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Imaginary characteristic roots of neutral systems with commensurate delays
Systems & Control Letters 127 (2019) 19–24
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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Imaginary characteristic roots of neutral systems with commensurate delays Fernando de Oliveira Souza Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-010, Belo Horizonte, MG, Brazil
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a b s t r a c t This paper concentrates on stability analysis of neutral linear time-invariant (LTI) delay systems with multiple commensurate time-delays. A new numerical procedure is offered for the determination of purely imaginary characteristic roots of neutral delay systems, which plays a crucial role in assessing the system stability. Based on simple linear algebra it is shown that the imaginary characteristic roots of such a system can be found by calculating the generalized eigenvalues of an associated matrix pair. If the system is of retarded type we just need to determine the eigenvalues of a single matrix. The results extend previously known work on neutral delay systems subject to a single delay for neutral systems subject to multiple commensurate delays. In the light of the main result of this paper we present a new method to determine the largest first time-delay interval for which a neutral delay system preserves its stability. The paper is closed by showing numerical examples that illustrate the applicability and effectiveness of the proposed method. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 6 June 2018 Received in revised form 2 March 2019 Accepted 19 March 2019 Available online xxxx Keywords: Neutral delay systems Multiple time-delays Stability Generalized eigenvalues First time-delay interval
1. Introduction
equation of the system (1) is given by
Stability analysis of delay-differential systems of neutral type is a subject of great practical and theoretical importance which has received considerable attention for decades. Comprehensive surveys on this subject can be found in the literature [1–5]. Practical examples of neutral delay-differential systems include lossless transmission lines [1,6], control processes [1,7], population ecology [8], etc. From a theoretical point of view, the solution to this problem is very complex due to the presence of exponential type transcendental terms in the system characteristic equation. Therefore the stability analysis of neutral linear time-invariant (LTI) delay systems with multiple commensurate time-delays has been a very active research topic [1,3,4,9–11]. In this paper we consider the general class of neutral LTI multi-delay systems of the form x˙ (t) −
m ∑
Bk x˙ (t − kτ ) = A0 x(t) +
k=1
m ∑
Ak x(t − kτ ),
(1)
k=1
where x(t) ∈ Rn , Ak , Bk ∈ Rn×n , k = 1, . . . , m. The system is of retarded type when all Bk = 0. Clearly, stability of neutral delay systems proves to be more complex, due to the fact that the system involves the derivative of the delayed state. The characteristic E-mail address: [email protected]. https://doi.org/10.1016/j.sysconle.2019.03.011 0167-6911/© 2019 Elsevier B.V. All rights reserved.
m m ) (( ∑ ) ∑ Ak z k ∆(s, z) := det s I − Bk z k − k=1
(2)
k=0
with z = e−sτ and τ ≥ 0. The system is infinite dimensional and as such it possesses infinitely many characteristic roots. Therefore, since direct calculation of the roots of the bivariate polynomial ∆(s, z) is not practical, the literature is rich in methods that try to indirectly assess the location of its roots, see e.g. [10–18]. This paper proposes one such method. The idea is to establish a simple methodology to compute the critical frequencies, ω ∈ R, where the roots of ∆(s, e−sτ ) = 0 cross the imaginary axis, i.e. ∆(jω, e−jωτ ) = 0. The determination of purely imaginary characteristic roots of neutral linear time-invariant (LTI) delay systems plays a crucial role in assessing the system stability, since under common assumptions on nonsingularity, stability switching only takes place where the system has such roots. Concerning retarded systems with multiple commensurate delays case in [12] was presented a 2n2 m × 2n2 m matrix having spectrum containing all system unitary modulus characteristic roots, i.e. all zi = e−jωi τi such that ∆(jωi , zi ) = 0. Again for the case of a retarded system, in [13] was introduced a pair of matrix pencils, to be checked for generalized eigenvalues with unitary modulus. The dimension of the highest order matrix pencil is also 2n2 m × 2n2 m. Further, concerning both neutral and retarded systems with single delay, in [14] another
20
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
elegant theoretical result provided a different 2n2 × 2n2 matrix having spectrum containing all imaginary characteristic roots of the delay system, i.e. all jωi such that ∆(jωi , zi ) = 0. Later in [11] the technique of [12] was extended to consider neutral systems with commensurate delays, where the resulting matrix having spectrum containing all system unitary modulus characteristic roots is also of dimension 2n2 m × 2n2 m. Addressing neutral and retarded delay systems with a single delay the method in [14] was improved in [15] where the reported method requires just the solution of an n2 × n2 generalized eigenvalue problem. In addition a very interesting method regarding neutral and retarded systems subject to multiple commensurate and distributed delays was proposed in [17,18]. This method is based on the observation that the characteristic roots of the time-delay system are related with the spectrum of an auxiliary delay free system used for the computation of Lyapunov matrices (see e.g. [3] for more results in this direction). This methodology differs from the others mentioned by the fact that it is strongly connected to time domain properties. In the case of neutral systems subject to multiple commensurate delays the method also involves a 2n2 m × 2n2 m matrix pair [17]. This paper proposes one such method. The present paper using a different methodology extends the earlier result in [14] for neutral systems with commensurate delays. In order to understand the proposed result in a more transparent light we recall that the neutral system with multiple commensurate delays in (1) can be rewritten as a higher dimensional neutral system subject to a single delay [11] x˙ (t) − B1 x˙ (t − τ ) = A0 x(t) + A1 x(t − τ )
(3)
where x(t) = xT (t), xT (t − τ ), . . . , xT (t − (m − 1)τ ) , B1 ⎢I
B1 = ⎢ ⎣ .. .
... ··· .. .
Bm−1 0
0
···
I
⎡
0 ⎢0
⎡
and A1 = ⎢ ⎣ ..
.
0
.. .
0 0
.. . ···
Bm 0⎥
⎤
⎢ .. ⎥ ⎦ , A0 = ⎣ .. . .
0
0
··· ··· .. . 0
A0 ⎢0
⎡
A1 0
.. . ···
··· ··· .. . 0
2. Notation and preliminaries some basic notations. Let the scalar j = √ First we introduce −1 and Rm×n (Cm×n ) be the set of m × n real (complex) matrices. R>0 (R<0 ) and C+ denote the set of positive (negative) real numbers and the closed right half plane of the complex plane, respectively. If s is a complex number, s∗ denotes its complex conjugate and |s| its complex modulus. Let Cϵ := {s ∈ C : s + s∗ ≥ ϵ, ϵ ∈ R<0 }. For a matrix X denote its determinant by det(X ), its spectrum by σ (X ), and its spectral radius by ρ (X ). For a matrix pair (A, B), denote the set of all generalized eigenvalues by σ (A, B) := {λ ∈ C : det(A − λB) = 0} and the ith generalized eigenvalue by σi (A, B). The notation ⊗ denotes the Kronecker product. Let In (0n ) be the identity (zero) matrix with dimension n × n and 0n×m the zero matrix with dimension n × m; throughout the paper such subscripts will be suppressed, whenever the dimension is clear from the context. Throughout the paper, we shall make use of some linear standard results. They are repeated here for convenience (see for instance [19]): Fact 1. Let A, B, C, and D be matrices of appropriate dimensions, such that the following operations are well defined, then:
]T
[
from that adopted in [14,17,18]. This constitutes the main benefit of the proposed result. Then, taking into consideration the main result of this paper we present a new computable procedure that allows one to determine the largest first time-delay interval for which a neutral delay system preserves its stability. The applicability and effectiveness of the main result will be illustrated by examples in Section 4.
Am−1 0 ⎥
⎤
(a) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD).
.. ⎥ ⎦, .
0
(b) det(D) det(A − BD−1 C) = det (c) det(A) det(D − CA−1 B) = det
Am 0⎥
⎤
(
A C
(
A C
)
B D
if D is nonsingular.
)
B D
if A is nonsingular.
The notion of stability of the neutral system with multiple commensurate delays in (1) is defined as follows [11].
.. ⎥ ⎦. .
