<p> Numerical Inverse Laplace Transform ... [a,ns,nd] </p>

5d3fc9fd-08f3-4504-99da-a15a61460003 16 3 12 decimal

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

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

s F t invLaplace t 1 el 105 -^: t.ini 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 e a^1 - (n^* - : 21 line for bdif 1 el nd 1 + Gamma nd 2 + 1 - Gamma / 1 Gamma / (: β2 β ns 2 + (ns 1 + nd + (range 11 range el : n 2 nd range bdif n el nd 1 + Gamma nd 2 + n - Gamma / n Gamma / (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 kt 1 pts range tt t kt el : s α tt / : βt β tt / : n 1 α length range βtF n el βt n el s n el F * : for RβtF βtF Re : ft kt el RβtF eval sum : 61 line for OUT t ft augment : 91 line :

Modeling a plant closed control loop

This work sheet will demonstrate the construction and analysis of proportional [P], integral [I] and derivative [D] controllers and how to use them to obtain a desired system response. In this application, we will consider the following unity feedback system.

The XFR function of the PID controller can be represented in the Laplace algebra. where: Kp => Proportional gain, Ki => Integral gain, Kd => Derivative gain

s Kp Kd Ki G Kd s 2^* Kp s * + Ki + s / :

<= The plant XFR s F 1 s s 2^α s * + β + (* / :

user range/mesh t.ini 0 : t.end 1.5 : pts 100 : 13 mat

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

<= Dynamics of the system α 10 : β 20 : 12 mat

XFR s F t invLaplace :

User XFR parameters [α, β]

XFR 0.05 2 1 sys

Proportional s F Kp s s 2^α s * + β + Kp + (* / :

Kp 360 : Kd 49 : Ki 338 : 31 mat user model parameters ≡

P s F t invLaplace :

User Proportional [α, β, Kp]

P 1 2 1 sys

Proportional + Derivative s F Kd s * Kp + s s 2^α Kd + (s * + β + Kp + (* / :

PD s F t invLaplace :

User PD [α, β, Kp, Kd]

PD 1 2 1 sys

Proportional + Integral s F Kp s * Ki + s s 3^α s 2^* + β Kp + (s * + Ki + (* / :

PI s F t invLaplace :

User PI [α, β, Kp, Ki]

PI 1 2 1 sys

s G Kp s * Kd s 2^* + Ki + s / : PID controller Xfr ≡

s H 1 s 2^α s * + β + / : Plant Xfr ≡

3 modes PID s F s G s H * s 1 s G s H * + (* / :

PID s F t invLaplace :

User PID [α, β, Kp, Ki, Kd]

PID 1 2 1 sys

x y XY x y / - :

P PI PD PID XY 1 6 1 sys