Uniform Ultimate Boundedness of Lur’e Systems with Switchings and Delays Текст научной статьи по специальности «Медицинские технологии»

у. О

2024. Т. 49. С. 63-77

Онлайн-доступ к журналу: http: / / mathizv.isu.ru

Серия «Математика»

Research article

УДК 517.935.4 MSC 34K99, 34D40

DOI https://doi.org/10.26516/1997-7670.2024.49.63

Uniform Ultimate Boundedness of Lur'e Systems with Switchings and Delays

NatalyaR. Andriyanova1^

1 Saint Petersburg University, Saint Petersburg, Russian Federation И [email protected]

Abstract. The paper investigates a hybrid system consisting of Lur'e subsystems with constant delays and time-dependent switching. It is assumed that nonlinearities from the right side of the systems have degrees less than unity. An analysis of such a property of the system as the uniform ultimate boundedness of all its solutions is conducted. The linear part of the system is supposed to be asymptotically stable. As is known, this means that there is a correspondent homogeneous Lyapunov function. Using this function, a common Lyapunov-Krasovskii functional is constructed which makes it possible to find sufficient conditions for the uniform ultimate boundedness with arbitrary choices of positive delays and switching laws. Moreover, delays can occur during switching, for example, when generating feedback. The derived conditions are found to be less conservative in the case of asynchronous switching compared to synchronous ones. The validity of the theoretical results is confirmed through numerical modeling.

Keywords: uniform ultimate boundedness, delay, synchronous and asynchronous switching

Acknowledgements: My special thanks are due to my research supervisor, Prof. Alexander Aleksandrov for his support.

For citation: AndriyanovaN. R. Uniform Ultimate Boundedness of Lur'e Systems with Switchings and Delays. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 49, pp. 63-77. https://doi.org/10.26516/1997-7670.2024.49.63

Научная статья

Равномерная предельная ограниченность решений систем Лурье с переключениями и запаздываниями

Н. Р. Андриянова1®

1 Санкт-Петербургский государственный университет, Санкт-Петербург, Российская Федерация ® [email protected]

Аннотация. Исследуются гибридные системы, состоящие из подсистем Лурье с постоянными запаздываниями и переключениями во времени. Предполагается, что нелинейности из правых частей систем имеют степени меньшие единицы. Проводится анализ такого свойства системы, как ограниченность всех ее решений. Линейная часть системы является общей для всех подсистем и предполагается асимптотически устойчивой. Как известно, это означает, что существует соответствующая однородная функция Ляпунова. С помощью этой функции строится общий для всех подсистем функционал Ляпунова - Красовского, позволяющий найти достаточные условия равномерной предельной ограниченности решений при произвольном выборе положительных запаздываний и законе переключений. Более того, при формировании обратной связи могут возникать задержки, ведущие к возникновению запаздывания в законе переключений. Установлено, что полученные условия в случае таких асинхронных переключений оказываются менее жесткими, чем при синхронных. Достоверность теоретических результатов подтверждена посредством численного моделирования.

Ключевые слова: равномерная предельная ограниченность решений, запаздывание, синхронное и асинхронное переключения

Благодарности: Автор выражает благодарность научному руководителю, профессору Александру Александрову за поддержку.

Ссылка для цитирования: АндрияноваН. Р. Равномерная предельная ограниченность решений систем Лурье с переключениями и запаздываниями // Известия Иркутского государственного университета. Серия Математика. 2024. Т. 49. С. 6377.

https://doi.org/10.26516/1997-7670.2024.49.63

1. Introduction and problem statement

Consider a hybrid system with positive delays r and h

x(t) = By(t) + Af) f (x(t)) + Af) f (x(t - r)), y(t) = Dy(t) + Cf(t)f (x(t)) + C^(t)f (x(t - h)),

where x(t) is an n-dimensional and y(t) is v-dimensional vectors, matrix parameters have the corresponding dimensions. Vector function f (x) = (x^1 ,..,xn)T has positive rational degrees with odd denominators and numerators. Without loss of generality, assume that < ... < The matrix coefficients preceding the nonlinearities change over time. This alteration is governed by a switching law a(t) — a piecewise-constant right continuous function which maps the time interval [0, +rc>) into the

set {1,..., N} of subsystems' numbers. Moreover, within each finite time period no more than a finite number of switchings can occur.

