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2017
ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА Т. 13. Вып. 4 ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ_
ПРОЦЕССЫ УПРАВЛЕНИЯ
UDC 517.929
M. Gomez 1, A. V. Egorov 2, S. Mondié 1
A LYAPUNOV MATRIX BASED STABILITY CRITERION FOR A CLASS OF TIME-DELAY SYSTEMS*
1 CINVESTAV-IPN, 2508, Av. Instituto Politécnico Nacional, Mexico city, 07360, United Mexican States
2 St. Petersburg State University, 7—9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
This paper is devoted to the stability analysis of linear time-invariant systems with multiple delays. First, we recover some basic elements of our research. Namely, we introduce the complete type functionals, the delay Lyapunov matrix, and a space of special functions that allow to present a family of necessary stability conditions. Then, we prove a sufficient stability condition (instability condition) in terms of a quadratic Lyapunov—Krasovskii functional. Summarizing these results, we finally obtain an exponential stability criterion for a class of linear time-delay systems. The criterion requires only a finite number of mathematical operations to be tested and depends uniquely on the delay Lyapunov matrix. Refs 15.
Keywords: time-delay system, Lyapunov matrix, stability criterion.
1 СШУЕЯТАУЛР^ Мексиканские Соединенные Штаты,
07360, Мехико, Проспект Национального Политехнического института, 2508
2 Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7—9
Gomez Marco — postgraduate student; [email protected]
Egorov Alexey Valerievich — PhD of physical and mathematical sciences, associate professor; [email protected]
Mondie Sabine — PhD of physical and mathematical sciences, professor; [email protected]
Гомез Марко — аспирант; [email protected]
Егоров Алексей Валерьевич — кандидат физико-математических наук, доцент; [email protected]
Мондьё Сабин — кандидат физико-математических наук, профессор; [email protected]
* This work was supported by Project CONACYT N 180725 and by Russian Foundation for Basic Research (RFBR), research project N 16-31-00436 мол_a (Работа выполнена при финансовой поддержке CONACYT (проект № 180725) и Российского фонда фундаментальных исследований (проект № 16-31-00436 мол_a)).
© Санкт-Петербургский государственный университет, 2017
М. Гомез 1, А. В. Егоров 2, С. Мондьё 1
i
КРИТЕРИЙ УСТОЙЧИВОСТИ ДЛЯ ОДНОГО КЛАССА СИСТЕМ С ЗАПАЗДЫВАНИЕМ, ОСНОВАННЫЙ НА МАТРИЦЕ ЛЯПУНОВА*
*
Статья посвящена анализу устойчивости линейных стационарных систем с несколькими запаздываниями. Сначала мы вводим известные элементы, на которых базируется наше исследование, а именно: квадратичные функционалы Ляпунова—Красовского полного типа, матрицу Ляпунова для систем с запаздыванием и пространство некоторых функций, которые позволяют получить семейство необходимых условий устойчивости. Эти функции строятся на основе значений фундаментальной матрицы на некотором конечном отрезке. Затем доказываем достаточное условие устойчивости (условие неустойчивости), выраженное через специальный квадратичный функционал Ляпунова—Красовского: показываем, что для неустойчивой системы найдется точка из некоторого компактного бесконечномерного множества, значение функционала в которой отрицательно. Суммируя данные результаты, в итоге получаем критерий экспоненциальной устойчивости для некоторого класса линейных систем с запаздыванием. Проверка критерия требует лишь конечного числа математических операций, а проверяемые условия зависят только от матрицы Ляпунова для систем с запаздыванием. Библиогр. 15 назв.
Ключевые слова: система с запаздыванием, матрица Ляпунова, критерий устойчивости.
1. Introduction. The importance of the Lyapunov—Krasovskii functionals [1] in the stability analysis of time-delay systems is well known. The main ideas in the construction of this class of functionals lies in the early contributions by [2-4]. A decisive impulse was given in the past decade in [5], where the so-called Lyapunov—Krasovskii functionals of complete type were introduced. The main characteristics of this class of functionals is that they have a quadratic lower bound, and they are determined by the delay Lyapunov matrix, solution of the dynamic, symmetry and algebraic properties. For a deeper study on functionals of complete type and the delay Lyapunov matrix, the reader is referred to the book [6].
