55:036 Embedded Systems and Systems Softwares-iihr64.iihr.uiowa.edu/MyWeb/Teaching/ece_55036_2013/Lectures/PID... · F. Vahid and T. Givargis, Embedded System Design —A Unified

PID Control , Slide 1 Embedded Systems and Software, 55:036. The University of Iowa, 2013

Embedded Systems and Software

A Simple Introduction to Embedded Control Systems (PID Control)


Acknowledgements

• The material in this lecture is adapted from: – F. Vahid and T. Givargis, Embedded System Design—A Unified

Hardware/Software Introduction, John Wiley & Sons, 2002 (Chapter 9)

– T. Wescott, “PID Without a PhD” http://www.embedded.com/2000/0010/0010feat3.htm


Control System Terminology

• Control physical system (plant) output – By setting plant’ input

• Examples – Cruise control, thermostat, disk drive, chemical processes

• Difficulty due to – Disturbance: wind, road, tire, brake; opening/closing door… – Human interface: feel good, feel right…


Control System Tracking

Reference Input a.k.a. Set Point where we want the output to be

For example, here the user turned up the thermostat

This how the system tracks the reference input or set point


Control System Tracking

Reference Input a.k.a. Set Point where we want the output to be

For example, here the user turned up the thermostat

This how the system tracks the reference input or set point


• Plant – Physical system to be controlled: car, plane, disk, heater,…

• Actuator – Device to control the plant: throttle, wing flap, disk motor,…

• Controller – Designed product to control the plant

Open-Loop Control Systems


• Output – What we are interested in: speed, disk location, temperature

• Reference Input / Set Point – Desired output: speed, desired location, desired temperature

• Disturbance – Uncontrollable input to the plant imposed by environment

Wind, bumping the disk drive, door opening

Open-Loop Control Systems

7


• Feed-forward control • Delay in actual change of the output • Controller doesn’t know how well things are going • Simple • Best use for predictable systems

– Examples?

Characteristics of Open Loop Control


Closed Loop Control Systems

Sensor measures plant output

Error detector. Can, and often is as simple as computing difference between Set Point and plant output

Use error to adjust controller

Concept: feedback control systems

Goal: minimize tracking error


• Develop a model of the plant • Develop a controller • Analyze the controller • Consider Disturbance • Determine Performance • Example: Open Loop Cruise Control System

Designing Open Loop Control System


• May not be necessary – Can be done through experimenting and tuning

• But, – Can ease design, and be useful for deriving the controller

Model of the Plant


Model the car speed at t +1 as a linear combination of the speed at t (i.e., vt) and the throttle at t (i.e., ut)

vt+1 = Avt + But

Next, experimentally determine the model parameters A and B. On flat surface at 50 mph, open the throttle to 40 degrees. Then measure the speed after one time unit. Assume the speed is 55 mph. Then

55 = 50A + 40B

Repeat the experiment, but now start at 65 mph and with the throttle at 50 degrees. Assume the resulting speed after one time unit is 70.5 mph. Then

70.5 = 65A + 50B

One can now solve these two equations for the two unknowns A and B, to find A = 0.7 and B = 0.5.

One can repeat the experiment many times and determine, say, average values for A and B. (There are more optimum techniques for dealing with multiple As and Bs). Regardless, we now have model for the car (plant).

vt+1 = 0.7vt + 0.5ut

Example


Designing the Controller

Assuming we want to use a simple linear function ut = F(rt) = P × rt

(Recall that rt is the desired speed or Set Point)

Linear proportional controller vt+1 = 0.7vt + 0.5ut = 0.7vt + 0.5P×rt

Let vt+1= vt at steady state (vss) vss= 0.7vss+0.5P×rt

At steady state, we want vss= rt. To achieve this, we need P = 0.6, or ut=0.6×rt

PID Control , Slide 14 Embedded Systems and Software, 55:036. The University of Iowa, 2013 14

Analyzing the Controller Let v0 = 20 mph, rt = 50 mph vt+1 = 0.7vt + 0.5(0.6)rt = 0.7vt + 0.3×50 = 0.7vt + 15 Throttle position is 0.6×50 = 30 degrees


Analyzing the Controller Let v0 = 20 mph, r0 = 50 mph vt+1 = 0.7vt + 0.5(0.6)rt = 0.7vt + 0.3×50 = 0.7vt + 15 Throttle position is 0.6×50 = 30 degrees








