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Stability analysis of linear time-varying systems: Improving conditions by adding more information about parameter variation
Systems & Control Letters 60 (2011) 338–343
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Stability analysis of linear time-varying systems: Improving conditions by adding more information about parameter variation Leonardo Amaral Mozelli a , Reinaldo Martinez Palhares b,∗ a
Campus Alto Paraopeba, Federal University of São João del-Rei, Rod. MG 443 Km 7, 36420-000, Ouro Branco, MG, Brazil
b
Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-010, Belo Horizonte, MG, Brazil
article
info
Article history: Received 27 July 2010 Received in revised form 20 January 2011 Accepted 22 February 2011 Available online 27 March 2011 Keywords: Robust stability Time-varying systems Parameter dependent Lyapunov functions LMIs
abstract In this paper a new Lyapunov function is proposed for stability analysis of linear time-varying systems. This new function carries more information regarding parameter variation leading to less conservative conditions. Using Finsler’s lemma and a suitable form to describe the high-order time-derivatives of the parameters, finite sets of LMIs are obtained which are progressively less conservative as a pair of parameters grow. Previous results can be seen as a special case and numerical examples are carried out for the sake of illustration. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nonlinearities, unknown parameters and unmodeled dynamics appear quite often when the issue is modeling physical systems. Therefore, robust stability analysis of time-varying uncertain systems is a major concern that has been handled with Lyapunov functions and LMIs over the past decades. When no bounds are available for the parameter variation strategies for time-invariant systems can be applied at the cost of some conservatism; see [1–5] and references therein. In the context of time-varying linear systems a usual approach to assess robust stability is to consider a Lyapunov function that depends on the time-varying parameters. In these approaches the Lyapunov functions require the information about the rate of parameter variation which can be included by means of its first time-derivative. In the literature such approaches based on this kind of strategy range from affine [6–8] to polynomial [9–11] parameter dependence. To the reader interested in such polynomial approaches is recommended reading [12]. More recently, [13] proposed an alternative quadratic in the state Lyapunov function with a special polynomial parameter dependence on the uncertain parameters, which was previously studied in the context of time-invariant polytopic linear systems [5]. An improvement of
∗
Corresponding author. Tel.: +55 31 3409 3457; fax: +55 31 3409 4850. E-mail addresses: [email protected] (L.A. Mozelli), [email protected] (R.M. Palhares). 0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.02.010
the approach in [13] is the fact that high-order time-derivatives of the states are considered and indirectly the high-order timederivatives of the parameters are also taken into account leading to less conservative results. In this paper a new parameter dependent Lyapunov function is proposed. Unlike in [13], the information regarding high-order time-derivatives of the parameters can be explicitly incorporated into the Lyapunov function, a proposal in this paper. Since the high-order time-derivatives of the states can also be considered in our approach some results in [13] can be seen as a special case. Starting from this new Lyapunov function, a first step is to use Finsler’s lemma to formulate new sufficient stability conditions (Theorem 1). In what follows, choosing a suitable form to represent the successive high-order time-derivatives of the time-varying parameters, finite sets of LMIs to check the new stability conditions are presented (Theorem 2). In these new LMI tests two parameters can be combined to progressively achieve less conservative conditions, whereas in [13] only the parameter related to the high-order of the states is available. Numerical experiments are presented to illustrate the less conservative results. 2. Problem formulation In this paper we study the robust stability of time-varying continuous-time linear systems in the form x˙ (t ) = A(α(t ))x(t ),
(1)
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
where x(t ) ∈ Rn is the state vector, α(t ) ∈ RN is the vector of timevarying unknown parameters with elements belonging to the unit simplex
ΛN , λ ∈ R : λi ≥ 0, c λi (t ) = 1 , N
T
(2)
T
where c = [1 1 · · · 1]. Given constant real matrices A1 , A2 , . . . , AN of proper dimension system (1) can be described in a polytopic fashion as A(α(t )) =
N −
αi (t )Ai .
i=1
Considering the notation for time-derivatives x(j) (t ) ,
dj x(t ) dt j
we assume that the uncertain parameters are differentiable α(t ) ∈ C L , α(t ) ∈ ΛN with bounded time-derivatives
|αi(l) (t )| ≤ rli ,
rli ∈ R+ , l = 1, 2, . . . , L, i = 1, 2, . . . , N .
Once we have the constraint c T α(t ) = 1, these successive timederivatives belong to closed polyhedral sets denoted by
ΩMl , co hl1 , hl2 , . . . , hlMl = hli ∈ RN : |hlij | ≤ rli , c T hli = 0,
(3)
∀i = 1, 2, . . . , Ml , ∀j = 1, 2, . . . , N } in such a manner that Ml −
α (l) (t ) ∈ ΩMl =
βli hli .
(4)
These matrices have the following structure P11 (α (j) )
P12 (α (j) )
j (j) P21 (α ) P (α ) , .. .
P22 (α (j) )
Pκ 1 (α (j) ) j
where Aκ (α, . . . , α T
Aκ (α, . . . , α T
V = x¯ T(κ−1) Pη x¯ (κ−1)
(5)
x¯ T(κ−1)
(2)
η
(η)
) + P (α ) + · · · + P (α ),
(6)
x(κ−1)
For this function, P 0 (α) is an uncertain time-varying symmetric matrix satisfying P 0 (α) =
N −
αi Pi0 ,
i=1
(j)
and P (α ) ∀j = 1, 2, . . . , η are also symmetric matrices depending linearly on the successive time-derivatives of α , i.e., j
(j)
P (α ) , j
N − dj αi i=1
j P dt j i
.
j
.. . j Pκκ (α (j) )
.
(7)
) is obtained recursively for κ ≥ 1 as
) Aκ−1 (α, . . . , α (κ−2) )
× Aκ−1 (α, . . . , α
0n×(κ−1)n
(κ−2)
)A(α) +
In
(1) Aκ−1 (α, . . . , α (κ−2) )
with AT0 (α) , In . A new time-varying robust stability condition is given in the following. As pointed out in [13, Lemma 1] (8)
where Bκ is defined in (14). This is a key property used in the proof of the following theorem. As occurs in [13], Finsler’s lemma is used [14,15].
.
P2κ (α (j) )
ATκ (α, . . . , α (κ−1) ) = B⊥ κ,
where 2
(κ−1)
(κ−1)
,
In the following we omitted some vectors dependency of time to avoid clutter. The new Lyapunov function proposed in this paper to assess the stability of (1) has the following structure
x (1) x x(2) , . . .
