Robust Absolute Stability of lumping Stochastic Systems

Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997

ROBUST ABSOLUTE STABILITY OF JUMPING STOCHASTIC SYSTEMS P. V. Pakshin' *Nizhny Novgorod State Technical Unive'rsity at Arzamas. 19, Kalinina Str., Arzamas, 607220, Russia. E-mail: [email protected]

Abstract. The control systems represented by a finite family of continuous time stochastic models are considered. Each individual model of this family is described by the differential equation of Ito type with parametrical white noise and sector bounded nonlinearities. The discontinuous (jumping) transitions between the individual models is described by a homogeneous Markov chain. An algebraic condition of exponential stability in the mean square of considered system for given parametrical noise intensities, arbitrary nonlinearities of the class above. and arbitrary transition probabilities of the Markov chain are presented. Generalization on the discrete time systems is also given .

•

Key Words. Stochastic systems, Jump process, Markov models, Robust stability, Absolute stability

in the mean square sense (Khas'minskii, 1980) for such systems for given noise intensities, arbitrary sector bounded nonlinearities and arbitrary transition probabilities of the Markov chain, We define these general property of the considered family of stochastic models as robust absolute stability. The considered problem is a generalization of the well know absolute stability problem on stochastic jumping systems. This problem was considered earlier in nonrobust statement, see (Barkin, et al., 1992), where stability conditions depended not only on parametrical noise intensities but also on transition probabilities. Note that a deterministic version of robust absolute stability was considered by Grujic and Petkovski (1987).

1. INTRODUCTION A wide variety of physical systems have variable structures , subject to random changes , which may result from the abrupt phenomena, such as component and interconnection failure, parameter shifting, tracking and the time required to measure some of the variables at different stages. Systems with this character may be modelled as hybrid ones, that is, the state space of the systems contains both discrete and continuous components. Among this kind of systems jumping systems have been a subject of great practical importance that has attracted a lot of interest for last three decades. In jumping systems the dynamics of the discrete and continuous states are modelled respectively by a finite state Markov chain and stochastic differential (or difference) equations subject to the discrete process.

The paper is organized as follows. In section 2 we formulate and solve the problem of robust absolute stability of the jumping stochastic systems in continuous time case. In section 3 generalization of the obtained results on discrete time systems is given. In conclusion, in section 4 the computational and analytical aspects of arising nonlinear algebraic problems are discussed.

The stability theory of jumping systems began to developed since the pioneering works of Kats and Krasovskii (1960). Now many important results in this area have been obtained , see (Barkin , et aI. , 1992; Mariton , 1990; Pakshin, 1994; Yaz, 199.5) and references therein. Recently interest increased appreciably to application of the jumping systems theory to research of complex dynamical systems, see (Siljak, 1991).

2 ROBUST ABSOLUTE STABILITY OF CONTINUOUS TIME SYSTEMS

The present paper considers a class of jumping systems in which each individual model is described by a differential equation of Ito type with parametrical white noise and sector bounded nonlinearities. The purpose of this paper is to fornlUlate a sufficient conditions of stochastic stability

2.1. Problem Statement

We consider a control system described by the finite family of stochastic differential equations

dx(t) = [A(itJx(t)

159

+ B(ir)u(t)]dt +

N

+ L O"I(it)AIUt)X(t)dwl(t), u(t)

We introduce the Hamiltonian matrix

(1)

= f(y(t) , it) , y(t) = -C(idx(t) ,

x(t) is n-dimensional state vector ; W1(t)(l = 1, 2, ... , N) are independent standard Wiener processes ; A(i) , Al(i)(l = 1, 2, .... , N) are n x n matrices; O"I(i)(I = 1, 2, ... , N) is positive where

of 2n x 2n dimension whose eigenvalues are placed symmetric with respect to the imaginary axis .

It follows from the last property that there exists the polynomial ~().) of n degree satisfying the factorization relation:

scalars, which characterize the intensity of components w1(1 = 1, 2, ... , N); it is Markov chain with discrete set of states IN = {I , 2, ... , l!} and with matrix of transition probabilities P( r) =

~(>.)( - I t ~(-).)

[Pij(r)]1' = exp(Qr) , Pij(r) = P{i(t + r) = j I i(r) = i} (i , j E IN) ; Q = [%]1' , % ~ OU "1= j) , qii = - L~;ij % ; f(y , i) is nonlinear vector

Now we define the matrix polynomial

~(H) = [~ll ~12]

function , whose components have form

f1(y , i) = f1(Y1 , i),f1(O , i) = 0 , 1= 1, ... , m

= det(U -H).

