ~~l
e(t-a-ih)A B i u(G")dG"](4)
t-h
This is indeed a particular case of t.he finite spectrum assignment problem introduced above. The motivation for using such control laws is their simplicity: the state feedback is linear and static,
I Supported by Conacyt, Mexico 31951-A and by GDRAutomatique: systemes a retards, France. 2 On leave at Heudyasic, UMR CNRS 6599, France.
201
and the distributed component consists of linear combination of elementary distributed delays. As shown below, it permit not only toassign a finite spectrum, but also finite invariant factors. The motivation for assigning invariant factors with a finite number of roots is the same as the one for spectrum assignment: the invariant factors give and additional freedom in shaping the dynamics. If they have a finite number of roots, they can be readily analyzed. The question is then whether or not the freedom, that exists in the general case, is restricted when such simpler laws are used. The problem under consideration is then the following.
also possible to assign prescribed invariant factors with a finite number of roots to the closed loop characteristic matrix.
Lemma 1. Consider a spectrally controllable linear multivariable system with delay in the input described by x (t)
= Ax(t) + E~oBi11.(t -
ih)
(5)
where A E m: nxn . B E JRnxm and the delays in the input are commensurate to h 2: O. Then the following problems are equivalent. (i) The control law
Problem Consider a spectrally controllable input time delay system described by (3). Under what conditions does static state feedback distributed control laws (4) exists, such that the closed loop has prescribed invariant factors with finite number of roots.
J t
u(t)
= I\[x(t) + ~;~o
e(t-17-ih)A Bi11.(O")dO"] (6)
t-ih assigns to the closed loop system (.5-6) a prescribed set of invariant factors
The solution to this problem is organized as follows. Some preliminary results are shown or recalled in Section 2. Then the main result is established in Section 3 and is discussed in the light of the Bezout domain E in Section 4. An illustrative example is presented in Section 5 and some concluding remarks end the paper.
o~(s), o~(s),···. o~(s) .
(i) The control law 11.(t) controllable system
Y (t) = Ay(t)
= Ky(t)
assigns to the
+ ~~oe-ihA B;11.(t)
(7)
a closed loop system with a prescribed set of invariant factors o~ (s), o~(s), ... ,o~ (s).
Notation: IR [s] denotes the ring of polynomials over JP'., the field of reals. The degree of its elements, o(s), is denoted deg (o(s)). JP'. (s) stands for the field of rational functions over m:, while the rings of proper and strictly proper rational functions are denoted. respectively, by IRp(s) and JIl2 sp (s). Further, l!l'.mxnand l!l'.mxn [s] .... , denote the sets of m x n matrices having elements in IR, lR [s], ... , respectively. Units of the ring JP'.mxm [s] are called unimodulars and those of the ring lP'.;,xm(s) bipropers. The set E is a Bezout domain whose elements are fractions of the form h o(s , e- hs ) -- n(s,e') where all the zeros of d(s) , d(s) E lR [s] are zeros of n(s,e- hs ) E IR [s.e- sh ]
Proof. Consider the transformation t
y(t)
J
= x(t) + ~~o
e(t-17-;h)A Bi11.(O")dO" (8)
t-ih The derivative of (8) is
y(t)
= x(t) + E~oe-ihA Biu(t) _E~oe(h-ih)ABi11.(t
- ih)
t
the set of quasipolynomials. Notice that 11+: [s] C E. Units of the Bezout domain E are called unimodulars in E. (for detailed information on these sets and their properties, see Gantmacher (1959), Kailath (l 980), Brethe and Loiseau (1998), and Loiseau (2000, 2001)).
+E~oA
J
e(t-17-ih)A Bi11.(O")dO"
t-ih
=Ax(t) + E~oBi11.(t -
ih)
+E~oe-ihABi11.(t) - E~oBi11.(t - ih)
J t
+~~oA
e(t-17-ih)A B;11.(O")dCJ
t-ih
2. PRELIMINARY RESULTS
= Ay(t)
""e now give a new interpretation of the result established in Manitius and Olbrot (1979), Artstein (1982) that has been extensively used in the literature, in the light of n-assignability, namely the ability to assign a finite spectrum closed loop system, provided that the system is spectrally controllable. Indeed, there is more to say: it is
+ ~~oe-ihA BiU(t).
It follows that the control law (6) assigns the
closed loop
y(t)
= (A + E~oe-ihA B;I\)y(t).
In order to establish the equivalence with (ii), consider the description of the system (5) and the
202
that assigns a closed loop whose invariant factors are monic polynomials o:~ (s), o:~ (s), ... , 0:1. (s) where o;(s) divides O:;_l(s),i = 2, ... ,m, if and only if
closed loop (6) in the frequency domain. They are respectively (sIn -A)x(s) = Ef::oBie-ihsu(s) and (lrn - K(E~oe-ihA(sl - A)-Ill - e-ih(.I-AJ]Bi)U(S)
= Kx(s).
Let
The closed loop characteristic matrix is (
SIn-A -B ) -I{ I m - I\.M .
_ ((SIn - A)(In - M 1\") - BI{
0
*)
Im
i=l
i=l
= m,
and o:;(s)
(11)
= 1, i = m +
Proof. According to Lemma 2, the pair (A, Ef::oe- ihA B i ) is controllable. Let Cl, C2, ... , Cm denote its controllability indices. From the Rosenbrock Control Structure Theorem recalled in the appendix, there exists a control law (lO) that assigns to the system (7) a closed loop with invariant factors o:~ (s), 0:; (s), ... , o:~ (s) where 0:; (s) divides 0:;_1 (s), i = 2, ... , m, if and only if conditions (11) hold. Finally, the result follows from Lemma 1. •
SIn-A -B )(I-M)(IO) -I\. I m -!\.M 0 I I{ I
-
j
with equality for j 1, ... , n.