.. . j Pκ 2 (α (j) )
j
V (x(t )) = xT Xx
3. New Lyapunov function and new stability conditions
Pη , P (α) + P (α
j
P1κ (α (j) )
··· ··· .. . ···
When we set η = 0 this function reduces to the same case presented in [13]. The stability tests provided in [13] made use of more information about the time-varying uncertainty by taking high-order derivatives of the states, i.e., making κ bigger, which ultimately leads to less conservative stability conditions. This new function enables one to use another parameter (η) to make this information available without necessarily taking high-order timederivatives of the states. For instance, if we take κ = 1 we have a Lyapunov function quadratic in the states and affine in the parameters, as occurs in many approaches in the literature [7, 8], and only the first time-derivative of uncertainty is available due to the time-derivative of the Lyapunov function. If κ = 2 the first time-derivative appears twice: as usual due to the Lyapunov function and again due to a higher-order time-derivative of the state. In [13], the information of the second time-derivative appears only if κ ≥ 3. In the proposed function keeping κ = 1 and letting η = 1 the second time-derivative of the uncertainty will appear without the need to further increase κ . Therefore, we conjecture that combining different sets of values for κ and η a compromise between potentially less conservative results and numerical complexity could exist. Notice that the proposed Lyapunov function can be rewritten highlighting its direct dependency on the state vector
(1)
j
= xT ATκ−1 (α, . . . , α (κ−2) )Pη Aκ−1 (α, . . . , α (κ−2) )x
Therefore, in this paper we investigate if the origin is the stable fixed point of the uncertain system (1) for α(t ) ∈ ΛN and α (l) ∈ ΩMl , l = 1, 2, . . . , L.
1
j
(j)
j
i=1
0
339
Lemma 1 (Finsler). Let y ∈ Rq , Q ∈ Rq , and B ∈ Rp×q such that rank (B ) < q. The following statements are equivalent 1. 2. 3. 4.
yT Qy < 0, ∀B y = 0, x ̸= 0. B ⊥T QB ⊥ < 0. ∃µ ∈ R : Q − µB T B < 0. ∃F ∈ Rn×m : Q + F B + B T F T < 0.
Theorem 1. Given κ ≥ 1 and η ≥ 0, the origin x = 0 is a globally asymptotically stable fixed point for the uncertain linear system (1) if there exists a parameter dependent symmetric matrix Pη ∈ Rnκ×nκ given by (6) and any matrices F ∈ Rn(κ+1)×nκ , Y ∈ Rn(κ)×n(κ−1) such that the following conditions hold Pη > YB(κ−1) + BT(κ−1) Y T
(9)
Q(κ,η) + FBκ + BTκ F T < 0
(10)
340
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
A(α) (1) A(α ) A(α (2) ) A(α (3) ) Bκ , .. . A(α (κ−2) ) A(α
(κ−1)
−I A(α) 2A(α (1) ) 3A(α (2) )
0
···
−I A(α) 3A(α (1) )
0 −I A(α)
··· ··· .. .
0 (1) ς(κ−2)(κ−1) A(α ) (2) ς(κ−2)κ A(α )
··· ···
)
···
0 −I
ς(κ−1)(κ−1) A(α) (1) ς(κ−1)κ A(α )
.. . −I ςκκ A(α)
0 0 0 0
.. .
0
(14)
−I
Box I.
4. Finite-dimensional tests
with
Q(κ,η) , Wκ Ψ ⊗ Pη Wκ + MκT Υ ⊗ P˙ η Mκ [ ] [ ] 0 1 1 0 Ψ , , Υ , , T
1
0
[
1
Wκ ,
Wκ
0
]
Wκ2
Wκ1 , Iκ n
[
,
Mκ ,
0κ n×n ,
(11)
0
1
Wκ
] (12)
Mκ2 Wκ2 , 0κ n×n
Iκ n ,
Mκ2 , 0κ n×(κ+1)n
(13)
and B0 , 0n×n and Bκ is given as in Box I. where
ςij ,
j−1 i−1
=
j − 1! . (i − 1)!(j − i)!
(15)
Proof. Consider the candidate function (5) rewritten in the form (7). To guarantee its positiveness the kernel matrix X must be positive definite: 0 < X
H i , hi1
hi2
···
hiMl .
The element of the jth row and kth column is denoted by Hjki . A set of nested summations is denoted by
= ATκ−1 (α, . . . , α (κ−2) )Pη Aκ−1 (α, . . . , α (κ−2) ) T ⊥ = B⊥ κ−1 Pη Bκ−1 .
In order to obtain finite-dimensional tests to verify Theorem 1 a suitable form to represent the successive time-derivatives must be chosen. The form in (1)–(4) proposed by [8,16] to evaluate the first time-derivative captures the main features of these bounded time-derivatives and its choice for this paper. Other forms were also employed in previous works as in [6,17] where the vectors hli are chosen without satisfying the constraint c T hli = 0, leading to more computational burden. In [18,19], a single upper bound is selected, leading to conservative results with lower computational demand. In [8], a model of differential inclusions is proposed which is the least conservative. However, as the reader can follow in the discussion in [13, Section 4.2], the construction of such a model is very labored hindering analysis as successive time-derivatives are considered. In the sequence some notation is described. Consider the following matrices H i ∈ RN ×Ml that accumulate the vectors of a given polyhedral set defined by (3)
(16)
Then applying statements 2 and 4 from Lemma 1 it follows that (16) is equivalent to (9). Taking the time-derivative of (5) with respect to the system trajectory it follows that T
V˙ = x˙¯ κ−1 Pη x¯ κ−1 + x¯ Tκ−1 Pη x˙¯ κ−1 + x¯ Tκ−1 P˙ η x¯ κ−1 P˙ η = P 0 (α (1) ) + P 1 (α (2) ) + · · · + P η−1 (α (η) ) + P η (α (η+1) ). The time-derivative of V can be rewritten in a quadratic form taking an augmented vector that includes one more timederivative of the states as follows
Mi −
Wkii ,
M1 − M2 − k1 =1 k2 =1
ki =1 1≤i≤L
···
ML −
Wk11 Wk22 · · · WkLL .
(17)
kL =1
Theorem 2. Let the Lth time-derivative of α be available. Given 1 ≤ κ ≤ L + 1 and 0 ≤ η ≤ L − 1, then system (1) is asymptotically stable j if there exist symmetric matrices Pi ∈ Rnκ×nκ and any matrices F ∈ n(κ+1)×nκ nκ×n(κ−1) R and Y ∈ R such that the following conditions hold ∀i = 1, 2, . . . , N:
V˙ = x¯ Tκ Q(κ,η) x¯ κ
P¯ (i,η) > Y B¯ (i,κ−1) + B¯ T(i,κ−1) Y T ,
(18)
where Q(κ,η) is defined in (11) and
Q¯ (i,κ,η) + F B¯ (i,κ) + B¯ T(i,κ) F T < 0,
(19)
[ x¯ κ =
]
x¯ (κ−1) . x(κ)
As can be seen in the discussion on [13, Section 3.2] the following is true Bκ x¯ κ = 0, where Bκ is defined as in (14). According to Lemma 1 once condition (10) is satisfied it follows that statement 1 is also true thereby guaranteeing that V˙ < 0 which completes the proof. Remark 1. If we replace Y and F in Theorem 1 by matrices depending on α such as Y (α) and F (α) and set η = 0 such conditions fall in the same case proposed in Theorem 2 in [13].
