ABOUT AN ESTIMATION PROBLEM OFA LINEAR SYSTEM WITH DELAY OF INFORMATION Текст научной статьи по специальности «Математика»

ДИНАМИЧЕСКИЕ СИСТЕМЫ И ОПТИМАЛЬНОЕ

УПРАВЛЕНИЕ

DYNAMIC SYSTEMS AND OPTIMAL CONTROL

Серия «Математика» 2022. Т. 42. С. 3—16

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

ИЗВЕСТИЯ

Иркутского государственного университета

Research article

УДК 517.977

MSC 93E10,62L12,34G25

DOI https://doi.Org/10.26516/1997-7670.2022.42.3

About an Estimation Problem of a Linear System with Delay of Information

Boris I. Ananyev1®, Polina A. Yurovskikh1

1 N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russian Federation ® abi@imm.uran.ru

Abstract. The problem of guaranteed estimation with geometrically bounded initial states and integrally limited disturbances is considered under delay information in the measurement equation. At the additional assumptions the problem is reduced to the creation of the reachable set of a special system. A discrete multistage system is specified for which the information set converges in Hausdorff's metric to the corresponding information set of the continuous system when the diameter of partition is reduced. A numerical example is given.

Keywords: guaranteed estimation, information set, reachable set

Acknowledgements: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2022-874).

For citation: AnanyevB. I., YurovskikhP. A. About an Estimation Problem of a Linear System with Delay of Information. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 42, pp. 3-16. https://doi.org/10.26516/1997-7670.2022.42.3

Научная статья

О задаче оценивания линейной системы с запаздыванием информации

Б. И. Ананьев1^, П. А. Юровских1

1 Институт математики и механики им. Н. Н. Красовского УрО РАН, Екатеринбург, Российская Федерация И abi@imm.uran.ru

Аннотация. Рассмотрена задача гарантированного оценивания с геометрически ограниченными начальными состояниями и интегрально ограниченными возмущениями при запаздывании информации в уравнении измерения. При дополнительных предположениях проблема сведена к построению области достижимости специальной системы. Указана дискретная многошаговая система, для которой информационное множество сходится в метрике Хаусдорфа к соответствующему информационному множеству непрерывной системы при уменьшении диаметра разбиения отрезка наблюдения. Приведен численный пример.

Ключевые слова: гарантированное оценивание, информационное множество, область достижимости

Благодарности: Работа выполнена в рамках исследований, проводимых в Уральском математическом центре при финансовой поддержке Министерства науки и высшего образования Российской Федерации (номер соглашения 075-02-2022-874).

Ссылка для цитирования: AnanyevB.I., YurovskikhP. A. About an Estimation Problem of a Linear System with Delay of Information // Известия Иркутского государственного университета. Серия Математика. 2022. Т. 42. C. 3-16. https://doi.org/10.26516/1997-7670.2022.42.3

1. Introduction

The article is a continuation of works [1;2] and follows the approach from the monograph [7]. A linear stationary system with an unknown state vector is considered. The measurement equation contains delays in the state, which is expressed by an integral with a function of bounded variation. In addition, both the system and the measurement equation additively include an uncertain and integrally bounded disturbance. It is assumed that the system state is bounded by the geometric inclusion at the initial instant. It is required to find an information set which with guarantee contains the true state of the system at a given instant. We note that the similar problem without delay were investigated in detail in the papers cited above. Guaranteed estimation problems for systems with delays were also considered in the works [3;4;6] and in many others. The difference of the work is that we do not use the space of infinite dimension and reduce the problem to the construction of reachable sets of a special finite-dimensional system. Various computational methods for the approximation of reachable sets for linear and nonlinear dynamic systems were investigated in [5; 10]. Note that we do not consider geometric restrictions on disturbances. For

further researches of similar problems with geometric restrictions there can be useful results of works [8; 9].