0 (4)
Hence, based on the higher dimensional system in (3) the methods in [14] and [15] can be applied directly. In which case, however, the dimension of the corresponding problem rises, thus leading to a higher, undesirable computation load. Namely, the methods in [14] and [15] will require the computation of the generalized eigenvalues of a 2(nm)2 × 2(nm)2 and (nm)2 × (nm)2 matrix pair, respectively. On the other hand, via a different methodology, the present paper proposes a similar method of that in [14] involving a 2n2 m × 2n2 m matrix pair, which demands less computational load than the direct application of the methods in [14] and [15] on the higher dimensional system (3) for m > 2. To conclude, we highlight that for retarded systems the proposed result is reduced to the standard eigenvalue problem, a similar feature with the one in [14]. Concisely, this paper revisits the problem of assessing the imaginary characteristic roots of neutral systems with commensurate delays. The result proposed extends the previous one regarding linear neutral delay systems subject to a single delay [14], presenting a similar matrix pair having spectrum containing all imaginary characteristic roots of neutral systems with commensurate delays. We emphasize that the proposed technique employs a simple linear algebra methodology rooted in Kronecker products and Schur complement formulas different
Definition 1. Let ∆(s, z) be given in (2). System (1) is stable for a given τ ≥ 0 if ∆(s, z) ̸ = 0 with z = e−sτ , ∀s ∈ Cϵ . Furthermore, it is said to be stable independent of delay if ∆(s, z) ̸ = 0 with z = e−sτ , for all ∀s ∈ Cϵ and τ ≥ 0. It is worth mentioning that a necessary condition for the system (1) to be stable concerns the difference equation x(t) −
m ∑
Bk x(t − kτ ) = 0.
(5)
k=1
Namely, for the system (1) to be stable, it is necessary that the difference Eq. (5) be stable for all τ ≥ 0 [2,20]. In addition, we point out that the following result gives a simple necessary and sufficient condition to assert the stability of (5) for all τ ≥ 0. Lemma 1 (Gu et al. [2, Lemma 3.18]). The system in (5) is stable for all τ ≥ 0 if and only if
ρ (B1 ) < 1
(6)
where B1 is given in (4). Fortunately, an important fact is that under the assumption that the difference equation (5) is stable, the condition in Definition 1 needs to hold only for C+ ; that is Cϵ can be replaced by C+ , as stated in the next definition [11].
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
Definition 2. Let ∆(s, z) be given in (2) and suppose that condition (6) holds. System (1) is stable for a given τ ≥ 0 if ∆(s, z) ̸= 0 with z = e−sτ , ∀s ∈ C+ . Furthermore, it is said to be stable independent of delay if ∆(s, z) ̸ = 0 with z = e−sτ , for all ∀s ∈ C+ and τ ≥ 0.
In this section we provide the main results of the paper which require only the computation of eigenvalues and generalized eigenvalues. First, we present the following auxiliary lemma needed to prove the main result.
⎛ m ∑
) Pk z k
k=1
⎡
P1 ⎢ In
P2 0
⎜ ⎜ ⎢ = det ⎜Inm − ⎢ . ⎝ ⎣ ..
..
. ...
0
−Inm ⊗ A0
Pm 0
0n2 m×(nm)2 Inm ⊗ B¯
0n2 m×(nm)2
= det(I) det I − P1 z − P2 z +
m ∑
In
0
⎥ ⎟ ⎥ ⎟ ⎥ z⎟ . ⎦ ⎠
.. .
−P2 z −
m ∑
k=1
] )
(
m )−1 ∑ (
Ak + λBk e−λkτ
det I − λI − A0
(
Pk z
−zI ⎛ [ ⎜ I − P1 z = det(I) det ⎝ −zI
I
m ∑
Pk z
det(I − H(λ)e−λτ ) =
⎞
⎤ k−2
⎥[ ⎦ 0
k=4
∑ k=4
I
0 I
⎛ ⎡ ⎜ I − P1 z ⎜⎣ = det(I) det ⎜ −zI ⎝ 0
−P2 z I
−zI
=
Pk z k−2 ⎥⎟
−P4 z −
m ∑
0
nm ∏ [
1 − σi (H(λ)) e−λτ
I
0
0
−zI
I
0
0
−zI
I
⎞ 0
Therefore, we know
]⎟ ⎟ zI ⎟ ⎠
∆(λ, e−λτ ) = 0 ⇔ ∃i : |σi (H(λ))| = 1.
∑m
Note that for λ = 0 we have that ∆(λ, e−λτ ) = 0 ⇔ k=0 Ak is singular. Now recall that the eigenvalues of the Kronecker product of H(λ) with its matrix of conjugates are all possible pairwise products of the eigenvalues of the matrix H(λ) with their conjugates [19, Proposition 7.1.10]. Now |σi (H(λ))| = 1 for some i,
⎤⎞ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ . ⎥⎟ ⎦⎠
0 = det Inm ⊗ Inm − H(λ) ⊗ H(−λ)
(
The result then follows by applying this procedure repeatedly.
□
The next theorem presents the procedure proposed to compute the purely imaginary characteristic roots of the neutral delay system (1). Theorem 1. Consider ∆(s, z) in (2), the of the system (1), assume that A0 has no define the matrices A¯ = [A1 A2 . . . Am ], C T = [In 0n×n(m−1) ],
[ D=
[ E=
0n×n(m−1) In(m−1) I(nm)2 Inm ⊗ D
]
0n 0n(m−1)×n
D ⊗ Inm I(nm)2
]−1 [
characteristic equation eigenvalue on jR, and B¯ = [B1 B2 . . . Bm ],
)
(
( )( )) = det Inm ⊗ Inm − H(λ) ⊗ Inm Inm ⊗ H(−λ) ([ ]) Inm ⊗ Inm H(λ) ⊗ Inm = det Inm ⊗ H(−λ) Inm ⊗ Inm ([ ]) Inm ⊗ Inm [C (λI − A0 )−1 B(λ) + D] ⊗ Inm = det −1
[Fact 1.(b)]
= det(D − CA−1 B)
[Fact 1.(a)]
Inm ⊗ [−C (λI + A0 )
B(−λ) + D]
(λI − A0 ) ⊗ Inm A= 0n2 m
[
]
0(nm)2 ×n2 m , Inm ⊗ C
[ B=
0
−Inm ⊗ B(−λ)
]
Inm
0n2 m , ⊗ (λI + A0 )
B(λ) ⊗ Inm , 0
]
[Fact 1.(a)]
Inm ⊗ Inm
with
,
C ⊗ Inm 0(nm)2 ×n2 m
]
∆(λ, e−λτ ) = 0 ⇔ {∃i : σi (H(λ))e−λτ = 1}.
Pk z k−3
k=5
)]
i=1
I
⎛⎡
(
it follows that
⎥⎟ ⎥⎟ ⎦⎠
⎤ ⎡ m ∑ ⎤ k− 3 Pk z −P3 z ⎥[ ⎢ ⎥ ⎢ 0 ⎦ − ⎢ k=4 ⎥ 0 ⎦ ⎣ 0
−P3 z
1 − σi H(λ)e−λτ
⎤⎞
0
−P2 z
nm ∏ [ i=1
]⎟
zI ⎠
0
−zI
B(λ) + D
to one with a single delay. Since
⎢P3 z + −⎣
−P3 z −
is defined as
⎥⎟ ⎦⎠
m
−P2 z
= det(I − H(λ)e−λτ ) = 0
⎤⎞
⎡ −P2 z
)
¯ The last equality is obtained by an application with B(λ) = A¯ +λB. of Lemma 2. The definition of H(λ) reduces the stability problem
k−1
⎛⎡
⎜⎢I − P1 z ⎜⎢ ⎜⎢ = det ⎜⎢ −zI ⎜⎢ ⎝⎣ 0
−1
I
]
)
k=1
H(λ) := C (λI − A0 )
Pk z k−1 z
k=3
k=1
Under the assumption that A0 has no eigenvalue on jR, then (jωI ±A0 )−1 is well defined for all ω ∈ R. Therefore ∆(λ, e−λτ ) = 0 if and only if
k=3
⎛⎡
(7)
m m ( ( ) ∑ ∑ ) ∆(λ, e−λτ ) = det λ I − Ak e−λkτ Bk e−λkτ − A0 −
where H(λ) ∈ C [
0
E.
m (( ) ) ∑ ( ) = det λI − A0 − Ak + λBk e−λkτ .
k=1
⎜⎢I − P1 z ⎜⎢ = det ⎜⎢ ⎝⎣ −zI
E , and
Proof. Set λ = jω, then
nm×nm
⎜⎢I − P1 z = det ⎝⎣
]
0n2 m×(nm)2
]
jΩ = {jωi : jωi ∈ σ (F , G)}.
