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STABILITY AND ROBUST STABILITY OF LINEAR TIME-INVARIANT DELAY DIFFERENTIAL-ALGEBRAIC

NGUYEN HUU DU†, VU HOANG LINH†, VOLKER MEHRMANN‡, AND

DO DUC THUAN§

Abstract Necessary and sufficient conditions for exponential stability of linear time-invariant

delay differential-algebraic equations are presented The robustness of this property is studied when the equation is subjected to structured perturbations and a computable formula for the structured stability radius is derived The results are illustrated by several examples.

Key words delay differential-algebraic equation, strangeness-free DAE, exponential stability,

spectral condition, restricted perturbation, stability radius

AMS subject classifications 06B99, 34D99, 47A10, 47A99, 65P99 DOI 10.1137/130926110

1 Introduction In this paper we present the stability analysis of homogeneous

linear time-invariant delay differential-algebraic equations (DDAEs) of the form

(1.3) E ˙x(t) + F ˙x(t − τ ) = Ax(t) + Dx(t − τ ).

However, by introducing a new variable, (1.3) can be rewritten into the form (1.1)with double dimension; see [10] For this reason, here we only consider (1.1)

The stability and robust stability analyses for DAEs are quite different from that

of ordinary differential equations (ODEs) (see, e.g., [23]), and has recently received

∗Received by the editors June 24, 2013; accepted for publication (in revised form) by W.-W Lin

September 24, 2013; published electronically December 5, 2013 The second and fourth authors were supported by IMU Berlin Einstein Foundation Program (EFP).

http://www.siam.org/journals/simax/34-4/92611.html

†Faculty of Mathematics, Mechanics, and Informatics, Vietnam National University, Thanh Xuan,

Hanoi, Vietnam (dunh@vnu.edu.vn, linhvh@vnu.edu.vn) The first author was partially supported

by the NAFOSTED grant 101.02–2011.21.

‡Institut f¨ur Mathematik, MA 4-5, TU Berlin, D-10623 Berlin, Germany (mehrmann@math.

tu-berlin.de) This author’s work was supported by Deutsche Forschungsgemeinschaft through

Son-derforschungsbereich 910 Control of self-organizing nonlinear systems: Theoretical methods and

application concepts.

§School of Applied Mathematics and Informatics, Hanoi University of Science and Technology,

Hanoi, Vietnam (ducthuank7@gmail.com).

1631

a lot of attention; see, e.g., [5, 6, 12, 26, 29, 32, 37, 38] and [11] for a recent survey.

In contrast to this, the stability and robust stability analyses for ODEs with delay(DDEs) is already well established; see, e.g., [20, 21, 22, 24, 35]

As an extension of both these theories, in this paper, we discuss DDAEs Suchequations, containing both algebraic constraints and delays arise, in particular, in thecontext of feedback control of DAE systems (where the feedback does not act instan-taneously) or as the limiting case for singularly perturbed ordinary delay systems; seee.g., [1, 2, 7, 8, 10, 31, 34, 43] In sharp contrast to the situation for DDEs and DAEs,even the existence and uniqueness theory of DDAEs is much less well established;see [17, 18] for a recent analysis and the discussion of many of the difficulties Thisunsatisfactory situation is even more pronounced in the context of (robust) stabilityanalysis for DDAEs Most of the existing results are only for linear time-invariantregular DDAEs [13, 41] or DDAEs of special form [1, 30, 44] Many of the resultsthat are known for DDEs do not carry over to the DDAE case Even the well-knownspectral analysis for the exponential stability or the asymptotic stability of lineartime-invariant DDAEs (1.1) is much more complex than that for DAEs and DDEs;see [10, 39, 43] for some special cases

The stability analysis is usually based on the eigenvalues of the nonlinear function

associated with the Laplace transform of (1.1), i.e., the roots of the characteristic

function

Let us define the spectral set σ(H) = {s : p H (s) = 0} and the spectral abscissa

α(H) = sup{Re s : p H (s) = 0} For linear time-invariant DDEs, i.e., if E = I n, the

exponential stability is equivalent to α(H) < 0 (see [20]) and the spectral set σ(H) is

bounded from the right However, for linear time-invariant DDAEs, the spectral set

σ(H) may not be bounded on the right as the following example shows.

