CC BY
14
YflK 517.977
V. M. Marchenko, professor; Z. Zaczkiewicz, magistr
ON THE WEAK OBSERVABILITY OF SMALL SOLUTIONS OF DIFFERENTIAL-ALGEBRAIC SYSTEMS WITH DELAYS1
The pa per c onsiders t he problem of ob servability of s mall s olutions f or hybrid time in variant differential-difference dynamic systems, i. e. linear stationary differential-algebraic systems with delays (DAD systems). Several types of observability of small solutions are defined and the corresponding parametric c riteria are gi ven. S pectral o bservability is c onsidered and relation of th e s pectral observability to the observability of small solutions is discussed.
Introduction. The behaviour of a number of real physics pr ocess c onsists of a c ombination of dynamic ( differential) a nd a lgebraic (functional) dependencies. These processes are described by differential-algebraic (DAE) systems. In that sense these s ystems a re hy brid s ystems. It s hould b e noted that the term «hybrid systems» has been widely used in the literature in various senses [1].
The paper deals with the weak obs ervability of small solutions of DAD systems; it is an extension of t he work [ 2]. T he s mall s olution is a s olution that goes to zero faster than any exponential function. Existence of such solutions for linear retarded systems was pr oved by H enry [3] a nd later b y Kappel [4] for linear neutral type systems. Lunel [5] gave explicit characterization of the smallest po ssi-ble time for which small solutions vanish. Observability of small solutions for the retarded time delay system c ase w as f irst studied by M anitius [ 6] and for general neutral system by Salomon [7].
1. Preliminaries. Let us consider DAD system in the form
x1(t) = A11x1(t) + A12x2(t), t > 0,
(1)
X2 (t) = A21X (t) + A22X2 (t - h), t > 0, (2)
with output
y(t) = B x(t) + B x2(t), (3)
Here x1(t) e R"1, x2 (t) e R"2, y(t) e R"
t > 0;
A11 e R "1 "",
A12 e R "1 ""2,
A22 e R"2""2,
B e Rr
B2 e Rrx« are constant matrices h is a constant delay, h >0. We regard an absolute continuous ^-vector functions Xj(-), an d a p iecewise co ntinuous x2 (•) «2-vector functions as the solutions of systems ( J)-(3), if t hey s atisfy t he equation (J) f or a lmost a ll t > 0 and (2) for all t > 0. System (I)-(3) should be completed with initial conditions in the form
Xj(0+) = Xj(0) = Xjo,
(4)
X2OO = v(t), Te[-h, 0),
where ye PC ([-h, 0), Rm ) and PC ([-h, 0), Rm ) is a s et of p iecewise c ontinuous m-vector f unctions i n [-h, 0].
Let E(g) denote t he exponential t ype of g : C ^ C, assuming g is an entire function of order 1. Then
E ( g ) = limsup
log I g (s)|
For g : C ^ Cq the e xponential type o f g is defined by
E ( g ) = max E ( g} ), where g = [ g ... g
1<j<q J L
J.
Let A(p) be the characteristic matrix function
A( p) =
PI"1 - A11
—Ati
A12
42 - A22e
- ph
The matrix function A(p) appears by applying the L aplace t ransform t o s ystem ( 1)-(3). L et detA(p) be the determinant of A(p). It follows from t he a bove t hat t he exponential t ype of det A(p) is less or equal n2h. Define s by
E(det A(p)) = n2h - s.
Let adj A(p) be the matrix function of cofac-tors of A(p). Since t he c ofactors C j are (n + n2 -1)(nj + n2 -1) subdeterminants of A(p), the exponential type of the cofactors is less or equal n2 h. Define a by
max E(C„) = n2h - a.
1<i, j <nj +n2 1
We have [2].
Proposition 1. For x1(-), x2(-) being solutions of system (1)-(3) the following implications hold:
i) if V£ e Z x1 (t)ekt ^ 0 as t ^ +œ, then x1 (t) = 0 for all t > e - c;
ii) if Vk e Z x2 (t)ekt ^ 0 as t ^ +œ,
(5)
then x2 (t) = 0 for all t > s - c. (6)
Definition 1. We say that a solution Xj(-), x2(-) is small, if there exists T > 0 such that xx(t) = 0, x2(t) = 0 for t > T. A small solution is trivial, if it is zero for t > 0.
s
s
1 The work is written in the framework of the cooperation with Bialystok Technical University.
Definition 2. We say that system ( 1)-(3) has a nontrivial small solutions, if there exists a solution x1(-), x2() such that conditions (5), (6) hold and at least x1(-) or x2()is not trivial.
