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Stability crossing set for systems with two scalar-delay channels
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Stability crossing set for systems with two scalar-delay channels Mohammad Naghnaeian ∗ Keqin Gu ∗∗ ∗
Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801-2906, USA (e-mail: [email protected]). ∗∗ Department of Mechanical and Industrial Engineering Southern Iliinois Univeristy Edwardsville Edwardsville, IL 62026-1805, USA (e-mail: [email protected]). Abstract: This article studies the stability crossing set of linear time-invariant systems with two scalar-delay channels. This study is crucial to the complete stability analysis along the idea of D-subdivision. The characteristic quasipolynomial of such systems contains an exponential term with the sum of two delays (cross term) in its exponent. A complete parameterization and geometric characterization of the stability crossing set is conducted. It was found instrumental to relate it to an associated quasipolynomial without such a cross term. Keywords: Delay, stability 1. INTRODUCTION
very general class of systems was given in Jarlebring (2009).
This article studies the stability crossing set of systems with characteristic quasipolynomial of the form
As argued in Gu (2010), systems with two scalar-delay elements may be modeled as one with two scalar-delay channels. In this model, a system consists of a forward system without delay
∆(s) = p0 (s) + p1 (s)e−τ1 s +p2 (s)e−τ2 s + p12 (s)e−(τ1 +τ2 )s ,
(1)
which is defined as the set of all (τ1 , τ2 ) such that the above quasipolynomial has at least one root on the imaginary axis. In the above expression, p0 (s), p1 (s), p2 (s) and p12 (s) are polynomials with real coefficients. The stability crossing set has been recognized as critical in one school of stability analysis since Neimark proposed the D-subdivision method (Neimark (1949), El’Sgol’ts & Norkin (1973), Kolmanovskii & Nosov (1986)). Examples of stability analysis of various such systems abound in the literature, see, for example, St´ep´an (1989), Hale & Huang (1993), Ruan & Wei (2003) and the references therein. In recent years, the study of stability crossing set of a class of systems has received substantial attention. This is in contrast to earlier works that typically study one specific system with variable parameters mentioned in the previous paragraph. For system (1) without the last term, Sipahi & Olgac (2003) seems to be the first such study, and a more complete parameterization and geometric characterization was given in Gu, Niculescu, & Chen (2005). Studies on more general systems, including the system (1), may be found in Fazelinia, Sipahi, & Olgac (2007), Olgac, Vyhl’idal, & Sipahi (2008), Sipahi & Olgac (2006a), and Sipahi & Olgac (2006b). However, none of these works include a complete parameterization and geometric characterization parallel to Gu, Niculescu, & Chen (2005). It is worth mentioning that an interesting complete parameterization of stability crossing set of a 978-3-902661-93-7/11/$20.00 © 2011 IFAC
x(t) ˙ = Ax(t) + Bu(t),
(2)
y(t) = Cx(t) + Du(t), (3) where , x(t) ∈ R , u1 (t) u(t) = ∈ R2 , u2 (t) y1 (t) y (t) = ∈ R2 , y2 (t) and a feedback consisting of two scalar-delay channels u1 (t) y1 (t − τ1 ) = . (4) u2 (t) y2 (t − τ2 ) The characteristic quasipolynomial of this system is −τ s e 1 0 sI − A −B −τ2 s 0−τ se . ∆(s) = det 1 e 0 −C I − D −τ2 s 0 e An application of the Schur determinant complement yields −τ s e 1 0 sI − A −B ∆(s) = det 0 e−τ2 s 0 M (s) n
= det (sI − A) · det [M (s)] ,
(5)
where
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M (s) = I − D + C (sI − A)
−1
B
e−τ1 s 0 0 e−τ2 s
. (6)
10.3182/20110828-6-IT-1002.01733
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
The characteristic function given in (5) is a quasipolynomial of the form given in (1). Therefore, even though (1) is less general than the system discussed in some recent literature cited above, its thorough analysis is of great practical importance. The purpose of this article is to extend the results of Gu, Niculescu, & Chen (2005) to the more general system described by (1). Due to space limitation, all proofs are omitted. These proofs are contained in Naghnaeian & Gu (2011). 2. PRELIMINARIES The ultimate purpose of such an analysis is to evaluate stability. In this article, by stability, we mean asymptotic stability. First the following non-triviality assumptions are made, Assumption 1. Existence of principal term: ord [p0 (s)] ≥ max{ord [p1 (s)], ord [p2 (s)], ord [p12 (s)]}, where ord [·] is the order of the polynomial concerned. Assumption 2. Zero frequency restriction: p0 (0) + p1 (0) + p2 (0) + p12 (0) 6= 0. Assumption 3. Restriction on difference operator: |p1 (s)| + |p2 (s)| + |p12 (s)| < 1. lim s→∞ |p0 (s)| These assumptions are made to exclude some obviously trivial cases. If Assumption 1 is violated, then the quasipolynomial does not have a principal term, and the system is unstable for any τ1 ≥ 0, τ2 ≥ 0 (Bellman & Cooke (1963)). If Assumption 2 is violated, then 0 is always a root of ∆(s) for any τ1 ≥ 0, τ2 ≥ 0, and therefore, it is impossible for the system to be stable. Assumption 3 ensures the asymptotic stability for arbitrary (τ1 , τ2 ) of the difference equation y1 (t) y1 (t − τ1 ) =D , (7) y2 (t) y2 (t − τ2 )
which is obtained by a substitution of (3) by (4). This also guarantees the existence of a vertical line on the left of the imaginary axis such that all the roots to the right of this vertical line are continuous with respect to the delay parameters (τ1 , τ2 ) ∈ R2+ . Assumption 3 may be replaced by the less restrictive stability conditions for the difference equation given in Carvalho (1996) or Fridman (2002) (which are equivalent as shown in Li & Gu (2010)). A naive approach is to use a method similar to the one given in Walton & Marshall (1987) to eliminate the last term of (1) and reduce it to the case discussed in Gu, Niculescu, & Chen (2005). This may proceed as follows: As all the coeffificnets of ∆ are real, if ∆(jω) = 0, (8) then ∆(−jω) = 0. (9) Multiply the equation (8) by p0 (−jω) and the equation (9) by e−jω(τ1 +τ2 ) p12 (jω), then subtract the resulting equations, the following equation is obtained ∆a (jω) = 0, (10) where ∆a (s) = P0 (s) + P1 (s)e−τ1 s + P2 (s)e−τ2 s , (11) and the coefficient polynomials of (11) are
P0 (s) = p12 (s)p12 (−s) − p0 (s)p0 (−s),
(12)
P1 (s) = p12 (s)p2 (−s) − p1 (s)p0 (−s),
(13)
P2 (s) = p12 (s)p1 (−s) − p2 (s)p0 (−s).