At each moment a subsystem of Lur'e-type is obtained [5; 7]. Such systems are particularly used in the theory of automatic system control and neural networks [10].

To get a solution (xT(t, t0, <p); yT(t, t0, y0))T of the system (1.1) we must specify an initial moment t0, value y0 = y(t0), and function ^ from a set of piece-wise continuous on [—t;0], t = max{r, h} functions with values in Rn. For the differential-difference systems the state is denoted by xt(0) = x(t + 0), where 0 € [—t ;0]. The uniform norm of some state ^ is defined with the aid of euclidean one as follows: ||^>||r = sup0e[-r;0] ||^>(0)||.

The purpose of this work is to find the conditions for the system (1.1) to be uniformly ultimately bounded.

Definition 1. [8] The system is uniformly ultimately bounded when there is R > 0 such that for any q > 0 there exists T(q) > 0 for which ||x(t, t0, ||2 + ||y(t,t0, y0)||2 < R2 will hold whenever ||^|2 + |M|2 < q2 for all t > t0+T, t0 > 0.

Notice that delays in the system can occur both in nonlinear functions and switching law, e.g., while generating feedback or receiving information about the moment, when the current subsystem was changed. Therefore along with the system (1.1), a system with asynchronous switching [11-13]

x(t) = By(t) + Af) f (x(t)) + Af-r) f (x(t — r)) (1 2)

y(t) = Dy(t) + Cf(i) f (x(t)) + C2(t-h) f (x(t — h))

will also be investigated. Here a(t) must be defined on [—t,

2. Analysis of the system with synchronous switching

In [1] a Lur'e system with homogeneous nonlinearities, delays, and asynchronous switching is studied via Lyapunov-Krasovskii approach. The found conditions of stability and ultimate boundedness are independent of the delay and the switching law. In this paper similar results are extended to the case of multiple homogeneous nonlinearities.

Assumption 1. Let ^n < 1. Suppose also that there is a positive definite diagonal matrix A such that for all s,p, q = 1,..., N the inequalities

fT A[4s) + 4p) - BD-1(cf + cf)]/ < - £ ß«xf(fc+1) (2.1)

fT A[4((S^^2 — uu -r ^2 >\J - ~ / ^0

i=1

are valid; here vector fa = {x^1,... ,xTn)T, rational k > — has odd

(i)

numerator and denominator, and ^0 > 0, i = 1,..., n.

Remark 1. For condition (2.1) to be satisfied it is enough to require for the matrix in its left side to be Hurwitz and Metzler [2].

Theorem 1. If assumption 1 is fulfilled and D is Hurwitz matrix, when the system (1.1) with any positive fixed delays is uniformly ultimately bounded.

Proof. Since the system y(t) = Dy(t) is asymptotically stable, a homogeneous Lyapunov function V0(y) of order m0 > 1 exists [14]. With a matrix A = diag(Ai,..., An) from Assumption 1, the Lyapunov-Krasovsky functional can be chosen:

fXi(t) M

H,y)=7 Ai/ dt - fTABD-1 C2

r-Xi(l) rU

V(xt,y) = Y, M - fl^BD-1 cft+e+h) f(x(t + 9))d9+

i=1 JU J-h

f U

+Vu(y) - fJABD-1y + / (в1 + Yi(r + d))\\f (x(t + 9))\\mid0+

J — r

' —r

rU M

+ffcTA / Af+e+r)f (x(t + 0))dJ9 + (f32 + 72 (h + 6))\\f (x(t + e))\\m1 de, J-r J-h

(2.2)

where m1,fi1,fi2,Y1,72 are any positive numbers. Some approaches to constructing this type of functionals can be found in [3; 4].

We denote (i,j) element of matrix M by (M)j, and its i-th row by (M )i.

