Dad dynamical systems: stability, reachability and observability Текст научной статьи по специальности «Математика»

yak: 517.977

msc2010: 34k35, 93c23, 93c99

DAD DYNAMICAL SYSTEMS: STABILITY, REACHABILITY AND

OBSERVABILITY

© V. M. Marchenko

Belarusian State Technological University

Department of Higher Mathematics ul Sverdlova, 13a, Minsk, 220050, Belarus e-mail: vladimir.marchenko@gmail.com

Bialystok University of Technology Faculty of Computer Science department of mathematics ul. Wiejska 45A ,15-351, Poland

The paper deals with linear differential-algebraic systems with delay (DAD systems) consisting of differential and difference equations. We study the stability of solutions of DAD systems and derive necessary and sufficient conditions for their asymptotic and exponential stability. A determining equation system is introduced and a number of algebraic properties of the determining equation solutions is established, in particular, the well-known Cayley-Hamilton matrix theorem is generalized to the solutions of determining equation. As a result, an effective parametric reachability and observability rank criteria are given.

Key words: differential-algebraic systems, time-delay, stability, reachability, observability.

Introduction

Scientific-technical progress, in particular, the widespread use of microprocessors in industry, subjects the development of new control systems and the support of existing ones to new, higher requirements related to the necessity of a more adequate description of these systems and the use of their specific properties, which often leads to hybrid dynamical systems. However, note that there is no common viewpoint to the notion of "hybrid systems"(see the papers [1]-[14]) and the bibliography therein). From our viewpoint, being hybrid means, in general, being inhomogeneous in the nature of the considered process or in its investigation methods. The notion "hybrid systems"can be used for systems that describe processes or objects with essentially distinct characteristics, for example, containing continuous and discrete variables (signals) in the basic dynamics, deterministic and random variables or inputs, and so on, which, in the end, defines the character (nature) of hybrid systems.

In the paper, we consider differential-algebraic time-delay (DAD) systems to which, in particular, some standard types of discrete-continuous and systems with retarded argument of neutral type can be reduced (Section 1). Such systems can be qualified as hybrid difference-differential systems or quite regular DAD systems which, in turn, a special case of descriptor (singular, implicit) systems with after-effect (for the entire collection and application see, e.g. [4], [7] [11], and references therein).

1. the motivation: examples of dad systems

Consider a linear neutral type time-delay equation d

— (x(t) - dx(t - h)) = ax(t) + aix(i - h),t > 0, (1)

dt

and a hybrid discrete-continuous system of the form

x(t) = a11x(t) + a12y[k], t e [kh, (k + 1)h), (2)

y[k + 1] = a21x(kh) + a22y[k], k = 0,1,... (3)

x(0) = xo, y[0] = yo. (4)

Here a, a1, d, a11, a12, a21, a22, h are given real numbers, d = 0, h > 0; y[k] denotes a function of integer k; the initial data x0,y0 are real numbers.

The objects (1)-(3) have usually been considered separately. Below we propose an unified approach to study them by reducing to DAD systems.

There are various approaches to the statement of initial-valued problem for Equation (1), in particular,

х(т) = <(т), т e [-h, 0], where < is usually of class C1 ; some approaches require that

<¿(0) = a<(0) + a^(-h) + d<(-h),

the other do not, and what is more, admit jump discontinuities of function <. Theoretically, we may consider an initial-valued problem of the form

х(т) = <(т), х(т) = ^(т), т e [-h, 0], x(+0) = x0,

where ф need not be the derivative of <. The similar observation was made [15, p. 187188 in the russian edition and p.169 in the english one] while reducing a second order time-delay equation to a system of equations of the first order.

