Stabilization of a scalar equation with delay in the state and control variables Текст научной статьи по специальности «Математика»

UDC 517.929.4 Vestnik of St. Petersburg University. Serie 10. 2014. Issue 4

L. V. Shayakhmetova, V. L. Kharitonov

STABILIZATION OF A SCALAR EQUATION WITH DELAY IN THE STATE AND CONTROL VARIABLES

St. Petersburg State University, 7/9, Universitetskaya embankment, St. Petersburg, 199034, Russian Federation

The contribution is dedicated to the stabilization problem of systems with input and state delay. We derive a stabilizing control law for the case of a scalar equation with several state and two input delays. We start with a control law of the form, where the right hand side contains future values of the state predicted by Cauchy formula. The presented control law is of the form of an integral equation. It is shown that the characteristic function of the closed-loop system consists of two factors. Special conditions that guarantee the exponential stability of the closed-loop system are derived. Exponential estimates for the solutions of the closed-loop system are given. An illustrative example is presented. Bibliogr. 9. Keywords: time-delay systems, stabilization.

Л. В. Шаяхметова, В. Л. Харитонов

СТАБИЛИЗАЦИЯ СКАЛЯРНОГО УРАВНЕНИЯ С ЗАПАЗДЫВАНИЯМИ В СОСТОЯНИИ И УПРАВЛЕНИИ

Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7/9

Данная статья посвящена задаче стабилизации систем с задержками одновременно и в состоянии, и в управлении. Построено стабилизирующее управление для случая скалярного уравнения с несколькими запаздываниями в состоянии и двумя запаздываниями в управлении. В соответствии с приведенным алгоритмом поиск регулятора осуществляется с использованием предсказанных по формуле Коши значений состояний в фиксированные моменты в будущем. Показано, что полученное стабилизирующее управление описывается интегральным уравнением, а характеристическая функция соответствующей замкнутой системы состоит из двух множителей. Выведены специальные условия, гарантирующие экспоненциальную устойчивость замкнутой системы. Построены экспоненциальные оценки состояния и управления. Приведен иллюстративный пример. Библиогр. 9 назв. Ключевые слова: системы с запаздыванием, стабилизация.

1. Introduction. Stabilization problem for time-delay systems has a long history. For the case of the systems with only input delay a set of stabilizing control laws has been derived in [1]. It has been shown there that these control laws are of the form of integral equations. A detailed account of the current research on stabilization of systems with input delay can be find in the recently published book [2].

For the stabilization problem in the case of systems with only state delay see book [3], and references therein. Up to now there are few results concerning the stabilization problem in the case when delay appears both in the state and control variables (see [4, 5]).

In this paper we study the problem for the case of scalar equation with several state and two input delays. In Section 2 we introduce the time-delay equation, provide basic

Shayakhmetova Liliya Vladimirovna — post-graduate student; e-mail: lilia.v.shayakhmetova@ gmail.com

Kharitonov Vladimir Leonidovich — doctor of physical and mathematical sciences, professor; e-mail: khar@apmath.spbu.ru

Шаяхметова Лилия Владимировна — аспирант; e-mail: lilia.v.shayakhmetova@gmail.com Харитонов Владимир Леонидович — доктор физико-математических наук, пpофессор; e-mail: khar@apmath.spbu.ru

notations, and present auxiliary results. In particular, we define the fundamental solution, and derive an explicit expression for the future values of the equation solutions in terms of the present and past values of the state and the control variables.

Section 3 starts with sufficient conditions from [6] for the exponential stability of a time-delay equation. Then a formal definition of the stabilization problem is given. A set of stabilizing controllers are derived. It is shown that these stabilizing controllers are also in the form of integral equations.

Section 4 is devoted to the computation of the characteristic function of the closed-loop system. It is shown that the characteristic function of the closed-loop system admits a factorization, where the first factor depends on the input coefficients, while the second factor depends on the choice of the desired control law. Some conditions that guarantee the exponential stability of the closed-loop system are given.

In Section 5 we derive exponential estimates for the solutions of the closed-loop system.

