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Automatica, Vol. 20, No. 1, pp. 51 .57. 1984 Printed in Great Britain

A Singular Perturbation Model of Reliability in Systems Control* R. K R T O L I C A t

Analysis and design of control systems that provide reliable behaviour in the presence of small input and output random perturbations. Key Words--Singular perturbation; reliability; stochastic control; mean exit-time; decentralized control; Suboptimality. Abstract--In many engineering systems, the intensity of the input noise is small and the state trajectory is mainly due to the deterministic part of the system structure. When this is the case for a white input noise, the mean exit-time from the reliability region may be represented as a solution to a partial differential equation with a small parameter multiplying the highest derivatives,. Hence, the evaluation of the mean exit-time from the reliability region reduces to the solution of a singularly perturbed partial differential equation for which an asymptotic solution can be evaluated analytically in a fairly general case. The mean exit-time is then used to measure the reliability of some decentralized control policies for linear stochastic systems.

region of reliable functioning of the system, called shortly the reliability region. When the free (deterministic) system has an exponentially stable equilibrium in the reliability region, the mean exittime depends mainly on the behaviour of the system inside the region and the given geometry of the region. In that case the mean exit-time assumes for dynamic systems the role that mean-time-to-failure ( M T T F ) plays in classical reliability theory. In m a n y engineering systems, the intensity of the input noise is small and the state trajectory is mainly due to the deterministic part of the system structure (Ventcel and Freidlin, 1979; Schuss, 1980; Meyer and Parter, 1980). W h e n this is the case for a white input noise, the mean exit-time from the reliability region m a y be represented as a solution to a partial differential equation with a small parameter multiplying the highest derivatives, (Ventcel and Freidlin, 1972; M a t k o w s k y and Schuss, 1977; Kamin, 1978). Hence, the evaluation of the mean exit-time from the reliability region reduces to the solution of a singularly perturbed partial differential equation for which an asymptotic solution can be evaluated analytically in a fairly general case. The mean exit-time is then used to measure the reliability of some decentralized control policies for linear stochastic systems (Krtolica and Siljak, 1982; Krtolica, 1981).

1. INTRODUCTION RELIABILITYof dynamic systems is often related to stability (Siljak, 1980). This characterization may be extended to stochastic systems when the input noise becomes small near the equilibrium state, (Ladde and Siljak, 1981). In general, however, this is not the case even for small input noise. F o r instance, the states of a time invariant system with an additive, stationary, Gaussian input noise are almost certainly drifting far off the equilibrium even if the variance of the noise is arbitrarily small. In this case, the conventional concepts of stochastic stability (stability in the mean and the like) cannot be used to describe the behaviour of the system. However, as long as the state of the system remains in some n e i g h b o u r h o o d of the equilibrium, it corresponds to some intuitive idea of satisfactory behaviour (which m a y be experienced in analog integration with k n o w n characteristics of the drift of the integrator). To pursue that idea, reliability of dynamic systems is characterized by the mean exittime from a given n e i g h b o u r h o o d of an equilibrium state of the system. This n e i g h b o u r h o o d is the

2. EVALUATION OF THE MEAN EXIT-TIME Let the state x = 0 in R" be the single asymptotically stable (in the large) equilibrium of the system ~t = f(xt)

* Received 20 August 1982; revised 22 May 1983.The original version of this paper was presented at the IFAC Workshop on Singular Perturbations and Robustness of Control Systems which was held in Ohrid, Yugoslavia during July 1982. The published proceedings of this IFAC meeting may be ordered from Pergamon Press Ltd, Headington Hill Hall, Oxford OX30BW, U.K. This paper was recommended for publication in revised form by associate editor M. Jamshidi under the direction of editor A. Sage. t Mathematical Institute, Knez Mihailova 35, 11000Beograd, Yugoslavia.