To expand the solutions, we reduce time-delay equation (1) to a DAD system. Introduce real a11, a12, a21, a22 by a22 = d, a11 + a12 = a,a11a22 = -a1, and denote

x1(t) = x(t) - a22x(t - h), x2(t) = x(t), t > 0. (5)

By (1), we have

¿i(t) = X(t) — dX(t — h) = ax(t) + aix(i — h) = (an + a12)X(t) — ana22 — h) = an(x(i)-—a22X(t — h)) + a12x(t) = a11x1(t) + a12X2(t), and, taking into account the first equation in (5), we obtain a DAD system of the normal form: ¿1(t) = anx1(t) + a12X2(t), X2(t) = X1(t) + a22X2(t — h), t > 0, X1(0) = X10, X2(r) = <^(r), T G [—h, 0), where function ^ is admitted to be of more general class than C1. Now, turning back to System (2)-(4), denote

x(t) = xi(i), x2(t)

x(kh)

for t G [kh, (k + 1)h), k = 0,1,

Applying the Cauchy formula x(kh+h) = eail(fch+h-fchx(kh)+/fcfchh+h eail(fch+h-rV2y[k]d = e"1lhx(kh) + e0h e"1l(h-s)dsai2 y[k], k = 0,1,..., to Equation (2), it is not difficult to see that System (2)-(4) can be represented as a DAD system of the symmetric form: xi(t) = Aiixi(t) + A12X2 (t), X2(t + h) = A2ixi(t) + A22X2(t), t > 0,

with Aii = an, Ai2 = [0, ai2], A2i = 0, A22 =

and the initial conditions are given by

xi(0) = xi(0+) = xio X2 (t ) = <^(r ) =

e«iih j0h eaii(h-r)drai2

a2i

a22

Xo yo

We believe that examples from above provic of DAD systems.

e the motivation for futher consideration

2. system description

For z G R, [z] denotes the integer part of real number z; C denotes the field of complex numbers. Symbol x(t) will be usually used to denote the value of function x(-) = x at the point t; | ■ | denotes a given norm in vector space Rn or Cn and || ■ || is used for a norm in functional spaces. PC(I, M) is the set of piecewise continuous functions on I C R with values in M. This is a linear normed space if we equip it with the sup norm M1 PC = supieJ |p(t)|.

In sequel we pay attention to the simplest DAD control and observation system of normal form:

X1(t) = Anx1(t) + A12X2CO + B1u(t), (6)

X2(t) = A21 X1(t) + A22X2(t — h) + B2u(t), t > 0, (7)

with the initial conditions

x1(0) = x10 e Rni, x2(т) = ф(т), т e [-h, 0), (8)

and the output

y(t) = C1x1(t) + C2x2(t),t > 0, (9)

where A11 e RniXni, A12 e RniXn2, B1 e RniXr, A21 e Rn2Xni, A22 e Rn2Xn2, B2 e Rn2Xr, C1 e RmXni, C2 e RmXn2,ф e PC([-h, 0],Rn2); the external action u(t) for t > 0 is a piecewise continuous r-vector function (admissible control).

We regard an absolutely continuous function x1(-) and a piecewise continuous function x2(-) as a solution of System (6)-(8) if it satisfies the initial conditions (8), it satisfies the equation (7) for t > 0 and Equation (6) almost everywhere (a. e.) for t > 0. If Equation (6) is satisfied for all t > 0 with right-hand value at t = 0 then we consider the solution x1(-), x2(-) as a strong solution of the system.

Computing the solution x1(t) = x1(t,x10,ф,u), x2(t) = x2(t, x10, ф, u), t > 0, of the system (6)-(8) by "step by step"one can prove that it exists, is unique, and its growth rate does not exceed an exponential one for any admissible control having no higher than the exponential rate of growth. This permits to apply the Laplace transform to the system.

3. stability

3.1. Problem Statement. Consider System (6), (7) for the switched-off control:

u(t) = 0, t > 0. (10)

Following the classical statement of the problem of stability for time-delay systems we give definitions of stability of DAD systems.

