<p> CPV PID </p>

2ff6a233-d63d-45aa-9283-ac2fa0affe5a 15 3 5 decimal

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

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 :

The classical CPV PID

'm' ramp length
'sp' setpoint final m 40 : sp 60 : 21 sys

X j 1 m range X j el j 3 / : for X 21 line :

U j 1 sp range U j el X X rows el : for U 21 line :

vy X U stack :

vy Max 13.333

unarySP 1 vy Max / : u unarySP U * : 21 sys

SP i 1 vy rows range vx i el i : for vx unarySP vy * augment 21 line :

P 1 :

I 0.25 :

Too much 'I'[>1.75] goes haywild, crashes Smath !

D 0 :

sets the Y2 value ideal for unarySP μ 0.5 :

<p> CPV PID </p>

Track CPV recursive PID Y 1 el 0 : Y 2 el μ u 1 el * : i 3 u rows range Y i el Y i 1 - el P u i el u i 1 - el - (* + I u i 1 - el Y i 1 - el - (* + D u i el 2 u i 1 - el (* - u i 2 - el + (* + : for i 1 Y rows range x i el i : for x Y augment 61 line :

a b c spec . 8 red 13 mat : sp SP a b c spec pts : 21 sys

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

Title 16 0.25 CPV PID tracking ramp SP 11 black 15 mat :

algo sp Title 3 1 sys

The CPV PID . . . CPV: CompositePastValue

is simple to implement. The discrete recursive form

reproduces 1/1 the PID analog/pneumatic ancestors.

The P, I, D parameters correspond to the analog.

Not yet a 1/1 replacement as the most governing

parameter is missing in the demo, i.e: the loop scan

frequency. Too frequent for the birds, too slow the

loop will crash the system.

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<p> XFR multistage </p>

XFR 00 0.25 6.611711 106 -^* 0.5 0.000172 0.75 0.001065 1 0.00366 1.25 0.009124 1.5 0.018576 1.75 0.032902 2 0.052653 2.25 0.078014 2.5 0.108822 2.75 0.144621 3 0.184737 3.25 0.228347 3.5 0.274555 3.75 0.322452 4 0.371163 4.25 0.419882 4.5 0.467896 4.75 0.514602 5 0.559507 5.25 0.602226 5.5 0.642482 5.75 0.680089 6 0.714943 6.25 0.747015 6.5 0.776328 6.75 0.802957 7 0.827008 7.25 0.848618 7.5 0.867938 7.75 0.885132 8 0.900368 8.25 0.913814 8.5 0.925636 8.75 0.935993 9 0.945036 9.25 0.952907 9.5 0.959737 9.75 0.965647 10 0.970747 10.25 0.975137 10.5 0.978906 10.75 0.982135 11 0.984895 11.25 0.98725 11.5 0.989253 11.75 0.990956 12 0.9924 12.25 0.993622 12.5 0.994654 512 mat :

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 :

There is a scale factor to be applied.
Introduced in the CPV PID recursive. scale XFR rows XFR 1 col XFR rows row / :

P 1.5 :

Convenient proportional

I 0.0675 :

Little integral

D 7 :

Not too much derivative [XFR is low decimals]

sets the Y2 value ideal for unarySP μ 0.5 :

u XFR 2 col :

<p> CPV PID </p>

Track CPV recursive PID Y 1 el 0 : Y 2 el μ u 1 el * : i 3 u rows range Y i el Y i 1 - el P u i el u i 1 - el - (* + I u i 1 - el Y i 1 - el - (* + D u i el 2 u i 1 - el (* - u i 2 - el + (* + : for i 1 Y rows range x i el i : for x scale / Y augment 61 line :

a b c spec . 8 red 13 mat : xfr XFR a b c spec pts : 21 sys

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

Title 1 0.25 CPV PID tracking ramp SP 11 black 15 mat :

algo xfr 1 Title 4 1 sys

NOTE:

This demo tracks a Laplace XFR multistage.

The CPV was expected to advance the XFR.

An admissible closed control loop scheme

w/o overshoot.

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β 2 :

E v 010 0.125 range : i 1 v rows range v i el i 1 - 10 / : erf i el v i el erf β^eval : 21 line for v erf augment 31 line :

E rows 81

1. Proportional
2. Integral
3. Derivative P I D 31 mat 2 0.25 0.5 0.75 1 1.25 1.5 16 mat transpose 031 mat :

k 2 :

u E 2 col :

scale E rows E 1 col E rows row / : 10.125 11 mat

n u rows : 81

I k el 0.5

<p> CPV PID </p>

f too much Integral action will cause the loop to windup y 1 el u 1 el : y 2 el u 2 el : i 3 n range y i el y i 1 - el P u i el u i 1 - el - (* + I k el u i 1 - el y i 1 - el - (* + D u i el 2 u i 1 - el * - u i 2 - el + (* + : for i 1 y rows range u i el i 1 - : for u scale / y augment 61 line :

'f' the CPV out is short of two values. The red 'E' is under blue 'f'

f E 2 1 sys

n 9 :

T 1 :

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

XFR of 'n' multistage system of
=  Unit Time Constant [1] t nth 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - 11 line :

x nth 1 1 sys