Post multiplication by well defined unimodular matrices leads to
(
j
LCi ~ Ldego:;, j = 1, ... ,m
~N B. -ihs B .'- .::..Ji=O t e , M := E~oe-ihA(sI - A)-l[I _ e-ih(sI-A)]Bi.
'
the symbol * standing for a nonspecified matrix. Then, the invariant factors of the closed loop are those of (sIn - A)(I - MI\") - BI\", that is
Remark 2. Clearly, the use of simpler control laws has a cost: the degrees of the invariant factors cannot be assigned arbitrarily.
~N -ihA (I ( S'In - A) . (In - ~i=Oe s - .4)-1 . (I - e-ih(sI-A))B;I\") - ~f::oBie-ihs I{ = sIn - A - ~!:oe-ihA[I - e-ih(sI-AJ]Bi!\.) ~N B. -ihsJ" -~i=O ,e \ -- S I n - .4 _ ~N -ihAB·J" ~i=Oe , \
and the result follows . •
Remark 1. The proof follows closely the steps of the proof in Manitius and Olbrot (1979) for nassignability. We recall it here to show that it permits to conclude on the invariant factors of the closed loop matrix, not only on its spectrum. Lemma 2. (Olbrot, 1978) The input delay system (5) is spectrally controllable if and only if the linear system (7) is controllable.
3. MAIN RESULT
Remark 3. Lemma 1 implies that a simple static state feedback distributed control law described by (10) permits to assign to the input delay system (9) a set of desired invariant factors that satisfies the inequalities (11). The parameter I{ of the control law , can be calculated in a straightforward manner as the solution to the problem of a static state feedback invariant factors assignment for the linear system with no delay (7). Notice that toolboxes for the analysis and design of linear systems are available (The Polynomial Toolbox, 1998). If a set of desired invariant factors does not satisfies the inequalities (11), the results obtained in Loiseau (2001) guarantee that there exists a control law described by (2) that allows to assign them. This control law if fully determined by hi,i= 1,2, ... ,/(T) andg(T).
We are now able to state our main result.
Theorem 3. Consider a spectrally controllable linear multivariable system with delay in the input described by x (i)
= Ax(t) + E;'{=oBiU(i -
ih),
4. INTERPRETATION OF THE RESULT IN THE BEZOUT DOMAIN £
(9)
where A E w:. nxn , B i E rn: nxm and h 2: 0 is the delay. Let Cl, C2, ... , Cm be the controllability indices of the pair (A, E!:oe- ihA B;). Then there exist a control law
J t
uti)
= K[x(i) + ~!:o
e(t-a-ih)A Biu(IT)dIT](lO)
t-h
203
The Bezout domain, which definition and basic properties are recalled in the notations. section 1, has proved to be a powerful tool for the study of commensurate time delay systems, by providing a solid algebraic framework that allows the generalization of many results established for linear systems without delays (Brethe, 1997). The following remarks enlightens some subtleties of the machinery in £, in connexion with our problem. First of all, we establish the following clue fact.
Fact 1. The invariant factors in the Bezout domain E of a polynomial matrix coincide with its invariant factor over the ring JR[s].
Consider indeed a polynomial matrix D( s) E JRnxm[s] of rank r. It is well known (Gantmacher, 1959) that there exist unimodular polynomial matrices U(s) E JRnxn[s] and VIs) E JRmxm[s] so that D( s) can be factored as D(s)
= U(s) (~iag{O'i(S).i = 1, ... ,r} ~)
Vis).
(12)
where Qi(S), i = 1, ... , r are unique monic polynomials such that Qi(S) divides Qi-ds) called the invariant factors of D( s). Further, one can show (Brethe, 1987) that E is also a so-called invariant factor domain. Every matrix D(s) over E can also be factored in the form (12), were now the matrices U (s) and V (s) and their inverses are over E, and the Qi(S), i = 1, ... , r are elements of E. which are also called the invariant factors of D(s). Now, since lR[s] C E, it is clear that when D(s) is polynomial, the polynomial factorisation (12) is also a factorisation over E: U (s) and V (s) are indeed matrices over E, which are unimodular over E, the rank of D(s) is rover E, and, since such a factorisation over E is unique. the O'i (s), i = 1, .... r, are also the uniquely defined invariant factors of D( s) over E. Fact 2. If the input delay system (5) is spectrally controllable, then its transfer matrix can be factared in the form (sIn - A)-l B = N(s, e- hs )D- l (s) ,
and
(Irn - lC(Ef':oe-ihA(sI - A)-I. (I - e-ih(sI-A))B;)u(s) = I\x(s). We first observe that a natural factorization also exists for the closed loop system. Let indeed (lv·I\(s,e-s),DK(s.e- S)) be a coprime factorization over [. of the transfer of the closed loop system. The closed loop denominator is given by Dl\(s,e- S ) = «(Im _/\'(~;"~'le-'hA(s1- A)-I (I - e-ih(sI-A))B;))D(s) -1-.,:N(s) = D(s) - 1\'N(s) -I{(~~I e-· hA (s1 - .4)-1 (I - e-·h(sI-AI )B, )D(s) = D(s) - /\'N(s) - /\'(s1 - A)-I ~~I e-· hA B;D(s) +h:(sI - A)-I ~;"~I e- ihs B,D(s)
Observe that the matrix
= Et': 0 (sIn - A)-l Bie-ihs )D-1( ) = "N --'i=Oe -ihsN( i ss,
(I m - j \'( SI - .4)-1"N --'i=Oe -ihAB·) , is a biproper matrix in IR:(s), hence DK(s,e- S ) is actually rational in the variable s, and, since it has no pole, the result follows.