(14)
The equation (11) may be analyzed using the method discussed in Gu, Niculescu, & Chen (2005). However, as will be shown later, there are two crucial issues that require special attention. First, there are some differences between the stability crossing set of (11) and that of (1). Second, the asssumption in Gu, Niculescu, & Chen (2005) that no imaginary zeros in the crossing frequency set for the coefficient polynomial P0 (s) is no longer appropriate due to its special structure. It turns out that the key to both issues lies in a singular set. We will refer to ∆ expressed in (1) as the original (characteristic) quasipolynomial, and refer to ∆a (s) expressed in (11) as the associated (characteristic) quasipolynomial. Similarly, ∆(s) = 0 and ∆a (s) = 0 are referred as the original (characteristic) equation and the associated (characteristic) equation, respectively. Sometimes, we write ∆(s) and ∆a (s) as ∆(s, τ1 , τ2 ) and ∆a (s, τ1 , τ2 ), respectively, in order to emphasize their dependence on the delays. Definition 4. A pair (τ1 , τ2 ) ∈ R2+ is said to be a crossing point if ∆(s, τ1 , τ2 ) = 0 has at least one solution for s on the imaginary axis. The set of all crossing points is known as the stability crossing set, and is denoted as T . An ω > 0 is known as a crossing frequency if there exists at least one pair (τ1 , τ2 ) such that ∆(jω, τ1 , τ2 ) = 0. The set of all crossing frequencies is known as the crossing frequency set Ω. Parallel concepts defined on ∆a (s, τ1 , τ2 ) are denoted as the T a and Ωa , and known as the associated stability crossing set and the associated crossing frequency set. Notice, it is not necessary to consider ω < 0 as all the roots of ∆(s) appear in complex conjugate pairs. Also, Assumption 2 implies that ω = 0 cannot be a crossing frequency, and therefore can be excluded from consideration. In Gu, Niculescu, & Chen (2005), stability crossing set is known as stability crossing curves. The change of terminology is to make it consistent with the three-delay case where the stability crossing set consists of surfaces. For a given ω, let Tω = {(τ1 , τ2 ) | ∆(jω, τ1 , τ2 ) = 0} . Of course, Tω is empty if ω ∈ / Ω. Then [ Tω . T = ω∈Ω
For any set Φ, it is also convenient to define [ TΦ = Tω . ω∈Φ
For a real scalar α, we also denote \ T¯[α−1/n,α)∪(α,α+1/n] Tˆα = n=1,2,...
where (·) represents the closure of the set. The typical context of using Tˆα is when the stability crossing set
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
is parameterized as continuous curves (τ1l,m (ω), τ2l,m (ω)), l, m = 0, ±1, ±2, . . ., in which case, ! [ l,m l,m lim τ (ω), ω→α lim τ2 (ω) . Tˆα = ω→α 1 l,m
ω∈Ω
ω∈Ω
Similarly, for a given ω, we define Tωa = {(τ1 , τ2 ) | ∆a (jω, τ1 , τ2 ) = 0} . Then [ Tωa , T a=
and
In this section, the crossing frequency set Ω and stability crossing set T of the original quasipolynomial ∆(s) will be studied without referring to the associated quasipolynomial ∆a . Proposition 5. The crossing frequency set Ω consists of all ω > 0 that satisfy 2 |P1 (jω)| 2 2 ≥ P0 (jω) − |p1 (jω)| − |p2 (jω)| ,
(15)
2 |P (jω)| 2 2 2 ≥ P0 (jω) + |p1 (jω)| − |p2 (jω)| ,
(16)
P1 (jω) 6= 0,
(17)
or equivalently,
P2 (jω) 6= 0, (18) all the corresponding crossing points (τ1 , τ2 ) ∈ Tω have the expression (19)
k1 ,k2 τ2 = τ2± (ω) 1 = [∠(p2 (jω) + p12 (jω)e−jωτ1 ) ω −∠(p0 (jω) + p1 (jω)e−jωτ1 )
(20)
k2 = 0, ±1, ±2, . . . , where α = ∠P1 (jω),
f (ω) = 2 |P1 (jω)| 2 2 2 − P0 (jω) − |p1 (jω)| − |p2 (jω)| , (25)
f (ω) = 0 f ′ (ω) = 0. Assumption 8. No singular crossing. No ω satisfies P0 (jω) = 0 and 2 |P (jω)| 1 2 2 = P0 (jω) − |p1 (jω)| − |p2 (jω)| ,
(26)
(27)
Proposition 5 completely parameterizes the stability crossing set under the non-degeneracy Assumptions 6 to ??. The cases where any of these three non-degeneracy assumption is violated are very rare. Indeed, in the terminology of dynamical systems theory, the systems where the three non-degeneracy assumptions are satisfied are generic (Guckenheimer & Holmes (1983)). Also, the cases when one of these three assumptions is violated can be easily handled individually. In the remaining part of the article, the three nondegeneracy assumptions and the three non-triviality assumptions will be assumed to be satisfied unless explicitly stated otherwise. The above development leads to the following conclusion. Proposition 9. The crosssing fequency set Ω consists of a finite number of intervals of finite nonzero lengths. At any interior point of such an interval, the inequalities (15) and (16) are satisfied strictly. At any nonzero end point, (15) and (16) becomes equalities, and P0 (jω) 6= 0. (28) For any ω ∈ Ω, Tω = Tˆω . (29) 4. ASSOCIATED CROSSING SET
(21) 2
(24)
simultaneously.
where P0 , P1 , and P2 are given in (12) to (14). For a given ω ∈ Ω that satisfies
+(2k2 + 1)π],
P2 (jω) 6= 0
f (0) 6= 0. Furthermore, no ω > 0 simultaneously satisfies
3. DIRECT PARAMETERIZATION
α ± β + 2k1 π , ω k1 = 0, ±1, ±2, . . . ,
(23)
Assumption 7. Transversality. Let
then
and Tˆαa are defined analogously.
k1 ,k2 τ1 = τ1± (ω) =
P1 (jω) 6= 0
2
ω∈Ωa
TΦa
Assumption 6. No imaginary zeros. For any ω ∈ Ω,
2
|p1 (jω)| − |p2 (jω)| − P0 (jω) . (22) 2 |P1 (jω)| In calculating ∠(·) in the above expressions, we will maintain the continuity as ω varies, which means that the values may not be restricted within any given 2π range. β = cos−1
To simplify presentation, the following non-degeneracy assumptions are made.