⎤ ⎞
Pk z k
(
0n2 m×(nm)2 Inm ⊗ A¯
B¯ ⊗ Inm
[ G = I2n2 m +
−A¯ ⊗ Inm
[ +
k=1
... ... .. .
Proof. The proof, inspired by Chen et al. [21], consists in applying the Schur determinant formula (Fact 1.b) as follows ( ) m ∑ det I −
]
0n2 m
Let z ∈ C and matrices Pk ∈ Cn×n . Then
Lemma 2.
(
A0 ⊗ Inm 0n2 m
Then all purely imaginary characteristic roots of ∆(s, e−sτ ) belong to the following set:
3. The main results
det In −
[ F =
21
22
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
C=
[ −C ⊗ Inm
]
0
0
−Inm ⊗ C
, and D =
[
Inm ⊗ Inm Inm ⊗ D
]
D ⊗ Inm . Inm ⊗ Inm
This is true if and only if 0 = det(A) det(D − CA
−1
([ B) = det
A C
(
])
B D
= det(D) det(A − BD−1 C)
det sI − (I −
[Fact 1.(c)] [Fact 1.(b)]
= det(A − BD C).
(8)
The latter steps hold because det(D) = 1, i.e. using Facts 1.(a) and 1.(b) we have det(D) = det(Inm ⊗ Inm ) det(Inm ⊗ Inm − D ⊗ D) and the last factor is the determinant of a lower triangular matrix with ones on the main diagonal. Therefore, recalling the definitions in the theorem we have that (8) can be rewritten as 2m
det(F − λG),
which completes the proof. □ The previous theorem states that all purely imaginary characteristic roots of the neutral delay system (1) when σ (A0 ) ∩ jR = ∅, are contained in the set of generalized eigenvalues of a 2n2 m × 2n2 m matrix pair (F , G). For retarded systems B¯ = 0 ⇒ G = I, and the generalized eigenvalue problem is reduced to the standard eigenvalue problem. Also, it is worth mentioning that a necessary condition for the neutral delay system (1) to be stable independently of delay is that A0 is Hurwitz [2]. Note also, if A0 does have eigenvalues on the imaginary axis, that we may search for pure imaginary eigenvalues of (1) by applying Theorem 1 to matrices A0 + ϵ I for some (arbitrarily small) scalar ϵ . Since based on Theorem 1 the complete set of purely imaginary characteristic roots of the neutral system in (1) can be found, one can deploy the method proposed in [22] which results in the determination of an exact and complete set of delay intervals where the system is stable. However, for illustration of the applicability of the previous theorem in the following we present a computable procedure that allows one to determine the largest first time-delay interval for which a neutral delay system preserves its stability. That is, under the condition in Lemma 1, to characterize whether there exists some τmax > 0 for which all roots of the transcendental function ∆(s, e−sτ ) in (2) are in the open left half of the complex plane for all values of τ ∈ [0, τmax ). Corollary 1. Consider the set Ω given in Theorem 1 and the matrices A0 , A1 , and B1 given in (4). Suppose that σ (A0 ) ∩ jR = ∅ and∑ the condition in Lemma 1 holds, i.e. ρ (B1 ) < 1. Then the matrix m I − k=1 Bk is nonsingular. Suppose that the ( )(1) is stable ∑also ∑system m m at τ = 0, that is, the matrix (I − k=1 Bk )−1 k=0 Ak is Hurwitz. Define
τmax
⎧ ∞ if Ω ∩ R>0 = ∅. Otherwise, ⎪ ⎪ ⎪ θi ⎪ ⎪ ⎨ min where the pair (ωi , θi ) belongs to the set i ω i := Θ := {(ωi , θi ) : ωi ∈ Ω ∩ R>0 and ⎪ ⎪ ⎪ ⎪ e−jθi ∈ σ (jωi I − A0 , A1 + jωi B1 ) ⎪ ⎩ with θi ∈ [0, 2π )} .
Then the system is stable for all τ ∈ [0, τmax ). Proof. Initially ρ (B1 ) < 1 implies that
(
det I −
m
∑ k=1
Bk
)
= det(I − B1 ) ̸= 0.
m ∑
Bk )−1
k=1
−1
0 = (−1)2n
Additionally, since the system (1) is stable at τ = 0, the Hurwitz condition holds: ∆(s, 1) is nonzero for all s ∈ C+ , i.e. ∄s ∈ C+ such that m ∑
Ak
)
= 0.
k=0
Hence ∆(0, 1) ̸ = 0. Consider the case τmax = ∞, with Ω ∩ R>0 = ∅, and let τ ≥ 0. Theorem 1 implies that for every ω > 0, we have ∆(jω, e−jωτ ) ̸= 0. We know that ∆(jω, e−jωτ ) ̸= 0 with ω = 0, hence there does not exist τ ≥ 0 such that any root of ∆(s, e−sτ ) intersects the imaginary axis. Since the system (1) is stable at τ = 0, it is stable for all τ ∈ [0, ∞); that is, the system is stable independently of delay. Suppose now that τmax < ∞. Then Ω ∩ R>0 ̸ = ∅. By Theorem 1 and under the assumptions that the system is stable at τ = 0 and ρ (B1 ) < 1, we find that
∆(jωi , e−jωi τi ) = ∆(jωi , e−jθi ) = 0 for some ωi > 0, τi > 0. By definition, θi ∈ [0, 2π );
(9)
choose θi ≤ ψ if ∆(jωi , e−jψ ) = 0 with ψ ≥ 0. We have ∆(jωi , e−jωi τi ) = 0 with τi = θi /ωi , and ∆(jωi , e−jωi τ ) ̸ = 0 for any τ ∈ [0, τi ). With τmax as defined, we see now that the system (1) is stable for all τ ∈ [0, τmax ). Finally, in order to compute e−jθi in (9) associated with ωi we invoke the characteristic equation of the high dimensional neutral system subject to a single delay in (3)
((
ˆ (s, z) := det s I − B1 z − A0 + A1 z ∆
) (
))
( ( ) ) = det sI − A0 − A1 + sB1 z . (10)
Therefore, it is clear that for any real ωi the matrix pair (jωi I − A0 , A1 + jωi B1 ) has a unit magnitude generalized eigenvalue, e−jθi , if and only if ∆(jωi , e−jθi ) = 0. The proof is now completed. □ The previous corollary constitutes a readily computable procedure to determine the first time-delay interval for which a neutral system preserves its stability. In terms of the computational demands, only computations on matrices with constant entries are required. 4. Numerical examples In order to illustrate the applicability and efficiency of the results presented, numerical examples are provided, some drawn from the literature. Moreover, we point out that the same conclusions in the examples of this section can be recovered by the methods presented in [17,18]. Example 1. A0 =
[ −2 0 0.5 0.75
[ B1 =
Consider the system in (1) with data
]
[ −1 A1 = −1
0 , −1 0
− 0.25
]
]
0 , −1
and
.
First following Theorem 1 we compute
σ (F , G) = {±j0, ± 0.8382, ± 1.0604, ± 2} that yields Ω = {0, 0}. Then accordingly with Corollary 1 we check that the condition in Lemma 1 holds, i.e.∑ ρ (B1 ) < ( ) 1, Ω ∩ R>0 = ∅ and we compute ∑ m −1 σ (I − m B ) A = {−1.6, − 6}, which means that k k k=1 k=0 the system is stable independently of delay.
