Explode the system

dac57e8f-d225-4f08-a46d-662021c61d51 32 4 5 decimal

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

Advanced composite CPV control scheme ... on 4 'Z' accumulated values

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

CPV Composite Past Value ≡

<= 'm'   length SetPoint initial m 60 :

<= 'sp' length SetPoint final sp 60 :

X j 1 m range X j el 1 : for X 21 line :

U j 1 sp range U j el 0.5 : for U 21 line :

u X U stack :

XY u X U stack : j 1 u rows range v j el j : for v u augment 31 line :

XY XY :

a 0.34468 1.1408 0.2369 31 mat :

b 2.2752 1.79063 0.47236 31 mat :

P 1 : Δ 0.025 : δ Δ P * : 31 sys

E recursive discrete algorithm s 1 el 0 : s 2 el δ a 1 el * : s 3 el δ a 1 el b 1 el * a 1 el a 2 el + (+ (* : s 4 el δ a 1 el a 2 el + a 3 el + (b 1 el b 1 el (2^+ b 2 el - (a 1 el * + a 2 el b 1 el * + (* : k 5 u rows range s k el Δ a 1 el u k 1 - el * a 2 el u k 2 - el * + a 3 el u k 3 - el * + (* 1 b 1 el s k 1 - el * - b 2 el s k 2 - el * + b 3 el s k 3 - el * - (* - : for i 1 s rows range x i el i : for x s augment 81 line :

a b c spec . 5 red 13 mat : sp XY a b c spec pts : 21 sys

a b c spec . 8 blue 13 mat : algo E a b c spec pts : 21 sys

'sp' is the sytem response

algo sp 2 1 sys

This advanced recursive CPV . . .

Once the vector XY is established, and the corresponding

'Δ' ... this algorithm is adjusted by only ONE parameter 'P'.

It reacts on the 2nd step change from the last accumulated

'Z=0' . . . 'P' shifts the algo response, an often desired

feature from the control scheme.

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setpoint X XY 1 col stack sp XY 2 col stack 12 mat :

response X E 1 col stack algo E 2 col stack 12 mat :

system setpoint response augment :

Explode the system

system X 1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798991001011021031041051061071081091101111121131141151161171181191201211 mat sp 111111111111111111111111111111111111111111111111111111111111 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1211 mat X 1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798991001011021031041051061071081091101111121131141151161171181191201211 mat algo 0 0.0086 0.0567 0.1567 0.3021 0.4766 0.6605 0.835 0.9854 1.1018 1.1799 1.22 1.2265 1.2064 1.1679 1.1195 1.0686 1.0214 0.9824 0.9539 0.9368 0.9304 0.9331 0.9425 0.956 0.9714 0.9864 0.9996 1.0098 1.0167 1.0202 1.0206 1.0187 1.0151 1.0106 1.006 1.0017 0.9981 0.9955 0.994 0.9934 0.9937 0.9945 0.9958 0.9972 0.9986 0.9998 1.0007 1.0013 1.0016 1.0017 1.0015 1.0012 1.0007 1.0003 0.9999 0.9996 0.9994 0.9992 0.9992 0.9992 0.995 0.971 0.9212 0.8486 0.7615 0.6696 0.5824 0.5072 0.449 0.41 0.3899 0.3866 0.3966 0.4158 0.44 0.4654 0.489 0.5085 0.5227 0.5313 0.5345 0.5332 0.5285 0.5217 0.5141 0.5066 0.5 0.4949 0.4914 0.4897 0.4895 0.4904 0.4922 0.4944 0.4968 0.4989 0.5007 0.502 0.5028 0.5031 0.5029 0.5025 0.5019 0.5012 0.5005 0.4999 0.4994 0.4991 0.4989 0.4989 0.499 0.4992 0.4994 0.4996 0.4998 0.5 0.5001 0.5001 0.5002 1211 mat 14 mat

Ramp tracking control system

a 0.894839 - : b 1.790649 : c 0.104191 : d 0.06125 : 41 sys

0.894839 - 1.790649 0.104191 0.417733 41 mat

user u 5 :

up dn locate vert j 1 up range v j el j dn + : u j el locate j 0 * + : 21 line for u v augment 21 line :

X j 160 range X j el j u / : for X 21 line :

f s 1 el 0 : s 2 el 1 : n 360 range s n el a s n 2 - el * b s n 1 - el * + c X n 2 - el * + d + : for i 1 s rows range x i el i : for x s augment 31 line 31 line :

Vert 305 - 60 vert :

x u / f X X rows el Vert 4 1 sys

Recursive behaviour 2nd order system

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

a 0.894839 - : b 1.790648 : c 0.104191 : 31 sys

a b + c + 1

SetPoint SP 1 :

