Feedback Control Roundup

• Where is this class going?• Why control?• What is feedback?• What are those block diagrams, and

how do they relate to the equations?• How does PID work?• How do controllers go out of tune, how

can I recognize them, and what do I do?

Where is this class going?

• Things I want you to learn– “Assume steady state” does not work in real life.– Engineering is a synthesis of what you have

learned, not a collection of separate topics– How to analyze and describe a dynamic system– How to understand control systems:

• Anything called a “regulator”

– Trade-offs in control design– How to save the day when the irresponsible

control engineer goes on vacation

Where is this class going?

• Content remaining– Control applications: feedforward, cascade,

decoupling– Ways to analyze the effect of control on

closed loop stability– Frequency domain analysis of dynamic

systems– State space (very good for mechanical

systems)

Why control?

• We use control to:– Keep a value close to its desired value:

• Room temperature, thermostat• Car speed, cruise control• Boiler drum level• Boiler fire-side pressure

– Trade off variation between controlled and manipulated variables

• Torque and position or speed• Flow and level in a surge tank• Temperature and heat stresses induced by heat

Where are we, as mechanical engineers, going to encounter

control?• Any time something moves, that motion must

be regulated.– Equipment: motors and guides– Fluids: flow rate, pressure and temperature

• Control is implemented through mechanical equipment– Motors, valves, dampers– Limitations or faults in this equipment will affect

control (and thus system) performance.

What is Feedback?

Speed

SpeedSensor

Speed in km/h

SpeedError in km/h

Desired speed in km/h

-+

CruiseControl

DesiredThrottlePosition Throttle

Actuator

ActualThrottlePosition Engine,

Transmission,Wheels

Disturbances - wind, slope, bad fuel

1. The controlled variable is measured.

2. It is then compared to a desired value, or setpoint.

3. The difference between setpoint and measurement is called the error.

4. The controller calculates a control output depending on the error.

5. The control output is sent to some physical equipment, a valve or similar, to affect the process. This physical equipment is called the Final Control Element.

6. The process responds to the change made by the final control element.

7. or 1. The process response is measured

What are those block diagrams?

• Block diagrams are a visual representation of the actual dynamic system and its governing equations.

• Each line is a signal - something that varies over time, and can be measured or calculated.

• Each block is a “functional element” - something that does something, and is governed by differential, integral or algebraic equations, and describable as a transfer function.

Block Diagram

CruiseControl

Engine,Transmission,

Wheels

ThrottleActuator

SpeedSensor

Speed

Speed in km/h

Desired speed in km/h

SpeedError in km/h

-+

DesiredThrottlePosition

ActualThrottlePosition

Disturbances - wind, slope, bad fuel

Standard Nomenclature• PV, or y : controlled variable• SP or ysp : setpoint, desired value• OP or u (text uses p): controller output• e : error, ysp - y• d or l : disturbance or load (wild) variable• ym : measured value of y• G : a transfer function

– Gp : process transfer function– Gc : controller transfer function– Gv : final control element (valve) transfer function– Gm : measurement element transfer function– Gd or GL : Disturbance (Load) transfer function– Gcl : closed loop transfer function

• K : transfer function gain : time constant : time delay (dead time) : damping ratio (also called damping factor)

Block Diagram

CruiseControl

Engine,Transmission,

Wheels

ThrottleActuator

SpeedSensor

Speed

Speed in km/h

Desired speed in km/h

SpeedError in km/h

-+

DesiredThrottlePosition

ActualThrottlePosition

Disturbances - wind, slope, bad fuel

ysp e y

ym

uGc Gv Gp

Gm

DisturbanceProcessGd or GL

dThe purpose of the C.L.T.F. is to describe the entire system with a single transfer function that can be analyzed.

Block Diagram

CruiseControl

Engine,Transmission,

Wheels

ThrottleActuator

SpeedSensor

Speed

Speed in km/h

Desired speed in km/h

SpeedError in km/h

-+

DesiredThrottlePosition

ActualThrottlePosition

Disturbances - wind, slope, bad fuel

ysp e y

ym

uGc Gv Gp

Gm

DisturbanceProcessGd or GL

d

Closed Loop System

Closed LoopSetpoint

Transfer Function

Closed LoopDisturbance

Transfer Function

Block Diagram

CruiseControl

Engine,Transmission,

Wheels

ThrottleActuator

Speed

Speed in km/h

Desired speed in km/h

SpeedError in km/h

-+

DesiredThrottlePosition

ActualThrottlePosition

Disturbances - wind, slope, bad fuel

ysp e y

ym

uGc Gv Gp

Gm

Gd or GL

d

•The block diagram is used to figure out the governing equations directly.

•Each transfer function is multiplied by its input to get its output:

u = Gc * eym = Gm * y

•Summers (circle with x inside) add inputs, or, if the input is marked with a - sign, that input is subtracted

e = ysp - ym

•These rules are applied around the block diagram:

y = Gd * d + Gp * Gv * uu = Gc * ee = ysp - ymym = Gm * y

y = Gd * d + Gp * Gv * u = Gd * d + Gp * Gv * Gc * e = Gd * d + Gp * Gv * Gc * (ysp - ym) = Gd * d + Gp * Gv * Gc * (ysp - Gm * y) = Gd * d + Gp * Gv * Gc * ysp - Gp * Gv * Gc * Gm * yy + Gp * Gv * Gc * Gm * y = Gd * d + Gp * Gv * Gc * yspy ( 1 + Gp * Gv * Gc * Gm) = Gd * d + Gp * Gv * Gc * yspy = (Gd/(1 + Gp Gv Gc Gm)) * d + ( Gp Gv Gv/(1 + Gp Gv Gc Gm)) * ysp

Block Diagram

CruiseControl

Engine,Transmission,

Wheels

ThrottleActuator

Speed

Speed in km/h

Desired speed in km/h

SpeedError in km/h

-+

DesiredThrottlePosition

ActualThrottlePosition

Disturbances - wind, slope, bad fuel

ysp e y

ym

uGc Gv Gp

Gm

Gd or GL

d

Closed Loop System

Closed LoopSetpoint

Transfer Function

Closed LoopDisturbance

Transfer Function

Gp Gv Gc 1 + Gp Gv Gc Gm

Gd 1 + Gp Gv Gc Gm

How does PID work?

• PID control acts on the error.• The control action is the sum of three terms:

– Proportional: Kc * the error now– Integral: Kc/TauI * the integral of error since the

controller was turned on– Derivative: Kc*TauD * the rate of change of the

error now

P animation

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

PI animation

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Animation

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Comparison

Comments

• P never reaches setpoint.

• PID responds before PI does because of the Derivative term.