While it is not difficult to completely parameterize T directly from the procedure given in the last setion, it is not easy to obtain geometric characterization. On the other hand, as the associated characteristic quasipolynomial is in a form that has already been completely characterized in Gu, Niculescu, & Chen (2005), significant insight can be gained from studying the relationship between T and T a . The following theorem proves important for this purpose.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Theorem 10. If a parameter combination (ω, τ1 , τ2 ) ∈ R×R2+ satisfies ∆(jω, τ1 , τ2 ) = 0, (30) then it also satisfies ∆a (jω, τ1 , τ2 ) = 0. (31) The reverse is also true provided that P0 (jω) 6= 0. (32) In other words, if (31) and (32) are satisfied, then (30) is also satisfied. It should be cautioned that the equivalence stated in the above theorem is valid only for stability crossing sets. No claim is made regarding the equivalence between the stability of the two quasipolynomials. It is important to study those ω that satisfy the equation P0 (jω) = p12 (jω)p12 (−jω) − p0 (jω)p0 (−jω) = 0. (33) Such an ω is known as a singular frequency. The set of singular frequencies is known as the singular frequency set, and is denoted as Ωs . Due to the structure of P0 (s), it can be easily shown that P0 (jω) is a polynomial of ω 2 with real coefficients. Therefore, the existence of imaginary roots of P0 (s) (corresponding to a real negative ω 2 ) is a rather common phenomenum, and a careful treatment of this case is essential for applications, in contrast to the treatment in Gu, Niculescu, & Chen (2005) where such cases are excluded. It will be shown later on in Proposition 11 that Ωs ⊂ Ωa . This, together with Theorem 10, may be succinctly expressed as Ωa = Ω ∪ Ωs ,
(34)
Tω = Tωa for all ω ∈ Ωa \Ωs ,
(35)
Tω ⊂ Tωa for all ω ∈ Ωs .
(36)
and
The following proposition describes Tωa for ω ∈ Ωs . Proposition 11. The singular frequency set Ωs satisfies Ωs ⊂ Ωa . Furthermore, given an ω ∈ Ωs , Tωa can be expressed as follows: Case I. If P1 (jω) = 0,
Theorem 12. The stability crossing set T can be expressed as ! [ [ [ a a T = Tˆω . Tω 5. ALTERNATIVE PARAMETERIZATION
Taking advantage of the relationship between the stability crossing set T and the associated stability crossing set T a , we will provide an alternative parameterization of T in this section. This parameterization is based on T a , and is modified from those given in Gu, Niculescu, & Chen (2005). Based on this parameterization, it is rather easy to obtain geometric insight of the stability crossing set. We start with characterizing the crossing frequency set. Proposition 13. The set Ω\Ωs = Ωa \Ωs consists of all ω that satisfy the following three conditions P0 (jω) 6= 0; f2 (ω) ≥ 1 or f2 (ω) ≤ −1;
s
In order to obtain T from T , we observe that Ω consists of a finite number of points. Apply Proposition 9 and the relation (35), we conclude that for any ω ∈ Ω ∩ Ωs , Tω = Tˆωa , from which we arrive at the following theorem.
(40) (41)
where |P1 (jω)| |P2 (jω)| − , P0 (jω) P0 (jω) |P1 (jω)| |P2 (jω)| f2 (ω) = + . P0 (jω) P0 (jω)
f1 (ω) =
(42) (43)
The above proposition means that we may identify Ω by plotting f1 (ω) and f2 (ω) against ω. As indicated by Proposition 9, Ω consists of intervals Ωk , k = 1, 2, . . . , N , N [ Ω= Ωk . k=1
Without loss of generality, let these intervals be ordered from left to right. We may write Ωk = [ωkl , ωkr ] l if ωk 6= 0, and Ωk = (ωkl , ωkr ] otherwise. Furthermore, we have the following conclusion. Proposition 14. An endpoint of these intervals, ω = ωkl or ωkr must satisfy one and only one of the following five equations
Case II. If
a
(39)
−1 ≤ f1 (ω) ≤ 1;
then all Tωa = R2+ . P1 (jω) 6= 0, then Tωa consists of all (τ1 , τ2 ) that satisfy the equation θ + (2k + 1)π , k = 0, ±1, ±2, . . . (37) τ 1 = τ2 − ω where θ = ∠P2 (jω) − ∠P1 (jω). (38)
ω∈Ω∩Ωs
ω∈Ωa \Ωs
f1 (ω) = −1,
(44)
f1 (ω) = 1,
(45)
f2 (ω) = −1,
(46)
f2 (ω) = 1,
(47)
ω = 0.
(48)
Denote T k = T Ωk . Then, Tk consists of continuous curves parameterized in (19) and (20). In order to obtain geometric parameterization, we will give the following alternative parameterization related to the one given in Gu, Niculescu, & Chen (2005),
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
of parameterization. This proves instrumental in obtaining geometric characterization pursued in the next section. 6. GEOMETRIC CHARACTERIZATION ± For the sake of convenience, let Tω,u,v denote the singleton u± v± {τ1 (ω), τ2 (ω)}. We will classify the end points ω = ωkl or ωkr of the interval Ωk into five types according to which condition is satisfied in Proposition 14.
|P1 (jω)|
|P2 (jω)|
Type 1 The equation (44) is satisfied, in other words
θ1
θ2
|P1 (jω)| − |P2 (jω)| = −P0 (jω).
P0 (jω)
From Figures 1 and 2, it can be easily concluded that θ1 = π, θ2 = 0. Therefore, − + Tω,u,v = Tω,u+1,v .
Fig. 1. θ1 and θ2 are the interior angles when P0 (jω) > 0.
Type 2 The equation (45) is satisfied. We may show that − + . Tω,u,v = Tω,u,v−1
Type 3 The equation (46) is satisfied. This is possible only when P0 (jω) < 0 in view of Assumption 6. In this case, θ1 = θ2 = π, and − + Tω,u,v = Tω,u+1,v−1 .
Type 4 The equation (47) is satisfied. This is possible only when P0 (jω) > 0. In this case,
|P1 (jω)|
|P2 (jω)|
θ1
+ − Tω,u,v = Tω,u,v .
Type 0 The equation (48) is satisfied. In this case, as ω → 0, + − Tω,u,v → ∞, Tω,u,v → ∞.
