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= Mathematical Analysis I =
−
= Control Theory =
  
−
* '''Course name''': Control Theory
 +
* '''Course name''': Mathematical Analysis I
 
* '''Code discipline''':
 
* '''Code discipline''':
−
* '''Subject area''': ['Introduction to Linear Control, Stability of linear dynamical systems', 'Controller design', 'Sensing, observers, Adaptive control']
 +
* '''Subject area''': ['Differentiation', 'Integration', 'Series']
   
 
== Short Description ==
  
== Short Description ==
Line 22: Line 22:
 
! Section !! Topics within the section
  
! Section !! Topics within the section
 
|-
  
|-
  +
| Sequences and Limits ||
−
| Introduction to Linear Control, Stability of linear dynamical systems ||
 
  +
# Sequences. Limits of sequences
−
# Control, introduction. Examples.
  
  +
# Limits of sequences. Limits of functions
−
# Single input single output (SISO) systems. Block diagrams.
  
  +
# Limits of functions. Continuity. Hyperbolic functions
−
# From linear differential equations to state space models.
  
−
# DC motor as a linear system.
  
−
# Spring-damper as a linear system.
  
−
# The concept of stability of the control system. Proof of stability for a linear system with negative real parts of eigenvalues.
  
−
# Multi input multi output (MIMO) systems.
  
−
# Linear Time Invariant (LTI) systems and their properties.
  
−
# Linear Time Varying (LTV) systems and their properties.
  
−
# Transfer function representation.
  
 
|-
  
|-
−
| Controller design. ||
+
| Differentiation ||
  +
# Derivatives. Differentials
−
# Stabilizing control. Control error.
  
  +
# Mean-Value Theorems
−
# Proportional control.
  
  +
# l’Hopital’s rule
−
# PD control. Order of a system and order of the controller.
  
  +
# Taylor Formula with Lagrange and Peano remainders
−
# PID control.
  
  +
# Taylor formula and limits
−
# P, PD and PID control for DC motor.
  
  +
# Increasing / decreasing functions. Concave / convex functions
−
# Trajectory tracking. Control input types. Standard inputs (Heaviside step function, Dirac delta function, sine wave).
  
−
# Tuning PD and PID. Pole placement.
  
−
# Formal statements about stability. Lyapunov theory.
  
−
# Types of stability; Lyapunov stability, asymptotic stability, exponential stability.
  
−
# Eigenvalues in stability theory. Reasoning about solution of the autonomous linear system.
  
−
# Stability proof for PD control.
  
−
# Stability in stabilizing control and trajectory tracking.
  
−
# Frequency response. Phase response.
  
−
# Optimal control of linear systems. From Hamilton-Jacobi-Bellman to algebraic Riccati equation. LQR.
  
−
# Stability of LQR.
  
−
# Controllability.
  
 
|-
  
|-
  +
| Integration and Series ||
−
| Sensing, observers, Adaptive control ||
 
  +
# Antiderivative. Indefinite integral
−
# Modelling digital sensors: quantization, discretization, lag.
  
  +
# Definite integral
−
# Modelling sensor noise. Gaussian noise. Additive models. Multiplicative models. Dynamic sensor models.
  
  +
# The Fundamental Theorem of Calculus
−
# Observability.
  
  +
# Improper Integrals
−
# Filters.
  
  +
# Convergence tests. Dirichlet’s test
−
# State observers.
  
  +
# Series. Convergence tests
−
# Optimal state observer for linear systems.
  
  +
# Absolute / Conditional convergence
−
# Linearization of nonlinear systems.
  
  +
# Power Series. Radius of convergence
−
# Linearization along trajectory.
  
  +
# Functional series. Uniform convergence
−
# Linearization of Inverted pendulum dynamics.
  
−
# Model errors. Differences between random disturbances and unmodeled dynamics/processes.
  
−
# Adaptive control.
  
−
# Control for sets of linear systems.
  
−
# Discretization, discretization error.
  
−
# Control for discrete linear systems.
  
−
# Stability of discrete linear systems.
  
 
|}
 
|}
 
== Intended Learning Outcomes (ILOs) ==
  
== Intended Learning Outcomes (ILOs) ==
   
 
=== What is the main purpose of this course? ===
  
=== What is the main purpose of this course? ===
  +
understand key principles involved in differentiation and integration of functions, solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities, become familiar with the fundamental theorems of Calculus, get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation.
−
Linear Control Theory is both an active tool for modern industrial engineering and a prerequisite for most of the state-of-the-art level control techniques and the corresponding courses. With this in mind, the Linear Control course is both building a foundation for the following development of the student as a learner in the fields of Robotics, Control, Nonlinear Dynamics and others, as well as it is one of the essential practical courses in the engineering curricula.
  