~21

(2)

~22

'

appropriate to ~(H), whose allocated blocks have dimensions n x n. Let ~l(H) denote the matrix polynomial appropriate to ~l().) = (-lt~(-).)· We assume that H matrix has not purely imaginary eigenvalues.

and satisfy restrictions

We assume that the initial state (xo , io) of system (1) does not depend on the augmented Wiener process w(t) = [Wl(t) , ... , wn(t)V . Then it is possible to construct a stochastic process as the solution of equation (1) in the sense of Ito at given realization it. The set of all possible pairs {x(t) , i(t), t ~ O} will define Markov process {X , I} = {x(t) , i(t) , t ~ O} . We notice , incidentally, that separately considered process {X} = {x(t) , t ~ O} is not Markov .

Under these assumptions the following statement is valid . Lemma 1. The matrices X , 5 , r satisfy the equations of Lur'e (4) if and only if the matrix X is a solution of linear algebraic equation

(5) or equivalent to it

As the main result of the presented paper we determine sufficient conditions of exponential stability in the mean square (ESMS) , see (Khas'minskii , 1980) , of system (1)-(3) at given parametrical noise intensities 0"1(1 = 1, ... N) and arbitrary transition intensities qij (i, j E IN).

[-X

I]~l(H)

= [0

0] .

(6)

Proof: Necessity . It follows from (4) that the X matrix is a solution of the quadratic equation

AT X

+ XA + (XB -

D)R-l(XB-

-Df + Q = O.

We fulfil the similarity transformation of matrix 1i using the matrix

2 .2. Generalization of the Bass method for' solution of Lur 'e equations

L=[i

An important role in the sequel is played by the special algebraic method of the solution of Lur 'e equations . This method is generalized well known Bass result , see (Kwakernaak and Sivan , 1972). Let matrices X , 5 , r satisfy the equations of Lur 'e ATX

+ XA + Q

HI

= L-1HL = [_~ ~] 1i _ [AI

(4)

-

rTr = R ,

=

~] .

As a result , taking into account (7) , we obtain

= -55T ,

XB-D=5r ,

(7)

0

[i

~] =

1

B R- BT ]

-(Alf

'

where AI = A - BR-1(DT - BT X). From this relation it is easy to see , that

=

where Q QT and R RT are n x n and m x m nonnegative and positive definite matrices , D is n x m matrix.

~().)

160

= det(U -

AI) .

According to the Cayley - Hamilton theorem (Horn and Johnson, 1986) ~(A') = 0, hence

hence taking into account that PI I matrix is nonsingular, P 21 = -P22 x.

(9)

By analogy to the proof of necessity we have where ~~2 and dimensions .

~~2

are nonzero blocks of n x n

Since ~(1l') is the matrix polynomial of 1l' and 1l' = L- I 1lL, we have According to (9) p-l matrix can be presented in following form Substituting the expressions for matrices Land in this relation we easily come to (5). Repeating the reasons above for the polynomial ~l (1l) we obtain (6).

Calculating H21 matrix and taking into consideration (8) , (9) we obtain

2] = p- I 1lP = [ ~ll i!.1 Hn '

H21

and det(M-H ll ) = ~(A) , det(M-H 22 ) = ~1(A). By analogy to the proof of necessity we have

=p-I~(1l)p=

[0o

= Pn[-X I)1l [ XI ] P l l = O.

Since R matrix is positive definite then there exists nonsingular (and moreover , positive definite) matrix r such that

= ~(p-l1lp) =

~(it)

~12] ,

R

~22

= rrT.

(10)

Hence we can define S = (X B - D)r- l , then

therefore

~(1l) [ ~~~

_0 ] [

P22

Since p- l matrix is nonsingular P22 matrix is also nonsingular. Then equation (6) is valid .

Sufficiency. According to Schur theorem (Horn and Johnson , 1986) there exists P matrix of 2n x 2n dimension such that

it

= [Io

p- l

~(1l')

~

] = [

XB - D

] .

= sr.

(11)

Taking into account (10), (11) it follows from (7) that

Since P matrix is nonsingular , then it follows, that PI I matrix is also nonsingular. It follows from existence of the solution of equation (5) that the previous equality will be true if

So, Lur 'e equations (4) are valid.

(8)

•

2.:3 . Robust stability conditions

since, taking into account , the assumption , concerning 1l matrix , the defect of ~(1l) matrix is equal to n.