To check whether (2.2) is positive, one can write down its lower bound using Holder's inequality for the second and third integrals. This yields

nn

V(xt,y) a^i+1(t)+ an+i\\y(t)\P - £ i=1 i=1

а'г\хг{Щк^ ( ||/(*(* + 0))\Г<Ю) ^ -

m1-l П / fO \

- h~ j h ii f{x{t+в))\\тчв) +

/U r- U

\\f (x(t + e))\\m1 dd + в J \\f (x(t + e))\\m1 dd -r J—h

UU

—h where

д. I v

ai = T,—jTT' an+l = ,P\}n > 1 =

+ 1 ПуП=1 1Mb11 j=

ai = max ||(AAj))iII, ai = - max ||(ABD—1C^j))iII, i = l,...,n. j=1,...,w j=1,...,N

Now introduce the vector

o=( 1 1 J_ J_ J_

I fc^l+l' ' ' ' ' k^n + V mo' mi ' mi

and define a generalized homogeneous norm = Iz%ISi [6] for

vector

t_

Z — I X1; Xn ü

/•0

11/(x(t+0))||mi dö

' —r

1

m

0

/ 11/(*(t+0))ir dö '—h

l

m-j^

Hence, the latest functional assessment can be expressed in terms of vector s:

n+1

J_ " i

n+3

i=1 i=1

1 n . n

m 1-1 . , -J--1 ml-1 , -J--1

-r mi zra+3.

i=1 i=1

Here nonnegative terms can be estimated as follows:

n+1 1 1 1

i=1,...,n+1

V+/?i<;+22 +Ä<;+33 > a(mm) IMU «(mm) = min {<*»,

f ^ 1 1 i= 1 n+1

i=1

Applying the property of homogeneous functions to the remaining terms and using the homogeneous norm of the (si,sn+j) class, we obtain

±_i

ctij = max \zi\Si zn+j, i = l,...,n, j = 1,2,3.

\\zi,zn+j \\(si,sn + j ) = 1

The last inequality can be enhanced:

\zi\~i~1 zn+j < ctijWzWl'3^3^', i = l,...,n, j = 1,2,3. Consequently, the assessment of the functional (2.2) can be written as:

n

V(xt,y) > a(min) ||z|s - an+i+1ai1|zis

i=1

|1 —Si+Sn + 1_

mi — 1

n

1 —1 m1

i=1 i=1

-r mi > OLMio\\z\\\ si+s"+z-h mi y^afa^lUI11 ■Si+Sn+3

Taking into account sn+2 = sn+3, denote

mj -1 mi —1

ân = an+i+iaitl, âi2 = r mi + h mi a-'a:^

and get

n n

¿=1 ¿=1

It is necessary to satisfy sn+j- < Sj, i = 1,..., n, j = 1,2 for the functional to be positive within some region \\z\\s > H which will be specified further. That is

mj > fc^i + 1, i = 1,..., n, j = 0,1. (2.3)

Note that situations may arise when conditions (2.3) turn into equalities for some (i, j) from a set I. To ensure the positivity of the functional in an arbitrary domain, we need to check additional conditions on the parameters:

аупшЧ > Щ.

v(mm)

In case when the inequalities (2.3) are strict, the conditions

2n \ -4-sn+j ' a(min)

entail that the general value Ho = maxi;j Hj defines the domain \\z\\s > Ho of the functionals positivity. Therefore, the right side of the estimate

V(xt, y) > a\\z\\s (2.4)

has a positive coefficient

2n

a = a(mira) - ■

j=1i=1

In a similar way an upper bound for (2.2)

V(xt,y) < y\\z\s (2.5)

can be obtained in the region \\z\\s > H0; here

2n

7 = а{тах) + ^ ^7T:?//,; \ а(таж) = max {од, /5Ь /52}.

Tsn + j Si

i=1,...,n+1 j=1¿=1

1

Derivative of the functional (2.2) along the solution of system (1.1) can be represented as

V = fT(x(t))A[Af) + Af+r) - BD-1(Cf(i) + C2CT(i+h))]f (x(t)) + V0 -~[By{t) + A^fixit)) + Af'>frfc2MlABD-iyit) + [By(i) +

+Af)f{x{t)) + AfJ° Af+d+r)f(x(t + -

-BD

+

-1

I-h

c2(i+ö+h)f (x(t + 0))d0

- Allfr

imi

+

ßl + Yir + ß2 + Y2hj ||f (x(t)) r1 - ß2 IlfhF1 -

/0 r-0

If (x(t + 0))||mid» - 72/ If(x(t + 0))||mid»,

r -h

where fr = f(x(t — r)), fh = f(x(t — h)), Jacobian matrix

dx

/dhjxjt)) dh (x(t))\

dxi

\ ctei

dx„

dfn(x(t)) dxn

/

Let us separately evaluate the following terms of the derivative:

(By + Af V + Af V.) > - E A

Z^ |x1 1=1

K1 1E ai

aiixi

kßi-11

v2 ^i

i=1

-Eßi)xjMi Lynn./ri

i=1

i=1

where

ß(i) = |a2i) IL(B)i

(i) , a2 = kßi max

]T(ABD-1)ij yj j=1

ß(i) = |a2i)| max ||(A2j))i||, ai = max|(A1j)^a^.

