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  +
= Control Theory =
−
= Mathematical Analysis I =
  
−
* '''Course name''': Mathematical Analysis I
 +
* '''Course name''': Control Theory
 
* '''Code discipline''':
 
* '''Code discipline''':
−
* '''Subject area''': ['Differentiation', 'Integration', 'Series']
 +
* '''Subject area''': ['Introduction to Linear Control, Stability of linear dynamical systems', 'Controller design', 'Sensing, observers, Adaptive control']
   
 
== Short Description ==
  
== Short Description ==
Line 22: Line 22:
 
! Section !! Topics within the section
  
! Section !! Topics within the section
 
|-
  
|-
  +
| Introduction to Linear Control, Stability of linear dynamical systems ||
−
| Sequences and Limits ||
 
  +
# Control, introduction. Examples.
−
# Sequences. Limits of sequences
  
  +
# Single input single output (SISO) systems. Block diagrams.
−
# Limits of sequences. Limits of functions
  
  +
# From linear differential equations to state space models.
−
# Limits of functions. Continuity. Hyperbolic functions
  
  +
# 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.
 
|-
  
|-
−
| Differentiation ||
+
| Controller design. ||
  +
# Stabilizing control. Control error.
−
# Derivatives. Differentials
  
  +
# Proportional control.
−
# Mean-Value Theorems
  
  +
# PD control. Order of a system and order of the controller.
−
# l’Hopital’s rule
  
  +
# PID control.
−
# Taylor Formula with Lagrange and Peano remainders
  
  +
# P, PD and PID control for DC motor.
−
# Taylor formula and limits
  
  +
# Trajectory tracking. Control input types. Standard inputs (Heaviside step function, Dirac delta function, sine wave).
−
# Increasing / decreasing functions. Concave / convex functions
  
  +
# 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.
 
|-
  
|-
  +
| Sensing, observers, Adaptive control ||
−
| Integration and Series ||
 
  +
# Modelling digital sensors: quantization, discretization, lag.
−
# Antiderivative. Indefinite integral
  
  +
# Modelling sensor noise. Gaussian noise. Additive models. Multiplicative models. Dynamic sensor models.
−
# Definite integral
  
  +
# Observability.
−
# The Fundamental Theorem of Calculus
  
  +
# Filters.
−
# Improper Integrals
  
  +
# State observers.
−
# Convergence tests. Dirichlet’s test
  
  +
# Optimal state observer for linear systems.
−
# Series. Convergence tests
  
  +
# Linearization of nonlinear systems.
−
# Absolute / Conditional convergence
  
  +
# Linearization along trajectory.
−
# Power Series. Radius of convergence
  
  +
# Linearization of Inverted pendulum dynamics.
−
# Functional series. Uniform convergence
  
  +
# 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? ===
  +
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.
−
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 ===
  
=== ILOs defined at three levels ===
Line 55: Line 78:
 
==== 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 ...
  +
* methods for control synthesis (linear controller gain tuning)
−
* Derivative. Differential. Applications
  
  +
* methods for controller analysis
−
* Indefinite integral. Definite integral. Applications
  
  +
* methods for sensory data processing for linear systems
−
* Sequences. Series. Convergence. Power Series
  
   
 
==== 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 ...
  +
* State-space models
−
* Derivative. Differential. Applications
  
  +
* Eigenvalue analysis for linear systems
−
* Indefinite integral. Definite integral. Applications
  
  +
* Proportional and PD controllers
−
* Sequences. Series. Convergence. Power Series
  
  +
* How to stabilize a linear system
−
* Taylor Series
  
  +
* 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 ...
  +
* Turn a system of linear differential equations into a state-space model.
−
* Take derivatives of various type functions and of various orders
  
  +
* Design a controller by solving Algebraic Riccati eq.
−
* Integrate
  
  +
* Find if a system is stable or not, using eigenvalue analysis.
−
* Apply definite integral
  
−
* Expand functions into Taylor series
  
−
* Apply convergence tests
 
 
== Grading ==
  
== Grading ==
   
Line 81: Line 110:
 