Considering the Disturbance

19

Assume road grade can affect the speed from –5 mph to +5 mph Recall our model: vt+1 = 0.7vt + 15 + wt With: wt = -5 and +5, the bounds are

vt+1 = 0.7×vt + 10 vt+1 = 0.7×vt + 20



20


vt+1 = 0.7×vt + 10 vt+1 = 0.7×vt + 20



21


vt+1 = 0.7×vt + 10 vt+1 = 0.7×vt + 20



22


vt+1 = 0.7×vt + 10 vt+1 = 0.7×vt + 20



23


vt+1 = 0.7×vt + 10 vt+1 = 0.7×vt + 20



24


vt+1 = 0.7×vt + 10 vt+1 = 0.7×vt + 20


Determining Performance

25

vt+1 = 0.7vt + 0.5P×r0 - w0 // Our Model

v1 = 0.7v0 + 0.5P×r0-w0 // Speed at time 0

v2 = 0.7(0.7×v0+0.5P×r0-w0) + 0.5P×r0-w0 // Speed at time 1

= 0.7×0.7×v0 + (0.7+1.0) × 0.5P×r0 -(0.7+1.0)w0

vn = (0.7n)v0 + (0.7n-1 + 0.7n-2 +…+ 0.7 + 1.0)(0.5P×r0-w0) // Speed at time n

Coefficient of vt determines rate of decay of v0. Bigger coefficient will result in slower response. >1 or <-1, vt will grow without bound <0, vt will oscillate


Designing Closed Loop Control System


Closed-Loop Control

[ ] BABBAvABBABvAABvAv +++=+++=+= 20

30

223

( ) ttt

ttttt

wPr.vP..wvrPvv−+−=

−−−=+

505070)(5.07.01

( )ttt vrPu −−=tttt wu.vv −+=+ 507.01

BvAv tt +=+1

( ) 005.0,5.07.0 wPrBPA −=−=

BvAv += 01

[ ] BABvABBvAABvAv ++=++=+= 02

012

( )BAAvAv nnnn 121

0 =+++= −−

Consider fixed set point and fixed disturbance

( ) 001 505070 wPr.vP..v tt −+−=+

( ) ( ) ( )( )( )0021

0 5.015.07.05.07.05.07.0 wPrPPvPv nnnn −+−+−+−= −−

Plant model Control Law (feedback)


Tuning Performance: Finding Proper P

( ) ( ) ( )( )( )0021

0 5.015.07.05.07.05.07.0 wPrPPvPv nnnn −+−+−+−= −−

To obtain convergence requires ( ) 15.07.0 <− P 4.36.0 <<−⇒ P

To avoid oscillation requires 05.07.0 ≥− P 4.1≤⇒ P

Reduce the effect of initial conditions (v0), make (0.7-0.5P) as small as possible 4.1=⇒ P

( ) 001 505070 wPr.vP..v tt −+−=+ ( ) 450 <−= tt vrPu



Steady state means vss = vt + 1 = vt ( ) 00505070 wPr.vP..v ssss −+−=

Pwr

PPvss 5.03.05.03.0

5.0 00 +−

+=

To get vss as close as possible to r0, (perfect tracking) let P ∞

( ) ( ) ( )( )( )0021

0 5.015.07.05.07.05.07.0 wPrPPvPv nnnn −+−+−+−= −−

( ) 001 505070 wPr.vP..v tt −+−=+ ( ) 450 <−= tt vrPu



To obtain convergence requires ( ) 15.07.0 <− P 4.36.0 <<−⇒ P

To avoid oscillation requires 05.07.0 ≥− P 4.1≤⇒ P

Reduce the effect of initial conditions (v0), make (0.7-0.5P) as small as possible 4.1=⇒ P

To get vss as close as possible to r0, (perfect tracking) let P ∞

Finally, pick 3.3=P stable, fast, good tracking, but some oscillation


0 10 20 30 40 50 600

10

20

30

40

50

60

70

Control Point (i.e., time)

Spee

d (m

ph)

Analyze Controller Performance 0 mph,50 mph,20 00 === wrv t

( ) 01 505070 Pr.vP..v tt +−=+

( ) 01 )3.3(50)3.3(5070 r.v..v tt +−=+

5.8295.01 +−=+ tt vv

( )tt vrPu −= 0 ( )tt vu −= 503.3

Valid range for controller is 0-45o. Thus, with this P, the actuator saturates.

