Introduction to Control Theory

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# Introduction to Control Theory - PowerPoint PPT Presentation

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

COMP417

Instructor: Philippe Giguère

Actuators
• Earlier lecture presented actuators
• Electrical motor
• Now how do we use them in an intelligent manner?
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!

speed w

no torque at max speed

stall torque

torque t

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
• 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
• 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…
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
• Oscillations (variability)
• Assessing stability of 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

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
• Regulation
• thermostat
• Tracking
• robot movement, adjust TCP window to network bandwidth
• Optimization
• best mix of chemicals, minimize response times
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

Name

f(t)

F(s)

Impulse d

1

Step

Ramp

Exponential

Sine

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

Exponential

• Step Response illustrates how a system response can be decomposed into two components:
• Transient
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
• Proportional Action
• Simplest Controller Function
• Integral Action
• Can cause oscillations
• Derivative Action (“rate control”)
• Effective in transient periods
• Provides faster response (higher sensitivity)
• Never used alone

Control Performance (2nd O), P-type

Kp = 20

Kp = 50

Kp = 500

Kp = 200

Control Performance, PI - type

Kp = 100

Ki = 50

Ki = 200

You’ve been integrated...

Kp = 100

instability & oscillation

Control Performance, PID-type

Kp = 100

Kd = 5

Kd = 2

Ki = 200

Kd = 10

Kd = 20

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
• Robotic systems often have a layered approach to control:

movie

Multiple Loops

Inner loops are generally “faster” that outer loop.

• 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
• Is a controller + actuator in a small package.
• Not expensive 10\$-100\$.
• Those in 417 will use some of these.
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.