! Grade !! Range !! Description of performance
  
! Grade !! Range !! Description of performance
 
|-
  
|-
−
| A. Excellent || 90-100 || -
 +
| A. Excellent || 85-100 || -
 
|-
  
|-
−
| B. Good || 75-89 || -
 +
| B. Good || 70-84 || -
 
|-
  
|-
−
| C. Satisfactory || 60-74 || -
 +
| C. Satisfactory || 50-69 || -
 
|-
  
|-
−
| D. Poor || 0-59 || -
 +
| D. Poor || 0-49 || -
 
|}
  
|}
   
Line 96: Line 125:
 
! Activity Type !! Percentage of the overall course grade
  
! Activity Type !! Percentage of the overall course grade
 
|-
  
|-
−
| Labs/seminar classes || 20
 +
| Labs/seminar classes || 30
 
|-
  
|-
−
| Interim performance assessment || 30
 +
| Interim performance assessment || 20
 
|-
  
|-
 
| Exams || 50
  
| Exams || 50
Line 109: Line 138:
   
 
=== Open access resources ===
  
=== Open access resources ===
  +
* Williams, R.L. and Lawrence, D.A., 2007. Linear state-space control systems. John Wiley & Sons.
−
* Zorich, V. A. “Mathematical Analysis I, Translator: Cooke R.” (2004)
  
  +
* Ogata, K., 1995. Discrete-time control systems (Vol. 2, pp. 446-480). Englewood Cliffs, NJ: Prentice Hall.
   
 
=== Closed access resources ===
  
=== Closed access resources ===
Line 126: Line 156:
 
| Homework and group projects || 1 || 1 || 1
  
| Homework and group projects || 1 || 1 || 1
 
|-
  
|-
−
| Midterm evaluation || 1 || 1 || 0
 +
| Testing (written or computer based) || 1 || 0 || 0
 
|-
  
|-
−
| Testing (written or computer based) || 1 || 1 || 1
 +
| Reports || 1 || 1 || 1
 
|-
  
|-
−
| Discussions || 1 || 1 || 1
 +
| Midterm evaluation || 0 || 1 || 0
  +
|-
  +
| Discussions || 0 || 1 || 0
 
|}
 
|}
 
== Formative Assessment and Course Activities ==
  
== Formative Assessment and Course Activities ==
Line 142: Line 174:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
−
| Question || A sequence, limiting value || 1
 +
| Question || What is a linear dynamical system? || 1
 
|-
  
|-
−
| Question || Limit of a sequence, convergent and divergent sequences || 1
 +
| Question || What is an LTI system? || 1
 
|-
  
|-
−
| Question || Increasing and decreasing sequences, monotonic sequences || 1
 +
| Question || What is an LTV system? || 1
 
|-
  
|-
−
| Question || Bounded sequences Properties of limits || 1
 +
| Question || Provide examples of LTI systems. || 1
 
|-
  
|-
−
| Question || Theorem about bounded and monotonic sequences || 1
 +
| Question || What is a MIMO system? || 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
−
| Question || Cauchy sequence The Cauchy Theorem (criterion) || 1
  
  +
|}
  +
==== Section 2 ====
  +
{| class="wikitable"
  +
|+
 
|-
  
|-
  +
! Activity Type !! Content !! Is Graded?
−
| Question || Limit of a function Properties of limits || 1
  
 
|-
  
|-
−
| Question || The first remarkable limit || 1
 +
| Question || What is stability in the sense of Lyapunov? || 1
 
|-
  
|-
−
| Question || The Cauchy criterion for the existence of a limit of a function || 1
 +
| Question || What is stabilizing control? || 1
 
|-
  
|-
−
| Question || Second remarkable limit || 1
 +
| Question || What is trajectory tracking? || 1
 
|-
  
|-
  +
| 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 sequence || 0
  
 
|-
  
|-
  +
| Question || How can a PD controller for a second-order linear mechanical system can be re-written in the state-space form? || 1
−
| Question || Find a limit of a function || 0
  
