Snippet: WCA Toolkit

Worst Case Analysis e162a747-c8b2-4f40-95df-39d17e3aa6ae 132 Snippet: WCA Toolkit Arie Lyles This snippet contains functions used in performing worst case anaylsis and RSS analysis on user defined functions. 3 3 true true 0 decimal

&[DATE] &[TIME] - &[FILENAME]

<content> snippet: varManager 2012.03.03 </content>

check if #var is an array #var IsArray #var IsString 0 ≡ #e #var num2str eval : #e mat(findstr 1 + (2^0 > 21 line 0 if 11 line :

check if #var is a number #var IsScalar #var IsArray ¬ (#var IsString ¬ (& 11 line :

returns rows and columns of an arrayreturns 1 if #var is a numberreturns "text" if #var is a string #var size #var IsString ¬ #var num2str mat(findstr stack 1 el 1 ≡ #var rows #var cols 21 mat eval 1 if text if :

returns summation of M.D.A. elements #var sum.1 #size #var size : #size text ≡ 0 #size 1 ≡ #var #output 0 : #i 1 #size 1 el range #j 1 #size 2 el range #var #i #j el size 1 ≡ #output #output #var #i #j el + : #output #output #var #i #j el sum.1 + : if for for #output eval 31 line if if 21 line :

compare #var1 and #var2, using  a text #bool parameter (≡,<,>,≤,≥,≠)works with numbers, strings, arrays and multi dimensional arrays #var1 #bool #var2 compare #bool #bool = ≡ (≡ #bool if : bools.allowed ≡ < > ≤ ≥ ≠ 16 mat transpose : bools.status 0 : n 1 bools.allowed length range #bool bools.allowed n el ≡ bools.status bools.status 1 + : 0 if for 41 line bools.status 1 ≡ #v1 #var1 : #v2 #var2 : #size1 #v1 size : #size2 #v2 size : #size1 text ≡ (#size2 text ≡ (| #bool ≡ ≡ #v1 #v2 ≡ #bool ≠ ≡ #v1 #v2 ≠ I can't determine if a string is larger or smaller than a number/matrix 021 line if if #size1 1 ≡ (#size2 1 ≡ (& #v1 num2str #bool #v2 num2str concat str2num #size1 1 ≡ (#size2 1 ≡ (| #size1 1 ≡ #v1 #v1 #size2 1 el #size2 2 el matrix + : #size1 #size2 : 21 line #v2 #v2 #size1 1 el #size1 2 el matrix + : #size2 #size1 : 21 line if 0 if #size1 #size2 ≡ #output 0 : #i 1 #size1 1 el range #j 1 #size1 2 el range #v1 #i #j el size 1 ≡ (#v2 #i #j el size 1 ≡ (& #output #i #j el #v1 #i #j el num2str #bool #v2 #i #j el num2str concat str2num : #output #i #j el #v1 #i #j el #bool #v2 #i #j el compare : if for for #output eval 31 line I can't compare matrix with different sizes 021 line if 21 line if if 51 line unallowed boolean: #bool concat if 21 line :

<content> Index Definitions </content>

MIN 1 :

NOM 2 :

MAX 3 :

<content> Supporting Function Definitions </content>

Sort Input takes a single input vector I# whose elements are either 3x1 vectors or scalar. It then sorts each individual 3x1 element of I#. If the nth element is a scalar, it creates a 3x1 vectorof the containing only the nth element in each row. The function returns the number of elements (rows) in I# (n#), the now sorted sub-elements of I#, and the n#-by-1 vector containing the second element of each subelement. I# SortInput n# I# rows : i 1 : i n# ≤ i i 1 + : I# i el IsArray I# i el size 3121 mat ≠ Each element in I must either be scalar or a 3x1 vector error 11 line I.NOMINAL i el I# i el sort NOM el : I i el I# i el sort : 21 line if 11 line I# i el I# i el 3 Ones * : I.NOMINAL i el I# i el (NOM el : 21 line if 11 line for n# I# I.NOMINAL 31 sys 31 line :