Definition 1. The zero solution of System (6), (7), (10) is said to be:

i) stable (in Lyapunov sense) if, given e > 0, there exists a 5 > 0 such that every solution x1(-), x2(-) of the system satisfying

|x101 + \\ф\\рс < 5 will also satisfy

max (\M\pc + \\x2t\\pc) < e, 0<i<+^

where x^) = x1(t + т), x2t^) = x2(t + т), т e [-h, 0];

ii) asymptotically stable (in Lyapunov sense) if it is stable and every solution x1(-), x2(-) of the system will also satisfy the relation

lim xi(t) = 0, lim x2(t) = 0; t—t—

iii) exponentially stable if there are positive numbers M and a such that every solution x1(-), x2(-) satisfies the inequality

max{|xi(t)|, |x2(t)|} < M(|xio| + U\\pc)e-at, t > 0.

3.2. Variation-of-Constants Formulas. Introduce matrix-valued functions X*1(-), X*2(-), and Z*(-) as the solutions of the following adjoint system:

X1(t) = X*i(i)An + X*(t)A2i, t e (jh, (j + 1)h), (11)

Xi2(t) = Xi(t)Ai2 + X*2(t - h)A22, t > 0, (12)

X* (kh + 0) - Xii(kh - 0) = Z* [k]A2i, (13)

z*[k] = Z*[k - 1]A22, k = 1,...,Tt, (14) with initial conditions of the form: X*2(t) = 0, t < 0; i = 1, 2; j = 0,1,...;

X*i(0) = X*i(+0) = Ini, Z* [0] = 0; (15)

X*i(0) = X*i(+0) = A2i, Z2*[0] = In2. (16)

Here and throughout the following, the symbol stands for the identity k by k matrix.

It is not difficult to check that X*i(t) and X*2(t) - X*2(t - h)A22 are continuous for t > 0.

Then the solution xi(t) = xi(t,xi 0,^), x2(t) = x2(t,xi o,-0) of System (6), (7), (10) can be computed by [12]

h

X i (t) = X*(t - 0)xio + J X*2(t - T)A22^(t - h)dT, (17)

0 h

X2(t) = X*(t - 0)xio + J X2*2(t - T)A22^(t - h)dT+

0

+Z2*[Tt]A22^(t - Tth - h),t > 0, (18)

where Tt = [h] is the integer part of |.

3.3. Stability Criteria. Using the method due originally to Euler, we attempt to find a solution of the form

x i(t) = eAtCn x2(t) = eAtc2, (19)

where A e C, ci e Cni, c2 e Cn2, and | + |c2| = 0. Then A, c i and c2 must satisfy the equation

A/n1 — A11 —A12 C1 0

-A21 /n2 - e-AhA22 C2 0

If the solution of this algebraic system is to be nontrivial A must be a root of the characteristic equation

det

A/m — Aii -A21

n 2

-A12

- e-AhA22

A(A) = 0.

(20)

Definition 2. The roots (complex, in general) of the characteristic equation (20) will be called the characteristic roots (values) of System (6), (7), (10).

Then we state:

i) the condition that all characteristic roots must have non-positive real parts is necessary for each kind of stability considered above;

ii) the condition

Re A < 0 (21)

for A G C such that A(A) = 0 is necessary for both asymptotic and exponential stability of System (6), (7), (10).

If we take the initial conditions (8) as follows

I T = - h

± TO2 > ' )

0, T G (-h, 0),

xio = 0 G Rni, ^(T)

we conclude: if System (6), (7), (10) is asymptotically stable then all roots of the equation

det(I„2 - e-AhA22) = 0 ( 22)

have negative real parts, i. e. A22 is a Schur matrix.

Using the representations (17), (3.2) and the Laplace transforms for the matrix-valued functions Xl (■), X*2(-), one can prove (the details are in [12]) that functions X^-), X22(-) are exponentially decreasing and as a result the conditions (21) and (7) are sufficient for asymptotic stability and exponential one as well.

Theorem 1. The following statements are equivalent:

i) A22 is a Schur matrix and all characteristic roots of System (6), (7), (10) have negative real parts, i. e.

