fd51e33b-0bde-4598-b7f2-a76d57f5236c 136 3 5 decimal

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vx vy char size color plotG program to collect points 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 51 line :

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 :

This is the PID CPV discrete iterative control algorithm. CPV = Composite Past Value The single line diagram helps visualising what's going on. The field input is converted from Analog/Digital, then passed into the PID CPV, then converted back to Analog. The D/A to field may be optional, some field controlling devices are fit to discrete signal.

Recursive setup of a damped 2nd order system

t0 1 time :

a 0.825 - :

b 1.5 :

This is the response of a perturbed system under initial 's' and it's own dynamics 'a, b' ... returning to rest position.

XY user recursive s 1 el 1 : s 2 el 0 : n 350 range s n el b s n 1 - el * a s n 2 - el * + : for i 1 s rows range x i el i : for x s augment 61 line :

a b c spec . 10 red 13 mat :

Pts XY a b c spec pts :

Title 2 1.5 damped 2nd order perturbed UnitImpulse 11 black 15 mat :

XY Pts Title 3 1 sys

Recursive behaviour of a 2nd order system

a 0.894839 - :

b 1.790648 :

c 0.104191 :

ε 1 :

a b + c + 1

Title 2 0.375 - response 2nd order under unit forcing [ε] 11 black 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 ε * + : for i 1 s rows range x i el i : for x s augment 51 line :

a b c spec . 8 blue 13 mat :

Pts UnitSP a b c spec pts :

The a+b+c = 1 is a particular property of a 2nd order correctly modeled for recursive. Observe that the system recovers more quickly with smaller [ε]

Pts ε Title 3 1 sys

Does the above demo come out of the blue ? Answer: NO ... here is the recursive model

9 y t 2 diff * y t diff + y + 8 x t 2 diff * 3 x t diff * + x + ≡

a1 9 :

b1 1 :

a2 8 :

b2 3 :

Δt 1 :

--------------------------------------------------------

r1 b1 - b1 2^4 a1 * - sqrt + 2 a1 * / :

r2 b1 - b1 2^4 a1 * - sqrt - 2 a1 * / :

r1 r2 21 mat 0.0556 - 0.3287 i * + 0.0556 - 0.3287 i * - 21 mat

α a1 a2 - (b1 b1 b2 - (* - Δt 2^/ :

β b1 b2 - (Δt / :

γ r1 r2 + (Δt * exp :

δ r1 Δt * exp r2 Δt * exp + :

γ 0.8948

δ 1.7906

--------------------------------------------------------

c1 γ - :

c2 δ :

α β γ δ 41 mat 32 - 0.894839 1.790649 41 mat

c3 γ α β 2 / - (1 γ + δ - (* - β 1 γ - (* - :

c4 δ - 2 α * 1 γ + δ - (* + 2 β * 1 γ - (* + :

c5 1 α β 2 / + (1 γ + δ - (* - β 1 γ - (* - :

c1 c2 c3 c4 c5 51 mat 0.894839 - 1.790649 0.688398 1.586147 - 1.00194 51 mat

Note that the recursive 3 coefficients collects the entire solution and that the final recursive settles to ε, either a setpoint [SP] change or perturbation from initial conditions s1,s2 = 0,0

--------------------------------------------------------

Fundamental property

The recursive coefficients

c1 c2 + c3 + c4 + c5 + 1

a c1 :

b c2 :

c c3 c4 + c5 + :

c3 c4 + c5 + 0.104191

d c4 2 c5 * + :

a b c d 41 mat 0.894839 - 1.790649 0.104191 0.417733 41 mat

a b + c + 1

In the control room, operators have long learned how to minimise overshoot from changing SetPoint brutally. Instead, they ramp or move SetPoint slowly. Here is the 'ramping' response, scaled wrt the 'm = 40' & 'sp =60' of this typical demo. Observe the ramped SetPoint settles a bit faster than the UnitStepPoint.

m 40 :

<= 'm' ramp length

sp 60 :

<= SetPoint final

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

a 0.894839 - :

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

b 1.790649 :

c 0.104191 :

vx X U stack :

ramp s 1 el 0 : s 2 el 1 : n 3 vx rows range s n el a s n 2 - el * b s n 1 - el * + c vx n 2 - el * + : for i 1 s rows range x i el i : for x s augment 51 line :

scale 1 vx max / :

scale 0.075 :

X ramp 1 col :

Y ramp 2 col scale * :

rampOUT X Y augment :

a b c spec . 10 black 13 mat :

ramp rampOUT a b c spec pts :

Title 16 0.2 damped 2nd order under ramped forcing 11 black 15 mat :

Line ramp 1 col vx scale * augment :

UnitStep & ramp SetPoint 2nd order response

UnitSP Line Title ramp 1 5 1 sys

UnitSP UnitStep SetPoint ≡ Line ramp SetPoint ≡ Title ramp ramp response ≡ 151 sys

The CPV PID digital controller -------

Example: 0

m 60 :

<= 'm' length SetPoint initial

sp 60 :

<= 'sp' length SetPoint final

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 U X stack :

u X U stack :

u 0.5 u * :

v j 1 u rows range v j el j : for v 21 line :

XY v u augment :

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 * :

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 :

sp XY 1 col XY 2 col . 5 red plotG :

algo E 1 col E 2 col . 8 blue plotG :

algo sp 2 1 sys

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 one 'Z=0' last accumulated value.