An illustrative example is given in Section 6.

2. Preliminaries. In this contribution a scalar retarded type equation of the form

d m

= Y^ aix(t ~ hi) + bMt - n) + b2u(t - r2) (1)

¿=0

is studied. Here coefficients a¿, i = 0,1,...,m; b1, b2 are real constants, and the system delays are ordered as follows:

0 = ho <h\ < ... < hm < ti < T2.

Let p : [-h, 0] ^ R1 be an initial function from the space PC([-h, 0],R1) of piecewise continuous functions, x(t, p) stands for the solution of the initial value problem x(0, p) = p(0), 0 G [-h, 0]. We will use the shorthand notation x(t) instead of x(t,p) when no confusion may arise. By xt (p) we denote the restriction of the solution on the segment [t - h,t], i.e., xt(p) : 0 ^ x(t + 0,p), 0 G [-h, 0].

The uniform norm,

\\p\\h = sup ||pC0)||,

ee[-hfi]

is used for elements of the space PC([-h, 0], R1).

In the following we need the fundamental solution of equation (1).

Definition 1 [7]. Function k(t) is the fundamental solution of (1) if it satisfies the equation

dk(t)

dt

i=0

a,jk(t — hi), t ^ 0,

and the initial condition, k(t) = 0 for t < 0, k(0) = 1.

Let x(t) be a solution of (1), then for a > 0 the solution satisfies the equality [8]

m 0

x(t + a) = k(a)x(t) + Y / k(a — hi — 0)aix(t + 0)d0 + (2)

i=i ■{ — hi

t+a t+a

+ J k(t + a — £)&iu(£ — ri)d£ + J k(t + a — Ç)b2u(Ç — r2)d£.

3. Problem formulation. We assume that real constants fo, fi, ..., fm are such that the equation

dx(t) ,,

= ]T(o, + fi)x(t - hi) (3)

i=0

is exponentially stable.

Definition 2 [7]. The equation (3) is said to be exponentially stable if there exist Z ^ 1 and a > 0, such that any solution x(t, p) of the equation satisfies the inequality

\\x(t,^)\\ < Ce-atM\h, t > 0.

The appropriate choice of fo, f1, ..., fm can be based on the following sufficient condition.

Theorem 1 [6]. Equation (3) is exponentially stable if the inequalities

ao + fo < 0,

m

\ao + fo\ > £ \ai + fi\,

i=i

hold.

Problem. Find a control law under which equation (1) coincides with (3) for t ^ t2 . An evident solution of the problem is a control law of the form

m

biu(t) + b2u(t + ti — T2)= fox(t + ti) + fix(t + ti - hi). (4)

i=i

Unfortunately, this control law can not be realized since the right hand side of (4) depends on the future values of x(t). Applying (2) we can rewrite (4) in the form where the state and control variables appear at the present and past time instants:

biu(t) + b2u(t + Ti — T2) = fok(Ti)x

(t) + J2 fpk(Ti — h p)x(t) + p=i

o -hp

m m m

+ Y, fp^ / k(Ti — hp — hi — e)aix(t + e)de + ^ fp k— — hp)biu(t + n)dn +

P=o i=i-hi p=o -T!

m T!-T2-hP

+ y^ fp k(Ti — T2 — hp — Y)b2u(t + j)dj.

--n J

p=o

F -T2

Equations (1) and (4) form the closed-loop system.

4. Characteristic function of the closed-loop system. The case m = 1 has been studied in [9]. For simplicity we assume that m = 2, and the delay are such that hi = 1 < h2 < t1 < 2 < t2 < 1 + h2 < 3. For this case the fundamental solution of equation (1) is

0, for t < 0,

eaot, for t e [0,hi),

() * eaot + aieao(t-hl)(t — hi), for t e [hi,h2),

eaot + a1eao(t-hl)(t — hi) + a2eao(t-h2)(t — h2), for t e [h2, 2).