Ni(s) = (sIn - A)-l BiD(S) , then the transfer of the system without delay (7) can be expressed as 4)-1 e -ihAB·t -_
= ~~OBie-ihsu(s)
= D(s) - E(sI - A)-l~t':l e- ihA BiD(s) = (Im - I\'(sI - A)-l~~le-ihA B;)D(s).(13)
whith
~
(sIn - A)x(s)
DK(s,e- S)
We can further state that if
"N """"i=O (1 S n -
We are now able to revisit the problem in the framework of E. The system and the control law are respectively described in the frequency domain by
and it follows from (??) that
where D(s) and N(s,e- hS ) are right coprime matrices that are respectively polynomial in the variables sand s, e- hs .
(sIn - A)-l B
that D(s)U(s,e- hs ) = D'(s), for some matrix U(s,e- hs ) which is unimodular over E. We can see that U(s,e- hs ) = D-l(s)D'(s) is actually rational in the variable s. Since it is also a matrix over E, an analytic function without pole, we conclude that it is actually polynomial in s, which shows the result. In the sequel, we shall call such a factorization with a polynomial denominator the natural factorization of the system.
One can further notice that a left product by a biproper matrix does not modify the column degrees. Thus the column degrees of D K (s, e- S ) are those of D( s).
("N "( )) D- 1 ( ) ":"'z=oe -ihA lV t s s .
Both systems have the same denominator. which is of course polynomial in s. As it is usual for systems without delays, we can assume that D(s) is comumn reduced, with column degrees Cl, C2, ...• Cm. We can notice in addition that the column degrees of such a column reduced polynomial denominator of the transfer of the system (5) are uniquely defined. Any other polynomial denominator, say D'(s) is indeed so
The natural coprime factorization bases a design algorithm for the finite spectrum assignment, which consists of the following steps. (i) Calculate a natural factorization (N(s,e- Sh ),
D(s)) of the transfer matrix of system (5). D(s) is column reduced with column degrees Cl,CZ,···,C m ·
204
(ii) Choose a set of monic polynomials O'~ (s), where O':(s) divides O':_l(S) for i = 2, ... , m" and take O'~+ds) = ... = O'~(s).
It is possible to design a control law that assigns prescribed invariant factors with finite roots, of degree {2, O}. A solution is given by
(iii) Providing that (11) hold, there exist a polynomial matrix DK(S), that is column reduced, with column degrees Ci and invariant factors 0: (s), for i = 1, ... , m, and constant matrices X and Y of convenient dimensions, X being invertible, so that
(h - F((s,e-S))D(s,e- S) - G(s,e-S)N(s,e- S)
O'~(s), ···,a~(s)
Algoritms to compute DK(S), X, and Y are provided by Kucera (1991). The feedback 1": = _X- 1 Y assigns to the closed loop system (3-4) the invariant factors (s), for i = 1, ... , n.
0':
The situation is quite different when considering factorizations over E. As shown in Loiseau (2001), contrarily to what happens in ][{(s), post multiplication of elements of E by unimodular over E modify the column degrees. An immediate consequence of this fact is the non unicity of the column degrees of the denominator of a left coprime factorization in E. Moreover, there is complete freedom in choosing the column degrees of such a factorization.
= DK(s,e-
ll
1
into (13), gives
-1) (15)
= (s'2-kkllS+kk1'2S-kk1'2 '2IS
+
2'2S -
1
'2'2
.
Now, if we want to assign invariant factors of degrees {I, I}, for instance both invariant factors taking the value s + 1, let us consider according to the design procedure given in Loiseau (2001) a coprime factorization with column degrees {l, 1} for the denominator. This new factorization is obtained as
(Irn - F(s, e-Sh)D(s, e- sh ) + G(s, e-Sh)N(s,e- sh ) = Z(e-hS)diag{o:;(s)} .
D' (s, e- S) = D(s)U(s, e- S ), N'(s, e- S) = N(s, e-S)U(s, e- S ),
This shows that a control law of the form (2) permits to assign the invariant factors (s) to the closed loop system (3-2). It should be clear that, in that case, a simpler control law of the form (4) cannot get this assignment.
0':
where U(s, e- S ) is given by U(s,e- S )=
2
1
s~:n~ -~ -0
( -s
-0
)
.
s+ln2
This leads to
5. ILLUSTRATIVE EXAMPLE
(Z(e- S ) - F(s, e-s))D'(s,e- S ) + G(s, e-S)N'(s, e- S ) = diag{ s + 1, s + 1} ,
The above results and design procedures are illustrated with the two dimensional academic example 0 0
(kk'21 kkn'2)
The invariant factors of degree {2, O}, namely {S2 + (-k l l +k 1'2 -k'21 +kd - (k 12 +k'2'2), I} , can indeed be assigned to arbitrary finite locations by choosing 1{. We have in this simple case obtained a parametrization of all the invariant factors that can be obtained with the control laws under consideration.
0':
(11)
=
K
-
0':
= (00) 1 0 x(t) +
(14)
).