(1)

with the initial condition x0 at t = O. We assume that the vector function f(x) is continuously differentiable in x and satisfies f(O) = O. When the system (1) is perturbed by a small Gaussian white noise, the resulting stochastic process x, m a y be represented as a solution to the stochastic differential equation dx~ = f(x~)dt + eG(x~.)dvt 51

(2)

52

R. KRTOLICA

with the same initial condition x~ = x, where the n x 1 vector v, denotes a standard Wiener process with Ev, = 0, E(dvt dye) = Idt (I denotes the n x n identity matrix) and G(xt) is a symmetric positive definite n x n matrix. The positive number e is a small parameter. It is well known that for some fixed number ~: > 0, the process xf will leave any neighbourhood of x = 0 with probability one, if the matrix G(x) does not degenerate (Khasminskii, 1969). Thus, the time spent by state trajectories (2) in the neighbourhood of the deterministic equilibrium x = 0 may be used to characterize the behaviour of the perturbed system. The neighbourhood F c R" of x = 0, in which we consider the states to be close enough to the equilibrium for some practical purpose, is referred to as the reliability region of the system (2). The corresponding boundary is denoted by 6F, (F = F w 6F). In order to characterize the reliability of the system (2), we consider the first exit-time of x . (Xo = x e F), from the reliability region F z~ = inf{t:xe F , x ~ F }

(3)

and its mathematical expectation E~z ~. It is well known (Dynkin, 1965) that v~(x)= E~z~ is the solution of the Dirichlet boundary value problem

~tr/G (x)(~x2]+ ? ~ - x f ( X ) = - 1 , C2

t

2

~21)a /

~D ~

v~(x) = 0,

xeF

x ~ aF

?~v

(4)

[ av

~v

where x = ( x l , x2 . . . . . x")T' ~--CX= I?~Xt , ~Xz , . . . , --

&.l

,--=

&2

~

,(i,j=l,

lax~c,xj]

2 ....

(5)

(6)

e~0

where ),v = inf V(x) xet~V

(7)

and the function V(x) satisfies the partial differential

(8)

in the reliability region x e F, with V(0) = 0. 3V Equation (8) implies V(x) > 0, and ~ # 0 for UX

x # 0. If there exists a smooth function ~(x) and a smooth vector function/(x) such that f(x) = - ~ G

(x) t ? x ]

+/(x)

c3mxx/(x)= 0

(9)

then the function V(x) is given explicitly by V(x) = @(x)

(10)

(Matkowsky and Schuss, 1977). Turning our attention back to equations (6) and (7), we remark that the function V(x) does not depend on the region F, while the number )~r > 0 is independent of the initial state x~ = x. Only the points of contact between the boundary t~F and the largest equipotential surface of the function V(x) affect the value of )w. Hence, )Lr characterizes the behaviour of the field f(x) in the presence of small diffusion within all neighbourhoods of x = 0, included by, and having points of contact with one equipotentential surface of the function V(x). This characterization is not restricted to the case when the equilibrium of (1) is asymptotically stable. Indeed, it follows from (5) and (6) that lim e,21nE0 z~ = 0

In general, there is no explicit solution to this problem. Whenever the equilibrium x = 0 of (1) is asymptotically stable, it is possible to find an expression for the leading term in the asymptotic expansion of the solution to the boundary value problem (4) (Ventcel and Freidlin, 1979, Ch. 4, Theorem 4.1 ) lim e 2 In E~z ~ = )-r

1 ~V 2 [('~Vl"f (qV f ( x ) = 0 2cTxG (x) v3x! + ¢?x

n).

When F = {x:llxll < g}, (0 < R < +Gc, II'll denotes the Euclidian norm of the indicated vector), f(x) = 0, G(x) = I, the explicit solution v~(0) is, (e.g. It6 and McKean, 1965), given by Eo z~ = R2/(nc2).

equation

( 11 )

~0

for the system (2) in the reliability region Ilxll < R, when f(x) = 0 and G(x) = I. This result should have been expected, since the equilibrium of the system does not oppose any attractive force to the diffusion. As the systems with unstable equilibrium cannot be better in this respect, we infer that when the equilibrium x = 0 of the system (1) is not asymptotically stable, the number 2r in (6) is always equal to zero. The rigorous proof for this fact can be found in Ventcel and Freidlin, 1979 (Ch. 6, Theorem 5.3, p. 262). Hence, we are motivated to introduce the following: Definition 1

The stochastic system (2) is reliable with degree )~ in the reliability region F if there exists a positive number 2 such that l i m F.2 e~O

In Ex v': ~> 2

holds for all initial states x e F.