Q¯ (i,κ,η)
, WκT Ψ ⊗ P¯ (i,η) Wκ + MκT Υ ⊗ P˙¯ (i,η) Mκ
(20)
where Wκ , Mκ , Ψ , Υ are the same as in (12) and (13) and P¯ (i,η) , Pi0 +
N −
η
η
Hjk1 1 Pj1 + · · · + Hjkη Pj
(21)
j =1
P˙¯ (i,η) ,
N − j =1
B¯ (i,κ) ,
N −
η+1
η
Hjk1 1 Pj0 + · · · + Hjkη+1 Pj
B˘ (i,κ)
j=1
with B˘ (i,κ) are given as in Box II.
(22)
(23)
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
B˘ (i,κ)
Ai Hjk1 1 Aj Hjk2 2 Aj Hjk3 3 Aj
, .. . (κ−2) Hjk Aj
−I
0
···
Ai 2Hjk1 1 Aj 3Hjk2 2 Aj
−I
0 −I Ai
Ai 3Hjk1 1 Aj
341
0 0 0 0
··· 0 −I
··· .. . 1
(κ−2) (κ−1) Hjk A (κ−1) j
0
··· 0
ς(κ−2)(κ−1) Hjk1 Aj
ς(κ−1)(κ−1) Ai
2 ς(κ−2)κ Hjk2 Aj
ς(κ−1)κ Hjk1 Aj
.. . −I
1
ςκκ Ai
(24)
−I
where ςij is given as in (15). Box II.
it follows that P˙ η assumes the form
These conditions must be satisfied for all kq such that if κ − 1 < η + 1 : k1 = 1, 2, . . . , M1 k2 = 1, 2, . . . , M2 kκ−1
Mq −
.. . = 1, 2, . . . , Mκ−1 .. .
Similar steps used to prove that (18) ⇒ (9) can be applied to condition (10) as follows: in (11) replace Pη and P˙ η by the left-hand side of (25) and by (26) respectively; in (10) replace He{FBκ } by the right-hand side of (25), considering κ and F instead of κ − 1 and Y , respectively. Finally, due to the positivity of α and βik it is sufficient i to guarantee LMIs (19) completing the proof.
if η + 1 < κ − 1 : k1 = 1, 2, . . . , M1 k2 = 1, 2, . . . , M2
5. Example 1
.. . = 1, 2, . . . , Mη+1 .. .
Consider the following system [13] x˙ (t ) =
kκ−1 = kL = 1, 2, . . . , ML ,
[
Proof. Recall condition (9):
i=1
N −
αj(1) Pj1 +
N −
j =1
αj(2) Pj2 + · · · +
N −
j=1
(η) η
αj Pj
j =1
> He YB(κ−1) . Replacing the successive time-derivatives of α by their bounds via the polyhedral form in (4), we have
− i=1
− αi Pi0 + βqkq Hjkq q Pjq kq =1 1≤q≤η
>
Mq
N
N − i =1
αi He
Mq −
kq =1 1≤q≤κ−1
˘ βqkq F B(i,κ−1)
(25)
according to notation defined in (17). Taking into account the positivity of α and βik it is sufficient that LMIs (18) hold ∀i = i 1, 2, . . . , N and for all vertices combinations to guarantee (9). Since P˙ η =
N − j=1
αj(1) Pj0 +
N − j =1
]
0
−2 − p(t )
0 A1 = −2
.. . kL = 1, 2, . . . , ML .
αi Pi0 +
[
1 x(t ), −1
where |˙p(t )| ≤ γ and p(t ) ∈ [0, ρ] is the time-varying uncertainty. This system can be represented in the form (1) with two vertices
if η + 1 = κ − 1 : k1 = 1, 2, . . . , M1 k2 = 1, 2, . . . , M2
N −
(26)
kq =1 1≤q≤η+1
kη+1 = kL = 1, 2, . . . , ML ,
kη+1
βqkq Hjkq q Pjq−1 .
αj(2) Pj1 + · · · +
N − j =1
(η+1) η
αj
Pj ,
]
1 , −1
[ A2 =
0
−2 − ρ
]
1 −1
and considering |α˙1 | = |α˙2 | = |˙p(t )|/ρ . The results obtained with Theorem 2 are indicated by T(κ,η) and are compared with the results of Theorem 2 from [13], indicated by C(κ) . To take into account the information about the second timederivative it is mandatory to use C(3) in the conditions proposed in [13]. Concerning this paper other options are available. If η = 1 we have T(1,1) , T(2,1) , T(3,1) . If η = 0, we fall back to the same case in [13], thus we are limited to T(3,0) . The objective in this example is to compute the largest value of γ for several values of ρ ∈ [4, 20]. The bound of the second time-derivative is considered as |α¨i | ≤ τ |α˙i |, i = 1, 2. In Fig. 1 we compare the results obtained by T(3,1) (solid line), T(3,0) (dashed line, ∗) and C(3) (dashed line, ◦). As expected (see Remark 1) conditions T(3,0) and C(3) give the same result. Also it can be noticed an improvement achieved by setting η = 1, i.e., by introducing more information about the second time-derivative. Fig. 2 depicts a comparison among different results obtained with Theorem 2: T(3,1) (solid line), T(2,1) (dashed line, ∗) and T(1,1) (dashed line, ◦). Noticed that until ρ ≈ 10 all conditions establish the same results. With the increase of ρ the performance of T(2,1) and T(1,1) degrades. Finally, Fig. 3 shows a comparison between the least computational demanding conditions from Theorem 2 and Theorem 3 in [13] that are able to take into account the second time-derivative information, i.e., T(1,1) (solid line) against C(3) (dashed line, ◦). Notice that until ρ ≈ 12 condition T(1,1) prevails whereas for ρ > 12 the opposite occurs. The computational demand is evaluated in terms of scalar variables (S), LMI rows (R) and solving time (normalized) considering ρ = 12, τ = 1.5, and γ = 16.8 as shown in Table 1.
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L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
Fig. 1. Stability analysis for example 1 when the information about the second time-derivative is available. In this case τ = 1.5 and the results of Theorem 2 with κ = 3 and η = 1 (solid line) and with κ = 3 and η = 0 (dashed line marked with ∗) are shown and compared to the results of Theorem 3 in [13] with κ = 3 (dashed line marked with ◦).
