Consider PD-Control with set-point weighting by setting ki = 0 in Equation 6.4.1. Determine the closed-loop transfer function from the position reference, r, to the angular position output, θ in radians, i.e., Y(s)/R(s), in terms of K, τ, bsp, kp and kd/ then, Determine kp and kd in terms of the closed-loop natural frequency, ωn, damping ratio, ζ, motor time constant, τ, and motor gain constant, K. Tetwo-uagiee ur iCCaonITCompelsauor with sUL pomt wCigntng Is.r()y(t)sp[6.4.1]dsd

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Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

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Consider PD-Control with set-point weighting by setting k_i = 0 in Equation 6.4.1. Determine the closed-loop transfer function from the position reference, r, to the angular position output, θ in radians_,i.e., Y(s)/R(s), in terms of K, τ, b_sp, k_pand k_d/then, Determine k_pand k_d in terms of the closed-loop natural frequency, ω_n, damping ratio, ζ, motor time constant, τ, and motor gain constant, K.

Tetwo-uagiee ur iCCaonIT
Compelsauor with sUL pomt wCigntng Is.
r()
y(t)
sp
[6.4.1]
d
sd

Expert Solution

Step 1

Part 1:

Write the expression for the two-degree of freedom PID compensator with set point weighting and Substitute 0 for k_i,

d
dt
dt
d
u(t)k, (bgr(t)-y(1)) + k b
dt
dt

Step 2

Take the Laplace transform,

U(s) ,(bR(s)-Y(s)) + k,s(bR(s) -Y (s)
U(s)=(k,bg kb)R(s) - (k, -k,s)Y(s)
.(1)

Step 3

Write the expression for transfer function for the motor,

Y (s)
U (s)
1
m
R
Ja
Rm
Y(s)
U (s)
1
eq*m s+1
k s
m
(
J R
-s +1 Y(s)
.(2)
U (s) ks
X

Step 4

Equate the equation (1) and equation (2),

JR
eq
-s 1
(k,bg +kb)R(s)-(k, -k,s)Y(s) =k_|
k2
(JR
(k,bg +kbsR(s)=k|
"s+1+k, -ks Y (s)
eqm
(k,bg + k,b)
Y(s)
R(s)
J R
eq
- ki s+1+k
k2

Step 5

This expression is the open loop transfer function. Here, there is a negative unity feedback system.

Thus,

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