We consider a linear autonomous system with delay in the measurement equation:

x(t) = Ax(t) + Bv(t), t > 0, x(t) e Rra, v(t) e R1, (1.1)

0

y(t) = Gx(t)+ Hv(t), y(t) e Rm, Qx(t) = J dG(s)x(t + s), (1.2)

-(hAi)

where the given matrices A, B, H have the corresponding dimension, and the matrix function G(s) of bounded variation in the operator Q from (1.2), where h > 0, is left continuous to [-h, 0], G(s) =0 Vs > 0, and G(s) = G(-h) Vs < -h; h A t = min{h, t}.

We assume that the uncertain function v(-) e Ll2[0, to) and the unknown initial state x0 e Rra are bounded by:

(1.3)

(1.4)

where X0 is a convex compact set; | ■ | is the Euclidean norm. We assume that the state x(T) of system (1.1) can be realized for any v(-), x0 satisfying the constraints (1.3) and (1.4) at any instant T > h.

2. System conversion and problem statement

We assume that the matrix H e Rmxi has rank H = m, which implies the condition

HH' > 0, (2.1)

where ' means transpose and the relation Q > 0 for square matrix Q means x'Qx > 0 for every x = 0. Let C = (HH')-1 and let Cx = It - H'CH is orthogonal projection onto the subspace ker H. Then v(t) = H'CHv(t) + C1v(t) and Hv(t) = y(t) — (Qx)(t). Therefore, constraint (1.3) takes the form

JT (|»(i) -Qx(t)H + Wt)l2Cl) dt < 1 (2.2)

due to orthogonality. Hereinafter, the symbol |«|p denotes the quadratic form x'Px, where the matrix P satisfies the condition P' = P > 0, |«|p = Vx'Px. We assume = |^|2, where P = I is the identity matrix. Substituting the decomposition of the function v(t) into (1.1) we have

x(t) = Ax(t) + B (Civ(t) + H'C (y(t) - Qx(t))) .

C lv(t)l2dt < 1, xq G xq,

The resulting relation is a differential equation with a distributed delay. In the paper, we will not consider such equations and suppose

BH' C f dG(s)x(t + s) = 0 (2.3)

[-(ftAt),0)

for all continuous n-dimensional functions x(t), t > 0. Because of Q x(t) = -G(0)x(t) + J dG(s)x(t + s),

[-(hAt),0)

we come to the system

x(t) = Ax(t) + by(t) + BCiv(t), (2 4)

A = A + bG(0), b = BH 'C, ( )

without delay.

Remark 1. Condition (2.3) is always satisfied if BH' = 0. In this case, the equation in (2.4) has the form x(t) = Ax(t) + BC1v(t) and does not depend on the signal y(t).

Remark 2. Similarly to [2], we lower the dimension of the uncertain disturbances v(t) in relations (2.2), (2.4). Since ker H©imH' = R, imC1 = ker H and dim(im H') = m, then rank C1 = I — m. By a well-known linear algebra theorem, we represent C1 as C1 = T(J1T', where T is an orthogonal matrix, TT' = T'T = Ii and (71 is a diagonal matrix with zeros and ones since C1 is a projection matrix, Cf = C1. Removing m zero columns from C1 and denoting the resulting matrix by D1, we get C1 = D1D1 and C1 = ^1^1, where D1 = TD1. Next, we set u(t) = D[v(t) e whence

we have

C1v(t) = Dm(t), D1 e Rlx(l-m), rankD1 = I — m.

In relations (2.2), (2.4), the Remark 1 equation and further, we will use the function D1u(t) instead of C1v(t). Note that D'1D1 = Ii-m. If I = m, then C1 = 0 and the function v(t) = H-1 (y(t) — Qx(t)) becomes known. In that case we set u(t) = 0.

Let us introduce a definition.

Definition 1. A family of state vectors Xt(y) = {xt} is called an information set (IS) if, for any xt e Xt(y), there exists a function v(-) and an initial state x0 satisfying constraints (1.3), (1.4) and such that equalities (1.1), (1.2) hold almost everywhere on an interval [0,T] with a boundary condition x(T) = xt■

Considering (2.2) and the reasoning above we come to a statement.