Shrink/Expand ε 0.75 :

Title 2 0.375 - response 2nd order under SetPoint 12 blue 15 mat :

UnitSP s 1 el 0 : s 2 el 0 : n 3100 range s n el a s n 2 - el * b s n 1 - el * + c SP * + : for i 1 s rows range x i el i : for x ε * s augment 51 line :

a b c spec . 8 blue 13 mat : Pts UnitSP a b c spec pts : 21 sys

Pts SP Title 3 1 sys

Fibonacci and nodes labels

plotG(vx,vy,a,b,c)

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

Fibonnacci incremental algorithm λ Fib set Exponential threshold data n 1 + el MCD ORIGIN 0 ≡ 21 sys n 1 λ range Fib n 0 + el 15 sqrt + 2 / (n^15 sqrt - 2 / (n^- (5 sqrt 5 / * : idx n 0 + el n : 21 line for idx Fib 0 round vectorize augment 31 line :

Create your own vector size n Vect n 1 ≡ size ct 1 : j 1 n range a ct 1 el size : ct ct 1 + : 21 line for v a 1 el : k 2 a rows range v v a k el stack : for v 51 line if :

15 sqrt + 2 / (n^15 sqrt - 2 / (n^- (5 sqrt 5 / * 15 sqrt + (n^15 sqrt - (n^- 5 sqrt 2 n^* / ≡

13 Fib transpose 123456789101112131123581321345589144233213 mat

λ 10 :

Fibonnacci infinitesimal sequence
OPTIMIZ  ... Numeric vx λ Fib 1 col : vy λ Fib 2 col : 12 mat x f vx vy x cinterp : x c 11 x ≤ (x λ 1 + ≤ (& cases : data vx vy . 14 black plotG : 41 sys

nodes 12345678910110 mat transpose : NodesLabel nodes var2str vectorize : X vx : Y vy : 12 mat n X rows : Graph nodes Y augment : 51 line

nodes_label X Y NodesLabel 10 n Vect red n Vect augment :

Graph 1121324355687138219341055102 mat

x f x c * data 1 25 1 25 - 2 2 mat nodes_label 4 1 sys

plot arguments x f x c * data 125125 - 22 mat nodes_label 41 sys

Solving difficult Electron function

a 816 π * sqrt * : 351.67021

Range discretizer ...
Mesh N  within the bounds L, H L H N xd X 0 : i 1 N range X i el L H L - N / i * + : for X 31 line :

x y Φ 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + :

L 1.7857 : H 14.245529 : N 199 : 13 mat U L H N xd : 21 line

U 1 el L :

==================================================

For solvability, the low limit 'L' is short of some decimals. The high limit 'H' is lot more complaisant .

t0 1 time :

sols i 1 U rows range data 1 i el U i el y Φ y solve eval : for v data 1 el : j 2 data cols range v v data j el augment : for v transpose 41 line :

Isolate new sols eval :

branch1 U new 1 col augment : branch2 U new 2 col augment : 21 sys

L < u < H ... escape end-points u 10 :

x pairwise user x y Φ y solve : x user 12 mat 21 line :

pairs u pairwise 12 el : 4.1749 - 4.1749 21 mat

pair wise u pairwise 10 4.174903 - 4.174903 21 mat 12 mat

1 time t0 - min 1.9

branch1 branch2 u pairs 1 el u 0 2 2 mat u pairs 2 el u 0 2 2 mat 1.7857 0 + 100 black 1 5 mat 14.2455275 0 + 100 black 1 5 mat u 0 . 14 black 1 5 mat 7 1 sys

branch1 1 row 1.7857 0.1696 - 12 mat

branch1 12 el 0.0001 - :

branch1 1.7857 0.0001 - 1.9109 0.378 - 1.9735 0.4678 - 2.0361 0.5457 - 2.0988 0.6162 - 2.1614 0.6815 - 2.224 0.743 - 72 mat

branch2 1.7857 0.0001 1.9109 0.378 1.9735 0.4678 2.0361 0.5457 2.0988 0.6162 2.1614 0.6815 2.224 0.743 72 mat

Fibonacci in-situ command line

Fibonnacci incremental algorithm command line λ Fib n 1 λ range Fib n 1 + el 15 sqrt + 2 / (n^15 sqrt - 2 / (n^- (5 sqrt 5 / * : idx n 1 + el n : 21 line for idx Fib 0 round vectorize augment 21 line :

15 sqrt + 2 / (n^15 sqrt - 2 / (n^- (5 sqrt 5 / * 15 sqrt + (n^15 sqrt - (n^- 5 sqrt 2 n^* / ≡

n 21 :

n Fib n 1 + row 211094612 mat

n Fib transpose 012345678910111213141516171819202101123581321345589144233377610987159725844181676510946222 mat