Re A < 0 for A G C such that A(A) = 0;

ii) System (6), (7), (10) is asymptotically stable;

iii) System (6), (7), (10) is exponentially stable.

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3.4. Scalar Case. Consider System (6), (7), (10), where

Aj = aj, i = 1,2; j = 1,2, (23)

are real numbers and the characteristic equation is of the form

A(A) = A + a + «2 e-Ah + «3 Ae-Ah = 0, (24)

where a = —an — ai2a2i, a2 = ana22, a3 = —a22.

By using the method of D-partitions for localization of the characteristic roots, one can prove (the details are in [12])

Theorem 2. The following statements are equivalent:

i) System (6), (7), (10), (23) is asymptotically stable;

ii) System (6), (7), (10), (23) is exponentially stable;

iii) |a22| < 1 and all roots the characteristic equation (24) have negative real parts;

iv) |a22| < 1 and at least one of the following conditions is satisfied: a1 > |a21 or a2 > |a1|, h < h*, where h* is given by

I 1 - «j , a1 + a2 a3, , .

h = W _2-2 arccos (--|-). (25)

3

aj — a2 + a1 a3 '

For a simplest hybrid difference-differential system in the scalar case, Theorem 2 provides stability conditions explicitly expressed via its original parameters, which permits one to analyze problems of robust stability of such systems and estimate the limit delay preserving the property of stability.

4. REACHABILITY

4.1. Reachability and Controllability. For a given q x (n1 + n2) matrix H and a time moment t* > 0, System (6), (7) is said to be relatively H — t*-reachable if for any final system state (x1,x2) € x Rn2 there exists a piecewise-continuous control u = u(-) such that the corresponding solution to the system with zero initial state possesses the property

H x1(t*, 0, 0, u) =H o^* 1

X2(t*, 0, 0, u) o^* x2

In the case H = /ni+n2, the system is called relatively t *-reachable; for H = [Ini, 0], it is relatively t *-reachable with respect to x1; and, for H = [0, In2, it is relatively t *-reachable with respect to x2.

If, in the previous definitions, the initial state is not zero, we come to the controllability concepts (zero-controllability, for the zero final state).

The time t* above can be omitted if the corresponding ¿^-reachability (controllability) takes place for at least one value t* > 0. Note that the reachability and controllability notions given above are not equivalent.

4.2. Determining Equations. Introduce matrix-valued functions Xk(t), X|(t), as the solutions of the following determining equations of System (6), (7):

X+i(t) = AnX(t) + A12Xfc2(t) + BiU (t), (26)

Xfc2(t) = A2iXl(t) + A22Xfc2(t - h) + B2U(t), (27)

k = -1, 0,1, 2,...; t > 0; with the initial conditions

Xl(t) = 0, X2(t) = 0 for k < 0 or t < 0;

Uo(0) = Ir; Ui(r) = 0 for i2 + t2 = 0 . ()

It is easy to see that = 0 for t = kh, k = 0,1, 2,... We introduce the notation:

P(w) = det(/„2 - A22W), C(w) = det(/ni - Anw),

nM = det(/„2CM - C(w)A2i(1ni - Aiiw)-1Ai2w),

A(w) = An + Ai2(/„2 - A22w)-iA2i,

-i ^ ,

Dm = (/„2 - A2i(1ni - au^)-1aI2w)-1a22,

B(w) = Bi + Ai2(/„2 - A22^)-1B2,

B(w) = B2 + A2i(1ni - (All + Ai2^2l)w)-1(Ai2B2 + Bi)w.

Then the characteristic equation of the matrix A(w):

0 = det(A/„i - AM) = №))-ni ЕП=0 ЕП=Г rijAni-V

and the characteristic equation of the matrix D(w):

0 = det(A/„2 - DM) = (nM)-n2 ЕП=0 ЕП=П PijAn2-V = 0

are equivalent to the following ones:

rai«,2 ni nm

Ani = - £ rojAni-V - £ £ rijAni-V, (29)

j=l i=l j=0 nm ni nm

Ani = - £ rojAni-V - £ £ rjAni- V, (30)

j=l i=l j=0 respectively for |w| < where is a sufficiently small real number.