θ2 P0 (jω)
Fig. 2. θ1 and θ2 are the exterior angles when P0 (jω) < 0. ∠P1 (jω) ± θ1 + (2u − 1)π , ω ∠P2 (jω) ∓ θ2 + (2v − 1)π τ2 = τ2v± (ω) = , ω u, v = 0, ±1, ±2, . . . τ1 = τ1u± (ω) =
(49) (50)
where 2
θ1 = cos
−1
θ2 = cos
−1
P0 (jω) + |p2 (jω)| − |p1 (jω)| 2 |P1 (jω)| 2
P0 (jω) + |p1 (jω)| − |p2 (jω)| 2 |P2 (jω)|
2
2
!
, (51)
!
. (52)
Proposition 15. The equations (49) and (50) continuously parameterize Tk for ω ∈ Ωk . It is interesting to observe that θ1 and θ2 are the interior angles of the triangle shown in Figure 1 when P0 (jω) > 0, and they are the exterior angles of the triangle shown in Figure 2. Although the parameterization given in (49) and (50) was motivated from the associate characteristic quasipolynomial, it automatically remove the spurious curves corresponding to the singular set Ωs and maintains continuity
An interval Ωk with type l left end and type r right end is said to be of type lr. Similar to Gu, Niculescu, & Chen (2005), we may conclude the following theorem from a careful analysis of the connection patterns of the curves [ + Tω,u,v . ω∈Ωk
. Theorem 16. The geometric characteristics of the stability crossing set Tk corresponding to the interval Ωk is determined by the type of the interval. In a coordinate system with horizontal τ1 -axis and vertical τ2 -axis, the specific geometry of Tk are summarized as follows:
i) If Ωk is of type lr with l = r, i.e., if it is of type 11, 22, 33, or 44, then Tk consists of closed curves; ii) If Ωk is of type lr with l 6= 0 and l 6= r, then Tk consists of spiral-like curves. The axis of the spiral-like curves are horizontal if Ωk is of type 14, 41, 23, or 32. The axis is vertical if the type is 13, 31, 24 or 42. The axis is 45◦ if the type is 12 or 21. The axis is −45◦ if the type is 34 or 43. iii) If Ωk is of type 0r for any r, then Tk consists of openended curves with both ends approaching ∞. Notice, a spiral-like curve with axis in −45◦ direction is not possible without the cross term. An example is presented below to illustrate such a curve. Example 17. Consider the following characteristic equation (1) with
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
f (ω)
8
1
f (ω) 2
6 4 2 0 −2 −4 −6 −8 0
0.5
1
1.5 ω
2
2.5
3
Fig. 3. f1 (ω) and f2 (ω) vs ω in Example 17 100 90 80 70
τ
2
60 50 40 30 20 10 0 0
10
20
30
40
50 τ
60
70
80
90
100
1
Fig. 4. A spiral-like curve with axis along −45◦ in Example 17 p0 (s) = s5 + s4 + 3s2 + s + 1, p1 (s) = s2 + 2s + 3, p2 (s) = s3 + s2 + 4s + 1, p12 (s) = s3 + 5s2 + 4.5. The crossing frequency set is identified from Figure 3 as Ω = Ω1 = [ω1 , ω2 ], where ω1 = 0.3023, ω2 = 1.6603. It can be seen from Figure 3 that Ω1 is of type 43, and the corresponding stability crossing set consists of spiral-like curves with axis in the −45◦ direction. One such curve is plotted in Figure 4. 7. CONCLUSIONS The stability crossing set of systems with two scalar delays is analyzed. The characteristic quasipolynomial of such a system generally contains a cross term containing a sum of two delays in the exponent. A new parameterization based on an associated characteristic quasipolynomial is used to obtain geometric characterization. REFERENCES Bellman, R. E., & Cooke, K. L. (1963). Differential Difference equations. Academic Press, New York.
Carvalho, L. A. V. (1996). On quadratic Liapunov functional for linear difference equations. Linear Algebra and its Applications, 240, 41–64. El’Sgol’ts, L. E., & Norkin, S. B. (1973). Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic Press, New York. Fazelinia, H., Sipahi, R., & Olgac, N. (2007). Stability Analysis of Multiple Time Delayed Systems Using ‘Building Block’ Concept. IEEE, T. on Automatic Control, 52(5), 799–810. Fridman, E. (2002). Stability of linear descriptor systems with delay: a Lyapunov-based approach. Journal of Mathematical Analysis and Applications, 273(1), 24–44. Gu, K., Niculescu, S.-I., & Chen, J. (2005). On stability of crossing curves for general systems with two delays. J. Math. Anal. Appl., 311(1), 231-253. Gu, K. (2010). Stability Problem of Systems with Multiple Delay Channels. Automatica, 46(4), 743–51. Guckenheimer, J. & Holmes, P. (1983). Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag, New York. Hale, J., & Huang, W. (1993) Global geometry of the stable regions for two delay differential equations. J. Math. Anal. Appl., 178(2), 344–362. Jarlebring, E. (2009) Critical delays and polynomial eigenvalue problems. Journal of Computational and Applied Mathematics, 224(1), 296-306. Kolmanovskii, V. B., & Nosov, V. R. (1986). Stability of functional differential equations. Academic Press, New York. Li, H., & Gu, K. (2010) Discretized Lyapunov-Krasovskii functional for coupled differential-difference equations with multiple delay channels. Automatica, 46(4), 902– 909. Neimark, Ju. I. (1949) D-subdivision and spaces of quasipolynomials. Prikl. Mat. Mech. 13, 349-380 (In Russian.). Naghnaeian, M., & Gu, K. (2011) Full version of this paper. Olgac, N., Vyhl’idal, T., & Sipahi, R. (2008). A new perspective in the stability assessment of neutral systems with multiple and cross-talking delays. SIAM Journal on Control and Optimization, 47(1), 327–344. Ruan, S., & Wei, J. (2003). On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamics of Continuous, Discrete and Impulsive Systems, 10(6), 863–874. St´ep´an, G. (1989). Retarded dynamical systems: stability and characteristic function. Wiley, New York. Sipahi, R., & Olgac, N., (2003). Stability analysis of multiple time delay systems using the direct method. Paper # 41495, ASME IMECE, Washington, D.C., Nov. 2003. Sipahi, R., & Olgac, N., (2006a). A unique methodology for the stability robustness of multiple time delay systems. Systems and Control Letters, 55(10), 819–825. Sipahi, R., & Olgac, N. (2006b). Complete stability analysis of neutral type first order — two time delay systems with cross-talking delays. SIAM J. of Control and Optimization, 45(3), 957-971. Walton, K., & Marshall, J. E., (1987). Direct method for TDS stability analysis. IEE Proceedings D.: Control Theory and Applications, 134(2), 101–107.