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
Example 2.
Example 4. Consider the system in (1) with matrices in (11) and let
Consider the system in (1) with data
1.5 −0.25 [ −0.75 B1 = 0.25
0.25 , 2
[
]
A0 =
] −0.25 0
[ A1 =
−3.5 0.5
−0.5 , −3
]
0.02 ⎢ 0 B1 = ⎣ 0 0
⎡
and
;
First following Theorem 1 we compute
Then accordingly with Corollary 1 we check that ρ (B1 ) < 1, the system is stable for τ = 0,
Θ = {(4.1654, 1.9809), (2.3834, 0.8911)} and } { 0.8911 1.9809 = 0.4756, = 0.3739 , τmax = min 4.1654 2.3834
0
Accordingly with Corollary 1 we check that ρ (B1 ) < 1, the system is stable for τ = 0,
Θ = {(3.3264, 0.47807), (1.3873, 2.0565), (1.1338, 0.80057), (0.8680, 2.9574), (0.3160, 1.76), (0.5809, 3.328)} { } τmax = min 0.14372, 1.4824, 0.70611, 3.4073, 5.5699, 5.7288
In [21] it is shown that the retarded type system
which means that the system is stable for all τ ∈ [0, 0.14372), as expected. Example 5. Consider the system in (1) with matrices in (11) and let B1 = A1 , B2 = A2 , and B3 = 12 A3 . First following Theorem 1 we compute Ω = {±3.1652, ± 1.7418, ± 1.3434, ± 0.77282, ± 0.37593, ± 0.54353}.
1 0 0 −3
−2 ⎡ −0.05 ⎢ 0.005 A1 = ⎣ 0
0 1 0 −5
Then accordingly with Corollary 1 we check that ρ (B1 ) < 1, the system is stable for τ = 0,
⎤
0 0 ⎥ , 1 ⎦ −2
(11)
0.005 0.005 0 0
0.25 0 0 −0.5
0 0⎥ , 0⎦ 0
0.005 ⎢ 0 A2 = ⎣ 0 −1
0.0025 0 0 −0.5
0 0.05 0 −0.5
0 0 ⎥ , 0.0005⎦ 0
0.0375 ⎢ 0 A3 = ⎣ 0.05 0
0 0.05 0.05 −2.5
0.075 0.05 0 0
0.125 0 ⎥ , 0 ⎦ −1
−1
⎡
⎡
⎤
0 0 ⎥ . 0 ⎦ 0.25
and
which means that the system is stable for all τ ∈ [0, 0.3739). It is worthwhile to mention that in [14] the same spectrum σ (F , G) provided above was computed, and accordingly the same conclusion about the system’s stability was obtained.
⎢0 A0 = ⎣ 0
0.03 0 0 .5 0
Ω = {±3.3264, ±1.3873, ±1.1338, ±0.8680, ±0.3160, ±0.5809}.
σ (F , G) = {±j4.1654, ± 1.4524 ± j2.4206, ± j2.3834} that yields Ω = {±4.1654, ± 2.3834}.
⎡
0 0.01 0 0
In [10] by an analytical method it was shown that this system is stable within the delay interval τ ∈ [0, 0.14372). Then using Theorem 1 we compute
as in [14].
Example 3. with data
23
⎤
Θ = {(3.1652, 0.97987), (1.7418, 1.1974), (1.3434, 1.1175), (0.77282, 3.1707), (0.37593, 1.8432), (0.54353, 3.496)} and
τmax = min {0.30958, 0.68748, 0.83185, 4.1028, 4.903, 6.4321} .
⎤
⎤
is not stable independent of delay. Additionally, in [10,23] by employing algebraic criteria it was shown that this system is stable within the delay interval τ ∈ [0, 0.47767). Therefore, applying Theorem 1 we have that
Ω = {±1.2507, ± 0.31479} and thus Ω ∩ R>0 ̸ = ∅, then accordingly with Corollary 1 we see that the system is not stable independent of delay, as expected. Moreover, we check that ρ (B1 ) < 1, the system is stable for τ = 0,
which means that the system is stable for all τ ∈ [0, 0.30958). 5. Conclusions This paper has revisited the problem of assessing the stability of linear time-invariant (LTI) neutral systems with multiple commensurate delays. The main result demonstrates that the imaginary characteristic roots of such a system can be found by calculating the generalized eigenvalues of an associated matrix pair. If the system is of retarded type we just need to determine the eigenvalues of a single matrix. This result extends a previously known work on neutral delay systems subject to a single delay, and shows that a similar result exists for neutral systems subject to multiple commensurate delays. Then, as a second contribution, in the light of the main result in this paper we present a new method to determine the largest first time-delay interval for which a neutral delay system preserves its stability. Applications and the effectiveness of the proposed methods are illustrated by numerical examples.
Θ = {(1.2507, 0.59742), (0.31479, 1.7588)}
Acknowledgments
and
The author is grateful to the reviewers for an insightful, complete, and thoughtful review of the manuscript. This research has been supported by the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior — CAPES, Brazil (Finance Code 001), Conselho Nacional de Desenvolvimento Científico e Tecnológico — CNPq, Brazil (Grants 311574/2017-3 and
τmax = min 0.47767, 5.5873
{
}
yielding the same conclusion in [10,23], that is the system is stable for all τ ∈ [0, 0.47767).
24
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
429819/2018-8), and Fundação de Amparo à Pesquisa do Estado de Minas Gerais - FAPEMIG, Brazil (Grant TEC - APQ-00543-17). Conflict of interest None. References [1] S.I. Niculescu, Delay effects on stability: a robust control approach, in: Lecture Notes in Control and Information Sciences, vol. 269, Springer, Berlin, Germany, 2001. [2] K. Gu, V. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhäuser, Boston, MA, 2003. [3] V.L. Kharitonov, Time-delay Systems: Lyapunov Functionals and Matrices, Birkhäuser, Boston, MA, 2013. [4] E. Fridman, Introduction to time-delay systems: analysis and control, in: Systems & Control: Foundations & Applications, first ed., Birkhäuser Basel, 2014. [5] L. Pekař, Q. Gao, Spectrum analysis of LTI continuous-time systems with constant delays: a literature overview of some recent results, IEEE Access 6 (2018) 35457–35491. [6] R.K. Brayton, Bifurcation of periodic solutions in a nonlinear differencediferrential equation of neutral type, Quart. Appl. Math. 24 (3) (1966) 215–224. [7] D. Salamon, Control and Observations of Neutral Systems, Pitman Advanced Publishing Program, Boston, MA, 1984. [8] Y. Kuang, Delay Differential Equations: with Applications in Population Dynamics, Vol. 191, Academic, San Diego, CA, 1993. [9] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Spring-Verlag, New York, 1993. [10] P. Wei, Q. Guan, W. Yu, L. Wang, Easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems, Systems Control Lett. 57 (2) (2008) 165–174.
