Robust Failure Detection and Isolation in Non-Linear Control Systems

Copyright © IFAC Algorithms and Architectures for Real- Time ControL SeouL Korea, 1992

ROBUST F AlLURE DETECI10N AND ISOLATION IN NON-LINEAR CONTROL SYSTEMS Alexey Ye. Shumsky Department of Radio EleclrIJnics, Far Eastern PO/l'Iechnical Institute , Vladil'OslOk, Russia

Abslracl. Using the oonoept of redundanoy relations a solution is given to the problem o~ robust failure deteotion and is?lation in non-linear disorete-bme oontrol systems. To obtam robust redundanoy relations the method of non-linear transformation of system and interval analysis is used. The method for optimal ohoosing redundanoy relations is developed.

Failure deteotion: non-linear systems: robustness; non-linear transformations: optimization; interval analysis.

KeY"'ords.

()ptimization problem. In Low and others (-1986}9.!1 al terna t i ve measure of optimum for redundancy relati,)ns, leading to far simpler optimization procedure involving a single singular value decomposition, was developed. '1'he ori terion taking into acr~o)mt relative oontribution of the effects of disturban'jes due to the model errors, noise and the failures was deseribed in Frank (1990) • It is different the 0riterion used in Lou and others (1986) which does not guarantee independence of the residual of each system input. The optimization problem, following from this criterion. reduces to a general ized eigenvalue-eigenvector problem. Solutions of robust residual generation problem described in Low and others (1986) and in Frank (1990) may be considered only as approximation to optimal d116 to their indepen,lence of operating points. In this work an optimal solution is aGhieved by using non-linear redlmdancy relai;}r)ns (even i.t' ini tial system i:::: linear) whidt are adaptive to behavi,)r of inputs of system. As rule, realization 01' Sll,}h solution does not need a r)omplex on-line optimization procedure.

INTRODUCTION An aotive approach to the design of fault tolerant oontrol systems is to first deteot and isolate a failed 00mponent and next reconfigurate the ()ontrol system to accommodate the failure, hence, an integral part of such a system is failure detection and isolation (FDI). A wide variety of techniques has been proposed in recent years to solve FDI problem (see, for example, surveys by Iserman. 1984, and Frank, 1990). In one way or another. the procedure of FDl essentially consists of two stages. The first stage ~s the generation of signals. called reslduals. whose values are nominally zero or close to zero when no failure in 5ystem is presented and must be disti~lishably different from zero under failure condi t ions. The second stage involves using residuals to make the approp~ia te der~isions. The procedures for resldual generation depend in turn on models relating the measured variables and can use full or reduoed-order observers, detection filters, Kalma.'1 filters or follow from the unified so-called parity space approach. As model errors and noise disturbance also influenoe on the residuals it leads in the issue to robust FDI, Le. the design of diagnostie prooedure that is maximally sensitive to the effeots of failures and minimally sensitive to model errors and noise disturbanee. The well known solutions of such a problem primarily 00ncern the stage of residual generation and deal with linear systems. In Chow and Willsky (1984) the design of a robust residual generation pro,~ess was form).llated aE a minimax optimization problem. Proposed criterion speoifies robustness with respect to a particular operating point, thereby allowing the possibility of adaptive choosing the best redundancy relations. However. a drawback of this approach is that. it le~ds to an extremely oomplex on-llne

PROBLEM

FORMULATION

Let system subjeeted to diagnosis described by rliserete-time model

is

x(t+l )=f(x(t),u(t),p) (1)

y(t)=h(x(t),p)

where \lent is the input veGtor, xeIt is the directly lmobservat·le state vector. YEnS is the output veetor, f and h in ')ommon case are non-linear vector fune t ions. During the normal operation the vector of uncertain parameters p takes values in a speeified subset F of Failures result in moving parameters from their toleranees. I t is supposed that measured veotors of input and output are

nm.

