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, z transform

PID Controller, z transform

Setup

Time domain interval

t.i t.e 12 mat 02012 mat :

Inverse Laplace setup

P 6201913 mat :

a ns nd 13 mat

N 25 :

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

Desired output, and it's Laplace transform.

t u 1 :

z Uz z z 1 - / :

t u t z ztrans maple z 1 - z + /

K 0.5 :

z Hz z K + z 2^z - K + / :

System transfer function

Convert to Laplace domain

T 1 :

s U 2 s T * + 2 s T * - / Uz :

s H 2 s T * + 2 s T * - / Hz :

https://en.wikipedia.org/wiki/Z-transform#Relationship_to_Laplace_transform

Open loop response

To de desired output

Yz z Hz z Uz * :

z Y.ol Yz :

Symbolic form

t y.sym 341 t +^* 21 t +^1 i - (1 t +^* + 421 t +^* 1 i - (t^* - 41 t +^/ :

Yz z t invztrans allvalues maple 341 t +^* 21 t +^1 i - (1 t +^* + 421 t +^* 1 i - (t^* - 41 t +^/ 341 t +^* 21 t +^1 i + (1 t +^* + 421 t +^* 1 i + (t^* - 41 t +^/ 21 mat

Numeric form

t y.ol s Y.ol t ILap :

To de step, numeric form

s Y.olu s H s / :

t y.olu s Y.olu t ILap :

x u x y.sym abs x y.ol x y.olu 4 1 sys

PID Controller Setup

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 2000 : t y.P1 s Y t ILap : 21 sys

Kp 3000 : t y.P2 s Y t ILap : 21 sys

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

Sintonize values:

Kd 0 : Ki 0 : Kp 2500 : 31 sys

Alvaro