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CONSTRUCTION OF ACCEPTABILITY REGIONS FOR PARAMETRIC RELIABILITY OPTIMIZATION
O.V.Abramov, Y.V.Katueva and D.A.Nazarov
Institute for Automation and Control Processes, Far Eastern Division of the Russian Academy of Sciences, 5 Radio Street, Vladivostok, 690041, Russia e-mail: abramov@iacp.dvo.ru
Abstract
The problem of representation and analysis of analog technical devices and systems acceptable regions is introduced. This problem occurs during designing and controlling in view of parametric dithering. The algorithms of constructing circumscribed parallelepiped, representation of acceptable region as a set of non-overlapping parallelepipeds are offered. The algorithm of acceptable region centre of mass computation is offered as the example of utilizing the acceptable region representation.
J. Introduction
The engineering system parameters are subject to random variations and the variations may be considered as non-stationary stochastic processes. The conventional methods for choosing parameters (parametric synthesis) generally do not take account of parameters field deviations from their design values. As a result the engineering systems designed in such a manner are not optimal in the sense of their gradual failure reliability.
Construction of acceptable region is necessary for solving problems usually arising in reliability theory. The following topics are among them:
• evaluation of the system's working capacity with account of parameters deviations from their design values,
• evaluation of parameters sensitivity and emphasizing the key input parameters,
• assigning tolerance values for parameters,
• choosing the most fit (optimal) nominal values for parameters of system components.
• reducing calculation time for parametric synthesis problem when the characteristics of random processes X(t) of system parameters variations are known [1].
The acceptable region construction in the space of input parameters is the one of steps in parametric synthesis solution. The task often can be cumbersome due to large dimension of the varying parameters space.
Most of known acceptable region construction methods can be applied on 3D space only. Some methods are efficient for a priori convex and simple connected regions. Other methods need information about form and orientation of the acceptable region.
2. Parametric reliability optimization problem
Suppose that we have a system which depends on a set of n input parameters x=(x1,..., xn). We will say that system is acceptable if Y(x) satisfy the conditions (1):
a <Y <b (1)
where Y, a and b are m-vectors of system responses (output parameters) and their specifications, e.g. Y1 (x) - average power, Y2(x) -delay, Y3(x) - gain.
The inequalities (1) define a region Dx in the space of input (system) parameters
Dx = {x | a <Y <b} (2)
Dx is called the region of acceptability for the system. Figure 1 illustrates such a region. The values given for a and b are the specifications for the system.
Fig. 1. Region of acceptability Dx defined by system response functions
The engineering system parameters are subject to random variations (aging, wear, temperature) and the variations may be considered as stochastic processes:
X (t) = { Xi(/),..., Xn (t)}
In general parametric optimization (optimal parametric synthesis) problem can be stated as follows [1].
Given the characteristics of random processes X(t) of system parameters variations, conditions of acceptability (1) and a service time T, find such a deterministic vector of parameter ratings (nominal values) xr=(x1r ,...,xnr) that the reliability
Pr(xr, T) = Pr {[(t),X2(t),...,Xn(t)] g Dx, Vt g [0,T]} = max (3)
was maximal.
The practical algorithm of the stochastic criterion calculation is based on the conventional Monte Carlo method.
At the beginning, the random vector of parameters is generated (this vector means random manufacturing device realization), and then the internal parameters degradation is simulated using degradation model. For example, parameters variations can be approximated as follows
m
X (t) = £ xruk (t)
k=0
where xk is a random variable; {uk (t)}mk=0 are continuous deterministic functions of time.
The Monte Carlo method approximates Pr(xr,T) by the ratio of number of acceptable realizations (falling in region Dx)-Na to the total number of trials - N.
p = V
Pr /N,
Unfortunately, often the region Dx is unknown. It is given only implicitly through system's equations and the system response functions. If we do not know the region Dx a Monte Carlo evaluation of probability Pr(xr,T) at particular nominal value xr requires N system analyses for each trial set of parameter xr. Typically, hundreds of trials are required to obtain a reasonable estimation
for Pr(xr,T).
Optimization requires evaluation of the probability Pr(xr,T) for many different values of parameters nominal values xr.
The acceptable region constructing problem can be stated as follows.
The first step in constructing acceptable region is narrowing the search area in a space of internal parameters. A circumscribed parallelepiped is constructed for this purpose. The next step consists in approximation of Dx with a set of non-overlapping n-dimensional parallelepipeds within circumscribed parallelepiped.