The availability of the analogous of the Lyapunov matrix for time-delay systems has allowed the extension of well-known results for delay free systems to the time-delay case (see, Section 2.12 in [6]). One of the most interesting problems is the extension of the stability criterion for linear systems, which depends on the positivity of the Lyapunov matrix V, solution of the Lyapunov equation ATV + V A = —W. The extension of this result to the time-delay case has been object of study in numerous contributions in the last years. See, for instance, the necessary stability conditions depending on the delay Lyapunov matrix introduced in [7] for systems with pointwise delays, in [8] for distributed delays and in [9] for the neutral-type case, also the stability criteria in [10, 11], for retarded and neutral type scalar equations, respectively.
A first criterion depending uniquely on the delay Lyapunov matrix systems with multiple pointwise delays was presented in [12] (see also [13]). However, the sufficiency part is only theoretical, as one has to make an infinite number of mathematical operations in order to check the stability condition (in this case, adopting the terminology used in [14], we say that the stability criterion is infinite). A first attempt to overcome this problem was recently reported in [14]. There, a new stability criterion depending on the delay Lyapunov matrix and the fundamental matrix of the system is introduced. The main distinctive feature of this criterion is that it is finite, i. e. a finite number of mathematical operations is required to test the stability condition. Nonetheless, the introduction of the fundamental matrix demands a greater computational effort in the test of the conditions.
Inspired by the results in [14], we present a new finite necessary and sufficient stability conditions for a class of retarded type systems depending uniquely on the delay Lyapunov matrix. The new criterion is constrained to systems with sufficiently small parameter values and delays. However, despite such restriction, it differs from the previously mentioned results in two aspects: (i) in contrast with [12], the stability criterion presented in this
contribution is finite and (ii) unlike [14], it does not require the computation of the fundamental matrix, as it only depends on the delay Lyapunov matrix.
The rest of the contribution is organized as follows. The Lyapunov—Krasovskii framework and some basic definitions are introduced in the next section. In Section 3 the problem formulation is presented. Some auxiliary results concerning instability conditions are provided in Section 4. In Section 5 the main result of the contribution is presented: necessary and sufficient stability condition for a class of retarded type systems. Finally we conclude with some remarks and future work.
Throughout the paper, PC ([—H, 0], Rn) and C(1) ([—H, 0], Rn) denotes the space of piecewise continuous and continuously differentiable vector functions defined on the interval [-H, 0], respectively. The maximum and minimum eigenvalues of a matrix Q are represented by Amax(Q) and Amin(Q), respectively. The notation Q > 0 means that the matrix Q is positive definite and || • || stands for the standard Euclidian vector norm. The square block matrix with j-th row and j-th column element Qij is represented by [Qij ]r j=r
2. Preliminary results. Consider the retarded type system
m
x(t) = ^2 Aix(t — hi), (1)
i=0
where 0 = h0 < h1 < ... < hm = H and Ai G Rnxn. The initial function ^ belongs to the space PC ([-H, 0], Rn). The restriction of the solution x(t,^>), t > —H, of system (1) (such that x(0,p) = y(0), 0 G [-H, 0]) on the interval [t — H,t], t > 0, is represented by
xt(^) : 0 ^ x(t + 0,V), 0 G [-H, 0]. We equip the set of piecewise continuous initial functions with the norm
Mh = sup M0)||.
ee[-H, o]
Definition 1. System (1) is exponentially stable if there exist constants n > 0 and a > 0 such that
Hx(t,V)H < ^Mh, t > 0.
The fundamental matrix of system (1), denoted by K, satisfies the equation
m
K(t) = £ AiK(t — hi), t> 0,
i=0
with the initial conditions K(0) = I and K(0) = 0 for 0 < 0.
Let us introduce the delay Lyapunov matrix.
Definition 2. The delay Lyapunov matrix U(t), t g [—H,H], associated with a symmetric matrix W is a continuous matrix function that satisfies the following equations:
m
U'(t)=y, U(t — hi)Ai, t G (0,H),
i=0
UT(t) = U(—t), t G [—H,H], U '(+0) — U'(—0) = —W,
called the dynamic, symmetry and algebraic properties, respectively. Here U'(+0) and U'(-0) denote the corresponding one-sided limits.