−
|}
 
−
==== Section 2 ====
  
−
{| class="wikitable"
  
−
|+
 
 
|-
  
|-
  +
| Question || Write a closed-loop dynamics for an LTI system with a proportional controller. || 1
−
! Activity Type !! Content !! Is Graded?
  
 
|-
  
|-
  +
| Question || Give stability conditions 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 || the asymptotes of this curve; || 1
 +
| Question || Provide an example of a LTV system with negative eigenvalues that is not stable. || 1
 
|-
  
|-
−
| Question || the derivative <math>{\textstyle y'_{x}}</math> || 1
 +
| Question || Write algebraic Riccati equation for a standard 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 algebraic Riccati equation for a given additive quadratic cost. || 1
 
|-
  
|-
−
| Question || Differentiation techniques: inverse, implicit, parametric etc || 0
 +
| Question || Derive differential Riccati equation for a standard additive quadratic cost. || 1
 
|-
  
|-
  +
| Question || What is the meaning of the unknown variable in the Riccati equation? What are its property in case of LTI dynamics. || 1
−
| Question || Find a derivative of a function || 0
  
 
|-
  
|-
−
| Question || Apply Leibniz formula || 0
 +
| Question || What is a frequency response? || 1
 
|-
  
|-
−
| Question || Draw graphs of functions || 0
 +
| Question || What is a phase response? || 1
 
|-
  
|-
−
| Question || Find asymptotes of a parametric function || 0
 +
| Question || Design control for an LTI system using pole placement. || 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 196: Line 232:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
  +
| Question || What are the sources of sensor noise? || 1
−
| Question || Find the indefinite integral <math>{\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}</math> || 1
  
  +
|-
  +
| Question || How can we combat the lack of sensory information? || 1
 
|-
  
|-
  +
| Question || When it is possible to 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 || How can we combat the sensory noise? || 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 || Integration techniques || 0
 +
| Question || What is an Observer? || 1
 
|-
  
|-
−
| Question || Integration by parts || 0
 +
| Question || What is a filter? || 1
 
|-
  
|-
−
| Question || Calculation of areas, lengths, volumes || 0
 +
| Question || How is additive noise different from multiplicative noise? || 1
 
|-
  
|-
−
| Question || Application of convergence tests || 0
 +
| Question || Simulate an LTI system with proportional control and sensor noise. || 0
 
|-
  
|-
−
| Question || Calculation of Radius of convergence || 0
 +
| Question || Design an observer for an LTI system with proportional control and lack of sensory information. || 0
 
|}
 
|}
 
=== Final assessment ===
  
=== Final assessment ===
 
'''Section 1'''
  
'''Section 1'''
  +
# Convert a linear differential equation into a state space form.
−
# Find limits of the following sequences or prove that they do not exist:
  
  +
# Convert a transfer function into a state space form.
−
# <math>{\displaystyle a_{n}=n-{\sqrt {n^{2}-70n+1400}}}</math> ;
  
  +
# Convert a linear differential equation into a transfer function.
−
# <math>{\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}</math> ;
  
  +
# What does it mean for a linear differential equation to be stable?
−
# <math>{\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}</math>
  
 
'''Section 2'''
  
'''Section 2'''
  +
# You have a linear system: <math>{\displaystyle {\dot {x}}=Ax+Bu}</math> and a cost function: a) <math>{\textstyle J=\int (x^{\top }Qx+u^{\top }Iu)dt}</math> b) <math>{\textstyle J=\int (x^{\top }Ix+u^{\top }Ru)dt}</math> Write Riccati eq. and find LQR gain analytically.
−
# Find a derivative of a (implicit/inverse) function
  
  +
# You have a linear system a) <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> b) <math>{\textstyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}-2&1\\2&40\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}</math> Prove whether or not it is stable.
−
# 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>
  
  +
# You have a linear system a) <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> b) <math>{\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}-2&1\\2&40\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}+{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}}</math> Your controller is: a) <math>{\textstyle {\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}={\begin{bmatrix}100&1\\1&20\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}</math> b) <math>{\textstyle {\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}={\begin{bmatrix}7&2\\2&5\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}</math> Prove whether the control system is stable.
−
# Draw graphs of functions
  