Definition 3. We say that system ( 1)-(3) has a nontrivial s mall s olution with r espect to x1, if there exists a solution x1(-), x2() such that condition (5) holds and x1 (•) is not trivial.
Definition 4. We say that system ( 1)-(3) has a nontrivial small solution with respect to x2, if there exists a solution x1(^), x2() such that condition (6) holds and x2() is not trivial.
2. Observability.
2.1. Observability of small solutions.
Definition 5. We say that nontrivial small solutions of s ystem ( 1)-(3) a re o bservable, i f every nontrivial s mall s olution has nonzero output f or some t > 0. This means that
x1(t ) = 0Vt > T 3T > 0 x2(t) = 0Vt > T y(t) = 0Vt > 0
x,(t ) = 0, x2(t ) = 0, Vt > 0.
Theorem 1. Nontrivial small solutions of system ( 1)—(3) are observable, if and only if the following conditions hold:
i) max rank
AEC
A11 -AI, A12 0
A21 - A22
0 A22 0
B1 B2 0
= n + n2 + rankA-
"22 '
(7)
ii) rank
B2 ^22
B2 (42 )2
B2 (^ )n
. (A22 )"
= rank
B2 A-22 B2 ( A22 )2
B2 ( A22 )"
( A22 )"
A
(8)
Proof. The proof is similar to theorem 2 and it can be omitted.
2.2. Spectral observability.
Definition 6. System ( 1)-(3) is i nfinite-time observable, if for all initial data for which y(t) = 0 for t e[0, <x>) there exists t1 such that x1(t) = 0 and x2 (t) = 0 for t e [tj, Do
Definition 7. System (1)-(3) is finite-time observable at t2, if for a ll initial data, f or w hich y(t) = 0 for 2 e [0, d), x^t) = 0 and x2(t) = 0 for
t e [t2, D)o
Definition 8. System (1)-(3) is spectrally observable, i f all its e igenvalues are o bservable. A n eigenvalue X is o bservable if t he c orresponding eigensolution of the f orm Xj (t) = exp(Xt)Xj (0),
x2(t) = exp(Xt)x2(0), xj(0) ^ 0, x2(0) ^ 0, obtains y(t) = 0 for t e [0, £0 . We have [2].
Proposition 2. System (1)-(3) is spectrally observable if and only if
rank
iX- A11
- A21 B
A12 ^ - A22^ B
= "1 + n2, (9)
for all complex X.
Proposition 3. System (1)-(3) is spectrally observable if and only if system (1)-(3) is infinite-time observable.
Corollary 1. System (1)-(3) is spectrally observable if and only if system (1)-(3) is finite-time observable at s - c.
Proof. By proposition 1 and proposition 3.
3. Relative observability of small solutions.
Definition 9. Nontrivial s mall s olutions with respect to x2 of system (1)-(3) are weakly observable, if every nontrivial small solution with r e-spect to x2 has nonzero output for t > 0 and x1 is a zero solution, i. e.
x1(t ) = 0 Vt > 0 3T > 0 x2(t) = 0 Vt > T y (t ) = 0 Vt > 0
• x2(t) = 0, Vt > 0
Theorem 2. Nontrivial small solutions with respect to x2 of system (1)-(3) are observable if and only if the following condition holds:
rank
B2 A-22 B2CA22)2
B2( A^)"2
= rank
B2 A22
B^f
B2 ( A22 )" A
(10)
Proof. The necessary condition. We assume
Î0, xe (-h, 0),
that x1(t) = 0, t > 0, 9(1) = <| Then
[9 T = -h.
equation ( 1) is satisfied for almost all t > 0 and weak observability with respect to x2 of system (1)—(3) means t hat conditions B2A2290 = 0, ..., B2(A21)k 90 = 0 for k = 1,2,... implies A2290 = 0 that by the Cayley - Hamilton theorem is equivalent to condition (10).