Fig. 4. Stability analysis for example 2 when the information about the second time-derivative is available. In this case τ = 0.5 and the results of Theorem 2 with κ = 3 and η = 1 (solid line) and with κ = 3 and η = 0 (dashed line marked with ∗) are compared to the result of Theorem 3 in [13] with κ = 3 (dashed line marked with ◦). Table 2 Computational demand of several methods in terms of scalar variables S, number of LMI rows R, and time taken to obtain solution (normalized) for example 2 when k = 1.3, τ = 0.5, and γ = 0.732. Method
Time
S
R
T(3,1) C(3) T(3,0) T(2,1) T(1,1)
3.321 2.950 2.930 1.011 1.000
756 522 522 344 312
1944 1944 1944 1440 1440
6. Example 2 A higher-order system is now considered, borrowed from [9,12] Fig. 2. Stability analysis for example 1 when the information about the second time-derivative is available. In this case τ = 1.5 and the results obtained by Theorem 2 with different values of κ and η are shown: κ = 3 and η = 1 (solid line); κ = 2 and η = 1 (dashed line marked with ∗); finally κ = 1 and η = 1 (dashed line marked with ◦).
x˙ (t ) = A(α(t ), k)x(t ),
α(t ) ∈ ΛN , α( ˙ t ) ∈ ΩM1
3
A(α(t ), k) =
−
αi (t )A¯ i (k),
A¯ i (k) , A0 + kAi
i =1
where the matrices are
A0
A1
A2 Fig. 3. Stability analysis for example 1 when the information about the second time-derivative is available. In this case τ = 1.5 and T(1,1) (solid line) is compared to C(3) (dashed line marked with ◦). Table 1 Computational demand of several methods in terms of scalar variables S, number of LMI rows R, and time taken to obtain solution (normalized) for example 1 when ρ = 12, τ = 1.5, and γ = 16.8. Method
Time
S
R
C(3) T(3,0) T(3,1) T(2,1) T(1,1)
2.331 1.881 1.325 1.018 1.000
114 114 156 72 64
88 88 88 64 64
A3
−2.4 0.7 = 0.5 −0.6 1.1 −0.8 = −1.9 −2.4 0.9 −3.4 = 1.1 −0.4 −1.0 2.1 = 0.4 1.5
−0.6 −2.1 2.4 2.9
−1.7 −2.6 −5.0 −2.0
3.1 −3.6 , −1.6 −0.6
−0.6 0.2 0.8 −3.1
−0.3 −1.1 −1.1 −3.7
3.4 −1.4 2.0 0.5
1.7 1.3 −1.5 2.3
−1.4 0.6 −1.4 0.9
−0.7 −0.1 1.3 0.4
−0.1 2.8 , 2.0 −0.1 1.5 1.4 , −3.4 1.5 −0.7 −2.1 . 0.7 −0.5
In [9,12] one of the parameters of this polytopic system is time-invariant. Instead we will considered in this example all parameters as time-varying. Following the ideas of the previous example the objective here is to find the maximum variation rate γ for different values of k, where |αi (t )| ≤ γ . The bound of the second time-derivative is also considered as |α¨ i | = τ |α˙ i |, i = 1, 2, 3. In Fig. 4, we show the main result concerning this example, namely the comparison among the conditions T(3,1) (solid line),
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
T(3,0) (dashed line, ∗) and C(3) (dashed line, ◦). Once again a less conservative result is found by setting η to one, exploiting the additional features of the new Lyapunov function proposed in this paper. Corroborating with Remark 1 conditions T(3,0) and C(3) give the same result. For the sake of brevity, the comparison between T(1,1) and C(3) is omitted because the results are very similar. Nevertheless we observe the same behavior seen in example 1. For a certain range of k condition T(1,1) prevails over C(3) which requires almost 3 times more computational demand than T(1,1) , check Table 2. This happens for k ∈ [1.1, 1.4]. For k ∈ [1.6, 2.0] the opposite occurs with C(3) being slightly better. In the remaining interval both conditions are tied up. References [1] G. Chesi, Time-invariant uncertain systems: a necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions, Automatica 46 (2010) 471–474. [2] J. Lavaeia, A.G. Aghdamb, Performance improvement of robust controllers for polynomially uncertain systems, Automatica 46 (2010) 110–115. [3] E.N. Goncalves, R.M. Palhares, R.H.C. Takahashi, R.C. Mesquita, New approach to robust D -stability analysis of linear time-invariant systems with polytopebounded uncertainty, IEEE Trans. Automat. Control 51 (10) (2006) 1709–1714. [4] E.N. Goncalves, R.M. Palhares, R.H.C. Takahashi, R.C. Mesquita, New strategy for robust stability analysis of discrete-time uncertain systems, Systems Control Lett. 56 (7) (2007) 516–524. [5] Y. Ebihara, D. Peaucelle, D. Arzelier, T. Hagiwara, Robust performance analysis of linear time-invariant uncertain systems by taking higher-order timederivatives of the state, in: Proc. 44th IEEE Conf. Decision and Control— European Control Conf., Seville, Spain, 2005, pp. 5030–5035. [6] V.F. Montagner, P.L.D. Peres, A new LMI condition for the robust stability of linear time-varying systems, in: Proc. 42nd IEEE Conf. Decision and Control, Hawaii, USA, 2003, pp. 6133–6138.
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[7] P. Gahinet, P. Apkarian, M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Trans. Automat. Control 41 (3) (1996) 436–442. [8] J.C. Geromel, P. Colaneri, Robust stability of time varying polytopic systems, Systems Control Lett. 55 (1) (2006) 81–85. [9] G. Chesi, A. Garulli, A. Tesi, A. Vicino, Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions, Automatica 43 (2) (2007) 309–316. [10] L. Jetto, V. Orsini, LMI conditions for the stability of linear uncertain polynomially time-varying systems, IEEE Trans. Automat. Control 54 (7) (2009) 1705–1709. [11] V.F. Montagner, R.C. Oliveira, P.L. Peres, P.A. Bliman, Stability analysis and gainscheduled state feedback control for continuous-time systems with bounded parameter variations, Internat. J. Control 82 (6) (2009) 1045–1059. [12] G. Chesi, A. Garulli, A. Tesi, A. Vicino, Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems, in: Lecture Notes in Control and Information Sciences, vol. 390, Springer, 2009. [13] R.C.L.F. Oliveira, M.C. de Oliveira, P.L.D. Peres, Special time-varying Lyapunov function for robust stability analysis of linear parameter varying systems with bounded parameter variation, IET Control Theory Appl. 3 (10) (2009) 1448–1461. [14] R.E. Skelton, T. Iwasaki, K.M. Grigoriadis, A unified algebraic approach to linear control design, Taylor & Francis, 1998. [15] M.C. de Oliveira, R.E. Skelton, On stability tests for linear systems, in: Proc. 15th IFAC Triennial World Congress, Barcelon, Spain, 2002. [16] G. Chesi, A. Garulli, A. Tesi, A. Vicino, Parameter-dependent homogeneous Lyapunov functions for robust stability of linear time-varying systems, in: Proc. 43rd IEEE Conf. Decision and Control, Paradise Island, Bahamas, 2004, pp. 4095–4100. [17] A. Jadbabaie, A reduction in conservatism in stability and L2 gain analysis of Takagi–Sugeno fuzzy systems, in: Proc. of IFAC World Congress, Beijing, China, 1999. [18] L.A. Mozelli, R.M. Palhares, F.O. Souza, E.M.A.M. Mendes, Reducing conservativeness in recent stability conditions of TS fuzzy systems, Automatica 45 (6) (2009) 1580–1583. [19] L.A. Mozelli, R.M. Palhares, E.M.A.M. Mendes, Equivalent techniques, extra comparisons, and less conservative control design for TS fuzzy system, IET Control Theory Appl. 4 (12) (2010) 2813–2822.
