Cubic solver in there

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This project models the control modes PI & PID. First, you need a cubic solver that works. If you change any of the parameters, you will have to reinstall the calculated working stuff: I, z1, z2, z3, α, β, γ At this point PI(t), PID(t) will follows. The other way to mdel PID is via completely symbolic, easy to manipulate but dependent. The suite P, P+D is given for completing the model visit.

P+I control model

L H N xd N intervals U 0 : dx H L - N / : i 1 N range U i el L dx i * dx - (+ : for U 51 line :

t0 1 time :

system dynamics α 11 : β 20 : 12 mat

L 0 : H 3 : N 300 : 13 mat U L H N xd : 21 sys

P 20 :

I 40 :

a I β P + α 31 mat :

a 40401131 mat

Cubic solver in there

The cubic roots solver -----------------------

q 13 / a 2 el * 19 / a 3 el (2^* - : r 16 / a 2 el a 3 el * 3 a 1 el * - (* 127 / a 3 el (3^* - : 12 mat u1 r q 3^r 2^+ sqrt + (13 /^: u2 r q 3^r 2^+ sqrt - (13 /^: 12 mat z1 u1 u2 + (a 3 el 3 / - : z2 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* + : z3 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* - : 51 line

r 404011141 mat polyroots : 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat

z1 z2 z3 31 mat 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat

composite dynamic  coefficients α I P z1 * + z1 z1 z2 - (* z1 z3 - (* / : β I P z2 * + z2 z2 z1 - (* z2 z3 - (* / : γ I P z3 * + z3 z3 z1 - (* z3 z2 - (* / : 31 sys 0.3962 - 3.5963 1017 -^* i * - 0.3019 - 1.0276 i * + 0.3019 - 1.0276 i * - 31 sys

r r reverse : x Lagrange x r n el - (n 13 product : 21 line

x Lagrange

re-assgin/isolate the working stuff

z1 1.6051 - : z2 4.6974 - 1.6894 i * + : z3 4.6974 - 1.6894 i * - : 31 sys

α 0.3962 - : β 0.3019 - 1.0276 i * + : γ 0.3019 - 1.0276 i * - : 31 sys

t PI I - z1 z2 * z3 * / α z1 t * exp * + β z2 t * exp * + γ z3 t * exp * + :

PI ii 1 U rows range m ii el U ii el PI : for U m augment 21 line :

Title 0.75 0.45 UnitStep response PI 10 blue 15 mat :

Proportional+Integral UnitStep response

PI 1 Title 3 1 sys

P+I+D control model

system dynamics α 11 : β 20 : 12 mat

P 270 :

I 350 :

D 55 :

a I β P + α D + 31 mat :

a 3502906631 mat

Cubic solver in there

The cubic roots solver -----------------------

q 13 / a 2 el * 19 / a 3 el (2^* - : r 16 / a 2 el a 3 el * 3 a 1 el * - (* 127 / a 3 el (3^* - : 12 mat u1 r q 3^r 2^+ sqrt + (13 /^: u2 r q 3^r 2^+ sqrt - (13 /^: 12 mat z1 u1 u2 + (a 3 el 3 / - : z2 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* + : z3 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* - : 51 line

r 35029066141 mat polyroots : 2.3164 - 0.5813 i * + 2.3164 - 0.5813 i * - 61.3673 - 31 mat

z1 z2 z3 31 mat 61.3673 - 2.3164 - 0.5813 i * + 2.3164 - 0.5813 i * - 31 mat

composite dynamic coefficients α I z1 P D z1 * + (* + z1 z1 z2 - (* z1 z3 - (* / : β I z2 P D z2 * + (* + z2 z2 z1 - (* z2 z3 - (* / : γ I z3 P D z3 * + (* + z3 z3 z1 - (* z3 z2 - (* / : 31 sys 0.8921 - 0.054 - 0.0061 i * - 0.054 - 0.0061 i * + 31 sys

r r reverse :

re-assgin/isolate the working stuff

x Lagrange x r n el - (n 13 product :

z1 61.3673 - : z2 2.3164 - 0.5813 i * + : z3 2.3164 - 0.5813 i * - : 31 sys

α 0.8921 - : β 0.054 - 0.0061 i * - : γ 0.054 - 0.0061 i * + : 31 sys

x Lagrange

t PID I - z1 z2 * z3 * / α z1 t * exp * + β z2 t * exp * + γ z3 t * exp * + :

PID ii 1 U rows range m ii el U ii el PID : for U m Re vectorize augment 21 line :

pid 0.25 0.7 UnitStep response PID 12 blue 15 mat : pi 0.5 0.4 UnitStep response PI 12 red 15 mat : 21 sys

PID PI pid pi 1 5 1 sys

Technical note: That PID responds faster than PI does not indicate the best choice in all cases. Many systems have time delay/dead time and PI may be a better choice. Note that PID is ~ PD w/o Offset.

1 time t0 - 20 s *

suite ... P, P+D

This document suite illustrates the UnitStep response of a 2nd order system under control P & PD analogic algorithm ... from Mathcad 11 Instrumemtation repository.

L H N xd N intervals U 0 : dx H L - N / : i 1 N range U i el L dx i * dx - (+ : for U 51 line :

t0 1 time :

Proportional K.p 200 :

Derivative K.d 60 :

system dynamics α 11 : β 20 : 12 mat

---------- Proportional [P] ----------

z1 α - 2 / α 2 / (2^β K.p + (- sqrt + :

z2 α - 2 / α 2 / (2^β K.p + (- sqrt - :

z1 z2 21 mat 5.5 - 13.775 i * - 5.5 - 13.775 i * + 21 mat

z1 5.5 - 13.775 i * - :

z2 5.5 - 13.775 i * + :

t P 1 z1 z2 * / K.p * K.p z1 z1 z2 - (* / z1 t * exp * + K.p z2 z1 z2 - (* / z2 t * exp * - :

Proportional [P]+ Derivative [D]... PD

r α K.d + β K.p + 21 mat :

z1 r 1 el - 2 / r 1 el 2 / (2^r 2 el - sqrt + :

z2 r 1 el - 2 / r 1 el 2 / (2^r 2 el - sqrt - :

z1 z2 21 mat 3.2471 - 67.7529 - 21 mat

z1 3.2471 - :

z2 67.7529 - :

t PD 1 z1 z2 * / K.p * z1 K.d * K.p + z1 z1 z2 - (* / z1 t * exp * + z2 K.d * K.p + z2 z1 z2 - (* / z2 t * exp * - :

L 0 : H 2 : N 200 : 13 mat U L H N xd : 21 sys

Title 0.25 0.5 UnitStep response P & PD 14 black 15 mat :

P ii 1 U rows range P ii el U ii el P : for U P augment 21 line :

PD ii 1 U rows range PD ii el U ii el PD : for U PD augment 21 line :

Offset K.p z1 z2 * / : 0.909

P 1 PD Offset Title 5 1 sys

1 Offset - 0.09

1 time t0 - 5 s *

jmGiraud: 2015/08/03