   
 
=== ILOs defined at three levels ===
  
=== ILOs defined at three levels ===
Line 78: Line 55:
 
==== Level 1: What concepts should a student know/remember/explain? ====
  
==== Level 1: What concepts should a student know/remember/explain? ====
 
By the end of the course, the students should be able to ...
  
By the end of the course, the students should be able to ...
  +
* Derivative. Differential. Applications
−
* methods for control synthesis (linear controller gain tuning)
  
  +
* Indefinite integral. Definite integral. Applications
−
* methods for controller analysis
  
  +
* Sequences. Series. Convergence. Power Series
−
* methods for sensory data processing for linear systems
  
   
 
==== Level 2: What basic practical skills should a student be able to perform? ====
  
==== Level 2: What basic practical skills should a student be able to perform? ====
 
By the end of the course, the students should be able to ...
  
By the end of the course, the students should be able to ...
  +
* Derivative. Differential. Applications
−
* State-space models
  
  +
* Indefinite integral. Definite integral. Applications
−
* Eigenvalue analysis for linear systems
  
  +
* Sequences. Series. Convergence. Power Series
−
* Proportional and PD controllers
  
  +
* Taylor Series
−
* How to stabilize a linear system
  
−
* Lyapunov Stability
  
−
* How to check if the system is controllable
  
−
* Observer design
  
−
* Sources of sensor noise
  
−
* Filters
  
−
* Adaptive Control
  
−
* Optimal Control
  
−
* Linear Quadratic Regulator
  
   
 
==== Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios? ====
  
==== Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios? ====
 
By the end of the course, the students should be able to ...
  
By the end of the course, the students should be able to ...
  +
* Take derivatives of various type functions and of various orders
−
* Turn a system of linear differential equations into a state-space model.
  
  +
* Integrate
−
* Design a controller by solving Algebraic Riccati eq.
  
  +
* Apply definite integral
−
* Find if a system is stable or not, using eigenvalue analysis.
 
  +
* Expand functions into Taylor series
  +
* Apply convergence tests
 
== Grading ==
  
== Grading ==
   
Line 110: Line 81:
 
! Grade !! Range !! Description of performance
  
! Grade !! Range !! Description of performance
 
|-
  
|-
−
| A. Excellent || 85-100 || -
 +
| A. Excellent || 90-100 || -
 
|-
  
|-
−
| B. Good || 70-84 || -
 +
| B. Good || 75-89 || -
 
|-
  
|-
−
| C. Satisfactory || 50-69 || -
 +
| C. Satisfactory || 60-74 || -
 
|-
  
|-
−
| D. Poor || 0-49 || -
 +
| D. Poor || 0-59 || -
 
|}
  
|}
   
Line 125: Line 96:
 
! Activity Type !! Percentage of the overall course grade
  
! Activity Type !! Percentage of the overall course grade
 
|-
  
|-
−
| Labs/seminar classes || 30
 +
| Labs/seminar classes || 20
 
|-
  
|-
−
| Interim performance assessment || 20
 +
| Interim performance assessment || 30
 
|-
  
|-
 
| Exams || 50
  
| Exams || 50
Line 138: Line 109:
   
 
=== Open access resources ===
  
=== Open access resources ===
  +
* Zorich, V. A. “Mathematical Analysis I, Translator: Cooke R.” (2004)
−
* Williams, R.L. and Lawrence, D.A., 2007. Linear state-space control systems. John Wiley & Sons.
  
−
* Ogata, K., 1995. Discrete-time control systems (Vol. 2, pp. 446-480). Englewood Cliffs, NJ: Prentice Hall.
  
