Numerical Inverse Laplace Transform

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

1 F t.i t.e N P ILap a ns nd 13 mat P : m ns nd + 1 + : 12 mat t N 1 matrix : ft t : t.o 105 -^t.i 21 mat max eval : Δt t.e t.o - N 1 - / : 14 mat α m 1 matrix : β α : γ nd 1 matrix : 13 mat n 1 m range : α n el a n 1 - (π * i * + eval : β n el 1 - (n 1 +^: 13 mat n h nd 1 + Γ 2 nd^nd 2 + n - Γ * n Γ * / eval : n 2 nd range : γ 1 el 1 h : γ n el n h γ n 1 - el + : 13 mat n ns 3 + (m range : β 1 el 0.5 β 1 el * : β n el β n el γ m n - 1 + el * : 13 mat β e a^β * eval : k 1 N range t k el t.o k 1 - (Δt * + : st α t k el / : βt β t k el / : 13 mat ft k el βt st F * Re vectorize eval sum : 21 line for t ft augment eval 101 line :

Γ.κ 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 :

ζ Γ z ζ 1 - : zh 1 - 2 ζ * + 2 / : zg zh 607128 / + : 13 mat 2 π * sqrt zg 0.5 zh *^(2^* e zg^/ Γ.κ 1 el Γ.κ κ 1 + el z κ + / κ 114 sum + (* eval 21 line :

Valid for Re(ζ)>0

Heaviside x Φ 1 x sign + 2 / :

PID Controller

PID Controller

Based on Mathcad QuickSheet PID Controller, and web site http://www.engin.umich.edu/group/ctm/PID/PID.html

Setup

Time domain interval

t.i t.e 12 mat 0312 mat :

Inverse Laplace setup

P 6201913 mat :

a ns nd 13 mat

N 20 :

1 F t ILap F s# F t.i t.e N P ILap : F 1 col F 2 col t cinterp 21 line :

System

Laplace transform, by maple (disabled) Heaviside isn't in SMath, so, can't call here u(t)

Desired output, and it's Laplace transform.

t u t 12 / - Φ t 4 / cos * :

t 12 / - Heaviside t 4 / cos * t s laplace maple 4 s 2 / - exp * 4 s * 18 / cos * 18 / sin - (* 116 s 2^* + /

s U 4 e s 2 / -^* 4 s * 18 / cos * 18 / sin - 116 s 2^* + / * eval :

Notice that this function is in the form

A e s 2 / -^* B s * C - D s 2^* 1 + / *

System transfer function: mass, spring and damper system

m 1 :

b 10 :

k 20 :

s H 1 m s 2^* b s * + k + / :

this is:

m x'' * b x' * + k x * + F ≡

Open loop response

To de desired output

s Y.ol s H s U * :

t y.ol s Y.ol t ILap :

To de step

s Y.ol s H s / :

t y.olu s Y.ol t ILap :

x u x y.ol x y.olu 3 1 sys

Proportional, Integral and Derivative transfers functions

s P Kp :

s I Ki s / :

s D Kd s * :

Transfer function of the whole PID controller

s G s P s I + s D + :

Closed-loop transfer function for the overall system

s T s G s H * 1 s G s H * + / :

System output in the Laplace domain

s Y s T s U * :

P Controller

Kd 0 : Ki 0 : 21 sys

Kp 100 : t y.P1 s Y t ILap : 21 sys

Kp 200 : t y.P2 s Y t ILap : 21 sys

x u x y.P1 x y.P2 3 1 sys

Sintonize values:

PD Controller

Kp 300 : KI 0 : 21 sys

Kd 5 : t y.PD1 s Y t ILap : 21 sys

Kd 20 : t y.PD2 s Y t ILap : 21 sys

x u x y.PD1 x y.PD2 3 1 sys

Sintonize values:

PI Controller

Kp 30 : Kd 0 : 21 sys

Ki 70 : t y.PI1 s Y t ILap : 21 sys

Ki 100 : t y.PI2 s Y t ILap : 21 sys

x u x y.PI1 x y.PI2 3 1 sys

Sintonize values:

PID Controller

Kp 350 : Kd 320 : Ki 50 : 13 mat t y.PID1 s Y t ILap : 21 sys

Kp 120 : Kd 420 : Ki 30 : 13 mat t y.PID2 s Y t ILap : 21 sys

x u x y.PID1 x y.PID2 3 1 sys

Sintonize values:

Alvaro