(12)

Singular perturbation model of reliability Hence, a dynamic system with unstable equilibrium is always unreliable when perturbed by small diffusion. The system is always reliable with some degree 2 ~<,~r in the neighbourhood of an asymptotically stable equilibrium. When the degree of reliability 2 is chosen within a safe margin below the upper bound 2r, it is not significantly affected either by small displacements of the contact points, or by small variations of the field fix) provided the equilibrium remains stable. The degree of reliability can be evaluated explicitly for all linear time-invariant stochastic systems, as shown by the following:

Theorem 1. The linear time-invariant stochastic system dx~ = Ax~dt + eGdvt

(13)

with x~ = x ~ F is reliable with degree 2 ~< 2r, where 2r is evaluated according to (7), V(x) is given by 1 T

1

V(x) = ~x H - x

(14)

and H is the symmetric positive definite solution to the Liapunov equation AH + HA T = - G 2.

(15)

This theorem specifies the results of Matkowsky and Schuss (1977) for a linear system. When the region of reliability F is given as

I'={x :~x 1 vD-Ix < 1}

(16)

with a positive definite matrix D, the degree of reliability of the system (13) is given immediately by •~v = 2m(D H - ')

(17)

where '~m denotes the minimal eigenvalue of the indicated matrix. For D = ½R21, the reliability region is the same as in (5) and 2r = ~R22~1(H)

53

When h ( x ) = p(x), where p(x) is the probability density of the initial random state vector x~ for the system (13), the probability Pr(F) that the initial state x~ will be found in F is equal to the total cost Pr(F) = fr h(x) dx. In that case, among all initially equally probable neighbourhoods of the asymptotically stable equilibrium state, the one in (19) allows the largest degree of reliability for the system (13). The mean exit-time Exz ~depends then upon the condition that the process x~ in (13) starts at x~ = x. With this interpretation o r E : ~, we may consider also the case when any initial state x is penalized by a nonnegative cost function r(x), (r standing for risk). If we now put h(x) = r(x).p(x),

:r h(x)dx-- 7(F)

we can think of 7(F) as of the average risk of starting the process in the region F. Then, among all domains F with the same initial risk (and a stable equilibrium as an interior point), I~ is the one that allows the largest degree of reliability of the process (13). We mention the special cases r(x) = 1 and h(x)

=- l, where frh(X)dx = Pr(F), frh(X)dx = area (F) respectively. The average initial risk is measured only by the size of the region of reliability F in the latter case. In any case, the region of reliability I~, that allows the largest degree of reliability for the process (13), remains unchanged, and in this sense we say that f" is proper to the system (13) with the stable equilibrium in 1~. Moreover, there is always a positive number eo = eo(F) such that whenever < %, the mean exit-time Exr[~ from the region I~ is larger than the mean exit-time E,,z~- from a region F, for all allowable F # r" (Ventcel and Freidlin, 1972). The largest degree of reliability allowed in the region 1~ proper to the system (13) satisfies

(18) 2r = 1

where 2M denotes the maximal eigenvalue of the indicated matrix. An interesting reliability region is obtained from (16) when D = H. Suppose that all states x in every allowable reliability region are penalized by a cost function h(x), and that we are looking for a region 1~ that provides the maximal degree of reliability for some fixed total cost / h(x)dx. Then, according to Fd

(20)

which means that the upper boundary 2r for the degree of reliability of any other allowable reliability region with the same initial risk 7(F) satisfies 2r ~< 1.