A NOT compliant recursive project

x f x abs sin 0.5 * x 2^(* :

u 3 π * - x ≤ (x 3 π * ≤ (* :

x f u / 1 1 sys

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

clean the system L H N Clear 1

One branch discrete solver/inverser export f L H N Lo Hi solveDiscrete Lo/Hi are the range of the solver ω 0 : sol 0 : 12 mat refresh the solver before use ≡ dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x Lo Hi solve eval : for sol 61 line :

f x f u / :

N 400 :

set upper solver a bit higher than in x ≤ 3*π solveHigh f 0 π N 03 π * 0.05 + solveDiscrete :

solveHigh 1 row 0 3.1416 6.2832 9.4248 41 mat 11 mat

solveHigh solveHigh rows row 6.4351 9.3531 21 mat 11 mat

V solveHigh Unest : i 1 V rows 5 range : j 2 V rows 6 range : branches V i el V j el augment : idx 1 branches rows range : sols idx branches augment : sol_1 sols 1 col sols 2 col augment : sol_2 sols 1 col sols 3 col augment : 81 line

sol_1 sol_2 0 π o 7 black 1 5 mat 3 1 sys

sol_1 rows sol_2 rows 21 sys 32632621 sys

sol_1 branches rows row sol_2 branches rows row 21 sys 326 6.4348 12 mat 326 9.3533 12 mat 21 sys

pairs sol_1 sol_2 augment :

The recursive inverse demo ...

x f x 2 π * / 2 π * x / tanh * sqrt :

Inversing a function

t0 1 time :

0.001 f 0.0126

In Mathematics, an InverseFunction is the solution of this function. f(x) is the model of difficulty, documented in Mathematica 4.0. The seeded solve scanner is of no gain versus the direct "Sol". The 'roots' can't be applied wrt the derivative that does not take discrete vector of values. In lieu, the 'solve' block concludes pretty well.. a bit slow. _____________________________ Note: the function starts existing around 'x = 0.02, f(0.02) = 0.0564' our choice to solve is wise , down to lower possible limit. The upper limit 0.99 is practical. In fact, at 1, the solution is infinite.

RHS  branch discrete solver/inverser export f L H N Lo Hi solveDiscrete Lo/Hi are the range of the solver ω 0 : sol 0 : 12 mat refresh the solver before use ≡ dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x Lo Hi solve eval : for ω sol augment 61 line :

compliant solve Sol f 0.033 0.99 150 0.00001 20 solveDiscrete :

Magnify the function to see

50 x f * Sol 2 1 sys

The native function

x f

This exercise is purely didactic/demo. No further exploration will be applied. It aims at setting up the solver wisely. The f(x) does not plot part of the low region, this default is due to Smath too granular in this region.

x f x 2 π * / 2 π * x / tanh * sqrt :

The other way around undocumented "root"

w sol t f w - t root : w 2 sqrt * π sqrt * - t 2 π * t / tanh * sqrt + 2 sqrt π sqrt * / t root

Maple expression var root ≡ (is live in Smath 6179 but undocumented ≡

For solving, we equate the Maple 'root' to 0 and include in the Smath solve block c/w the 'roots' range. Indirectly, Mathcad/Mathsoft "tracking" implements a version of Maple "root" [or similar to Maple], but active by clicking in the curve for quick coordinates.

The symmetric solver within the ± range

40 f 0.9959

Given 'y' ...
solve for 'x' y x Sol y 2 sqrt * π sqrt * - x 2 π * x / tanh * sqrt + 2 sqrt π sqrt * / 0 ≡ x 40 - 40 solve :

ordinate(s) datum y 0.75 :

within the 'x' range ± 40 x 0.99 ≤

y x Sol 3.804022 - 3.804021 21 mat

up dn locate line u 0 : v 0 : 12 mat j 1 up range v j el j dn + : u j el locate : 21 line for u v augment 31 line :

Neg 21 - y x Sol 1 el line :

Pos 21 - y x Sol 2 el line :

Curve tracking Mathcad/Mathsoft style

x f y Neg Pos 4 1 sys

3.8042 ± f 0.75 0.75 i * 0.75 i * 0.75 41 sys

3.80402 ± f Re 0.75 00 0.75 41 sys

y y x Sol 12 mat 0.75 3.804 - 3.804 21 mat 12 mat

jmGiraud: 2015_08_23

Part 1: simple and cumulative Σ(M,s,e,col)

v algo v sum :

v 369121551 mat :

u 1234551 mat :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

v algo u algo 21 sys 451521 sys

M s e col Σ vy M s e col col submatrix : Σ 1 el vy 1 el : i 2 vy rows range Σ i el vy i el Σ i 1 - el + : for Σ 41 line :

