Keywords: linear differential-algebraic equations, index of matrix pencil, asymptotic stability, structured perturbation, complex stability radius.. Introduction *.[r]
Trang 118
Some Remarks on the Complex Stability Radius of
Differential-Algebraic Equations
Le Huy Hoang*
Faculty of Informatics, National University of Civil Engineering, Dong Tam, Hanoi, Vietnam
Received 05 December 2012 Revised 28 February 2013; accepted 15 March 2013
Abstract This paper is concerned with the robust stability of linear differential-algebraic
equations (DAEs) A system of linear DAEs subjected to structured perturbation is considered Computable formulas of the complex stability radius are given and analysed A comparison of our formula to previous results is given
Keywords: linear differential-algebraic equations, index of matrix pencil, asymptotic stability, structured perturbation, complex stability radius
1 Introduction *
Differential-algebraic equations (DAEs) play an important roles in mathematical modeling of real-life problems arising in a wide range of applications, for example, multibody mechanics, prescribed path control, eletrical design, biology, biomedicine, see [1, 2] and references therein On the other hand, the robustness issue is a crucial problem for the application of control theory, for example, one
of the basic goal of feedback control is to enhance system robustness Robust stability is also an important topic in linear algebra as well as in numerical analysis
Consider a linear DAE
where A E, ∈ Kn n× , K= o r The leading coefficient matrix E is singular
Definition 1.1 (see [1]) The matrix pencil { , }E A is said to be regular if there exists t ∈ such that the determinant of (A tE− ), denoted by det(A tE− ), is different from zero We also say that (1.1) is regular Otherwise, if det(A tE− )=0, ∀ ∈ t , we say that { , }E A is irregular
_
*Tel.: +84-973633002
Email: lehuyhoangxd@gmail.com
Trang 2If { , }E A is regular, then a complex number t is called a (generalized finite) eigenvalue of { , }E A
if det(A tE− )=0 The set of all eigenvalues is called the spectrum of the pencil { , }E A and denoted
by σ( , )E A
If E is singular and { , }E A is regular, then we say that { , }E A has the eigenvalue ∞
Suppose that the matrix pencil { , }E A is regular Then the pairs can be transformed to Kronecker
canonical form i.e there exist nonsingular matrices P , Q such that
r
n r
J O
I O
O I
where N is a nilpotent matrix of index k (see [1, 2]) If N is a zero matrix, then k = 1 Furthermore,
we may assume without loss of generality, that N and J are upper triangular If { , }E A is regular, then the nilpotency index of N in (1.2) is called the index of matrix pencil { , }E A and we write
}
ndex{
i E A, =k
In particular, a regular index-one system can be given by the form
11 12
11 12
21 22
E E
A A
where A22 and E11−E A A12 22−1 21 are square and of full rank (or invertible matrices)
Now, we give the definition of asymptotic stability of the solution of (1.1), see [2]
Definition 1.2 Suppose that {E, A} is regular Let Q be a projector onto the subspace of consistent
initial conditions Let P = I - Q We say that the null solution of (1.1) is stable if for any ε > 0, there exists δ > 0 such that for an arbitrary vector y0∈ n satisfying ||y0||<δ , the solution of the initial value problem
0
Ey x Ay x x
P y y
exists uniquely and the estimate || ( ) ||y x < ε holds for all x ≥ 0
The null solution is said to be asymptotically stable if it is stable and lim || ( ) || 0
→∞ = for solution y
of (1.1) If the null solution, of (1.1) is asymptotically stable, we say that system (1.1) is asymptotically stable
Theorem 1.1 (see [3]) The null solution, of system (1.1) is asymptotically stable if and only if the
eigenvalues of the matrix pencil {E, A} all have negative real part
If the eigenvalues of the matrix pencil {E , A } all have negative real part, then the matrix pencil { E , A } is said to be stable
Now, let us suppose that system (1.1) is asymptotically stable and consider the perturbed system
(E + B E A E C)y’ = (A + B A A A C)y, (1.3)
Trang 3where B A∈ Kn p× 1, B E∈ Kn p× 2, Kq n
C∈ × are given matrices, ∆ ∈A Kp1×q, ∆ ∈E Kp2×q (K=
or K= ) are uncertain perturbations B A∆A C, B E∆E C are called structured perturbations Denote
A
E
∆
∆
, we define the set of "bad" (destabilizing) perturbations
1 2
K
.