(21)

3. SYNTHESIS OF THE MOST RELIABLECONTROL Let us consider a linear time-invariant stochastic system dx~ = Ax~ d t + u(y~)dt + eGdvt

Ventcel and Freidlin (1972), 1 XH - I x < 1~. 1~ = {x:~x

J

dy~ = Cx~ dt + eG ,, dwt

(19)

(22)

with the initial state x~ = x~ F and the constant

54

R. KRTOL1CA

matrices A, G, C, G ,, of the size n × n, n × n, q × n and q × q, respectively. The matrices G and G w are symmetric and positive definite. The n × 1 control vector u depends on the q × 1 output vector y~, which is corrupted by an independent vector Wiener process of size q × 1. Both the input and the output noise are considered to be small. The problem of control synthesis consists in finding the n × q constant matrix K in the linear control law u(y,)dt = - Kdy,

(23)

which maximizes the largest allowable degree of reliability 2r of the system (22) in the reliability region F. We shall refer to this control law as to the most reliable control law. This problem is solved by the following: T h e o r e m 2.

If the matrix pair (A, C) is observable, the most reliable linear control law for the system (22) in the reliability region F is given by uO(y,)dt = _ pCTG ~2 dy,

(24)

where matrix P is the unique positive definite solution of the algebraic Riccati equation AP + PA T - p C T G ~ 2 C p + G 2 = 0.

(25)

The upper b o u n d of the degree of reliability for the system (22) with the control law (24) is given by 2 ° = inf if(x)

(26)

xe?V

where 1 T

V(x) = ~x P

Xx

(27)

the system matrix A - KoC, (Ko = pCTG,~ 2) iS stable (Hurwitz). For the reliability region in (16), the largest allowable degree of reliability is given by ,~o = /.m(DP- 1)

(28)

R2 and when D = ~ - ' 1 20 = 1 R Z 2 ~ l(p).

(29)

2

Let us now use the degree of reliability to measure the reliability of decentralized control policies (Krtolica and Siljak, 1982; Krtolica, 1981). It is assumed that the system (22) can be d e c o m p o s e d into s interconnected subsystems described by ni × 1 state and input noise vectors x~, vi, and by qi x 1 output and output noise vectors yi, w~, (i = 1, 2,.. s), where x x = [xT, x ~X. . . . . x~X3, Vx = [ v I , .,

v~ . . . . . v~T ], yV = [y~, W I = [w[, y~, .... y[], w], .... wsr], n = n l + n z + ' " + n s , q=ql+q2 + " " + qs. (The shorthand notation x and y is used here instead of xt and Yr.) We denote A = AD + A o where AD = diag {A1,A2 . . . . . A~}, Ac = (Aifl, and the size of the matrices Ai, Aij, (i,j = 1, 2 . . . . . s), is n i x ni and ni x n~, respectively. We assume that the initial subsystem states are mutually uncorrelated, and that G = diag {G~, G2 . . . . . Gs}, G , v = d i a g {Gwl, G w2 . . . . . G wsl, C = diag {C1, C2 . . . . . C~}, where the size of the matrices Gi, G wi, Ci, (i = 1, 2 . . . . . s), is ?li ?K i,li, qi × qi, and qi × hi, respectively. Observability of both (AD,C) and (A,C) is assumed. When the system (22) is disconnected (Ac = 0), (i.e. when (22) becomes a set of isolated subsystems), the control law that minimizes the steady-state variance (t ~ + ~ ) is given by equations (24) and (25), (e.g. K w a k e r n a a k and Sivan, 1972), where the matrices G and G w are replaced by eG and e.G ~, so that the corresponding steady-state covariance 1 matrix P~D and the control gain matrix KD = ~.