[11] P. Fu, S.I. Niculescu, J. Chen, Stability of linear neutral time-delay systems: exact conditions via matrix pencil solutions, IEEE Trans. Automat. Control 51 (6) (2006) 1063–1069. [12] J. Chen, G. Gu, C.N. Nett, A new method for computing delay margins for stability of linear delay systems, Systems Control Lett. 26 (2) (1995) 107–117. [13] S.I. Niculescu, Stability and hyperbolicity of linear systems with delayed state: a matrix-pencil approach, IMA J. Math. Control Inform. 15 (4) (1998) 331–347. [14] J. Louisell, A matrix method for determining the imaginary axis eigenvalues of a delay system, IEEE Trans. Automat. Control 46 (12) (2001) 2008–2012. [15] J. Louisell, Matrix polynomials similar operators and the imaginary axis eigenvalues of a matrix delay equation, SIAM J. Control Optim. 53 (1) (2015) 399–413. [16] J. Louisell, Imaginary axis eigenvalues of matrix delay equations with a certain alternating coefficient structure, Systems Control Lett. 110 (2017) 49–54. [17] G. Ochoa, S. Mondié, V.L. Kharitonov, Computation of Imaginary Axis Eigenvalues and Critical Parameters for Neutral Time Delay Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012, pp. 61–72. [18] G. Ochoa, V.L. Kharitonov, S. Mondié, Critical frequencies and parameters for linear delay systems: a Lyapunov matrix approach, Systems Control Lett. 62 (9) (2013) 781–790. [19] D.S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas, second ed., Princeton University Press, New Jersey, 2009. [20] J.K. Hale, E.F. Infante, F.-S.P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl. 105 (2) (1985) 533–555. [21] J. Chen, H. Latchman, Frequency sweeping tests for stability independent of delay, IEEE Trans. Automat. Control 40 (9) (1995) 1640–1645. [22] N. Olgac, R. Sipahi, A practical method for analyzing the stability of neutral type LTI-time delayed systems, Automatica 40 (5) (2004) 847–853. [23] N. Gu, M. Tan, W. Yu, An algebra test for unconditional stability of linear delay systems, in: Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228), vol. 5, 2001, pp. 4746–4747.
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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Imaginary characteristic roots of neutral systems with commensurate delays Fernando de Oliveira Souza Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-010, Belo Horizonte, MG, Brazil
article
info
a b s t r a c t This paper concentrates on stability analysis of neutral linear time-invariant (LTI) delay systems with multiple commensurate time-delays. A new numerical procedure is offered for the determination of purely imaginary characteristic roots of neutral delay systems, which plays a crucial role in assessing the system stability. Based on simple linear algebra it is shown that the imaginary characteristic roots of such a system can be found by calculating the generalized eigenvalues of an associated matrix pair. If the system is of retarded type we just need to determine the eigenvalues of a single matrix. The results extend previously known work on neutral delay systems subject to a single delay for neutral systems subject to multiple commensurate delays. In the light of the main result of this paper we present a new method to determine the largest first time-delay interval for which a neutral delay system preserves its stability. The paper is closed by showing numerical examples that illustrate the applicability and effectiveness of the proposed method. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 6 June 2018 Received in revised form 2 March 2019 Accepted 19 March 2019 Available online xxxx Keywords: Neutral delay systems Multiple time-delays Stability Generalized eigenvalues First time-delay interval
1. Introduction
equation of the system (1) is given by
Stability analysis of delay-differential systems of neutral type is a subject of great practical and theoretical importance which has received considerable attention for decades. Comprehensive surveys on this subject can be found in the literature [1–5]. Practical examples of neutral delay-differential systems include lossless transmission lines [1,6], control processes [1,7], population ecology [8], etc. From a theoretical point of view, the solution to this problem is very complex due to the presence of exponential type transcendental terms in the system characteristic equation. Therefore the stability analysis of neutral linear time-invariant (LTI) delay systems with multiple commensurate time-delays has been a very active research topic [1,3,4,9–11]. In this paper we consider the general class of neutral LTI multi-delay systems of the form x˙ (t) −
m ∑
Bk x˙ (t − kτ ) = A0 x(t) +
k=1
m ∑
Ak x(t − kτ ),
(1)
k=1
where x(t) ∈ Rn , Ak , Bk ∈ Rn×n , k = 1, . . . , m. The system is of retarded type when all Bk = 0. Clearly, stability of neutral delay systems proves to be more complex, due to the fact that the system involves the derivative of the delayed state. The characteristic E-mail address: [email protected]. https://doi.org/10.1016/j.sysconle.2019.03.011 0167-6911/© 2019 Elsevier B.V. All rights reserved.
m m ) (( ∑ ) ∑ Ak z k ∆(s, z) := det s I − Bk z k − k=1
(2)
k=0
with z = e−sτ and τ ≥ 0. The system is infinite dimensional and as such it possesses infinitely many characteristic roots. Therefore, since direct calculation of the roots of the bivariate polynomial ∆(s, z) is not practical, the literature is rich in methods that try to indirectly assess the location of its roots, see e.g. [10–18]. This paper proposes one such method. The idea is to establish a simple methodology to compute the critical frequencies, ω ∈ R, where the roots of ∆(s, e−sτ ) = 0 cross the imaginary axis, i.e. ∆(jω, e−jωτ ) = 0. The determination of purely imaginary characteristic roots of neutral linear time-invariant (LTI) delay systems plays a crucial role in assessing the system stability, since under common assumptions on nonsingularity, stability switching only takes place where the system has such roots. Concerning retarded systems with multiple commensurate delays case in [12] was presented a 2n2 m × 2n2 m matrix having spectrum containing all system unitary modulus characteristic roots, i.e. all zi = e−jωi τi such that ∆(jωi , zi ) = 0. Again for the case of a retarded system, in [13] was introduced a pair of matrix pencils, to be checked for generalized eigenvalues with unitary modulus. The dimension of the highest order matrix pencil is also 2n2 m × 2n2 m. Further, concerning both neutral and retarded systems with single delay, in [14] another
20
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
elegant theoretical result provided a different 2n2 × 2n2 matrix having spectrum containing all imaginary characteristic roots of the delay system, i.e. all jωi such that ∆(jωi , zi ) = 0. Later in [11] the technique of [12] was extended to consider neutral systems with commensurate delays, where the resulting matrix having spectrum containing all system unitary modulus characteristic roots is also of dimension 2n2 m × 2n2 m. Addressing neutral and retarded delay systems with a single delay the method in [14] was improved in [15] where the reported method requires just the solution of an n2 × n2 generalized eigenvalue problem. In addition a very interesting method regarding neutral and retarded systems subject to multiple commensurate and distributed delays was proposed in [17,18]. This method is based on the observation that the characteristic roots of the time-delay system are related with the spectrum of an auxiliary delay free system used for the computation of Lyapunov matrices (see e.g. [3] for more results in this direction). This methodology differs from the others mentioned by the fact that it is strongly connected to time domain properties. In the case of neutral systems subject to multiple commensurate delays the method also involves a 2n2 m × 2n2 m matrix pair [17]. This paper proposes one such method. The present paper using a different methodology extends the earlier result in [14] for neutral systems with commensurate delays. In order to understand the proposed result in a more transparent light we recall that the neutral system with multiple commensurate delays in (1) can be rewritten as a higher dimensional neutral system subject to a single delay [11] x˙ (t) − B1 x˙ (t − τ ) = A0 x(t) + A1 x(t − τ )
(3)
where x(t) = xT (t), xT (t − τ ), . . . , xT (t − (m − 1)τ ) , B1 ⎢I
B1 = ⎢ ⎣ .. .
... ··· .. .
Bm−1 0
0
···
I
⎡
0 ⎢0
⎡
and A1 = ⎢ ⎣ ..
.
0
.. .
0 0
.. . ···
Bm 0⎥
⎤
⎢ .. ⎥ ⎦ , A0 = ⎣ .. . .
0
0
··· ··· .. . 0
A0 ⎢0
⎡
A1 0
.. . ···
··· ··· .. . 0
2. Notation and preliminaries some basic notations. Let the scalar j = √ First we introduce −1 and Rm×n (Cm×n ) be the set of m × n real (complex) matrices. R>0 (R<0 ) and C+ denote the set of positive (negative) real numbers and the closed right half plane of the complex plane, respectively. If s is a complex number, s∗ denotes its complex conjugate and |s| its complex modulus. Let Cϵ := {s ∈ C : s + s∗ ≥ ϵ, ϵ ∈ R<0 }. For a matrix X denote its determinant by det(X ), its spectrum by σ (X ), and its spectral radius by ρ (X ). For a matrix pair (A, B), denote the set of all generalized eigenvalues by σ (A, B) := {λ ∈ C : det(A − λB) = 0} and the ith generalized eigenvalue by σi (A, B). The notation ⊗ denotes the Kronecker product. Let In (0n ) be the identity (zero) matrix with dimension n × n and 0n×m the zero matrix with dimension n × m; throughout the paper such subscripts will be suppressed, whenever the dimension is clear from the context. Throughout the paper, we shall make use of some linear standard results. They are repeated here for convenience (see for instance [19]): Fact 1. Let A, B, C, and D be matrices of appropriate dimensions, such that the following operations are well defined, then:
]T
[
from that adopted in [14,17,18]. This constitutes the main benefit of the proposed result. Then, taking into consideration the main result of this paper we present a new computable procedure that allows one to determine the largest first time-delay interval for which a neutral delay system preserves its stability. The applicability and effectiveness of the main result will be illustrated by examples in Section 4.