is either irregular or unstab e
K
l
V
Definition 1.3. Let, the system (1.1) be asymptotically stable The structured stability radius for (1.1) is defined by rK =inf{∆ ∆ ∈ V| K},where · is a matrix norm induced by vector norms in
1 2
( )
K p+p ×q
Depending on K= or K= , we talk about the complex or the real stability radius, respectively Obviously, we have the estimate r≤rThe problem of computing the stability radius for ODEs was introduced in [4-7] Later, the result was extended to DAEs in [8-12], The aim of this paper is to compute a general formula of the complex stability radius Moreover, a comparison of our formula to previous results is given
The outline of this paper is as follows Firstly, the complex structured stability radius for (1.1) is given Furthermore, we show the details on stability and structure robustness of systems of index one Finaly, we show that the formula of the complex stability radius by Byers and Nichols in [8] can be obtained as a special case of our result
2 Main result
Theorem 2.1. The complex structured stability radius for (1.1) is given by
i
t
−
∈
=
where G t1( )= −C A tE( − )−1B A, G t2( )=tC A tE( − )−1B E
Proof To prove this theorem, we use the technique that is same as in [9] First, we prove that
1
0
t R
−
≥
=
To this end, we prove that
1
0
t R
−
≥
≥
There are two cases in which {(E+B E∆E C), (A+B A∆A C)} is either unstable or irregular
Trang 4The first case Let {(E+B E∆E C), (A+B A∆A C)} be unstable, then there exists
t∈σ E+B ∆ C A+B ∆ C and Re( )t ≥0 There exists x ≠ 0 satisfying,
(A+B A∆A C x) =t E( +B E∆E C x)
⇔ (A tE x− ) = −( B A∆A C+tB E∆E C x)
)
E
x A tE − B tB ∆ Cx
∆
Cx = C A tE − − − B tB ∆ Cx
Given u = Cx, we have, u = [ G t1( ) G t2( ) ] ∆ u
Re 0
t
G t G t
−
≥
∆ ≥
or
Re 0
t
−
≥
≥
The second case Let {(E+B E∆E C), (A+B A∆A C)} be irregular which means that for any t ∈
we have det((A+B A∆A C)−t E( +B E∆E C))=0 Given t such that Re( )t ≥0, then there exists
0
x ≠ satisfying
t E+B ∆ C x= A+B ∆ C x
Similarly to the first case, we obtain
Re 0
t
G t G t
−
≥
∆ ≥
It is clear that, in any case,
1
0
t
G t G t
−
≥
Now, we prove the inverse inequality
Re 0
t
−
≥
≤
Indeed, for any ε > 0, there exists t0 having Re( )t0 ≥0 such that
1
Re 0
t
−
−
≥
Trang 5We construct a destabilizing perturbation A
E
∆
∆
such that ∆ = [G t1( )0 G t2( )0 ]−1 There exists a vector p1 p2
x∈ + , x = 1 such that [G t1( )0 G t2( )0 ]x = [G t1( )0 G t2( )0 ] Invoking a corollary of the Hahn-Banach theorem, there exists a functional y*∈ q, y* =1 such that
*
1( )0 2( )0 1( )0 2( )0 1( )0 2( ) 0
1( )0 2( )0
1( )0 2( )0 1( )0 2( )0 1( )0 2( )0
= [G t1( )0 G t2( )0 ]−1x G t[ 1( )0 G t2( ) 0 ]
We deduce ∆ ≥ [G t1( )0 G t2( )0 ]−1
On the other hand, from the definition of ∆, we have ∆ ≤ [G t1( )0 G t2( )0 ]−1
Thus, ∆ = [G t1( )0 G t2( )0 ]−1
We show that the perturbed system will be either unstable or irregular
1 0 2 0
1
−
Multiplying both sides with 1[ ]
( A t E − )− − BA t BE from the left, denoting
1
u = A t E − − − B t B x, we obtain
1
0
A
E
−
∆
∆
We have u ≠ 0 because x ≠ 0 It is obtained that eithert0∈σ((E+B E∆E C), (A+B A∆A C)) or { ( E + BE∆EC ), ( A B + A∆AC ) } is irregular
Because Re( )t0 ≥0 then the perturbed system is either unstable or irregular It is clear that
1
Re 0
t
−
−
≥
deduce
Trang 6[ ]
Re 0
t
−
≥
≤
From (2.5) and (2.6) we have
Re 0
t
−
≥
=
To complete this proof, we note that [ G t1( ) G t2( ) ] is analytic in \ −, due to the maximum principle, their least upper bound is attained in i (at a finite point or at infinity) Hence,
Re 0
=
Finaly, we have { [ ] } 1
i
t
−
∈
=
where G t1( )= −C A tE( − )−1B A, G t2( )=tC A tE( − )−1B E □
Remark 2.1
If E is nonsingular matrix, then
t
→+∞
If E is singular matrix, then
1
1
1 1
1
1
1
r
k
n r i
r
i
n r i
t
−
−
=
−
−
−
−
=
−
+
∑
∑
Thus, the complex structured stability radius for (1.1) is strictly positive only if the
index E A = Moreover, for index E A{ , } 1= , the radius is positive when the structured perturbation of the leading term has influence only on the differential part And from now on, we consider the system having index one with suitable structured perturbations of the leading term
Lemma 2.1. Let, {E, A} be regular
If index E A{ , } 1≤ then deg det( [ A tE − ) ] = rank E
Trang 7If index{ , } 1E A > then deg det( [ A tE − ) ] < rankE
Proof Let pencil {E, A} have canonical form,
r
n r
J O
I O
O I
O N
−
• If k = 0 then E=P I Q−1 n −1, A=P JQ−1 −1 It means that rankE = n
1 1
det(A tE− )=det(P Q− − ) det(J−tI n)
Because det(J−tI n) is polynomial of degree n, so deg det( [ A tE − ) ] = rankE
• If k = 1, then N is null matrix, and 1 r 1.