  +
# You have linear dynamics:
−
# Find asymptotes
  
  +
a) <math>{\textstyle 2{\ddot {q}}+3{\dot {q}}-5q=u}</math>
−
# Apply l’Hopital’s rule
  
−
# Find the derivatives of the following functions: <math>{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}</math>
 +
b) <math>{\textstyle 10{\ddot {q}}-7{\dot {q}}+10q=u}</math>
−
# <math>{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}</math> ;
 +
c) <math>{\textstyle 15{\ddot {q}}+17{\dot {q}}+11q=2u}</math>
−
# <math>{\textstyle y(x)}</math> that is given implicitly by <math>{\textstyle x^{3}+5xy+y^{3}=0}</math>
 +
d) <math>{\textstyle 20{\ddot {q}}-{\dot {q}}-2q=-u}</math>
  +
If <math>{\textstyle u=0}</math> , which are stable (a - d)?
  +
Find <math>{\textstyle u}</math> that makes the dynamics stable.
  +
Write transfer functions for the cases <math>{\textstyle u=0}</math> and <math>{\textstyle u=-100x}</math> .
  +
# If <math>{\textstyle u=0}</math> , which are stable (a - d)?
  +
# Find <math>{\textstyle u}</math> that makes the dynamics stable.
  +
# Write transfer functions for the cases <math>{\textstyle u=0}</math> and <math>{\textstyle u=-100x}</math> .
  +
# What is the difference between exponential stability, asymptotic stability and optimality?
 
'''Section 3'''
  
'''Section 3'''
  +
# Write a model of a linear system with additive Gaussian noise.
−
# Find the following integrals:
  
  +
# Derive and implement an observer.
−
# <math>{\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}</math>
  
  +
# Derive and implement a filter.
−
# <math>{\textstyle \int 2^{2x}e^{x}\,dx}</math>
  
−
# <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 12:06, 20 April 2022

Control Theory

  • Course name: Control Theory
  • Code discipline:
  • Subject area: ['Introduction to Linear Control, Stability of linear dynamical systems', 'Controller design', 'Sensing, observers, Adaptive control']

Short Description

Prerequisites

Prerequisite subjects

Prerequisite topics

Course Topics

Course Sections and Topics
Section Topics within the section
Introduction to Linear Control, Stability of linear dynamical systems
  1. Control, introduction. Examples.
  2. Single input single output (SISO) systems. Block diagrams.
  3. From linear differential equations to state space models.
  4. DC motor as a linear system.
  5. Spring-damper as a linear system.
  6. The concept of stability of the control system. Proof of stability for a linear system with negative real parts of eigenvalues.
  7. Multi input multi output (MIMO) systems.
  8. Linear Time Invariant (LTI) systems and their properties.
  9. Linear Time Varying (LTV) systems and their properties.
  10. Transfer function representation.
Controller design.
  1. Stabilizing control. Control error.
  2. Proportional control.
  3. PD control. Order of a system and order of the controller.
  4. PID control.
  5. P, PD and PID control for DC motor.
  6. Trajectory tracking. Control input types. Standard inputs (Heaviside step function, Dirac delta function, sine wave).
  7. Tuning PD and PID. Pole placement.
  8. Formal statements about stability. Lyapunov theory.
  9. Types of stability; Lyapunov stability, asymptotic stability, exponential stability.
  10. Eigenvalues in stability theory. Reasoning about solution of the autonomous linear system.
  11. Stability proof for PD control.
  12. Stability in stabilizing control and trajectory tracking.
  13. Frequency response. Phase response.
  14. Optimal control of linear systems. From Hamilton-Jacobi-Bellman to algebraic Riccati equation. LQR.
  15. Stability of LQR.
  16. Controllability.
Sensing, observers, Adaptive control
  1. Modelling digital sensors: quantization, discretization, lag.
  2. Modelling sensor noise. Gaussian noise. Additive models. Multiplicative models. Dynamic sensor models.
  3. Observability.
  4. Filters.
  5. State observers.
  6. Optimal state observer for linear systems.
  7. Linearization of nonlinear systems.
  8. Linearization along trajectory.
  9. Linearization of Inverted pendulum dynamics.
  10. Model errors. Differences between random disturbances and unmodeled dynamics/processes.
  11. Adaptive control.
  12. Control for sets of linear systems.
  13. Discretization, discretization error.
  14. Control for discrete linear systems.
  15. Stability of discrete linear systems.