Example 1.1 Consider the DDAE from [9]

x1(t − 1) = x1(t), which is of advanced type Thus, x1(t) = x (m)1 (t − m) for m − 1 ≤

t < m, m ∈ N Therefore, the solution is discontinuous in general and cannot be

extended on [0, ∞) unless the initial function is infinitely often differentiable.

In some special cases, [31, 40], it has been shown that the exponential stability

of DDAEs is equivalent to the spectral condition that α(H) < 0 In general, however

this spectral condition is only necessary, but not sufficient, as the following exampleshows

Example 1.2 Consider (1.1) with

Therefore, p H (s) = det H(s) = −(1 + s)(1 − e −2sτ /2), the eigenvalues are s = −1

and s = (− ln 2 + 2kπi)/2τ, k ∈ Z, and hence all eigenvalues are in the open left half

complex plane, which would suggest the exponential stability of the system, i.e., thatall nontrivial solutions would be exponentially decaying However, we will see thatthe asymptotic behavior (and even the existence) of the solutions depends strongly

on the smoothness and the behavior of the initial function φ.

Setting x = [x1, x2, x3, x4]T, the system reads

This underlying neutral delay ODE has the characteristic function −p H (s), so its

spectral set is the same as that of the original system The spectral condition ensures

the exponential stability of the underlying equation for x1; see [20] However, x2

and x3 are just the second and the first derivatives of x4(t) = x1(t − τ )/2 Thus, if the first component of φ is not differentiable on (−τ, 0) or it is differentiable (almost

everywhere) but the derivative is unbounded, then the solution does not exist or is

unbounded For example, the function φ1(t) = t sin(1/t) is continuous on [−τ, 0],

differentiable on (−τ, 0), but the derivative is obviously unbounded.

Example 1.2 shows that linear time-invariant DDAEs may not be exponentiallystable although all roots of the characteristic function are in the open left half com-plex plane To characterize when the roots of the characteristic function allow theclassification of stability, in this paper we derive necessary and sufficient conditionsthat guarantee that for time-invariant DDAEs exponential stability is equivalent to

the condition that all eigenvalues of H have a negative real part and thus extend

recent results of [31]

With a characterization of exponential stability at hand we also study the question

of robust stability for linear time-invariant DDAEs, i.e., we discuss the structured

stability radius of maximal perturbations that are allowed to the coefficients so that

the system keeps its exponential stability These results extend previous results onDDEs and DAEs in [5, 6, 12, 11, 24, 35]

The paper is organized as follows In the next section we introduce the basicnotation and present some preliminary results Then, in section 3, we characterizeexponential stability for general linear time-invariant DDAEs In section 4, we willintroduce allowable perturbations for two different classes of systems (1.1) and present

a formula for the structured stability radius for DDAEs In section 5, some conclusionsand open problems close the paper

2 Preliminaries In the following, we denote by I n ∈ C n,nthe identity matrix,

by 0∈ C n,n the zero matrix, by AC(I,C n) the space of absolutely continuous functions,

as-Note that instead of seeking solutions in AC([0, ∞), C n), alternatively we often

consider the space C pw1 ([0, ∞), C n) In fact, (1.1) may not be satisfied at (countably

many) points, which usually arise at multiples of the delay time τ

Definition 2.2 System (1.1)–(1.2) is called exponentially stable if there exist constants K > 0, ω > 0 such that

for all t ≥ 0 and all consistent initial functions φ, where φ ∞= sup−τ≥t≥0 φ(t).

Note that one can transform (1.1) in such a way that a given solution x(t; φ) is

mapped to the trivial solution by simply shifting the arguments

Definition 2.3 A matrix pair (E, A), E, A ∈ C n,n is called regular if there exists s ∈ C such that det(sE−A) is different from zero Otherwise, if det(sE−A) = 0 for all s ∈ C, then we say that (E, A) is singular.

If (E, A) is regular, then a complex number λ is called a (generalized finite)

eigenvalue of (E, A) if det(λE − A) = 0 The set of all (finite) eigenvalues of (E, A)

is called the (finite) spectrum of the pencil (E, A) and denoted by σ(E, A) If E is singular and the pair is regular, then we say that (E, A) has the eigenvalue ∞.