Am−1 0 ⎥
⎤
(a) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD).
.. ⎥ ⎦, .
0
(b) det(D) det(A − BD−1 C) = det (c) det(A) det(D − CA−1 B) = det
Am 0⎥
⎤
(
A C
(
A C
)
B D
if D is nonsingular.
)
B D
if A is nonsingular.
The notion of stability of the neutral system with multiple commensurate delays in (1) is defined as follows [11].
.. ⎥ ⎦. .
0 (4)
Hence, based on the higher dimensional system in (3) the methods in [14] and [15] can be applied directly. In which case, however, the dimension of the corresponding problem rises, thus leading to a higher, undesirable computation load. Namely, the methods in [14] and [15] will require the computation of the generalized eigenvalues of a 2(nm)2 × 2(nm)2 and (nm)2 × (nm)2 matrix pair, respectively. On the other hand, via a different methodology, the present paper proposes a similar method of that in [14] involving a 2n2 m × 2n2 m matrix pair, which demands less computational load than the direct application of the methods in [14] and [15] on the higher dimensional system (3) for m > 2. To conclude, we highlight that for retarded systems the proposed result is reduced to the standard eigenvalue problem, a similar feature with the one in [14]. Concisely, this paper revisits the problem of assessing the imaginary characteristic roots of neutral systems with commensurate delays. The result proposed extends the previous one regarding linear neutral delay systems subject to a single delay [14], presenting a similar matrix pair having spectrum containing all imaginary characteristic roots of neutral systems with commensurate delays. We emphasize that the proposed technique employs a simple linear algebra methodology rooted in Kronecker products and Schur complement formulas different
Definition 1. Let ∆(s, z) be given in (2). System (1) is stable for a given τ ≥ 0 if ∆(s, z) ̸ = 0 with z = e−sτ , ∀s ∈ Cϵ . Furthermore, it is said to be stable independent of delay if ∆(s, z) ̸ = 0 with z = e−sτ , for all ∀s ∈ Cϵ and τ ≥ 0. It is worth mentioning that a necessary condition for the system (1) to be stable concerns the difference equation x(t) −
m ∑
Bk x(t − kτ ) = 0.
(5)
k=1
Namely, for the system (1) to be stable, it is necessary that the difference Eq. (5) be stable for all τ ≥ 0 [2,20]. In addition, we point out that the following result gives a simple necessary and sufficient condition to assert the stability of (5) for all τ ≥ 0. Lemma 1 (Gu et al. [2, Lemma 3.18]). The system in (5) is stable for all τ ≥ 0 if and only if
ρ (B1 ) < 1
(6)
where B1 is given in (4). Fortunately, an important fact is that under the assumption that the difference equation (5) is stable, the condition in Definition 1 needs to hold only for C+ ; that is Cϵ can be replaced by C+ , as stated in the next definition [11].
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
Definition 2. Let ∆(s, z) be given in (2) and suppose that condition (6) holds. System (1) is stable for a given τ ≥ 0 if ∆(s, z) ̸= 0 with z = e−sτ , ∀s ∈ C+ . Furthermore, it is said to be stable independent of delay if ∆(s, z) ̸ = 0 with z = e−sτ , for all ∀s ∈ C+ and τ ≥ 0.
In this section we provide the main results of the paper which require only the computation of eigenvalues and generalized eigenvalues. First, we present the following auxiliary lemma needed to prove the main result.
⎛ m ∑
) Pk z k
k=1
⎡
P1 ⎢ In
P2 0
⎜ ⎜ ⎢ = det ⎜Inm − ⎢ . ⎝ ⎣ ..
..
. ...
0
−Inm ⊗ A0
Pm 0
0n2 m×(nm)2 Inm ⊗ B¯
0n2 m×(nm)2
= det(I) det I − P1 z − P2 z +
m ∑
In
0
⎥ ⎟ ⎥ ⎟ ⎥ z⎟ . ⎦ ⎠
.. .
−P2 z −
m ∑
k=1
] )
(
m )−1 ∑ (
Ak + λBk e−λkτ
det I − λI − A0
(
Pk z
−zI ⎛ [ ⎜ I − P1 z = det(I) det ⎝ −zI
I
m ∑
Pk z
det(I − H(λ)e−λτ ) =
⎞
⎤ k−2
⎥[ ⎦ 0
k=4
∑ k=4
I
0 I
⎛ ⎡ ⎜ I − P1 z ⎜⎣ = det(I) det ⎜ −zI ⎝ 0
−P2 z I
−zI
=
Pk z k−2 ⎥⎟
−P4 z −
m ∑
0
nm ∏ [
1 − σi (H(λ)) e−λτ
I
0
0
−zI
I
0
0
−zI
I
⎞ 0
Therefore, we know
]⎟ ⎟ zI ⎟ ⎠
∆(λ, e−λτ ) = 0 ⇔ ∃i : |σi (H(λ))| = 1.
∑m
Note that for λ = 0 we have that ∆(λ, e−λτ ) = 0 ⇔ k=0 Ak is singular. Now recall that the eigenvalues of the Kronecker product of H(λ) with its matrix of conjugates are all possible pairwise products of the eigenvalues of the matrix H(λ) with their conjugates [19, Proposition 7.1.10]. Now |σi (H(λ))| = 1 for some i,
⎤⎞ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ . ⎥⎟ ⎦⎠
0 = det Inm ⊗ Inm − H(λ) ⊗ H(−λ)
(
The result then follows by applying this procedure repeatedly.
□
The next theorem presents the procedure proposed to compute the purely imaginary characteristic roots of the neutral delay system (1). Theorem 1. Consider ∆(s, z) in (2), the of the system (1), assume that A0 has no define the matrices A¯ = [A1 A2 . . . Am ], C T = [In 0n×n(m−1) ],
[ D=
[ E=
0n×n(m−1) In(m−1) I(nm)2 Inm ⊗ D
]
0n 0n(m−1)×n
D ⊗ Inm I(nm)2
]−1 [
characteristic equation eigenvalue on jR, and B¯ = [B1 B2 . . . Bm ],
)
(
( )( )) = det Inm ⊗ Inm − H(λ) ⊗ Inm Inm ⊗ H(−λ) ([ ]) Inm ⊗ Inm H(λ) ⊗ Inm = det Inm ⊗ H(−λ) Inm ⊗ Inm ([ ]) Inm ⊗ Inm [C (λI − A0 )−1 B(λ) + D] ⊗ Inm = det −1
[Fact 1.(b)]
= det(D − CA−1 B)
[Fact 1.(a)]
Inm ⊗ [−C (λI + A0 )
B(−λ) + D]
(λI − A0 ) ⊗ Inm A= 0n2 m
[
]
0(nm)2 ×n2 m , Inm ⊗ C
[ B=
0
−Inm ⊗ B(−λ)
]
Inm
0n2 m , ⊗ (λI + A0 )
B(λ) ⊗ Inm , 0
]
[Fact 1.(a)]
Inm ⊗ Inm
with
,
C ⊗ Inm 0(nm)2 ×n2 m
]
∆(λ, e−λτ ) = 0 ⇔ {∃i : σi (H(λ))e−λτ = 1}.