3. Circumscribed box Construction 3.1. Definition of circumscribed box
Let information about variations of internal parameters is given as limits of their possible values, i.e.
^min < ^ < ^max , * = 1 n (5)
The region inside of space of internal parameters is a n-dimensional orthogonal parallelepiped called tolerance box Bd:
B , = {x g Rn | x. • < x. < x. , i = 1,n} d 1 1 i min i i max }
with the volume
Vd = ^T ( X' max Xi min )
(6)
(7)
The box B0 circumscribed about acceptability region Dx is the n-dimensional orthogonal parallelepiped [2] within the tolerance box Bd.
Bo = {x g Rn | a0 < xt < b° Vi = l~n} (8)
The circumscribed box's volume is represented as
where a° = min xi, b° = max xi.
xg Dx xeD_
Vo =n(b- -a,0)s
i=1
(9)
x2
Circumscribed Box
Acceptability Region
xi
Fig. 2: The circumscribed box
The construction of the circumscribed box may narrow the area for construction of the acceptable region Dx and decreases computational cost of stochastic estimations. There's no need to
i=1
perform expensive scheme's simulations at the points outside the circumscribed box because the conditions (1) are not satisfied at those points.
The construction of circumscribed box and acceptability region's volume with the method of statistical testing is similar to computation of multiple integral with Monte-Carlo method.
The algorithm of circumscribed box construction
Let the tolerance box's boundaries (5) are known and the conditions of acceptability (1) are
given.
The task consists in overlapping of the tolerance region with uniform points and testing satisfaction of the condition (1) at the each point.
The random point coordinates is a set of uniform numbers \ = (<%,...,%„) distributed on
x. <£. < x , i = 1, n .
i min ^ i i max > >
To find components of the vectors aO, bO we set the initial values a° = x.max and b° = x.min, i = 1, n, Ng = 1. Ng is amount of the points where conditions (1) are satisfied.
An acceptability of the system (satisfaction of the conditions (1)) is checked at the every test. Depending on result of the test, the value of the indicator function g is set.
[1, if conditions (1) are satisfied [0, if conditions (1) aren't satisfied If acceptability conditions (1) are satisfied at the point % the following values are set: aO = min(aO), b° = max(bf), i = M, Ng = Ng +1 .
Having performed N point simulations, we compute acceptability region's volume estimation:
1 N
vg / vo = N 2 g£1) (10)
N 1=1
or
- Ng
^ = Vo-f, (11)
where VO - the circumscribed box's BO volume, Ng - amount of points inside of the acceptability region.
This method enables us to solve several problems of practical importance:
a) Region Dx is multiply connected region.
b) Region Dx fall outside the limits of the tolerance box Bd.
c) Region Dx include the tolerance box (B0 g Dx, Ng = N).
d) Region Dx and tolerance box Bd are mutually disjoint (Dx nB0 = 0, Ng = 0).
This method can be easily implemented on parallel computers with MPP architecture. The main condition for result accuracy is absence of interprocessor correlations [2, 3]. In the beginning of the process, there is an initialization of parallel random-number generator, then each process is initialized with the amount of tests to perform. Upon termination of calculations each of the processes passes computed ratings to the main processor which forms a final rating.
Following the given necessary tests amount N and processors amount k, necessary tests amount m is assigned to each processor (the sample volume on each processor), thereby k ■ m = N. The final estimation of the acceptable region is computed with the formula:
V* = V0 Z Nj /N, (12)
j=1
where N - necessary tests amount, NG - the amount of points within the region of acceptability for each of the processors.
The boundaries of the circumscribed box BO are computed with the formulas: Vi = 1, n a0 = min(a° ), j = 1, k,
— O O — , (13)
Vz = 1, n bO = max(bO), j = 1, k
where aO = min xz, bO = max xz for each of k processors.
J xeDx J xeDx
The communication overheads are not considerable because they are reduced to primary initialization of the random numbers generator and obtaining of final estimations.
K. The multidimensional probing method
The region of acceptability within the tolerance box Dx e Bd can be defined with the
methods based on multidimensional probing of the tolerance box Bd.
Let's consider the algorithm of acceptability region construction. It is based on the concept of matrix tests on reliability.
The circumscribed box BO is divided into a set of non-overlapping elementary boxes. Let's perform separation of circumscribed box BO with the hyperplanes parallel to its bounds into l = (/j,..., ln) slices per each coordinate.
