Stability of linear systems with general sawtooth delay
IMA Journal of Mathematical Control and Information Page 1 of 18 doi:10.1093/imamci/dnq023
Stability of linear systems with general sawtooth delay
[Received on 4 February 2010; revised on 2 June 2010; accepted 31 August 2010] It is well known that in many particular systems, the upper bound on a certain time-varying delay that preserves the stability may be higher than the corresponding bound for the constant delay. Moreover, sometimes oscillating delays improve the performance (Michiels, W., Van Assche, V. & Niculescu, S. (2005) Stabilization of time-delay systems with a controlled time-varying delays and applications. IEEE Trans. Automat. Control, 50, 493–504). Sawtooth delays τ with τ˙ = 1 (almost everywhere) can posses this property (Louisell, J. (1999) New examples of quenching in delay differential equations having time-varying delay. Proceedigns of the 5th ECC, Karlsruhe, Germany). In this paper, we show that general sawtooth delay, where τ˙ 6= 0 is constant (almost everywhere), also can posses this property. By the existing Lyapunov-based methods, the stability analysis of such systems can be performed in the framework of systems with bounded fast-varying delays. Our objective is to develop ‘qualitatively new methods’ that can guarantee the stability for sawtooth delay which may be not less than the analytical upper bound on the constant delay that preserves the stability. We suggest two methods. One method develops a novel input–output approach via a Wirtinger-type inequality. By this method, we recover the result by Mirkin (2007, Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Trans. Automat. Control, 52, 1109–1112) for τ˙ = 1 and we show that for any integer τ˙ , the same maximum bound that preserves the stability is achieved. Another method extends piecewise continuous (in time) Lyapunov functionals that have been recently suggested for the case of τ˙ = 1 in Fridman (2010, A refined input delay approach to sampled-data control. Automatica, 46, 421–427) to the general sawtooth delay. The time-dependent terms of the functionals improve the results for all values of τ˙ , though the most essential improvement corresponds to τ˙ = 1. Keywords: time-varying delay; Lyapunov-based methods; LMI.
1. Introduction Over the past decades, much effort has been invested in the analysis and design of uncertain systems with time-varying delays (see, e.g. Kolmanovskii & Myshkis, 1999; Niculescu, 2001; Kharitonov & Niculescu, 2002; Fridman & Shaked, 2003; Richard, 2003; Gu et al., 2003; He et al., 2007; Park & Ko, 2007). The delay under consideration has been either differentiable with a known upper bound 0 6 τ˙ 6 d < 1 or piecewise continuous without any constraints on the delay derivative (fast-varying delay) (Fridman & Shaked, 2003). In the existing Lyapunov-based methods, the maximum delay bound that preserves the stability corresponds to d = 0 and this bound is usually a decreasing function of d. However, it is well known (see examples in Louisell, 1999 and discussions on ‘quenching’ in Papachristodoulou et al., 2007 as well as Example 1 below) that in many particular systems, the upper bound on a certain time-varying delay that preserves the stability may be higher than the corresponding c The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
K UN L IU , V LADIMIR S UPLIN AND E MILIA F RIDMAN∗ Department of Electrical Engineering-Systems, School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel [email protected] [email protected] ∗ Corresponding author: [email protected]
2 of 18
K. LIU ET AL.
kφkWn [a,b) = max |φ(θ)| + θ ∈[a,b]
Z
a
b
˙ |φ(s)| ds 2
12
.
We also denote W = Wn [−h, 0) and xt (θ ) = x(t + θ )(θ ∈ [−h, 0]). 2. Problem formulation Consider the system (
x(t) ˙ = Ax(t) + A1 x(t − τ (t)),
x(t) = φ(t),
t ∈ [−h, 0],
(2.1)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
bound for the constant delay. Moreover, sometimes oscillating delays improve the performance (Michiels et al., 2005). Recently, a discontinuous Lyapunov function method was introduced to sampled-data control systems (corresponds to the sawtooth delay with τ˙ = 1 (almost everywhere)) in Naghshtabrizi et al. (2008). This method improved the existing Lyapunov-based results and it inspired a piecewise continuous (in time) Lyapunov–Krasovskii functional (LKF) approach to sampled-data systems (Fridman, 2010). The LKFs in the latter paper are time dependent and they do not grow after the sampling times. The introduced novel discontinuous terms of Lyapunov functionals in Fridman (2010) lead to qualitatively new results, allowing a superior performance under the sampling, than the one under the constant delay. The input delay approach to sampled-data control was also recently revised by using the scaled small gain theorem and a tighter upper bound on the L 2 -induced norm of the uncertain term (Mirkin, 2007). In the present paper, stability of systems with bounded sawtooth delay τ is analysed, where τ˙ 6= 0 is piecewise constant. By the existing Lyapunov-based methods, the stability analysis of such systems can be performed in the framework of systems with bounded fast-varying delays. Our objective is to develop ‘qualitatively new methods’ that can guarantee the stability for sawtooth delay which may be not less than the analytical upper bound on the constant delay that preserves the stability. We suggest two methods. One method develops a novel input–output (I–O) approach via a Wirtinger-type inequality. By this method, we recover the result by Mirkin (2007) for τ˙ = 1 and we show that for any integer τ˙ , the same maximum bound that preserves the stability is achieved. Another method extends direct Lyapunov approach to systems with a general form of sawtooth delay. By constructing appropriate discontinuous LKFs, we obtain sufficient delay-dependent conditions that guarantee the exponential stability of systems in terms of linear matrix inequalities (LMIs). The time-dependent terms of LKFs improve the results for all values of τ˙ , though the most essential improvement corresponds to τ˙ = 1. A conference version of discontinuous LKF approach was presented in Liu & Fridman (2009). Notations. Throughout the paper, the superscript ‘>’ stands for matrix transposition, Rn denotes the n-dimensional Euclidean space with vector norm k ∙ k, Rn×m is the set of all n × m real matrices and the notation P > 0, for P ∈ Rn×n means that P is symmetric and positive definite. The symmetric elements of the symmetric matrix will be denoted by ∗. L 2 is the space of square integrable functions R∞ v: [0, ∞) → Rn with the norm kvk L 2 = [ 0 kv(t)k2 dt]1/2 . The space of functions φ: [a, b] → Rn , which are absolutely continuous on [a, b), have a finite limθ →b− φ(θ ) and have square integrable firstorder derivatives is denoted by Wn [a, b) with the norm
3 of 18
GENERAL SAWTOOTH DELAY
FIG. 1. Case A1.
where x(t) ∈ Rn is the state vector, A and A1 denote the constant matrices, φ(t) is the initial function, τ (t) ∈ [0, h] is the time-varying delay. It is assumed that the delay function has the form of sawtooth (see Figs 1 and 2), satisfying either A1 or A2 below: A1: τ (t) = d(t − tk ),
A2: τ (t) = d(tk+1 − t),
where d > 0 and tk =
t ∈ [tk , tk+1 ), k = 0, 1, 2 . . . ,
t ∈ [tk , tk+1 ), k = 0, 1, 2 . . . ,
(2.2) (2.3)
kh d .
It is clear that under A1, we have τ˙ = d > 0 and under A2, we have τ˙ = −d < 0. Both cases can be analysed by using time-independent Lyapunov functionals corresponding to systems with fast-varying delays. Our objective is to derive delay-dependent stability criteria for system (2.1) that improve the recent results for fast-varying delays (see, e.g. Park & Ko, 2007).
3. I–O approach via Wirtinger-type inequality We recall the following Wirtinger-type inequality (Hardy et al., 1934): let z ∈ W1 [a, b) be a scalar function with z(a) = 0. Then, Z
a
b
z 2 (ξ )dξ 6
4(b − a)2 π2
Z
b
a
z˙ 2 (ξ )dξ.
(3.1)
This inequality is trivially extended to the vector case. L EMMA 3.1 Let z ∈ Wn [a, b). Assume that z(a) = 0. Then, for any n × n-matrix R > 0, the following inequality holds: Z
a
b
4(b − a)2 z (ξ )Rz(ξ )dξ 6 π2 >
Z
a
b
z˙ > (ξ )R z˙ (ξ )dξ.
(3.2)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
FIG. 2. Case A2.
4 of 18
K. LIU ET AL.
Proof. We have Z
a
b
4(b − a)2 z (ξ )Rz(ξ )dξ 6 π2
Z
4(b − a)2 π2
Z
>
=
b
a
b
a
System (2.1) can be rewritten as follows: x(t) ˙ = ( A + A1 )x(t) − A1
Z
t t−τ (t)
x(s)ds. ˙
(3.4)
We present the latter as the following forward system: (
x˙ = ( A + A1 )x(t) + A1 u(t) ˙ y(t) = x(t),
(3.5)
with the feedback u(t) = −
Z
t
y(s)ds.
(3.6)
t−τ (t)
Assume that A + A1 is Hurwitz and y(t) = 0 for t 6 0.
L EMMA 3.2 Assume that the time delay is given by (2.2), where d ∈ N . Then the following holds: kuk L 2 6
2h kyk L 2 . π
(3.7)
Proof. Defining u s (t) = −
Z
t
y(s)ds, tk
u st (t) = −
Z
tk
y(s)ds, t−τ
t ∈ [tk , tk+1 ),
we note that u(t) = u s (t) + u st (t). We will prove next the following bounds: 2h kyk L 2 , πd 2h 6 (d − 1) kyk L 2 , πd
ku s k L 2 6
(3.8)
ku st k L 2
(3.9)
which imply (3.7) since kuk L 2 6 ku s k L 2 + ku st k L 2 6
2h kyk L 2 . π
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
2 dp > z (ξ )Rz(ξ ) dξ dξ !2 Z 4(b − a)2 b > z˙ > (ξ )Rz(ξ ) p dξ 6 z˙ (ξ )R z˙ (ξ )dξ. (3.3) π2 a z > (ξ )Rz(ξ )
5 of 18
GENERAL SAWTOOTH DELAY
By using (3.3), we obtain ku s k2L 2 =
∞ Z X
tk+1
k=0 tk
− x(t))>
(x(tk ) − x(t))> (x(tk ) − x(t))dt 6
Z ∞ X 4(tk+1 − tk )2 π2
k=0
tk+1 tk
d (x(tk ) dt
∞ Z d 4h 2 X tk+1 T 4h 2 (x(tk ) − x(t))dt 6 2 2 ˙ 2L 2 . x˙ (t)x(t)dt ˙ = 2 2 kxk dt π d π d tk
(3.10)
k=0
ku st k2L 2 =
∞ Z X
tk+1
k=0 tk
6 (d − 1)
2
(x(t − d(t − tk )) − x(tk ))> (x(t − d(t − tk )) − x(tk ))dt
Z ∞ X 4(tk+1 − tk )2 π2
k=0
tk+1 tk
x˙ T (t − d(t − tk ))x(t ˙ − d(t − tk ))dt.
Using the following change of variables
we arrive to
s = t − d(t − tk ) ds = (1 − d)dt, * + t = tk t = tk+1 , s = tk s = (1 − d)tk+1 + dtk ku st k2L 2 6 (d − 1) 6 (d − 1)
Z ∞ X 4(tk+1 − tk )2 π2
k=0
tk (1−d)tk+1 +dtk
d = 2, 3, . . .
x˙ > (s)x(s)ds ˙
∞ Z 2 4h 2 X kh T 2 4h x ˙ (s) x(s)ds ˙ = (d − 1) kxk ˙ 2L 2 . π 2d 2 π 2d 2 (k+1)h−dh k=0
R EMARK 3.1 For d = 1, the bound of Lemma 3.2 coincides with the bound of Mirkin (2007), where sampled-data control with variable sampling tk+1 − tk 6 h was analysed by using the lifting technique. We note that the bounds in (3.10) are valid also for tk+1 − tk 6 h, i.e. for d = 1, we recover result of Mirkin (2007). Moreover, Lemma 3.2 strengthens the result of Mirkin (2007), showing that the same bound holds for any d ∈ N if tk+1 − tk = h. It follows that stability of (3.5) can be verified by using the small gain theorem (Gu et al., 2003). Namely, (3.5) is stable if there exists non-singular M ∈ Rn×n
such that kM G M −1 k∞ < 1,
(3.11)
where G(s) = s[s I − ( A + A1 )]−1 ∙
2h A1 . π
(3.12)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
Similarly, we have
6 of 18
K. LIU ET AL.
We can verify the condition (3.11) via the following LMI: > P(A + A1 ) + (A + A1 )> P 2h π P A1 (A + A1 ) R 2h > −R ∗ 0, R > 0.