According to Theorem 3, a control law where G(s.e- S) K, and F(s,e- S) 1":(s1 hA A)-l(e- e- hs I)B does the job, and substituting D(s), N(s, e- S), A, B, and
DK(s,e- S )
If a set of invariant factors (s) does not satisfy the inequalities (11), exept for j = m, one can find a factorization of the transfer where D(s, e- Sh ) is column reduced, with column degrees deg (s), i = I, ... , m, and there exist a unimodular matrix in e- hs , say Z(e- hs ) and matrices F(s, e- sh ) and G(s, e- sh ) over E, such that
. x(t)
S
with G( s, e
u(t - h).
-S)
= ((I+ln2)(2-e2 _ e- s
S 1 2 - eS Z(e- )= ( 13-e- s
S )
In2(2-e-
S ))
0
)
'
e S Fll(s,e- S ) = (s(I+ln2))+ln2) C-s - )'2 ,
This system is spectrally controllable because the pair (A. exp( -hA)B) is controllable (See Lemma 2). Its controllability indices are {2,0}.
(1 - e- s )2
F 12 (s,e- S )=(I+ln2) F2ds,e- S ) = (l_e- S)2,
The natural coprime factorization for this system IS
s
+
1 - e- s
,
s
and 1 N(s,e ") = (se-so) e-S 0 ,D(s) = (s2_ 0 1 ) .
Fn(s,e- S ) = (l-e-
_0
s
205
S
f
S
+(I_ln2)2-es + In 2
Of course this calculation is uneasy, and so is the expression of this control law in the time domain (Brethe, 1997), (Brethe and Loiseau,1998), since there is no specialized toolbox to deal with E.
6. CONCLUSION
It is shown that static state feedback distributed control laws for input delay systems that allow the assignment of a finite spectrum, can be used to assign as well invariant factors with a finite number of roots. This feature is indeed of interest in shaping the dynamic of the closed loop system. It is shown that the multiplicities of the invariant factors cannot be assigned freely: they are restricted by Rosenbrock inequalities by a set of well defined integers. The result is explained in the framework of the Bezout domain E. where more general control laws allow a complete freedom in assigning the multiplicities of the invariant factors. Clearly, static state feedback distributed control laws can be designed in a straightforward manner, but at a cost: the freedom in the assignment of the multiplicities of the invariant factors is restricted.
7. APPENDIX We recall here the Theorem of Rosenbrock (for instance from Kucera (1991)), which is a clue for our proof. Lemma 4. Consider a linear controllable system described by
= Ax(t) + Bu(t)
x (t)
where A E IP: nxn . B E IP;nxm and such that the controllability indices of the pair (A. B) are Cl, C2, ... , Cm' There exists a static state feedback u (t)
= I{ x (t )
such that the invariant factors of the closed loop system are O'~ (s), O'~ (s), ... , O'k(s) where O';(s) divides 0';_1 (s), i = 2, ... , rn, if and only if j
L
j
Ci :::;
i=l
L deg (O';),j = 1. ... ,
rn
i=l
with equality for j
= rn,
and O';(s)
= 1. i = m +
1, ... , n.
REFERENCES Artstein, Z. (1982). Linear systems with delayed controls: a reduction. IEEE Transation on Automatic Control, Vo!. AC-27, No. 4, 869-879.
206
Bellman, R. and K.L. Cooke (1963). Differential Difference Equations, Academic Press, London. Brethe, D. (1997). Contribution a. l'etude de la stabilisation des systemes lineaires a. retards, These de Doctorat, IRCYN, Nantes, France. Brethe D. and Loiseau .1 ..1. (1998). An effective algorithm for finite spectrum assignment of single-input systems with delays, Mathematics in computers and simulation, 45, 339-348. Gantmacher, F. R. (1959). The theory of matrix, Vol LAMS Chelsea Publishing, New York. Kamen E.\\7, Khargonekar P.P. and A. Tannenbaum (1986). Proper stable bezout factorizations and feedback control of linear time delay systems, Int. 1. Contr., Vo!. 43, No. 3,837-857. Kailath T. (1980). Lmear Systems, Prentice Hall. Kolmanovski, V B. and V.R. Nosov (1986). Stability of functional differential equations, Academic Press, New York. Kucera. V. (1991). Analysis and design of dzscrete linear control systems, Prentice- Hall, London, and Academia, Prague. Loiseau .1 ..1. (2000). Algebraic tools for the control and stabilization of time-delay systems, Anmwl Reviews in Control 24, 135-149. Loiseau .1 ..1. (2001). Invariant factors assignment for a class of time-delay systems, Xybernetika, Volume 37, Number 3, Pages 265-275. rvIanitius. A. Z. and A.\V Olbrot (1979). Finite Spectrum Assignment problem for Systems with Delays, IEEE Trans. Autom. Cont1'., Vo!. AC-24, No. 4, 541-553. Mondie S.. Dambrine M. and Santos O. (2001), Approximation of control laws with distributed delays: a necessary condition for stability, IFAC Conference on Systems. Structure and Control, Prague. Czek Republic. Niculescu S. (2001). Delays effects on stability. A robust control approach. Springer, Heidelberg. Morse, A.S. (1976). Ring Models for DelayDifferential Systems. A utomatica. Vo!. 12, 529531. Olbrot, A. (1978). Stabilizability, Detectability, and spectrum assignment for linear autonomous systems with general time delays, IEEE Trans. Autom. Contr., Vo!. AC-23, No. 5, 887-890. The polynomial toolbox, Polynomial Methods for Systems, Signal and Control (1998). Vidyasagar. M. (1985). Control System Synthesis, The MIT Press, USA.