   
 
=== Closed access resources ===
  
=== Closed access resources ===
Line 156: Line 126:
 
| Homework and group projects || 1 || 1 || 1
  
| Homework and group projects || 1 || 1 || 1
 
|-
  
|-
−
| Testing (written or computer based) || 1 || 0 || 0
 +
| Midterm evaluation || 1 || 1 || 0
 
|-
  
|-
−
| Reports || 1 || 1 || 1
 +
| Testing (written or computer based) || 1 || 1 || 1
 
|-
  
|-
−
| Midterm evaluation || 0 || 1 || 0
 +
| Discussions || 1 || 1 || 1
−
|-
  
−
| Discussions || 0 || 1 || 0
  
 
|}
 
|}
 
== Formative Assessment and Course Activities ==
  
== Formative Assessment and Course Activities ==
Line 174: Line 142:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
−
| Question || What is a linear dynamical system? || 1
 +
| Question || A sequence, limiting value || 1
 
|-
  
|-
−
| Question || What is an LTI system? || 1
 +
| Question || Limit of a sequence, convergent and divergent sequences || 1
 
|-
  
|-
−
| Question || What is an LTV system? || 1
 +
| Question || Increasing and decreasing sequences, monotonic sequences || 1
 
|-
  
|-
−
| Question || Provide examples of LTI systems || 1
 +
| Question || Bounded sequences Properties of limits || 1
 
|-
  
|-
−
| Question || What is a MIMO system? || 1
 +
| Question || Theorem about bounded and monotonic sequences || 1
 
|-
  
|-
  +
| Question || Cauchy sequence The Cauchy Theorem (criterion) || 1
−
| Question || Simulate a linear dynamic system as a higher order differential equation or in state-space form (Language is a free choice, Python and Google Colab are recommended Use built-in solvers or implement Runge-Kutta or Euler method || 0
  
−
|}
 
−
==== Section 2 ====
  
−
{| class="wikitable"
  
−
|+
 
 
|-
  
|-
  +
| Question || Limit of a function Properties of limits || 1
−
! Activity Type !! Content !! Is Graded?
  
 
|-
  
|-
−
| Question || What is stability in the sense of Lyapunov? || 1
 +
| Question || The first remarkable limit || 1
 
|-
  
|-
−
| Question || What is stabilizing control? || 1
 +
| Question || The Cauchy criterion for the existence of a limit of a function || 1
 
|-
  
|-
−
| Question || What is trajectory tracking? || 1
 +
| Question || Second remarkable limit || 1
 
|-
  
|-
  +
| Question || Find a limit of a sequence || 0
−
| Question || Why the control for a state-space system does not include the derivative of the state variable in the feedback law? || 1
  
 
|-
  
|-
  +
| Question || Find a limit of a function || 0
−
| Question || How can a PD controller for a second-order linear mechanical system can be re-written in the state-space form? || 1
  
  +
|}
  +
==== Section 2 ====
  +
{| class="wikitable"
  +
|+
 
|-
  
|-
  +
! Activity Type !! Content !! Is Graded?
−
| Question || Write a closed-loop dynamics for an LTI system with a proportional controller || 1
  
 
|-
  
|-
  +
| Question || A plane curve is given by <math>{\displaystyle x(t)=-{\frac {t^{2}+4t+8}{t+2}}}</math> , <math>{\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}</math> Find <math>{\textstyle y'_{x}}</math> || 1
−
| Question || Give stability conditions for an LTI system with a proportional controller || 1
  
 
|-
  
|-
−
| Question || Provide an example of a LTV system with negative eigenvalues that is not stable || 1
 +
| Question || the asymptotes of this curve; || 1
 
|-
  
|-
−
| Question || Write algebraic Riccati equation for a standard additive quadratic cost || 1
 +
| Question || the derivative <math>{\textstyle y'_{x}}</math> || 1
 
|-
  
|-
−
| Question || Derive algebraic Riccati equation for a given additive quadratic cost || 1
 +
| Question || Derive the Maclaurin expansion for <math>{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}</math> up to <math>{\textstyle o\left(x^{3}\right)}</math> || 1
 
|-
  
|-
−
| Question || Derive differential Riccati equation for a standard additive quadratic cost || 1
 +
| Question || Differentiation techniques: inverse, implicit, parametric etc || 0
 
|-
  
|-
  +
| Question || Find a derivative of a function || 0
−
| Question || What is the meaning of the unknown variable in the Riccati equation? What are its property in case of LTI dynamics || 1
  
 
|-
  
|-
−
| Question || What is a frequency response? || 1
 +
| Question || Apply Leibniz formula || 0
 
|-
  
|-
−
| Question || What is a phase response? || 1
 +
| Question || Draw graphs of functions || 0
 
|-
  
|-
−
| Question || Design control for an LTI system using pole placement || 0
 +
| Question || Find asymptotes of a parametric function || 0
−
|-
  