The existence and uniqueness of matrix U depends on the Lyapunov condition. This result is recalled in the next theorem (see [6]).
Theorem 1. The delay Lyapunov matrix U associated with a matrix W exists and is unique if and only if the Lyapunov condition holds, i. e. if the spectrum of system (1)
Л
det isI Ai€-shA =0
does not contain a root 's such that —'s also belongs to the spectrum. In [5] a functional, such that
^-v0(xt(p)) = -xT{t,p)Wx{t,p), W > 0, p G PC([—if, 0], R"), (2)
dt
is introduced and is determined by the delay Lyapunov matrix as follows:
m 0
v0(v) = fT(0)u(0)^(0) + 2<pT(0)£ / u(—hi — e)Ai^(e)de +
i=1-hi
0 0 mm /■ /■
+ ^^ / / fT(0i)ATU(0i — 02 + hi — hj)Ajf(02)d02d0i, i=1 j=1-hi -hj
f G PC ([—H, 0], R").
A functional based on v0 that plays a key role for obtaining necessary and sufficient stability condition depending on the delay Lyapunov matrix is presented next [7, 12]:
0
vi(f)= vo(f)+j fT (0)Wf(0)d0, f ePC ([—H, 0], R"). (3)
-H
Its derivative along the solutions of system (1) satisfies
^v(xt(<p)) = -xT(t - H, <p)Wx(t - H, cp), ^ e PC([-ff, 0], R"). We introduce now the following bilinear functional:
z(<p1,<p2) = pT (0)U (0)^2(0) +
0 0 m m
+ pT (0)£ / U(-hi - 0)Aip2(O)d0 + ^ / pT (0)AJ U (hi + 0)d0p2 (0) +
i=1-Îi i=1-h 0 0
mm
+ Y.Y, J J pT (01 )AT U (01 - 02 + hi - hj )Aj P2(02 )d02d0i + i=1 j=1-hi -hj 0
+ J pT(0)Wp2(0)d0, P1 ,P2 G PC ([-H, 0], Rn).
-H
s
Notice that this bilinear functional is related to the functional v1 by the equality v1(y) = z(y, y). In the next lemma (see [14]) we recall some results concerning the quadratic upper bounds of the functional v1 and the bilinear functional z.
Lemma 1. There exists a positive number 3 such that
V1 (y) < 3MH, y GPC ([—H, 0], Rn), \z(yhy2)\ < 3|y1 Uh ||y2||H, y1 ,y2 G PC ([—H, 0], Rn).
3. Problem statement. Consider the following function that depends on the fundamental matrix:
Vv(0} = X) K(Ti + в)л, в e [-H, 0],
(4)
where 7» e R™, n e [0, if], i = T~r.
Based on new properties that connect the fundamental matrix K with the delay Lyapunov matrix U, and the introduction of the functional z(:, •), the next equality has been provided in [7]:
vi(Vr} = [U(Tj - п}]1
Y
(5)
with y = (yt ... 7T) .
Take equidistant points from the segment [0,H]:
г - 1
r — 1
-H, i = 1, r.
For these points introduce block-matrix
Kr = [U(Tj - Ti}]
i,j=1
U
3 ~ » r1
H
i,j=1
£2 =
for r > 2 and separately K1 = U(0). To make the notation more clear, we present three particular cases:
U (0) U (H)
* U (0)
(U(0) U(H/2) U(H) \ K3 = ( * U(0) U(H/2) I \ * * U(0) )
(U(0) U(H/3) U(2H/3) U(H) \
U (0) U (H/3) U (2H/3)
* U (0) U (H/3)
\ * * * U(0) J
Stars here denote the blocks, which are obvious, as we deal with the symmetric matrices. In [12] an infinite stability criterion for system (1) is introduced. Theorem2. Assume that the Lyapunov condition holds. System (1) is exponentially stable if and only if for every natural number r, the following holds:
K4
Kr > 0.
(6)
r
r
The result is based on the fact that one can make an arbitrarily close approximation of any continuous function by the function ^r given by (4) for sufficiently large r. This criterion can be successfully used as a necessary stability condition to discard unstable systems, but checking of the stability requires an infinite number of mathematical operations, which leads us to formulate the problem addressed in this contribution.