•
Condition (3.11) holds, where G is given by (3.12).
•
There exist positive n × n-matrices P, R such that LMI (3.13) is feasible.
4. Lyapunov–Krasovskii approach 4.1
Lyapunov-based exponential stability
D EFINITION 4.1 The system (2.1) is said to be exponentially stable if there exists constants μ > 0 and δ > 0 such that kx(t)k 6 μ e−δ(t−t0 ) kφkW for t > t0 . L EMMA 4.1 (Fridman, 2010). Let there exist positive numbers β, δ and a functional V : R × W × L 2 [−h, 0] → R such that ˙ 6 δkφk2W . β|φ(0)|2 6 V (t, φ, φ)
(4.1)
Let the function Vˉ (t) = V (t, x t , x˙t ) is continuous from the right for x(t) satisfying (2.1) absolutely continuous for t 6= tk and satisfies lim Vˉ (t) > Vˉ (tk ).
(4.2)
t→tk−
Given α > 0, if along (2.1) V˙ˉ (t) + 2α Vˉ (t) 6 0,
(4.3)
almost for all t,
then (2.1) is exponentially stable with the decay rate α. 4.2
Exponential stability: Case A1
We start with the case, where τ satisfies τ˙ = d > 0. We consider separately 0 < d 6 1 and d > 1. (i) When 0 < d 6 1, we are looking for the functional of the form V (t, xt , x˙t ) = Vˉ (t) = V0 (xt , x˙t ) + where V0 (xt , x˙t ) = x > (t)P x(t) +
Z
t t−h
2 P
i=1
Vi (t, xt , x˙t ) + V3 (xt ),
e2α(s−t) x > (s)Sx(s)ds +
1 h
Z
0 −h
Z
t t+θ
(4.4)
e2α(s−t) x˙ > (s)R x(s)ds ˙ dθ, (4.5)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
T HEOREM 3.1 For d ∈ N , (3.5)–(3.6) is I–O stable (and, thus, (2.1)–(2.2) is asymptotically stable) if one of the following conditions is satisfied:
7 of 18
GENERAL SAWTOOTH DELAY
h−τ > ξ (t) V1 (t, xt , x˙t ) = h 1
Z
h−τ V2 (t, xt , x˙t ) = h Z
t
2
∗
t t−τ
−X + X 1
> −X 1 − X 1> + X +X 2
ξ1 (t),
(4.6)
e2α(s−t) x˙ > (s)U x(s)ds, ˙
(4.7)
e2α(s−t) x > (s)Qx(s)ds,
(4.8)
t−τ
with ξ1 (t) = col{x(t), x(t − τ )}, α > 0, P > 0, S > 0, R > 0, U > 0, Q > 0. The above functional coincides with the one introduced in Fridman (2010) for S = R = Q = 0, where τ˙ = 1 was considered. The positive terms, depending on S, R, Q, guarantee that the results will be not worse than for the case of time-varying delays, where the above functional with U = X = X 1 can be applied. We note that the term V3 is non-negative before the jumps at t = tk and it becomes zero just after the jumps (because t|t=tk = (t − τ )|t=tk ). The time-dependent terms V1 and V2 vanish before the jumps (because τ = h) and after the jumps (because τ = 0 and thus x(t − τ ) = x(t)). Thus, Vˉ does not increase after the jumps and the condition limt→t − Vˉ (t) > Vˉ (tk ) holds. k To guarantee that V > 0 in the sense that V satisfies (4.1), we assume that # " > X1 − X P + X +X 2 > 0. (4.9) > −X 1 − X 1> + X +X ∗ 2 Differentiating Vˉ , we find along (2.1)
h−τ ˙ > > ˉ ˉ U x(t) ˙ + 2α[x > (t)P x(t)] V (t) + 2α V (t) 6 2x (t)P x(t) ˙ + x˙ (t) R + h Z Z 1 −2αh t > d −2αh t > x˙ (s)R x(s)ds ˙ − e x˙ (s)U x(s)ds ˙ − e h h t−h t−τ
+ x > (t)[S + Q]x(t) − x > (t − h)e−2αh Sx(t − h) − (1 − d)x˙ > (t − τ ) h−τ ˙ − τ ) − (1 − d)x > (t − τ )e−2αh Qx(t − τ ) × U e−2αh x(t h >
X +X −X + X 1 d − 2α(h − τ ) > 2 ξ1 (t) − ξ1 (t) > h ∗ −X 1 − X 1> + X +X 2 h−τ + x˙ > (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x(t − τ ) h
˙ − τ) +2(1 − d)x > (t)(−X + X 1 )x(t X + X> x(t − τ ) . +2(1 − d)x˙ > (t − τ ) −X 1 − X 1> + 2
Following He et al. (2007), we employ the representation Z Z t Z t−τ x˙ > (s)R x(s)ds ˙ =− x˙ > (s)R x(s)ds ˙ − − t−h
t−h
t t−τ
x˙ > (s)R x(s)ds. ˙
(4.10)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
V3 (xt ) =
X +X >
8 of 18
K. LIU ET AL.
We apply the Jensen’s inequality (Gu et al., 2003) Z Z t Z t 1 t > x˙ > (s)[R + U ]x(s)ds ˙ > x˙ (s)ds[R + U ] x(s)ds, ˙ τ t−τ t−τ t−τ Z t−τ Z t−τ Z t−τ 1 x˙ > (s)R x(s)ds ˙ > x˙ > (s)ds R x(s)ds. ˙ h − τ t−h t−h t−h
(4.11)
we obtain h−τ ˙ > > ˉ ˉ U x(t) ˙ + 2α[x > (t)P x(t)] V (t) + 2α V (t) 6 2x (t)P x(t) ˙ + x˙ (t) R + h τ > h−τ > [R + dU ]v11 − e−2αh − e−2αh v11 v12 Rv12 + x > (t)[S + Q]x(t) h h h−τ − x > (t − h)e−2αh Sx(t − h) − (1 − d)x˙ > (t − τ ) ˙ − τ) U e−2αh x(t h − (1 − d)x > (t − τ )e−2αh Qx(t − τ ) " # X +X > d − 2α(h − τ ) > −X + X 1 2 − ξ1 (t) ξ1 (t) > h ∗ −X 1 − X 1> + X +X 2 h−τ + x˙ > (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x(t − τ ) h ˙ − τ) +2(1 − d)x > (t)(−X + X 1 )x(t X + X> > > x(t − τ ) . +2(1 − d)x˙ (t − τ ) −X 1 − X 1 + 2
(4.12)
Following He et al. (2004), we insert free-weighting n × n-matrices by adding the following expressions to V˙ˉ : 0 = 2[x > (t)Y1> + x˙ > (t)Y2> + x > (t − τ )T > ][−x(t) + x(t − τ ) + τ v11 ],
0 = 2[x > (t)Z 1> + x˙ > (t)Z 2> ][−x(t − τ ) + x(t − h) + (h − τ )v12 ].
(4.13)
We use further the descriptor method (Fridman, 2001), where the right-hand side (RHS) of the expression 0 = 2[x > (t)P2> + x˙ > (t)P3> ][Ax(t) + A1 x(t − τ ) − x(t)], ˙ with some n × n-matrices, P2 , P3 is added into the RHS of (4.12).
(4.14)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
Here, for τ = 0, we understand by Z Z 1 t 1 t x(s)ds ˙ = lim x(s)ds ˙ = x(t). ˙ τ →0 τ t−τ τ t−τ R t−τ 1 ˙ is defined similarly as x(t For h − τ = 0, the vector h−τ ˙ − h). Then, denoting t−h x(s)ds Z Z t−τ 1 t 1 v11 = x(s)ds, ˙ x(s)ds, ˙ v12 = τ t−τ h − τ t−h
9 of 18
GENERAL SAWTOOTH DELAY
Setting η1 (t) = col{x(t), x(t), ˙ − τ )}, ˙ x(t − h), v11 , v12 , x(t − τ ), x(t we obtain that V˙ˉ (t) + 2α Vˉ (t) 6 η1> (t)Ψ1 η1 (t) < 0,
Ψ1 =
Φ11 −
2h
∗
(X + X > )
> Φ12 + h−τ 2h (X + X )
Φ22 + h−τ h U
Z 1>
τ Y1>
Z 2>
τ Y2>
∗
−S e−2αh
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
(h − τ )Z 1>
∗
Φ17
(h − τ )Z 2>
Φ26
0
0
0
0 0
0 − τh [R + dU ] e−2αh
Φ16
0
τT
∗
−2αh − h−τ h Re
0
0
Φ66
Φ67
∗
∗
∗
Φ77
∗
∗
< 0
(4.16)
holds, where Φ11 = A> P2 + P2> A + 2α P + S + Q − Y1 − Y1> , Φ12 = P − P2> + A> P3 − Y2 , d − 2α(h − τ ) (X − X 1 ), Φ16 = Y1> − Z 1> + P2> A1 − T + h h−τ (−X + X 1 ), Φ17 = (1 − d) h > Φ22 = −P3 − P3 + R, h−τ Φ26 = Y2> − Z 2> + P3> A1 − (X − X 1 ), h X + X > − 2X 1 − 2X 1> Φ66 = −(1 − d)Q e−2αh + T + T > − [d − 2α(h − τ )] , 2h h−τ (X + X > − 2X 1 − 2X 1> ), Φ67 = (1 − d) 2h h−τ Φ77 = −(1 − d) U e−2αh . h The latter inequality for τ → 0 and τ → h leads to the following LMIs: d−2αh X +X > > > > Ψ11
=
Φ11 −
2h
∗
(X + X )
∗
Φ12 +
Φ22 + U ∗
∗
∗
∗
∗
∗
Ψ12
=
2
∗ ∗
∗
∗
Z 2>
−S e−2αh
Φ12 Φ22 ∗
∗
∗
h Z1
h Z 2>
Φ16 |τ =0
Φ17 |τ =0
Φ26 |τ =0
0
0
0
0
∗
−R e−2αh
0
0
∗
∗
∗
∗
d (X + X > ) Φ11 − 2h
Z1
∗
∗
Z 1>
hY1>
−S e−2αh
0
Z 2> ∗
∗
Φ66 |τ =0
hY2> −[R + dU ] e−2αh ∗
Φ67 |τ =0
Φ77 |τ =0
Φ16 |τ =h
Φ26 |τ =h 0
hT
Φ66 |τ =h
< 0,
< 0.