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Stability crossing set for systems with two scalar-delay channels Mohammad Naghnaeian ∗ Keqin Gu ∗∗ ∗
Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801-2906, USA (e-mail: [email protected]). ∗∗ Department of Mechanical and Industrial Engineering Southern Iliinois Univeristy Edwardsville Edwardsville, IL 62026-1805, USA (e-mail: [email protected]). Abstract: This article studies the stability crossing set of linear time-invariant systems with two scalar-delay channels. This study is crucial to the complete stability analysis along the idea of D-subdivision. The characteristic quasipolynomial of such systems contains an exponential term with the sum of two delays (cross term) in its exponent. A complete parameterization and geometric characterization of the stability crossing set is conducted. It was found instrumental to relate it to an associated quasipolynomial without such a cross term. Keywords: Delay, stability 1. INTRODUCTION
very general class of systems was given in Jarlebring (2009).
This article studies the stability crossing set of systems with characteristic quasipolynomial of the form
As argued in Gu (2010), systems with two scalar-delay elements may be modeled as one with two scalar-delay channels. In this model, a system consists of a forward system without delay
∆(s) = p0 (s) + p1 (s)e−τ1 s +p2 (s)e−τ2 s + p12 (s)e−(τ1 +τ2 )s ,
(1)
which is defined as the set of all (τ1 , τ2 ) such that the above quasipolynomial has at least one root on the imaginary axis. In the above expression, p0 (s), p1 (s), p2 (s) and p12 (s) are polynomials with real coefficients. The stability crossing set has been recognized as critical in one school of stability analysis since Neimark proposed the D-subdivision method (Neimark (1949), El’Sgol’ts & Norkin (1973), Kolmanovskii & Nosov (1986)). Examples of stability analysis of various such systems abound in the literature, see, for example, St´ep´an (1989), Hale & Huang (1993), Ruan & Wei (2003) and the references therein. In recent years, the study of stability crossing set of a class of systems has received substantial attention. This is in contrast to earlier works that typically study one specific system with variable parameters mentioned in the previous paragraph. For system (1) without the last term, Sipahi & Olgac (2003) seems to be the first such study, and a more complete parameterization and geometric characterization was given in Gu, Niculescu, & Chen (2005). Studies on more general systems, including the system (1), may be found in Fazelinia, Sipahi, & Olgac (2007), Olgac, Vyhl’idal, & Sipahi (2008), Sipahi & Olgac (2006a), and Sipahi & Olgac (2006b). However, none of these works include a complete parameterization and geometric characterization parallel to Gu, Niculescu, & Chen (2005). It is worth mentioning that an interesting complete parameterization of stability crossing set of a 978-3-902661-93-7/11/$20.00 © 2011 IFAC
x(t) ˙ = Ax(t) + Bu(t),
(2)
y(t) = Cx(t) + Du(t), (3) where , x(t) ∈ R , u1 (t) u(t) = ∈ R2 , u2 (t) y1 (t) y (t) = ∈ R2 , y2 (t) and a feedback consisting of two scalar-delay channels u1 (t) y1 (t − τ1 ) = . (4) u2 (t) y2 (t − τ2 ) The characteristic quasipolynomial of this system is −τ s e 1 0 sI − A −B −τ2 s 0−τ se . ∆(s) = det 1 e 0 −C I − D −τ2 s 0 e An application of the Schur determinant complement yields −τ s e 1 0 sI − A −B ∆(s) = det 0 e−τ2 s 0 M (s) n
= det (sI − A) · det [M (s)] ,
(5)
where
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M (s) = I − D + C (sI − A)
−1
B
e−τ1 s 0 0 e−τ2 s
. (6)
10.3182/20110828-6-IT-1002.01733
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
The characteristic function given in (5) is a quasipolynomial of the form given in (1). Therefore, even though (1) is less general than the system discussed in some recent literature cited above, its thorough analysis is of great practical importance. The purpose of this article is to extend the results of Gu, Niculescu, & Chen (2005) to the more general system described by (1). Due to space limitation, all proofs are omitted. These proofs are contained in Naghnaeian & Gu (2011). 2. PRELIMINARIES The ultimate purpose of such an analysis is to evaluate stability. In this article, by stability, we mean asymptotic stability. First the following non-triviality assumptions are made, Assumption 1. Existence of principal term: ord [p0 (s)] ≥ max{ord [p1 (s)], ord [p2 (s)], ord [p12 (s)]}, where ord [·] is the order of the polynomial concerned. Assumption 2. Zero frequency restriction: p0 (0) + p1 (0) + p2 (0) + p12 (0) 6= 0. Assumption 3. Restriction on difference operator: |p1 (s)| + |p2 (s)| + |p12 (s)| < 1. lim s→∞ |p0 (s)| These assumptions are made to exclude some obviously trivial cases. If Assumption 1 is violated, then the quasipolynomial does not have a principal term, and the system is unstable for any τ1 ≥ 0, τ2 ≥ 0 (Bellman & Cooke (1963)). If Assumption 2 is violated, then 0 is always a root of ∆(s) for any τ1 ≥ 0, τ2 ≥ 0, and therefore, it is impossible for the system to be stable. Assumption 3 ensures the asymptotic stability for arbitrary (τ1 , τ2 ) of the difference equation y1 (t) y1 (t − τ1 ) =D , (7) y2 (t) y2 (t − τ2 )
which is obtained by a substitution of (3) by (4). This also guarantees the existence of a vertical line on the left of the imaginary axis such that all the roots to the right of this vertical line are continuous with respect to the delay parameters (τ1 , τ2 ) ∈ R2+ . Assumption 3 may be replaced by the less restrictive stability conditions for the difference equation given in Carvalho (1996) or Fridman (2002) (which are equivalent as shown in Li & Gu (2010)). A naive approach is to use a method similar to the one given in Walton & Marshall (1987) to eliminate the last term of (1) and reduce it to the case discussed in Gu, Niculescu, & Chen (2005). This may proceed as follows: As all the coeffificnets of ∆ are real, if ∆(jω) = 0, (8) then ∆(−jω) = 0. (9) Multiply the equation (8) by p0 (−jω) and the equation (9) by e−jω(τ1 +τ2 ) p12 (jω), then subtract the resulting equations, the following equation is obtained ∆a (jω) = 0, (10) where ∆a (s) = P0 (s) + P1 (s)e−τ1 s + P2 (s)e−τ2 s , (11) and the coefficient polynomials of (11) are
P0 (s) = p12 (s)p12 (−s) − p0 (s)p0 (−s),
(12)
P1 (s) = p12 (s)p2 (−s) − p1 (s)p0 (−s),
(13)
P2 (s) = p12 (s)p1 (−s) − p2 (s)p0 (−s).