Copyright © IF AC Time Delay Systems, New Mexico, USA, 2001
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Publications www.elsevier.comllocate/ifac
FINITE SPECTRUM ASSIGNMENT FOR INPUT DELAY SYSTEMS 1 S. Mondie·,2 and J.J. Loiseau·· • Departamento de Control Automcitico C1NVESTA \'-IPN, Av. 1PN 2508, A.P. 14-740, 07300 Mexico, D.F., Mexico, [email protected]. mx •• 1nstitut de Recherche en Nantes, UMR CNRS 6597, de Nantes, Ecole des Mines Cedex 03, France.
Communications et Cybernetique de Ecole Centrale de Nantes, Unit1ersite de Nantes. BP 92101. 44321 Nantes [email protected]'
Abstract: The problem of polynomial invariant. factors assignment. of input delay systems with classical spectrum assigment control laws with distributed delays is adressed. The multiplicities of the invariant fadors are shown to be restricted by specified Rosenbrock type inequalities. The results are proved with t.he help of an equivalent linear assignment problem with no delay, and within the Bezout domain E. A bidimensional illustrative example is given. Copyright © 2001 1FAC Keywords: delay systems, finite spectrum assignment, invariant factors. 1. I TRODUCTION
loop, are crucial for the closed loop dynamics. The question that. arises is to characterize the freedom in assigning the invariant. factors.
Consider a linear system with delay described by K
x(t)
=L
K
Aix(t - ih)
i=O
+L
The answer to this query was given in the recent result in Loiseau (2001): the limitation is that the sum of the degrees of the invariant factors must. be equal to n.
BiU(t - ih) . (1)
i=O
where the control input u(t) E lE. m and the inst.antaneous state x(t) E m. n for t 2: 0, and the family of control laws described by integral Volterra equations of the second kind,
J +J Kh
u(t)
=
f(r)u(t - r)dr +
K
L giX(t -
o
In this work, we restrict our attention to input delay systems. This subfamily of the systems introduced above includes the models of a wide class of applications where the delay is due to transport phenomenons. time consuming information processing, sensors design among others. These systems are described by
ih)
1=0
Kh
g(r)x(t - r)dr.
x(t)
= Ax(t) + ~~oBiU(t -
ih).
(3)
(2)
As shown in Manitius and Olbrot (1979), Olbrot (1978), these systems are n-assignable by control laws that are simpler that those described by (2), those of the form
o Such a control law was introduced in Olbrot (1978). where the spectral controllability of system (1) is shown to be a necessary and sufficient. condition for the control law (2) to freely assign a closed loop finite spectrum. However, not only the location of the roots. but also their multiplicit.ies, or equivalently the invariant. factors of the closed
J t
u(t)
= I\'[x(t) + ~~l
e(t-a-ih)A B i u(G")dG"](4)
t-h
This is indeed a particular case of t.he finite spectrum assignment problem introduced above. The motivation for using such control laws is their simplicity: the state feedback is linear and static,
I Supported by Conacyt, Mexico 31951-A and by GDRAutomatique: systemes a retards, France. 2 On leave at Heudyasic, UMR CNRS 6599, France.
201
and the distributed component consists of linear combination of elementary distributed delays. As shown below, it permit not only toassign a finite spectrum, but also finite invariant factors. The motivation for assigning invariant factors with a finite number of roots is the same as the one for spectrum assignment: the invariant factors give and additional freedom in shaping the dynamics. If they have a finite number of roots, they can be readily analyzed. The question is then whether or not the freedom, that exists in the general case, is restricted when such simpler laws are used. The problem under consideration is then the following.
also possible to assign prescribed invariant factors with a finite number of roots to the closed loop characteristic matrix.
Lemma 1. Consider a spectrally controllable linear multivariable system with delay in the input described by x (t)
= Ax(t) + E~oBi11.(t -
ih)
(5)
where A E m: nxn . B E JRnxm and the delays in the input are commensurate to h 2: O. Then the following problems are equivalent. (i) The control law
Problem Consider a spectrally controllable input time delay system described by (3). Under what conditions does static state feedback distributed control laws (4) exists, such that the closed loop has prescribed invariant factors with finite number of roots.
J t
u(t)
= I\[x(t) + ~;~o
e(t-17-ih)A Bi11.(O")dO"] (6)
t-ih assigns to the closed loop system (.5-6) a prescribed set of invariant factors
The solution to this problem is organized as follows. Some preliminary results are shown or recalled in Section 2. Then the main result is established in Section 3 and is discussed in the light of the Bezout domain E in Section 4. An illustrative example is presented in Section 5 and some concluding remarks end the paper.
o~(s), o~(s),···. o~(s) .
(i) The control law 11.(t) controllable system
Y (t) = Ay(t)
= Ky(t)
assigns to the
+ ~~oe-ihA B;11.(t)
(7)
a closed loop system with a prescribed set of invariant factors o~ (s), o~(s), ... ,o~ (s).
Notation: IR [s] denotes the ring of polynomials over JP'., the field of reals. The degree of its elements, o(s), is denoted deg (o(s)). JP'. (s) stands for the field of rational functions over m:, while the rings of proper and strictly proper rational functions are denoted. respectively, by IRp(s) and JIl2 sp (s). Further, l!l'.mxnand l!l'.mxn [s] .... , denote the sets of m x n matrices having elements in IR, lR [s], ... , respectively. Units of the ring JP'.mxm [s] are called unimodulars and those of the ring lP'.;,xm(s) bipropers. The set E is a Bezout domain whose elements are fractions of the form h o(s , e- hs ) -- n(s,e') where all the zeros of d(s) , d(s) E lR [s] are zeros of n(s,e- hs ) E IR [s.e- sh ]
Proof. Consider the transformation t
y(t)
J
= x(t) + ~~o
e(t-17-;h)A Bi11.(O")dO" (8)
t-ih The derivative of (8) is
y(t)
= x(t) + E~oe-ihA Biu(t) _E~oe(h-ih)ABi11.(t
- ih)
t
the set of quasipolynomials. Notice that 11+: [s] C E. Units of the Bezout domain E are called unimodulars in E. (for detailed information on these sets and their properties, see Gantmacher (1959), Kailath (l 980), Brethe and Loiseau (1998), and Loiseau (2000, 2001)).