The closed-loop system takes the form dx(t)

-= aoxit) + a\x(t — hi) + a2x(t — h2) + b\u(t — t\) + b2u(t — t2), (5)

dt

biu(t) + b2u(t + Ti — T2) = [fok(ri) + fik(Ti — hi) + f2k(ri — h2)] x(t) + (6)

0 0

+ f0 J k(ri — hi — e)aix(t + e)dd + f0 J k(ri — h2 — 0)a2x(t + 9)d9 +

— hi —h2 0 Ti —T2

+ f0 J k(—rj)biu(t + n)dn + f0 j k(ri — T2 — 7)b2u(t + 7)dq +

— Ti —T2

0 0

+ h J k(Ti — 2hi — e)aix(t + e)de + h J k(Ti — hi — h2 — e)a2x(t + e)de +

-hi -h2 -hi Ti T2 hi

+ fi J k(—n — hi)biu(t + n)dn + fi J k(Ti — T2 — hi — 7)b2u(t + 7)d7 +

— Ti —T2

0 0

+ f2 J k(Ti — hi — h2 — e)aix(t + e)de + f2 j k(Ti — 2h2 — e)a2x(t + e)de +

-hi -h2 — h2 Ti T2 h 2

+ f2 j k(—n — h2)biu(t + n)dn + f2 j k(Ti — T2 — h2 — 7)b2u(t + 7)dq.

— Ti —T2

To find the characteristic function of the closed-loop system (5), (6) we look for a non trivial solution of the closed-loop system of the form

x(t) = aest, u(t) = (3est.

Substituting this solution in (5) and cancelling the non zero factor est, we obtain a linear equation for coefficients a and 3

a(s — a0 — aie—shi — a2e—sh2 ) = 3(bi e—STi + b2e—ST2 ). (7)

Then, we substitute this solution in equation (6), and cancelling factor est we get one more equation for the coefficients

ari(s) = 3r2(s), (8)

where

ri(s) = f0 eaoTi+f0ai eao(Ti—hi)(Ti — hi)+f0a2eao(Ti—h2)(Ti — h2)+fieao(Ti—hi)+f2eao(Ti—h2) +

+ -foai

s — a0

gaoCn-M _ g-^lgaoT! + ai-3_eS(r1-2fe1) + ^ -^-tn-hz)

s — a0 s — a0

11

s — a0 ) V s — a0

- ailr1-h1 + —— ) e-shleao{-Tl-h^ - a2 n - h2 +- e-»fcieao(n-fc2)

+

+

s — ao

-foa2

oa0(ri-h2) _ e-sh2 ea0Ti +

a i -

s — ao

3s(r1-h1-h2) a2-e.

_J_es(T1-2h2) _

s — ao

1

- ai ( n - ht + — ) e-^gao^-M _ alTl_h2 +

s — ao J \ s — ao

1

e-sh2 eao(Ti-h2)

+

+

1

-fiai

s — ao H---—/201

es(ri-2hi) _ e~shi eao(Ti-hi)

+ -

1

fi a2

es(ri-hi-h2) _ e-sh2 ea0(ri-hi)

s — ao

es(ri-hi-h2) _ e-shi ea0(ri-h2)

s — ao

H----/2 02

s — ao

+

es(Ti-2h2) _ e-sh2 eao(Ti-h2)

and

T2(s

- ai T\ - hi +

1

s — ao~

(s)=(b1+b2e^Tl-T2n 1--fo 1 - e-Tl(s-a°) +04-e-shl+a2-e~ah2

s — ao

-Ti(s-ao) e-aohi

— aW Ti — h2 +

1

s — ao 1

s — ao

s — ao

e-Ti(s-ao ) e-aoh2

s — ao

fi

e-shi e-Ti(s-ao)e-aohi

s — ao

f2

e-sh2 _ e-Ti(s-ao) e-aoh2

System (7), (8) admits a nontrivial solution, (a,ß) = (0,0), if and only if the following condition holds:

^ A.(s — ao — aie-shi — aie-sh2 bie-sTi + b2e-sT2 N q(s) = det , , , , =0.

V ri(s) r'2(s) J

Function q(s) is the desired characteristic function of the closed-loop system (5), (6). By tedious but direct computations one can check that the determinant is of the form

q(s)= (bi + b2es(Ti-T2)j (s — (ao + fo) — (ai + fi)e-shi — (aa + f2)e-sh2) .