Lemma 1. Let conditions (2.1) and (2.3) be satisfied. Then IS XT(y) is the reachable set of the equation

x(t) = Ax(t) + by(t) + BDm(t) (2.5)

at the instant T > 0 over all disturbances u(-) and initial states x0 satisfying constraints

J(T,X0,u,y) = fT (\y(t) -Gx^l + Ki)|2) dt < 1 (2.6)

and (1.4). Here the matrix D1 and the disturbances u(t) e are defined according to the Remark 2.

Our aim is to describe the IS XT(y) using support function and also to construct an approximating multistage system whose IS converges to XT(y) as the diameter of the partition of the segment [0, T] decreases. In addition, the issues of ellipsoidal approximation of IS are considered, similar to how it is done in [2] for systems without information delay.

3. Support function for the information set

Since the IS by its construction is a convex compact set, it is uniquely described by its support function

p(l^T(y)) = max l'x, I e Rra (3.1)

xexT (y)

Let us represent the solution of the system (2.5) as the sum x(t) = x(t) + x(i), where

x(t) = Ax + by(t), x0 e X0, X(t) = Ax + BDm(t), x(0) = 0.

We put y(t) = y(t) - Qx(t) on [0, T]. Then instead of (2.6) we have the inequality

J(T,xQ,u,y) = J(T,x0,y) (IQx(t)H + lu(t)l2-

2y'(t)Cgx(i))dt < 1, where J(T,xQ,y) = ft Iy(t)I2cdt.

(3.2)

We fix xq and select in (3.2) the full square in u(-), for which we introduce operator K according to the relation

fT , L,/rtl2

Si dgx(t)ic + wm dt = (u, Ku),

where u e L2 m[0, T] is an arbitrary function. Let us also introduce the matrix function

G(i)= / dG(s)eA(t+s), (3.3)

-(hAt)

with which we represent Qx(t) = ku(i) = f0 G(t—s)BD\u(s)ds. Therefore, the operator K is written as

K = Id + k*Ck, or

T T

Ku(t) = u(t) + D'B J J G'(s — t)CG(s — T)dsBDiu,(T)dT,

0 tVr

where Id is the identity operator in Ll2~m[0, T] and k* is the operator adjoint to k. When choosing a full square in (3.2), it is necessary to solve the equation

T

£u(0 = /(•), where f(t) = k*Cy(t) = D[B' J G'(s — t)Cy(s)ds, (3.4)

t

which is a Fredholm integral equation of the 2-nd kind with non-negative symmetric kernel, which has a unique solution in space Ll,-m[0,T], as is well known. Therefore, the inequality (3.2) becomes

J(T, X0, y) + <u — K-lf, K, (u — K-1D) — {f, K-1D < 1. (3.5)

Hereinafter (U,V) = J^U'(t)V(t)dt e R9Xr, where U(t) e RpX9, V(t) e Rpxr are matrix functions with elements from L2[0,T]. For a fixed vector I e Rra and an initial state x0, we find the value

R(l,T,xo,y) = maxl'x(T) = l'x(T) + (lT, K-1 f) + (lT, K-1 lT)

O

x ^1 + {f, K-lf) — J (T,xo, y), It (t) = D[B 'eA'(T-t)l. Finally, we find the support function for the IS:

(3.6)

p(l \XT(y))= max R(l,T,xo, y). (3.7)

x0ex0, \+(f,K.-lf)>j(T,x0,y)

Let us summarize.

Theorem 1. The IS Xt(y) is a convex compact set under conditions (2.1) and (2.3). The IS support function (3.1) is defined by formulas (3.6), (3.7) with parameters given by relations (3.2)—(3.5).