+E~oA
J
e(t-17-ih)A Bi11.(O")dO"
t-ih
=Ax(t) + E~oBi11.(t -
ih)
+E~oe-ihABi11.(t) - E~oBi11.(t - ih)
J t
+~~oA
e(t-17-ih)A B;11.(O")dCJ
t-ih
2. PRELIMINARY RESULTS
= Ay(t)
""e now give a new interpretation of the result established in Manitius and Olbrot (1979), Artstein (1982) that has been extensively used in the literature, in the light of n-assignability, namely the ability to assign a finite spectrum closed loop system, provided that the system is spectrally controllable. Indeed, there is more to say: it is
+ ~~oe-ihA BiU(t).
It follows that the control law (6) assigns the
closed loop
y(t)
= (A + E~oe-ihA B;I\)y(t).
In order to establish the equivalence with (ii), consider the description of the system (5) and the
202
that assigns a closed loop whose invariant factors are monic polynomials o:~ (s), o:~ (s), ... , 0:1. (s) where o;(s) divides O:;_l(s),i = 2, ... ,m, if and only if
closed loop (6) in the frequency domain. They are respectively (sIn -A)x(s) = Ef::oBie-ihsu(s) and (lrn - K(E~oe-ihA(sl - A)-Ill - e-ih(.I-AJ]Bi)U(S)
= Kx(s).
Let
The closed loop characteristic matrix is (
SIn-A -B ) -I{ I m - I\.M .
_ ((SIn - A)(In - M 1\") - BI{
0
*)
Im
i=l
i=l
= m,
and o:;(s)
(11)
= 1, i = m +
Proof. According to Lemma 2, the pair (A, Ef::oe- ihA B i ) is controllable. Let Cl, C2, ... , Cm denote its controllability indices. From the Rosenbrock Control Structure Theorem recalled in the appendix, there exists a control law (lO) that assigns to the system (7) a closed loop with invariant factors o:~ (s), 0:; (s), ... , o:~ (s) where 0:; (s) divides 0:;_1 (s), i = 2, ... , m, if and only if conditions (11) hold. Finally, the result follows from Lemma 1. •
SIn-A -B )(I-M)(IO) -I\. I m -!\.M 0 I I{ I
-
j
with equality for j 1, ... , n.
Post multiplication by well defined unimodular matrices leads to
(
j
LCi ~ Ldego:;, j = 1, ... ,m
~N B. -ihs B .'- .::..Ji=O t e , M := E~oe-ihA(sI - A)-l[I _ e-ih(sI-A)]Bi.
'
the symbol * standing for a nonspecified matrix. Then, the invariant factors of the closed loop are those of (sIn - A)(I - MI\") - BI\", that is
Remark 2. Clearly, the use of simpler control laws has a cost: the degrees of the invariant factors cannot be assigned arbitrarily.
~N -ihA (I ( S'In - A) . (In - ~i=Oe s - .4)-1 . (I - e-ih(sI-A))B;I\") - ~f::oBie-ihs I{ = sIn - A - ~!:oe-ihA[I - e-ih(sI-AJ]Bi!\.) ~N B. -ihsJ" -~i=O ,e \ -- S I n - .4 _ ~N -ihAB·J" ~i=Oe , \
and the result follows . •
Remark 1. The proof follows closely the steps of the proof in Manitius and Olbrot (1979) for nassignability. We recall it here to show that it permits to conclude on the invariant factors of the closed loop matrix, not only on its spectrum. Lemma 2. (Olbrot, 1978) The input delay system (5) is spectrally controllable if and only if the linear system (7) is controllable.
3. MAIN RESULT
Remark 3. Lemma 1 implies that a simple static state feedback distributed control law described by (10) permits to assign to the input delay system (9) a set of desired invariant factors that satisfies the inequalities (11). The parameter I{ of the control law , can be calculated in a straightforward manner as the solution to the problem of a static state feedback invariant factors assignment for the linear system with no delay (7). Notice that toolboxes for the analysis and design of linear systems are available (The Polynomial Toolbox, 1998). If a set of desired invariant factors does not satisfies the inequalities (11), the results obtained in Loiseau (2001) guarantee that there exists a control law described by (2) that allows to assign them. This control law if fully determined by hi,i= 1,2, ... ,/(T) andg(T).
We are now able to state our main result.
Theorem 3. Consider a spectrally controllable linear multivariable system with delay in the input described by x (i)
= Ax(t) + E;'{=oBiU(i -
ih),
4. INTERPRETATION OF THE RESULT IN THE BEZOUT DOMAIN £
(9)
where A E w:. nxn , B i E rn: nxm and h 2: 0 is the delay. Let Cl, C2, ... , Cm be the controllability indices of the pair (A, E!:oe- ihA B;). Then there exist a control law
J t
uti)
= K[x(i) + ~!:o
e(t-a-ih)A Biu(IT)dIT](lO)
t-h
203
The Bezout domain, which definition and basic properties are recalled in the notations. section 1, has proved to be a powerful tool for the study of commensurate time delay systems, by providing a solid algebraic framework that allows the generalization of many results established for linear systems without delays (Brethe, 1997). The following remarks enlightens some subtleties of the machinery in £, in connexion with our problem. First of all, we establish the following clue fact.