Then we have non-overlapping elementary boxes with bounds parallel to corresponding bounds of BO. The length of elementary box's rib parallel to z-th coordinate axis is lz times smaller than the corresponding rib of BO. So we have:
l1 l2 ln
BO=UU...UBk1,k2,..kn 04)
k=1 k2 =1 kn =1
The total amount of elementary boxes is
R = fpz . (15)
i=1
The step (i.e. the length of elementary box's rib) on the each parameter axis is
h = (b0 - a0)//t, i = m. (16)
Let's assign the central point of each elementary box as a point-" representative". The coordinates of point-representative for the elementary box Bk, k k are computed as follows:
xz = a° + ^^, z = 1~n. (17)
If acceptability conditions (1) are satisfied at the point-representative of the elementary box Bk, k k this elementary box belongs to a set of "good" elementary boxes, otherwise it belongs to a
set of "bad" elementary boxes. Note also that
lim-^ = VDx / VBo , (18)
R—^
where Mg is the amount of "good" elementary boxes, R is the total amount of elementary boxes. VD / VB is the relation of acceptability region volume to the volume of circumscribed box BO.
The figure consisting of "good" elementary boxes is defined as D°x. The figure D°x approximates the acceptable region Dx.
The «-dimensional array C[l1,12,...,ln ] is corresponded to the circumscribed box BO divided
into elementary boxes Bk, k k . Each element of the array stores the state of corresponding
elementary box (i.e. "good" or "bad", accordingly 1 or 0). The total amount of array elements can be found using (15). This array also can be called as n-dimensional matrix. In the text below it is called matrix.
Now consider the representation of n-dimensional array using 1-dimensional array by placing its elements sequentially in a row.
In common case the elements of that array A[R] are placed as follows: A = {(1,1,1,...,1), (2,1,1,...,1),...,(/1,1,1,...,1),
(1,2,1,...,1), (2,2,1,...,1),...,(l1,2,1,...,1),
(1,3,1,...,1),(2,3,1,...,1),...,(l1,3,1,...,1),..., . (19)
(1,12 ,1,...,1), (2, 12,1,. . .,1),. . .,(l1,12 ,1,...,1),...,
(1,12,1,.., l«-1 ,1), (2, 12 ,...,2),..., (l] , 12 ,..., ln-!, ln )}
The index p of the array element, corresponding to the elementary box Bk, k k is computed as follows:
n—1
p = k + k(k2 —1)+l^k —1)+...+nz- (kn —1)—1.
(20)
¿=1
In some kinds of tasks it is necessary to compute a set of indexes (kj, k2, k3,..., kn) having the array index p and the amount of quanta (l1,l2,l3,...,ln) per each coordinate axis. To compute a set of indexes (k1, k2, k3,..., kn) it is necessary to do the following sequentially:
k =
№
+1,
k„—1 =
p—kn-nz
_i=1
n—2
n l
+1,
(21)
k2 =
k =
p—k n — kn—1 -nl —...—k, n
h
+1,
p—K n — kn—1 n —...—k, n — k2 -1
+1
For example in the case of 3D parameter space the (21) looks as follows:
k3 =
k2 =
1 - h
+1,
p — k3 -11 -12
h
+1,
k1 =[p — k3 • l1 • l2 — k2 • l1]+ 1
i=1
i=1
i=1
i=1
i=1
i=1
i=1
i=1
The procedure of the array A[R] filling consists in follows.
Let the bounds aO, bO of the circumscribed box and the amount of quanta per each coordinate is known. Initially the amount of "good" elementary boxes is Mg = 0.
(17).
For the each index in = 1, ln; in-1 = 1, ln-l ; ...; i1 = 1, l1 :
1. Compute the coordinates of the elementary box's xz z z representative using the formula
2. Compute the indexp of the array A unit using the formula (20).
3. Compute the output system parameters y(xz ).
4. Check the acceptability conditions (1).
5. If conditions (1) are satisfied A[p] = 1, Mg = Mg +1.
6. Else A[ p] = 0.
As the result we have an array A[R] that represents circumscribed box BO divided into
Mg Vg
elementary boxes and the amount Mg of "good" elementary boxes. If
RV
the array A[R]
represents the circumscribed box BO and the acceptable region Dx.
Figure 3 illustrates rendered elementary boxes that represent approximation of the acceptable region.
Fig. 3: Rendered "good" elementary boxes
The procedure of the array A[R] filling can be performed in parallel mode. This algorithm can be presented as a distributed process that requires l} processors for implementation. The algorithm of calculating the center of gravity for acceptable region constructed of elementary boxes.