159

u(t)=U(t)+C(t),

y(t)=y(t)+~(t)

equal to nominal the right-hand side ot relation (7) is e1ual to zero. With the exoeption of Xi' =1,2, ... ,k, from (3), (4) we turn to input-output relation for estimation y_(t) in the next torm

(2)

where C( t ) and ~ ( t ) are zero-mean Gaussian noise veotors with famous covariance matrices. Under this oonditions the speotrum of the next problems is developed (1) defining robust redundanoy relations for noise-free conditions; redundancy (2) optimal choosing selective relations which have sensi tivity to distortions of defini fe parameters; (3) expansion of robust redundancy relations for noise conditions. Because to obtain robust redundancy relations for non-linear systems the special nonequivalent non-linear transformation of initial model is required the next questions are discussed in addition to ones mentioned above (4) special mathematical techniques; (5) speoial kind of nonequivalent non-linear transformation of model (1). ROBUST

REDUNDANCY

y.(t)=F
Let now PEP and real vector functions Fmin (t-l ). if (t-l » and FmQx cl' (t-l ). if(t-l» be the lower and upper boundary vector functions for interval vector funotion F(i'(t-l ).if(t-l ).p). Le. functions Fmin(i' (t-l ),if(t-l» and Fmax(i'(t-1 ).if(t-1» give the best approximation for all functions F(i'(t-1 ).if(t-1 ).p). pep. such that Fm,n (i' (t-l ).if (t-1 ) )~F (if (t-l ). i' (t-l ). p»~FmQX(i'(t-l ).if(t-l ». Therefore, during the normal operation of system (under peP) Fmin(i' (t-l ). if(t-l» ~ ~tP(y(t»~FmQX(i'(t-1 ).if(t-l ». The last inequali ty leads to the pair of robust redundanoy relations in the form of inequali ties

et

In order to develop a clear pioture of using redundancy for residual generation in non-linear systems. oonsider the case of following deterministio model Xl (t+1 )=f. (u(t),y(t),p), (3)

rmax (t)=q>-(t)-Fmin(i' (t-l ),if(t-l »~O

1=2,3, ... k

rmin(t)=q>-(t)-Fmax(i' (t-l ).U·(t-1 ))~O (4)

where function tP-(t) is determined as follows tP-(t)=q>(y(t». Checking these relations composes the maintenance of robust diagnostio procedure. Violation of one of these relations results from the moving at least one of the parameters from its toleranoe. Functions tP·(t)-Fmin(i'(t-l). if (t-l ». tP* (t )-Fmax (i' (t-l ).if (t-l » we call upper redundancy function (URF) and lower redundancy function (LRF) acoordingly. To resume mentioned above we formulate robust diagnostic procedure for noise-free oonditions as one of calculating URF and LRF of measured inputs and outputs taken over a finite time window and then comparison obtained values with zero threshold. The stage of transition from interval function to lower and upper boundary functions is assential for obtaining LRF and UHF • To get such functions for concrete interval function the well known procedures may be used. The most simple solution of this problem one can have when oonsidered interval function is polynomial one.

where x one may consider as the subvector of X. y_(t)=y(t). The simple way of residual ~eneration for some nominal value of p 1S the next r(t)=y(t)-y.(t) aooording which residual r(t) is generated as the difference between output of system y(t) and its estimation y. (t) obtained by open-loop generation based on the model (3), (4). Th~e to the struoture of this model behavior of the residual is independent of initial state of system for all t~k. To obtain suoh property for oommon oase of model (1) it must be transformed to the form (3 ) • (4). Because the equivalent ( i.e. keeping the input-output map ) transformation of model (1) to model (3), (4) in non-linear oase may be impossible. we assume vectors X., .•. ,Xk • y. depend on vectors X and y accordingly Xi(t)=CPi(X(t» y. (t )=q>(y(t»

y' (t-2) •..••

y'(t-k»)'. if(t-l)=[u'(t-l).u'(t-2) •.•.• u'(t-k»)' and mark ' denote the transposition.