Our choice of fo, fi,f2 guarantees that any root of the second factor,

q2(s) = s — (ao + fo) — (ai + fi)e-shl — (a^. + f2)e-h, has negative real part. It is evident that the roots of the first factor,

bi + b2es(Tl-T2),

lie in the open left half complex plane if and only if the following inequality holds:

\b2\ < \bi\.

5. Exponential estimates. Any solution of the closed-loop system (5), (6) is defined by the corresponding initial functions

x(t,p,^) = p(t), t e [—h2,0], p e PC([—h2,0],Ri),

u(t,p,^) = ->Kt), t e [—T2,0], ^ e PC([—T2,0],Ri).

In the following we denote solution (x(t, p, ^), u(t, p, ^)) by (x(t), u(t)).

For t e [0,t2] there exists v such that function x(t) admits an upper estimate of the

form

\x(t,p,^)\ < V (\\p\\h2 + M\t2 ) .

1

1

1

e

1

1

For t > t2 the first equation of the closed-loop system coincides with the exponentially stable equation (3). This means that there exist Z > 1, a > 0, such that

\x(t,p,^)\ < Ze-a(t-T2 )\xT2 (p,^)\h2 < Ye-at (\\p\\h2 +

),

where y = vZeaT2.

The solution of the second equation of the closed-loop system (5), (6) satisfies the equation

biu(t) + b2u(t + ti - t2) = fox(t + ti) + fix(t + ti - hi) + f2x(t + ti - h2).

b2

Let t = t2 — ti, b = — — , then the preceding equation takes the form b1

u(t) = bu(t - t)+ m(t), where function m(t) admits the estimation

\m(t)\ < je-^ (\\p\\h2 +

),

(9)

(10)

with

J = Y

e-<JTl +

e-^(Ti-hi) +

-a(Ti-h2)

Given t > 0, let us define k, such that t = kT + S, S G [0,t). Iterating difference equation (9) k times we arrive at the equality

u(t) = bk+1 u(t - (k + 1)t ) + bk m(t - kT) + bk-1m(t - (k - 1)t ) + ... + m(t).

As t - (k + 1)t G (-T2, 0], then u(t - (k + 1)t) = ^(t - (k + 1)t), and \b\ < 1 implies that there exists a1 > 0, such that \b\ = e-aiT. Applying inequality (10) we obtain

\u(t)\ < e-(k+1)aiT\\^\\t2 +e-kaiTje-a(t-kT)C+e-(k-1)aiTje-a(t-(k-1)T)C+.. .+je-aitC,

here C = \\p\\h2 + \\^\\T2. If we set a0 =min{a1,a}, then

\u(t)\ < e-(k+1)a0T\№\\t2 + kje-aotC.

As t < kT, and

< C we arrive at the inequality

\u(t)\ < e-(a0-E)t

t

1 + —/x ) e

C.

Now, let us select positive e, such that e < oo- Function ( 1 + —/x ) e et is bounded on [0, to), i. e., there exists x(e) for which

1 + -fi) e~Et < x(e), t > 0.

This means that the following inequality:

\u(t,p,^)\ < e-(°-t K(e)(\\p\\h2 + M\t2 ), t > 0,

holds.

e

et

Finally, we arrive at the following result.

Theorem 2. If the coefficients b1, b2 of the equation (1) satisfy the inequality \b2\ < |bi|, and parameters f0, fi,f2 are such that the equation (3) is exponentially stable, then the solutions of the closed-loop system (5), (6) admit an exponential estimate of the form

\x(t,v,n + \u(t,v,^)\ < (y + -E)t(M\h2 + №\T2), t > 0.

6. Example. Given the scalar equation

^^ = x(t - 1) - x(t - 1.2) + 2u(t - 1.6) + u(t - 2.1). (11)

The fundamental solution of the equation (11) is

{1, for t e [0,1), t, for t e [1,1.2), 1.2, for t e [1.2, 2).