Regular pairs (E, A) can be transformed to Weierstraß–Kronecker canonical form (see [4, 14, 15]), i.e., there exist nonsingular matrices W, T ∈ C n,n such that

r = n, i.e., the second diagonal block does not occur.

Definition 2.4 Consider a regular pair (E, A) with E, A ∈ C n,n in Weierstraß— Kronecker form (2.2) If r < n and N has nilpotency index ν ∈ {1, 2, }, i.e.,

N ν = 0, N i = 0 for i = 1, 2, , ν − 1, then ν is called the index of the pair (E, A) and we write ind(E, A) = ν If r = n, then the pair has index ν = 0.

For system (1.1) with a regular pair (E, A), the existence and uniqueness of

solutions has been studied in [7, 8, 9] and for the general case in [17] It follows from

Corollary 4.12 in [17] that (1.1)–(1.2) has a unique solution if and only if the initial

condition φ is consistent and p H (s) = det(H(s)) ≡ 0.

For a matrix triple (E, A, D) ∈ C n,n × C n,n × C n,n, there always exists a

nonsin-gular matrix W ∈ C n,n such that

⎡

⎣E010

⎤

⎦ , W −1 A =

⎡

⎣A A120

where E1, A1, D1 ∈ C d,n , A2, D2 ∈ C a,n , D3 ∈ C h,n with d + a + h = n, rank E1 =

rank E = d, and rank A2= a Then, system (1.1) can be scaled by W −1 to obtain

be the left unitary factor of the SVD of U ∗ A with rank U ∗ A = a Then, we define

W = U diag(I d , ˜ U ) It is easy to check that multiplying by W −1 = diag(I d , ˜ U ∗ )U ∗,the form (2.3) is obtained with

E1= U1∗ E, A1= U1∗ A, D1= U1∗ D, A2= ˜U2∗ U2∗ A, D2= ˜U2∗ U2∗ D, D3= ˜U3∗ U2∗ D.

We immediately see that to obtain solvability of the equation, the initial function has

to be in the set

S := {φ : φ ∈ AC([−τ, 0], C n ), A2φ(0)+D2φ(−τ ) = 0, D3φ(t) = 0 for all t ∈ [−τ, 0]}.

Shifting the time in the last equation of (2.4) by τ , we obtain

It is easy to show that, although the transformed form (2.3) is not unique (anynonsingular matrix that operates block-wise in the three block rows can be applied),

the strangeness-free property is invariant with respect to the choice of W If (1.1) is

strangeness free, then, setting

⎤

D =

⎡

⎣D010

⎤

F =

⎡

⎣−D020

⎤

⎦ , the implicit system of (2.6) is equivalent to the neutral linear time-invariant DDE

in which any further factor cancels out and which admits a unique solution thatsatisfies the consistent initial condition (1.2)

We conclude this section with two remarks The first one gives a characterization

of the class of strangeness-free equations In the second one, since the matrix W

in (2.3) is not unique, the relation between different such transformation matrices isestablished

Remark 2.6 Consider a strangeness-free equation (1.1) together with its

trans-formed coefficients (2.3) Only two cases are possible with the pair (E, A) If h = 0,

E vanishes, then the pair (E, A) is regular and of index at

most 1 Otherwise, the pair (E, A) is singular Consequently, the class of free equations and the class of equations with regular higher-index pair (E, A) are

strangeness-complementary

Remark 2.7 Suppose that (1.1) is strangeness free and W and W are two

nonsingular matrices that both transform the coefficients of the equation to the form

E i A i D i be the transformed blocks corresponding to W Define

⎤

⎦ =

⎡

⎣E010

⎤

⎦ , P

⎡

⎣A A120

⎤

⎦ =

⎡

⎣A A120

⎤

⎦

Due to the assumptions on the form (2.3), it is easy to verify that P is a block triangular matrix, i.e., P21, P31, and P32 are zero blocks Since P is nonsingular, the diagonal blocks P ii , i = 1, 2, 3, are nonsingular Thus, W = P W with

3 Exponential stability of linear DDAEs In this section we show that for

strangeness-free systems the spectral condition characterizes exponential stability.Theorem 3.1 Suppose that equation (1.1) is strangeness free Then (1.1) is exponentially stable if and only if α(H) < 0.