Problem 1. Find an estimate of the number r for which condition (6) of Theorem 2 is necessary and sufficient for the stability of system (1). 4. Auxiliary results. Consider the compact set [14]
5 ={p eC(1) ([-H, 0], Rn) Mh = ||p(0)|| = 1, W(0)11 < M, 0 e [—H, 0]},
m
where M = ^^ H^H. We introduce some instrumental results that allow us to obtain the
i=0
main contribution of the paper. The first one concerns the approximation of a function p of the compact set S by the function ^r given by (4). This approximation has been introduced in [14] and is included here for the sake of completeness. The second one is inspired by [15] and corresponds to an instability condition of system (1) based on the functional v1.
4-1- Approximation of the set S. Consider an arbitrary initial function p from the set S and the function ^r of the form (4), which satisfies the following equalities:
Vv(-Tj) = i = l,r,
with Tj = (i — 1 )H/(r — 1), i = 1 ,r. Such function always exists [14], and vectors 7¿, i = 1 ,r, can be computed iteratively. The next lemma (see [14]) gives an estimate of the approximation error between p eS and , denoted by Rr = p — . Lemma 2. For every p eS,
IIRrlH = Hp — ^rIIh < £r,
where
_ (M + L)e
£T
LH
1/5r + L H
Here 6r = -- and L is the Lipschitz constant for the fundamental matrix on [0, if],
r — 1
i. e. it is such that HK'(i)|| ^ L on [0,H] almost everywhere.
4-2. Instability condition. The next lemma is useful for proving the main result of this section.
Lemma 3. Let P and Q be real matrices. If det(P + iQ) = 0, then there exist two vectors Ci and C2 such that
1) (P + iQ)(Ci + iC2) =0;
2) ||Ci|| = 1;
3) ||C2|| < 1;
4) Cf C2 = 0. Here i is the imaginary unit.
Proof. Since det(P + iQ) = 0, there exists a complex vector £1 + = 0, such that
(P + iQ) (£1 + i&) = 0.
Let us introduce now the following vectors:
Ci = £1 + b£2,
C2 = —b^1 + £2, where b is a real number, and observe that
(P + iQ) (C1 + ¿C2) = 0.
Consider the product
^T C2 = (1 — b2)&£2 — b (nan2 —1|6||2).
We need to choose b to guarantee that CTC2 = 0. If £T£2 = 0 one can take b = 0, otherwise, one can take any real solution of the quadratic equation, which always has two:
+ 1 = 0.
Now, at least one of the vectors CC1 and CC2 is nonzero and the desired vectors C1 and C2 can be constructed as follows: if ||Ci|| > ||C*2||, we have
<?1 = -S-, C2 d2
and if jjCljj < jjC2jj,
!|Cil! ll<^iH'
n n
— ,, - ,, ' °2 — —"
n^2U U^2U
□
We provide an instability condition for system (1) depending on the functional v1. The result is based on [15], where the condition is obtained for the functional v0. Lemma4. Assume that system (1) is such that
where the matrix W is the one to which the delay Lyapunov matrix is associated. If system (1) is unstable, there exists y GS, such that
v1(y) < —a1,
herea1 = ^f^l-\max(W)H>0.
P r o o f. As system (1) is unstable, there exist an eigenvalue A = a + i/3 with a > 0, and two vectors C1, C2 G Rn that satisfy the conditions of Lemma 3 such that the following expression is a solution of system (1):
x(t,y) = eatfi(t), &(t) = cos(3tC1 — sin f3tC2, t G (—<x>, <x>),
which corresponds to the initial function
y(0)= x(0,y), 0 G [—H, 0].
We prove first that p e S. By Lemma 3, it is easy to see that ||p(0)|| = 1 and
max ||^(t)|| = 1.
t G R
Indeed, Urnf = HCiH'2 + (||C2|2 — ||Ci||2)sin2ft < 1. Thus,
||x(e)|| = ||#0)|| < 1, e < 0.