V 109875 π 61 mat :

M 12345678933 mat :

V 161 Σ 1019273439 42.1416 61 mat

V 151 Σ 101927343951 mat

U 191 Σ 1.7857 3.6966 5.6702 7.7063 9.8051 11.9664 14.1904 16.477 18.8262 91 mat

U M Unwrap : 14725836991 mat

U 391 Σ 79142225314071 mat

M sum 45

Part 2: recursive-cumulative Σ(M,s,e,col)

x 110 1.25 range :

Y x 1 x rows 1 Σ :

F x ln vectorize :

NUMERIC p x Y augment :

N x F augment :

P F 1 F rows 1 Σ :

Project x P augment :

p Project 2 1 sys N 2 1 sys

The Smath Gaussian 'Ksmooth' ...

ksmooth # plotG

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

σ 0.3708158988058244 :

The Gaussian ksmooth

x K 12 π * sqrt σ * / x 2^2 σ 2^* / - exp * :

Smath recognises the silent 'bw', something
like the Mathcad "global definition". j X Y Ksmooth X i el X j el - bw / K Y i el * (i 1 X rows sum X i el X j el - bw / K (i 1 X rows sum / :

Gaussian Ksmooth Kernel

x K

X Y bw 13 mat

W j 1 X rows range s j el j X Y Ksmooth eval : for s 21 line :

W 11 mat

raw samples XY 02116121212209932120421135209862067720718207092056102083112090122072132064142081152074162054172056182058192061202 mat :

Prepare and code the Smath project

t0 1 time :

X XY 1 col : Y XY 2 col : 21 line

bw 5 : (bandwidth ≡ W W : (isolate W ≡ XW X W augment : 31 line

W 11 mat

The working path: collect & return [X,Y]=>[bw]=>[Ksmooth(j,X,Y,bw)]=>[W]=>[XW]

XW X W o 7 red plotG : raw X Y + 7 black plotG : 21 line

===== Finish/collect the project =====

spline join/mesh the smoothed data set Cfit t 019 0.1 range : i 1 t rows range v i el X W t i el cinterp eval : for t v augment 31 line :

collect plot plot Cfit raw XW 31 sys :

1 time t0 - 3 s *

XY=>raw data, XW=>smoothed data, Cfit=> meshed spline export

plot

Exponential recursive smoothing

plotG(vx,vy,char,size,color)

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

XY 171270369468564665772878975107511751270122 mat :

X XY 1 col : Y XY 2 col : 21 line α 0.3 : 12 mat

x c 10 x ≤ (x 13 ≤ (& cases :

Exponential smoothing [input] S x 0 : S 1 el Y 1 el : 12 mat j 2 XY rows range S j el α Y j el * 1 α - (S j 1 - el * + : for i 1 S rows range x i el i : for x S augment 41 line :

vx S 1 col : vy S 2 col : 21 line

n Predict X vy n cinterp :

input X Y o 5 black plotG : smooth vx vy o 5 black plotG : 21 line

If you need 'D' in the PID mode, a window of 3 smoothed inputs is required.

plot XY S input smooth 41 sys :

Exponential smoothing of input data

plot 1 1 sys

S 1712 70.7 3 70.2 4 69.5 5 67.9 6677 68.5 8 71.4 9 72.4 10 73.2 11 73.7 12 72.6 122 mat

smooth x Predict x c * 2 1 sys

Ksmooth this ▼ data set

ksmooth ... plotG

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

σ 0.3708158988058244 :

The Gaussian ksmooth

x K 12 π * sqrt σ * / x 2^2 σ 2^* / - exp * :

Smath recognises the silent 'bw', something
like the Mathcad "global definition". j X Y Ksmooth X i el X j el - bw / K Y i el * (i 1 X rows sum X i el X j el - bw / K (i 1 X rows sum / :

Gaussian Ksmooth Kernel

x K

X Y bw 13 mat

W j 1 X rows range s j el j X Y Ksmooth eval : for s 21 line :

W 11 mat

Prepare and code the Smath project

raw 171270369468564665772878975107511751270122 mat :

t0 1 time :

X raw 1 col : Y raw 2 col : 21 line

bw 3 : (bandwidth ≡ W W : (isolate W ≡ XW X W augment : 31 line

W 11 mat

Original raw : XW X W . 10 black plotG : raw X Y + 7 black plotG : 31 line

collect plot plot Original raw XW 31 sys :

===== Finish/collect the project =====

1 time t0 - 1 s *

n Predict X W n cinterp :

x Predict x c * Original XW 3 1 sys