(4.17)
(4.18)
(4.19)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
if the following inequality d−2α(h−τ )
(4.15)
10 of 18
K. LIU ET AL.
˙ − τ )}(i = 1, 2), the latter two LMIs Denoting: η1i (t) = col{x(t), x(t), ˙ x(t − h), v1i , x(t − τ ), x(t imply (4.16) because h−τ > τ > (t)Ψ12 η11 (t) = η1> (t)Ψ1 η1 (t) < 0, η12 (t)Ψ11 η12 (t) + η11 h h and Ψ1 is thus convex in τ ∈ [0, h]. (ii) For d > 1, we consider the following LKF:
where V0 is given by (4.5) and h−τ > ξ (t) V1 (t, xt , x˙t ) = h 2
"
h−τ V2 (t, xt , x˙t ) = h
X +X > 2
t t− dτ
i=1
Vi (t, xt , x˙t ),
−X + X 1
−X 1 − X 1> +
∗
Z
2 P
X +X > 2
#
(4.20)
ξ2 (t),
(4.21)
e2α(s−t) x˙ > (s)U x(s)ds, ˙
(4.22)
with ξ2 (t) = col x(t), x t − dτ , α > 0, U > 0. To guarantee that V > 0, we assume (4.9). Since t|t=tk = t − dτ |t=t , we see that Vˉ does not grow after the jumps. k Differentiating Vˉ , we have along (2.1) h−τ ˙ > > ˉ ˉ ˙ + x˙ (t) R + V (t) + 2α V (t) 6 2x (t)P x(t) U x(t) ˙ + 2α[x > (t)P x(t)] h Z Z 1 −2αh t > d − 2αh t d − e x˙ (s)R x(s)ds ˙ − e x˙ > (s)U x(s)ds ˙ h h t−h t− dτ + x > (t)Sx(t) − x > (t − h)e−2αh Sx(t − h) " # X +X > −X + X 1 d − 2α(h − τ ) > 2 − ξ2 (t) ξ2 (t) > h ∗ −X 1 − X > + X +X 1
2
τ i h−τ h > x˙ (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x t − . + h d
We employ the representation −
Z
t
>
t−h
x˙ (s)R x(s)ds ˙ =−
Z
t−τ t−h
>
x˙ (s)R x(s)ds ˙ −
Z
t− dτ t−τ
>
x˙ (s)R x(s)ds ˙ −
Z
t t− dτ
x˙ > (s)R x(s)ds. ˙
Similar to (4.11), applying the Jensen’s inequality and then denoting v21 =
d τ
Z
t t− dτ
x(s)ds, ˙
v22 =
1 h−τ
Z
t−τ t−h
x(s)ds, ˙
v23 =
d h(d − 1)
Z
t− dτ t−τ
x(s)ds, ˙
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
V (t, xt , x˙t ) = Vˉ (t) = V0 (xt , x˙t ) +
11 of 18
GENERAL SAWTOOTH DELAY
we obtain
Similar to (4.13), we insert free-weighting n × n-matrices by adding the following expressions to V˙ˉ : h i τ >i h τ τ 0 = 2 x > (t)Y1> + x˙ > (t)Y2> + x > t − − x(t) + x t − T1 + v21 , d d d 0 = 2[x > (t)Z 1> + x˙ > (t)Z 2> + x > (t − τ )T2> ][−x(t − τ ) + x(t − h) + (h − τ )v22 ], h τ h(d − 1) i + x(t − τ ) + v23 . 0 = 2[x > (t)M1> + x˙ > (t)M2> ] − x t − d d
Similar to (4.14), the same expression is added into the RHS of (4.2). Setting η2 (t) = col x(t), x(t), ˙ x(t − h), v21 , v22 , v23 , x t − dτ , x(t − τ ) , we obtain that V˙ˉ (t) + 2α Vˉ (t) 6 η2> (t)Ψ2 η2 (t) < 0,
(4.23)
if the inequality Ψ2 =
) > Ω11 − d−2α(h−τ (X + X > ) Φ12 + h−τ Z 1> 2h 2h (X + X ) > ∗ Φ22 + h−τ U Z h 2 ∗ −S e−2αh ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
τ > d Y1 τ > d Y2
0 Ω44 ∗ ∗ ∗ ∗
(h − τ )Z 1> (h − τ )Z 2>
h(d−1) > M1 d h(d−1) > M2 d
0 0 0 0 −2αh − h−τ 0 h Re −2αh ∗ − d−1 d Re ∗ ∗ ∗ ∗
Ω17
Ω18
Ω27 Ω28 0 T2 τ 0 d T1 0 (h − τ )T2 0 0 0 Ω77 > ∗ T2 + T2
P2 + P2> A + 2α P + S − Y1 − Y1> , d − 2α(h − τ ) (X − X 1 ), Ω17 = Y1> − M1> − T1 + h Ω18 = M1> − Z 1> + P2> A1 , h−τ Ω27 = Y2> − M2> − (X − X 1 ), h Ω28 = M2> − Z 2> + P3> A1 , τ R −2αh − 2αh d +U e Ω44 = − , e h d X + X > − 2X 1 − 2X 1> Ω77 = T1 + T1> − [d − 2α(h − τ )] . 2h
(4.25)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
h−τ ˙ > > ˉ ˉ U x(t) ˙ + 2α[x > (t)P x(t)] V (t) + 2α V (t) 6 2x (t)P x(t) ˙ + x˙ (t) R + h 2αh τ > R −2αh > −2αh d−1 v > Rv e − v21 + U e− d v21 − e−2αh h−τ 23 23 h v22 Rv22 − e d h d +x > (t)Sx(t) − x > (t −"h)e−2αh Sx(t − h) # X +X > d − 2α(h − τ ) > −X + X 1 2 − ξ2 (t) ξ2 (t) > h ∗ −X 1 − X 1> + X +X 2 h−τ > x˙ (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x t − dτ . + h
12 of 18
K. LIU ET AL.
The latter inequality for τ → 0 and τ → h leads to the following LMIs: > Ψ21
X +X > Ω11 − d−2αh 2 2h (X + X ) Φ12 + ∗ Φ22 + U
∗
∗
∗
=
∗
d > 11 − 2h (X + X ) Φ12
∗ ∗ ∗
∗
∗
∗
Φ22 ∗ ∗ ∗
∗
∗
h Z 2> 0
0
∗
−R e−2αh
0
∗
∗
Z 2>
−S e−2αh
Ω28
0
T2
0
hT2
0
0
∗
h > d Y1 h > d Y2
h(d−1) > M1 d h(d−1) > M2 d
Ω17 |τ =h Ω27 |τ =h
Ω28
0
0
0
T2
0
h d T1
0
0
0
Ω44 |τ =h
∗
∗
∗
∗
∗
−2αh − d−1 d Re
∗
∗
∗
Ω18
Ω27 |τ =0
∗
∗ ∗
Ω17 |τ =0
−2αh − d−1 d Re
∗
Z 1>
h(d−1) > M1 d h(d−1) > M2 d
∗
∗
∗
∗
Ω
Ψ22
∗
∗
Z 2>
−S e−2αh
∗
∗
h Z 1>
Ω77 |τ =0 ∗
Ω77 |τ =h ∗
Ω18
0 T2 + T2>
Similarly, we can obtain that Ψ2 is also convex in τ ∈ [0, h]. We summarize the results in the following theorem.
0 T2 + T2>
< 0 and
(4.26)
< 0.
(4.27)
T HEOREM 4.2 (i) Given α > 0, 0 < d 6 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, Q > 0, X, X 1 , T, P2 , P3 , Yi and Z i (i = 1, 2) such that the LMIs (4.9), (4.18), (4.19) with notations given in (4.17) are feasible. Then, (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A1. (ii) Given α > 0, d > 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, X, X 1 , P2 , P3 , Ti , Yi , Z i and Mi (i = 1, 2) such that the LMIs (4.9), (4.26), (4.27) with notations given in (4.25) are feasible. Then, (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A1. R EMARK 4.1 LMIs of Theorem 4.2 with X = X 1 = U = 0 give sufficient conditions for exponential stability of (2.1) with τ (t) ∈ [0, h]: (i) for all slowly varying delays and (ii) for fast-varying delays. In the numerical examples, these conditions lead to the same results as the results of Park & Ko (2007), however, they posses a fewer number of slack matrices. 4.3 Exponential stability: Case A2 For τ˙ = −d < 0, we also differ between 0 < d 6 1 and d > 1. (i) When 0 < d 6 1, we are looking for the functional of the form: V (t, xt , x˙t ) = Vˉ (t) = V0 (xt , x˙t ) + where V0 is given by (4.5) and τ V1 (t, xt , x˙t ) = ξ3> (t) h
"
X +X > 2
∗
2 P
i=1
Vi (t, xt , x˙t ) + V3 (xt ),
−X + X 1
−X 1 − X 1> +
X +X > 2
#
ξ3 (t),
(4.28)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
=
Z 1>
13 of 18
GENERAL SAWTOOTH DELAY
τ V2 (t, xt , x˙t ) = h Z V3 (xt ) =
Z
t t−(h−τ )
t
e2α(s−t) x˙ > (s)U x(s)ds, ˙
e2α(s−t) x > (s)Qx(s)ds,
t−(h−τ )
V (t, xt , x˙t ) = Vˉ (t) = V0 (x t , x˙t ) +
2 P
i=1
Vi (t, xt , x˙t ),
(4.29)
where V0 is also given by (4.5) and " # X +X > −X + X 1 τ > 2 V1 (t, xt , x˙t ) = ξ4 (t) ξ4 (t), > h −X 1 − X 1> + X +X ∗ 2 Z τ t V2 (t, xt , x˙t ) = e2α(s−t) x˙ > (s)U x(s)ds, ˙ h t− h−τ d , α > 0, U > 0 and (4.9) also satisfied to guarantee that V > 0. with ξ4 (t) = col x(t), x t − h−τ d ˉ Also in this case, V does not grow after the jumps. Similar to Theorem 4.2, we obtain the following theorem. T HEOREM 4.3 (i) Given α > 0, 0 < d 6 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, Q > 0, X, X 1 , P21 , P31 , P22 , P32 , Yi j , Mi j , Z i j and Ti j (i, j = 1, 2) such that (4.9) and the following four LMIs: (4.30), for τ → 0 and τ → h2 , and (4.31), for τ → h2 and τ → h, Θ
11
∗
τ (X + X > ) Θ12 + 2h
Θ22 + τh U
> M11
> τ Y11
> M12
> τ Y12
> (h − 2τ )Z 11
> τ M11
Θ17
> − M> Θ Θ18 + Z 11 19 11
> (h − 2τ )Z 12
> τ M12
Θ27
> − M> Θ28 + Z 12 12
0 0
∗
∗
−S e−2αh
0
0
0
0
0
∗
∗
∗
− τh [R + dU ]e−2αh
0
0
τ T11
0
0
∗
∗
∗
∗
Θ55
0
0
0
∗
∗
∗
∗
∗
− τh R e−2αh
(h − 2τ )T12
0
0
0
∗
∗
∗
∗
∗
∗
> T11 + T11
−T12
0
∗
∗
∗
∗
∗
∗
∗
> Θ88 + T12 + T12
Θ89
∗
∗
∗
∗
∗
∗
∗
∗
Θ99
< 0,
(4.30)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
with ξ3 (t) = col{x(t), x(t − (h − τ ))}, α > 0, U > 0, Q > 0 and (4.9) also satisfied to guarantee that V > 0. We note that V3 is non-negative before the jump at t = tk and it becomes zero just after the jump (because t|t=tk = (t − (h − τ ))|t=tk ). The time-dependent terms V1 and V2 vanish before the jumps (because τ = 0) and after the jumps (because τ = h and thus x(t − (h − τ )) = x(t)). Hence, the condition limt→t − Vˉ (t) > Vˉ (tk ) holds. k (ii) For d > 1, the discontinuous Lyapunov functional is modified as follows:
14 of 18
τ (X + X > ) Ξ11 Ξ12 + 2h
∗
∗
∗ ∗
> Y21
∗
> Y22 −2αh −S e
∗
∗
Ξ22 + τh U ∗
∗
> (h − τ )Y21
> (2τ − h)Z 21
> (h − τ )M21 > (h − τ )M22
0
0
0
> (h − τ )Y22 −2αh − h−τ h Re
0
0
0
Ξ66
0
∗
∗
∗
∗
∗
∗
> −T21 − T21
∗
∗
∗
0
(h − τ )T21
∗ ∗
0
0
∗
∗
0
T21
0
∗
∗
Ξ27
0
∗
∗
∗
∗
where > A + 2α P + S + Q − Y − Y > − Θ11 = A> P21 + P21 11 11
> + A> P − Y , Θ12 = P − P21 31 12
> − Z> Θ Θ18 + M21 19 21 > − Z> Θ28 + M22 22
− 2τh−h R e−2αh
∗
∗
Ξ17
∗
∗
∗
> (2τ − h)Z 22
∗ ∗
(2τ − h)T22 0
0 0
T22
0
> Θ88 − T22 − T22
Θ89
∗
Θ99
< 0,
(4.31)
d − 2ατ (X + X > ), 2h
> − Z> − T + P> A , Θ17 = Y11 11 11 21 1 d − 2ατ (X − X 1 ), Θ18 = h τ Θ19 = (1 − d) (−X + X 1 ), h > + R, Θ22 = −P31 − P31 > − Z> + P> A , Θ27 = Y12 12 31 1 τ Θ28 = − (X − X 1 ), h h − 2τ [R + dU ]e−2αh , Θ55 = − h
X + X > − 2X 1 − 2X 1> Θ88 = −(1 − d)Q e−2αh − (d − 2ατ ) , 2h τ Θ89 = (1 − d) (X + X > − 2X 1 − 2X 1> ), 2hτ Θ99 = −(1 − d) U e−2αh , h > A + 2α P + S + Q − M − M > − d − 2ατ (X + X > ), > Ξ11 = A P22 + P22 21 21 2h > > Ξ12 = P − P22 + A P32 − M22 , > − Y > + P> A , Ξ17 = Z 21 21 22 1 > + R, Ξ22 = −P32 − P32
> − Y > + P> A , Ξ27 = Z 22 22 32 1 h−τ Ξ66 = − [R + dU ]e−2αh , h
are feasible. Then (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A2. (ii) Given α > 0, d > 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, X, X 1 , P21 , P31 , P22 , P32 , Yi j , Mi j , Z i j and Ti j (i, j = 1, 2) such that (4.9) and the following four LMIs: (4.32), for h h , and (4.33), for τ → 1+d and τ → h, τ → 0 and τ → 1+d