Below we establish some algebraic properties of the determining equations solutions (the details are in [9]).

Lemma 1. The following identities hold:

(AM)fc BM = E+=° (jhV;

(In2 - A22^)-1A21 (AM)fc B(w) = E+=°° Xfc2+i(jh)^j;

(/„2 - A22^)-1B2 = E+°° X02(jh)^j; k = 0,1, 2,...;

(DM)j B(w) = Xfc2(jh)^k, j = 0,1, 2,...;

(/ni - A^)-1^ (D(^))j B(w) = E+To Xfc1(jh)^fc, j = 1, 2,...;

(/rai - (An + A12^21)w)-1(A12B2 + B^ =

E+°o X^(0)wfc, where |w| < ^1.

Lemma 2 (generalized Cayley-Hamilton theorem). The solution Xk(t) of the determining equations (26) - (28) satisfies the characteristic equations (29), (30) and, as a result, we have (|w| < w1):

min{fc,raira2} ni min{fc,nin2}

XV(kh) = - £ rojXV(kh - jh) -- £ £ ryXV-i(kh - jh), j=1 i= 1 j=0

k = 0,1, 2,...; 7 = n1 + 1,n1 + 2,..., v = 1, 2;

min{fc,nin2}

XV ((7+2-v )h) = - £ poj XV-j ((7+2-v )h)-

j=1

k = 0,1, 2,..., y = n2,n2 + 1,n2 + 2,..., v = 1, 2.

n2 min{fc,raira2}

e E - xv

i=1 j=0

Pu XV-j ((Y+2-v-i)h),

4.3. Solution Series Expansions. Using the Laplace transform and Lemma 1, one can prove (the details are in [9]) that the solution of System (6),(7) can be represented as follows

. t—ih

" (t - T - zh)fc

xi(t) = EE XkVi(ih) /

_n + J

fc=0 ih< t t—ih

k!

u(T)dT + x1(t, x10, 0),

(31)

- (t _ T _ ih)k

X2(t)^E EXfc+1(ihW --k-- u(r)dr + E X02(ih)u(t - ih)+ a^XM,^, 0).

fc=0 ih< t 0 ih< t

(32)

4.4. Reachability Criteria. Using the solution representations (31),(32) and taking into account Lemma 2, which claims that in the linear span of column-solutions to the determining equations there is a finite set of generators, we obtain Theorem 2. System (6),(7) is i) relatively H - t *-reachable if and only if

rank

H

0

X2(jh)

H

X1 (t)

X2 (t)

H,

t G [0,t*) n [0,n2h],k = 1,2,..., n1, j = 0,1,...,min{[h] }

rank

H

X02(jh)

H

X(t)

X2(t)

t G [0,t*) n [0,n2h],k = 1, 2,...,m, j = 0,1,..., min{ [|] , n2}

ii) relatively reachable if and only if

rank

0

Xo2(jh)

X(t)

X2(t)

t G [0,n2h],k = 1, 2,...,m, j = 0,1,

,^2

ni + n ;

iii) relatively reachable with respect to x if and only if rank[X1(jh), k = 1, 2,..., n^ j = 0,1,..., n2] = n^

iv) relatively reachable with respect to x2 if and only if rank[X2(jh), k = 0,1, 2,..., n1, j = 0,1, ...,n2] = n2;

v) relatively reachable if it is relatively reachable with respect to x1 and, additionally, the following condition holds:

rank[B2, A22B2, ..., (A22)n2-1B2] = n2.

5. OBSERVABILITY

For the simplicity, we concentrate on the problem of relative observability with respect to x1.