Applying S - procedure, see (Barkin et al. , 1992 ; Boyd et al. , 1994) we replace inequalities (3) by the matrix inequality

It follows from (8) that P matrix can be presented in the form

]

P= [IX PP12] [ o ~ 22 Pll

where .

K(i) = diag[K/(i)];n , T( i ) = diagh(i)];n , T/(i)

Then it is possible to write

p- l

= [Pl-;l o

0] [ Pf!1l I

21

We introduce the family of Hamiltonian matrices

f!.12 P ]. n

AK(i) G( 1') -- [ -Mc(i)

I

Calculating p- P we obtain

- I Pn] [P:l

[I] X

P ll

> 0, 1= 1, ... , m.

B(i)K(i)T-I(i)BT(i) ] -[AK(i)V '

where AK(i) = A(i) - ~B(i)K(i)G(i)], Mc(i) = M(i) + tGT(i)T(i)K(i)C(i) , M(i) > 0, i E IN .

= 0, 161

= x T (AT (i)H + H A(i))x + 2xT (H B(i) -

The eigenvalues of these matrices are placed symmetriC wIth respect to the imaginary axis. AccordIng to It We define polynomials di ()..) from the factonzaton relati:>ns det[)..J - G( i)] == di ( )..)( -1

rd

1 -2CT (i)T(i))u

N

+ xT[I>-? AT (i)H A/(i) + /=1

11

+qii H + L %H]x = -[ST(i)x - r(i)uf x

i ( -)..) .

i~j

Now we .construct the matrix polynomials di ( G(i)) appr~p~~ate. to di ()..). We denote as H (i) the matriX , ::,a Isfymg the lInear algebraic equation

H~i) ] == [ ~ ] .

di(G(i)) [

where £ is expectation operator. According to lemma 1 and (Barkin, et al., 1986) from the last relation the result of theorem follows.

•

T.

-2 C (Z)T(i)][K(i)T-l(i)][H B(i)-

_~CT(i)T(i)f],

3 . ROBUST ABSOLUTE STABILITY OF DISCRETE TIME SYSTEMS

(1:3) Consider a control system described by the family of stochastic difference equations

N

(j(i)

+xT[L (7f AT(i)H A/(i) - M(i)Jx :S /=1

= _[AT(i)H + HA(i) + [H B(i)1

= M(i) -

I>f AT (i)H A1(i), i E IN . (14) 1=1

Xk+l = A(ik)Xk

+ L(7/A/(ik )Xk V /(k) , uk

(15)

The sys~em of equations (15) is equivalent to the quadratIc matnx equation

In this section we determine conditions of ESMS of system (17) for given parametrical noise intensities (7/ (I = 1, . .. , N) and arbitrary transition probabilities Pij (i , j E IN).

AT(i)H(i) + H(i)A(i) + (H(i)B(i)T.

.

-2 C (l)T(z))I{(i)T- I (i)(H(i)B(i) 1

-2 C

T

.

(Z)T(i)f

+ M(i) = 0, i

E Ir\.

(16)

Let the matrices X , S, r satisfy the discrete equations of Lur'e

Iffor i = ~: there exists positive definite solution of the equatIOn (12) and for i =I k M (i) matrices are defined by formula (13) then there exists the solutIOn of the equation (16) and hence of the system of equatIOns (15) H(i) H(k) H. Then for stochastIc Lyapunov function V(x, i) = V(x) = x T H x > 0 we have

=

= f(Yk , ik) , Yk = -C(ik)Xk,

where f(y , i) is nonlinear vector function, whose components have form (2) and satisfy restrictions (:3) ; v/(k) (l = 1,2 , . . . , N) are components of Ndimensional vector v( k) of Gaussian white noises with identity covariance matrix; ik is Markov chain with discrete set of states IN = {I , 2, . .. , 11 } and stochastic matrix P = [Pij]L Pij = P{ik+l = j I ik = i} (i , j E IN). It is supposed that noise process v( k) is independent on the initial state of system (17) ; Markov chain ik is independent on Xk and v(k) (n = 0, 1, .. .).

AT(i)H(i) + H(i)A(i) + +M(i) = -S(i)ST (i) ,

~CT(i)T(i) = S(i)r(i), rT(i)f(i) = T(i)K-I(i).

(17)

/=1

Pr~or According to Lemma 1 the matrix H(i) , satIsfymg the equation (12), is a solution of Lur'e equatIOns

H(i)B(i) -

+ B(ik)Uk +

N

be positive. definite. Then system (1) - (3) is ES~S at gIven (J/ (l == 1, 2, ... , N) and arbitrary % (z,) E IN).