(14)
The equation (11) may be analyzed using the method discussed in Gu, Niculescu, & Chen (2005). However, as will be shown later, there are two crucial issues that require special attention. First, there are some differences between the stability crossing set of (11) and that of (1). Second, the asssumption in Gu, Niculescu, & Chen (2005) that no imaginary zeros in the crossing frequency set for the coefficient polynomial P0 (s) is no longer appropriate due to its special structure. It turns out that the key to both issues lies in a singular set. We will refer to ∆ expressed in (1) as the original (characteristic) quasipolynomial, and refer to ∆a (s) expressed in (11) as the associated (characteristic) quasipolynomial. Similarly, ∆(s) = 0 and ∆a (s) = 0 are referred as the original (characteristic) equation and the associated (characteristic) equation, respectively. Sometimes, we write ∆(s) and ∆a (s) as ∆(s, τ1 , τ2 ) and ∆a (s, τ1 , τ2 ), respectively, in order to emphasize their dependence on the delays. Definition 4. A pair (τ1 , τ2 ) ∈ R2+ is said to be a crossing point if ∆(s, τ1 , τ2 ) = 0 has at least one solution for s on the imaginary axis. The set of all crossing points is known as the stability crossing set, and is denoted as T . An ω > 0 is known as a crossing frequency if there exists at least one pair (τ1 , τ2 ) such that ∆(jω, τ1 , τ2 ) = 0. The set of all crossing frequencies is known as the crossing frequency set Ω. Parallel concepts defined on ∆a (s, τ1 , τ2 ) are denoted as the T a and Ωa , and known as the associated stability crossing set and the associated crossing frequency set. Notice, it is not necessary to consider ω < 0 as all the roots of ∆(s) appear in complex conjugate pairs. Also, Assumption 2 implies that ω = 0 cannot be a crossing frequency, and therefore can be excluded from consideration. In Gu, Niculescu, & Chen (2005), stability crossing set is known as stability crossing curves. The change of terminology is to make it consistent with the three-delay case where the stability crossing set consists of surfaces. For a given ω, let Tω = {(τ1 , τ2 ) | ∆(jω, τ1 , τ2 ) = 0} . Of course, Tω is empty if ω ∈ / Ω. Then [ Tω . T = ω∈Ω
For any set Φ, it is also convenient to define [ TΦ = Tω . ω∈Φ
For a real scalar α, we also denote \ T¯[α−1/n,α)∪(α,α+1/n] Tˆα = n=1,2,...
where (·) represents the closure of the set. The typical context of using Tˆα is when the stability crossing set
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
is parameterized as continuous curves (τ1l,m (ω), τ2l,m (ω)), l, m = 0, ±1, ±2, . . ., in which case, ! [ l,m l,m lim τ (ω), ω→α lim τ2 (ω) . Tˆα = ω→α 1 l,m
ω∈Ω
ω∈Ω
Similarly, for a given ω, we define Tωa = {(τ1 , τ2 ) | ∆a (jω, τ1 , τ2 ) = 0} . Then [ Tωa , T a=
and
In this section, the crossing frequency set Ω and stability crossing set T of the original quasipolynomial ∆(s) will be studied without referring to the associated quasipolynomial ∆a . Proposition 5. The crossing frequency set Ω consists of all ω > 0 that satisfy 2 |P1 (jω)| 2 2 ≥ P0 (jω) − |p1 (jω)| − |p2 (jω)| ,
(15)
2 |P (jω)| 2 2 2 ≥ P0 (jω) + |p1 (jω)| − |p2 (jω)| ,
(16)
P1 (jω) 6= 0,
(17)
or equivalently,
P2 (jω) 6= 0, (18) all the corresponding crossing points (τ1 , τ2 ) ∈ Tω have the expression (19)
k1 ,k2 τ2 = τ2± (ω) 1 = [∠(p2 (jω) + p12 (jω)e−jωτ1 ) ω −∠(p0 (jω) + p1 (jω)e−jωτ1 )
(20)
k2 = 0, ±1, ±2, . . . , where α = ∠P1 (jω),
f (ω) = 2 |P1 (jω)| 2 2 2 − P0 (jω) − |p1 (jω)| − |p2 (jω)| , (25)
f (ω) = 0 f ′ (ω) = 0. Assumption 8. No singular crossing. No ω satisfies P0 (jω) = 0 and 2 |P (jω)| 1 2 2 = P0 (jω) − |p1 (jω)| − |p2 (jω)| ,
(26)
(27)
Proposition 5 completely parameterizes the stability crossing set under the non-degeneracy Assumptions 6 to ??. The cases where any of these three non-degeneracy assumption is violated are very rare. Indeed, in the terminology of dynamical systems theory, the systems where the three non-degeneracy assumptions are satisfied are generic (Guckenheimer & Holmes (1983)). Also, the cases when one of these three assumptions is violated can be easily handled individually. In the remaining part of the article, the three nondegeneracy assumptions and the three non-triviality assumptions will be assumed to be satisfied unless explicitly stated otherwise. The above development leads to the following conclusion. Proposition 9. The crosssing fequency set Ω consists of a finite number of intervals of finite nonzero lengths. At any interior point of such an interval, the inequalities (15) and (16) are satisfied strictly. At any nonzero end point, (15) and (16) becomes equalities, and P0 (jω) 6= 0. (28) For any ω ∈ Ω, Tω = Tˆω . (29) 4. ASSOCIATED CROSSING SET
(21) 2
(24)
simultaneously.
where P0 , P1 , and P2 are given in (12) to (14). For a given ω ∈ Ω that satisfies
+(2k2 + 1)π],
P2 (jω) 6= 0
f (0) 6= 0. Furthermore, no ω > 0 simultaneously satisfies
3. DIRECT PARAMETERIZATION
α ± β + 2k1 π , ω k1 = 0, ±1, ±2, . . . ,
(23)
Assumption 7. Transversality. Let
then
and Tˆαa are defined analogously.
k1 ,k2 τ1 = τ1± (ω) =
P1 (jω) 6= 0
2
ω∈Ωa
TΦa
Assumption 6. No imaginary zeros. For any ω ∈ Ω,
2
|p1 (jω)| − |p2 (jω)| − P0 (jω) . (22) 2 |P1 (jω)| In calculating ∠(·) in the above expressions, we will maintain the continuity as ω varies, which means that the values may not be restricted within any given 2π range. β = cos−1
To simplify presentation, the following non-degeneracy assumptions are made.