P~DCTG~, 2 are block-diagonal P~D = diag {P~D~, P~D2 . . . . . P,:D~}, KD = diag {KD1, KD2 . . . . . KDfl~, (the size of the submatrices P~oi and KD~, (i = 1, 2 . . . . . st, being n~ × n~ and n~ × qg, respectively). If we denote by PD = diag {PD1, PD2 . . . . . Pos} the solution of the unaltered equation (25), we remark that P~D = e2pD and that the control gain matrix KD = PDCTG w 2 remains unchanged in both cases and does not depend on ~. In fact, there is no difference between the steady-state m i n i m u m variance control law u(y,)dt = - K D d y , , and the most reliable control law (24). By applying the decentralized control law u(yf)dt =-KDd~ to the interconnected system (22), (Ac # 0), we obtain a state covariance matrix P+ and a corresponding weighted mean square of steady-states (a + )z which differ from the matrix PD and the mean square of steady-states (0-o)2 calculated for the set of isolated optimal closed-loop subsystems. When a u t o n o m y of each subsystem is regarded as essential, the optimal disconnected system m a y be used as reference, and the degree of suboptimality, (Krtolica and Siljak, 1982) of the control law - KD dy~ for the system (22) is defined as a positive n u m b e r p such that the inequality (a+)2 ~< I~- 1 (ao)2 holds for all weighting vectors a for which (a+)2 = aTp+a, lao)2 = aVpDa. T h e o r e m (2.9) in (Krtolica, 1981) shows that the control law - K o d y ~ for the system (22) is suboptimal with degree/~o = )-M ~ (PD l p ~ ), if and only if the matrix P+ is finite. It can be shown that ~t ~< #o, i.e. that #o is the maximal admissible degree of suboptimality of the decentralized control law u(y0dt = - K D d y t for the interconnected system (22), (Ac # 0).

Singular perturbation model of reliability We observe now that the degree of reliability of the connected closed-loop system (22) with decentralized control u(~)dt = - K o d y ~ in the reliability region f" =

applying Theorem 1, and using (15), we obtain

5.8333]'

tt = [4.2083 -4.8542

4.2083-1 H -1

x:~x P~ x < 1 proper to

the most reliable disconnected system, is bounded from above by 2r = 2~ 1 ( p ~ l p + ) (see Fig. 1). We conclude that the upper bounds for the degree of suboptimality and the degree of reliability (defined as above) are identical #o = 2r. (30) To illustrate various aspects involved in analysis and synthesis of reliable control, we consider a twodimensional system (22) with

55

=

[

-0.3968

-0.39681 0.4577 J'

When the reliability region F is given as TF = ~ x( : ~ x1D

D=~L 0

1x < l ~ ,tw h e r e

0]

(33)

b2

we use (17) to calculate

A 1,0,1 2 - 2 ,C = [0:01 1.4142

~r = 0.1375 a2 + 0.1145b 2 - x/(0.0189 a 4 + 0.0131 b4 + 0.0079 a2b2).

G = [2.3452

2.50951,Gw = [0.70071 021

(32)

(31)

(34)

When a = b = R, we have

and begin with reliability analysis for u - - 0 . By

2r = 0.0523 R 2.

(35)

xvr~R~(P~) f'~M(p')

x~po-'x =i,(aI% II.-

XD

XT(P*)-'X =X ~c xT( p+)-lx =1

X Fc ^ = X-~(po-, p')

~,/'~U( P D) /

/~c~-~-m(P • ) X^

FIG. 1. Graphical illustration of the two-dimensional case.

56

R. KRTOLICA

We note that ,~-r > 0, which is in agreement with the fact that the matrix A is Hurwitz, (2~(A) = -2.618, 22(A) = -0.382). According to (19), the reliability region proper to the system (22), (and specified by u - 0 and (31)), is bounded by the ellipse

with the state covariance and control gain matrices

0.55x 2 - 0.7936XlX2 + 0.4577x 2 = 1.

When the decentralized control u(y~)dt = - KD dy', is applied to the disconnected system (optimal case), the state-feedback matrix AvD = AD -- KDC is given as

(36)

In order to design the most reliable control (23) for the system (22) with parameters (31), we remark that the matrix pair (A,C) is observable and that Theorem 2 applies. By solving (25), we obtain

-0.3333

0.6667 4 .(37)

01

01

1.3852 'KD =

0.4897 ' (44)

Avo=

I

- 1.94 0

0 ] -2.69 "

In the reliability region F = { x : ~ x l

(45) D ix

The gain matrix K = K0 in (23), and the closed-loop state-feedback matrix AF are calculated according to (24) and AF = A - KoC:

where the matrix D is given by (33), we evaluate the largest allowable degree of reliability as 2FD =

min (0.9365 a 2, 0.6926 b e ).