RELATIONS

x~t+1 )=f~u(t).y(t),xit)""Xi_Jt).P)

i'(t-l )=[y' (t-l).

(5 ) (6)

MATHEMATICAL TECHNIQUES

where CPi' 1=1,2, ••. k, and tP are vector functions. As result the rule of residual generation is modified as follows

Now we give the brief introduction into special mathematical techniques. used for determination of functions ~. tP (Zhirabok. Shumsky 1 987. 1989 ). Le t G denote a set of vector funotions on Rn. A pair of vector functions a{x), ~(x)€G is said to be oonneoted by partial order relation ~ (a~~) iff there exists vector function 1 determined on the set of

(7 )

Clearly, during the normal operation of system when value of veotor p is

160

transformed to the next form

values of funotion a suoh that ~=ra. To examine the fulfillment of this relation the rank criterion of funotional dependenoe may be used. It a~~ and ~~a then a and ~ are equivalent. denoted ~~. beoause relation ~ is reflexive. symmetrio and transitive. The equivalent transformations of veotor funotions may be used for their simplifioation. Let binary operation x denote

~f

where a o ' 8 t , ••• ,a., are arbitrary veotor functions, tor which there are such values of input u~c., u=c., ••• ,u-~ that the next oonditions are fulfilled ~r (X. C•• p h~r (X, Cz,P h

~x~~[~]

aa/ ax a1/ ax

Rank [

a~/ ax

] =Rank [

lJ =Rank [

a1/ ax aa/ax. a~/ax

aa/ax

a~/ax

2: Let the funotion f3eG have one oomponent and the function ~f(x.u.p) be written as " ~f (x. u.p )=ao (x.P )+bo (u)+ E a . (x.p)b. (u)

Theorem

j=t

]

]

Theorem

Then

and if

then

f. (h(x). u.P)=
From (1), (4) and (6) we get h.(
ax~~a, ax~5~

i f
(3)

if a~~

(4)

if

(5 ) (6 )

a:s~

then then then

(9)

To expel unknown funotions fi, i=1,2 ••••• k. let us puss from (8).(9) to the funotional inequalities oonoerning veotor functio~
1: For arbitrary functions a,f3eG it is true:

Theorem

(2)

(8)

fi (h(x) .u'
a5M(~)

(a, m(a»E6 and if (a,~)E6 then m(a)~~ where TIu' TIu(x,U)=u, and TIx' TIx(X,u)=x, are projeotions. f is the vector funotion from model (1).

(1 )

(~TIxXTIU)Df~f.

Let us find the necessary and sufficient conditions in whioh nonequivalent non-linear transformation ot model (1) to model (3).(4) exists. Substituting right-hand side of relation (3) to left-hand side of (5) and right-hand side of the first relation (1) to righthand side of (5) and replacing veotor y with function h (acoording seoond relation (1» we oome to equations

TIu x ~ 5~f (a'~)E6

3: Let

m(~)~

TRANSFORMATION OF INITIAL MODEL

aD~=Ax

(M(P).~)E6

J

By USing the property (5) of operator M the result of last theorem may be spread for a multioomponent oase. The next theorem gives the rule for calculating operator m.

where A is the matrix with maximum rank whose eaoh row is linear dependent on rows as both Jacobian matrix da/dX and Jacobian matrix d~/dX. We define the binary relation 6. operators M and m as follows iff

J

where the system of funotions 1. b. (U) ••••• bs (u) is linear independent. Then M(~)~aoxa.x •.• xa... Proof: see Zhirabok and Shumsky (1987).

where and a,J/ox are the Jacobian matrices. These equalities leads to partial differential equations concerning oomponent 1 J • Its integration in common case may be diffioult. The essential simplifioation of this problem may be achieved by restriotions which are put on set G. Thus. in linear oase we have

(~'~)E6

... )(~:rcX. er,p )~a~

i=1.2 ••.. ,s. Then M(~)~oxatx ••• xa,,' Since the examination ot last oonditions may be difficult, the next rule of operator M oaloulation oan be used in series of oases.

