Util ... Alvaro

d7b2569b-6e7a-45a6-8214-47f2b1d1c8e3 6 15 12 decimal

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

Util ... Alvaro

Euler Mascheroni constant γ 0.577215664901532860606 :

X Cols Ans# 0 : k# 1 k# range Ans# k# el Ans# k# col : for Ans# 31 line :

x# IsZero x# IsDefined x# 0 ≡ (& 11 line :

E# x# DIFF diff(E# x#_DIFF# EcuRep num2str,_DIFF#) concat str2num_DIFF# x# EcuRep 11 line :

E# x# n# DIFF diff(E# x#_DIFF# EcuRep num2str,_DIFF#, n# num2str) concat str2num_DIFF# x# EcuRep 11 line :

XY x cinterp X Y 12 mat XY 1 col XY 2 col 12 mat : U X 1 el UnitsOf Y 1 el UnitsOf 12 mat : U 2 el X U 1 el / Y U 2 el / x U 1 el / cinterp * X min eval x ≤ (* x X max eval ≤ (* 31 line :

Gamma function. Valid for Re(ζ)>0 ζ Γ G 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat : zg ζ 543128 / + : κ Clear 12 mat 2 π * sqrt zg 0.5 ζ 0.5 - (*^(2^* e zg^/ G 1 el G κ el ζ κ + 2 - / κ 215 sum + (* eval 31 line :

Runge-Kutta 2nd and 3rd order. 2 D to uo N E RK23 n uo length : u uo 1 n range el : U u : t to 1 el : T t 11 mat : v 1 n range : 16 mat ho to 2 el to 1 el - N / eval : h ho : n 1 : 13 mat t u k K h t u D * eval h k * eval try 11 line : t to 2 el < k.1 t u u K : k.2 t h + u k.1 + k.1 K : k.3 t 0.5 h * + u 0.25 k.1 * + 0.25 k.2 * + k.2 K : d k.1 2 k.3 * - k.2 + 3 / normi : (q E u normi ho 12 mat max * : (≤ t t h + : T n n 1 + : el t : 12 mat u u k.1 4 k.3 * + k.2 + 6 / + eval : U v n el u v el : 12 mat 21 line if d 0 ≠ h ho 0.9 h * q d / 3 nthroot * 12 mat min eval : if 51 line while T U transpose augment eval 51 line :

Numerical Inverse Laplace Transform Alvaro

Based on algorithm of Oscar: https://en.smath.info/forum/yaf_postst1341_Signals-and-Systems.aspx

Hi Basile, The method is based on work of J. Valsa and L. Brancik: Approximate Formulae for Numerical Inversion of Laplace Transforms, Int. Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 11, (1998), pp. 153-166

a ns nd ILT.αβ m ns nd + 1 + : α m 1 matrix : β α : γ nd 1 matrix : 14 mat μ 1 m range : nδ 2 nd range : nσ ns 3 + (m range : 13 mat α μ el a μ 1 - (π * i * + eval : β μ el 1 - (μ 1 +^e a^* eval : 12 mat γ 1 el 2 nd -^: γ nδ el nd 1 + Γ 2 nd^nd 2 + nδ - Γ * nδ Γ * / γ nδ 1 - el + eval : 12 mat β 1 el 0.5 β 1 el * : β nσ el β nσ el γ m nσ - 1 + el * eval : 12 mat α β μ 13 mat 61 line :

Inverse Laplace transform of F(s) 1 F t.i t.e N ILT t N 1 matrix : ft t : Fs αL rows 1 matrix : 13 mat t.o 105 -^t.e t.i - UnitsOf * t.i 12 mat max eval : Δt t.e t.o - N 1 - / : 12 mat n 1 N range t n el t.o n 1 - (Δt * + : Fs μL el βL μL el t n el / αL μL el t n el / F * eval : 12 mat ft n el Fs sum Re eval : 21 line for t ft augment eval 41 line :

α and β values αL βL μL 13 mat 62019 ILT.αβ eval :

3 appVersion 0.98.6179 ≡ Timing 4 min ≡ 21 mat

Numerical Inverse Laplace Transform Oscar ... [a,ns,nd]

s F t invLaplace Original Oscar in program t 1 el 105 -^: t 1 el 0 ≡ t cases [a,ns,nd] user comman line n 1 ns nd + 1 + (range α n el a n 1 - (π * i * + : β n el a exp 1 - (n^* - eval : 21 line for Γ(x) faster than Gamma(x) bdif 1 el nd 1 + Γ nd 2 + 1 - Γ / 1 Γ / (: β2 β ns 2 + (ns 1 + nd + (range 11 range el : n 2 nd range bdif n el nd 1 + Γ nd 2 + n - Γ / n Γ / (bdif n 1 - el + : for bdif bdif reverse 2 nd^/ : n 1 nd range β2 n el β2 n el bdif n el * : for β β 1 ns 1 + (range 11 range el β2 stack : β 1 el β 1 el 2 / : 41 line k 1 t rows range u t k el eval : s α u / : βt β u / : 12 mat n 1 α length range βtF n el βt n el s n el F * eval : for RβtF βtF Re vectorize : ft k el RβtF eval sum : 51 line (for OUT t ft augment : 111 line :

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

s R SP s / :

s Z 1 s / δ s * - exp * :

s H3 KI s * :

s H 1 :

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

s G1 KP :

Numeric s F s G1 s G2 * s R * s G2 s Z * + 1 s G1 s G2 * s H * + / :

====================================================================================

user range/mesh t.ini 0 : t.end 20 : pts 200 : 13 mat

in-situ incremental t t.ini t.end t.end t.ini - (pts 1 - / range :

The respective parameters of this project

α 1.25 :

SP 0.5 :

<= SetPoint

Turbo dynamics

β 0.25 :

KP 30 :

<= Proportional gain

δ 10 :

<= Delay perturb

KI 11 :

<= Integral gain

G.P 1.25 :

<= Process gain

in program ... user a 6 : ns 20 : nd 19 : 13 mat

TurboGen s F t invLaplace :

Steady TurboGen TurboGen rows row (: 20 0.533 12 mat

User XFR parameters [H, γ, T]

TurboGen SP δ 0.5625 δ 0.475 2 2 mat 8 0.533 25 0.533 2 2 mat 4 1 sys

α β δ 31 mat G.P SP KP KI 41 mat 21 mat 1.25 0.25 1031 mat 1.25 0.5 301141 mat 21 mat