Proof Necessity Suppose that (1.1) is exponentially stable, i.e., inequality (2.1)

holds with positive constants K and ω, but α(H) ≥ 0 Then there exists an eigenvalue

λ ∈ σ(H) with Re λ > −ω Let v = 0 be an eigenvector associated with λ, i.e.,

(λE − A − e −λτ D)v = 0, then obviously x(t) = e λt v is a solution of (1.1), but it does

not satisfy (2.1) This is a contradiction and thus α(H) < 0.

Sufficiency Suppose that α(H) < 0 and consider a solution x of (1.1) As seen

in the previous section, x also satisfies the neutral delay ODE system (2.7), whose

v = 0 But since 0 ∈ σ(H), this implies that

∗ (t) Thus, (1.1) is exponentially stable.

H) always

holds Thus, if system (1.1) is strangeness free, then the spectral set σ(H) is bounded from the right, or equivalently the spectral abscissa satisfies α(H) < ∞.

Now we consider the case when the pair (E, A) (1.1) is regular and it is

trans-formed into the Weierstraß–Kronecker canonical form (2.2) Setting

From the explicit solution formula for linear time-invariant DAEs (see [7, 25]), thesecond equation of (3.5) implies that

It follows that φ needs to be differentiable at least ν times if the coefficients D21and

D22do not satisfy further conditions Extending this argument to t ∈ [τ, 2τ ), [2τ, 3τ ),

etc., the solution cannot be extended to the full real half-line unless the initial function

φ is infinitely often differentiable or the coefficient associated with the delay is highly

structured

Corollary 3.3 Consider the DDAE (1.1)–(1.2) with a regular pair (E, A),

ind(E, A) ≤ 1, and its associated spectral function H Then (1.1) is exponentially

stable if and only if α(H) < 0.

Proof If ind(E, A) ≤ 1, then the system is obviously strangeness free in the sense

of Definition 2.5 with d + a = n and h = 0 Thus, by Theorem 3.1, the system is exponentially stable if and only if α(H) < 0.

We note that the result of Corollary 3.3 is obtained in [31] by a direct proof

Let us now consider exponential stability for the case that ind(E, A) > 1 In order

to avoid an infinite number of differentiations of φ induced by (3.8), it is reasonable

to assume that for a system in Weierstraß–Kronecker form (2.2) with transformed

matrices as in (3.4) the allowable delay condition N D 2i = 0, i = 1, 2, holds Note that this condition is trivially true for the index-1 case, since then we have N = 0.

In terms of the original coefficients for (1.1) for a regular pair (E, A) with arbitrary

index this allowable delay condition can be described as follows

Choose any fixed ˆs ∈ C such that det(ˆ sE − A) = 0 and set

(3.9) E = (ˆˆ sE − A) −1 E, D = (ˆˆ sE − A) −1 D.

Proposition 3.4 Consider a DDAE of the form (1.1) with a regular pair (E, A)

of arbitrary index, let ˆ s ∈ C be such that det(sE − A) = 0, and consider the system

(2.2) after transformation to Weierstraß–Kronecker canonical form Then the

allow-able delay conditions N D21 = 0 and N D22 = 0 are simultaneously satisfied if and

only if

where ˆ E D denotes the Drazin inverse of ˆ E.

Proof From (2.2) it follows that

ˆ

E = T

(ˆsI r − J) −1 0

ˆ

E D = T

(ˆsI r − J) −1 0

bility for DDAEs with regular pair (E, A).

Theorem 3.5 Consider the DDAE (1.1)–(1.2) with a regular pair (E, A) fying (3.10) Then (1.1) is exponentially stable if and only if α(H) < 0.

satis-Proof Necessity The proof is analogous to that of Theorem 3.1 and we conclude

that if (1.1) is exponentially stable, then α(H) < 0.