The binary operation 0 we introduce as follows: aof3eG and i t a~1. ~1. then ao~1 It follows from definition of partial order relation that expelling funotional dependent components is the equivalent transformation for vector funotions. As result to find the vector function ~~ it is !:.utricient to determine full set of its functional independent components Ijsuch the next equali ties are fulfiled Rank [

(x. u,p )=g(ao (x) ,at (x), ••• ,a.. (x). u,p)

ax~~ m(a.):sm(~) M(a):sM(~)

m(h ):s
M(ax~)~(a)xM(~)

ID (hx
):S
(10)

i=2.3 ••••• k

Q:SM[m(a) ] 4\c:SIPh

Proof: see Zhirabok and Shumsky (1987). Let us consider the common way of operator M caloulation (Zhirabok and Shumsky, 1989). Let the funotion ~f be

(11 )

Taking into account the property (1) of operation x and property (3) of

161

operator m, we obtain from (10)q>k ~ 41c-1 ~ ... ~ q>1 At last, referring to the property (2) of operation x, we ~rQm(10)

ooma

2.

5.

fulfilling

i=2, ••• ,k

the

(12)

The

ot

determL~1ng

OPTIMAL REDUNDANCY RELATIONS We have seen earlier. one may take scalar function ~ as a function of basis functions and to obtain the pair of robust redundanoy relations. The useful expansion of the set of possible redundancy relations may be aohieved if to oonstruct funotion ~ as a function of basis funotions taken in different instants of time ~o(t), ~0(t-1 ), .•• , ~o(t-rn) Le ~(t)=Ill(~o(t)'~0(t-1), ... , ~o (t-rn) ) ) • In this oase oorresponding UHF and LRF may be determined under interval funotion Fcf'(t-1 ),if(t-1 ),p) III (Fdyki (t-1 ), if1 (t-1 )'P), ••• ,F.('t~(t-1 ).if~(t-1 ),p), ••. , F1CT'1(t-rn),if t (t-m),p), .•• , F. (t's (t-rn) ,ifS (t-rn) ,p)) j where Fj er) (t-'t) , lt (t-'t) , p} is the interval function that corresponds to basis function
,

\.-j.

'1>,,0

procedure

functions ~j is the base of Algorithm 2 ( step 1 and step 2). The possibility to determine them for eaoh j follows immediately from properties (4) and (6) of operators. The oonver~enoe of Algorithm 2 follows from the fInal value of k.

1. Take i:=2, 4li,O := m(h). Caloulate ~. 0:= m($_t oxh). 3. If function q>',0 is not equal to q> ,0 , then increment i and go to the step 2, otherwise go to the next step. 4. Determine function ~o as a minimum vector function. satisfying inequality L

For

~,~-1' ••• ,qij+1' End.

functional

Algorit..hm 1

\.,

the step

each j=k,k-1 , ••. ,1 take where ~j.is obtained from ~j by exoepting those oomponents, whioh are depend~ on oomponents of funotions

inequali ties (12), (11) is the necessary 00ndition for the existence of model (1) transformation. The sufficiency of this condition follows from the equivalence of (10). (11) and (8). (9) and possibility to pass from last equations to (3). (4) by replacing functions h, ~L wi th corresponding vectors y, XL' Among possible transformations of (1) the most interested is such that allows to get the minimum vector function~, denoted as ~o.The oomponents of ~o make form the basis for all functions ~, satisfying inequality (11). To determ~e function ~o' the next algorithm, based on resolution of (11), (12) is proposed

2.

Take j:=j+1 and go to

~1c-j+1:=~j.'

to the next inequaltiea

m(h)~q>1,m(hx~i_1)'

Thus,

and h.