This function performs a brute force Extreme Value Analysis for a user-defined function and it's inputs.For a function with n inputs, it will calculate the result of function with 2^n permutations of it's inputs. The function returns the Minimum, Nominal and Maximum results._1 f I# WC.EVA n# I# I.NOM# 31 mat I# SortInput : i 1 n# range I# i el IsArray I.EVA# i el I# i el min I# i el max 21 mat : j 1 : j i < I.EVA# i el I.EVA# i el I.EVA# i el stack : j j 1 + : 21 line while I.EVA# i el I.EVA# i el sort : j i 1 + (n# range i n# < I.EVA# i el I.EVA# i el I.EVA# i el stack : continue if 11 line for 51 line I.EVA# i el 2 n#^Ones I# i el * : 11 line if 11 line for i 12 n#^range I.TEMP# n# Zeros : k 1 n# range I.TEMP# k el I.EVA# k el (i el : 11 line for result# i el I.TEMP# f : 31 line for 11 line result# min I.NOM# f result# max 31 mat 41 line :

stack(...) - Returns an array formed by placing arguments top to bottom. Arguments are matrices/

vectors having the same number of columns, or they are scalars and vectors. (A vector is a column matrix.)

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Ones("number") - Returns a vector containing "1:number" values equal to 1.

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I# n# param1 i 1 n# range I# i el IsArray param# i el I# i el min I# i el max 21 mat : j 1 : j i < param# i el param# i el param# i el stack : j j 1 + : 21 line while param# i el param# i el sort : j i 1 + (n# range i n# < param# i el param# i el param# i el stack : continue if 11 line for 51 line param# i el 2 n#^Ones I# i el * : 11 line if 11 line for param# 21 line :

<content> WC.DIFFEVA </content>

The function WC.DIFFEVA accepts an arbitrary function, f, and an array of input arguments, I. The array 'I' is an Nx1 array of inputs to the function 'f', where N N is the required number of inputs for 'f' i.e. f(x1, x2,..., xN). Each argument in 'I' must either be a scalar, or a 3x1 vector containing the min,nom, and max values for that variable.  WC.DIFFEVA performs the worst case calculation of the function 'f' for each variablein 'I' and returns a 2x1 array. Rather than computing results for all permutations of inputs,this function performs the partial differential w.r.t. each input variable and computes the,max, min and nominal results. The first result in the 2x1 array is a 3x1 array containingthe Min, Nom, Max result of 'f' for the inputs in 'I'. The second result in the 2x1 array is an Nx1 array containing the differential coefficients_1 f I# WC.DIFFEVA n# I# I.NOM# 31 mat I# SortInput : i 1 : i n# ≤ i i 1 + : diff# i el I.NOM# f I.NOM# i el diff (: diff# i el diff# i el diff# i el UoM / : diff_sign# i el diff# i el sign : I.MAX# i el I# i el (diff_sign# i el 2 + el : I.MIN# i el I# i el (diff_sign# i el 2 - abs el : 51 line for I.MIN# f I.NOM# f I.MAX# f 31 mat diff_sign# diff# I.NOM# 31 mat 21 mat 11 line 31 line :

UoM("variable") - Returns the variable SMath base Unit of Measurement; returns 1 if variable have no unit