−
| Question || Design control for an LTI system using Riccati (LQR) || 0
  
−
|-
  
−
| Question || Simulate an LTI system with LQR controller || 0
  
 
|}
 
|}
 
==== Section 3 ====
  
==== Section 3 ====
Line 232: Line 196:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
  +
| Question || Find the indefinite integral <math>{\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}</math> || 1
−
| Question || What are the sources of sensor noise? || 1
  
−
|-
  
−
| Question || How can we combat the lack of sensory information? || 1
  
 
|-
  
|-
  +
| Question || Find the length of a curve given by <math>{\textstyle y=\ln \sin x}</math> , <math>{\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}</math> || 1
−
| Question || When it is possible to combat the lack of sensory information? || 1
  
 
|-
  
|-
  +
| Question || Find all values of parameter <math>{\textstyle \alpha }</math> such that series <math>{\textstyle \displaystyle \sum \limits _{k=1}^{+\infty }\left({\frac {3k+2}{2k+1}}\right)^{k}\alpha ^{k}}</math> converges || 1
−
| Question || How can we combat the sensory noise? || 1
  
 
|-
  
|-
−
| Question || What is an Observer? || 1
 +
| Question || Integration techniques || 0
 
|-
  
|-
−
| Question || What is a filter? || 1
 +
| Question || Integration by parts || 0
 
|-
  
|-
−
| Question || How is additive noise different from multiplicative noise? || 1
 +
| Question || Calculation of areas, lengths, volumes || 0
 
|-
  
|-
−
| Question || Simulate an LTI system with proportional control and sensor noise || 0
 +
| Question || Application of convergence tests || 0
 
|-
  
|-
−
| Question || Design an observer for an LTI system with proportional control and lack of sensory information || 0
 +
| Question || Calculation of Radius of convergence || 0
 
|}
 
|}
 
=== Final assessment ===
  
=== Final assessment ===
 
'''Section 1'''
  
'''Section 1'''
  +
# Find limits of the following sequences or prove that they do not exist:
−
# Convert a linear differential equation into a state space form
  
  +
# <math>{\displaystyle a_{n}=n-{\sqrt {n^{2}-70n+1400}}}</math> ;
−
# Convert a transfer function into a state space form
  
  +
# <math>{\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}</math> ;
−
# Convert a linear differential equation into a transfer function
  
  +
# <math>{\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}</math>
−
# What does it mean for a linear differential equation to be stable?
  
 
'''Section 2'''
  
'''Section 2'''
  +
# Find a derivative of a (implicit/inverse) function
−
# <math>{\displaystyle {\dot {x}}=Ax+Bu}</math>
  
  +
# Apply Leibniz formula Find <math>{\textstyle y^{(n)}(x)}</math> if <math>{\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}</math>
−
# <math>{\textstyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}1&10\\-3&4\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}</math>
  
  +
# Draw graphs of functions
−
# <math>{\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}1&10\\-3&4\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}+{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}}</math>
  
  +
# Find asymptotes
−
# You have linear dynamics: <math>{\textstyle 2{\ddot {q}}+3{\dot {q}}-5q=u}</math> <math>{\textstyle u=0}</math>
  
  +
# Apply l’Hopital’s rule
−
# <math>{\textstyle u=0}</math>
  
−
# <math>{\textstyle u}</math>
 +
# Find the derivatives of the following functions: <math>{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}</math>
−
# <math>{\textstyle u=0}</math>
 +
# <math>{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}</math> ;
  +
# <math>{\textstyle y(x)}</math> that is given implicitly by <math>{\textstyle x^{3}+5xy+y^{3}=0}</math>
−
# What is the difference between exponential stability, asymptotic stability and optimality?
  
 
'''Section 3'''
  
'''Section 3'''
  +
# Find the following integrals:
−
# Write a model of a linear system with additive Gaussian noise
  
  +
# <math>{\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}</math>
−
# Derive and implement an observer
  
  +
# <math>{\textstyle \int 2^{2x}e^{x}\,dx}</math>
−
# Derive and implement a filter
  
  +
# <math>{\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}</math>
  +
# Use comparison test to determine if the following series converge <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {3+(-1)^{k}}{k^{2}}}}</math>
  +
# <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}}</math>
  +
# Find the sums of the following series:
  +
# <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {1}{16k^{2}-8k-3}}}</math>
  +
# <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}}</math>
   
