COMP417Introduction to Robotics and Intelligent Systems

Lecture 5: Feedback Control and PID control

Guest lecture by José Ruiz Ascencio

On slides by

Professor Gregory Dudek

Open-loop?• We want to spin a motor at a given angular

velocity. We can apply a fixed voltage to it, and

never check to see if it is rotating properly.

• Called "open loop control", but there isn't any

actual control because there is no feedback.

Key notion

Open-loop?• We want to spin a motor at a given angular

velocity. We can apply a fixed voltage to it, and

never check to see if it is rotating properly.

• Called open loop.

Open-loop?

• What if there is a changing load on the motor?

– Our output velocity will change!

torque t

speed wno torque at max speed

stall torque

Closing the loop• Let’s measure the actual angular velocities.

• Now we can compensate for changes in load by

feeding back some information.

Early Example of Feedback

System

• James Watt’s

“Centrifugal

Governor” in 1788.

• Regulates the steam

engine speed.

Other Simple Examples

• Body temperature regulation– If cold, shiver (muscles produce heat)

– If hot, sweat (evaporation takes away heat)

• Maintaining social peace– If a crime is found (sensor), the guilty party is

punished (actuator).

• Cruise control in cars– You set a speed, CC will increase fuel intake

uphill, and decrease it downhill.

• Etc…

Control Theory• Roots in another science: Cybernetics

Cybernetics is the study of feedback and derived

concepts such as communication and control in

living organisms, machines and organisations.

• Expression was coined by

Norbert Weiner in 1948.

Why Control Theory

• Systematic approach to analysis and design– Transient response

– Consider sampling times, control frequency

– Taxonomy of basic controllers (PID, open-loop, Model-based, Feedforward…)

– Select controller based on desired characteristics

• Predict system response to some input– Speed of response (e.g., adjust to workload changes)

– Oscillations (variability)

• Assessing stability of system

• JR

Characteristics of Feedback

System

• Power amplification

– Compare neural signal power (mW) vs muscle power

output (tens of W)

– Means it is an active system, as opposed to passive.

• Actuator

• Feedback

– measurement (sensor)

• Error signal

• Controller

Classic Feedback Diagram

Control System Goals

• Regulation

– thermostat

• Tracking

– robot movement, adjust TCP window to

network bandwidth

• Optimization

– best mix of chemicals, minimize response

times

• Each of these is slightly different:

understand and remember how (ask now if

necessary!)

Approaches to System Modeling

• First Principles

– Based on known laws.

– Physics, Queueing theory.

– Difficult to do for complex systems.

• Experimental (System Identification)

– Requires collecting data

– Is there a good “training set”?

Common Laplace TransfomName f(t) F(s)

Impulse d

Step

Ramp

Exponential

Sine

1

Damped Sine

Transfer Function

• Definition: H(s) = Y(s) / X(s)

• Relates the output of a linear system to its input.

• Describes how a linear system responds to an

impulse… called impulse response

– (remember! Convolving with impulse yields the same

thing. Hence probing a system with and impulse is

finding its transfer function…)

• All linear operations allowed

– Scaling, addition, multiplication

Block Diagram

• Expresses flows and relationships between

elements in system.

• Blocks may recursively be systems.

• Rules

– Cascaded elements: convolution.

– Summation and difference elements

Laplace Transform of Classic

Feedback Sysem

Key Transfer Functions

First Order System

Feedback Control

Main question: what are the controls that will take the system from state A to B?

Example: wall/line following at fixed speed

cross-track error at time t = e(t)

You have to control the angular velocity

Cross-track error:CTEHow far from the wall you are.

Why bother with wall/line following?

• Because it enables arbitrary path following where the line is the local tangent of the (curvy) path

Idea 1: bang-bang control

What’s wrong with this?

Idea 2: proportional (P-)control

Will the car reach the target line?

Will the car overshoot the target line?

Is the asymptotic (steady-state) error zero?

Idea 2: proportional (P-)control

Will the car reach the target line? YES

Will the car overshoot the target line? YES

Is the asymptotic (steady-state) error zero? NO

Addressing the oscillation problem

• Need to reduce turning rate well before the line is approached

• Idea: have a small proportional gain

Problem: that means the car doesn’t turn very much

• Idea: need to predict the error in the near future

This is good, as long as the error does not oscillate at a very high

frequency

Idea 3: proportional-derivative (PD-)control

How do we set the gains?

What if there are systematic errors/biases?

What if the error estimate is very noisy?