]]> <content> WC.RSS </content>

The function WC.RSS accepts an arbitrary function, f, and an array of input arguments, I. The array 'I' is an Nx1 array of inputs to the function 'f', where N N is the required number of inputs for 'f' i.e. f(x1, x2,..., xN). Each argument in 'I' must either be a scalar, or a 3x1 vector containing the min,nom, and max values for that variable.  WC.RSS performs the RSS calculation of the function 'f' for each variablein 'I' and returns a 2x1 array. The first result in the 2x1 array is a 3x1 array containingthe Min, Nom, Max RSS result of 'f' for the inputs in 'I'. The second result is the overal % tolerance calculated.  I assume that all input variables have 3σ tolerances that result in a 3σ result._1 f I# WC.RSS n# I# I.NOMINAL# 31 mat I# SortInput : TOL.RSS_N# 0 : TOL.RSS_P# 0 : f.NOM# I.NOMINAL# f : i 1 : i n# ≤ i i 1 + : diff# i el I.NOMINAL# f I.NOMINAL# i el diff (: diff# i el diff# i el diff# i el UoM / : diff_sign# i el diff# i el sign : 31 line for i 1 : i n# ≤ i i 1 + : diff_sign# i el 0 > ΔI1# I# i el (MAX el I# i el (NOM el - : ΔI2# I# i el (NOM el I# i el (MIN el - : 21 line ΔI1# I# i el (NOM el I# i el (MIN el - : ΔI2# I# i el (MAX el I# i el (NOM el - : 21 line if TOL.RSS_P# TOL.RSS_P# I.NOMINAL# f I.NOMINAL# i el diff ΔI1# * (2^+ : TOL.RSS_N# TOL.RSS_N# I.NOMINAL# f I.NOMINAL# i el diff ΔI2# * (2^+ : 31 line for TOL.RSS_N# TOL.RSS_N# sqrt : TOL.RSS_P# TOL.RSS_P# sqrt : f.NOM# TOL.RSS_N# - f.NOM# f.NOM# TOL.RSS_P# + 31 mat TOL.RSS_N# f.NOM# / TOL.RSS_P# f.NOM# / 21 mat TOL.RSS_N# TOL.RSS_P# 21 mat diff_sign# 11 mat 41 mat 91 line :

La fonction prend les pires cas, pas forcément centré pour le calcul ce qui en cas de fonction non-normal pose problème.

The function WC.RSS accepts an arbitrary function, f, and an array of input arguments, I. The array 'I' is an Nx1 array of inputs to the function 'f', where N N is the required number of inputs for 'f' i.e. f(x1, x2,..., xN). Each argument in 'I' must either be a scalar, or a 3x1 vector containing the min,nom, and max values for that variable.  WC.RSS performs the RSS calculation of the function 'f' for each variablein 'I' and returns a 2x1 array. The first result in the 2x1 array is a 3x1 array containingthe Min, Nom, Max RSS result of 'f' for the inputs in 'I'. The second result is the overal % tolerance calculated.  I assume that all input variables have 3σ tolerances that result in a 3σ result._1 f I# WC.RSS n# I# I.NOMINAL# 31 mat I# SortInput : TOL.RSS# 0 : f.NOM# I.NOMINAL# f : i 1 : i n# ≤ i i 1 + : diff# i el I.NOMINAL# f I.NOMINAL# i el diff (: diff# i el diff# i el diff# i el UoM / : diff_sign# i el diff# i el sign : 31 line for i 1 : i n# ≤ i i 1 + : diff_sign# i el 0 > ΔI1# I# i el (MAX el I# i el (NOM el - : ΔI2# I# i el (NOM el I# i el (MIN el - : 21 line ΔI1# I# i el (NOM el I# i el (MIN el - : ΔI2# I# i el (MAX el I# i el (NOM el - : 21 line if TOL.RSS_P# TOL.RSS_P# I.NOMINAL# f I.NOMINAL# i el diff ΔI1# * (2^+ : TOL.RSS_N# TOL.RSS_N# I.NOMINAL# f I.NOMINAL# i el diff ΔI2# * (2^+ : 31 line for TOL.RSS_N# TOL.RSS_N# sqrt : TOL.RSS_P# TOL.RSS_P# sqrt : f.NOM# TOL.RSS_N# - f.NOM# f.NOM# TOL.RSS_P# + 31 mat TOL.RSS_P# f.NOM# / diff_sign# 31 mat 81 line :