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
K. LIU ET AL.
15 of 18
GENERAL SAWTOOTH DELAY
Υ
Θ22 + τh U
∗
> τ Y11
> (h − h−τ d )M11 > (h − h−τ d )M12
Θ17
> τ Y12
h−(1+d)τ > Z 11 d h−(1+d)τ > Z 12 d
> M12
0
0
0
0
∗
Σ44
0
0
τ T11
∗
∗
∗
∗
Σ55
0
0 0
0
∗
∗
∗
∗
Σ66
∗
∗
∗
∗
∗
> T11 + T11
−T12
∗
∗
∗ ∗
∗
11
∗
τ (X + X > ) Ξ12 + 2h
Ξ22 + τh U
∗
> Y21 > Y22
∗ ∗
Ξ17
> (h − τ )Y22
0
0
0
T21
0
(h − τ )T21
0
0
−2αh − h−τ h Re
∗
∗
∗
∗
0 R e−2αh − h1 (1+d)τ−h d
∗
∗
∗
∗
∗
∗
∗ ∗
∗ ∗
> Σ88 +T12 +T12
h−τ M > d 21 h−τ M > d 22
∗
∗
0
h−(1+d)τ T12 d
(1+d)τ−h > Z 21 d (1+d)τ−h > Z 22 d
−S e−2αh
∗
0
> (h − τ )Y21
∗
∗
∗
∗ ∗
∗ ∗
> − M> Θ28 + Z 12 12
Θ27
−S e−2αh
∗
∗
> −M > Θ18 + Z 11 11
∗
∗
∗
> M11
Ξ27
< 0,
> −Z > Θ18 + M21 21
0 0
(1+d)τ−h T22 d
0
0
∗
> −T21 −T21
T22
∗
> −Z> Θ28 + M22 22
Υ66 ∗
(4.32)
> Σ88 −T22 −T22
< 0,
(4.33)
where > A + 2α P + S − Y − Y > − d − 2ατ (X + X > ), Σ11 = A> P21 + P21 11 11 2h τ −2αh − 2αh + dU e d , Σ44 = − R e h 2αh 1 h − (1 + d)τ −2αh + dU e− d , Re Σ55 = − h d 1 h−τ Σ66 = − h− R e−2αh , h d X + X > − 2X 1 − 2X 1> Σ88 = −(d − 2ατ ) , 2h > A + 2α P + S − M − M > − d − 2ατ (X + X > ), Υ11 = A> P22 + P22 21 21 2h 1 h − τ −2αh 2αh − d Υ66 = − + dU e , Re h d
are feasible. Then (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A2. 5. Examples E XAMPLE 5.1 Consider the system from Yue et al. (2005) " # " # 0 1 0 x(t) ˙ = x(t) + K x(t − τ (t)), 0 −0.1 0.1 K = −[3.75 11.5]. The stability of this system was studied by many authors (see Naghshtabrizi et al., 2008 and the references therein). For the constant sampling, it was found in Naghshtabrizi et al. (2008) that the system
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
τ (X + X > ) Σ11 Θ12 + 2h
16 of 18
K. LIU ET AL.
E XAMPLE 5.2 We consider the following simple and much-studied problem (Papachristodoulou et al., 2007; Fridman, 2010): x(t) ˙ = −x(t − τ (t)).
(5.1)
It is well known that the equation x(t) ˙ = −x(t − τ ) with constant delay τ is asymptotically stable for τ < π/2 and unstable for τ > π/2, whereas for the fast-varying delay, it is stable for τ < 1.5 and there exists a destabilizing delay with an upper bound >1.5. The latter means that all the existing methods, that are based on time-independent Lyapunov functionals, corresponding to stability analysis of systems with fast-varying delays, cannot guarantee the stability for the samplings, which may be >1.5. It is easy to check, that the system remains stable for all constant samplings 2. Conditions of Naghshtabrizi et al. (2008) and of Mirkin (2007) guarantee asymptotic stability for all variable samplings up to 1.28 and 1.57, respectively. For different τ˙ , by applying our methods, we obtain the maximum value of h given in Table 2.We can also see that discontinuous terms of LKFs improve the performance and the I–O approach via Wirtinger-type inequality improves the result for d > 1, d ∈ N compared to time-dependent L–K approach. E XAMPLE 5.3 Consider the system (De Souza & Li, 1999) " # " # −2 0 −1 0 x(t) ˙ = x(t) + x(t − τ (t)). 0 −0.9 −1 −1 TABLE 1 Example 1: maximum value of h for different τ˙ h\τ˙ L–K approach X = X1 = U = 0
−1.2 1.14 1.04
−1 1.14 1.04
−0.9 1.11 1.04
0.5 1.24 1.04
0.9 1.61 1.04
1 1.69 1.04
1.1 1.34 1.04
h\τ˙ I–O approach
τ˙ ∈ N 1.3659
TABLE 2 Example 2: maximum value of h for different τ˙ h\τ˙ L–K approach X = X1 = U = 0
−1.2 1.41 1.33
−1 1.41 1.33
−0.9 1.36 1.33
0.5 1.47 1.33
0.9 1.89 1.33
1 1.99 1.33
1.1 1.61 1.33
1.2 1.54 1.33
h\τ˙ I–O approach
τ˙ ∈ N 1.57
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
remains stable for all constant samplings 1.7. Moreover, the above system with constant delay τ is asymptotically stable for τ 6 1.16 and becomes unstable for τ > 1.17. The latter means that all the existing methods, that are based on time-independent Lyapunov functionals, corresponding to stability analysis of systems with fast-varying delays, cannot guarantee the stability for the samplings with the upper bound >1.17. In Naghshtabrizi et al. (2008), the upper bound on the constant sampling interval that preserves the stability is found to be h = 1.3277, improving all the existing LMI-based results. For different τ˙ , by applying Lyapunov–Krasovskii(L–K) approach with α = 0 and I–O approach via Wirtinger-type inequality, we obtain the maximum value of h given in Table 1. Our results for X = X 1 = U = 0 coincide with the ones of Park & Ko (2007). We see that discontinuous terms of LKFs improve the performance. Moreover, the I–O approach via Wirtinger-type inequality improves the result for d > 1, d ∈ N compared to time-dependent L–K approach.
17 of 18
GENERAL SAWTOOTH DELAY
TABLE 3 Example 3: maximum value of h for different τ˙ h\τ˙ L–K approach X = X1 = U = 0
−1.2 1.95 1.86
−1 1.98 1.86
−0.9 1.90 1.86
0.5 2.34 2.33
0.9 2.06 1.87
1 2.53 1.86
1.1 1.91 1.86
h\τ˙ I–O approach
τ˙ ∈ N 1.57
1 1.64 1.06
1.1 1.26 1.06
h\τ˙ I–O approach
τ˙ ∈ N 0.8797
TABLE 4 Example 4: maximum value of h for different τ˙ −1.2 1.10 1.06
−1 1.11 1.06
−0.9 1.10 1.06
0.5 1.44 1.26
0.9 1.48 1.06
It is well known that this system is stable for constant delay τ 6 6.17, whereas in Park & Ko (2007), it was found that the system is stable for all fast-varying delays τ 6 1.86. For different τ˙ , by applying our methods, we obtain the maximum value of h given in Table 3. Our results for X = X 1 = U = 0 coincide with the ones of Park & Ko (2007). Also in this Example, the time-dependent terms of LKFs improve the performance. However, the I–O approach via Wirtinger-type inequality has not improved the results for d > 1, d ∈ N and the result is worse than the result for the fast-varying delays. E XAMPLE 5.4 Consider the system (Kharitonov & Niculescu, 2002): " # " # 0 1 0 0 x(t) ˙ = x(t) + x(t − τ (t)). −1 −2 −1 1 For different τ˙ , by applying our methods, we obtain the maximum value of h given in Table 4. R EMARK 5.1 Simulations in all the examples above show that our results are conservative (at least for d 6= 1) and that the value of maximum h that preserves the stability grows for growing d > 1. 6. Conclusions In this paper, two methods have been introduced to investigate delay-dependent stability problem for systems with sawtooth delay with constant τ˙ 6= 0. One method develops a novel I–O approach via a Wirtinger-type inequality. The result by Mirkin (2007) is recovered for τ˙ = 1 and for any integer τ˙ , the same maximum bound that preserves the stability is achieved. Another method improves stability criteria by constructing piecewise continuous (in time) Lyapunov functionals. Though the most essential improvement corresponds to τ˙ = 1, the time-dependent terms improve the results for all values of τ˙ . Funding Israel Science Foundation (754/10); China Scholarship Council. R EFERENCES D E S OUZA , C. E. & L I , X. (1999) Delay-dependent robust H∞ control of uncertain linear state-delayed systems. Automatica, 35, 1313–1321. F RIDMAN , E. (2001) New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Syst. Control Lett., 43, 309–319.
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
h\τ˙ Our method X = X1 = U = 0
18 of 18
K. LIU ET AL.
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
F RIDMAN , E. (2010) A refined input delay approach to sampled-data control. Automatica, 46, 421–427. F RIDMAN , E. & S HAKED , U. (2003) Delay-dependent stability and H∞ control: constant and time-varying delays. Int. J. Control, 76, 48–60. G U , K., K HARITONOV, V. & C HEN , J. (2003) Stability of Time-Delay Systems. Boston, MA: Birkhauser. H ARDY, G., L ITTLEWOOD , J. E. & P OLYA , G. (1934) Inequalities. Cambridge, UK: Cambridge University Press. H E , Y., WANG , Q. G., L IN , C. & W U , M. (2007) Delay-range-dependent stability for systems with time-varying delay. Automatica, 43, 371–376. H E , Y., W U , M., S HE , J. & L IU , G. (2004) Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst. Control Lett., 51, 57–65. K HARITONOV, V. & N ICULESCU , S. (2002) On the stability of linear systems with uncertain delay. IEEE Trans. Autom. Control, 48, 127-132. KOLMANOVSKII , V. & M YSHKIS , A. (1999) Introduction to the Theory and Applications of Functional Dierential Equations. Dordrecht, Netherlands: Kluwer Academic. L IU , K. & F RIDMAN , E. (2009) Discontinuous Lyapunov functionals for linear systems with sawtooth delays. Proceedings of the 8th IFAC Workshop on Time-Delay Systems, Sinaya, Romania. L OUISELL , J. (1999) New examples of quenching in delay differential equations having time-varying delay. Proceedings of the 5th European Control Conference. Karlsruhe, Germany. M ICHIELS , W., VAN A SSCHE , V. & N ICULESCU , S. (2005) Stabilization of time-delay systems with a controlled time-varying delays and applications. IEEE Trans. Automat. Control, 50, 493–504. M IRKIN , L. (2007) Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Trans. Automat. Control, 52, 1109–1112. NAGHSHTABRIZI , P., H ESPANHA , J. & T EEL , A. (2008) Exponential stability of impulsive systems with application to uncertain sampled-data systems. Syst. Control Lett., 57, 378–385. N ICULESCU , S. (2001) Delay Effects on Stability: A Robust Control Approach. Lecture Notes in Control and Information Sciences, vol. 269. London: Springer. PAPACHRISTODOULOU , A., P EET, M. & N ICULESCU , S. (2007) Stability analysis of linear systems with timevarying delays: delay uncertainty and quenching. Proceedings of the 46th IEEE CDC, New Orleans, USA. PARK , P. G. & KO , J. W. (2007) Stability and robust stability for systems with a time-varying delay. Automatica, 43, 1855–1858. R ICHARD , J. P. (2003) Time-delay systems: an overview of some recent advances and open problems. Automatica, 39, 1667–1694. Y UE , D., H AN , Q. L. & L AM , J. (2005) Networked-based robust H∞ control of systems with uncertainty. Automatica, 41, 640–644.
