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Introduction to Control Theory. COMP417 Instructor: Philippe Giguère. Actuators. Earlier lecture presented actuators Electrical motor Now how do we use them in an intelligent manner?. Open-loop?.

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introduction to control theory

Introduction to Control Theory

COMP417

Instructor: Philippe Giguère

actuators
Actuators
  • Earlier lecture presented actuators
    • Electrical motor
  • Now how do we use them in an intelligent manner?
open loop
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 loop4
Open-loop?
  • What if there is a changing load on the motor?
    • Our output velocity will change!

speed w

no torque at max speed

stall torque

torque t

closing the loop
Closing the loop
  • Let’s measure the actual angular velocities.
  • Now we can compensate for changes in load by feeding back some information.
control theory
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.
early example of feedback system
Early Example of Feedback System
  • James Watt’s “Centrifugal Governor” in 1788.
  • Regulates the steam engine speed.
other simple examples
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…
why control theory
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
characteristics of feedback system
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
controller design methodology
Controller Design Methodology

Start

System Modeling

Controller Design

Block diagram construction

Controller Evaluation

Transfer function formulation and validation

Objective achieved?

Stop

Y

Model Ok?

N

Y

N

control system goals
Control System Goals
  • Regulation
    • thermostat
  • Tracking
    • robot movement, adjust TCP window to network bandwidth
  • Optimization
    • best mix of chemicals, minimize response times
approaches to system modeling
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 transfom
Common Laplace Transfom

Name

f(t)

F(s)

Impulse d

1

Step

Ramp

Exponential

Sine

Damped Sine

transfer function
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
Block Diagram
  • Expresses flows and relationships between elements in system.
  • Blocks may recursively be systems.
  • Rules
    • Cascaded elements: convolution.
    • Summation and difference elements
response of system
Response of System
  • Impulse
  • Step

Exponential

steady state vs transient
Steady-State vs. Transient
  • Step Response illustrates how a system response can be decomposed into two components:
    • Steady-state part:
    • Transient
second order response
Second Order Response
  • Typical response to step input is:

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% or 5% of the final value

effect of controller functions
Effect of Controller Functions
  • Proportional Action
    • Simplest Controller Function
  • Integral Action
    • Eliminates steady-state error
    • Can cause oscillations
  • Derivative Action (“rate control”)
    • Effective in transient periods
    • Provides faster response (higher sensitivity)
    • Never used alone
slide27

Control Performance (2nd O), P-type

Kp = 20

Kp = 50

Kp = 500

Kp = 200

slide29

Control Performance, PI - type

Kp = 100

Ki = 50

Ki = 200

slide30

You’ve been integrated...

Kp = 100

instability & oscillation

slide31

Control Performance, PID-type

Kp = 100

Kd = 5

Kd = 2

Ki = 200

Kd = 10

Kd = 20

slide33

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.

Ziegler-Nichols Rules for Tuning PID Controller:

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

for P control

for PI control

for PID control

Kp = 0.5 K

Kp = 0.45 K

Kp = 0.6 K

Ziegler-Nichols Tuning for second or higher order systems

Ki = 1.2 / P

Ki = 2.0 / P

Kd = P / 8.0

layered approach to control
Layered Approach to Control
  • Robotic systems often have a layered approach to control:

movie

multiple loops
Multiple Loops

Inner loops are generally “faster” that outer loop.

advanced control topics
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.
rc servo
RC Servo
  • Is a controller + actuator in a small package.
  • Not expensive 10$-100$.
  • Those in 417 will use some of these.
slide38
PWM
  • PWM -- “pulse width modulation
  • Duty cycle:
    • The ratio of the “On time” and the “Off time” in one cycle.
    • Determines the fractional amount of full power delivered to the motor.
  • Why:
    • Transistor acts as a resistor when partially on.
    • Leads to energy being wasted  heat.