Tracking error ~ 8 mph

Significant oscillation

3.3=P


Analyze Controller Performance

0 10 20 30 40 50 600

10

20

30

40

50

60

70


Spee

d (m

ph)

Tracking error ~ 20 mph

No oscillation

Valid range for controller is 0-45o. Thus, with this P=1, the actuator does not saturate.

Set P = 1 to avoid oscillation


Analyzing the Controller

0 10 20 30 40 50 600

10

20

30

40

50

60

70


Spee

d (m

ph)

P = 1.0

P = 3.3 P = 2.5

Set point (where we want to be)


Closed-Loop Performance

0 10 20 30 40 50 600

10

20

30

40

50

60

70


Spee

d (m

ph)

P = 3.3, W =+5 P = 3.3, W =0

P = 3.3, W = - 5

0 mph,50 mph,20 000 ≠== wrv

( ) 01 505070 Pr.vP..v tt +−=+

( ) 01 )3.3(50)3.3(5070 r.v..v tt +−=+

01 5.8295.0 wvv tt ++−=+

Change is 5 mph compared to 66-33 = 33 mph for open-loop control


• Objective – Causing output to track a reference even in the presence of

Measurement noise Model error Disturbances

• Metrics – Stability

Output remains bounded – Performance

How well an output tracks the reference – Disturbance rejection – Robustness

Ability to tolerate modeling error of the plant

General Control System


Performance (generally speaking)

• Rise time – Time it takes

from 10% to 90%

• Peak time • Overshoot

– Percentage by which Peak exceed final value

• Settling time – Time it takes to

reach 1% of final value


• Difficult • May need to be done first • Plant is usually on continuous time

– Not discrete time. For example, car speed continuously reacts to throttle position, not at discrete interval

– Sampling period must be chosen carefully to ensure “nothing interesting” happens in between sampling intervals

• Plant is usually non-linear – For example, shock absorber response may need to be 8th order

differential equation • Iterative development of the plant model and controller

– Have a plant model that is “good enough”

Plant Modeling


Controller Design: “P”

( )tt vrPu −= 0Proportional control - multiplies the tracking error with a constant P

P affects

Stability

Tracking

Disturbance rejection

Large P can lead to overshoot and oscillation

Make P large to reduce tracking error

Make P large to improve disturbance rejection

( ) 001 505070 wPr.vP..v tt −+−=+Closed-loop model with linear plant


Controller Design: “PD”

Proportional and Derivative control - P control, and also consider error over time

Intuitively

Want to “push” more if error is not reducing fast enough

Want to “push” less if error is reducing really fast

( ) ( ) ( )( )11 −− −−−+−= ttttttt vrvrDvrPu

( )1−−+= tttt eeDPeu


PD Controller

( )1−−+= tttt eeDPeu

tttt wu.vv −+=+ 507.01 Plant model

Control Law (feedback)

( ) ( ) ( )( )( ) ttttttttt wvrvrDvrP.vv −−−−+−+= −−+ 111 507.0

ttt vre −=

( )( ) ( ) ttttttt wDrrDPDvvDP.vv −−++++−= −−+ 111 5.05.05.0507.0

05.07.015.0 r

PPvss

+−=

Assume reference input and disturbance are constant, then the steady-state speed is

P can be set for best tracking and disturbance control

Then D can be set to control transient behavior

Does not depend on D

Steady state means vss = vt + 1 = vt


PD Control Example


Controller Design: “PI” Proportional and Integral control P control, and also integrate (sum) error

Intuitively

Sum (integrate) error over time, and use this to “push”

Ensure that system reaches desired steady state, eventually

( ) ( ) ( )( )+−+−+−= −− 11 ttttttt vrvrIvrPu

( )++++= −− 21 ttttt eeeIPeu

P controls disturbance, I controls steady state convergence and convergence rate


Controller Design: “PID” Proportional , Integral and Derivative control - use all 3 types of control

Intuitively

P considers current error, controls disturbance

I controls long-term error, controls steady state convergence and convergence rate

( ) ( ) ( )( ) ( ) ( )( )1111 −−−− −−−++−+−+−= ttttttttttt vrvrDvrvrIvrPu

( ) ( )121 −−− −+++++= ttttttt eeDeeeIPeu

D considers how error is changing, controls transient response


Examples


• More than 90% of all controllers used in process industries are PID controllers.