(46)

When a = b = R, (46) reduces to 0.7071 j , AF = [0.6--6267 In the reliability region F =

03

)~rD = 0.6926R2.

x:~x D - ~ x < 1 ,

(47)

The reliability region proper to the most reliable disconnected system is bounded by the ellipse

where D is given by (33), we find according to (26), (27),)o = 2m(DP 1), that is 1.9353 )o = g1 [a 2 + b2 -- x/'(a 4 + b4 -- a2b2)].

(39)

When a = b = R, (39) reduces to )o = 0.1667R 2.

(40)

We note that reliable control design increases the largest allowable degree of reliability of the closedloop system with respect to the largest allowable degree of reliability of the open-loop system, (cf. relations (35) and (40)). The reliability region proper to the closed-loop system (22), (23) with (31) and (38) is bounded by the ellipse 0.3333(x 2 - x~x2 + x2z) = 1.

(41)

In order to illustrate on the same example the relation between reliable control design and suboptimal decentralized control design, we introduce the notation AD = [ ; 1

0 2 ] ,Ac = [02 0;5]

(42)

where A

= AD +

Ac.

(43)

We note that (AD, C) is still observable, and evaluate the decentralized control law (23) for the system (22) with (31 ) by minimizing the steady-state variance of the disconnected system (assuming Ac = 0). As mentioned earlier, the control law that minimizes the steady-state variance is given by (24) and (25)

+ tl'6644[

= I.

(48)

We apply now the decentralized control u(y~)dt = -KDdy~ to the interconnected system, (Ac is given by (42)), and evaluate the state-feedback matrix Arc = A - KDC as Avc=

[-1.9365 2

--

;;59

26

J

(49)

its eigenvalues are ,~l(Avc)=-3.3836, ,~2(Avc) = - 1.2455. To analyse the reliability of the closed-loop system obtained, we use Theorem 1 and evaluate the matrix P+ as the solution of the Liapunov equation AFcP + + P+A~c = - ( G 2 +

KDG2K~).

(50)

Hence we have p+

[2.0386 0.6415q + =L0.6415 1.8616J '(P ) ~ = [ 0.5502 -0.1896] [_-0.1896 0.6025j

In the reliability region F =

(51)

1 t } x:~x D ~x < 1 ,

where the matrix D is given by (33), we evaluate the largest allowable degree of reliability as 2rc = 0.1376a 2 + 0.1507 b 2 - x/(0.0189 a 4 + 0.0227 b~ - 0.0324 a2b 2)

(52)

which reduces to ,;~rc = 0.1924

R2

(53)

when a = b = R. The reliability region proper to the

Singular perturbation model of reliability interconnected closed-loop system with centralized control is bounded by the ellipse

de-

0.2751x 2 - 0.1896x~x2 + 0.3013x22 = I.

(54)

The degree of suboptimality is now used to measure the effect of the decentralized control law on the performance of the interconnected closed-loop system relative to the effect of the same control law on the performance of the referent disconnected closed-loop system. As mentioned earlier, we use the relation #o = 2~ ~(PD aP+) to evaluate the largest allowable degree of suboptimality /~o of the decentralized control law -KDdy~ for the interconnected system (22), (23), (31). We obtain /~o = 0.5089.

(55)

On the other hand, (17) implies that the largest allowable degree of reliability 2/-c of the interconnected closed-loop system (22), (23), (31) with the decentralized control law -KDdy~ in the reliability region proper to the disconnected closedloop system (defined by its boundary in (54)) is evaluated by the same formula )~tc = 2~ t (PD l p+). Hence, we also have 2i~c = 0.5089.

(56)