While it is not difficult to completely parameterize T directly from the procedure given in the last setion, it is not easy to obtain geometric characterization. On the other hand, as the associated characteristic quasipolynomial is in a form that has already been completely characterized in Gu, Niculescu, & Chen (2005), significant insight can be gained from studying the relationship between T and T a . The following theorem proves important for this purpose.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Theorem 10. If a parameter combination (ω, τ1 , τ2 ) ∈ R×R2+ satisfies ∆(jω, τ1 , τ2 ) = 0, (30) then it also satisfies ∆a (jω, τ1 , τ2 ) = 0. (31) The reverse is also true provided that P0 (jω) 6= 0. (32) In other words, if (31) and (32) are satisfied, then (30) is also satisfied. It should be cautioned that the equivalence stated in the above theorem is valid only for stability crossing sets. No claim is made regarding the equivalence between the stability of the two quasipolynomials. It is important to study those ω that satisfy the equation P0 (jω) = p12 (jω)p12 (−jω) − p0 (jω)p0 (−jω) = 0. (33) Such an ω is known as a singular frequency. The set of singular frequencies is known as the singular frequency set, and is denoted as Ωs . Due to the structure of P0 (s), it can be easily shown that P0 (jω) is a polynomial of ω 2 with real coefficients. Therefore, the existence of imaginary roots of P0 (s) (corresponding to a real negative ω 2 ) is a rather common phenomenum, and a careful treatment of this case is essential for applications, in contrast to the treatment in Gu, Niculescu, & Chen (2005) where such cases are excluded. It will be shown later on in Proposition 11 that Ωs ⊂ Ωa . This, together with Theorem 10, may be succinctly expressed as Ωa = Ω ∪ Ωs ,
(34)
Tω = Tωa for all ω ∈ Ωa \Ωs ,
(35)
Tω ⊂ Tωa for all ω ∈ Ωs .
(36)
and
The following proposition describes Tωa for ω ∈ Ωs . Proposition 11. The singular frequency set Ωs satisfies Ωs ⊂ Ωa . Furthermore, given an ω ∈ Ωs , Tωa can be expressed as follows: Case I. If P1 (jω) = 0,
Theorem 12. The stability crossing set T can be expressed as ! [ [ [ a a T = Tˆω . Tω 5. ALTERNATIVE PARAMETERIZATION
Taking advantage of the relationship between the stability crossing set T and the associated stability crossing set T a , we will provide an alternative parameterization of T in this section. This parameterization is based on T a , and is modified from those given in Gu, Niculescu, & Chen (2005). Based on this parameterization, it is rather easy to obtain geometric insight of the stability crossing set. We start with characterizing the crossing frequency set. Proposition 13. The set Ω\Ωs = Ωa \Ωs consists of all ω that satisfy the following three conditions P0 (jω) 6= 0; f2 (ω) ≥ 1 or f2 (ω) ≤ −1;
s
In order to obtain T from T , we observe that Ω consists of a finite number of points. Apply Proposition 9 and the relation (35), we conclude that for any ω ∈ Ω ∩ Ωs , Tω = Tˆωa , from which we arrive at the following theorem.
(40) (41)
where |P1 (jω)| |P2 (jω)| − , P0 (jω) P0 (jω) |P1 (jω)| |P2 (jω)| f2 (ω) = + . P0 (jω) P0 (jω)
f1 (ω) =
(42) (43)
The above proposition means that we may identify Ω by plotting f1 (ω) and f2 (ω) against ω. As indicated by Proposition 9, Ω consists of intervals Ωk , k = 1, 2, . . . , N , N [ Ω= Ωk . k=1
Without loss of generality, let these intervals be ordered from left to right. We may write Ωk = [ωkl , ωkr ] l if ωk 6= 0, and Ωk = (ωkl , ωkr ] otherwise. Furthermore, we have the following conclusion. Proposition 14. An endpoint of these intervals, ω = ωkl or ωkr must satisfy one and only one of the following five equations
Case II. If
a
(39)
−1 ≤ f1 (ω) ≤ 1;
then all Tωa = R2+ . P1 (jω) 6= 0, then Tωa consists of all (τ1 , τ2 ) that satisfy the equation θ + (2k + 1)π , k = 0, ±1, ±2, . . . (37) τ 1 = τ2 − ω where θ = ∠P2 (jω) − ∠P1 (jω). (38)
ω∈Ω∩Ωs
ω∈Ωa \Ωs
f1 (ω) = −1,
(44)
f1 (ω) = 1,
(45)
f2 (ω) = −1,
(46)
f2 (ω) = 1,
(47)
ω = 0.
(48)
Denote T k = T Ωk . Then, Tk consists of continuous curves parameterized in (19) and (20). In order to obtain geometric parameterization, we will give the following alternative parameterization related to the one given in Gu, Niculescu, & Chen (2005),
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
of parameterization. This proves instrumental in obtaining geometric characterization pursued in the next section. 6. GEOMETRIC CHARACTERIZATION ± For the sake of convenience, let Tω,u,v denote the singleton u± v± {τ1 (ω), τ2 (ω)}. We will classify the end points ω = ωkl or ωkr of the interval Ωk into five types according to which condition is satisfied in Proposition 14.
|P1 (jω)|
|P2 (jω)|
Type 1 The equation (44) is satisfied, in other words
θ1
θ2
|P1 (jω)| − |P2 (jω)| = −P0 (jω).
P0 (jω)
From Figures 1 and 2, it can be easily concluded that θ1 = π, θ2 = 0. Therefore, − + Tω,u,v = Tω,u+1,v .
Fig. 1. θ1 and θ2 are the interior angles when P0 (jω) > 0.
Type 2 The equation (45) is satisfied. We may show that − + . Tω,u,v = Tω,u,v−1
Type 3 The equation (46) is satisfied. This is possible only when P0 (jω) < 0 in view of Assumption 6. In this case, θ1 = θ2 = π, and − + Tω,u,v = Tω,u+1,v−1 .
Type 4 The equation (47) is satisfied. This is possible only when P0 (jω) > 0. In this case,
|P1 (jω)|
|P2 (jω)|
θ1
+ − Tω,u,v = Tω,u,v .
Type 0 The equation (48) is satisfied. In this case, as ω → 0, + − Tω,u,v → ∞, Tω,u,v → ∞.