n r
J tI O
−
−
Hence, det(A tE− )=det(P Q−1 −1) det(J−tI r)
Because pencil { , }E A is regular, so det(A tE− )≡/0, or det(J−tI r) is polynomial of degree r
On ther other hand, rankE = r, so deg det( [ A tE − ) ] = rank E
• If k > 1, then N is in Jordan form, N = dia g [ J1, , Jl], and
,
r
− − −
where MT is upper triangular matrix which have diagonal elements equal to one We deduce
1 1
det(A tE− )=det(P Q− − ) det(J−tI r)
Because pencil { , }E A is regular, so det(A tE− )≡/0, or det(J−tI r) is polynomial of degree
r On ther other hand, rankE > r, so deg det( [ A tE − ) ] < rank E □
To prove the following lemma, the technique is used in the proof of Theorem 2.1 can be applied Thus, we obtain the result
Lemma 2.2 Let M∈ n n× is nonsingular matrix If d =inf{∆ ,M +H T∆ is singular}, where H∈ n p× , ∆ ∈ p q× , T∈ q n× are given matrices, then d = TM H−1 −1
Without loss of generality, the system having index one can be simplified as follows
·
1 11 12 1
·
21 22 2 2
,
y
=
where A22 is invertible From the Remark 2.1, for index{ , } 1E A = , the radius is positive when the structured perturbation of the leading term has influence only on the differential part So that, we consider the structured perturbations with
Trang 81 2
A
B B B
=
B B O
=
, C = [ C1 C2]
Theorem 2.2 Let pencil { , }E A be regular and index one For any A (p1 p2 ) q
E
+ ×
∆
∆
satisfies ∆ < r, we have
deg det((
ind
e
Proof Consider the perturbed system
1 11 1 1 12 1 2 1
2 21 2 1 22 2 2 2
.
I B C B C
=
The perturbed system has index one if and only if A22+B2∆A C2, and
1
1 2( 22 2 2) ( 21 2 1)
I+ ∆B C − ∆B C A +B∆ C − A +B∆ C are invertible
Denote R1=inf{∆A ,A22+B2∆A C2 is singular} Using Lemma 2.2, we obtain
1 1
1 2 22 2
R = C A B− −
Next, we prove inequality
1 1
1 2 22 2
t i
G t C A B
−
−
−
∈
≤
i.e
2 22 2
t C A tE − B C A B−
We have (see [13,14])
11 12
21 22
A tI A
A tE
−
1 1
11 12 22 21
12 22
1
22 21 22
,
A tI A A A O
I A A
A A I
−
−
−
So
−
−
We deduce
1 2 22 21 11 12 22 21 1 12 22 2 2 22 2
C tE A − B C C A A− tI A A A A− − B A A B− C A B−
We have lim ( ) 1 A 2 221 2
t C A tE − B C A B−
Hence,
Trang 9{ }1
t i
−
∈
This means that A22+B2∆A C2 is nonsingular if ∆ < r
On the other hand, we have
21 2 1 22 2 2
1
2( 22 2 2)
−
1
1 2 22 2 2 21 2 1
22 2 2
A
−
22 2 2 21 2 1
It is easy to see that I+ ∆B E C1− ∆B E C A2( 22 +B2∆A C2) (−1 A21+B2∆A C1) is singular if and only if
21 2 1 22 2 2
21 22 2
E
+
∆
Denote
2 inf
21 22 2
+
Using Lemma 2.2, we have
1
2 2 22 2 ( 1 2 22 21)
R = C A B− C −C A A− B −
Now, we need to prove
2 22 2 1 2 22 21
t C A tE − B tC A tE − B C A B− C C A A− B
From
−
−
We deduce
1 2 22 21 11 12 22 21 1 12 22 2 2 22 2
1 2 22 21 11 12 22 21
tC A tE− − B =t C −C A A− A −tI−A A A− − B
We obtain
2 22 2 1 2 22 21
t C A tE − B tC A tE − B C A B− C C A A− B
i
t
−
∈
Trang 10
In conclusion, if ∆ < r, then index E{( +B E∆E C), (A+B A∆A C)} 1= Using Lemma 1.1, we have
deg det(( A B + A∆AC ) − t E ( + BE∆EC ) ) = ( E + BE∆EC ) = nk E □
The algebraic structure of an index one DAEs is characterized by the index and the number of the finite eigenvalues of the pencil We denote