Fact 1. The invariant factors in the Bezout domain E of a polynomial matrix coincide with its invariant factor over the ring JR[s].
Consider indeed a polynomial matrix D( s) E JRnxm[s] of rank r. It is well known (Gantmacher, 1959) that there exist unimodular polynomial matrices U(s) E JRnxn[s] and VIs) E JRmxm[s] so that D( s) can be factored as D(s)
= U(s) (~iag{O'i(S).i = 1, ... ,r} ~)
Vis).
(12)
where Qi(S), i = 1, ... , r are unique monic polynomials such that Qi(S) divides Qi-ds) called the invariant factors of D( s). Further, one can show (Brethe, 1987) that E is also a so-called invariant factor domain. Every matrix D(s) over E can also be factored in the form (12), were now the matrices U (s) and V (s) and their inverses are over E, and the Qi(S), i = 1, ... , r are elements of E. which are also called the invariant factors of D(s). Now, since lR[s] C E, it is clear that when D(s) is polynomial, the polynomial factorisation (12) is also a factorisation over E: U (s) and V (s) are indeed matrices over E, which are unimodular over E, the rank of D(s) is rover E, and, since such a factorisation over E is unique. the O'i (s), i = 1, .... r, are also the uniquely defined invariant factors of D( s) over E. Fact 2. If the input delay system (5) is spectrally controllable, then its transfer matrix can be factared in the form (sIn - A)-l B = N(s, e- hs )D- l (s) ,
and
(Irn - lC(Ef':oe-ihA(sI - A)-I. (I - e-ih(sI-A))B;)u(s) = I\x(s). We first observe that a natural factorization also exists for the closed loop system. Let indeed (lv·I\(s,e-s),DK(s.e- S)) be a coprime factorization over [. of the transfer of the closed loop system. The closed loop denominator is given by Dl\(s,e- S ) = «(Im _/\'(~;"~'le-'hA(s1- A)-I (I - e-ih(sI-A))B;))D(s) -1-.,:N(s) = D(s) - 1\'N(s) -I{(~~I e-· hA (s1 - .4)-1 (I - e-·h(sI-AI )B, )D(s) = D(s) - /\'N(s) - /\'(s1 - A)-I ~~I e-· hA B;D(s) +h:(sI - A)-I ~;"~I e- ihs B,D(s)
Observe that the matrix
= Et': 0 (sIn - A)-l Bie-ihs )D-1( ) = "N --'i=Oe -ihsN( i ss,
(I m - j \'( SI - .4)-1"N --'i=Oe -ihAB·) , is a biproper matrix in IR:(s), hence DK(s,e- S ) is actually rational in the variable s, and, since it has no pole, the result follows.
Ni(s) = (sIn - A)-l BiD(S) , then the transfer of the system without delay (7) can be expressed as 4)-1 e -ihAB·t -_
= ~~OBie-ihsu(s)
= D(s) - E(sI - A)-l~t':l e- ihA BiD(s) = (Im - I\'(sI - A)-l~~le-ihA B;)D(s).(13)
whith
~
(sIn - A)x(s)
DK(s,e- S)
We can further state that if
"N """"i=O (1 S n -
We are now able to revisit the problem in the framework of E. The system and the control law are respectively described in the frequency domain by
and it follows from (??) that
where D(s) and N(s,e- hS ) are right coprime matrices that are respectively polynomial in the variables sand s, e- hs .
(sIn - A)-l B
that D(s)U(s,e- hs ) = D'(s), for some matrix U(s,e- hs ) which is unimodular over E. We can see that U(s,e- hs ) = D-l(s)D'(s) is actually rational in the variable s. Since it is also a matrix over E, an analytic function without pole, we conclude that it is actually polynomial in s, which shows the result. In the sequel, we shall call such a factorization with a polynomial denominator the natural factorization of the system.
One can further notice that a left product by a biproper matrix does not modify the column degrees. Thus the column degrees of D K (s, e- S ) are those of D( s).
("N "( )) D- 1 ( ) ":"'z=oe -ihA lV t s s .
Both systems have the same denominator. which is of course polynomial in s. As it is usual for systems without delays, we can assume that D(s) is comumn reduced, with column degrees Cl, C2, ...• Cm. We can notice in addition that the column degrees of such a column reduced polynomial denominator of the transfer of the system (5) are uniquely defined. Any other polynomial denominator, say D'(s) is indeed so
The natural coprime factorization bases a design algorithm for the finite spectrum assignment, which consists of the following steps. (i) Calculate a natural factorization (N(s,e- Sh ),
D(s)) of the transfer matrix of system (5). D(s) is column reduced with column degrees Cl,CZ,···,C m ·
204
(ii) Choose a set of monic polynomials O'~ (s), where O':(s) divides O':_l(S) for i = 2, ... , m" and take O'~+ds) = ... = O'~(s).
It is possible to design a control law that assigns prescribed invariant factors with finite roots, of degree {2, O}. A solution is given by
(iii) Providing that (11) hold, there exist a polynomial matrix DK(S), that is column reduced, with column degrees Ci and invariant factors 0: (s), for i = 1, ... , m, and constant matrices X and Y of convenient dimensions, X being invertible, so that
(h - F((s,e-S))D(s,e- S) - G(s,e-S)N(s,e- S)
O'~(s), ···,a~(s)
Algoritms to compute DK(S), X, and Y are provided by Kucera (1991). The feedback 1": = _X- 1 Y assigns to the closed loop system (3-4) the invariant factors (s), for i = 1, ... , n.