1

[ST(i)x - r(i)u]- cI>(x, u) +

(12)

Theorem 1. ~et matrices A(i) (i E IN) be Hurwitz and there eXIts at least one index k E IN such that equation (12) with i == k has a positive definite solutIOn H (k) :::: H and matrices

M(i)

x N

ATXA - X +Q = -SST, ATXB - D = sr , BT X B - R = - rT r ,

=

=

(18)

=

where Q QT and R RT are n x n and m x m nonnegstive and positive definite matrices D is n x m matrix.

.cV(x , i) == lim _1 {£[V(x(t + .6.t). i(t + ~t-O .6.t . +.6.t)) I x(t) ::: x , i(t) = i]- V(x, i)} =

We suppose that matrix (A - BR- 1 DT) is non162

where M(i) (i E IN) are nonnegative definite matrices , and construct the family of Euler matrices

singular and define the following matrices Ell

= (A -

BR- 1DT) - BR- 1BT (A -BR- 1DT)-T (Q + DR- 1DT) ,

= BR- 1BT (A -

BR- 1DT)-T , E2 1 = -(A - BR-1DT)-T(Q + DR- 1D T ), E12

The eigenvalues of these matrices are symmetric with respect to the bound of the unit circle. According to it we define di(.A) polynomials from the factorizaton relations

En = (A - BR- 1 DT)-r.

Now we construct the Euler matrix Now we construct the matrix polynomials di(E( i») according to di (,A) scalar polynomials. Let H (i) matrix is a solution of linear algebraic equation

of 2n x 2n dimension , whose eigenvalues are placed symmetric with respect to the bound of the unit circle in the complex plane.

di (E(i) [

According to it we find the polynomial ~(,A) of n degree from the following factorization relation :

H~i)

] = [

~

] .

(19)

Theorem 2. Let eigenvalues of matrices A( i)( i E IN) are smaller than 1 in modulus , rank di(E(i)) = n( i E IN) and there exits at least one index k E IN such that equation (19) with i = k has a positive definite solution H (k) = H and matrices

We construct the matrix polynomial

M(i) = H(k) - AT(i)H(k)A(i) _[AT(i)H(k)B(i) -

_~CT(i)T(i)][BT(i)H(k)B(i) -

appropriate to ~(E) , where the allocated blocks have n x n dimensions. We denote the matrix polynomial appropriate to ~1(,A) = I~(,A-l ),An by ~1(,A).

_K-l(i)T(i)t l [AT(i)H(k)B(i) -

~CT (i)T( i)f] ,

(20)

N

Under these assumptions the following lemma is valid.

8(i)

= M(i) -

I:a·f Af(i)H(i)AI(i), 1=1

K-l(i)T(i) - BT(i)H(i)B(i) , i E IN.

Lemma 2. Let rank of ~(E) matrix is equal t.o n .The matrices X , S, r satisfy the Lur'e equations (18) such that detr i= 0 if and only if the matrix X is a solution of linear algebraic equation

be positive definite. Then system (17) is ESMS at given (71 (l = 1,2 , ... , N) and arbitrary Pij(i , j E IN) . Proof: According to lemma 2 H (i) matrix , satisfying the equation (19) , satisfies Lur 'e equations system

or equivalent to it

AT(i)H(i)A(i) - H(i)

= -S(i)ST (i) ,

satisfying to condition AT(i)H(i)B(i) -

BTXB-R
= A(i) - ~B(i)K(i)C(i) ,

Ell(i) = A.K(i)

-~B(i)K(i)T-l(i)

(21)

This system is equivalent to the quadratic matrix equation

x

4 x BT (i)[AK (i)t T C T (i)K( i)T( i)C( i) ,

Edi)

~CT(i)T(i) =

= S(i)r(i) , BT (i)H(i)B(i) - f{-l(i)T(i) = = -rT(i)r(i).

We introduce the following matrices AK(i)

+ M(i) =

= B(i )K(i)T-1(i)BT(i)[AK(i)]-T ,

H(i)

1 = -[AK(i)tT[M + 4CT(i) x T x K(i)T(i)C(i)], En(i) = [AK(i)t , i E E\

= AT(i)H(i)A(i) + [AT(i)H(i)B(i) - ~

x

xCT(i)T(i))[BT(i)H(i)B(i) - f{-l(i)T(i)rl x

E21(i)

x [AT(i)H(i)B(i) -

163

~CT(i)T(i)f] + M(i) .