is either irregular or unstable,
or its algebraic structure changes
p p q
+ ×
V
The algebraic structure of the pencil {(E+B E∆E C), (A+B A∆A C)} changes if its index, or the number of finite singular values, or both change Now, for a DAEs of index one, the complex structured stability radius can be redefined as following r=inf{∆ ∆ ∈| V}
By the Theorems 2.1 and 2.2, the following result immediately follows
Theorem 2.3. Using the same assumption as in the Theorem (2.1) and (2.2), we have
i
t
−
∈
(2.7)
where G t1( )= −C A tE( − )−1B A, G t2( )=tC A tE( − )−1B E
Example 2.1 For sake of simplicity, we use the maximum norm as the matrix norm
Consider
−
It is easy to verify that index E A{ , }=2, σ ( , ) E A = − { } 1 With C = I,
1 0 0
2 0 0
0 0 0
A
B
,
E
B
, we have
1 1
1
1
A
t
G t C A tE − B
1 2
0 0 1
E
t t
G t tC A tE − B
−
+
Trang 11Hence, [ 1 2 ]
1
t
∞
The least upper bound is attained at t = 0 So, 1
3
r = , we construct the destabilizing
perturbations
0
A
,
1
3
E
{ }
σ + ∆ + ∆ = which means that the system is unstable
In the case which the least upper bound attained at ∞, we consider arbitrary sequence t n
proceeding to ∞ Then, for each t n, we construct the destabilizing perturbations
n A n E
∆
The
sequence
n
A
n
E
∆
∆
which have limits being the stability radius But the perturbations that have norm
equal to the stability radius only change the algebraic structured To verify that, we consider following
Example 2.2 Consider
1
1
2
−
−
−
It is easy to verify that index E A{ , } 1= , 1
( , )
2
E A
With C = =I B A and
1 0
E
−
1 1
4 1
1 1 2
A
t t
G t C A tE − B
1 2
2 0
E
t
t
t
∞
+
Trang 12the least upper bound attained at t = ∞ Considering the sequence t n =in, for each t n, we construct
the destabilizing perturbations 2 2 2 2
n A
, ∆ =n E O It is easy to see that
n
A
n
E
∆
is a destabilizing perturbations i.e the pencil {( ), ( )}
E+B ∆ C A+B ∆ C is unstable On
the other hand, we have
n
, ∆ → ∆ =n E E O when n → ∞ But A
E
∆
∆
only
changes the algebraic structured because σ((E+B E∆n E C), (A+B A∆n A C))= ∅ This can be explained that the least upper bound is not attained at a finite point so the greatest lower bound of the destabilizing perturbations is not attained Althought, the limits of the sequence of perturbations is existence Now, we will show that our result can be used to obtain the result of Ralph Byers, N.K Nichols in [8] Firstly, we can repeat some results from [8]
Definition 2.1 ( see 8] ) The radius of stability of the stable regular pencil { , }A E is given by
or algebraic structure changes
F
where ||·||F denotes the Frobenius norm
Lemma 2.3.( see [8] )If { , }E A is regular and of index less than or equal to one, then there exists
an orthogonal matrix P and a permutation matrix Q such that
11 12
11 12
21 22
E E
A A
where rank(E11,E12)=rankE= , k rankA22 = − and n k 11 12 11 12
21 22
21 22
11 12 11 12
21 22
H
Theorem 2.4.( see [8] ) If { , }E A is stable, regular, and of index less than or equal to one, then