The function WC.MC accepts an arbitrary function (f), an array of input arguments(I) and the number of simulations to run (N) . The array 'I' is an nx1 array of inputs to the function 'f', where n n is the required number of inputs for 'f' i.e. f(x1, x2,..., xn). Each argument in 'I' must either be a scalar, or a 3x1 vector containing the min,nom, and max values for that variable. WC.MC performs a Monte Carlo simulation assuming uniform distribution. The first result in the 2x1 array is a 2x1 array containingthe Min, Max results of the simulation The second result in the 2x1 array is an Nx1 array containing results of the N simulations._1 f I# N# WC.MC n# I# I.NOMINAL# 31 mat I# SortInput : I.MC# I# length Zeros : results# N# Zeros : i 1 I# length range I.MC# i el I# i el (MIN el N# Random I# i el (MAX el (I# i el (MIN el (- (* + eval : 11 line for i 1 N# range I.TEMP# I# length Zeros : k 1 I# length range I.TEMP# k el I.MC# k el (i el : for result# i el I.TEMP# f eval : 31 line for result# Min result# Max 21 mat result# 21 mat 61 line :

_1 f I WC.MC_1 f I# 1000 WC.MC :

This set of functions performs various worst-case analyses on other user-defined functions. The user-defined functions and their inputs must defined in a particular way in order for these functions to work properly. Each function must only accept one input vector. However, that vector may be of any size n. Each user defined function must define it's input vector within the function. Consider the following simple example:

I'd like to define a function that represents the output of this circuit in terms of V1, R1, and R2.

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I# V.OUT Defining Input Vector I P28V I# 1 el : VD I# 2 el : VZ I# 3 el : R_LIM I# 4 el : I_CAP I# 5 el : B I# 6 el : Defining Function 1 P28V VD - VZ - R_LIM / I_CAP B / - 0 mA * if 11 line 91 line :

I# V.OUT Defining Input Vector I P28V I# 1 el : VD I# 2 el : VZ I# 3 el : R_LIM I# 4 el : I_CAP I# 5 el : B I# 6 el : Defining Function 1 P28V VD - VZ - R_LIM / I_CAP B / - 0.01 mA * if 11 line 91 line :

V_ALIM VD - VZ - R_LIM / I_CAP B / -

I.TEST 28 V * 27 V * 20 V * 27 V * 10 V * 10 V * 29 V * 14 V * 15 V * 10 ohm * 7 ohm * 10 ohm * 1 mA * 10 mA * 1 mA * 10201063 mat :

I.TEST 28 V * 27 V * 29 V * 10 ohm * 1 mA * 1061 mat :

P28V 20 V * >

I.TEST V.OUT 2.80 A * -

I.TEST 1 el 28.0 V *

I.TEST 2 el 27.0 V *

I.TEST 5 el 1.00 103 -^* A *

diff# 1 el I.TEST V.OUT I.TEST 1 el diff : 0.100 A 2^s 3^* kg m 2^* / * 11 mat

diff# i el diff# i el UoM /

diff# 1 el diff# 1 el diff# 1 el UoM / : 0.100 11 mat

Define inputs to V.OUT

. I will define the inputs as 3x1 vectors in the form of [MIN,NOM,MAX]. These inputs could be scalars and the Worst Case Analysis functions will handle them properly.

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Assuming R1 is 5% 4.99k resistor

Assuming R2 is 1% 39k resistor

R.1 4.99 kΩ * 95 % * 100 % * 105 % * 31 mat * :

R.2 39 kΩ * 99 % * 100 % * 101 % * 31 mat * :

Assuming V1 can vary between 4.5-5.5 volts

V.1 4.5 5 5.5 31 mat V * :

T.MIN 40 - (°C * :

P28V 1628 31.5 31 mat V * :

T.A 25 °C * :

T.MAX 70 °C * :

ΔT T.MAX T.A - T.A T.MIN - abs Max : 65 K *

R.DRIFT 100 ppm * Δ°C / :