Therefore, if for i = k there exists positive definite solution of equation (19) and for i -# J..~ matrices M (i) are defined by formula (20) then there exists the solution of the equations system (21) : H(i) =

Special algorithms for solution of systems of type (23) are developed (Gahinet and Nemirovskii , 1993). At the same time both the approach presented in this paper and LMI approach are not effective for obtaining stability region . These methods permit to calculate H matrix , but the condition of their applications remain unknown. In this connection the presented paper can be considered as the first step on the way of obtaining of more constructive solution of the stated problem.

H(k) = H > 0. So for function Vex , i) = Vex) have

= xT H x

> Owe

.cV(x , i) = xT(AT(i)HA(i) - H + N

+ L O"IAT (i)H AI(i))x +

Aknowledgements. Author thanks the Russian Foundation for Basic Research and the Ministry of General and Professional Education of Russian Federation for financial support. Author also thanks professor Dragoslav D . Siljak from Santa Clara University (Santa Clara, California) for valuable remarks and suggestions.

1=1

~CT(i)T(i))U +

+2xT (A T (i)H B(i) -

+uT[BT(i)HB(i) - J{-1(i)T(i)]u = -[ST(i).1: - r(i)uf[ST(i)x - r(i)ut

=

N

-~( x, u) + x T [LO"l AT(i)HAI(i)1=1

5. REFERENCES

-M(i)]x ;:; xTO(i)x < 0,

Barkin , A.I ., Zelentsovsky, A .L. and Pakshin , P.V . (1992). Absolute stability of deterministic and stochastic control systems. MAL Moscow (in Russian). Boyd , S. , El Ghaoui , E. , Feron , E., and Balakrishnan , V . (1994). Linear matrix inequalities in control and system theory. (SIAM studies in Applied Mathematics; vo1.15). SIAM . PA . Gahinet , P. and Nemirovskii , A . (1993) . LMI Lab: A Package for manipulating and solving LMls. INRIA. Grujic, L.T . and Petkovski , D .B . (1987). Robust absolutely stable Lurie systems. Int. 1. Control. 46 . No.I. 357-368 Horn , R .A. and Johnson , C .R. (1986). Matrix analysis. Cambridge University Press. Cambridge. Kats , I.Ya. and Krasovskii , N.N. (1960) . On the stability of systems with random parameters . Prikladnaya Matematika i Mekhanika . 27. No.5 . 809-82:3 (In Russian). Khas 'minskii , R.Z. (1980) . Stochastic stability of differential equations . Sijthoff & Nordhoff. Netherlands. Kwakernaak , H. and Sivan , R. (1972). Linear optimal control systems. Wiley-Interscience. New York . Mariton , M. (1990) . Jump linear systems in automatic control. Marcel Dekker. New York. Pakshin , P. V . (1994) . Discrete-time systems with random parameters and structure . Fizmatlit. Moscow (In Russian) . Siljak , D.D. (1991). Dezentralized control of complex systems. Academic Press , Cambridge, MA. Yaz , E.(1995) . Robust stability of discrete time randomly perturbed systems, In: Control and Dynamic Systems (Edited by C . T. Leondes) 73 , 89-113. Academic Press. New York .

where

.cV(x , i) = £[V(Xk+1 , ik+d

I Xk

=

X , ik

= i]-

-Vex , i) . From this inequality, according to lemma 2 and (Pakshin , 1994) the result of the theorem follows.

•

4. CONCLUSION The problem of the absolute stability analysis , considered in this paper is reduced to an algebraic problem of obtaining existence conditions of positive definite matrix H = HT , satisfying the system of quadratic matrix equations

At H + HA i + (H Bi - Di)Ri 1(H B; -Dif +Qi =0 , iEIN

°

for any matrices Qi = QT > and given matrices Ai , B j , D j and Ri Rt > 0 or their discrete analogs . It is easy to see, that this problem is equivalent to obtaining solvability conditions of the system of quadratic matrix inequalities

=

At H + H Aj

+ (H B j -

Dj)Ri 1(H B j -

-Djf < 0, i E IN .

(22)

The problem of the solution of inequalities of type (22) is intensively studied in the last years within the LMI approach (Boyd , et al., 1994) . According to this approach the system of quadratic inequalities (22) is equivalent to system of linear matrix inequalities

AT H + HAi [ BTH - D j

H Bi - DT] < o, z· E IN . (2:3) _RT

,

164