~ ~oh. End.

The number of iterations, required for fulfillment of Algorithm 1, can not exceed the number of functional independent components of function ~ • ',0 Le n. To obtain redundancy relations one may ohoose any scalar funotion ~ as a function depended on basis functions. To obtain functions f,.i=1,2, •..• k, h.it is necessary to solve equations (8),(9) under famous function ~. But to solve these equations functions ~, • i=1,2, ..• ,k. must be determined first from the next algorithm.

Clearly, there are many candidate redundancy relations for a given system. The maximum sensitivity of redundancy relation to distortions of definite parameter we consider as the base for ohoosing ooncrete version. The minimum length W, of interval such that the moving parameter from its range under other parameters in their toleranoes resul ~s in violation of redundanoy relatlons we use as the criterion to oharaoterize the sensitivity of redundancy relation to parameter p,' Let rj= [1'jmi.,., • l' jma.x] be th e tolerano e for

Al gorllhm 2.

1. Take j:=1. k:= number of iterations. required for fulfillment of Algorithm 1. ~.(X):= veotor funotion with minimum number of oomponents such that. firstly, eaoh of its oomponents is depended on ')omponents of th and. seoondly, 'IJc,o function ~h depends on function ~J' 2. Caloulate funotion M(~j)' 3. I t funD t ion M(~J) depends on h. take k:=i and go to the step 5, otherwise to the next step. 4. Determine veotor funotion '+'J+1 /1\ with minimum number of oomponents such that. firstly, each of its oomponents is depending on components of ~k -),0 and, seoondly, depends on funotions ~JH

parameter Pj, j=1,2, .•• ,m. The next theorem gives the rule for determining W, as a function of if(t-1), t'(t-1). 4: Let the next oonditions are fulfilled ( 1) 1'7 '''';; 1'* m,,., (t' (t-1 ), It (t-1 )} and

Theorem

1',.ma.x= 1'i-m",,(t'(t-1),if(t-1)) are the minimum and equations

162

maximum

solutions

of

Fer (t-1 ) ,u

DIAGNOSIS UNDER

k

(t-1 Lp)-F m,n . (ylc(t-1),lt'(t-1»=0 (13 ) le Fet, (t-1 ), u (t-1 ) ,p)-F (yle(t-1),lt'(t-1»=0

NOISE CONDITIONS

The proof of this theorem is omitted due to the limited volume of the paper.

Let measured input and output veotors satisfy (2). Substituting expressions for veotors u(t), y(t) to interval function F('i'(t-1 ),It'(t-1 ),p) one obtain function FCT(t-1 ),ft(t-1 ),C(t-1 ),T/(t-l ),p) depended on measured veotors 1« (t-1 )= = [y' (t -1 ) ,y' (t -2 ) , ..• ,y' (t - k) ] " ft (t -1 ) = =[u'(t-1),u'(t-2), •.. ,u'(t-k)}' and le noise vectors C (t-1 )=[C' (t-1 ),C' (t-2),. ..,C'(t-k)J', T/k(t-1)=(T/'(t-1),T/'(t-2), . .. ,T/' (t-k»)'. Its linearization concerning vectors Ck (t-1), T/1c(t-1) about zero values of these vectors gives

To reduce problem of choosing redundancy relations to optimization one it is proposed to use the next structure of function
F(TJ'(t-1 ),1«(t-1 ),C Ic (t-1 ),T/le(t-1 ),p)::: ::::F(if(t-1 ),1«(t-1 ),ph le le +D(U (t-l ),1«(t-1 ),p)[C (t-1 ),rt(t-1 »)'

me"

concernil"la parameter p , under p.er,i;tj; J J (2) function FCt'(t-1 ),It'(t-1 ),p) is the monotonous one for pt under pjerj, i;t j , ~..,

