Coursera Specialization on Embedding
Sensors and Motors
ESM_C4M1V3
Automotive Cruise Control Systems Use PID Control
[1]
2012 Honda Accord on cruise control
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Cruise control buttons on steering
wheel of 2012 Honda Accord
Proportional control – based on difference between setpoint and actual speed
Integral control – opens the throttle more to help a car get up a hill, and helps
settle the car into correct speed
Derivative control – senses deceleration uphill to open throttle early, senses
acceleration on steep downhills and may downshift the cars (on later models)
Loop Control Definitions: Proportional
Error
(e)
up(t)
Time Domain
up(t) = Kp e(t)
KI
[2]
from sensor
KD
Sensor
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Proportional (P) control: output = gain x error
Large P gain lets the output catch up quickly to setpoint
Setting the P gain too high causes overshoot
Loop output is negligible when P gain error is small
Laplace Domain
Up(s) = Kp E(s)
Proportional Control for Series RLC Circuit
output
Laplace transform for natural
solution:
s2 + (R/L) s + 1/LC = 0
[3]
input
Plant model P(s) for 2nd order plant:
Gp(s) = output / input = Vc(s) / Vs(s)
Gp(s) = (1/LC) / (s2 + (R/L) s + 1/LC)
substituting: ωn = 1 / √LC and α = R/L
ωn2 = 1 / LC
s2 + (R/L) s + 1/LC) = s2 + αωns + wn2= 0
Gp(s) = ωn2 / (s2 + αωns + wn2 )
Proportional Control for Series RLC Circuit
Plant model for 2nd order plant:
Gp(s) = ωn2 / (s2 + αωns + wn2 )
[3]
(RLC Circuit)
[4]
Sensor
Controller model:
U(s) = Up(s) E(s)
U(s) = Kp E(s)
Transfer Functions for Proportional Control of Series RLC Circuit
(RLC Circuit)
[4]
Sensor
Plant model: Gp(s) = ωn2 / (s2 + αωns + wn2 )
Controller model: Gc(s) = U(s)/ E(s) = Kp
Open Loop Transfer Function: Gp(s)Gc(s) = wn2 Kp / (s2 + αωns + wn2 )
Closed Loop Transfer Function = G(s) = Gp(s)Gc(s)/(1 + Gp(s)Gc(s))
Closed Loop Transfer Function for Proportional Control of Series RLC Circuit
(RLC Circuit)
[4]
Sensor
Closed Loop G(s) = [wn2 Kp / (s2 + αωns + wn2 )] / [1 + (wn2 Kp / (s2 + αωns + wn2 )]
= wn2 Kp / (s2 + αωns + (1 + Kp) wn2 )
[5]
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α
α
As Kp is increased, the closed loop natural frequency increases and the
closed loop damping ratio decreases
Setting Kp too high causes overshoot
Steady State Error for Proportional Control of Series RLC Circuit
G(s) =
closed loop
transfer
α
[6]
R = step input
Ess = steady state error
Ess(s) = lim(s→0) R – R G(s)
= R ( 1 – Kp / (1 + Kp)
= R / (1 + Kp)
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No way to reduce error to zero using solely proportional gain
This would result in a steady-state offset error = R / (1 + Kp)
You can reduce the error by making Kp large, but then you get overshoot
Loop Control Definitions: Integral
Error
(e)
KI
[2]
from sensor
ui(t)
KD
Sensor
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Integral (I) control: output = gain x integral of the error
The loop stores all measured error (∫ε), positive or negative
Functions best when absolute error is kept very small
Acts to reduce error to zero or very close
Control loop action at steady state is due to integral control
Contributes strongly to overshoot of the setpoint, needs
differential control to offset this effect
Time Domain
ui (t) = Ki
∫
0
t
e (t) dt
Laplace Domain
Ui(s) = (Ki / s) E(s)
Loop Control Definitions: Differential
Error
(e)
KI
[2]
from sensor
KD
ud(t)
Sensor
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Differential (D) control: output = gain x derivative of the error
Time Domain
Looks at the rate of change of the error
ud (t) = Kd d/dt e (t)
The more error changes or the longer the derivative time, the
larger the derivative factor becomes.
When error is large, the P and the I terms increase the output u(t) Laplace Domain
Ud(s) = (Kd s) E(s)
and this aggressive response can cause overshoot
The derivative then acts more aggressively to reduce the overshoot
Relative Effects of P, I, and D Control
[7]
High gain in proportional control reduces
the error quickly, but causes overshoot
High gain in integral control corrects errors faster,
but you need derivative control to prevent overshoot
Modest derivative control
resolves overshoot, but
excessive derivative control
can lead to system instability
PID Control for Series RLC Circuit
[6]
Plant model : Gp(s) = ωn2 / (s2 + 2αωns + wn2 )
(RLC Circuit)
[4]
Laplace transform for
natural solution:
s2 + (R/L) s + 1/LC = 0
substituting in:
ωN = 1 / √LC
α = R/L
gives: s2 + 2αωns + wn2= 0
Controller model:
U(s) = (Up(s) + Ui(s) + Ud(s)) x E(s)
U(s) = (Kp + Ki / s + Kd s ) x E(s)
Open Loop Transfer Function for PID Control of Series RLC Circuit
(RLC Circuit)
[4]
Plant model: Gp(s) = ωn2 / (s2 + αωns + wn2 )
Controller model: Gc(s) = U(s)/ E(s) = (Kp + Ki / s + Kd s)
Open Loop Transfer Function: Gp(s)Gc(s) = ωn2 (Kp + Ki / s + Kd s) / (s2 + αωns + wn2 )
multiply by s / s Gp(s)Gc(s) = ωn2 (Kp s + Ki + Kds2 ) / s (s2 + αωns + wn2 )
Closed Loop Transfer Function for PID Control of Series RLC Circuit
Open Loop Transfer Function: Gp(s)Gc(s) = (Kp s + Ki + Kds2)/ s (s2 + αω + wn2 )
Closed Loop Transfer Function = G(s) = Gp(s)Gc(s)/(1 + Gp(s)Gc(s))
G
α
α
α
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The closed loop transfer function is a third order system with two zeros
Kp, Ki, and Kd give complete control over the three poles of the system
This equation can be solved analytically using an iterative approach
First, let Ki = 0 and reduce the third order equation to second order
Then, solve for Kp and Kd to give satisfactory rise time and overshoot specs
Then, adjust Ki until a satisfactory setting time is met
[8]
Tuning the PID Controller for a Plant Model
[9]
• First, let Ki = 0 and reduce the third order equation to second order
• Then, solve for Kp and Kd to give satisfactory rise time and overshoot specs
• Then, adjust Ki until a satisfactory setting time is met
Ziegler Nichols Method for Tuning the PID Controller for a Plant Model
[10]
• Created in 1942 by John G. Ziegler and Nathaniel B. Nichols of Taylor
Instruments
• For PID control set Ki and Kd to zero, and increase Kp until the loop output
oscillates
• Second, document critical gain Kc and the period of output Pc
• Third, adjust Kp to 0.6Kc, Ki to 2Kp/Pc, and Kd to KpPc/8
• Adequate for loops where stability is achieved after ¼ wavelength of time
• Most common manual tuning method for all types of controllers
Citations
[1] www.motortrend.com
[2][7][10] www.machinedesign.com
[3] www.slideplayer.net
[4] www.elprocus.com
[5][6][8] www.maplesoft.com
[9] www.slideshare.net
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