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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 stabilised 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.

Setup in Laplace domain

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

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

s R SP s / :

s H3 KI s * :

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 α 1 - t α / - exp + (* + β t β / - 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. You can reduce G2 in more familiar "trig" .

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. Further processing is required to render the project exportable.

The respective parameters of this project

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

s C 201510 s * - exp + (* s s 45 s * + (4 s + (* 220 + (* 600 + (* /

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 * -

The Maple/Laplace Companion -------------------

t C s C s t invlaplace allvalues maple :

s C t 0 > assume s t invlaplace simplify allvalues maple

t 0 > assume NOT needed ≡

The Maple advanced expansion is in term of solutions "RootOf". The command "allvalues" restricts to explicit solutions wrt 't'. "simplify" is not needed either.

Maple stacks 3 solutions t C 1 el t C 2 el t C 3 el 31 mat ≡

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

Collect the solved system

The imaginary terms cancel each other, the solution is finally Re[C(t)] ... consequently suitable for Smath.

t F 13 / t C 1 el t C 2 el + t C 3 el + (* eval :

Project considerations

Differently than the "TurboGen Smath" in which the 1rst solution is inert, in the "TurboGen Signum" the first solution is a "mapping component" thus the 1/3 Σ sol(t). Maple freaks the Signum part of the solution, observe the reconstruction. The discrete plot is constructed in two parts, juxtaposed in the plot to complete/illustrate the project.

SP 0.5

x SP 0.5 :

bias SP 10.01 F Re - : 0.022218 -

x f x F Re 0 x ≤ (x 10 ≤ (& x F Re bias + 10 x ≤ (x 20 ≤ (& cases :

15 f 0.511091

Complete response

x f x SP 0.511 4 1 sys

Examine the subsystems

t Cfnct t C 1 el :

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

XFR t XFR mtp :

G t G2 mtp :

C1 t Cfnct mtp :

Delay δ SP 0.125 + δ SP 0.125 - 22 mat :

Collect XFR XFR : G G : C1 C1 : Out_0 Out_0 : Out_10 Out_10 : 51 line

Out_0 vx 010 0.1 range : j 1 vx rows range vy j el vx j el F Re eval : for vx vy augment 31 line :

Out_10 vx 10.01 15 10.125 range : j 1 vx rows range vy j el vx j el F Re eval : for vx vy augment 31 line :

TurboGen Dynamic XFR

XFR G.P 2 1 sys

G2 subsystem

G 0.0847 2 1 sys

The mapping 1rst solution

C1 SP Delay 3 1 sys Out_0 Out_10 bias + Delay SP 4 1 sys

Plot arguments Out_0 Out_10 bias + Delay SP 41 sys

0 C 0.0311 0.4845 0.1081 i * - 0.4845 0.1081 i * + 31 mat

The project is certified as is, as perfect as Maple/Mathcad 11

Mathcad 8Pro ... 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

jmG: 2015/10/23

Smath Bug:

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

Maple Bug

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

t f t a i b * + (* exp :

t f trig convert maple t a * cosh t a * sinh + (i t b * sin * t b * cos + (*

The 3 Maple solutions can be extracted they are voluminous !

t C 1 el t C 2 el t C 3 el 31 mat

If the perturbation term is absent

s L 20 s 15 + (* s 600 s 22045 s * + (4 s + (* + (* + (* / :

t C s L s t invlaplace allvalues maple :

t F 13 / t C 1 el 1 t C 2 el (* + 1 t C 3 el (* + (* eval :

t f t F Re 0 t ≤ (t 10 ≤ (& (cases :

x SP SP : 0.5

0 f 10 f 21 mat 0.333 0.5 21 mat

Response 0 => delay δ

x f x SP 2 1 sys