Arb 03 Pid Control

8/13/2019 Arb 03 Pid Control

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G52ARBAdvanced Robotics

Lecture 3

PID Control

University of Nottingham G52ARB 2

Overview PID principles

basic control algorithm

PID parameter effects Worked example

DC motor control

PID tuning manual methods

formal methods

PID implementation pseudo code


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University of Nottingham G52ARB

Navigation

control

3

the robotmust modulate

its motor outputs toachieve the desired trajectory


The Sense-Think-Act Cycle

4

Repeat! sense the current state! reduce difference between current state and goal state

Until! current state = goal state


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The Sense-Think-Act Cycle

5

state

goal

compareerror compute

action requiredto reduce error

controlaction


Compute the Control Action Present

the current error proportional

Past the sum of errors up to present time

integral

Future the rate of change of the error

derivative

6


4/14

CA (t ) = K p e (t ) + K i t

0

e ( )d + K dde (t )

dt


PID Control

A proportional-integral-derivative controller (PID

controller) simply combines these three terms ina weighted sum to obtain the control action

7


Appropriate Parameters Some applications may not require the use of all

three terms appropriate parameters can be set to zero in order to

remove unused terms PI, PD, P or I controllers

PI particularly common since the integral term is required in order to remove steady-state

error

the derivative term is very sensitive to measurement noise

8


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PID Block Diagram

9

state

setpoint

!error

P

controlactionI

D

!

+

-K i

t

0

e ( )d

K p e (t )

K dde

(t)dt

P out (t ) = K p e (t )


Proportional Term

A high proportional gain results in a large changein the output for a given error if the proportional gain is too high, the system can

become unstable if the proportional gain is too low, the system will be

very slow responding to changes Proportional term is often the dominant one

but pure proportional control will not settle at setpoint

10


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I out (

t) =

K i

t

0

e(

)d


Integral Term

Contribution is proportional to both themagnitude and the duration of the error sum of the instantaneous error over time

Integral term accelerates the process towardssetpoint and eliminates steady-state error but easily causes overshoot (the actual value

crosses over the setpoint and creates an error in theopposite direction)

11

D out (t ) = K dde (t )

dt


Derivative Term

Contribution is proportional to the rate of changeof the error over time first derivative of the instantaneous error

Used to slow the rate of change of the error effect is most noticeable near to the setpoint

Helps reduce overshoot caused by integral term

Differentiation amplifies signal noise and soderivative can cause instability with noise12


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PID Summary

Proportional gain: K p" K p faster response # instability

Integral gain: K i " K i elimination of steady state error # overshoot

Derivative gain: K d " K d decreases overshoot, slows transient response# instability

13

CA(

t) =

K p

e(

t) +

K i

t

0

e(

)d

+ K

d

de (t )dt

DC Motor

+

-

v emf (t)

! (t)

R

L

I

LLoadInertialLoad J

vapp (t)

-

+

" (t)

Kf! ( " )

Torqu e

Angu la r ra te

Vis cou sfriction

i ( t )


DC Motor Input: voltage vapp (t )

Output: angular velocity ! (t )


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0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


Open-Loop Controlvapp (t ) = 26.681V

! (t ) reaches approx. 1 rotation per second

0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


P ControlP = 20

P small: note the steady state error


9/14

0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


P ControlP = 100

P increases: the steady state error persists, and oscillations appear

0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


P ControlP = 800

P increases: oscillations becoming dominant


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0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


P ControlP = 900

P increases further : unstable behaviour

0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


PI ControlP = 100, I = 100

I non-zero: the steady state error is reduced


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0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


PI ControlP = 100, I = 1000

I increases: rise-time improves, but overshoots are significant

0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


PID ControlP = 100, I = 1000, D=20

D non-zero: oscillations are damped


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0 5 10 15 20 - 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

T

w


PID ControlP = 300, I = 1000, D=20

PID tuned: exceedingly close control


Parameter Tuning If PID parameters are chosen incorrectly

the process may be unstable its output may diverge, with/without oscillation

The parameters must be adjusted to achieve thedesired behaviour for a given application PID parameter tuning

The optimum behaviour is application dependent rise-time must be less than a specified time overshoot may not be allowed minimise the energy required to reach setpoint oscillations may not be permitted

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Effects of Increasing Parameters

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Parameter Rise-time Overshoot Settling

time

Steady-

state error

K p

K i

K d

decrease increasesmall

changedecrease

decrease increase increase eliminate

smallchange

decrease decrease none


Tuning Methods Manual heuristic method

set I and D to zero increase P until the output oscillates

set P to about half the critical oscillating value increase I until the steady-state error is eliminated inan appropriate amount of time for the application

increase D until overshoot is reduced to acceptable Automated (software) tuning

repeat automatically induce setpoint changes analyse the process characteristics automatically adjust parameters

until acceptable26


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Ziegler-Nichols Method Set I and D to zero

increase P until it reaches the critical gain, K c ,at which output oscillates note the oscillation period, P c

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ControlType

K p K i K d

P

PI

PID

0.5 K c - -

0.45 K c 1.2 K p / P c -

0.6 K c 2 K p / P c K p P c / 8

loop error[t]= setpoint - actual_position integral[t]= integral[t-1] + error[t]

derivative[t]= error[t] - error[t-1

action= Kp * error[t] + Ki * integral[t] * dt +

Kd * derivative[t] / dt

wait(dt)

end loop


Pseudo-Code

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Arb 03 Pid Control