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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Stability analysis of linear time-varying systems: Improving conditions by adding more information about parameter variation Leonardo Amaral Mozelli a , Reinaldo Martinez Palhares b,∗ a
Campus Alto Paraopeba, Federal University of São João del-Rei, Rod. MG 443 Km 7, 36420-000, Ouro Branco, MG, Brazil
b
Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-010, Belo Horizonte, MG, Brazil
article
info
Article history: Received 27 July 2010 Received in revised form 20 January 2011 Accepted 22 February 2011 Available online 27 March 2011 Keywords: Robust stability Time-varying systems Parameter dependent Lyapunov functions LMIs
abstract In this paper a new Lyapunov function is proposed for stability analysis of linear time-varying systems. This new function carries more information regarding parameter variation leading to less conservative conditions. Using Finsler’s lemma and a suitable form to describe the high-order time-derivatives of the parameters, finite sets of LMIs are obtained which are progressively less conservative as a pair of parameters grow. Previous results can be seen as a special case and numerical examples are carried out for the sake of illustration. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nonlinearities, unknown parameters and unmodeled dynamics appear quite often when the issue is modeling physical systems. Therefore, robust stability analysis of time-varying uncertain systems is a major concern that has been handled with Lyapunov functions and LMIs over the past decades. When no bounds are available for the parameter variation strategies for time-invariant systems can be applied at the cost of some conservatism; see [1–5] and references therein. In the context of time-varying linear systems a usual approach to assess robust stability is to consider a Lyapunov function that depends on the time-varying parameters. In these approaches the Lyapunov functions require the information about the rate of parameter variation which can be included by means of its first time-derivative. In the literature such approaches based on this kind of strategy range from affine [6–8] to polynomial [9–11] parameter dependence. To the reader interested in such polynomial approaches is recommended reading [12]. More recently, [13] proposed an alternative quadratic in the state Lyapunov function with a special polynomial parameter dependence on the uncertain parameters, which was previously studied in the context of time-invariant polytopic linear systems [5]. An improvement of
∗
Corresponding author. Tel.: +55 31 3409 3457; fax: +55 31 3409 4850. E-mail addresses: [email protected] (L.A. Mozelli), [email protected] (R.M. Palhares). 0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.02.010
the approach in [13] is the fact that high-order time-derivatives of the states are considered and indirectly the high-order timederivatives of the parameters are also taken into account leading to less conservative results. In this paper a new parameter dependent Lyapunov function is proposed. Unlike in [13], the information regarding high-order time-derivatives of the parameters can be explicitly incorporated into the Lyapunov function, a proposal in this paper. Since the high-order time-derivatives of the states can also be considered in our approach some results in [13] can be seen as a special case. Starting from this new Lyapunov function, a first step is to use Finsler’s lemma to formulate new sufficient stability conditions (Theorem 1). In what follows, choosing a suitable form to represent the successive high-order time-derivatives of the time-varying parameters, finite sets of LMIs to check the new stability conditions are presented (Theorem 2). In these new LMI tests two parameters can be combined to progressively achieve less conservative conditions, whereas in [13] only the parameter related to the high-order of the states is available. Numerical experiments are presented to illustrate the less conservative results. 2. Problem formulation In this paper we study the robust stability of time-varying continuous-time linear systems in the form x˙ (t ) = A(α(t ))x(t ),
(1)
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
where x(t ) ∈ Rn is the state vector, α(t ) ∈ RN is the vector of timevarying unknown parameters with elements belonging to the unit simplex
ΛN , λ ∈ R : λi ≥ 0, c λi (t ) = 1 , N
T
(2)
T
where c = [1 1 · · · 1]. Given constant real matrices A1 , A2 , . . . , AN of proper dimension system (1) can be described in a polytopic fashion as A(α(t )) =
N −
αi (t )Ai .
i=1
Considering the notation for time-derivatives x(j) (t ) ,
dj x(t ) dt j
we assume that the uncertain parameters are differentiable α(t ) ∈ C L , α(t ) ∈ ΛN with bounded time-derivatives
|αi(l) (t )| ≤ rli ,
rli ∈ R+ , l = 1, 2, . . . , L, i = 1, 2, . . . , N .
Once we have the constraint c T α(t ) = 1, these successive timederivatives belong to closed polyhedral sets denoted by
ΩMl , co hl1 , hl2 , . . . , hlMl = hli ∈ RN : |hlij | ≤ rli , c T hli = 0,
(3)
∀i = 1, 2, . . . , Ml , ∀j = 1, 2, . . . , N } in such a manner that Ml −
α (l) (t ) ∈ ΩMl =
βli hli .
(4)
These matrices have the following structure P11 (α (j) )
P12 (α (j) )
j (j) P21 (α ) P (α ) , .. .
P22 (α (j) )
Pκ 1 (α (j) ) j
where Aκ (α, . . . , α T
Aκ (α, . . . , α T
V = x¯ T(κ−1) Pη x¯ (κ−1)
(5)
x¯ T(κ−1)
(2)
η
(η)
) + P (α ) + · · · + P (α ),
(6)
x(κ−1)
For this function, P 0 (α) is an uncertain time-varying symmetric matrix satisfying P 0 (α) =
N −
αi Pi0 ,
i=1
(j)
and P (α ) ∀j = 1, 2, . . . , η are also symmetric matrices depending linearly on the successive time-derivatives of α , i.e., j
(j)
P (α ) , j
N − dj αi i=1
j P dt j i
.
j
.. . j Pκκ (α (j) )
.
(7)
) is obtained recursively for κ ≥ 1 as
) Aκ−1 (α, . . . , α (κ−2) )
× Aκ−1 (α, . . . , α
0n×(κ−1)n
(κ−2)
)A(α) +
In
(1) Aκ−1 (α, . . . , α (κ−2) )
with AT0 (α) , In . A new time-varying robust stability condition is given in the following. As pointed out in [13, Lemma 1] (8)
where Bκ is defined in (14). This is a key property used in the proof of the following theorem. As occurs in [13], Finsler’s lemma is used [14,15].
.
P2κ (α (j) )
ATκ (α, . . . , α (κ−1) ) = B⊥ κ,
where 2
(κ−1)
(κ−1)
,
In the following we omitted some vectors dependency of time to avoid clutter. The new Lyapunov function proposed in this paper to assess the stability of (1) has the following structure
x (1) x x(2) , . . .