θ2 P0 (jω)
Fig. 2. θ1 and θ2 are the exterior angles when P0 (jω) < 0. ∠P1 (jω) ± θ1 + (2u − 1)π , ω ∠P2 (jω) ∓ θ2 + (2v − 1)π τ2 = τ2v± (ω) = , ω u, v = 0, ±1, ±2, . . . τ1 = τ1u± (ω) =
(49) (50)
where 2
θ1 = cos
−1
θ2 = cos
−1
P0 (jω) + |p2 (jω)| − |p1 (jω)| 2 |P1 (jω)| 2
P0 (jω) + |p1 (jω)| − |p2 (jω)| 2 |P2 (jω)|
2
2
!
, (51)
!
. (52)
Proposition 15. The equations (49) and (50) continuously parameterize Tk for ω ∈ Ωk . It is interesting to observe that θ1 and θ2 are the interior angles of the triangle shown in Figure 1 when P0 (jω) > 0, and they are the exterior angles of the triangle shown in Figure 2. Although the parameterization given in (49) and (50) was motivated from the associate characteristic quasipolynomial, it automatically remove the spurious curves corresponding to the singular set Ωs and maintains continuity
An interval Ωk with type l left end and type r right end is said to be of type lr. Similar to Gu, Niculescu, & Chen (2005), we may conclude the following theorem from a careful analysis of the connection patterns of the curves [ + Tω,u,v . ω∈Ωk
. Theorem 16. The geometric characteristics of the stability crossing set Tk corresponding to the interval Ωk is determined by the type of the interval. In a coordinate system with horizontal τ1 -axis and vertical τ2 -axis, the specific geometry of Tk are summarized as follows:
i) If Ωk is of type lr with l = r, i.e., if it is of type 11, 22, 33, or 44, then Tk consists of closed curves; ii) If Ωk is of type lr with l 6= 0 and l 6= r, then Tk consists of spiral-like curves. The axis of the spiral-like curves are horizontal if Ωk is of type 14, 41, 23, or 32. The axis is vertical if the type is 13, 31, 24 or 42. The axis is 45◦ if the type is 12 or 21. The axis is −45◦ if the type is 34 or 43. iii) If Ωk is of type 0r for any r, then Tk consists of openended curves with both ends approaching ∞. Notice, a spiral-like curve with axis in −45◦ direction is not possible without the cross term. An example is presented below to illustrate such a curve. Example 17. Consider the following characteristic equation (1) with
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
f (ω)
8
1
f (ω) 2
6 4 2 0 −2 −4 −6 −8 0
0.5
1
1.5 ω
2
2.5
3
Fig. 3. f1 (ω) and f2 (ω) vs ω in Example 17 100 90 80 70
τ
2
60 50 40 30 20 10 0 0
10
20
30
40
50 τ
60
70
80
90
100
1
Fig. 4. A spiral-like curve with axis along −45◦ in Example 17 p0 (s) = s5 + s4 + 3s2 + s + 1, p1 (s) = s2 + 2s + 3, p2 (s) = s3 + s2 + 4s + 1, p12 (s) = s3 + 5s2 + 4.5. The crossing frequency set is identified from Figure 3 as Ω = Ω1 = [ω1 , ω2 ], where ω1 = 0.3023, ω2 = 1.6603. It can be seen from Figure 3 that Ω1 is of type 43, and the corresponding stability crossing set consists of spiral-like curves with axis in the −45◦ direction. One such curve is plotted in Figure 4. 7. CONCLUSIONS The stability crossing set of systems with two scalar delays is analyzed. The characteristic quasipolynomial of such a system generally contains a cross term containing a sum of two delays in the exponent. A new parameterization based on an associated characteristic quasipolynomial is used to obtain geometric characterization. REFERENCES Bellman, R. E., & Cooke, K. L. (1963). Differential Difference equations. Academic Press, New York.
Carvalho, L. A. V. (1996). On quadratic Liapunov functional for linear difference equations. Linear Algebra and its Applications, 240, 41–64. El’Sgol’ts, L. E., & Norkin, S. B. (1973). Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic Press, New York. Fazelinia, H., Sipahi, R., & Olgac, N. (2007). Stability Analysis of Multiple Time Delayed Systems Using ‘Building Block’ Concept. IEEE, T. on Automatic Control, 52(5), 799–810. Fridman, E. (2002). Stability of linear descriptor systems with delay: a Lyapunov-based approach. Journal of Mathematical Analysis and Applications, 273(1), 24–44. Gu, K., Niculescu, S.-I., & Chen, J. (2005). On stability of crossing curves for general systems with two delays. J. Math. Anal. Appl., 311(1), 231-253. Gu, K. (2010). Stability Problem of Systems with Multiple Delay Channels. Automatica, 46(4), 743–51. Guckenheimer, J. & Holmes, P. (1983). Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag, New York. Hale, J., & Huang, W. (1993) Global geometry of the stable regions for two delay differential equations. J. Math. Anal. Appl., 178(2), 344–362. Jarlebring, E. (2009) Critical delays and polynomial eigenvalue problems. Journal of Computational and Applied Mathematics, 224(1), 296-306. Kolmanovskii, V. B., & Nosov, V. R. (1986). Stability of functional differential equations. Academic Press, New York. Li, H., & Gu, K. (2010) Discretized Lyapunov-Krasovskii functional for coupled differential-difference equations with multiple delay channels. Automatica, 46(4), 902– 909. Neimark, Ju. I. (1949) D-subdivision and spaces of quasipolynomials. Prikl. Mat. Mech. 13, 349-380 (In Russian.). Naghnaeian, M., & Gu, K. (2011) Full version of this paper. Olgac, N., Vyhl’idal, T., & Sipahi, R. (2008). A new perspective in the stability assessment of neutral systems with multiple and cross-talking delays. SIAM Journal on Control and Optimization, 47(1), 327–344. Ruan, S., & Wei, J. (2003). On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamics of Continuous, Discrete and Impulsive Systems, 10(6), 863–874. St´ep´an, G. (1989). Retarded dynamical systems: stability and characteristic function. Wiley, New York. Sipahi, R., & Olgac, N., (2003). Stability analysis of multiple time delay systems using the direct method. Paper # 41495, ASME IMECE, Washington, D.C., Nov. 2003. Sipahi, R., & Olgac, N., (2006a). A unique methodology for the stability robustness of multiple time delay systems. Systems and Control Letters, 55(10), 819–825. Sipahi, R., & Olgac, N. (2006b). Complete stability analysis of neutral type first order — two time delay systems with cross-talking delays. SIAM J. of Control and Optimization, 45(3), 957-971. Walton, K., & Marshall, J. E., (1987). Direct method for TDS stability analysis. IEE Proceedings D.: Control Theory and Applications, 134(2), 101–107.
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