Dynamic Systems and Feedback

✍ Scribed by Packard, Horowitz, Poolla, Borrelli

• Tongue: English
• Leaves: 493
• Category: Library

✦ Table of Contents

Introduction
Structure of a closed-loop control system
Example: Temperature Control in Shower
Problems
Block Diagrams
Common blocks, continuous-time
Example
Summary
Problems
Mathematical Modeling and Simulation
Systems of 1st order, coupled differential equations
Remarks about Integration Options in simulink
Problems
State Variables
Definition of State
State-variables: from first order evolution equations
State-variables: from a block diagram
Problems
First Order, Linear, Time-Invariant (LTI) ODE
The Big Picture
Solution of a First Order LTI ODE
Free response
Forced response, constant inputs
Forced response, bounded inputs
Stable system, Forced response, input approaching 0
Linearity
Forced response, input approaching a constant limit
Forced response, Sinusoidal inputs
Forced response, input approaching a Sinusoid
First-order delay-differential equation: Stability
Summary
Problems
Feedback systems
First-order plant, Proportional control
Proportional Plant, first-order controller
Problems
Two forms of high-order Linear ODEs, with forcing
Problems
Jacobian Linearizations, equilibrium points
Jacobians and the Taylor Theorem
Equilibrium Points
Deviation Variables
Tank Example
Output Equations
Calculus for systems not in standard form
Another common non-standard form
Linearizing about general solution
Problems
Additional Related Problems
Linear Algebra Review
Notation
Determinants
Inverses
Solving Linear equations: Gaussian Elimination
Matrix functions of Time
Linear Systems and Time-Invariance
Linearity of solution
Time-Invariance
Matrix Exponential
Diagonal A
Block Diagonal A
Effect of Similarity Transformations
Solution To State Equations
Examples
Problems
Eigenvalues, eigenvectors, stability
Diagonalization: Motivation
Eigenvalues
Diagonalization Procedure
eAt as t
Complex Eigenvalues
Alternate parametrization with complex eigenvalues
Examples
Step response
Quick estimate of unit-step-response of 2nd order system
Problems
Frequency Response for Linear Systems: State-Space representations
Theory for Stable System: Complex Input Signal
MIMO Systems: Response due to real sinusoidal inputs
Experimental Determination
Steady-State response
Important special cases for designing closed-loop systems
Roots of 2nd-order monic polynomial
Setting the coefficients to attain certain roots
1st-order plant, 1st-order controller
2nd-order plant, constant-gain controller with derivative feedback
Problems
Step response
Quick estimate of unit-step-response of 2nd order system
Stabilization by State-Feedback
Theory
State-Feedback with Integral Control
Theory
Example
Problems
Linear-quadratic Optimal Control
Learning more
Single, high-order, linear ODES (SLODE)
Linear, Time-Invariant Differential Equations
Importance of Linearity
Solving Homogeneous Equation
Interpretation of complex roots to ODEs with real-coefficients
General Solution Technique
Behavior of Homogeneous Solutions as t
Response of stable system to constant input (Steady-State Gain)
Example
Stability Conditions for 2nd order differential equation
Important 2nd order example
Summary for SLODEs
Main Results
General Solution Technique
Behavior of Homogeneous Solutions as t
Stability of Polynomials
2nd order differential equation
Solutions of 2nd order differential equation
Problems
Frequency Responses of Linear Systems
Complex and Real Particular Solutions
Response due to real sinusoidal inputs
Problems
Derivatives appearing on the inputs: Effect on the forced response
Introduction
Other Particular Solutions
Limits approaching steps
Problems
Distributions
Introduction
Procedure to get step response
Summary: Solution of SLODEs with Derivatives on the inputs
Introduction
Problems
Transfer functions
Linear Differential Operators (LDOs)
Algebra of Linear differential operations
Feedback Connection
More General Feedback Connection
Cascade (or Series) Connection
Parallel Connection
General Connection
Systems with multiple inputs
Poles and Zeros of Transfer Functions
Problems
Arithmetic of Feedback Loops
Tradeoffs
Signal-to-Noise ratio
What's missing?
Problems
Robustness Margins
Gain Margin
Time-Delay Margin
Percentage Variation Margin
The Small-Gain Theorem
Necessary and Sufficient Version
Application to Percentage Variation Margin
Summary
Examples
Generic
Missile
Application to percentage variation margin
Problems
Gain/Time-delay margins: Alternative derivation
Sylvester's determinant identity
Setup
Gain margin
Time delay margin
Appendix
Connection between Frequency Responses and Transfer functions
Interconnections
Decomposing Systems into Simple Parts
Problems
Unfiled problems
Recent exams
Fall 2017 Midterm 1
Fall 2017 Midterm 2
Fall 2017 Final
Fall 2015 Midterm 1
Fall 2015 Midterm 2
Fall 2015 Final
Spring 2014, Midterm 1
Spring 2014, Midterm 2
Spring 2014 Final Exam
Older exams
Spring 2012, Midterm 1
Spring 2012, Midterm 2
Spring 2012, Final Exam
Spring 2009, Midterm 1
Spring 2009, Midterm 2
Spring 2009, Final Exam
Spring 2005 Midterm 1
Spring 2005 Midterm 2
Spring 2004 Midterm 1
Fall 2003 Midterm 1
Fall 2003 Midterm 2
Fall 2003 Final