We find a control law under which (11) for t > t2 coincides with the exponential stable equation

^ = -x(t). dt

To this end we set f0 = -1, f1 = -1, f2 = 1. Then, the control law is of the form

-0.5 0 -1.2

Mt) + - 05 = -j Ui, + - 2 j Ui, + - uj Ui, + -

-1.7 -1.2 -1.6

-1.7 -1 -1.5

- 1.2 J u(t + n)dn + 2 J Vu(t + n)dn + j (0.5 + n)u(t + n)dn - 1.2x(t) +

-2.1 -1.2 -1.7

-1 -0.6 -0.4 -0.6

+ 1.2 J x(t+e)de -0.2 J x(t+e)de - J (0.6 - o)x(t+e)de + j (0.4 - o)x(t+e)de. -1.2 -0.8 -0.6 -0.8

Equation (11) with the derived control law forms the closed-loop system with the characteristic function

q(s) = (2 + e-05) (s + 1).

All roots of the function have real part not greater than -1.

7. Conclusion. In this contribution the classical prediction scheme used for the computation of stabilizing control laws is extended to the case of scalar equations with several delays in the state variable and two delays in the control variable. It is worth of mentioning that the new control laws are described by difference integral equations, and belong to the class of neutral type time-delay equations.

References

1. Manitius A., Olbrot A. Finite spectrum assignment problem for system with delays. IEEE Transactions on Automatic Control, 1979, vol. 24, no. 4, pp. 541—553.

2. Krstic M. Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Boston: Birkhauser, 2009, 466 p.

3. Michiels W., Niculescu S.-I. Stability and Stabilization of Time-Delay Systems. Philadelphia: SIAM, 2007, 378 p.

4. Bekiaris-Liberis N., Krstic M. Stabilization of linear strict-feedback systems with delayed integrators. Automatica, 2010, vol. 46, no. 11, pp. 1902—1910.

5. Kharitonov V. L. An extension of the prediction scheme to the case of systems with both input and state delay. Automatica, 2014, vol. 50, no. 1, pp. 211—217.

6. Tsypkin Ya. Z. The systems with delayed feedback. Automatika i Telemechanika, 1946, vol. 7, no. 2, 3, pp. 107-129.

7. Kharitonov V. L. Time-Delay Systems. Boston: Birkhäuser, 2013, 311 p.

8. Bellman R. E., Cooke K. L. Differential Difference Equations. New York: Academic Press, 1963, 461 p.

9. Shayakhmetova L. V. Stabilization problem for a scalar equation with one state and two input delays. Control Processes and Stability, 2014, vol. 1 (17), pp. 70-75.

Литература

1. Manitius A., Olbrot A. Finite spectrum assignment problem for system with delays // IEEE Transactions on Automatic Control. 1979. Vol. 24, N 4. P. 541-553.

2. Krstic M. Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Boston: Birkhaäuser, 2009. 466 p.

3. Michiels W., Niculescu S.-I. Stability and Stabilization of Time-Delay Systems. Philadelphia: SIAM, 2007. 378 p.

4. Bekiaris-Liberis N., Krstic M. Stabilization of linear strict-feedback systems with delayed integrators // Automatica. 2010. Vol. 46, N 11. P. 1902-1910.

5. Kharitonov V. L. An extension of the prediction scheme to the case of systems with both input and state delay // Automatica. 2014. Vol. 50, N 1. P. 211-217.

6. Tsypkin Ya. Z. The systems with delayed feedback // Automatika i Telemechanika. 1946. Vol. 7, N 2, 3. P. 107-129.

7. Kharitonov V. L. Time-Delay Systems. Boston: Birkhauser, 2013. 311 p.

8. Bellman R. E., Cooke K. L. Differential Difference Equations. New York: Academic Press, 1963. 461 p.

9. Shayakhmetova L. V. Stabilization problem for a scalar equation with one state and two input delays // Control Processes and Stability. 2014. Vol. 1 (17). P. 70-75.

The article is received by the editorial office on June 26, 2014.

Статья поступила в редакцию 26 июня 2014 г.