Stability of linear systems with general sawtooth delay
[Received on 4 February 2010; revised on 2 June 2010; accepted 31 August 2010] It is well known that in many particular systems, the upper bound on a certain time-varying delay that preserves the stability may be higher than the corresponding bound for the constant delay. Moreover, sometimes oscillating delays improve the performance (Michiels, W., Van Assche, V. & Niculescu, S. (2005) Stabilization of time-delay systems with a controlled time-varying delays and applications. IEEE Trans. Automat. Control, 50, 493–504). Sawtooth delays τ with τ˙ = 1 (almost everywhere) can posses this property (Louisell, J. (1999) New examples of quenching in delay differential equations having time-varying delay. Proceedigns of the 5th ECC, Karlsruhe, Germany). In this paper, we show that general sawtooth delay, where τ˙ 6= 0 is constant (almost everywhere), also can posses this property. By the existing Lyapunov-based methods, the stability analysis of such systems can be performed in the framework of systems with bounded fast-varying delays. Our objective is to develop ‘qualitatively new methods’ that can guarantee the stability for sawtooth delay which may be not less than the analytical upper bound on the constant delay that preserves the stability. We suggest two methods. One method develops a novel input–output approach via a Wirtinger-type inequality. By this method, we recover the result by Mirkin (2007, Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Trans. Automat. Control, 52, 1109–1112) for τ˙ = 1 and we show that for any integer τ˙ , the same maximum bound that preserves the stability is achieved. Another method extends piecewise continuous (in time) Lyapunov functionals that have been recently suggested for the case of τ˙ = 1 in Fridman (2010, A refined input delay approach to sampled-data control. Automatica, 46, 421–427) to the general sawtooth delay. The time-dependent terms of the functionals improve the results for all values of τ˙ , though the most essential improvement corresponds to τ˙ = 1. Keywords: time-varying delay; Lyapunov-based methods; LMI.
1. Introduction Over the past decades, much effort has been invested in the analysis and design of uncertain systems with time-varying delays (see, e.g. Kolmanovskii & Myshkis, 1999; Niculescu, 2001; Kharitonov & Niculescu, 2002; Fridman & Shaked, 2003; Richard, 2003; Gu et al., 2003; He et al., 2007; Park & Ko, 2007). The delay under consideration has been either differentiable with a known upper bound 0 6 τ˙ 6 d < 1 or piecewise continuous without any constraints on the delay derivative (fast-varying delay) (Fridman & Shaked, 2003). In the existing Lyapunov-based methods, the maximum delay bound that preserves the stability corresponds to d = 0 and this bound is usually a decreasing function of d. However, it is well known (see examples in Louisell, 1999 and discussions on ‘quenching’ in Papachristodoulou et al., 2007 as well as Example 1 below) that in many particular systems, the upper bound on a certain time-varying delay that preserves the stability may be higher than the corresponding c The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
K UN L IU , V LADIMIR S UPLIN AND E MILIA F RIDMAN∗ Department of Electrical Engineering-Systems, School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel [email protected] [email protected] ∗ Corresponding author: [email protected]
2 of 18
K. LIU ET AL.
kφkWn [a,b) = max |φ(θ)| + θ ∈[a,b]
Z
a
b
˙ |φ(s)| ds 2
12
.
We also denote W = Wn [−h, 0) and xt (θ ) = x(t + θ )(θ ∈ [−h, 0]). 2. Problem formulation Consider the system (
x(t) ˙ = Ax(t) + A1 x(t − τ (t)),
x(t) = φ(t),
t ∈ [−h, 0],
(2.1)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
bound for the constant delay. Moreover, sometimes oscillating delays improve the performance (Michiels et al., 2005). Recently, a discontinuous Lyapunov function method was introduced to sampled-data control systems (corresponds to the sawtooth delay with τ˙ = 1 (almost everywhere)) in Naghshtabrizi et al. (2008). This method improved the existing Lyapunov-based results and it inspired a piecewise continuous (in time) Lyapunov–Krasovskii functional (LKF) approach to sampled-data systems (Fridman, 2010). The LKFs in the latter paper are time dependent and they do not grow after the sampling times. The introduced novel discontinuous terms of Lyapunov functionals in Fridman (2010) lead to qualitatively new results, allowing a superior performance under the sampling, than the one under the constant delay. The input delay approach to sampled-data control was also recently revised by using the scaled small gain theorem and a tighter upper bound on the L 2 -induced norm of the uncertain term (Mirkin, 2007). In the present paper, stability of systems with bounded sawtooth delay τ is analysed, where τ˙ 6= 0 is piecewise constant. By the existing Lyapunov-based methods, the stability analysis of such systems can be performed in the framework of systems with bounded fast-varying delays. Our objective is to develop ‘qualitatively new methods’ that can guarantee the stability for sawtooth delay which may be not less than the analytical upper bound on the constant delay that preserves the stability. We suggest two methods. One method develops a novel input–output (I–O) approach via a Wirtinger-type inequality. By this method, we recover the result by Mirkin (2007) for τ˙ = 1 and we show that for any integer τ˙ , the same maximum bound that preserves the stability is achieved. Another method extends direct Lyapunov approach to systems with a general form of sawtooth delay. By constructing appropriate discontinuous LKFs, we obtain sufficient delay-dependent conditions that guarantee the exponential stability of systems in terms of linear matrix inequalities (LMIs). The time-dependent terms of LKFs improve the results for all values of τ˙ , though the most essential improvement corresponds to τ˙ = 1. A conference version of discontinuous LKF approach was presented in Liu & Fridman (2009). Notations. Throughout the paper, the superscript ‘>’ stands for matrix transposition, Rn denotes the n-dimensional Euclidean space with vector norm k ∙ k, Rn×m is the set of all n × m real matrices and the notation P > 0, for P ∈ Rn×n means that P is symmetric and positive definite. The symmetric elements of the symmetric matrix will be denoted by ∗. L 2 is the space of square integrable functions R∞ v: [0, ∞) → Rn with the norm kvk L 2 = [ 0 kv(t)k2 dt]1/2 . The space of functions φ: [a, b] → Rn , which are absolutely continuous on [a, b), have a finite limθ →b− φ(θ ) and have square integrable firstorder derivatives is denoted by Wn [a, b) with the norm
3 of 18
GENERAL SAWTOOTH DELAY
FIG. 1. Case A1.
where x(t) ∈ Rn is the state vector, A and A1 denote the constant matrices, φ(t) is the initial function, τ (t) ∈ [0, h] is the time-varying delay. It is assumed that the delay function has the form of sawtooth (see Figs 1 and 2), satisfying either A1 or A2 below: A1: τ (t) = d(t − tk ),
A2: τ (t) = d(tk+1 − t),
where d > 0 and tk =
t ∈ [tk , tk+1 ), k = 0, 1, 2 . . . ,
t ∈ [tk , tk+1 ), k = 0, 1, 2 . . . ,
(2.2) (2.3)
kh d .
It is clear that under A1, we have τ˙ = d > 0 and under A2, we have τ˙ = −d < 0. Both cases can be analysed by using time-independent Lyapunov functionals corresponding to systems with fast-varying delays. Our objective is to derive delay-dependent stability criteria for system (2.1) that improve the recent results for fast-varying delays (see, e.g. Park & Ko, 2007).
3. I–O approach via Wirtinger-type inequality We recall the following Wirtinger-type inequality (Hardy et al., 1934): let z ∈ W1 [a, b) be a scalar function with z(a) = 0. Then, Z
a
b
z 2 (ξ )dξ 6
4(b − a)2 π2
Z
b
a
z˙ 2 (ξ )dξ.
(3.1)
This inequality is trivially extended to the vector case. L EMMA 3.1 Let z ∈ Wn [a, b). Assume that z(a) = 0. Then, for any n × n-matrix R > 0, the following inequality holds: Z
a
b
4(b − a)2 z (ξ )Rz(ξ )dξ 6 π2 >
Z
a
b
z˙ > (ξ )R z˙ (ξ )dξ.
(3.2)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
FIG. 2. Case A2.
4 of 18
K. LIU ET AL.
Proof. We have Z
a
b
4(b − a)2 z (ξ )Rz(ξ )dξ 6 π2
Z
4(b − a)2 π2
Z
>
=
b
a
b
a
System (2.1) can be rewritten as follows: x(t) ˙ = ( A + A1 )x(t) − A1
Z
t t−τ (t)
x(s)ds. ˙
(3.4)
We present the latter as the following forward system: (
x˙ = ( A + A1 )x(t) + A1 u(t) ˙ y(t) = x(t),
(3.5)
with the feedback u(t) = −
Z
t
y(s)ds.
(3.6)
t−τ (t)
Assume that A + A1 is Hurwitz and y(t) = 0 for t 6 0.
L EMMA 3.2 Assume that the time delay is given by (2.2), where d ∈ N . Then the following holds: kuk L 2 6
2h kyk L 2 . π
(3.7)
Proof. Defining u s (t) = −
Z
t
y(s)ds, tk
u st (t) = −
Z
tk
y(s)ds, t−τ
t ∈ [tk , tk+1 ),
we note that u(t) = u s (t) + u st (t). We will prove next the following bounds: 2h kyk L 2 , πd 2h 6 (d − 1) kyk L 2 , πd
ku s k L 2 6
(3.8)
ku st k L 2
(3.9)
which imply (3.7) since kuk L 2 6 ku s k L 2 + ku st k L 2 6
2h kyk L 2 . π
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
2 dp > z (ξ )Rz(ξ ) dξ dξ !2 Z 4(b − a)2 b > z˙ > (ξ )Rz(ξ ) p dξ 6 z˙ (ξ )R z˙ (ξ )dξ. (3.3) π2 a z > (ξ )Rz(ξ )
5 of 18
GENERAL SAWTOOTH DELAY
By using (3.3), we obtain ku s k2L 2 =
∞ Z X
tk+1
k=0 tk
− x(t))>
(x(tk ) − x(t))> (x(tk ) − x(t))dt 6
Z ∞ X 4(tk+1 − tk )2 π2
k=0
tk+1 tk
d (x(tk ) dt
∞ Z d 4h 2 X tk+1 T 4h 2 (x(tk ) − x(t))dt 6 2 2 ˙ 2L 2 . x˙ (t)x(t)dt ˙ = 2 2 kxk dt π d π d tk
(3.10)
k=0
ku st k2L 2 =
∞ Z X
tk+1
k=0 tk
6 (d − 1)
2
(x(t − d(t − tk )) − x(tk ))> (x(t − d(t − tk )) − x(tk ))dt
Z ∞ X 4(tk+1 − tk )2 π2
k=0
tk+1 tk
x˙ T (t − d(t − tk ))x(t ˙ − d(t − tk ))dt.
Using the following change of variables
we arrive to
s = t − d(t − tk ) ds = (1 − d)dt, * + t = tk t = tk+1 , s = tk s = (1 − d)tk+1 + dtk ku st k2L 2 6 (d − 1) 6 (d − 1)
Z ∞ X 4(tk+1 − tk )2 π2
k=0
tk (1−d)tk+1 +dtk
d = 2, 3, . . .
x˙ > (s)x(s)ds ˙
∞ Z 2 4h 2 X kh T 2 4h x ˙ (s) x(s)ds ˙ = (d − 1) kxk ˙ 2L 2 . π 2d 2 π 2d 2 (k+1)h−dh k=0
R EMARK 3.1 For d = 1, the bound of Lemma 3.2 coincides with the bound of Mirkin (2007), where sampled-data control with variable sampling tk+1 − tk 6 h was analysed by using the lifting technique. We note that the bounds in (3.10) are valid also for tk+1 − tk 6 h, i.e. for d = 1, we recover result of Mirkin (2007). Moreover, Lemma 3.2 strengthens the result of Mirkin (2007), showing that the same bound holds for any d ∈ N if tk+1 − tk = h. It follows that stability of (3.5) can be verified by using the small gain theorem (Gu et al., 2003). Namely, (3.5) is stable if there exists non-singular M ∈ Rn×n
such that kM G M −1 k∞ < 1,
(3.11)
where G(s) = s[s I − ( A + A1 )]−1 ∙
2h A1 . π
(3.12)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
Similarly, we have
6 of 18
K. LIU ET AL.
We can verify the condition (3.11) via the following LMI: > P(A + A1 ) + (A + A1 )> P 2h π P A1 (A + A1 ) R 2h > −R ∗ 0, R > 0.
•
Condition (3.11) holds, where G is given by (3.12).