Pk z k−3
k=5
)]
i=1
I
⎛⎡
(
it follows that
⎥⎟ ⎥⎟ ⎦⎠
⎤ ⎡ m ∑ ⎤ k− 3 Pk z −P3 z ⎥[ ⎢ ⎥ ⎢ 0 ⎦ − ⎢ k=4 ⎥ 0 ⎦ ⎣ 0
−P3 z
1 − σi H(λ)e−λτ
⎤⎞
0
−P2 z
nm ∏ [ i=1
]⎟
zI ⎠
0
−zI
B(λ) + D
to one with a single delay. Since
⎢P3 z + −⎣
−P3 z −
is defined as
⎥⎟ ⎦⎠
m
−P2 z
= det(I − H(λ)e−λτ ) = 0
⎤⎞
⎡ −P2 z
)
¯ The last equality is obtained by an application with B(λ) = A¯ +λB. of Lemma 2. The definition of H(λ) reduces the stability problem
k−1
⎛⎡
⎜⎢I − P1 z ⎜⎢ ⎜⎢ = det ⎜⎢ −zI ⎜⎢ ⎝⎣ 0
−1
I
]
)
k=1
H(λ) := C (λI − A0 )
Pk z k−1 z
k=3
k=1
Under the assumption that A0 has no eigenvalue on jR, then (jωI ±A0 )−1 is well defined for all ω ∈ R. Therefore ∆(λ, e−λτ ) = 0 if and only if
k=3
⎛⎡
(7)
m m ( ( ) ∑ ∑ ) ∆(λ, e−λτ ) = det λ I − Ak e−λkτ Bk e−λkτ − A0 −
where H(λ) ∈ C [
0
E.
m (( ) ) ∑ ( ) = det λI − A0 − Ak + λBk e−λkτ .
k=1
⎜⎢I − P1 z ⎜⎢ = det ⎜⎢ ⎝⎣ −zI
E , and
Proof. Set λ = jω, then
nm×nm
⎜⎢I − P1 z = det ⎝⎣
]
0n2 m×(nm)2
]
jΩ = {jωi : jωi ∈ σ (F , G)}.
⎤ ⎞
Pk z k
(
0n2 m×(nm)2 Inm ⊗ A¯
B¯ ⊗ Inm
[ G = I2n2 m +
−A¯ ⊗ Inm
[ +
k=1
... ... .. .
Proof. The proof, inspired by Chen et al. [21], consists in applying the Schur determinant formula (Fact 1.b) as follows ( ) m ∑ det I −
]
0n2 m
Let z ∈ C and matrices Pk ∈ Cn×n . Then
Lemma 2.
(
A0 ⊗ Inm 0n2 m
Then all purely imaginary characteristic roots of ∆(s, e−sτ ) belong to the following set:
3. The main results
det In −
[ F =
21
22
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
C=
[ −C ⊗ Inm
]
0
0
−Inm ⊗ C
, and D =
[
Inm ⊗ Inm Inm ⊗ D
]
D ⊗ Inm . Inm ⊗ Inm
This is true if and only if 0 = det(A) det(D − CA
−1
([ B) = det
A C
(
])
B D
= det(D) det(A − BD−1 C)
det sI − (I −
[Fact 1.(c)] [Fact 1.(b)]
= det(A − BD C).
(8)
The latter steps hold because det(D) = 1, i.e. using Facts 1.(a) and 1.(b) we have det(D) = det(Inm ⊗ Inm ) det(Inm ⊗ Inm − D ⊗ D) and the last factor is the determinant of a lower triangular matrix with ones on the main diagonal. Therefore, recalling the definitions in the theorem we have that (8) can be rewritten as 2m
det(F − λG),
which completes the proof. □ The previous theorem states that all purely imaginary characteristic roots of the neutral delay system (1) when σ (A0 ) ∩ jR = ∅, are contained in the set of generalized eigenvalues of a 2n2 m × 2n2 m matrix pair (F , G). For retarded systems B¯ = 0 ⇒ G = I, and the generalized eigenvalue problem is reduced to the standard eigenvalue problem. Also, it is worth mentioning that a necessary condition for the neutral delay system (1) to be stable independently of delay is that A0 is Hurwitz [2]. Note also, if A0 does have eigenvalues on the imaginary axis, that we may search for pure imaginary eigenvalues of (1) by applying Theorem 1 to matrices A0 + ϵ I for some (arbitrarily small) scalar ϵ . Since based on Theorem 1 the complete set of purely imaginary characteristic roots of the neutral system in (1) can be found, one can deploy the method proposed in [22] which results in the determination of an exact and complete set of delay intervals where the system is stable. However, for illustration of the applicability of the previous theorem in the following we present a computable procedure that allows one to determine the largest first time-delay interval for which a neutral delay system preserves its stability. That is, under the condition in Lemma 1, to characterize whether there exists some τmax > 0 for which all roots of the transcendental function ∆(s, e−sτ ) in (2) are in the open left half of the complex plane for all values of τ ∈ [0, τmax ). Corollary 1. Consider the set Ω given in Theorem 1 and the matrices A0 , A1 , and B1 given in (4). Suppose that σ (A0 ) ∩ jR = ∅ and∑ the condition in Lemma 1 holds, i.e. ρ (B1 ) < 1. Then the matrix m I − k=1 Bk is nonsingular. Suppose that the ( )(1) is stable ∑also ∑system m m at τ = 0, that is, the matrix (I − k=1 Bk )−1 k=0 Ak is Hurwitz. Define
τmax
⎧ ∞ if Ω ∩ R>0 = ∅. Otherwise, ⎪ ⎪ ⎪ θi ⎪ ⎪ ⎨ min where the pair (ωi , θi ) belongs to the set i ω i := Θ := {(ωi , θi ) : ωi ∈ Ω ∩ R>0 and ⎪ ⎪ ⎪ ⎪ e−jθi ∈ σ (jωi I − A0 , A1 + jωi B1 ) ⎪ ⎩ with θi ∈ [0, 2π )} .
Then the system is stable for all τ ∈ [0, τmax ). Proof. Initially ρ (B1 ) < 1 implies that
(
det I −
m
∑ k=1
Bk
)
= det(I − B1 ) ̸= 0.
m ∑
Bk )−1
k=1
−1
0 = (−1)2n
Additionally, since the system (1) is stable at τ = 0, the Hurwitz condition holds: ∆(s, 1) is nonzero for all s ∈ C+ , i.e. ∄s ∈ C+ such that m ∑
Ak
)
= 0.
k=0
Hence ∆(0, 1) ̸ = 0. Consider the case τmax = ∞, with Ω ∩ R>0 = ∅, and let τ ≥ 0. Theorem 1 implies that for every ω > 0, we have ∆(jω, e−jωτ ) ̸= 0. We know that ∆(jω, e−jωτ ) ̸= 0 with ω = 0, hence there does not exist τ ≥ 0 such that any root of ∆(s, e−sτ ) intersects the imaginary axis. Since the system (1) is stable at τ = 0, it is stable for all τ ∈ [0, ∞); that is, the system is stable independently of delay. Suppose now that τmax < ∞. Then Ω ∩ R>0 ̸ = ∅. By Theorem 1 and under the assumptions that the system is stable at τ = 0 and ρ (B1 ) < 1, we find that
∆(jωi , e−jωi τi ) = ∆(jωi , e−jθi ) = 0 for some ωi > 0, τi > 0. By definition, θi ∈ [0, 2π );
(9)
choose θi ≤ ψ if ∆(jωi , e−jψ ) = 0 with ψ ≥ 0. We have ∆(jωi , e−jωi τi ) = 0 with τi = θi /ωi , and ∆(jωi , e−jωi τ ) ̸ = 0 for any τ ∈ [0, τi ). With τmax as defined, we see now that the system (1) is stable for all τ ∈ [0, τmax ). Finally, in order to compute e−jθi in (9) associated with ωi we invoke the characteristic equation of the high dimensional neutral system subject to a single delay in (3)
((
ˆ (s, z) := det s I − B1 z − A0 + A1 z ∆
) (
))
( ( ) ) = det sI − A0 − A1 + sB1 z . (10)
Therefore, it is clear that for any real ωi the matrix pair (jωi I − A0 , A1 + jωi B1 ) has a unit magnitude generalized eigenvalue, e−jθi , if and only if ∆(jωi , e−jθi ) = 0. The proof is now completed. □ The previous corollary constitutes a readily computable procedure to determine the first time-delay interval for which a neutral system preserves its stability. In terms of the computational demands, only computations on matrices with constant entries are required. 4. Numerical examples In order to illustrate the applicability and efficiency of the results presented, numerical examples are provided, some drawn from the literature. Moreover, we point out that the same conclusions in the examples of this section can be recovered by the methods presented in [17,18]. Example 1. A0 =
[ −2 0 0.5 0.75
[ B1 =
Consider the system in (1) with data
]
[ −1 A1 = −1
0 , −1 0
− 0.25
]
]
0 , −1
and
.