The sufficient condition. If condition (10) is satisfied then there exists a matrix D e Rn2xrn2 such
that A22 = D
B2 A-22
B2(A,2)2
B2( A22)"
For a ny i nitial
function ф(х),те [-h, 0] for which B2(A22) ф(х) = 0, те [-h, 0], k = 1,..., n2 condition А22ф(т) = 0, те [-h, 0] is also satisfied that is equivalent to the weak obs ervability of nontrivial s mall s olutions of system (1)—(3) with respect to x2.
Definition 10. We s ay t hat x1(t), t > 0, x2(t), t > 0 is a strong solution of system ( 1)-(3), if equations (1)-(3) are satisfied for all t, t > 0 (the derivative in (1) we means right-hand derivative at t = 0).
Theorem 3. Nontrivial strong s mall solutions with respect to x2 for system ( 1)-(3), are o bservable if and only if the following condition holds:
rank
A A
12 22
A An2
12 22
B2 A22
, B2 A22
= rank
A A
12 22
A An2
12 22
B2 A22
B2 A22
22
(11)
Proof. Definition 10 is equivalent to
A12 x1(t) = 0, t > 0 x2 (t) = A22x2 (t - h), 0 J ^ x2 (t) = 0, t0> У(t) = B2 x2(t), t > 0
«
«
A,2( AJ ф(т) = 0j B2( A22)] ф(т) = 0 J
^ A22ф(т) = 0,
where
Те -h, 0), i = 1,..., П2; j = 1,..., П2]
«
« {фТ (т)[( AÏ2Ï42,( A22 ) ]B:
T
i = 1,..., П2; , =
П2]:
фТ (Т) A2T2 =0}.
It is equivalent to (11).
Corollary 2. If nontrivial strong small solutions with respect to x2 of system ( 1)-(3) are observable t hen nontrivial s mall s olutions w ith r e-spect to x2 of the system are also observable.
Definition 9. Nontrivial s mall s olutions with respect to x1 of system (1)-(3) are weakly observable i f e very non trivial small solution w ith respect to x1 has nonzero output for t > 0 and for x2 being zero solution, i. e.
x2(t) = 0 Vt > 0 3T > 0 x1(t) = 0 Vt > T y(t) = 0 Vt > 0
x1(t) = 0, Vt > 0
Theorem 4. Nontrivial small solutions with respect to x of system (1)-(3) are always observable.
Proof. Condition x2(t) = 0 for t > 0 implies x1 (t) = A11 x1 (t), and the system has solutions of the form x1 (t) = eAlltx1 (0), it proves that all small solutions with respect to x1 are trivial.
Conclusion. In this paper we investigated the problem of r elative weak obs ervability of nontrivial s mall s olutions of the hybrid differential-difference (HDR) systems. Weak observability of nontrivial small solution with respect to x2 and x1 are considered. Strong small solutions are defined and weak obs ervability of nontrivial s trong s mall solutions with respect to x2 is established. Other types of observability and relations between these types of observability are discussed.
References
1. Марченко, В. М. Некоторые нерешенные задачи в теории управляемых динамических ГДР систем / В. М. Марчгнко // Труды БГТУ. Сер. VI, Физ.-мат. науки и информ. - 2006. -Вып. XIV. - С. 3-6.
2. M archenko, V . M . O bservability of s mall solutions of l inear di fferential-algebraic s ystems with delays / V. M. Marchenko, Z. Zaczkiewicz // Control and Cybernetics. - 2006. - Vol. 35, № 4. -P.997-1013.
3. Henry, D. Small Solutions of Linear Autonomous F unctional D ifferential Equations / D. Henry // J. Differential Equations. - 1970. - Vol. 8. - P. 494-501.
4. Kappel, F . Laplace transform methods and linear autonomous functional differential equations / F. Kappel // Math. Institur, Univ. Graz, Bericht. -1976. - № 64.
5. Verduyn Lunel, S . M. A Sharp Version of Henry's Theorem on Small Solutions / S. M. Ver-duyn Lunel // J. Differential Equations. - 1986. -Vol. 62, № 2. - P. 266-274.
6. Manitius, A. F-controlability and observability of linear retarded systems / A. Manitius // Appl. Math. Optim. - 1982. - Vol. 9. - P. 73-95.
7. Salamon, D. On controllability and observability of time delay systems / D .Salamon // IEEE Trans. Aut omat. C ontr. - 1984. - Vol. 29. -P. 432-439.