•
There exist positive n × n-matrices P, R such that LMI (3.13) is feasible.
4. Lyapunov–Krasovskii approach 4.1
Lyapunov-based exponential stability
D EFINITION 4.1 The system (2.1) is said to be exponentially stable if there exists constants μ > 0 and δ > 0 such that kx(t)k 6 μ e−δ(t−t0 ) kφkW for t > t0 . L EMMA 4.1 (Fridman, 2010). Let there exist positive numbers β, δ and a functional V : R × W × L 2 [−h, 0] → R such that ˙ 6 δkφk2W . β|φ(0)|2 6 V (t, φ, φ)
(4.1)
Let the function Vˉ (t) = V (t, x t , x˙t ) is continuous from the right for x(t) satisfying (2.1) absolutely continuous for t 6= tk and satisfies lim Vˉ (t) > Vˉ (tk ).
(4.2)
t→tk−
Given α > 0, if along (2.1) V˙ˉ (t) + 2α Vˉ (t) 6 0,
(4.3)
almost for all t,
then (2.1) is exponentially stable with the decay rate α. 4.2
Exponential stability: Case A1
We start with the case, where τ satisfies τ˙ = d > 0. We consider separately 0 < d 6 1 and d > 1. (i) When 0 < d 6 1, we are looking for the functional of the form V (t, xt , x˙t ) = Vˉ (t) = V0 (xt , x˙t ) + where V0 (xt , x˙t ) = x > (t)P x(t) +
Z
t t−h
2 P
i=1
Vi (t, xt , x˙t ) + V3 (xt ),
e2α(s−t) x > (s)Sx(s)ds +
1 h
Z
0 −h
Z
t t+θ
(4.4)
e2α(s−t) x˙ > (s)R x(s)ds ˙ dθ, (4.5)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
T HEOREM 3.1 For d ∈ N , (3.5)–(3.6) is I–O stable (and, thus, (2.1)–(2.2) is asymptotically stable) if one of the following conditions is satisfied:
7 of 18
GENERAL SAWTOOTH DELAY
h−τ > ξ (t) V1 (t, xt , x˙t ) = h 1
Z
h−τ V2 (t, xt , x˙t ) = h Z
t
2
∗
t t−τ
−X + X 1
> −X 1 − X 1> + X +X 2
ξ1 (t),
(4.6)
e2α(s−t) x˙ > (s)U x(s)ds, ˙
(4.7)
e2α(s−t) x > (s)Qx(s)ds,
(4.8)
t−τ
with ξ1 (t) = col{x(t), x(t − τ )}, α > 0, P > 0, S > 0, R > 0, U > 0, Q > 0. The above functional coincides with the one introduced in Fridman (2010) for S = R = Q = 0, where τ˙ = 1 was considered. The positive terms, depending on S, R, Q, guarantee that the results will be not worse than for the case of time-varying delays, where the above functional with U = X = X 1 can be applied. We note that the term V3 is non-negative before the jumps at t = tk and it becomes zero just after the jumps (because t|t=tk = (t − τ )|t=tk ). The time-dependent terms V1 and V2 vanish before the jumps (because τ = h) and after the jumps (because τ = 0 and thus x(t − τ ) = x(t)). Thus, Vˉ does not increase after the jumps and the condition limt→t − Vˉ (t) > Vˉ (tk ) holds. k To guarantee that V > 0 in the sense that V satisfies (4.1), we assume that # " > X1 − X P + X +X 2 > 0. (4.9) > −X 1 − X 1> + X +X ∗ 2 Differentiating Vˉ , we find along (2.1)
h−τ ˙ > > ˉ ˉ U x(t) ˙ + 2α[x > (t)P x(t)] V (t) + 2α V (t) 6 2x (t)P x(t) ˙ + x˙ (t) R + h Z Z 1 −2αh t > d −2αh t > x˙ (s)R x(s)ds ˙ − e x˙ (s)U x(s)ds ˙ − e h h t−h t−τ
+ x > (t)[S + Q]x(t) − x > (t − h)e−2αh Sx(t − h) − (1 − d)x˙ > (t − τ ) h−τ ˙ − τ ) − (1 − d)x > (t − τ )e−2αh Qx(t − τ ) × U e−2αh x(t h >
X +X −X + X 1 d − 2α(h − τ ) > 2 ξ1 (t) − ξ1 (t) > h ∗ −X 1 − X 1> + X +X 2 h−τ + x˙ > (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x(t − τ ) h
˙ − τ) +2(1 − d)x > (t)(−X + X 1 )x(t X + X> x(t − τ ) . +2(1 − d)x˙ > (t − τ ) −X 1 − X 1> + 2
Following He et al. (2007), we employ the representation Z Z t Z t−τ x˙ > (s)R x(s)ds ˙ =− x˙ > (s)R x(s)ds ˙ − − t−h
t−h
t t−τ
x˙ > (s)R x(s)ds. ˙
(4.10)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
V3 (xt ) =
X +X >
8 of 18
K. LIU ET AL.
We apply the Jensen’s inequality (Gu et al., 2003) Z Z t Z t 1 t > x˙ > (s)[R + U ]x(s)ds ˙ > x˙ (s)ds[R + U ] x(s)ds, ˙ τ t−τ t−τ t−τ Z t−τ Z t−τ Z t−τ 1 x˙ > (s)R x(s)ds ˙ > x˙ > (s)ds R x(s)ds. ˙ h − τ t−h t−h t−h
(4.11)
we obtain h−τ ˙ > > ˉ ˉ U x(t) ˙ + 2α[x > (t)P x(t)] V (t) + 2α V (t) 6 2x (t)P x(t) ˙ + x˙ (t) R + h τ > h−τ > [R + dU ]v11 − e−2αh − e−2αh v11 v12 Rv12 + x > (t)[S + Q]x(t) h h h−τ − x > (t − h)e−2αh Sx(t − h) − (1 − d)x˙ > (t − τ ) ˙ − τ) U e−2αh x(t h − (1 − d)x > (t − τ )e−2αh Qx(t − τ ) " # X +X > d − 2α(h − τ ) > −X + X 1 2 − ξ1 (t) ξ1 (t) > h ∗ −X 1 − X 1> + X +X 2 h−τ + x˙ > (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x(t − τ ) h ˙ − τ) +2(1 − d)x > (t)(−X + X 1 )x(t X + X> > > x(t − τ ) . +2(1 − d)x˙ (t − τ ) −X 1 − X 1 + 2
(4.12)
Following He et al. (2004), we insert free-weighting n × n-matrices by adding the following expressions to V˙ˉ : 0 = 2[x > (t)Y1> + x˙ > (t)Y2> + x > (t − τ )T > ][−x(t) + x(t − τ ) + τ v11 ],
0 = 2[x > (t)Z 1> + x˙ > (t)Z 2> ][−x(t − τ ) + x(t − h) + (h − τ )v12 ].
(4.13)
We use further the descriptor method (Fridman, 2001), where the right-hand side (RHS) of the expression 0 = 2[x > (t)P2> + x˙ > (t)P3> ][Ax(t) + A1 x(t − τ ) − x(t)], ˙ with some n × n-matrices, P2 , P3 is added into the RHS of (4.12).
(4.14)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
Here, for τ = 0, we understand by Z Z 1 t 1 t x(s)ds ˙ = lim x(s)ds ˙ = x(t). ˙ τ →0 τ t−τ τ t−τ R t−τ 1 ˙ is defined similarly as x(t For h − τ = 0, the vector h−τ ˙ − h). Then, denoting t−h x(s)ds Z Z t−τ 1 t 1 v11 = x(s)ds, ˙ x(s)ds, ˙ v12 = τ t−τ h − τ t−h
9 of 18
GENERAL SAWTOOTH DELAY
Setting η1 (t) = col{x(t), x(t), ˙ − τ )}, ˙ x(t − h), v11 , v12 , x(t − τ ), x(t we obtain that V˙ˉ (t) + 2α Vˉ (t) 6 η1> (t)Ψ1 η1 (t) < 0,
Ψ1 =
Φ11 −
2h
∗
(X + X > )
> Φ12 + h−τ 2h (X + X )
Φ22 + h−τ h U
Z 1>
τ Y1>
Z 2>
τ Y2>
∗
−S e−2αh
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
(h − τ )Z 1>
∗
Φ17
(h − τ )Z 2>
Φ26
0
0
0
0 0
0 − τh [R + dU ] e−2αh
Φ16
0
τT
∗
−2αh − h−τ h Re
0
0
Φ66
Φ67
∗
∗
∗
Φ77
∗
∗
< 0
(4.16)
holds, where Φ11 = A> P2 + P2> A + 2α P + S + Q − Y1 − Y1> , Φ12 = P − P2> + A> P3 − Y2 , d − 2α(h − τ ) (X − X 1 ), Φ16 = Y1> − Z 1> + P2> A1 − T + h h−τ (−X + X 1 ), Φ17 = (1 − d) h > Φ22 = −P3 − P3 + R, h−τ Φ26 = Y2> − Z 2> + P3> A1 − (X − X 1 ), h X + X > − 2X 1 − 2X 1> Φ66 = −(1 − d)Q e−2αh + T + T > − [d − 2α(h − τ )] , 2h h−τ (X + X > − 2X 1 − 2X 1> ), Φ67 = (1 − d) 2h h−τ Φ77 = −(1 − d) U e−2αh . h The latter inequality for τ → 0 and τ → h leads to the following LMIs: d−2αh X +X > > > > Ψ11
=
Φ11 −
2h
∗
(X + X )
∗
Φ12 +
Φ22 + U ∗
∗
∗
∗
∗
∗
Ψ12
=
2
∗ ∗
∗
∗
Z 2>
−S e−2αh
Φ12 Φ22 ∗
∗
∗
h Z1
h Z 2>
Φ16 |τ =0
Φ17 |τ =0
Φ26 |τ =0
0
0
0
0
∗
−R e−2αh
0
0
∗
∗
∗
∗
d (X + X > ) Φ11 − 2h
Z1
∗
∗
Z 1>
hY1>
−S e−2αh
0
Z 2> ∗
∗
Φ66 |τ =0
hY2> −[R + dU ] e−2αh ∗
Φ67 |τ =0
Φ77 |τ =0
Φ16 |τ =h
Φ26 |τ =h 0
hT
Φ66 |τ =h
< 0,
< 0.
(4.17)
(4.18)
(4.19)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
if the following inequality d−2α(h−τ )
(4.15)