.. . j Pκ 2 (α (j) )
j
V (x(t )) = xT Xx
3. New Lyapunov function and new stability conditions
Pη , P (α) + P (α
j
P1κ (α (j) )
··· ··· .. . ···
When we set η = 0 this function reduces to the same case presented in [13]. The stability tests provided in [13] made use of more information about the time-varying uncertainty by taking high-order derivatives of the states, i.e., making κ bigger, which ultimately leads to less conservative stability conditions. This new function enables one to use another parameter (η) to make this information available without necessarily taking high-order timederivatives of the states. For instance, if we take κ = 1 we have a Lyapunov function quadratic in the states and affine in the parameters, as occurs in many approaches in the literature [7, 8], and only the first time-derivative of uncertainty is available due to the time-derivative of the Lyapunov function. If κ = 2 the first time-derivative appears twice: as usual due to the Lyapunov function and again due to a higher-order time-derivative of the state. In [13], the information of the second time-derivative appears only if κ ≥ 3. In the proposed function keeping κ = 1 and letting η = 1 the second time-derivative of the uncertainty will appear without the need to further increase κ . Therefore, we conjecture that combining different sets of values for κ and η a compromise between potentially less conservative results and numerical complexity could exist. Notice that the proposed Lyapunov function can be rewritten highlighting its direct dependency on the state vector
(1)
j
= xT ATκ−1 (α, . . . , α (κ−2) )Pη Aκ−1 (α, . . . , α (κ−2) )x
Therefore, in this paper we investigate if the origin is the stable fixed point of the uncertain system (1) for α(t ) ∈ ΛN and α (l) ∈ ΩMl , l = 1, 2, . . . , L.
1
j
(j)
j
i=1
0
339
Lemma 1 (Finsler). Let y ∈ Rq , Q ∈ Rq , and B ∈ Rp×q such that rank (B ) < q. The following statements are equivalent 1. 2. 3. 4.
yT Qy < 0, ∀B y = 0, x ̸= 0. B ⊥T QB ⊥ < 0. ∃µ ∈ R : Q − µB T B < 0. ∃F ∈ Rn×m : Q + F B + B T F T < 0.
Theorem 1. Given κ ≥ 1 and η ≥ 0, the origin x = 0 is a globally asymptotically stable fixed point for the uncertain linear system (1) if there exists a parameter dependent symmetric matrix Pη ∈ Rnκ×nκ given by (6) and any matrices F ∈ Rn(κ+1)×nκ , Y ∈ Rn(κ)×n(κ−1) such that the following conditions hold Pη > YB(κ−1) + BT(κ−1) Y T
(9)
Q(κ,η) + FBκ + BTκ F T < 0
(10)
340
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
A(α) (1) A(α ) A(α (2) ) A(α (3) ) Bκ , .. . A(α (κ−2) ) A(α
(κ−1)
−I A(α) 2A(α (1) ) 3A(α (2) )
0
···
−I A(α) 3A(α (1) )
0 −I A(α)
··· ··· .. .
0 (1) ς(κ−2)(κ−1) A(α ) (2) ς(κ−2)κ A(α )
··· ···
)
···
0 −I
ς(κ−1)(κ−1) A(α) (1) ς(κ−1)κ A(α )
.. . −I ςκκ A(α)
0 0 0 0
.. .
0
(14)
−I
Box I.
4. Finite-dimensional tests
with
Q(κ,η) , Wκ Ψ ⊗ Pη Wκ + MκT Υ ⊗ P˙ η Mκ [ ] [ ] 0 1 1 0 Ψ , , Υ , , T
1
0
[
1
Wκ ,
Wκ
0
]
Wκ2
Wκ1 , Iκ n
[
,
Mκ ,
0κ n×n ,
(11)
0
1
Wκ
] (12)
Mκ2 Wκ2 , 0κ n×n
Iκ n ,
Mκ2 , 0κ n×(κ+1)n
(13)
and B0 , 0n×n and Bκ is given as in Box I. where
ςij ,
j−1 i−1
=
j − 1! . (i − 1)!(j − i)!
(15)
Proof. Consider the candidate function (5) rewritten in the form (7). To guarantee its positiveness the kernel matrix X must be positive definite: 0 < X
H i , hi1
hi2
···
hiMl .
The element of the jth row and kth column is denoted by Hjki . A set of nested summations is denoted by
= ATκ−1 (α, . . . , α (κ−2) )Pη Aκ−1 (α, . . . , α (κ−2) ) T ⊥ = B⊥ κ−1 Pη Bκ−1 .
In order to obtain finite-dimensional tests to verify Theorem 1 a suitable form to represent the successive time-derivatives must be chosen. The form in (1)–(4) proposed by [8,16] to evaluate the first time-derivative captures the main features of these bounded time-derivatives and its choice for this paper. Other forms were also employed in previous works as in [6,17] where the vectors hli are chosen without satisfying the constraint c T hli = 0, leading to more computational burden. In [18,19], a single upper bound is selected, leading to conservative results with lower computational demand. In [8], a model of differential inclusions is proposed which is the least conservative. However, as the reader can follow in the discussion in [13, Section 4.2], the construction of such a model is very labored hindering analysis as successive time-derivatives are considered. In the sequence some notation is described. Consider the following matrices H i ∈ RN ×Ml that accumulate the vectors of a given polyhedral set defined by (3)
(16)
Then applying statements 2 and 4 from Lemma 1 it follows that (16) is equivalent to (9). Taking the time-derivative of (5) with respect to the system trajectory it follows that T
V˙ = x˙¯ κ−1 Pη x¯ κ−1 + x¯ Tκ−1 Pη x˙¯ κ−1 + x¯ Tκ−1 P˙ η x¯ κ−1 P˙ η = P 0 (α (1) ) + P 1 (α (2) ) + · · · + P η−1 (α (η) ) + P η (α (η+1) ). The time-derivative of V can be rewritten in a quadratic form taking an augmented vector that includes one more timederivative of the states as follows
Mi −
Wkii ,
M1 − M2 − k1 =1 k2 =1
ki =1 1≤i≤L
···
ML −
Wk11 Wk22 · · · WkLL .
(17)
kL =1
Theorem 2. Let the Lth time-derivative of α be available. Given 1 ≤ κ ≤ L + 1 and 0 ≤ η ≤ L − 1, then system (1) is asymptotically stable j if there exist symmetric matrices Pi ∈ Rnκ×nκ and any matrices F ∈ n(κ+1)×nκ nκ×n(κ−1) R and Y ∈ R such that the following conditions hold ∀i = 1, 2, . . . , N:
V˙ = x¯ Tκ Q(κ,η) x¯ κ
P¯ (i,η) > Y B¯ (i,κ−1) + B¯ T(i,κ−1) Y T ,
(18)
where Q(κ,η) is defined in (11) and
Q¯ (i,κ,η) + F B¯ (i,κ) + B¯ T(i,κ) F T < 0,
(19)
[ x¯ κ =
]
x¯ (κ−1) . x(κ)
As can be seen in the discussion on [13, Section 3.2] the following is true Bκ x¯ κ = 0, where Bκ is defined as in (14). According to Lemma 1 once condition (10) is satisfied it follows that statement 1 is also true thereby guaranteeing that V˙ < 0 which completes the proof. Remark 1. If we replace Y and F in Theorem 1 by matrices depending on α such as Y (α) and F (α) and set η = 0 such conditions fall in the same case proposed in Theorem 2 in [13].