Finally, since x(t, p) is a solution of system (1), we get
m
||p'(t)|| = ||x(t)|| ^ HAUnt — hi)! < M||p(0)| = M, t e [-H,0].
i=0
We focus next on the functional v0. Let us set T = 2n/f for f = 0 and T =1 for f = 0. It is easy to see that T is a period of ^>(t). By equation (2),
^v0{xt{ip)) = -xT{t, ip)Wx(t, ip), t > 0,
which implies that
T
vo(p) = vo(xt )+/xT (t,p)Wx(t,p)dt.
0
As x(T + 0) = eaTp(0),
vo(xf (p)) = e2aT vo (p),
hence,
T T
voM = /xT(t^)Wx(t^)dt < J \\x(t)\\2dt.
0 0
By Lemma 3,
T / T \ T
J ||x(t)||2dt = iy e2at (cos2(ft)HClH2 +sin2(ft)|C2|2) dt 1 ^ j e2at cos2(ft)dt.
0 0 0
Solving the right hand side integral, we obtain
T
/1 1
e2at cos2(f3t)dt = —(e2aT - 1) + --pr (e2aT(a cos(2/3T) + /3sin(2/3T)) - a) =
I f )
0
_ {2a2 + j32) {e2oT - I) e2aT-l ~ 4a(a2 + /32) ^ 4a '
Taking into account that the spectral abscissa can be estimated by M = ^™=0 ||Ai||, we arrive at
, Amin(W) ^ Amin(W) vo{<p) <--— <
Finally, from expression (3) for the functional v1 , it follows that
□
5. Necessary and sufficient conditions. We provide the main result of the contribution, namely, finite necessary and sufficient stability conditions for the system of the form (1) that satisfies inequality (7).
Theorem3. Assume that the Lyapunov condition holds and system (1) satisfies (7). System (1) is exponentially stable if and only if
for _
r = 1 + H (eLH(M + L)(a+ y/a{a + 1)) - i) , (8)
where a = ff/a1.
P r o o f. The necessity directly follows from Theorem 2. In order to prove the sufficiency, we assume by contradiction that system (1) is unstable but the Lyapunov condition and the conditions of the theorem hold. Consider y gS and notice that
v1 (y) = v1(Rr + ^r) = v1 (^r) + 2z(y, Rr) — v1(Rr). By Lemmas 1 and 4,
v1 (4>r) < —a1 + 2fURruh + fURruhh.
By using Lemma 2 and considering r given by (8), we have that
(M + L)eLH a1
RAh si , .„ . si
1 /Sr + L y/pip + aj + p' which implies that
—a1 + 2fURruh + fURrUH < 0. Finally, from the previous inequality and equation (5), we get
v1(^r) = YTKrY < 0.
The obtained contradiction finishes the proof. □
Remark. The number r given by (8) has the same form as the estimate obtained in [14]. The only difference lies in the computation of a as in this case a1 does not come from the quadratic lower bound of the functional v1, but from the instability result given in Lemma 4.
Although Theorem 3 partially solves Problem 1 by giving a finite stability criterion, it only works for a constrained class of systems of the form (1). However, in contrast with the finite stability criterion introduced in [14], here one does not need the computation of the fundamental matrix, which increases the computational effort and seems not to be a trivial task in the non-commensurate delays case. Obviously, the constraints introduced by inequality (7) can be relaxed if one considers a positive definite matrix W such that Amin(W) = Amax(W), or uses less conservative estimate on a1 in the proof of Lemma 4.
)Cr > 0
6. Conclusions. A new stability criterion for a class of time delay systems is introduced in the framework of the Lyapunov—Krasovskii functionals. The main characteristic of this criterion is that it depends exclusively on the delay Lyapunov matrix and only requires a finite number of mathematical operations to be checked. Future research directions include the generalization to systems without restrictions in the parameter values and the extension to the neutral-type case.
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For citation: Gomez M., Egorov A. V., Mondie S. A Lyapunov matrix based stability criterion for a class of time-delay systems. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2017, vol. 13, iss. 4, pp. 407-416. https://doi.org/10.21638/11701/ spbu10.2017.407
Статья рекомендована к печати проф. А. П. Жабко. Статья поступила в редакцию 2 сентября 2017 г. Статья принята к печати 12 октября 2017 г.