10 of 18
K. LIU ET AL.
˙ − τ )}(i = 1, 2), the latter two LMIs Denoting: η1i (t) = col{x(t), x(t), ˙ x(t − h), v1i , x(t − τ ), x(t imply (4.16) because h−τ > τ > (t)Ψ12 η11 (t) = η1> (t)Ψ1 η1 (t) < 0, η12 (t)Ψ11 η12 (t) + η11 h h and Ψ1 is thus convex in τ ∈ [0, h]. (ii) For d > 1, we consider the following LKF:
where V0 is given by (4.5) and h−τ > ξ (t) V1 (t, xt , x˙t ) = h 2
"
h−τ V2 (t, xt , x˙t ) = h
X +X > 2
t t− dτ
i=1
Vi (t, xt , x˙t ),
−X + X 1
−X 1 − X 1> +
∗
Z
2 P
X +X > 2
#
(4.20)
ξ2 (t),
(4.21)
e2α(s−t) x˙ > (s)U x(s)ds, ˙
(4.22)
with ξ2 (t) = col x(t), x t − dτ , α > 0, U > 0. To guarantee that V > 0, we assume (4.9). Since t|t=tk = t − dτ |t=t , we see that Vˉ does not grow after the jumps. k Differentiating Vˉ , we have along (2.1) h−τ ˙ > > ˉ ˉ ˙ + x˙ (t) R + V (t) + 2α V (t) 6 2x (t)P x(t) U x(t) ˙ + 2α[x > (t)P x(t)] h Z Z 1 −2αh t > d − 2αh t d − e x˙ (s)R x(s)ds ˙ − e x˙ > (s)U x(s)ds ˙ h h t−h t− dτ + x > (t)Sx(t) − x > (t − h)e−2αh Sx(t − h) " # X +X > −X + X 1 d − 2α(h − τ ) > 2 − ξ2 (t) ξ2 (t) > h ∗ −X 1 − X > + X +X 1
2
τ i h−τ h > x˙ (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x t − . + h d
We employ the representation −
Z
t
>
t−h
x˙ (s)R x(s)ds ˙ =−
Z
t−τ t−h
>
x˙ (s)R x(s)ds ˙ −
Z
t− dτ t−τ
>
x˙ (s)R x(s)ds ˙ −
Z
t t− dτ
x˙ > (s)R x(s)ds. ˙
Similar to (4.11), applying the Jensen’s inequality and then denoting v21 =
d τ
Z
t t− dτ
x(s)ds, ˙
v22 =
1 h−τ
Z
t−τ t−h
x(s)ds, ˙
v23 =
d h(d − 1)
Z
t− dτ t−τ
x(s)ds, ˙
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
V (t, xt , x˙t ) = Vˉ (t) = V0 (xt , x˙t ) +
11 of 18
GENERAL SAWTOOTH DELAY
we obtain
Similar to (4.13), we insert free-weighting n × n-matrices by adding the following expressions to V˙ˉ : h i τ >i h τ τ 0 = 2 x > (t)Y1> + x˙ > (t)Y2> + x > t − − x(t) + x t − T1 + v21 , d d d 0 = 2[x > (t)Z 1> + x˙ > (t)Z 2> + x > (t − τ )T2> ][−x(t − τ ) + x(t − h) + (h − τ )v22 ], h τ h(d − 1) i + x(t − τ ) + v23 . 0 = 2[x > (t)M1> + x˙ > (t)M2> ] − x t − d d
Similar to (4.14), the same expression is added into the RHS of (4.2). Setting η2 (t) = col x(t), x(t), ˙ x(t − h), v21 , v22 , v23 , x t − dτ , x(t − τ ) , we obtain that V˙ˉ (t) + 2α Vˉ (t) 6 η2> (t)Ψ2 η2 (t) < 0,
(4.23)
if the inequality Ψ2 =
) > Ω11 − d−2α(h−τ (X + X > ) Φ12 + h−τ Z 1> 2h 2h (X + X ) > ∗ Φ22 + h−τ U Z h 2 ∗ −S e−2αh ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
τ > d Y1 τ > d Y2
0 Ω44 ∗ ∗ ∗ ∗
(h − τ )Z 1> (h − τ )Z 2>
h(d−1) > M1 d h(d−1) > M2 d
0 0 0 0 −2αh − h−τ 0 h Re −2αh ∗ − d−1 d Re ∗ ∗ ∗ ∗
Ω17
Ω18
Ω27 Ω28 0 T2 τ 0 d T1 0 (h − τ )T2 0 0 0 Ω77 > ∗ T2 + T2
P2 + P2> A + 2α P + S − Y1 − Y1> , d − 2α(h − τ ) (X − X 1 ), Ω17 = Y1> − M1> − T1 + h Ω18 = M1> − Z 1> + P2> A1 , h−τ Ω27 = Y2> − M2> − (X − X 1 ), h Ω28 = M2> − Z 2> + P3> A1 , τ R −2αh − 2αh d +U e Ω44 = − , e h d X + X > − 2X 1 − 2X 1> Ω77 = T1 + T1> − [d − 2α(h − τ )] . 2h
(4.25)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
h−τ ˙ > > ˉ ˉ U x(t) ˙ + 2α[x > (t)P x(t)] V (t) + 2α V (t) 6 2x (t)P x(t) ˙ + x˙ (t) R + h 2αh τ > R −2αh > −2αh d−1 v > Rv e − v21 + U e− d v21 − e−2αh h−τ 23 23 h v22 Rv22 − e d h d +x > (t)Sx(t) − x > (t −"h)e−2αh Sx(t − h) # X +X > d − 2α(h − τ ) > −X + X 1 2 − ξ2 (t) ξ2 (t) > h ∗ −X 1 − X 1> + X +X 2 h−τ > x˙ (t)(X + X > )x(t) + 2x˙ > (t)(−X + X 1 )x t − dτ . + h
12 of 18
K. LIU ET AL.
The latter inequality for τ → 0 and τ → h leads to the following LMIs: > Ψ21
X +X > Ω11 − d−2αh 2 2h (X + X ) Φ12 + ∗ Φ22 + U
∗
∗
∗
=
∗
d > 11 − 2h (X + X ) Φ12
∗ ∗ ∗
∗
∗
∗
Φ22 ∗ ∗ ∗
∗
∗
h Z 2> 0
0
∗
−R e−2αh
0
∗
∗
Z 2>
−S e−2αh
Ω28
0
T2
0
hT2
0
0
∗
h > d Y1 h > d Y2
h(d−1) > M1 d h(d−1) > M2 d
Ω17 |τ =h Ω27 |τ =h
Ω28
0
0
0
T2
0
h d T1
0
0
0
Ω44 |τ =h
∗
∗
∗
∗
∗
−2αh − d−1 d Re
∗
∗
∗
Ω18
Ω27 |τ =0
∗
∗ ∗
Ω17 |τ =0
−2αh − d−1 d Re
∗
Z 1>
h(d−1) > M1 d h(d−1) > M2 d
∗
∗
∗
∗
Ω
Ψ22
∗
∗
Z 2>
−S e−2αh
∗
∗
h Z 1>
Ω77 |τ =0 ∗
Ω77 |τ =h ∗
Ω18
0 T2 + T2>
Similarly, we can obtain that Ψ2 is also convex in τ ∈ [0, h]. We summarize the results in the following theorem.
0 T2 + T2>
< 0 and
(4.26)
< 0.
(4.27)
T HEOREM 4.2 (i) Given α > 0, 0 < d 6 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, Q > 0, X, X 1 , T, P2 , P3 , Yi and Z i (i = 1, 2) such that the LMIs (4.9), (4.18), (4.19) with notations given in (4.17) are feasible. Then, (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A1. (ii) Given α > 0, d > 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, X, X 1 , P2 , P3 , Ti , Yi , Z i and Mi (i = 1, 2) such that the LMIs (4.9), (4.26), (4.27) with notations given in (4.25) are feasible. Then, (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A1. R EMARK 4.1 LMIs of Theorem 4.2 with X = X 1 = U = 0 give sufficient conditions for exponential stability of (2.1) with τ (t) ∈ [0, h]: (i) for all slowly varying delays and (ii) for fast-varying delays. In the numerical examples, these conditions lead to the same results as the results of Park & Ko (2007), however, they posses a fewer number of slack matrices. 4.3 Exponential stability: Case A2 For τ˙ = −d < 0, we also differ between 0 < d 6 1 and d > 1. (i) When 0 < d 6 1, we are looking for the functional of the form: V (t, xt , x˙t ) = Vˉ (t) = V0 (xt , x˙t ) + where V0 is given by (4.5) and τ V1 (t, xt , x˙t ) = ξ3> (t) h
"
X +X > 2
∗
2 P
i=1
Vi (t, xt , x˙t ) + V3 (xt ),
−X + X 1
−X 1 − X 1> +
X +X > 2
#
ξ3 (t),
(4.28)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
=
Z 1>
13 of 18
GENERAL SAWTOOTH DELAY
τ V2 (t, xt , x˙t ) = h Z V3 (xt ) =
Z
t t−(h−τ )
t
e2α(s−t) x˙ > (s)U x(s)ds, ˙
e2α(s−t) x > (s)Qx(s)ds,
t−(h−τ )
V (t, xt , x˙t ) = Vˉ (t) = V0 (x t , x˙t ) +
2 P
i=1
Vi (t, xt , x˙t ),
(4.29)
where V0 is also given by (4.5) and " # X +X > −X + X 1 τ > 2 V1 (t, xt , x˙t ) = ξ4 (t) ξ4 (t), > h −X 1 − X 1> + X +X ∗ 2 Z τ t V2 (t, xt , x˙t ) = e2α(s−t) x˙ > (s)U x(s)ds, ˙ h t− h−τ d , α > 0, U > 0 and (4.9) also satisfied to guarantee that V > 0. with ξ4 (t) = col x(t), x t − h−τ d ˉ Also in this case, V does not grow after the jumps. Similar to Theorem 4.2, we obtain the following theorem. T HEOREM 4.3 (i) Given α > 0, 0 < d 6 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, Q > 0, X, X 1 , P21 , P31 , P22 , P32 , Yi j , Mi j , Z i j and Ti j (i, j = 1, 2) such that (4.9) and the following four LMIs: (4.30), for τ → 0 and τ → h2 , and (4.31), for τ → h2 and τ → h, Θ
11
∗
τ (X + X > ) Θ12 + 2h
Θ22 + τh U
> M11
> τ Y11
> M12
> τ Y12
> (h − 2τ )Z 11
> τ M11
Θ17
> − M> Θ Θ18 + Z 11 19 11
> (h − 2τ )Z 12
> τ M12
Θ27
> − M> Θ28 + Z 12 12
0 0
∗
∗
−S e−2αh
0
0
0
0
0
∗
∗
∗
− τh [R + dU ]e−2αh
0
0
τ T11
0
0
∗
∗
∗
∗
Θ55
0
0
0
∗
∗
∗
∗
∗
− τh R e−2αh
(h − 2τ )T12
0
0
0
∗
∗
∗
∗
∗
∗
> T11 + T11
−T12
0
∗
∗
∗
∗
∗
∗
∗
> Θ88 + T12 + T12
Θ89
∗
∗
∗
∗
∗
∗
∗
∗
Θ99
< 0,
(4.30)
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
with ξ3 (t) = col{x(t), x(t − (h − τ ))}, α > 0, U > 0, Q > 0 and (4.9) also satisfied to guarantee that V > 0. We note that V3 is non-negative before the jump at t = tk and it becomes zero just after the jump (because t|t=tk = (t − (h − τ ))|t=tk ). The time-dependent terms V1 and V2 vanish before the jumps (because τ = 0) and after the jumps (because τ = h and thus x(t − (h − τ )) = x(t)). Hence, the condition limt→t − Vˉ (t) > Vˉ (tk ) holds. k (ii) For d > 1, the discontinuous Lyapunov functional is modified as follows:
14 of 18
τ (X + X > ) Ξ11 Ξ12 + 2h
∗
∗
∗ ∗
> Y21
∗
> Y22 −2αh −S e
∗
∗
Ξ22 + τh U ∗
∗
> (h − τ )Y21
> (2τ − h)Z 21
> (h − τ )M21 > (h − τ )M22
0
0
0
> (h − τ )Y22 −2αh − h−τ h Re
0
0
0
Ξ66
0
∗
∗
∗
∗
∗
∗
> −T21 − T21
∗
∗
∗
0
(h − τ )T21
∗ ∗
0
0
∗
∗
0
T21
0
∗
∗
Ξ27
0
∗
∗
∗
∗
where > A + 2α P + S + Q − Y − Y > − Θ11 = A> P21 + P21 11 11
> + A> P − Y , Θ12 = P − P21 31 12
> − Z> Θ Θ18 + M21 19 21 > − Z> Θ28 + M22 22
− 2τh−h R e−2αh
∗
∗
Ξ17
∗
∗
∗
> (2τ − h)Z 22
∗ ∗
(2τ − h)T22 0
0 0
T22
0
> Θ88 − T22 − T22
Θ89
∗
Θ99
< 0,
(4.31)
d − 2ατ (X + X > ), 2h
> − Z> − T + P> A , Θ17 = Y11 11 11 21 1 d − 2ατ (X − X 1 ), Θ18 = h τ Θ19 = (1 − d) (−X + X 1 ), h > + R, Θ22 = −P31 − P31 > − Z> + P> A , Θ27 = Y12 12 31 1 τ Θ28 = − (X − X 1 ), h h − 2τ [R + dU ]e−2αh , Θ55 = − h
X + X > − 2X 1 − 2X 1> Θ88 = −(1 − d)Q e−2αh − (d − 2ατ ) , 2h τ Θ89 = (1 − d) (X + X > − 2X 1 − 2X 1> ), 2hτ Θ99 = −(1 − d) U e−2αh , h > A + 2α P + S + Q − M − M > − d − 2ατ (X + X > ), > Ξ11 = A P22 + P22 21 21 2h > > Ξ12 = P − P22 + A P32 − M22 , > − Y > + P> A , Ξ17 = Z 21 21 22 1 > + R, Ξ22 = −P32 − P32
> − Y > + P> A , Ξ27 = Z 22 22 32 1 h−τ Ξ66 = − [R + dU ]e−2αh , h
are feasible. Then (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A2. (ii) Given α > 0, d > 1, let there exist n × n-matrices P > 0, R > 0, U > 0, S > 0, X, X 1 , P21 , P31 , P22 , P32 , Yi j , Mi j , Z i j and Ti j (i, j = 1, 2) such that (4.9) and the following four LMIs: (4.32), for h h , and (4.33), for τ → 1+d and τ → h, τ → 0 and τ → 1+d