Likewise,

x

n n n \ r 0

+ E K1£ O^-1 + E ß(i) xk"i-1 llfr Il) llf (x(t + 0))||d0, 1=1 i=1 i=1 ' J-r

0

with parameters a^i) = ai, вэ^ = a^ maxs \\(A[,s))i\, в!^ = a^i)\(B)i\\, and ail = maxs |(A1s))i1 |a^i). As above,

/Ю / n

— C2CT(t+6+h)f (x(t + 0))d0 > -( £ e6i)xfcMi—11

±t^Aabd-i J Cf+e+h)f{x{t + e))de > -(¿/^Ч^МК

n n n \ r U

+ £ К' l£ ai1'xk"i—1 + £e(i)xk^i—1\fr\\ / \\f(x(t + 6>))\\d0,

-h

where = fc^aj', ^ = |«4j) |\\(B)j\\, ^ = I«? I maxs H(A2s))i\, < = maxs |(A1s)|a4j).

Taking into account the assessment of homogeneous function: <

Yn+1\\y\\m'0-1, where Y'n+1 is a positive constant, we have

V0 = Vo\y=Dy + ("^7) (c^f + C^f(x(t-h))) <-(in+l\\y\r + + Yn +1 \y\m0-1(maxWcf^f \\ + max\\C<2s)\\\\f (x(t - h))\\). Now we can continue estimating the derivative of the functional:

n n 1

v < - £ ¡f^{k+V) + yU\\y\r~1 max ||cf || ( £ af') 5 -i=1 1=1

- A\\fr+ Yn +1\\y\P°-1 max \\c2s) \\ \\fh\\ - Y1 f° \\f (x(t + e))\\mi de -

s ./-r

n n n

- ^MP + £ ^xiw-1\\y\\2 + £ Ixf1 l£ -1\\y\\+

n

+ £ e1i)xf^i—1\y\\fr \\ + £ e4i) xkMi—1\y\+ £ |xf' IE ail xkMi—1 +

i=1 ^ i=1 г=1 i=1

U

+£e3i)xkMi—1\fr\\ / \f(x(t+6))\d»+Mre1i)xk^i—1\y\+ i=1 ^j—r v i=1 n n n \ r U

+ £ Ixf1 l£ai'/xk"i—1 + £e5i)xkMi—1\fr\\ / \\f(x(t + 6>))\\d0 + г=1 i=1 i=1 ' h

Г u

+ [в1 + 71Г + в2 + Y2h \\f \\m1 - e2\\fh\\m1 - Y2 / \\f (x(t + 0))\\mi d0.

—h

We introduce new variables

z-= (z, \\f (x(t - r))\\, \\f(x(t - Л))\\)T, s'' = (s',

1 1 '

т\ ' т\ ,

where

/Il(fc+1)' /in(fc+1) ' "10 ' nil' t»i

According to the properties of homogeneous functions, there are con-nt that

stants ßo > 0, ßj > 0, j = 1,..., 5 and ß0j ) > 0, j = 6,7,8 such

i=1 i=1

kin-1____n kin-1____mi

i ^«(^Msll«^1) I o(4)||Ç|| , o(6) ||Ç||fc+î I

+ / ,ßo,i ||zns'' + ßo,i ||zns» + ßo ||zHs'' +

i=1 i=1

n kji^ — 1 2 2mQ+mQfc —fc —1 __1___

I \ ^ û(5) II Z\ \ Mi(fc + 1) »1 I «(7)||S|| m0(fc + l) I «(8)|U|| ml m0

+ / yßo,i H^Hs» + ßo H^Hs» + ßo H^Hs» •

i=1

We can find conditions for the parameters k, m0, mi to ensure that in the last estimate of the derivative of the functional the norms' degrees do not exceed 1. Hence,

kfii- 12 + 1) .