System (6), (7), (9), (10) is said to be -observable with respect to x1 if for every x10,x10 G Rni the condition y(t, x10, 0) = y(t, x10, 0) for every ^ G PC([-h, 0), Rn2) and for t > 0 implies that x10 = x10 that, by linearity of the system, is equivalent to the implication:

y(t, x10, 0, 0) = 0, t > 0, x10 G Rni x10 = 0. (33)

For System (6), (7), (9), (10) of observation, we introduce its determining equations as follows:

X01(t) = AnX 0-i (t) + A12X021(t) + U _i(t)

(34)

(35)

(36)

X02(t) = A2iXo0i(t) + A22X02(t - h), Yfc(t) = CiX0i(t) + C2X02(t), t > 0, k = 0,1,...

with initial conditions

X0i(t) = 0,X02(t) = 0, Yfc(t) = 0 for t < 0 or k < 0; Uo(0) = , U(t) = 0 for t2 + k2 = 0.

Similarly to Lemma 1 and Lemma 2, we can establish some algebraic properties of X0i(t),X02(t),yfc(t), in particular,

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0

the following identities hold (|u| < u and the symbols A(u), D(u), r^),pj are the same as in Section 4.2:

(Ci + C2(/„2 - A22^)-1A2i)(A(^)^ = £ ,

j=0

(Ci(1ni - Aii^)-1Ai2u + C2) X (D(u))j

(A2i(1ni - (Aii + Ai2a2I)u)-1) = Y Z(jh)uk—i

fc=i n $7

""ij 1 k—i

n (Yh) = - £ roj Yk ((y - j )h)rij Yk—i((Y - j )h) j=i i=i j=0

i = 0,1,...; j = 1, 2,...; y = 0,1,... , k = ni + 1, ni + 2,... , where = min{Y, nin2};

and

Sk m Sfc

Yfc (Yh) = - Y PojYfc—j (Yh) - Y J^PijYfc—j ((Y - i)h) j=i i=i j=0

for k = 1, 2,..., y = n2 + 1, n2 + 2,..., where = min{k, nin2}.

Using the Laplace transformation and algebraic properties of the determining equations solutions Xk0(t), X|0(t), Yk (t), one can prove (details are in [8]) that the solution xi(t) = xi(t,xi0, 0, 0), x2(t) = xi(t,xi0,0, 0), t > 0, of System (6), (7), (9), (10) can be represented as follows

xi(t) = £ Y Xfci+i(jh)xi0, (37)

10

fc=0 t—jh>0

_ / + _ j^\k

"kT

*2(t) = £ E (tj-X0+i(jh)xi0. (38)

fc=0 t—jh>0 and as a result we have

Theorem 3. System (6), (7), (9), (10) is Rni-observable with respect to xi if and only if

rank[YfcT(jh), j = 0,..., n2; k = 1,..., ni]T = ni,

where symbol ()T means transposition.

Similarly as above one can obtain the other observability criteria and establish an "observability-reachability"duality principle.

6. CONCLUSION AND FUTURE WORK

In this paper, we have considered the simplest stationary linear differential-algebraic systems of observation and control with delay. For such systems, the problem of stability has been studied, in particular, we have proved that, unlike in the case of classical time-delay neutral type systems, the exponential stability is equivalent to asymptotic one. Necessary and sufficient conditions for asymptotic and exponential stability have been given. In the scalar case, these conditions have been refined and expressed via the original coefficients of the system in parametric form, which permits one to keep track of how the perturbations in the coefficients affect the solutions and to find the limiting value of the delay for which stability is preserved. We introduced a determining equation system and have established a number of algebraic properties of determining equations in order to obtain an effective rank condition for relative reachability and observability in terms of determining equation solutions. The results obtained can be generalized to the problem of stabilization of DAD systems, DAD systems with several state and control delays and to problems of functional observability and controllability that will permit to formulate a general "observability-reachability-controllability"duality principle. This will be the object of another paper.

The research was carried out in the framework of the scientific collaboration (project no. S/WI/2/2011) of the Bialystok University of Technology and the Belarusian State Technological University

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Статья поступила в редакцию 11.05.2015