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
K. LIU ET AL.
15 of 18
GENERAL SAWTOOTH DELAY
Υ
Θ22 + τh U
∗
> τ Y11
> (h − h−τ d )M11 > (h − h−τ d )M12
Θ17
> τ Y12
h−(1+d)τ > Z 11 d h−(1+d)τ > Z 12 d
> M12
0
0
0
0
∗
Σ44
0
0
τ T11
∗
∗
∗
∗
Σ55
0
0 0
0
∗
∗
∗
∗
Σ66
∗
∗
∗
∗
∗
> T11 + T11
−T12
∗
∗
∗ ∗
∗
11
∗
τ (X + X > ) Ξ12 + 2h
Ξ22 + τh U
∗
> Y21 > Y22
∗ ∗
Ξ17
> (h − τ )Y22
0
0
0
T21
0
(h − τ )T21
0
0
−2αh − h−τ h Re
∗
∗
∗
∗
0 R e−2αh − h1 (1+d)τ−h d
∗
∗
∗
∗
∗
∗
∗ ∗
∗ ∗
> Σ88 +T12 +T12
h−τ M > d 21 h−τ M > d 22
∗
∗
0
h−(1+d)τ T12 d
(1+d)τ−h > Z 21 d (1+d)τ−h > Z 22 d
−S e−2αh
∗
0
> (h − τ )Y21
∗
∗
∗
∗ ∗
∗ ∗
> − M> Θ28 + Z 12 12
Θ27
−S e−2αh
∗
∗
> −M > Θ18 + Z 11 11
∗
∗
∗
> M11
Ξ27
< 0,
> −Z > Θ18 + M21 21
0 0
(1+d)τ−h T22 d
0
0
∗
> −T21 −T21
T22
∗
> −Z> Θ28 + M22 22
Υ66 ∗
(4.32)
> Σ88 −T22 −T22
< 0,
(4.33)
where > A + 2α P + S − Y − Y > − d − 2ατ (X + X > ), Σ11 = A> P21 + P21 11 11 2h τ −2αh − 2αh + dU e d , Σ44 = − R e h 2αh 1 h − (1 + d)τ −2αh + dU e− d , Re Σ55 = − h d 1 h−τ Σ66 = − h− R e−2αh , h d X + X > − 2X 1 − 2X 1> Σ88 = −(d − 2ατ ) , 2h > A + 2α P + S − M − M > − d − 2ατ (X + X > ), Υ11 = A> P22 + P22 21 21 2h 1 h − τ −2αh 2αh − d Υ66 = − + dU e , Re h d
are feasible. Then (2.1) is exponentially stable with the decay rate α for all delays 0 6 τ 6 h satisfying A2. 5. Examples E XAMPLE 5.1 Consider the system from Yue et al. (2005) " # " # 0 1 0 x(t) ˙ = x(t) + K x(t − τ (t)), 0 −0.1 0.1 K = −[3.75 11.5]. The stability of this system was studied by many authors (see Naghshtabrizi et al., 2008 and the references therein). For the constant sampling, it was found in Naghshtabrizi et al. (2008) that the system
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
τ (X + X > ) Σ11 Θ12 + 2h
16 of 18
K. LIU ET AL.
E XAMPLE 5.2 We consider the following simple and much-studied problem (Papachristodoulou et al., 2007; Fridman, 2010): x(t) ˙ = −x(t − τ (t)).
(5.1)
It is well known that the equation x(t) ˙ = −x(t − τ ) with constant delay τ is asymptotically stable for τ < π/2 and unstable for τ > π/2, whereas for the fast-varying delay, it is stable for τ < 1.5 and there exists a destabilizing delay with an upper bound >1.5. The latter means that all the existing methods, that are based on time-independent Lyapunov functionals, corresponding to stability analysis of systems with fast-varying delays, cannot guarantee the stability for the samplings, which may be >1.5. It is easy to check, that the system remains stable for all constant samplings 2. Conditions of Naghshtabrizi et al. (2008) and of Mirkin (2007) guarantee asymptotic stability for all variable samplings up to 1.28 and 1.57, respectively. For different τ˙ , by applying our methods, we obtain the maximum value of h given in Table 2.We can also see that discontinuous terms of LKFs improve the performance and the I–O approach via Wirtinger-type inequality improves the result for d > 1, d ∈ N compared to time-dependent L–K approach. E XAMPLE 5.3 Consider the system (De Souza & Li, 1999) " # " # −2 0 −1 0 x(t) ˙ = x(t) + x(t − τ (t)). 0 −0.9 −1 −1 TABLE 1 Example 1: maximum value of h for different τ˙ h\τ˙ L–K approach X = X1 = U = 0
−1.2 1.14 1.04
−1 1.14 1.04
−0.9 1.11 1.04
0.5 1.24 1.04
0.9 1.61 1.04
1 1.69 1.04
1.1 1.34 1.04
h\τ˙ I–O approach
τ˙ ∈ N 1.3659
TABLE 2 Example 2: maximum value of h for different τ˙ h\τ˙ L–K approach X = X1 = U = 0
−1.2 1.41 1.33
−1 1.41 1.33
−0.9 1.36 1.33
0.5 1.47 1.33
0.9 1.89 1.33
1 1.99 1.33
1.1 1.61 1.33
1.2 1.54 1.33
h\τ˙ I–O approach
τ˙ ∈ N 1.57
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
remains stable for all constant samplings 1.7. Moreover, the above system with constant delay τ is asymptotically stable for τ 6 1.16 and becomes unstable for τ > 1.17. The latter means that all the existing methods, that are based on time-independent Lyapunov functionals, corresponding to stability analysis of systems with fast-varying delays, cannot guarantee the stability for the samplings with the upper bound >1.17. In Naghshtabrizi et al. (2008), the upper bound on the constant sampling interval that preserves the stability is found to be h = 1.3277, improving all the existing LMI-based results. For different τ˙ , by applying Lyapunov–Krasovskii(L–K) approach with α = 0 and I–O approach via Wirtinger-type inequality, we obtain the maximum value of h given in Table 1. Our results for X = X 1 = U = 0 coincide with the ones of Park & Ko (2007). We see that discontinuous terms of LKFs improve the performance. Moreover, the I–O approach via Wirtinger-type inequality improves the result for d > 1, d ∈ N compared to time-dependent L–K approach.
17 of 18
GENERAL SAWTOOTH DELAY
TABLE 3 Example 3: maximum value of h for different τ˙ h\τ˙ L–K approach X = X1 = U = 0
−1.2 1.95 1.86
−1 1.98 1.86
−0.9 1.90 1.86
0.5 2.34 2.33
0.9 2.06 1.87
1 2.53 1.86
1.1 1.91 1.86
h\τ˙ I–O approach
τ˙ ∈ N 1.57
1 1.64 1.06
1.1 1.26 1.06
h\τ˙ I–O approach
τ˙ ∈ N 0.8797
TABLE 4 Example 4: maximum value of h for different τ˙ −1.2 1.10 1.06
−1 1.11 1.06
−0.9 1.10 1.06
0.5 1.44 1.26
0.9 1.48 1.06
It is well known that this system is stable for constant delay τ 6 6.17, whereas in Park & Ko (2007), it was found that the system is stable for all fast-varying delays τ 6 1.86. For different τ˙ , by applying our methods, we obtain the maximum value of h given in Table 3. Our results for X = X 1 = U = 0 coincide with the ones of Park & Ko (2007). Also in this Example, the time-dependent terms of LKFs improve the performance. However, the I–O approach via Wirtinger-type inequality has not improved the results for d > 1, d ∈ N and the result is worse than the result for the fast-varying delays. E XAMPLE 5.4 Consider the system (Kharitonov & Niculescu, 2002): " # " # 0 1 0 0 x(t) ˙ = x(t) + x(t − τ (t)). −1 −2 −1 1 For different τ˙ , by applying our methods, we obtain the maximum value of h given in Table 4. R EMARK 5.1 Simulations in all the examples above show that our results are conservative (at least for d 6= 1) and that the value of maximum h that preserves the stability grows for growing d > 1. 6. Conclusions In this paper, two methods have been introduced to investigate delay-dependent stability problem for systems with sawtooth delay with constant τ˙ 6= 0. One method develops a novel I–O approach via a Wirtinger-type inequality. The result by Mirkin (2007) is recovered for τ˙ = 1 and for any integer τ˙ , the same maximum bound that preserves the stability is achieved. Another method improves stability criteria by constructing piecewise continuous (in time) Lyapunov functionals. Though the most essential improvement corresponds to τ˙ = 1, the time-dependent terms improve the results for all values of τ˙ . Funding Israel Science Foundation (754/10); China Scholarship Council. R EFERENCES D E S OUZA , C. E. & L I , X. (1999) Delay-dependent robust H∞ control of uncertain linear state-delayed systems. Automatica, 35, 1313–1321. F RIDMAN , E. (2001) New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Syst. Control Lett., 43, 309–319.
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
h\τ˙ Our method X = X1 = U = 0
18 of 18
K. LIU ET AL.
Downloaded from imamci.oxfordjournals.org at TEL AVIV UNIVERSITY on November 14, 2010
F RIDMAN , E. (2010) A refined input delay approach to sampled-data control. Automatica, 46, 421–427. F RIDMAN , E. & S HAKED , U. (2003) Delay-dependent stability and H∞ control: constant and time-varying delays. Int. J. Control, 76, 48–60. G U , K., K HARITONOV, V. & C HEN , J. (2003) Stability of Time-Delay Systems. Boston, MA: Birkhauser. H ARDY, G., L ITTLEWOOD , J. E. & P OLYA , G. (1934) Inequalities. Cambridge, UK: Cambridge University Press. H E , Y., WANG , Q. G., L IN , C. & W U , M. (2007) Delay-range-dependent stability for systems with time-varying delay. Automatica, 43, 371–376. H E , Y., W U , M., S HE , J. & L IU , G. (2004) Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst. Control Lett., 51, 57–65. K HARITONOV, V. & N ICULESCU , S. (2002) On the stability of linear systems with uncertain delay. IEEE Trans. Autom. Control, 48, 127-132. KOLMANOVSKII , V. & M YSHKIS , A. (1999) Introduction to the Theory and Applications of Functional Dierential Equations. Dordrecht, Netherlands: Kluwer Academic. L IU , K. & F RIDMAN , E. (2009) Discontinuous Lyapunov functionals for linear systems with sawtooth delays. Proceedings of the 8th IFAC Workshop on Time-Delay Systems, Sinaya, Romania. L OUISELL , J. (1999) New examples of quenching in delay differential equations having time-varying delay. Proceedings of the 5th European Control Conference. Karlsruhe, Germany. M ICHIELS , W., VAN A SSCHE , V. & N ICULESCU , S. (2005) Stabilization of time-delay systems with a controlled time-varying delays and applications. IEEE Trans. Automat. Control, 50, 493–504. M IRKIN , L. (2007) Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Trans. Automat. Control, 52, 1109–1112. NAGHSHTABRIZI , P., H ESPANHA , J. & T EEL , A. (2008) Exponential stability of impulsive systems with application to uncertain sampled-data systems. Syst. Control Lett., 57, 378–385. N ICULESCU , S. (2001) Delay Effects on Stability: A Robust Control Approach. Lecture Notes in Control and Information Sciences, vol. 269. London: Springer. PAPACHRISTODOULOU , A., P EET, M. & N ICULESCU , S. (2007) Stability analysis of linear systems with timevarying delays: delay uncertainty and quenching. Proceedings of the 46th IEEE CDC, New Orleans, USA. PARK , P. G. & KO , J. W. (2007) Stability and robust stability for systems with a time-varying delay. Automatica, 43, 1855–1858. R ICHARD , J. P. (2003) Time-delay systems: an overview of some recent advances and open problems. Automatica, 39, 1667–1694. Y UE , D., H AN , Q. L. & L AM , J. (2005) Networked-based robust H∞ control of systems with uncertainty. Automatica, 41, 640–644.