Handling systematic biases:the integral term

• Examples of systematic biases: • wheels are misaligned• car is much heavier on one side

• Add integral of the error from the beginning of time:

• Can the PI-controller have nonzero error asymptotically (in steady-state)?

• NO. In steady state both the control and the error must be constant. If the asymptotic error is nonzero then the control is not constant:

Potential problem: integrator windup

• What happens if the control variable reaches the actuator’s limits?

• I.e. the car can’t turn as fast as the controller commands it.

• Actuator may remain at its limit for a long time while the controller modifies its commands

• Error increases, integral term winds up while controller goes back to issuing commands in the feasible region.

Potential problem: integrator windup

• Heuristic fixes:• Limit integral error values

• Stop integral error while the commands are in the non feasible region

• Reduce gain of integral error term

PID controller

Perhaps the most widely used controller in industry and robotics.Perhaps the easiest to code.

You will also see it as:

Tips: how to implement PID

• Assume time is discrete

• Identify your error function e(t)= current_state(t)- target_state(t)

• Is the measurement of the state reliable?

• If the measurement of the current state is very noisy you might want to smooth/filter it, using:

• Moving average filter with uniform weights

Potential problem: the larger the window of the filter the slower it is going to register changes.

• Exponential filter

Potential problem: the closer is to 1 the slower it is going to register changes.

Tips: how to implement PID

• Approximate the integral of error by a sum

• Approximate the derivative of error by:• Finite differences

• Filtered finite differences, e.g.

• Limit the computed controls

• Limit or stop the integral term when detecting large errors and windup

Control Performance (2nd O), P-type

Kp = 20

Kp = 200

Kp = 50

Kp = 500

Steady-state Errors, P-type

Kp = 200 Kp = 50

Control Performance, PI - type

Kp = 100

Ki = 50 Ki = 200

You’ve been integrated...

Kp = 100

instability &

oscillation

Control Performance, PID-typeKp = 100

Ki = 200 Kd = 2

Kd = 10 Kd = 20

Kd = 5

PID final control

PID Tuning

How to get the PID parameter values ?

(1) If we know the transfer function, analytical methods can

be used (e.g., root-locus method) to meet the transient and

steady-state specs.

(2) When the system dynamics are not precisely known, we

must resort to experimental approaches.

Using only Proportional control, turn up the gain until the system

oscillates w/o dying down, i.e., is marginally stable. Assume that K

and P are the resulting gain and oscillation period, respectively.

Then, use

Ziegler-Nichols Tuning

for second or higher

order systems

for P control for PI control for PID control

Kp = 0.6 K

Ki = 2.0 / P

Kd = P / 8.0

Kp = 0.45 K

Ki = 1.2 / P

Kp = 0.5 K

Ziegler-Nichols Rules for Tuning PID Controller:

Layered Approach to Control

• Robotic systems often have a layered approach

to control:

movie

Multiple Loops

Inner loops are generally “faster” that outer loop.

Advanced Control Topics

• Adaptive Control– Controller changes over time (adapts).

• MIMO Control– Multiple inputs and/or outputs.

• Predictive Control– You measure disturbance and react before measuring

change in system output.

• Optimal Control– Controller minimizes a cost function of error and

control energy.

• Nonlinear systems– Neuro-fuzzy control.

– Challenging to derive analytic results.

Tips: how to tune the PID

• Manually:• First, use only the proportional term. Set the other gains to zero.

• When you see oscillations slowly add derivative term• Increasing increases the duration in which linear error prediction

is assumed to be valid

• Add a small integral gain

Tips: how to tune the PID

• Ziegler-Nichols heuristic:• First, use only the proportional term. Set the other gains to zero.

• When you see consistent oscillations, record the proportional gain and the oscillation period

Tips: how to tune the PID

• After manual or Z-N tweaking you might want to use coordinate ascent to search for a better set of parameters automatically:

See Sebastian Thrun’s online class“AI for robotics” on Udacity for more details on this. He calls the algorithm Twiddle and it is in Lesson 5.

When is PID insufficient?

• Systems with large time delays

• Controllers that require completion time guarantees• E.g. the system must reach target state within 2 secs

• Systems with high-frequency oscillations

• High-frequency variations on the target state

Example applications: self-driving cars

Cascading PID

• Sometimes we have multiple error sources (e.g. multiple sensors) and one actuator to control.

• We can use a master PID loop that sets the setpoint for the slave PID loop. Master (outer loop) runs at low rate, while slave (inner loop) runs at higher rate.

• One way of getting hierarchical control behavior.