H--< 1 rrij >--—, j = 0,1;

^i(k +1) mj J ^i + 1

1 + ~ I, + — < 1 ^ m3 > ßl{k + 1), ¿ = 0,1;

k + 1 ^(k + 1) m

j

^¿(k +1) mo mi mo mi ^¿(k +1) '

m0 < m1 < k + 1.

Combining the obtained inequalities with the conditions for V to be positive: mj > k^ + 1, j = 0,1, the total result is

2^i(k + 1)

—-- < m0 < mi < k + 1.

+ 1

Under such conditions there is a region p||s» > H, where

U(xt,y) < -Splls« is valid for some positive constant S• Moreover, the estimation

û(xt,y) <-S|z|s' (2.6)

will work in the region ||z||s' > H•

The previously obtained regions ||z||s > H0 of functional positivity and ||z||s' > H0 of its derivative's negativity follow from the common region ||z|| > H, where H = ^Jn + 3 max.,- {H^-jH^i}.

1

1

1

1

1

s

Then, differential inequality

U(xt,y) <-cV,y) (2.7)

could be obtained in the found unified region. Constant c is positive and

p = minj=i n i 3 ^ less then 1. The (2.7) follows from the fact that in

' ' Si

llzll > H

ri+3 ri+3 i

4 ^ ST^ ( ~\p ^ _ ii . up

|z|s' = 2^zi — H) -C2|z

where c2 > 0 and a» > 0 for all i = 1,..., n + 3.

Further we will use the approach from [1] for proving the boundedness of a solution (xT(t, t0yT(t, t0, y0))T. Differential inequality (2.7) works when ||z|| > H. Then it works all the more in

i 1

II II fV fV I H mins; ii

Lz L > H, H = max < , = , ,

Using the assessment (2.4) value A = aH is chosen so that for V(xt, y) — A the inequality (2.7) is still valid.

If at the initial moment the functional V(<£>, y0) < A, then it will remain in the domain V(xt,y) < A because of its decreasing outside. In the case when V(<£, y0) > A the moment T can be found when V(xt,y) < A for all t — t0 + T. For this purpose, we integrate the inequality (2.7) for t € [t0; t1] while V(xt, y) > A:

V(xt,y) < (V 1-V,yo) - c(1 - p)(t - to))

i

i-p

Now it can be noticed that the inequality V(xt,y) > A will be broken for t — t0 + T, where

T = 1

c(1 - p)

(V 1-p(^,yo) - A1-^ .

From (2.4) the region of ultimate boundedness for system (1.1) can be derived:

n A

£x^i+1(i) + ||y(t)|r° < —. (2.8)

1 a

г=1

Let A be any positive quantity and ш = supy^p +цпц2<д2 V(0, n). Then

for Ц-0Ц2 + ||nl|2 < A2 the solution (xT(t,t0,^>);yT(t,t0,y0))T of system (1.1) belongs to (2.8) for any t > max{0,i}, where

This completes the proof of Theorem 1. □

3. Analysis of a system with asynchronous switching

The subject of this section is the system with asynchronous switching (1.2).

Assumption 2. Let < 1 and suppose that there is a positive definite diagonal matrix A such that for every s = 1,..., N the inequalities

n

fTA[A1S) + A2S) - £D-1(cf + Cf)]f < - £^xf(fc+1)

i=1

(j)

hold, where ^0 > 0.

Theorem 2. If assumption 2 is fulfilled and D is Hurwitz matrix, when the system (1.2) with any positive fixed delays is uniformly ultimately bounded.

Proof. Based on (2.2) a modified functional can be constructed

n rXi r0

V(xt, y) = £ A J d£ + V0(y) - fTABD-1y + fTA / Af+0) x i=1 J0 J-r

t'0 i' 0

xf (x(t + 0))d0 - fTABD-1 / C2(t+0)f (x(t + 0))d0 + / (ft +

J-h J-r

+71 (r + 0))||f(x(t + 0))||mid0 + T (ft + Y2(h + 0))||f(x(t + 0))||mid0,

-h

(3.1)

here Aj are components of matrix A from Assumption 2.