In fact the distribution laws of system parameters variations and the characteristics of random parameters degradation processes X(t) are often unknown. In some cases the center of gravity for acceptable region may be the numerical solution of the parametric optimization problem.
Let's consider the acceptable region D° as a system ofM unit-mass material points.
The center of gravity coordinates C = (x^,..., x-n) of the body can be found as follows:
M
Z h (j)
x- = j1-,
Cz M
where M is normalized mass of Dx (amount of elementary boxes Dx consists of), Ii (j)are static moments with respect to corresponding axis of each of the elementary boxes.
Static moment I(Bk k k ) of the corresponding elementary box Bk, k k is calculated in
the following way:
kj, if K.,„ e Dx
1 ^kl,k2,...,kj -1
for j = 1, n . Since
center mass coordinates can be found on whole circumscribed box B0 using the formula:
l1 l2 ln
H..ZI i (Bkik2,...kn )
I] (Bkl,k2, . ,kn) 1 0, if Bki,...k £ D0
Since Do is represented with "good" elementary boxes inside the circumscribed box B0 its
x = ki =1 k2 =1 kn =
XCi ~
M
for ki = 1, ^ kn = 1, In .
When center of gravity is being calculated simultaneously with construction of acceptable region using one-dimensional array the procedure of filling the array C[l1,l2,...,ln] (A[RJ) and
calculating the center of gravity C = (x^,..., x^n) is in follows.
Let the boundaries aO, bO of the circumscribed box, the amount of quanta 1 = (l1, l2, l3,..., ln) on each of the axes and the lengths h = (h1v.., hn) of quanta on each of the axes are known. Let also the amount of "good" matrix elements M = 0 and SI = (SI1,..., SIn) is the array of static moments sum with respect to n coordinate axes.
L. For kn = 11„; kn-1 = 1 ln-1; ...;k1 = 1,l1 :
1.1.the coordinates xk1k2 kn of point-representative for the current elementary box are calculated:
h —
xi = a0 + hk, —-, i = 1, n . 2
1.2.The system response functions y(xk1k2 kn) are calculated.
1.3.The acceptability conditions (1) are checked. 1.4.If conditions (1) are satisfied
C[k1, k2,..., kn ] = 1, M = M +1,
SIt = SIt + k, i =
1.5.otherwise C[k1, k2,..., kn ]. 2. The gravity center coordinates for Dx0 are calculated:
SIt . 1-
^ = —, i = 1, n .
C'M
As the result the array C[l1, l2,...,ln] (A[RJ) is filled, the amount M of "good" elementary boxes and the coordinates of the center of gravity C for D° are obtained.
In the case when the array C[l1,l2,...,ln] (A[RJ) was filled previously the procedure of calculation of the gravity centre for D° consists in follows.
1. For kn = 1, ln; kn-1 = 1, ln-1; k = 1, A:
if A(, k2,..., kn ) = 1, then SI, = SI, + k,, z = 1, n. 2. The gravity center coordinates for Dx0 are calculated:
SIt . t-
x—. = —-, z = 1, n .
CzM
Hence it is necessary to perform exhaustive search of the array C[lj,l2,...,ln] (A[R]) elements for calculating of the gravity center of the body.
Having found the center of gravity for Dx0 the inverse mapping on acceptable region is performed the following way:
7 0 0
b - a j • r~ xCz =-x-, z = 1, n.
Cz 7 Cz5 5
lz
Conclusion
The circumscribed box may narrow the area for construction of the acceptable region and decreases computational cost of stochastic estimations. Computation of internal parameters ratings with lack of distribution density functions for both input and output parameters requires obtaining of acceptable region characteristics. On the basis of the proposed methods and the algorithms a computer-aided reliability-oriented design system called CARD has been developed. The CARD system states mathematical models and calculates ratings of component parameters with the highest precision, acceptability (manufacturing yield) or reliability of engineering systems under design.
References
[1] O.V.Abramov and K.S.Katuyev. "Effective methods for parametric syntheszs of stochastic systems'', Proc. First Asian Control Conference, Tokyo, vol. 3, pp. 587-589, 1994.
[2] O.V.Abramov, Y.V.Katueva, and A.A.Syponya . "Parallel and dzstributed algorithms for optimal parametric syntheszs", Proceedings of the 4-th Asian Control Conference, Singapore, pp.1573-1577, 2002.
[3] O.V.Abramov and Y.V.Katueva "Apptication of parallel computing technzques for stochastic optimization problems", Proceedings of the 5-th Asian Control Conference, Melbiurne, 2004, pp.1573-1577, 2004.