• A typical chemical plant has 100s or more PID controllers.

• PID controllers are widely used in: – Chemical plants – Oil refineries – Pharmaceutical industries – Food industries – Paper mills – Electronic equipments

Application of PID Control

http://upload.wikimedia.org/wikipedia/commons/a/a2/Harddisk-full.jpg


Off-The-Shelf PID Controllers

There are many off-the-shelf PID controllers available


Block Diagram View of PID Controller


Another Block Diagram View of PID Controller

Set desired input



Compare actual output against setpoint



Use part of the error signal



Use sum of the previous errors



Use rate of change of errors



Update the process, and repeat


Sidebar: Analog PID Controller

Integral

Proportional

Derivative

Summer


PID Pseudo Code

previous_error = 0

integral = 0

start:

get actual_position

error = setpoint - actual_position

integral = integral + error*dt

derivative = (error - previous_error)/dt

output = P*error + I*integral + D*derivative

previous_error = error

wait(dt)

goto start


• Main function loops forever, during each iteration – Read plant output sensor (may require ADC) – Read current desired reference input (set point) – Call a routine PidUpdate, to determine actuator value – Set actuator value (may require DAC)

Software Coding


Software Coding



Computation



• Analytically deriving P, I, D may not be possible – E.g., plant not is not available, or to costly to obtain

• Ad hoc method for getting “reasonable” P, I, D – Start with a small P, I = D = 0 – Increase D, until seeing oscillation, then reduce D a bit – Increase P, until seeing oscillation, then D a bit – Increase I, until seeing oscillation

• Iterate until one can change anything without excessive oscillation

PID Tuning


• PID Without a PhD by Tim Wescott • http://www.embedded.com/2000/0010/0010feat3.htm • http://www.embedded.com/story/OEG20020726S0044

Excellent Reference for PID Control

http://www.embedded.com/2000/0010/0010feat3.htm


• Quantization • Overflow • Aliasing • Computation Delay

Practical Issues with Computer-Based Control


Quantization & Overflow

• Quantization – E.g., can’t store 0.36 as 4-bit fractional number – Can only store 0.75, 0.59, 0.25, 0.00, -0.25, -050,-0.75, -1.00 – Choose 0.25 – Results in quantization error of 0.11

• Sources of quantization error – Operations, e.g., 0.50*0.25=0.125

Can use more bits until input/output to the environment/memory – ADC resolution

• Overflow – Can’t store 0.75 + 0.50 = 1.25 as 4-bit fractional number

• Solutions – Use fix-point representation/operations carefully

Time-consuming – Use floating-point co-processor

Costly


Aliasing Example

The points below are the samples from a sine wave

t

Volta

ge


Aliasing Example

Reconstruction of the sine wave

t

Volta

ge


Aliasing Example

But why not this reconstruction?

t

Volta

ge


Aliasing Example

Both (an many others) are valid reconstructions if we don’t exclude higher frequencies

In other words, if we don’t eliminate (filter out) high frequencies their samples become undistinguishable from low frequency samples. The high frequencies alias as low frequency signals.

t

Volta

ge


Aliasing Example

Sampling at 2.5 Hz (period = 0.4 s) the following signals are indistinguishable

( )tty π6sin)( =

( )tty πsin)( =

Frequency f = 3 Hz

Frequency f = 0.5 Hz

In fact, with sampling frequency of 2.5 Hz one can only correctly sample signal below the Nyquist frequency 2.5/2 = 1.25 Hz.


• Inherent delay in processing – Actuation occurs later than expected

• Need to characterize implementation delay to make sure it is negligible

• Hardware delay is usually easy to characterize – Synchronous design

• Software delay is harder to predict – Should organize code carefully so delay is predictable and

minimized – Write software with predictable timing behavior (be like hardware)

Time Trigger Architecture Synchronous Software Language and/or RTOS

Computation Delay


Documents

55:036 Embedded Systems and Systems Softwares-iihr64.iihr.uiowa.edu/MyWeb/Teaching/ece_55036_2013/Lectures/PID... · F. Vahid and T. Givargis, Embedded System Design —A Unified