4. Some ways of approximation of IS 4.1. Approximation by ellipsoids

Let us introduce the notation E(Q, c) = [x £ Rra : lx — cIq < 1} for nondegenerate ellipsoids, where the matrix Q = Q > 0, с £ Rra. Ellipsoid

support function p^^(Q,c)) = l'c + a/I'Q-11. We choose an arbitrary ellipsoid e(pq,xo) D Xo for the initial set Xo and consider the quadratic constraint

(1 - a^xo - x0^ + a fT ^(t^dt < 1, a e (0,1). (4.1)

Denote by X^'«(y) the IS for constraints (4.1). By definition, we have an inclusion for arbitrary specified ellipsoids:

X$'«(y) D Xt(y).

Repeating the reasoning of Section 2, we come to the inequality

J«(T,x0,u,y) = (1 - a^xo - x0^ + +« /oT (W) - Qx^c + M*)|2) dt < 1,

(4.2)

which is similar to (2.6) for the equation (2.5). Using (3.3), the expansion x(t) = x(t) + x(t) and denoting y(t) = y(t) + G(i)^o, rewrite (4.2)

(1 - a)|x0 - x0+ a (||y - G^\\2C + \\u\\l - 2 (y - G^, CQx)) < 1. We introduce an operator in the space Rra x Ll2-m[0,T]

Ka[xo; u] = [P«£o + a (D,u); a (D(-)®0 + fcu)], (4 3)

P« = (1 - a)P0 + a (G,CG), (4.3)

where D(i) = k*CG(t) = D[B' ¡T G'(s - t)CG(s)ds, and with its help we transform the last inequality to

\\N; ^HL + y« - 2 ([x«; af], [xo; u])0 < 1, where (44)

y« = (1 - a)^^ + allyHc, x« = (1 - a)P0X0 + a (G,Cy), (.)

and f(t) = k*Cy(t) = D[B' jf G'(s - t)Cy(s)ds. Here the symbol (•, ■)0 denotes the inner product in Rra x Ll-m[0,T].

Taking (4.4) into account, we find the support function

p^X^^^ I' £ eA(T-t)by(t)dt + ([eA'TI; It], K-1[x«; af])0

yj, (4.5)

where the function lT(■) is defined in (3.6).

To definite K-1, it is necessary to minimize an expression of the form

+ a\M\2 + 2ax' (D, u) - 2x'q - 2 (u, v)

with respect to [x;u]. The solution [x*;u*] is K—l[q;v] and the minimum is — ||[Q;v}\\t-i. We come to the relations

u*() = K-M) ^(0 = <) — aD(^)P-1q,

(4.6)

x* = P-1 (q — a (D, u*)), Ka = aK. — a2DP-1D'. The value |[<?;'w]|K-i = \q\%-1 + ||^||2—. The operator Ka is coercive

^a

and invertible. We have Ka = aId + k* (a-1cc' + (1 — a)"1GP0"1G*)-1 k. Here G means the operator G : Rra ^ L™[0,T] according to the equality Gx = G()x.

From the formula (4.5) it follows that IS Xrj^,a(y) is a non-degenerate ellipsoid. Let E denotes the family of ellipsoids E(P0,X0) with property E(P0,X0) D X0. Similar to the reasoning in the proof of Theorem 1 in [2] we obtain the statement.

Lemma 2. Let us define the set X^ (y) = n«e(0 ^ ^^(y), where E e E. Then '

p(l(y))= max R(l ,T,X0, y) Vie Rra (4.7)

xoEE, 1+(f,K.-1f)>J(T,x0,y)

The notation in the formula (4.7) is the same as in (3.7).

The next theorem describes the approximation by ellipsoids.

Theorem 2. The equality Xt(y) = f]Eee(y) holds.

Using Lemma 1 from [2], the Theorem 3 is proved similar to the proof of Theorem 1 from the cited paper.

