<p> Utilities </p>

7bde70bf-1dc8-4a15-a0a7-7cf61ade1748 24 3 5 decimal

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

<p> Utilities </p>

M Low High Scale High Low - (M M Min - M Max M Min - 106 -^+ / (* Low + :

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 :

XFR models => a ≥ 1 ... integer/fractional

Computes 7 decimals  ... a≥1
Plots on native canvas only.
Plots discrete as well ...  a≥1
Mathcad computes/plots any 'a' α x γ t α 1 -^t - exp * t 1012 -^x int :

7 decimals expected  ... a≥0
Plots on native canvas only.
Γ(a,x) upper limit 100 is specific
to Smath  ... from built-in ? α x Γ t α 1 -^t - exp * t x 100 int :

α 3 :

end 10 :

Optimiz: Symbolic γax x 1012 -^end 0.1 range : i 1 x rows range γ i el α x i el γ eval : for x γ augment 31 line :

Optimiz: Symbolic Γax x 1012 -^end 0.1 range : i 1 x rows range Γ i el α x i el Γ eval : for x Γ augment 31 line :

Optional scaling to meet your collected data set(s)

vx γax 1 col : vy γax 2 col 01 Scale : xy vx vy augment : 31 line

X Γax 1 col : Y Γax 2 col 01 Scale : XY X Y augment : 31 line

x f α x γ :

f a b Int Romberg singular limit cumulative Σ w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 1012 -^1 int * 41 line :

xy XY 2 1 sys

f 1012 -^end Int 14.007

α 2 :

end 5 :

x c 0 x ≤ (x end ≤ (* :

Expose the integrand α t f t α 1 -^t - exp * :

α x f x c *

Optimiz: Symbolic model x 0 end 0.05 range : i 1 x rows range y i el α x i el f eval : for x y augment 31 line :

model 00 0.05 0.048 0.1 0.09 0.15 0.129 0.2 0.164 0.25 0.195 62 mat

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 61 line :

x f x α 1 -^x - exp * : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

t0 1 time :

max f 0.5 1.5 y1 y2 ≥ 1012 -^Extrema :

max 1

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

1 e / 0.368

max f 0.368

1 time t0 - 0.2 s *

α x f x c * max max f + 30 red 1 5 mat 2 1 sys

sub xy 152512 submatrix :

sub 1 col sub 2 col Slope sub 1 col sub 2 col Intercept 21 mat 0.266 0.208 - 21 mat

I 0.24 - : S 0.275 : 12 mat

Tangent u 1 - 5 0.99 - range : i 1 u rows range T i el I S u i el * + eval : for u T augment 31 line :

a b c spec . 5 black 13 mat : Line Tangent a b c spec pts : 21 line

Reaction Curve model ... Instrumentation

Line xy 2 1 sys

Explain the project: Process Control & Instrumentation

In real industrial projects, some loops are not easy to configure

from the design board [pH, Temperature ...]. The graph at left is

called Reaction Curve that can be captured from step impulse

and reading from recorder. Optimum control parameters P,I,D

are concluded from the empirical Ziggler-Nichols method.

Those P, I, D paramaters relate in some ways with the reaction

curve captured from in-situ test. Up to this point of the captured

reaction curve, this document offers modelling via γax.

XFR = Transfer Function

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========== XFR [Transfer Function] ==========

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An 'n' stages system

The classical 2nd order system

n 9 :

T 1 :

α 1.25 :

β 0.5 :

K.p 1 :

XFR of 'n' multistage system of
Unit Time Constant [T= 1] t nth 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - :

XFR of the 2nd order classical system:
Kp [Process gain], α,β [Dynamics] t XFR K.p 1 α β α - (/ t - α / exp * + β β α - (/ t - β / exp * - (* :

==================== Reaction Rate =======================

Reaction rate of the nth system t Laplace t n 1 - (^n 1 - (! / t - T / exp T n^/ * :

Reaction rate 2nd order x XFR x diff K.p x β / - exp x α / - exp - (* β α - / ≡

x nth x Laplace 2 1 sys x XFR x XFR x diff 1.141 0.75 1.141 0 2 2 mat 3 1 sys

Source: Inst_PID Discrete Schema

x XFR x XFR x diff - x 12 solve 1.141

Laplace algebra of 2nd order under 
unit step ... Gp => Process gain s G.P α β G3 G.P s 1 α s * + (* 1 β s * + (* / :

The XFR(t) is the InverseLaplace  of open loop Laplace algebra... in Smath from the "maple companion" t G.P α β XFR s G.P α β G3 s t invlaplace maple : G.P β α 1 - t α / - exp + (* + β t β / - exp * - (* β α - /

G.P 1 : α 0.5 : β 1.25 : 31 line

x xfr 0 x ≤ x G.P α β XFR invalid if :

x xfr 1 2 1 sys

You can copy the RHS maple result and export. Always recommanded for isolation purpose.

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

n 9 :

T 1 :

XFR of 'n, T' multistage system t nth 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - 11 line :

x nth

n T 21 mat 9121 mat