Example: 1

m 30 :

<= 'm' length SetPoint initial

sp 30 :

<= 'sp' length SetPoint final

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

u 0 :

algo 0 :

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

u xx uu stack :

u uu xx stack :

u 0.5 u * :

v j 1 u rows range v j el j : for v 21 line :

XY v u augment :

P 0.75 :

I 0.25 :

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

D 0 :

[D,± 0.5] demo values

Y 0 :

OUT CPV recursive PID Y 1 el 0 : Y 2 el 0.5 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 :

sp XY 1 col XY 2 col . 8 red plotG :

algo OUT 1 col OUT 2 col . 8 blue plotG :

Title 16 1.375 The balistic CPV PID 11 black 15 mat :

algo sp Title 3 1 sys

This much simpler CPV needs less 'Z' accumulated initial values, recovers faster, NO overshoot. Most industrial controllers are of this CPV type. In this example, deirivative mode 'D = 0'. Y1, Y2 = 0 ... the system at rest is stable, the last two accumulated 'Y's' are '0' [no output]. For faster response, take 0.5 the step change.

u 0 :

algo 0 :

Example: 2

u Line 2 col :

P 0.875 :

I 0.5 :

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

D 0 :

[D,± 0.5] demo values

Y 0 :

Track CPV recursive PID Y 1 el 0 : Y 2 el 0.5 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 :

sp Line 1 col Line 2 col . 5 red plotG :

algo Track 1 col Track 2 col . 8 blue plotG :

Title 16 1.375 CPV PID tracking ramp SP 11 black 15 mat :

algo Line Title 3 1 sys

Note that Examples: 0, 1 are reversible wrt the signal 'u'

jmGiraud: 2015/08/03

XFR [Transfer Function] ... what is ? XFR is the graph response of a system moving from a steady state to another steady state w/o oscillation. Moving the load on the boom of a crane too brutally w/o control, will oscillate ; moving "along" the XFR will not oscillate.

Typical XFR of 2nd order dynamic α, β process gain Kp

x fnct mtp mtp,multitraceplot vx 1012 -^12.5 0.05 range : i 1 vx rows range vy i el vx i el fnct : 11 line for vx vy augment 41 line :

An 'nth order system of equal 'T' has representation:

The nth order system of = T has two direct equivalent expansions from the developpers

T n t nth n Γ n t T / Γ - n Γ / ≡ 1 t - T / exp 1 k! / t T / (k^* (k 1 n sum * - ≡

Mathcad 11/Maple 11

The nth order system of = T is also the indefinite integral of the InverseLaplace

1 T s * 1 + (n^/ InverseLaplace 1 n 1 - (! / t n 1 - (^* t - T / exp T n^/ * ≡

Mathcad 11/Maple 11

t n 1 - (^n 1 - (! / t - T / exp T n^/ * t int 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - ≡

Mathcad 11/Maple 11

t f t n 1 - (^n 1 - (! / t - T / exp T n^/ * t int :

Smath/Maple symbolic wrt WhittakerM has no interest in there

α 1.25 :

β 0.5 :

K.p 1 :

t XFR K.p 1 α β α - (/ t - α / exp * + β β α - (/ t - β / exp * - (* :

n 4 :

T 1 :

t nth 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - :

t Laplace t n 1 - (^n 1 - (! / t - T / exp T n^/ * :

<= Reaction rate

L x Laplace mtp :

xfr x XFR mtp :

Nth x nth mtp :

data xfr 1 col xfr 2 col Nth 2 col L 2 col augment :

plot xfr Nth L 31 sys :

plot

2nd order nth order Laplace

1 time t0 - 22 s *

Review the procedures

In Engineering/Scientific consulting, you can't be wrong, otherwise your bread will have no butter ! In community work, some smart buddy is precious checking your work. No matter the project you are involved and the !Help!, "Joe Blo" must be able to trace your work. Thus the procedures, because they don't come out of the blue. In this project, Smath/Maple are just tools you must use wisely.

n 5 :

T Clear

Does nothing

1 TT s * 1 + (n^/ s t invlaplace maple t 4^t TT / - exp * 24 TT 5^* /

Proceed with dummy TT

set n: 1, 2, 3, 5 conclude any 'n' 21 mat t T / - exp T / t t T / - exp * T 2^/ t 2^t T / - exp * 2 T 3^* / t 4^t T / - exp * 24 T 5^* / 14 mat ≡

1 T s * 1 + (n^/ InverseLaplace t n 1 - (^n 1 - (! / t - T n^/ exp * ≡

n 4 :

T 1 :

t f t n 1 - (^n 1 - (! / t - T n^/ exp * :

t q t 3^t T / - exp * 6 T 4^* / :

ON_OFF 0 :

set ON_OFF 0/1 to check that both plots agree

x f ON_OFF x q * 2 1 sys

______________________________________________

n 5 :

t f t n 1 - (^n 1 - (! / t - TT / exp TT n^/ * t int :

t f maple t TT / - exp t t t t 4 TT * + (* 12 TT 2^* + (* 24 TT 3^* + (* 24 TT 4^* + (* 24 TT 4^* / -

Sanity check wrt InverseLaplace integral 1. in there, the Integral InverseLaplace sums to UnitStep ! thus '1' preceeding both expansions [just some rule in the Laplace integration algebra]. 2. In th Σ, 'k' runs: 0, 1, 2, 3, 4 ... i.e: the n=5 in the maple(f(t)) ... just a nuance to catch.

n 4 :

T 1 :

t f 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - :

t g 1 t T / - exp t t t t 4 T * + (* 12 T 2^* + (* 24 T 3^* + (* 24 T 4^* + (* 24 T 4^* / - :

ON_OFF 1 :

set ON_OFF 0/1 to check that both plots agree

x f ON_OFF x g * 2 1 sys