4.2. Approximation with multistage systems

Exact and approximate methods for constructing IS using ellipsoids require solving integral equations of the form (3.4) and inverting operators of the form (4.3), (4.6). Let us propose a method for IS approximation with multistage systems. For simplicity, we assume the quantities T, h such that h = rA, T = N A, where r and N are natural numbers. Let tk = kA, k e 0 : N. Assuming the disturbances u(t) = uk to be constant on the half-intervals [tk-1, tk), k e 1 : N, of the partition, we obtain discrete system

Xk = axk_1 + Yk + Bufc, a = eAA, k e 1:N Yk = ¡A eAtby(tk — t)dt, B = f0A eAt dtBDX. ( . )

In the inequality (2.6) we assume Qx(t) = Qx(tk), t e [tk-1, tk), and

Qx(tk) = G(i)x0 + GYk + Ek= 1 Gkuz, where GYk = f0tk G(tk — s)by(s)ds, Gk = H^ G(tk — s) BD^s.

Then instead of (2.6) we have an inequality with the sum on the left side:

JN (X0,y,U1:N) = Ek=1 ftL ( |»(i) - Gx(tk )& + K ^dt < 1.

We consider the solution of the system (4.8) as the sum xk = xk+x^, where

xk = axk-1 + Yk, X0 e X0, xk = axfc-1 + Buk, x(0) = 0. We put y(t) = y(t) - G(t)x0 - GYfc on [tk-1, tk]. We come to the inequality

N k

„. „. \ _ JN¡^ „,\ 1 a V^ f\ V^ r^K. 12

JN (X0,y,U1: N) = JN (X0,y) + A^ 0 E Gi ^H + K H -

k=1 i=1

N k

2 E E Gi ^ < 1, (4.9)

k=1 i=1

N

/ ^(ty^dt, yk = / y(t)dt. Let us single k=1'' tk-l Jtk-1

out in (4.9) a full square on u1:n, for which we introduce the matrix K,A according to the relation

A T,k=1 (| TH=l G^H + ^ ^ = u[:N K*U1:N.

We assume that u1:n e

rN (l-m)

is a column vector. We also introduce

matrices according to relations

k£u1:N = YH=1 Gt Ui, with the help of which we obtain the representation for the matrix K,A:

K,A = A (lN(i-m) + kA) ,

where kA = EfcU(kA)'CkA e RN(l-m)xN(l-m) is a symmetrical matrix. When single out a full square in (4.9), it is necessary to decide algebraic equation

KAul:N = f, where f = Ei=1(kf)'Cyl. Therefore, the inequality (4.9) becomes

JN(X0,y) + | U1:N - (K,A)-lf | 2kA - | f | 2^A)-I < 1. For a fixed vector I e Rra and an initial state x0, we find the value Rn(l,X0,y) = maxl'xN = l'xN + l'N(KA)-1f + A/l' (fcA)-1lNx

ui.M V

^1 + | f | 2jca)-i - JN (X0,V), In = [aN-1B,..., B]' I.

(4.10)

Finally, we find the support function for the IS of discrete system:

p(l \XN (y)) = max Rn (I ,xo, y). (4.11)

xo^Xo, l+\f \2KA)-i>JN(x0,y)

Formulas (4.10), (4.11) are discrete analog of formulas (3.6), (3.7). It is proved in [1] that Xn(y) ^ Xt(y) as N ^ œ in the Hausdorff metric for systems without measurement delay. A similar theorem is also valid in the case under consideration. Its proof repeats the proof of Theorem 6 in [1] with minimal modifications.

Theorem 3. Let conditions (2.1), (2.3) be satisfied and the quantities T and h be commensurable, that is, T/h is a rational number. Then the IS Xn(y) of the discrete system (4.8) for constraints (4.9) and (1.4) converges to IS Xt(y) in the Hausdorff metric as N ^ œ.

5. Numerical example

We consider the motion of a material point along a straight line, subject to disturbances 1( ) and 2( )

X1 = x2 + v\t) — v2(t), X2 = v1^) — v2(t), 0 <t<T.