Sufficiency Suppose that α(H) < 0 Since the pair (E, A) is regular, it

fol-lows that (1.1)–(1.2) is equivalent to the system in canonical form (3.5) Under the

assumption (3.10), we have N D 2i = 0, i = 1, 2, and thus (3.7) is reduced to

This implies that det ˜H(s) = 0 if and only if det H(s) = 0, and hence α( ˜ H) = α(H) <

0 Thus, by Corollary 3.3, system (3.12) with initial condition (3.6) is exponentiallystable and hence system (1.1)–(1.2) is exponentially stable

For the system in Example 1.2 which has a regular pair (E, A) that is already in Weierstraß–Kronecker form, we have N D21 = 0 but ND22 = 0 and the system has

α(H) < 0 but the system is not exponentially stable The following example presents

the same observation for the case N D21= 0 but N D22 = 0.

Example 3.6 Consider (1.1) with

Therefore, det H(s) = −(1 + s)(2 + e −sτ)3, the eigenvalues are λ = −1 and s =

(− ln 2 + (2k + 1)π)/τ, k ∈ Z, and hence all eigenvalues are in the open left half

If the solution is defined for all t ≥ 0, it depends on the derivatives of the initial

function in general Thus, the system is not exponentially stable

We have seen that the spectral condition α(H) < 0 is necessary for the exponential

stability of (1.1), but in general it is not sufficient Introducing further restrictions onthe delays, we get that exponential stability is equivalent to the spectral condition

4 Robust exponential stability We have seen in the previous section that

under some extra conditions the exponential stability of a linear time-invariant DDAE

can be characterized by the spectral properties of the matrix function H(s)

Typi-cally, however, the coefficient functions are not exactly known, since they arise, e.g.,from a modeling, or system identification process, or as coefficient matrices from adiscretization process Thus, a more realistic scenario for the stability analysis is toanalyze the robustness of the exponential stability under small perturbations Toperform this analysis, in this section we study the behavior of the spectrum of the

triple of coefficient matrices (E, A, D) under structured perturbations in the matrices

where Δ i ∈ C p i ,q , i = 1, 2, 3, are perturbations and B i ∈ C n,p i , i = 1, 2, 3, C ∈ C q,n,are matrices that restrict the structure of the perturbations We could also consider

different matrices C i in each of the coefficients but for simplicity, see Remark 4.9below, we assume that the column structure in the perturbations is the same for allcoefficients Set

and p = p1+ p2+ p3 and consider the set of destabilizing perturbations

VC(E, A, D; B, C) = {Δ ∈ C p×q : (4.1) is not exponentially stable}.

Then we define the structured complex stability radius of (1.1) subject to structured

perturbations as in (4.1) as

(4.3) rC(E, A, D; B, C) = inf{Δ : Δ ∈ VC(E, A, D; B, C)},

where ·  is a matrix norm induced by a vector norm If only real perturbations Δ

are considered, then we use the term structured real stability radius but here we focus

on the complex stability radius

With H as in (1.4), we introduce the transfer functions

we obtain an explicit formula for the structured stability radius

Theorem 4.1 Suppose that system (1.1) is exponentially stable Then the tured stability radius of (1.1) subject to structured perturbations as in (4.1) satisfies the inequality

sup

Re λ≥0 G(λ)

−1

.

Proof Let be an arbitrary positive number and let λ0∈ ¯C+, where ¯C+={λ ∈

C, Re λ ≥ 0} is the closed right half-plane, be such that

G(λ0) −1 ≤

sup

Re λ≥0 G(λ)

−1

+ , and let u ∈ C n be such that u = 1 and

This relation implies that λ0 is a root of the characteristic function associated with

(4.1) Since Re λ0 ≥ 0, it follows that (4.1) is not exponentially stable Thus, Δ ∈

VC(E, A, D; B, C), which implies that

rC(E, A, D; B, C) ≤ Δ = G(λ0) −1 ≤

sup

Re λ≥0 G(λ)

−1

+ Since is arbitrary, it follows that

rC(E, A, D; B, C) ≤

sup

Re λ≥0 G(λ)

−1

,

and the proof is complete

For every perturbation Δ as in (4.2) we define

(4.8) H Δ (λ) = λ(E + B1Δ1C) − (A + B2Δ2C) − e −λτ (D + B3Δ3C)

and have the following proposition

Proposition 4.2 Consider system (1.1) and the perturbed system (4.1) If the associated spectral abscissa satisfy α(H) < 0 and α(H Δ)≥ 0, then we have

sup

Since α(H Δ)≥ 0, we have two cases.