Cubic solver in there

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About this Smath document This project models the control modes PI & PID. First, you need a cubic solver that works [polyroots fails]. If you change any of the parameters, you will have to reinstall the calculated working stuff: I, z1, z2, z3, α, β, γ as well as re-symbolic PI(t), PID(t) as the experiment will apply and re-plug the symbolic in the collapsed area. The discretiser xd(L,H,N) produces readable increments to ease exporting the project decently tabulated.

P+I control model

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

t0 1 time :

a0 11 :

b0 20 :

P 20 :

I 40 :

a I b0 P + a0 31 mat :

a 40401131 mat

L 0 :

H 3 :

N 300 :

U L H N xd :

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^* - :

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

u1 r q 3^r 2^+ sqrt + (13 /^:

u2 r q 3^r 2^+ sqrt - (13 /^:

z1 u1 u2 + (a 3 el 3 / - :

z1 1.6051 -

z2 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* + :

z2 4.6974 - 1.6894 i * +

z3 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* - :

z3 4.6974 - 1.6894 i * -

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 - (* / :

α 0.396 - 2.593 1017 -^* i * -

β 0.3019 - 1.0276 i * +

γ 0.3019 - 1.0276 i * -

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

PI j 1 U rows range PI j el U j el PI : for U PI augment 21 line :

<= Can NOT parse then get the symbolic

I 40 :

z1 1.6051 - :

α 0.396 - :

z2 4.6974 - 1.6894 i * + :

β 0.3019 - 1.0276 i * + :

z3 4.6974 - 1.6894 i * - :

γ 0.3019 - 1.0276 i * - :

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

collapse the hugly symbolic expansion

t PI 402500000000000000158904923487 - 8447 i * + (* 234878447 i * + (* 16051 t * 10000 / - exp * + (* 1605123487 - 8447 i * + (* 234878447 i * + (* 3019 - 10276 i * + (t 23487 - 8447 i * + (* 5000 / exp * - 301910276 i * + (t 234878447 i * + (* 5000 / - exp * + (* + 16051000023487 - 8447 i * + (* 234878447 i * + (* / - :

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

Note that by redefining coefficients, i.e: isolate, and assigning the symbolic, Smath has turned "Speedy Gonzalez", Ah !

Title 0.75 0.45 UnitStep response PI 10 black 15 mat :

Proportional+Integral UnitStep response

PI 1 Title 3 1 sys

P+I+D control model

a0 11 :

b0 20 :

P 270 :

I 330 :

D 55 :

a I b0 P + a0 D + 31 mat :

a 3302906631 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^* - :

u1 r q 3^r 2^+ sqrt + (13 /^:

u2 r q 3^r 2^+ sqrt - (13 /^:

z1 u1 u2 + (a 3 el 3 / - :

z1 2.2905 - 1.3501 1015 -^* i * +

z2 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* + :

z2 2.348 - 8.8613 1015 -^* i * -

z3 u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* - :

z3 61.3616 - 6.3712 1015 -^* i * +

... polyroots fails ----------------------

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 - (* / :

α 0.015 - 4.2621 1014 -^* i * +

β 0.0927 - 3.919 1014 -^* i * -

γ 0.8922 - 1.0171 1016 -^* i * -

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

To get the symbolic expansion, recast the system with isolated parameters. Click on the PID(t) below and see the symbolic expansion. At this point, expand the symbolic, copy/paste in a new definition [collapsed]

I 330 :

z1 2.2905 - :

α 0.015 - :

z2 2.348 - :

β 0.0927 - :

z3 61.3616 - :

γ 0.8922 - :

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

collapse this huge symbolic

t PID 156255068750000000001031276414974581 t * 2000 / - exp * - (* 31866441222573587 t * 250 / - exp * - (* 479221259081372000038351 t * 625 / - exp * - 5371231327968750000 / :

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

Note that by redefining coefficients, i.e: isolate, and assigning the symbolic, Smath has turned "Speedy Gonzalez", Ah !

pid 0.25 0.7 UnitStep response PID 10 blue 15 mat :

pi 0.5 0.4 UnitStep response PI 10 green 15 mat :

Proportional+Integral+Derivative UnitStep response

PID 1 PI pid pi 5 1 sys

What have we learned from this Smath document: _____________________________________________ 1. It reproduces Mathcad 11 in Smath style. 2. How to get the symbolic expansion by isolating the parameters and by same token minimises calculation. 3. Detected the Smath bug, i.e: in case of PI that is based on 'exp(complex)', the discretised compactum CAN NOT be indexed 'i', it freaks & kills Smath !!! XP Home, e-machines, 1.8 GHz computes worksheet 10 sec. Read more: The P mode has two complex z1, z2 exp(complex) is compatible with 'i' index. The PI, PID have two complex + one real root, 'i' index is not compatible ... an important nuance to avoid failure.

1 time t0 - 10 s *

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.

jmGiraud: 2015/08/03