V.Z 18 V * 15 % * - (1 (15 % * + (31 mat * : V 17.1 18 18.9 31 mat

β 6311131 mat * :

V.CAP 8 V * 11131 mat * :

R_LIM 1 kΩ * 1 R.DRIFT ΔT * - 1 % * - 100 % * 1 R.DRIFT ΔT * + 1 % * + 31 mat * : kΩ 0.98350 1.00000 1.01650 31 mat

R.LOAD 499 ohm * 1 R.DRIFT ΔT * - 1 % * - 100 % * 1 R.DRIFT ΔT * + 1 % * + 31 mat * : ohm 490.7665 499 507.2335 31 mat

I.CAP 4 mA * 4 mA * 20 mA * + 2 / 22.3 mA * 31 mat :

VD 0.6 V * 0.05 - (V * 0 V * 0.05 V * 31 mat + : V 0.55 0.6 0.65 31 mat

P28V 2828 31.5 31 mat V * :

T.A 25 °C * :

T.MAX 70 °C * :

ΔT T.MAX T.A - T.A T.MIN - abs Max : 65 K *

R.DRIFT 100 ppm * Δ°C / :

VD 0.6 V * 0.05 - (V * 0 V * 0 V * 31 mat + : V 0.55 0.6 0.6 31 mat

V.Z 18 V * 15 % * - (1 (1 (31 mat * : V 17.1 181831 mat

1.00 1.00 - 1.00 - 1.00 - 1.00 - 1.00 61 mat

R_LIM 1 kΩ * 1 R.DRIFT ΔT * - 1 % * - 100 % * 100 % * 31 mat * : kΩ 0.98350 1.00000 1.00000 31 mat

Sign to have the WCA

I.CAP 4 mA * 4 mA * 20 mA * + 2 / 4 mA * 20 mA * + 2 / 31 mat :

β 6311131 mat * :

Performing brute force EVA=

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V.EVA_1 V.OUT P28V VD V.Z R_LIM I.CAP β 61 mat WC.EVA (: mA 9.20952 9.20952 14.01887 31 mat

P28V 11 el 10 V * > 1.00

P28V 1 el 10 V * > 1.00

Performing partial differentiation based EVA. Note results correlate exactly for this example which is to be expected.

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V.DIFFEVA Result.DIFF 21 mat_1 V.OUT P28V VD V.Z R_LIM I.CAP β 61 mat WC.DIFFEVA (:

V.DIFFEVA mA 9.20952 9.20952 14.01887 31 mat

Result.DIFF

sign of partial differential w.r.t. each variable

unitless result of partial diff

Performing RSS

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V.RSS V.RSS% Δ.MLIN_MAX M.SIGN 41 mat_1 V.OUT P28V VD V.Z R_LIM I.CAP β 61 mat WC.RSS (:

V.RSS mA 9.2095 9.2095 12.8293 31 mat

V.RSS% % 0.00000 39.30456 21 mat

V.RSS% 0.00 0.393 21 mat

Δ.MLIN_MAX mA 0.00 3.62 21 mat

M.SIGN 1.00 1.00 - 1.00 - 1.00 - 1.00 - 1.00 61 mat 11 mat

Performing Monte Carlo Analysis.

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V.MC V.MC_DATA 21 mat_1 V.OUT P28V VD V.Z R_LIM I.CAP β 61 mat 1000 WC.MC (:

V.MC V #

Summary of Results

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Brute Force EVA

RSS Method

V.EVA 0.00921 A * 0.00921 A * 0.01402 A * 31 mat

V.RSS 0.00921 A * 0.00921 A * 0.01283 A * 31 mat

Partial Differential EVA

Monte Carlo Analysis

V.DIFFEVA 0.00921 A * 0.00921 A * 0.01402 A * 31 mat

V.MC V #

V.MC V.MC_DATA V.DIFFEVA Result.DIFF V.RSS V.RSS% V.EVA V.1 R.1 R.2_1 V.OUT Clear 1.0000000