Feedback Control

CS 478: Microcontroller SystemsUniversity of Wisconsin-Eau Claire Dan Ernst

Feedback ControlFeedback Control


A Simple AbstractionA Simple Abstraction

• An open loop control system• Goal/desired state

ControllerThe process

Under control

Desired output

Process outputProces

s input


Open Loop ControlOpen Loop Control

• Only for static environment• Accurate manipulation needed• Problem:

– Noisy environment (disturbance)– Inaccurate effector


Under control

Desired output

Process output

Process input

Disturbance


Open Loop ControlOpen Loop Control• May use expected disturbance

– Static environment


Under control

Desired output

Process output

Process input

Disturbance

Predicted disturbance


ExamplesExamples

• Putting book on a desk

• Activating an event– Start a sensor– Play a sound– Conduct a scripted movement

• Question: What about inserting a light bulb?


Feed Forward ControlFeed Forward Control• Disturbance measured on the fly• Problems?

– May not include all the parameters


Under control

Desired output

Process output

Process input

DisturbanceMeasured disturbance Sensors


Feedback Control (Closed Loop)Feedback Control (Closed Loop)

• Include all the parameters as included into the output.


Under control

Desired output

Process output

Process input

Disturbance

Sensors


Feedback Control Feedback Control

• Another diagram


Under control

Desired

output

Process output

Process input

Sensors

Σ+-


ExampleExample


ErrorsErrors

• Direction (sign)• Magnitude (distance)

• Frequent feedback is needed– Sensor rates can effect response

• Control may not be immediate– May be a delay from when you

decide to change, and when a change actually occurs!


A Wall Following RobotA Wall Following Robot

• How would you use feedback control to implement a wall-following behavior in a robot?

• What sensors would you use?• Would they provide magnitude and direction of the error?

• What will this robot's behavior look like?


Oscillation and the Set PointOscillation and the Set Point• Desired state is called the set point• Can we decrease oscillation?

– A range rather than a single value– Slower change

• Wall following example:– Larger turning angle– A range rather than fix distance


Sensor NoiseSensor Noise

• What happens when there is sensor noise in the system? • Example:

– Sensor tells the robot it is far from a wall, when it is close? – vice versa?

• How might we fix these problems?


Control TheoryControl Theory

• Studies the behavior of control systems

• Major basic controllers: – P: proportional control– PD: proportional derivative control– PID: proportional integral derivative control


P: Proportional ControlP: Proportional Control

• Error = measurement – setpoint– Setpoint = desired output

• Process input = Gain * error + bias – bias: manual reset (to fix any offset)


Under control

Desired

output

Process output

Process input

Sensors

Σ+-


P:P: Proportional ControlProportional Control

• Q: What happens if the gain is increased?• A: Loop may go unstable

• Q: What if the gain is decreased?• A: It takes along time to get close enough to the setpoint.

• Determining the gain: hard problem– analytically (mathematics)– empirically (trial and error)


Setting GainSetting Gain• Determining the gain depends on the physics of the system:

• Analytical approaches:– System should be understood well– System should be characterized mathematically.

• Trial and error (ad hoc, system-specific):– System should be tested extensively.– Can be done

• manually • Automatically by the system

• Wrong gain may put the system into oscillation!


OscillationOscillation

• Wrong gain may put the system into oscillation


DampingDamping• The process of systematically decreasing oscillation• Properly damped: Reduces and removes oscillation in a reasonable

amount of time.


P:P: Proportional ControlProportional Control

• Q: What happens if the system is very dynamic? – Example: Following another robot

• A: P control does not work well.– It senses the present time.

• Q: What happens close to the setpoint?• A:

– If gain is fixed: May not work for low errors• Leaves some offset

– If gain is high: Tends to overshoot

ExampleExample

Kp = 20

Kp = 200

Kp = 50

Kp = 500


Example: Analysis Example: Analysis

steady-state error

settling time

rise time

overshoot

overshoot -- % of final value exceeded at first oscillation

rise time -- time to span from 10% to 90% of the final value

settling time -- time to reach within 2% of the final value

ss error -- difference from the system’s desired value


D: Derivative ControlD: Derivative Control

• Predict the future:

• Adjust based on the rate of change– The speed of change

• Example: wall following robot– High derivative: Very fast toward the wall– Low derivative: Very slow toward the wall

• Output:– o = Kd * di/dt


Momentum of CorrectionMomentum of Correction

• Momentum of correction results in oscillation and instability– A result of a slow reaction time to the controller

• Momentum = mass * velocity • D to overcome oscillation

• Output = error * Gp + d(error)/dt * Gk– Example: Wall following:P and D are opposite each other– PD: Mostly used for industrial plants


PD ControlPD Control

Kd = 300

Kd = 3

Kd = 30


Integral ControlIntegral Control

• The controller output is proportional to the amount of time the error is present.

– Integrate all previous values.

• To overcome (eliminate) the offset– The longer the offset hangs around, the larger the I component becomes

• Output o = Kf * int i(t)dt


Example: PIExample: PI• SS error (offset) has been removed

Ki = 0 Ki = 2


PI Example: Draw backsPI Example: Draw backsKi = 20

Ki = 90

Ki = 200


PID ControllerPID Controller

• Proportional Integral Derivative Control is a combination of proportional, integral, and derivative control:

• output = Kp * i + Kd * di/dt + Kf * int i(t) dt


PID ControllerPID Controller


PID resultsKp = 100 Ki = 200

Kd = 2

Kd = 10 Kd = 20

Kd = 5


SimulationSimulation

http://newton.ex.ac.uk/cgi-bin/metaform?http://newton.ex.ac.uk/teaching/CDHW/Feedback/OvSimForm-gen.html


Choosing ParametersChoosing ParametersZiegler-Nichols Method

1. Adjust the set-point value, Ts, to a typical value for the system and turn off the derivative and integral actions (0). Select a safe value for the maximum power M and set the proportional gain to minimum.

2. Progressively increase the gain until suddenly decreasing or increasing Ts by about 5% induces oscillations that are just self-sustaining.

3. Call the gain at this stage Gu, and the period of the oscillations tu. Note the values of each quantity.

4. Set the controller parameters as follows:– P-Control: P=0.50*Gu, I=0, D=0. – PI-Control: P=0.45*Gu, I=1.2/tu, D=0. – PID-Control: P=0.60*Gu, I=2/tu, D=tu/8.

5. Check the overall performance of system is satisfactory under all normal conditions.


Is it Set Well?Is it Set Well?How do we measure system accuracy?• Elementary: The plant didn’t blow up• Informal:

– Optimum decay ratio (1/4 wave decay)

– Minimum Overshoot

– Maximum Disturbance Rejection


Is it Set Well?Is it Set Well?

• Mathematical:– Various integral definitions, such as:

• IAE - Integral of absolute value of error • ISE - Integral of error squared

– Mostly reserved for “academic” purposes

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Feedback Control

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