L1: Background Concepts Flashcards
Control:
Basic Definition
The word control is usually taken to mean
- Regulate
- Direct
- Command
Control is:
- A key enabling technology in all branches of engineering
- Used whenever some quantity, such as temperature, altitude or speed, must be made to behave in some desirable way over time
System:
Basic Definition
A System is defined as a collection of objects interacting with each other.
Examples
- Mechanical Systems:
- Robots, Airplanes, Cars, etc
- Electrical Systems
- Amplifiers, Filters, transformers, etc
- Economic Systems
- Human Body
- Environment
How is a system
represented mathematically?
Using mathematical expressions such as
Ordinary Differential Equations
that relate some input function, u(t)
to some output function, y(t)
General
System Characteristics
- Order: n
- Continuous/Discrete
- Linear/Nonlinear
- Time Invariant/Time Variant
* There are other characteristics that are not covered in this course
Control System
Definition
A Control System is
an arrangement of physical components
connected or related in such a manner as to
command, direct, or regulate itself or another system.
A controller provides a desired system response(output).
Representing a Control System:
Block Diagram
The Block Diagram relates some Input to some Output
-
Input
- The excitation or command applied to a control system
- Typically from an external energy source
- Usually in order to produce a specified response from the control system
-
Output
- The actual response obtained from a control system.
- May or may not be equal to the specified response implied by the input (some error)
Background Concepts:
Important Terms/Concepts
(15, not including subconcepts)
- Control
- System
- Control System
- Block Diagram
- Open-Loop Control
- Closed-Loop Control
- Feedback
- Modern Control Theory
- Classic Control Theory
- Time Domain
- Frequency Domain
- Complex Frequency
- Poles
- Zeros
- Laplace Transform
- Inverse Laplace Transform
- Transform Table
- Properties
- Important Test Waveforms
- Theorems
- Initial Value Theorem
- Final Value Theorem
- Finding Inverse Transforms
- Partial Fraction Expansion
- Three Forms
- Simple Poles
- Repeated Poles
- Complex Poles
- Solving ODE using Laplace
Open-Loop Control
An Open-Loop Control System
utilizes an actuating device to control the process directly, without using feedback
- Sometimes it is useful to evaluate a complex system in the open-loop configuration by temporarily removing feedback
Examples:
- Simple Timer
- Toaster
Open-Loop Control:
Characteristics/Benefits/Drawbacks
- Must be closely monitored
- No Feedback
- Difficult to control with accuracy
- Easy to design
- More economical
Closed Loop Control:
Description
A Closed-Loop Control System
uses a measurement of the output and feedback of this signal to compare it with the desired output.
- Must have feedback
- Sensor on output
- Continually adjusts the process
- Almost always negative feedback
- More difficult to design and costlier
- But produces more accurate output
Example: Room Temperature Control
Closed-Loop Control:
Roles of Feedback
- Reduce or eliminate error
- Reduce Sensitivity
- Enhance Robustness
- Disturbance Rejection or Elimination
- Improve Dynamic Performance
- Adjust transient response
Applications of Control:
Various Industries and types of
systems that Control Systems Theory
can be used for
- Industrial Plants
- Transportation
- Robotics
- Biological Systems
- Economic Systems
- Biomedical Systems
Applications of Control:
Use in Industrial Plants
- Manufacturing and Assembly Lines
- Machining
- Power Plants
Goals
- Maximize efficiency
- Minimize environmental impact
- Meet all quality specifications
Most modern industrial plants could not operate without control systems
Applications of Control:
Transportation Applications
- Automobiles
- Cruise control
- Lane keeping
- “Ecoboost”
- Fuel System
Applications of Control:
Robotics
- Dextrous manipulation
- Haptics
- RC cars and gliders
- Research submarines
- Smart wheelchairs
Applications of Control:
General Benefits of Control Systems
- Provides performance that would otherwise be unattainable
- Feedback amplifier
- Aircraft Autopilot
- Disk drives, CD players
- Cellular telephones
- ABS in cars
- Prosthetics
- Operate in environments that humans can’t tolerate
- Mars
- Satellites (including GPS)
- Underwater exploration and research
Applications of Control:
Biological Systems
- Cell regulation mechanisms
- Population Dynamics
- Epidemiology
Applications of Control:
Economic Systems
- Inflationary Mechanics
- Fiscal Policies
Applications of Control:
Biomedical Systems
- Bone development and morphology
- Sleep cycles
- Seasonal Affective Disorder
- Faulty feedback mechanisms in Parkinson’s disease
Modern Control
vs
Classcial Control
Modern Control
Classical Control
- Time Domain
- Integrals/Derivatives
- Easier for Computers
- MIMO Systems
- Internal States
- State Variable feedback
- Frequency Domain
- Simple Algebra
- Easier for Humans
- SISO systems
- Root Locus
- Bode Plots
Implementation of Control
- Embedded microprocessors observe signals from sensors and provide command signals to electromechanical actuators
- Designers use Computer-Aided design software
- MATLAB
- Design usually tested on simulations before implementation
- Control engineering requires a thorough understanding of the application area
Goals of this
Control Systems
Course
- Identify how control is used in engineering systems
- Identify benefits of feedback
- Analyze and predict common behaviors of dynamical systems with feedback
- Apply relevant mathematical theory
- Solve simple control design problems
- Use relevant computational tools
- Recognize difficult control problems
Important Test Waveforms/Functions
- Unit Impulse Function 𝛿(t)
- Unit Step Function u(t)
- Ramp Function tu(t)
- Parabola
- Sinusoid
What is the
Laplace Transform
used for?
Converts a mathematical expression
from the Time Domain to the Complex Frequency Domain.
The Inverse Laplace Transform
Converts an expression from the Frequency Domain to the Time Domain
Laplace Transform:
General Equation
Inverse Laplace Transform:
General Equation
Complex Frequency
s
Important Concepts
The Complex Frequency is represented by:
s = 𝝈 + jω
- 𝝈 represents the real component
- jω represents the imaginary component
- This is the Rectangular Form, representing the frequency as a vector in the complex plane
Poles and Zeros:
Basic Idea
A function in the complex domain can be represented as a ratio of two polynomials:
F(s) = N(s) / D(s)
- Zeros:
- Values of s that make N(s) = 0
- The roots of the Numerator, N(s)
- Poles:
- Values of s that make D(s) = 0
- Roots of the Denominator, D(s)
Initial and Final Value Theorems:
Basic Idea
Initial Value Theorem (IVT)
Final Value Theorem(FVT)
These theorems show that it is possible to get
the Initial Value, f(0)
and the Final Value, f(∞)
of a function directly from the Laplace Transfrom of the function
Converting Back to Time Domain
Steps to finding Inverse Laplace Transform
of F(s)
- Assume F(s) = N(s) / D(s)
- Both N(s) and D(s) are polynomials
- Decompose F(s) into simple terms
- Use Partial Fraction Expansion
- Find the inverse of each individual term using basic transform pairs
- F(s) can have three possible forms:
- Simple Poles
- Repeated Poles
- Complex Poles
Inverse Laplace Transform:
Simple Poles Case
- Finding Residues:
- Mutliply function by denominator of the associated term
- Substitute the value of s that makes it zero
- Perform for each residue term
Inverse Laplace Transform:
Repeated Poles Case
On decomposition, one or more of the terms will have a denominator raised to some power:
(s + p)n and other terms like (s + p)n-m
- The residue kn for the (s+p)n is found in the normal way
- Where residues kn-m for terms (s+p) are found using a derivative and factorial (see image for formula)
Inverse Laplace Transform:
Complex Poles Case
F(s) has a part that includes complex poles
roots to a denominator of form: (s2 + as + b)
- Solve this by Completing the Square
Laplace Transform Pairs:
Unit Impulse Function 𝛿(t)
F(s) = 1