0':
The situation is quite different when considering factorizations over E. As shown in Loiseau (2001), contrarily to what happens in ][{(s), post multiplication of elements of E by unimodular over E modify the column degrees. An immediate consequence of this fact is the non unicity of the column degrees of the denominator of a left coprime factorization in E. Moreover, there is complete freedom in choosing the column degrees of such a factorization.
= DK(s,e-
ll
1
into (13), gives
-1) (15)
= (s'2-kkllS+kk1'2S-kk1'2 '2IS
+
2'2S -
1
'2'2
.
Now, if we want to assign invariant factors of degrees {I, I}, for instance both invariant factors taking the value s + 1, let us consider according to the design procedure given in Loiseau (2001) a coprime factorization with column degrees {l, 1} for the denominator. This new factorization is obtained as
(Irn - F(s, e-Sh)D(s, e- sh ) + G(s, e-Sh)N(s,e- sh ) = Z(e-hS)diag{o:;(s)} .
D' (s, e- S) = D(s)U(s, e- S ), N'(s, e- S) = N(s, e-S)U(s, e- S ),
This shows that a control law of the form (2) permits to assign the invariant factors (s) to the closed loop system (3-2). It should be clear that, in that case, a simpler control law of the form (4) cannot get this assignment.
0':
where U(s, e- S ) is given by U(s,e- S )=
2
1
s~:n~ -~ -0
( -s
-0
)
.
s+ln2
This leads to
5. ILLUSTRATIVE EXAMPLE
(Z(e- S ) - F(s, e-s))D'(s,e- S ) + G(s, e-S)N'(s, e- S ) = diag{ s + 1, s + 1} ,
The above results and design procedures are illustrated with the two dimensional academic example 0 0
(kk'21 kkn'2)
The invariant factors of degree {2, O}, namely {S2 + (-k l l +k 1'2 -k'21 +kd - (k 12 +k'2'2), I} , can indeed be assigned to arbitrary finite locations by choosing 1{. We have in this simple case obtained a parametrization of all the invariant factors that can be obtained with the control laws under consideration.
0':
(11)
=
K
-
0':
= (00) 1 0 x(t) +
(14)
).
According to Theorem 3, a control law where G(s.e- S) K, and F(s,e- S) 1":(s1 hA A)-l(e- e- hs I)B does the job, and substituting D(s), N(s, e- S), A, B, and
DK(s,e- S )
If a set of invariant factors (s) does not satisfy the inequalities (11), exept for j = m, one can find a factorization of the transfer where D(s, e- Sh ) is column reduced, with column degrees deg (s), i = I, ... , m, and there exist a unimodular matrix in e- hs , say Z(e- hs ) and matrices F(s, e- sh ) and G(s, e- sh ) over E, such that
. x(t)
S
with G( s, e
u(t - h).
-S)
= ((I+ln2)(2-e2 _ e- s
S 1 2 - eS Z(e- )= ( 13-e- s
S )
In2(2-e-
S ))
0
)
'
e S Fll(s,e- S ) = (s(I+ln2))+ln2) C-s - )'2 ,
This system is spectrally controllable because the pair (A. exp( -hA)B) is controllable (See Lemma 2). Its controllability indices are {2,0}.
(1 - e- s )2
F 12 (s,e- S )=(I+ln2) F2ds,e- S ) = (l_e- S)2,
The natural coprime factorization for this system IS
s
+
1 - e- s
,
s
and 1 N(s,e ") = (se-so) e-S 0 ,D(s) = (s2_ 0 1 ) .
Fn(s,e- S ) = (l-e-
_0
s
205
S
f
S
+(I_ln2)2-es + In 2
Of course this calculation is uneasy, and so is the expression of this control law in the time domain (Brethe, 1997), (Brethe and Loiseau,1998), since there is no specialized toolbox to deal with E.
6. CONCLUSION
It is shown that static state feedback distributed control laws for input delay systems that allow the assignment of a finite spectrum, can be used to assign as well invariant factors with a finite number of roots. This feature is indeed of interest in shaping the dynamic of the closed loop system. It is shown that the multiplicities of the invariant factors cannot be assigned freely: they are restricted by Rosenbrock inequalities by a set of well defined integers. The result is explained in the framework of the Bezout domain E. where more general control laws allow a complete freedom in assigning the multiplicities of the invariant factors. Clearly, static state feedback distributed control laws can be designed in a straightforward manner, but at a cost: the freedom in the assignment of the multiplicities of the invariant factors is restricted.
7. APPENDIX We recall here the Theorem of Rosenbrock (for instance from Kucera (1991)), which is a clue for our proof. Lemma 4. Consider a linear controllable system described by
= Ax(t) + Bu(t)
x (t)
where A E IP: nxn . B E IP;nxm and such that the controllability indices of the pair (A. B) are Cl, C2, ... , Cm' There exists a static state feedback u (t)
= I{ x (t )
such that the invariant factors of the closed loop system are O'~ (s), O'~ (s), ... , O'k(s) where O';(s) divides 0';_1 (s), i = 2, ... , rn, if and only if j
L
j
Ci :::;
i=l
L deg (O';),j = 1. ... ,
rn
i=l
with equality for j
= rn,
and O';(s)
= 1. i = m +
1, ... , n.
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