•

•

p,e[1'imin' 1'im,,,J, Pie(r\ mex,r\ mex J and in the neighborhood of "Yi.*

1'l*

mi.n'

ma><.

mln

4>*(t)=ll j

•

a J 't(t)4>o l(y(t-'H1»

where D (if (t-l ),1« (t-1 ) ,p)= le dF(U (t-1 ),Y" (t-1 ),C" (t-1 ),T/le (t-1 ),p)

(14)

'(

under coeffioients aj'((t) determined from demand on minimum of l!)' at each time step. As result we come to the following minimax problem min max w, (It (t-1 ), 'i' (t-1 ) ,aH Pjer j

, ••

d ( Ck (t -1 ) ,T/" (t -1 ) ) under Ck (t-1 )=0, T/le(t-1 )=0. Also, function ~(t) may be performAas a function 0/ measured vectors y(t), y(t-1 ), .•• ,y(t-m) and noise vectors T/(t), T/(t-1 ), ...• T/(t-m). Its linearization about zero values of T/(t). T/(t-1 ), ... ,T/(t-m) gives

,a"m ,p)

Qj'J;

Sinoe it has a trivial solution (OJ'(=O), wi thout loss of generality we can restrict a J,(

4>(Y"'(t-1 ),Y(t),T/m(t-1 ),T/(t»::: :::
II a~'(t)=l J

,y

'J;

equations (13) admi t a closed form solution one may obtain relations for calculating optimal coeffioients a j '( at the stage of synthesis of diagnostic procedure in the form of functions a".(t)=a,..('i' (t-1 ),Ule (t-1» (15)

where ~(Y'" (t-1 )

In this Gase optimal function 4>* (t) is obtained by substituting (15) to (14). Otherwise. determining optimal coefficients needs on-line solution of optimization problem and may lead to extensive oomputation under diagnostio process. Thus, to develop the optimal robust ii iagnostic procedure one can carry out the next actions (1) to obtain function 4Jo (from algorithm 1) and to realize transformation of initial model (using algorithm 2 and relations (8),(9» for each basis function: (2) to get input-output relations for estimations Y*i' corresponded to basis functions 4>o,(y(t», i=1,2, ... ,m, in the form of interval functions; • (3) to find optimal funotions 4> (t) by solving minimax problem for some fixed index m and for each parameter of system and then to obtain oorresponding URF and LRF.

Define

It

le·

d 4>

,y( t-1 ) )=

er (t -1 ) ,y (t ) , T/m (t -1 ) , T/ (t ) )

d (T/m(t-1 ),T/(t» m under T/ (t-1 )=0, T/(t)=O.

le

T~(t)=maX{T0(t,P)'

PeP}

where positive real number T~(t,P) determines the boundary of interval [-T~(t'P),T~(t,P)] such that the value of random funotion [HUle (t-1 ) ,1« (t -1 ), Y(t ) , Ck (t-1 ), T/k (t-1 ), T/(t) ,P)= =D(1«(t-1 ),if(t-1 ),p)[C le (t-l ),T/K(t-1 )J't ~
,y

163

. . le

Ale

Fmin(Y (t-1l'U (t-1))-T~(t)~ :54> (t)~ Ale A)c :5Fmox(Y (t-1 ),U (t-1

))+T~(t)

It gives the robust redundanoy relations

in the form

rmClX (t )=ljt (t )-I'mLn ?;T(.I(t)

necessary computations under diagnostic prooess.

er (t-1 ),ir' (t-1 ))~

CONCLUSION

(16) rmLn (t)=Ij>*(t)-Fma)(CT (t-1 ),ir'(t-1 ))~ I-'

This

~ (t )=max{~ (t ,p), peP}

where ~(t,P) is the varianoe of variate 'T)

d'k ( t -1 ), ir' (t -1 ), Y(t I , Ck(t -1 ) , 'Tt (t -1 ) ,

(t),p) determined by the next relation " k

" k

le'"


,. ) dP.