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

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

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

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

  • methods for control synthesis (linear controller gain tuning)
  • methods for controller analysis
  • methods for sensory data processing for linear systems

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 ...

  • State-space models
  • Eigenvalue analysis for linear systems
  • Proportional and PD controllers
  • 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?

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

  • Turn a system of linear differential equations into a state-space model.
  • Design a controller by solving Algebraic Riccati eq.
  • Find if a system is stable or not, using eigenvalue analysis.

Grading

Course grading range

Grade Range Description of performance
A. Excellent 85-100 -
B. Good 70-84 -
C. Satisfactory 50-69 -
D. Poor 0-49 -

Course activities and grading breakdown

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

Recommendations for students on how to succeed in the course

Resources, literature and reference materials

Open access resources

  • 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

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
Testing (written or computer based) 1 0 0
Reports 1 1 1
Midterm evaluation 0 1 0
Discussions 0 1 0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type Content Is Graded?
Question What is a linear dynamical system? 1
Question What is an LTI system? 1
Question What is an LTV system? 1
Question Provide examples of LTI systems. 1
Question What is a MIMO system? 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

Activity Type Content Is Graded?
Question What is stability in the sense of Lyapunov? 1
Question What is stabilizing control? 1
Question What is trajectory tracking? 1
Question Why the control for a state-space system does not include the derivative of the state variable in the feedback law? 1
Question How can a PD controller for a second-order linear mechanical system can be re-written in the state-space form? 1
Question Write a closed-loop dynamics for an LTI system with a proportional controller. 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 Write algebraic Riccati equation for a standard additive quadratic cost. 1
Question Derive algebraic Riccati equation for a given additive quadratic cost. 1
Question Derive differential Riccati equation for a standard additive quadratic cost. 1
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 What is a phase response? 1
Question Design control for an LTI system using pole placement. 0
Question Design control for an LTI system using Riccati (LQR). 0
Question Simulate an LTI system with LQR controller. 0

Section 3

Activity Type Content Is Graded?
Question What are the sources of sensor noise? 1
Question How can we combat the lack of sensory information? 1
Question When it is possible to combat the lack of sensory information? 1
Question How can we combat the sensory noise? 1
Question What is an Observer? 1
Question What is a filter? 1
Question How is additive noise different from multiplicative noise? 1
Question Simulate an LTI system with proportional control and sensor noise. 0
Question Design an observer for an LTI system with proportional control and lack of sensory information. 0

Final assessment

Section 1

  1. Convert a linear differential equation into a state space form.
  2. Convert a transfer function into a state space form.
  3. Convert a linear differential equation into a transfer function.
  4. What does it mean for a linear differential equation to be stable?

Section 2

  1. You have a linear system: and a cost function: a) b) Write Riccati eq. and find LQR gain analytically.
  2. You have a linear system a) b) Prove whether or not it is stable.
  3. You have a linear system a) b) Your controller is: a) b) Prove whether the control system is stable.
  4. You have linear dynamics:

a) b) c) d) If , which are stable (a - d)? Find that makes the dynamics stable. Write transfer functions for the cases and .

  1. If , which are stable (a - d)?
  2. Find that makes the dynamics stable.
  3. Write transfer functions for the cases and .
  4. What is the difference between exponential stability, asymptotic stability and optimality?

Section 3

  1. Write a model of a linear system with additive Gaussian noise.
  2. Derive and implement an observer.
  3. Derive and implement a filter.

The retake exam

Section 1

Section 2

Section 3

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