The derivative of the functional (3.1) along the solution of system (1.2) takes the form

V = fT(x(t))A[Af) + Af) - BD-1(Cf(t) + C2(t))]f (x(t)) + V0 -~[By(t) + A^f(x(t)) + Af~^f(x(t - r))]T^ABD-'y(t) +

+ [5y(t) + Af V(®(i)) + Af~r)f(x(t - r)fMAx

'0 Af+0)f(x(t + 0))d0 - BD-1 T C2(t+0)f (x(t + 0))d0

'-h

+

ft +

+71 r + A + Y2hj ||f(x(t))||mi - ft ||f (x(t - r))||mi - ft||f(x(t - h))||mi -

/0 r-0

||f (x(t + 0))|nd^ - 72 / ||f(x(t + 0))|pd0.

r -h

In the same way as it was obtained for assessments (2.4), (2.5) and (2.6) one can show the existence of positive values H0, H0 a, ft and 7 such that

a||z||s ^ V(xt, y) ^ 7||z||s, for ||z||s > H0

X

Figure 1. Solution of (1.1) in phase space.

Figure 2. Solution of (1.2) in phase space.

and

U(xt,y) < ||z||s', for ||z||s' > H0.

Hence, the subsequent proof follows the same logical reasoning as presented in Theorem 1. □

4. Example

To illustrate the obtained results, consider an example. Suppose the systems (1.1) and (1.2) consist of two subsystems with scalar parameters

B = -1, D = -2, A(11) = -1, A21) = -5, Cj1) = -3, C21) = 5,

r = 5, h = 3, ¡1 = 42) = 3, 42) = -5, c[2) = -2, Cf ]] = 3. 5

It can easily be checked that Assumptions 1 and 2 are valid. Initial data: t0 = 0, xt0 = 100 for [-5;0], y(t0) = 160. The switching law is given by

a(t) = 1 for t € [-5; 1) U

i(i-l).i(i + l)\

a(t) = 2 for t €

2 ' 2

i(i - 1)_ i(i + 1) 2 ' 2

i = 2, 4,

The Runge-Kutta method of fourth-order with accuracy 0.01 is applied for solving the systems. Charts are plotted in phase spaces. Figure 1 presents the solution of the system (1.1) with synchronous switching and Figure 2 corresponds to the case of system (1.2) with asynchronous switching.

References

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2. Aleksandrov A.Y. On the existence of a common Lyapunov function for a family of nonlinear positive systems. Systems Control Lett., 2021, vol. 147, p. 104832. https: //doi.org/10.1016/j .sysconle.2020.104832

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Aleksandrov A., Andriyanova N., Efimov D. Stability analysis of Persidskii time-delay systems with synchronous and asynchronous switching // Internat. J. Robust Nonlinear Control. 2022. Vol. 32, N 6. P. 3266-3280.https://doi.org/10.1002/rnc.5660

Андриянова Н. Р. Устойчивость системы типа Лурье с асинхронными и синхронными переключениями и постоянными запаздываниями // Вестник Санкт-Петербургского университета. Прикладная математи-

ка. Информатика. Процессы управления. 2023, Т. 19, № 3. С. 320-336. https://doi.org/10.21638/11701/spbu10.2023.302

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6. Efimov D., Aleksandrov A. Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals // Internat. J. Robust Nonlinear Control. 2021. Vol. 31, N 9. P. 3730-3746. https://doi.org/10.1002/rnc.5115

7. Guiver C., Logemann H. The circle criterion for a class of sector-bounded dynamic nonlinearities // Math. Control Signals Systems. 2022. Vol. 34, N 3. P. 461-492. https://doi.org/10.1007/s00498-022-00324-3

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Об авторах

Андриянова Наталья Романовна, аспирант, Санкт-Петербургский государственный университет, Санкт-Петербург, 199034, Российская Федерация, [email protected], https://orcid.org/0000-0002-6152-3572

About the authors Natalya R. Andriyanova,

Postgraduate, Saint Petersburg State University, Saint Petersburg, 199034, Russian Federation, st040174@student .spbu .ru, https://orcid.org/0000-0002-6152-3572

Поступила в 'редакцию / Received 20.12.2023 Поступила после рецензирования / Revised 05.02.2024 Принята к публикации / Accepted 07.02.2024