dplc)

has

developed

methods

for

for FDI in non-linear dynamic systems. Two oases of noise-free and low noise conditions wereoonsidered. In both oases diagnostic procedure inoludes oalculating non-linear redundanoy functions of measured inputs and outputs of system taken over a finite time window and threshold testing obtained values. Under noise-free conditions threshold is equal to zero. Under low noise conditions threshold is determined at eaoh time step by variance that may be calculated analogous to redundancy relations as the funotion of measured inputs and outputs. Method guarantees the fulfilling redundancy relations (with set probability for low noise case) during normal operation of system. Calcula tion of covariance of noise veotors in the expression for variance allows to make diagnostio prooedure adaptive to alternation of noise .)haraoteristios. The optimization procedure was developed to obtain redundancy relations with selective (maximally high) sensitivity to definite failures. It allows t o organize not only detection but isolation of failures in system too. Using optimal redundancy relations for noise oase needs in resolving on-line optimization problem. Without signifioant loosing optimality one may overcome this drawback by using redundanoy relations which are optimal for noise-free case.

Thl!-S. robust diagnostic procedure for nOlse conditions consists in calculating URF and LRF and comparison obtained values with nonzero thresholds Tp(t) and -T (t) oorrespondingly. ca~oulating threshold T~(t) at eaoh time step one may realize by varianoe

II

work

deternlining robust redundanoy relations

:5-T~(t)

(t)

(t)

where the noise veotor pk(ti=['Y]' (t), YJ~ (t-1 ), ..•• 'T)' (t-k), C' (t -1 ) , .. ,C' (t -k) J , P (t) is the corresponding component of v~otor pk (t), K'j is the element of oovarianoe matrix for noise veotor k p (t). Other ,yvords. the expression for caloulating ()(t) may be obtain in the form of upper bovndary function for interval funotion ()(t,p). Therefore one may oalculate the rating of the variance a2 (t) at each time step by the measured veotors lr(t-1), Y«(t-1), yet) under famous covarianoe matrix.

REFERENCES

Chow, E.Y. and A.S. WilIsky. (1984). Analytical redundanoy and the design of robust failure detection systems. IEEE Trans.AuL.Conlrol. AC-29. 603-614. Frank. P.M. (1990). Fault diagnosis in dynamic systems using analytioal and knowledge-based redundanoy a survey and some new resul ts. AULOID4Lica, 26, 453-474. Isermann, R. (1984) . PrOC8SS faul t deteotion bas8d on modeling and estimation methods a survey. Automalica, 20, 387-404. Lou, x.e., A.S. Willsky and G.L. Verghese. (1986). Optimally robust redundancy relations for failure deteotion in unoertain systems. AULomaLica, 22 , 333-344. Zhirabok, A.N. and A.Ye. Shumsky. (1987) Functional diagnosis of continuous dynamio systems desoribed by equations whose right-hand side is polynomial. AULomaLion and remote control, No 8, p.p. 154-164. Zhirabok, A.N. and A.Ye. Shumsky. (1989) Funotional diagnosis of nonstationary dynamic systems. Automation and remote control, No11, p.p. 146-154.

To get optimal redundanoy relations under noise oondi t ions one mus t take into aooount the influenoe of threshold on w,' To determine optimal ooefficients ai', (t) in this oase the it era ti ve procedure is proposed. At the first its step the determining ooeffioients aj~(t) is fulfilled by resol ving lnlnlma."'{ problem under noise-free oonditions. At the each next step (1) the new value of threshold is caloulated under values of ooeffioients aJ~(t) determined at the preoeding step and (2) the new values of ooefficients are obtained under equa t ions (13) whose right-hand sides are correoted by substituting thresholds T~ (t) and -Tp (t) oorrespondingly. Prooedure is finished i t value of W, deternlined at the some

step is not less then one obtained at the preoeding step. Beoause such optimization prooedure must be realized on-line. it leads to inoreasing 164