Let the sum of the disturbances also have an additive effect on the measurement equation containing the delay

y(t) = x1 (i) +x1(t — 1) + x2(t — 1) + v1^) + v2(t).

System parameters:

A =

0 1 0 0

в =

1 -1 1 -1

\-1

G(s) = -[1, 0]x( s) - [1,1]x( 1 - s),

H = [1,1], С = (HH')-1 = 1/2, C1 = I2 -H'CH =

1 -1 11

/2,

X(s) =

¡0:

1, ifs > 0,

0, if s < 0.

The condition (2.3) is met because BH' = 0. The dimension of the disturbance can be reduced, by setting u( ) = ( 1 ( ) — 2 (i)/\/2. Then D1 = [1; —1]/^2 and bDx = [1; 1]y/2. We have

C1V (t) = D\u(t), \v (t)\2Cl =u2(t).

The system (2.5) will take the form

X1 =x2 + V2u(t), X2 = V2u(t), 0 <t<T,

-2

Figure 1. Discrete approximation of IS, T = 2.

with constraints (2.6)

J(T, X0, u, y) = f0T ((y(t) - xl(t) - x1 (t - 1) - x2(t - 1))2 /2+

+u2(t))dt < 1.

Here we put x(t) = 0 if t < 0. Let X0 = {x e R2 : | < 1, |®2l < 1}, and the signal was realized at

xQ = [0.5; -0.5], v(t) = 0.9[cos(i);sin(i)]/Vr and T = 2.

The matrix function (3.3) has the form

G(i) = [1,i](1 + X(t - 1)).

Let us find the IS support function using discretization. For discretization we set N = 20, A = T/N = 0.1. The formula (4.8) will take a form

xk = axk-i + B uk, a =

1 A' 0 1

B

A A2/2 0A

fee 1 : N,

BD1

Further, using the formulas (4.10), (4.11), we find the support function of the discrete IS. The corresponding IS was obtained by intersection the support half-spaces and is shown in Fig. 1, where the star represents the true state at T = 2.

B. I. ANANYEV, P. A. YUROVSKIKH 6. Conclusion

In this work, a problem of the guaranteed state estimation of a linear system is considered with geometrical restrictions for initial states and integral restrictions on perturbations with a delay of information in the measurement equation. Under additional assumptions on the system, the arising problem is reduced to the creation of reachable sets for a special system. Two ways of approximation of the information set for the original system are proposed: by means of ellipsoids and by a method with multistage systems. The information sets of discrete multistep systems converges in the Hausdorff metric to the corresponding information set of the continuous system when the partition diameter of the observation interval is reduced. A numerical example is given.

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Список источников

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10. Filippova T. F. Control and estimation for a class of impulsive dynamical Systems // Ural Math. J. 2019. Vol. 5, N 2. P. 21-30. https://doi.org/10.15826/umj.2019.2.003

Об авторах

Ананьев Борис Иванович, д-р

физ.-мат. наук, с.н.с., Институт математики и механики им. Н. Н. Красовского УрО РАН, Российская Федерация, 620108, г. Екатеринбург, abi@imm.uran.ru,

https://orcid.org/0000-0002-1378-0240

About the authors Boris I. Ananyev, Dr. Sci.

(Phys.-Math.), Senior Res., N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, 620108, Russian Federation, abi@imm.uran.ru, https://orcid.org/0000-0002-1378-0240

Юровских Полина Александровна, аспирант, Институт математики и механики им. Н. Н. Красовского УрО РАН, Российская Федерация, 620108, г. Екатеринбург, polina2104@list.ru, https://orcid.org/0000-0001-8051-3428

Polina A. Yurovskikh,

Postgraduate, N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, 620108, Russian Federation, polina2104@list.ru,

https://orcid.org/0000-0001-8051-3428

Поступила в 'редакцию / Received 27.08.2022 Поступила после рецензирования / Revised 05.10.2022 Принята к публикации / Accepted 10.10.2022