fbf079fc-191a-409d-b832-69631229fbbd 65 4 5 decimal

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

UnitStep =>

s φ m s / δ s * - exp * :

m δ t φ s φ s t invlaplace maple :

m δ t φ m 1 t δ - sign + (* 2 /

m 0.5 :

δ 5 :

x shift 0.25 :

x datum 0 :

UnitSep => magnitude => delay => ± shift

0 x ≤ 0.5 5 x φ x shift + Undefined if x datum 2 1 sys

This document can be used to model a system under control. The proposed model is purely dynamic, thus it has no D [derivative]. In "TurboMachines" , besides Sp, KP, KI we are most intereste by 'δ', the stabised time laps to step up the system. It's about not possible to extract a complete exportable analytic expression. Instead, we offer a discrete export of the model

In Control Engineering UnitStep is by convention : 1, δ from datum = 0 It can be used more generally.

s G3 G.P s 1 α s * + (* 1 β s * + (* / :

s R SP s / :

s H3 KI s * :

Watch this: you must redefine 'δ' wrt your TurboGen model. If you don't, it takes 'δ' from above introduction.

δ 10 :

s H 1 :

s Z 1 s / δ s * - exp * :

s G1 KP :

t XFR s G3 s t invlaplace maple : G.P β 1 t β / - exp - (* α 1 - t α / - exp + (* + (* β α - /

s G2 s G3 1 s G3 s H3 * + / :

The G2 subsystem will come out below, after it has digested all what it needs to be productive. Unfortunately, you can't convert that crappy expansion into "sin/cos trig". Smath ?, Maple ?

s C s G1 s G2 * s R * s G2 s Z * + 1 s G1 s G2 * s H * + / :

C(s) is the complete Laplace system. Its inversion will solve the system back in the time domain. It will solve completely wrt all involved parameters. The nuance and annoyance start here @ the symbolic solution of C(s). The Maple version Smath has access, does solve uniquely in complex form and whereas Smath does not "QuickPlot" complex, we will be confined to plot discrete. Apparently, by "Conjugating" one member of the solution [the solved system is the Σ of two members] we get an identical quick plot than the discrete plot. This "Inverse Engineer" ad absurdum proof is left to the user.

α 1.25 :

SP 0.5 :

<= SetPoint

β 0.25 :

KP 30 :

<= Proportional gain

δ 10 :

<= Delay

KI 11 :

<= Integral gain

G.P 1.25 :

<= Process gain

In Maple, imaginary is 'I'[just stupid !]. So, if you subscript 'I' something, it will scrap and freak your work !

Bug: 1

t G2 s G2 s t invlaplace simplify maple :

t G2 5259 sqrt * 259 sqrt 112 t * 5 / - exp 2 t * 259 sqrt * 5 / cos * - (* 612 t * 5 / - exp * 2 t * 259 sqrt * 5 / sin * - (* 15281 /

By wisely working individual members of G2(t) we arrive at a readable formulation identical to Mathcad 11.

20 G2 0.0847

0.0847 0.0847 2.4 t * - exp * 6.44 t * cos * - 0.0316 2.4 t * - exp * 6.44 t * sin * -

t C s C s t invlaplace simplify allvalues maple :

s C s t invlaplace simplify allvalues evalf maple memo ≡

s C s t invlaplace simplify maple expands in terms of Sum RootOf ≡

s C s t invlaplace simplify allvalues maple stack [sum] the expanded RootOf ≡

Bug: 2

Smath does not access more than 21 rows of the Maple internal vertical code.

t C 1 el real, not used ≡

t C 2 el t C 3 el 12 mat complex ≡

t C2 t C 2 el Re i t C 2 el * - :

<= Conjugate for sol2

Smath Bug: 3

The working solution

Index 'i' is not isolated from previous use. 'i' freaks the program with imaginary. It tabulates garbage that will crash Smath.

t F 2 sqrt 2 / t C2 Re (2^t C2 Im (2^+ sqrt * :

x fnct mtp mtp,multitraceplot vx 020 0.1 range : i 1 vx rows range vy i el vx i el fnct : 11 line for vx vy augment 41 line :

Out vx 0.1 20 0.125 range : j 1 vx rows range vx j el vx j el : vy j el vx j el F eval : 21 line for vx vy augment 31 line :

t Cfnct t C 1 el :

XFR x XFR mtp :

G x G2 mtp :

C1 x Cfnct mtp :

Turbogen Dynamic XFR

XFR G.P 2 1 sys

G2 subsystem

G 0.0847 2 1 sys

1rst solution ?

C1 SP 2 1 sys

Responses to SetPoint & perturbation.

Out SP 0.533 4 1 sys

t Model t C 2 el Im :

TurboGen vx 020 0.1 range : j 1 vx rows range vx j el vx j el : vy j el vx j el Model eval : 21 line for vx vy augment 31 line :

For what it may be useful, we can line up both responses to highlight the presumed defect of the imaginary C(t)2, C(t)3

The project is acceptable as is, not as perfect as Maple/Mathcad.

n 2 :

0 x ≤ ( x 15 ≤ ( & ( x C n el Im if 0 x ≤ ( x 15 ≤ ( & ( x C n el Re if 2 1 sys 0 x ≤ ( x 15 ≤ ( & ( x C 1 el Im if 0 x ≤ ( x 15 ≤ ( & ( x C 1 el Re if 2 1 sys TurboGen 0 2 1 sys

0 x ≤ (x 15 ≤ (& (x C n el Im if 0 x ≤ (x 15 ≤ (& (x C n el Re if 21 sys

0 x ≤ (x 15 ≤ (& (x C 1 el Im if 0 x ≤ (x 15 ≤ (& (x C 1 el Re if 21 sys

0 C Im 0 0.1081 - 0.1081 31 mat

Mathcad 8Pro. up Mathcad 11

Sanity check

G.P SP KP KI 41 mat 1.25 0.5 301141 mat

α β δ 31 mat 1.25 0.25 1031 mat