 
=== The retake exam ===
  
=== The retake exam ===

Revision as of 22:47, 19 April 2022

Mathematical Analysis I

  • Course name: Mathematical Analysis I
  • Code discipline:
  • Subject area: ['Differentiation', 'Integration', 'Series']

Short Description

Prerequisites

Prerequisite subjects

Prerequisite topics

Course Topics

Course Sections and Topics
Section Topics within the section
Sequences and Limits
  1. Sequences. Limits of sequences
  2. Limits of sequences. Limits of functions
  3. Limits of functions. Continuity. Hyperbolic functions
Differentiation
  1. Derivatives. Differentials
  2. Mean-Value Theorems
  3. l’Hopital’s rule
  4. Taylor Formula with Lagrange and Peano remainders
  5. Taylor formula and limits
  6. Increasing / decreasing functions. Concave / convex functions
Integration and Series
  1. Antiderivative. Indefinite integral
  2. Definite integral
  3. The Fundamental Theorem of Calculus
  4. Improper Integrals
  5. Convergence tests. Dirichlet’s test
  6. Series. Convergence tests
  7. Absolute / Conditional convergence
  8. Power Series. Radius of convergence
  9. Functional series. Uniform convergence

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

understand key principles involved in differentiation and integration of functions, solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities, become familiar with the fundamental theorems of Calculus, get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

  • Derivative. Differential. Applications
  • Indefinite integral. Definite integral. Applications
  • Sequences. Series. Convergence. Power Series

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

  • Derivative. Differential. Applications
  • Indefinite integral. Definite integral. Applications
  • Sequences. Series. Convergence. Power Series
  • Taylor Series

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

  • Take derivatives of various type functions and of various orders
  • Integrate
  • Apply definite integral
  • Expand functions into Taylor series
  • Apply convergence tests

Grading

Course grading range

Grade Range Description of performance
A. Excellent 90-100 -
B. Good 75-89 -
C. Satisfactory 60-74 -
D. Poor 0-59 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Labs/seminar classes 20
Interim performance assessment 30
Exams 50

Recommendations for students on how to succeed in the course

Resources, literature and reference materials

Open access resources

  • Zorich, V. A. “Mathematical Analysis I, Translator: Cooke R.” (2004)

Closed access resources

Software and tools used within the course

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section
Learning Activities Section 1 Section 2 Section 3
Homework and group projects 1 1 1
Midterm evaluation 1 1 0
Testing (written or computer based) 1 1 1
Discussions 1 1 1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type Content Is Graded?
Question A sequence, limiting value 1
Question Limit of a sequence, convergent and divergent sequences 1
Question Increasing and decreasing sequences, monotonic sequences 1
Question Bounded sequences Properties of limits 1
Question Theorem about bounded and monotonic sequences 1
Question Cauchy sequence The Cauchy Theorem (criterion) 1
Question Limit of a function Properties of limits 1
Question The first remarkable limit 1
Question The Cauchy criterion for the existence of a limit of a function 1
Question Second remarkable limit 1
Question Find a limit of a sequence 0
Question Find a limit of a function 0

Section 2

Activity Type Content Is Graded?
Question A plane curve is given by , Find 1
Question the asymptotes of this curve; 1
Question the derivative 1
Question Derive the Maclaurin expansion for up to 1
Question Differentiation techniques: inverse, implicit, parametric etc 0
Question Find a derivative of a function 0
Question Apply Leibniz formula 0
Question Draw graphs of functions 0
Question Find asymptotes of a parametric function 0

Section 3

Activity Type Content Is Graded?
Question Find the indefinite integral 1
Question Find the length of a curve given by , 1
Question Find all values of parameter such that series converges 1
Question Integration techniques 0
Question Integration by parts 0
Question Calculation of areas, lengths, volumes 0
Question Application of convergence tests 0
Question Calculation of Radius of convergence 0

Final assessment

Section 1

  1. Find limits of the following sequences or prove that they do not exist:
  2.  ;
  3.  ;

Section 2

  1. Find a derivative of a (implicit/inverse) function
  2. Apply Leibniz formula Find if
  3. Draw graphs of functions
  4. Find asymptotes
  5. Apply l’Hopital’s rule
  6. Find the derivatives of the following functions:
  7.  ;
  8. that is given implicitly by

Section 3

  1. Find the following integrals:
  5. Use comparison test to determine if the following series converge
  7. Find the sums of the following series:

The retake exam

Section 1

Section 2

Section 3

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