First following Theorem 1 we compute
σ (F , G) = {±j0, ± 0.8382, ± 1.0604, ± 2} that yields Ω = {0, 0}. Then accordingly with Corollary 1 we check that the condition in Lemma 1 holds, i.e.∑ ρ (B1 ) < ( ) 1, Ω ∩ R>0 = ∅ and we compute ∑ m −1 σ (I − m B ) A = {−1.6, − 6}, which means that k k k=1 k=0 the system is stable independently of delay.
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
Example 2.
Example 4. Consider the system in (1) with matrices in (11) and let
Consider the system in (1) with data
1.5 −0.25 [ −0.75 B1 = 0.25
0.25 , 2
[
]
A0 =
] −0.25 0
[ A1 =
−3.5 0.5
−0.5 , −3
]
0.02 ⎢ 0 B1 = ⎣ 0 0
⎡
and
;
First following Theorem 1 we compute
Then accordingly with Corollary 1 we check that ρ (B1 ) < 1, the system is stable for τ = 0,
Θ = {(4.1654, 1.9809), (2.3834, 0.8911)} and } { 0.8911 1.9809 = 0.4756, = 0.3739 , τmax = min 4.1654 2.3834
0
Accordingly with Corollary 1 we check that ρ (B1 ) < 1, the system is stable for τ = 0,
Θ = {(3.3264, 0.47807), (1.3873, 2.0565), (1.1338, 0.80057), (0.8680, 2.9574), (0.3160, 1.76), (0.5809, 3.328)} { } τmax = min 0.14372, 1.4824, 0.70611, 3.4073, 5.5699, 5.7288
In [21] it is shown that the retarded type system
which means that the system is stable for all τ ∈ [0, 0.14372), as expected. Example 5. Consider the system in (1) with matrices in (11) and let B1 = A1 , B2 = A2 , and B3 = 12 A3 . First following Theorem 1 we compute Ω = {±3.1652, ± 1.7418, ± 1.3434, ± 0.77282, ± 0.37593, ± 0.54353}.
1 0 0 −3
−2 ⎡ −0.05 ⎢ 0.005 A1 = ⎣ 0
0 1 0 −5
Then accordingly with Corollary 1 we check that ρ (B1 ) < 1, the system is stable for τ = 0,
⎤
0 0 ⎥ , 1 ⎦ −2
(11)
0.005 0.005 0 0
0.25 0 0 −0.5
0 0⎥ , 0⎦ 0
0.005 ⎢ 0 A2 = ⎣ 0 −1
0.0025 0 0 −0.5
0 0.05 0 −0.5
0 0 ⎥ , 0.0005⎦ 0
0.0375 ⎢ 0 A3 = ⎣ 0.05 0
0 0.05 0.05 −2.5
0.075 0.05 0 0
0.125 0 ⎥ , 0 ⎦ −1
−1
⎡
⎡
⎤
0 0 ⎥ . 0 ⎦ 0.25
and
which means that the system is stable for all τ ∈ [0, 0.3739). It is worthwhile to mention that in [14] the same spectrum σ (F , G) provided above was computed, and accordingly the same conclusion about the system’s stability was obtained.
⎢0 A0 = ⎣ 0
0.03 0 0 .5 0
Ω = {±3.3264, ±1.3873, ±1.1338, ±0.8680, ±0.3160, ±0.5809}.
σ (F , G) = {±j4.1654, ± 1.4524 ± j2.4206, ± j2.3834} that yields Ω = {±4.1654, ± 2.3834}.
⎡
0 0.01 0 0
In [10] by an analytical method it was shown that this system is stable within the delay interval τ ∈ [0, 0.14372). Then using Theorem 1 we compute
as in [14].
Example 3. with data
23
⎤
Θ = {(3.1652, 0.97987), (1.7418, 1.1974), (1.3434, 1.1175), (0.77282, 3.1707), (0.37593, 1.8432), (0.54353, 3.496)} and
τmax = min {0.30958, 0.68748, 0.83185, 4.1028, 4.903, 6.4321} .
⎤
⎤
is not stable independent of delay. Additionally, in [10,23] by employing algebraic criteria it was shown that this system is stable within the delay interval τ ∈ [0, 0.47767). Therefore, applying Theorem 1 we have that
Ω = {±1.2507, ± 0.31479} and thus Ω ∩ R>0 ̸ = ∅, then accordingly with Corollary 1 we see that the system is not stable independent of delay, as expected. Moreover, we check that ρ (B1 ) < 1, the system is stable for τ = 0,
which means that the system is stable for all τ ∈ [0, 0.30958). 5. Conclusions This paper has revisited the problem of assessing the stability of linear time-invariant (LTI) neutral systems with multiple commensurate delays. The main result demonstrates that the imaginary characteristic roots of such a system can be found by calculating the generalized eigenvalues of an associated matrix pair. If the system is of retarded type we just need to determine the eigenvalues of a single matrix. This result extends a previously known work on neutral delay systems subject to a single delay, and shows that a similar result exists for neutral systems subject to multiple commensurate delays. Then, as a second contribution, in the light of the main result in this paper we present a new method to determine the largest first time-delay interval for which a neutral delay system preserves its stability. Applications and the effectiveness of the proposed methods are illustrated by numerical examples.
Θ = {(1.2507, 0.59742), (0.31479, 1.7588)}
Acknowledgments
and
The author is grateful to the reviewers for an insightful, complete, and thoughtful review of the manuscript. This research has been supported by the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior — CAPES, Brazil (Finance Code 001), Conselho Nacional de Desenvolvimento Científico e Tecnológico — CNPq, Brazil (Grants 311574/2017-3 and
τmax = min 0.47767, 5.5873
{
}
yielding the same conclusion in [10,23], that is the system is stable for all τ ∈ [0, 0.47767).
24
F.O. Souza / Systems & Control Letters 127 (2019) 19–24
429819/2018-8), and Fundação de Amparo à Pesquisa do Estado de Minas Gerais - FAPEMIG, Brazil (Grant TEC - APQ-00543-17). Conflict of interest None. References [1] S.I. Niculescu, Delay effects on stability: a robust control approach, in: Lecture Notes in Control and Information Sciences, vol. 269, Springer, Berlin, Germany, 2001. [2] K. Gu, V. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhäuser, Boston, MA, 2003. [3] V.L. Kharitonov, Time-delay Systems: Lyapunov Functionals and Matrices, Birkhäuser, Boston, MA, 2013. [4] E. Fridman, Introduction to time-delay systems: analysis and control, in: Systems & Control: Foundations & Applications, first ed., Birkhäuser Basel, 2014. [5] L. Pekař, Q. Gao, Spectrum analysis of LTI continuous-time systems with constant delays: a literature overview of some recent results, IEEE Access 6 (2018) 35457–35491. [6] R.K. Brayton, Bifurcation of periodic solutions in a nonlinear differencediferrential equation of neutral type, Quart. Appl. Math. 24 (3) (1966) 215–224. [7] D. Salamon, Control and Observations of Neutral Systems, Pitman Advanced Publishing Program, Boston, MA, 1984. [8] Y. Kuang, Delay Differential Equations: with Applications in Population Dynamics, Vol. 191, Academic, San Diego, CA, 1993. [9] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Spring-Verlag, New York, 1993. [10] P. Wei, Q. Guan, W. Yu, L. Wang, Easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems, Systems Control Lett. 57 (2) (2008) 165–174.
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