Q¯ (i,κ,η)
, WκT Ψ ⊗ P¯ (i,η) Wκ + MκT Υ ⊗ P˙¯ (i,η) Mκ
(20)
where Wκ , Mκ , Ψ , Υ are the same as in (12) and (13) and P¯ (i,η) , Pi0 +
N −
η
η
Hjk1 1 Pj1 + · · · + Hjkη Pj
(21)
j =1
P˙¯ (i,η) ,
N − j =1
B¯ (i,κ) ,
N −
η+1
η
Hjk1 1 Pj0 + · · · + Hjkη+1 Pj
B˘ (i,κ)
j=1
with B˘ (i,κ) are given as in Box II.
(22)
(23)
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
B˘ (i,κ)
Ai Hjk1 1 Aj Hjk2 2 Aj Hjk3 3 Aj
, .. . (κ−2) Hjk Aj
−I
0
···
Ai 2Hjk1 1 Aj 3Hjk2 2 Aj
−I
0 −I Ai
Ai 3Hjk1 1 Aj
341
0 0 0 0
··· 0 −I
··· .. . 1
(κ−2) (κ−1) Hjk A (κ−1) j
0
··· 0
ς(κ−2)(κ−1) Hjk1 Aj
ς(κ−1)(κ−1) Ai
2 ς(κ−2)κ Hjk2 Aj
ς(κ−1)κ Hjk1 Aj
.. . −I
1
ςκκ Ai
(24)
−I
where ςij is given as in (15). Box II.
it follows that P˙ η assumes the form
These conditions must be satisfied for all kq such that if κ − 1 < η + 1 : k1 = 1, 2, . . . , M1 k2 = 1, 2, . . . , M2 kκ−1
Mq −
.. . = 1, 2, . . . , Mκ−1 .. .
Similar steps used to prove that (18) ⇒ (9) can be applied to condition (10) as follows: in (11) replace Pη and P˙ η by the left-hand side of (25) and by (26) respectively; in (10) replace He{FBκ } by the right-hand side of (25), considering κ and F instead of κ − 1 and Y , respectively. Finally, due to the positivity of α and βik it is sufficient i to guarantee LMIs (19) completing the proof.
if η + 1 < κ − 1 : k1 = 1, 2, . . . , M1 k2 = 1, 2, . . . , M2
5. Example 1
.. . = 1, 2, . . . , Mη+1 .. .
Consider the following system [13] x˙ (t ) =
kκ−1 = kL = 1, 2, . . . , ML ,
[
Proof. Recall condition (9):
i=1
N −
αj(1) Pj1 +
N −
j =1
αj(2) Pj2 + · · · +
N −
j=1
(η) η
αj Pj
j =1
> He YB(κ−1) . Replacing the successive time-derivatives of α by their bounds via the polyhedral form in (4), we have
− i=1
− αi Pi0 + βqkq Hjkq q Pjq kq =1 1≤q≤η
>
Mq
N
N − i =1
αi He
Mq −
kq =1 1≤q≤κ−1
˘ βqkq F B(i,κ−1)
(25)
according to notation defined in (17). Taking into account the positivity of α and βik it is sufficient that LMIs (18) hold ∀i = i 1, 2, . . . , N and for all vertices combinations to guarantee (9). Since P˙ η =
N − j=1
αj(1) Pj0 +
N − j =1
]
0
−2 − p(t )
0 A1 = −2
.. . kL = 1, 2, . . . , ML .
αi Pi0 +
[
1 x(t ), −1
where |˙p(t )| ≤ γ and p(t ) ∈ [0, ρ] is the time-varying uncertainty. This system can be represented in the form (1) with two vertices
if η + 1 = κ − 1 : k1 = 1, 2, . . . , M1 k2 = 1, 2, . . . , M2
N −
(26)
kq =1 1≤q≤η+1
kη+1 = kL = 1, 2, . . . , ML ,
kη+1
βqkq Hjkq q Pjq−1 .
αj(2) Pj1 + · · · +
N − j =1
(η+1) η
αj
Pj ,
]
1 , −1
[ A2 =
0
−2 − ρ
]
1 −1
and considering |α˙1 | = |α˙2 | = |˙p(t )|/ρ . The results obtained with Theorem 2 are indicated by T(κ,η) and are compared with the results of Theorem 2 from [13], indicated by C(κ) . To take into account the information about the second timederivative it is mandatory to use C(3) in the conditions proposed in [13]. Concerning this paper other options are available. If η = 1 we have T(1,1) , T(2,1) , T(3,1) . If η = 0, we fall back to the same case in [13], thus we are limited to T(3,0) . The objective in this example is to compute the largest value of γ for several values of ρ ∈ [4, 20]. The bound of the second time-derivative is considered as |α¨i | ≤ τ |α˙i |, i = 1, 2. In Fig. 1 we compare the results obtained by T(3,1) (solid line), T(3,0) (dashed line, ∗) and C(3) (dashed line, ◦). As expected (see Remark 1) conditions T(3,0) and C(3) give the same result. Also it can be noticed an improvement achieved by setting η = 1, i.e., by introducing more information about the second time-derivative. Fig. 2 depicts a comparison among different results obtained with Theorem 2: T(3,1) (solid line), T(2,1) (dashed line, ∗) and T(1,1) (dashed line, ◦). Noticed that until ρ ≈ 10 all conditions establish the same results. With the increase of ρ the performance of T(2,1) and T(1,1) degrades. Finally, Fig. 3 shows a comparison between the least computational demanding conditions from Theorem 2 and Theorem 3 in [13] that are able to take into account the second time-derivative information, i.e., T(1,1) (solid line) against C(3) (dashed line, ◦). Notice that until ρ ≈ 12 condition T(1,1) prevails whereas for ρ > 12 the opposite occurs. The computational demand is evaluated in terms of scalar variables (S), LMI rows (R) and solving time (normalized) considering ρ = 12, τ = 1.5, and γ = 16.8 as shown in Table 1.
342
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
Fig. 1. Stability analysis for example 1 when the information about the second time-derivative is available. In this case τ = 1.5 and the results of Theorem 2 with κ = 3 and η = 1 (solid line) and with κ = 3 and η = 0 (dashed line marked with ∗) are shown and compared to the results of Theorem 3 in [13] with κ = 3 (dashed line marked with ◦).