4. CONCLUSIONS

Relying on some recent results in the asymptotic evaluation of mean exit-times for stochastic processes with small diffusion, the concepts of reliability region and of the appropriate degree of reliability are proposed. In system analysis they are used to measure the instability of the perturbed system in the given domain. The synthesis of a most reliable control is demonstrated to be feasible. For large, linear, weakly coupled systems, the degree of suboptimality of the decentralized control is interpreted as an appropriate degree of reliability of the system. Acknowledgements The author would like to acknowledge the reviewers helpful suggestions for the improvement of the paper. The research reported herein was supported by the Community for Scientific Research in Beograd, Yugoslavia, under the Project on System Theory, Control, and Numerical Mathematics No. 5354, at the Mathematical Institute, Beograd. REFERENCES Dynkin, E. B. (1965). Markov Processes. Academic Press, New York. It6, K. and H. P. McKean (1965). Diffusion Processes and their Sample Paths. Springer, Berlin. Kamin, S. (1978). Elliptic perturbation of a first order operator with a singular point of attracting type. Indiana Univ. Math. J., 27, 935. Khasminskii, R. Z. (1969). Stability of systems of differential equations with random parameters (in Russian). Nauka, Moscow. Krtolica, R. (1981). Suboptimality of the decentralized state estimator. In J. E. Marshall, W. D. Collins, C. J. Harris and D. H. Owens (Eds.), Third IMA Conference on Control Theory. Academic Press, London, pp. 881-899. Krtolica, R. and D. D. Siljak (1982). Correction to Suboptimality of decentralized stochastic control and estimation. IEEE Trans. Aut. Control. AC-27, 515.

57

Kwakernaak, H. and R. Sivan (1972). Linear optimal Control Systems. John Wiley, New York. Ladde, G. S. and D. D. Siljak (1981). Multiplex control systems: stochastic stability and dynamic reliability. 20th Conference on Decision and Control, San Diego, CA, 16 18 December. Matkowsky, B. J. and Z. Schuss (1977). The exit problem for randomly perturbed dynamical systems. SIAM J. appl. Math., 33, 365.

Meyer, R. E. and S. V. Parter (Eds.) (1980). Singular Perturbations and Asymptotics. Academic Press, New York. Schuss, Z. (1980). Theory and Applications of Stochastic Differential Equations. John Wiley, New York. Siljak, D. D. (1980). Reliable control using multiple control systems. Int. J. Control, 31,303. Ventcel, A. D. and M. J. Freidlin (1972). Some problems concerning stability under small stochastic perturbations. Theory Probab. Applic., 17, 269. Ventcel, A. D. and M. J. Freidlin (1979). Fluctuations in dynamical systems subject to small random perturbations (in Russian). Nauka, Moscow. Wonham, W. M. (1968). On the matrix Riccati equation of stochastic control. SIAM J. Control, 6, 681. APPENDIX

Proof of Theorem 1 Equation (15) implies

and therefore, if we choose V(x) according to (14), we get

0v I

0 x Ax + 2

G2/°v/T1 / 0 x / J = 0.

(A.2)

Hence, equations (8) and (9) are satisfied by V(x). As (14) implies also V(0) = 0, the function V(x) fulfills the conditions required by Matkowsky and Schuss (1977, Sections 3 and 4) for the evaluation of 2r in (7).

Proof of Theorem 2. To find the optimal linear control law, we rearrange (22) and (23) to obtain dx~ = (A - KC)x~ dt + e(G dv~ - KG~dwz)

(A3)

and remark that Edit = 0, E(d~td~/r) = (G 2 + KG~KT)dt when d~ = G d v l - KGwdwt, due to the independence of the Wiener processes v, and Wr Using the representation dx~ = (A - KC)x~dt + e(G 2 + K G 2 K r ) 1/2 d~',

E~ = O, Ed~d~ x : I dr.

(A4)

Theorem 1 and equation (7), we obtain 2 r = min V(x) x~t~F

1

T ~

V(x)=-x 2

1

P- x

(A - KC)~ + ~(A - KC) T + K G ~ K T + G 2 = 0.

(A5)

As the matrix pair (A,G) is controllable, the matrix A - KC is stable, and the matrix P - P is positive semidefinite, (Wonham, 1968), when the matrix P satisfies (25). The observability of the pair (A, C) implies positive definiteness of the matrix P and of the

matrix P, thereafter. We observe now that OF does not depend on control, and hence the minimization and maximization on the right-hand side of the relation 2°r = m a x m i n V(x) u

(A6)

xc~F

may be interchanged. We denote l?(x) = m a x V(x)

(A7)

u

and remark that when K is equal to [iCTG-,,2 in the control law (23), equation (A5) reduces to (25) with solution P = P which maximizes the quadratic form V(x), as prescribed by (A7).

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