Fig. 4. Stability analysis for example 2 when the information about the second time-derivative is available. In this case τ = 0.5 and the results of Theorem 2 with κ = 3 and η = 1 (solid line) and with κ = 3 and η = 0 (dashed line marked with ∗) are compared to the result of Theorem 3 in [13] with κ = 3 (dashed line marked with ◦). Table 2 Computational demand of several methods in terms of scalar variables S, number of LMI rows R, and time taken to obtain solution (normalized) for example 2 when k = 1.3, τ = 0.5, and γ = 0.732. Method
Time
S
R
T(3,1) C(3) T(3,0) T(2,1) T(1,1)
3.321 2.950 2.930 1.011 1.000
756 522 522 344 312
1944 1944 1944 1440 1440
6. Example 2 A higher-order system is now considered, borrowed from [9,12] Fig. 2. Stability analysis for example 1 when the information about the second time-derivative is available. In this case τ = 1.5 and the results obtained by Theorem 2 with different values of κ and η are shown: κ = 3 and η = 1 (solid line); κ = 2 and η = 1 (dashed line marked with ∗); finally κ = 1 and η = 1 (dashed line marked with ◦).
x˙ (t ) = A(α(t ), k)x(t ),
α(t ) ∈ ΛN , α( ˙ t ) ∈ ΩM1
3
A(α(t ), k) =
−
αi (t )A¯ i (k),
A¯ i (k) , A0 + kAi
i =1
where the matrices are
A0
A1
A2 Fig. 3. Stability analysis for example 1 when the information about the second time-derivative is available. In this case τ = 1.5 and T(1,1) (solid line) is compared to C(3) (dashed line marked with ◦). Table 1 Computational demand of several methods in terms of scalar variables S, number of LMI rows R, and time taken to obtain solution (normalized) for example 1 when ρ = 12, τ = 1.5, and γ = 16.8. Method
Time
S
R
C(3) T(3,0) T(3,1) T(2,1) T(1,1)
2.331 1.881 1.325 1.018 1.000
114 114 156 72 64
88 88 88 64 64
A3
−2.4 0.7 = 0.5 −0.6 1.1 −0.8 = −1.9 −2.4 0.9 −3.4 = 1.1 −0.4 −1.0 2.1 = 0.4 1.5
−0.6 −2.1 2.4 2.9
−1.7 −2.6 −5.0 −2.0
3.1 −3.6 , −1.6 −0.6
−0.6 0.2 0.8 −3.1
−0.3 −1.1 −1.1 −3.7
3.4 −1.4 2.0 0.5
1.7 1.3 −1.5 2.3
−1.4 0.6 −1.4 0.9
−0.7 −0.1 1.3 0.4
−0.1 2.8 , 2.0 −0.1 1.5 1.4 , −3.4 1.5 −0.7 −2.1 . 0.7 −0.5
In [9,12] one of the parameters of this polytopic system is time-invariant. Instead we will considered in this example all parameters as time-varying. Following the ideas of the previous example the objective here is to find the maximum variation rate γ for different values of k, where |αi (t )| ≤ γ . The bound of the second time-derivative is also considered as |α¨ i | = τ |α˙ i |, i = 1, 2, 3. In Fig. 4, we show the main result concerning this example, namely the comparison among the conditions T(3,1) (solid line),
L.A. Mozelli, R.M. Palhares / Systems & Control Letters 60 (2011) 338–343
T(3,0) (dashed line, ∗) and C(3) (dashed line, ◦). Once again a less conservative result is found by setting η to one, exploiting the additional features of the new Lyapunov function proposed in this paper. Corroborating with Remark 1 conditions T(3,0) and C(3) give the same result. For the sake of brevity, the comparison between T(1,1) and C(3) is omitted because the results are very similar. Nevertheless we observe the same behavior seen in example 1. For a certain range of k condition T(1,1) prevails over C(3) which requires almost 3 times more computational demand than T(1,1) , check Table 2. This happens for k ∈ [1.1, 1.4]. For k ∈ [1.6, 2.0] the opposite occurs with C(3) being slightly better. In the remaining interval both conditions are tied up. References [1] G. Chesi, Time-invariant uncertain systems: a necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions, Automatica 46 (2010) 471–474. [2] J. Lavaeia, A.G. Aghdamb, Performance improvement of robust controllers for polynomially uncertain systems, Automatica 46 (2010) 110–115. [3] E.N. Goncalves, R.M. Palhares, R.H.C. Takahashi, R.C. Mesquita, New approach to robust D -stability analysis of linear time-invariant systems with polytopebounded uncertainty, IEEE Trans. Automat. Control 51 (10) (2006) 1709–1714. [4] E.N. Goncalves, R.M. Palhares, R.H.C. Takahashi, R.C. Mesquita, New strategy for robust stability analysis of discrete-time uncertain systems, Systems Control Lett. 56 (7) (2007) 516–524. [5] Y. Ebihara, D. Peaucelle, D. Arzelier, T. Hagiwara, Robust performance analysis of linear time-invariant uncertain systems by taking higher-order timederivatives of the state, in: Proc. 44th IEEE Conf. Decision and Control— European Control Conf., Seville, Spain, 2005, pp. 5030–5035. [6] V.F. Montagner, P.L.D. Peres, A new LMI condition for the robust stability of linear time-varying systems, in: Proc. 42nd IEEE Conf. Decision and Control, Hawaii, USA, 2003, pp. 6133–6138.
343
[7] P. Gahinet, P. Apkarian, M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Trans. Automat. Control 41 (3) (1996) 436–442. [8] J.C. Geromel, P. Colaneri, Robust stability of time varying polytopic systems, Systems Control Lett. 55 (1) (2006) 81–85. [9] G. Chesi, A. Garulli, A. Tesi, A. Vicino, Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions, Automatica 43 (2) (2007) 309–316. [10] L. Jetto, V. Orsini, LMI conditions for the stability of linear uncertain polynomially time-varying systems, IEEE Trans. Automat. Control 54 (7) (2009) 1705–1709. [11] V.F. Montagner, R.C. Oliveira, P.L. Peres, P.A. Bliman, Stability analysis and gainscheduled state feedback control for continuous-time systems with bounded parameter variations, Internat. J. Control 82 (6) (2009) 1045–1059. [12] G. Chesi, A. Garulli, A. Tesi, A. Vicino, Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems, in: Lecture Notes in Control and Information Sciences, vol. 390, Springer, 2009. [13] R.C.L.F. Oliveira, M.C. de Oliveira, P.L.D. Peres, Special time-varying Lyapunov function for robust stability analysis of linear parameter varying systems with bounded parameter variation, IET Control Theory Appl. 3 (10) (2009) 1448–1461. [14] R.E. Skelton, T. Iwasaki, K.M. Grigoriadis, A unified algebraic approach to linear control design, Taylor & Francis, 1998. [15] M.C. de Oliveira, R.E. Skelton, On stability tests for linear systems, in: Proc. 15th IFAC Triennial World Congress, Barcelon, Spain, 2002. [16] G. Chesi, A. Garulli, A. Tesi, A. Vicino, Parameter-dependent homogeneous Lyapunov functions for robust stability of linear time-varying systems, in: Proc. 43rd IEEE Conf. Decision and Control, Paradise Island, Bahamas, 2004, pp. 4095–4100. [17] A. Jadbabaie, A reduction in conservatism in stability and L2 gain analysis of Takagi–Sugeno fuzzy systems, in: Proc. of IFAC World Congress, Beijing, China, 1999. [18] L.A. Mozelli, R.M. Palhares, F.O. Souza, E.M.A.M. Mendes, Reducing conservativeness in recent stability conditions of TS fuzzy systems, Automatica 45 (6) (2009) 1580–1583. [19] L.A. Mozelli, R.M. Palhares, E.M.A.M. Mendes, Equivalent techniques, extra comparisons, and less conservative control design for TS fuzzy system, IET Control Theory Appl. 4 (12) (2010) 2813–2822.